added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/BNF_Comp.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,20 @@
+(* Title: HOL/Codatatype/BNF_Comp.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Composition of bounded natural functors.
+*)
+
+header {* Composition of Bounded Natural Functors *}
+
+theory BNF_Comp
+imports Basic_BNFs
+keywords
+ "bnf_of_typ" :: thy_decl
+uses
+ "Tools/bnf_comp_tactics.ML"
+ "Tools/bnf_comp.ML"
+ "Tools/bnf_fp_util.ML"
+begin
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/BNF_Def.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,22 @@
+(* Title: HOL/Codatatype/BNF_Def.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Definition of bounded natural functors.
+*)
+
+header {* Definition of Bounded Natural Functors *}
+
+theory BNF_Def
+imports BNF_Library
+keywords
+ "print_bnfs" :: diag
+and
+ "bnf_def" :: thy_goal
+uses
+ "Tools/bnf_util.ML"
+ "Tools/bnf_tactics.ML"
+ "Tools/bnf_def.ML"
+begin
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/BNF_GFP.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,20 @@
+(* Title: HOL/Codatatype/BNF_GFP.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Greatest fixed point operation on bounded natural functors.
+*)
+
+header {* Greatest Fixed Point Operation on Bounded Natural Functors *}
+
+theory BNF_GFP
+imports BNF_Comp
+keywords
+ "bnf_codata" :: thy_decl
+uses
+ "Tools/bnf_gfp_util.ML"
+ "Tools/bnf_gfp_tactics.ML"
+ "Tools/bnf_gfp.ML"
+begin
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/BNF_LFP.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,20 @@
+(* Title: HOL/Codatatype/BNF_LFP.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Least fixed point operation on bounded natural functors.
+*)
+
+header {* Least Fixed Point Operation on Bounded Natural Functors *}
+
+theory BNF_LFP
+imports BNF_Comp
+keywords
+ "bnf_data" :: thy_decl
+uses
+ "Tools/bnf_lfp_util.ML"
+ "Tools/bnf_lfp_tactics.ML"
+ "Tools/bnf_lfp.ML"
+begin
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/BNF_Library.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,826 @@
+(* Title: HOL/Codatatype/BNF_Library.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Library for bounded natural functors.
+*)
+
+header {* Library for Bounded Natural Functors *}
+
+theory BNF_Library
+imports
+ "../Ordinals_and_Cardinals_Base/Cardinal_Arithmetic"
+ "~~/src/HOL/Library/List_Prefix"
+ Equiv_Relations_More
+begin
+
+lemma subset_Collect_iff: "B \<subseteq> A \<Longrightarrow> (B \<subseteq> {x \<in> A. P x}) = (\<forall>x \<in> B. P x)"
+by blast
+
+lemma subset_CollectI: "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> Q x \<Longrightarrow> P x) \<Longrightarrow> ({x \<in> B. Q x} \<subseteq> {x \<in> A. P x})"
+by blast
+
+lemma mem_Collect_eq_split: "{(x, y). (x, y) \<in> X} = X"
+by simp
+
+lemma image_comp: "image (f o g) = image f o image g"
+by (rule ext) (auto simp only: o_apply image_def)
+
+lemma empty_natural: "(\<lambda>_. {}) o f = image g o (\<lambda>_. {})"
+by (rule ext) simp
+
+lemma Union_natural: "Union o image (image f) = image f o Union"
+by (rule ext) (auto simp only: o_apply)
+
+lemma in_Union_o_assoc: "x \<in> (Union o gset o gmap) A \<Longrightarrow> x \<in> (Union o (gset o gmap)) A"
+by (unfold o_assoc)
+
+lemma comp_single_set_bd:
+ assumes fbd_Card_order: "Card_order fbd" and
+ fset_bd: "\<And>x. |fset x| \<le>o fbd" and
+ gset_bd: "\<And>x. |gset x| \<le>o gbd"
+ shows "|\<Union>fset ` gset x| \<le>o gbd *c fbd"
+apply (subst sym[OF SUP_def])
+apply (rule ordLeq_transitive)
+apply (rule card_of_UNION_Sigma)
+apply (subst SIGMA_CSUM)
+apply (rule ordLeq_transitive)
+apply (rule card_of_Csum_Times')
+apply (rule fbd_Card_order)
+apply (rule ballI)
+apply (rule fset_bd)
+apply (rule ordLeq_transitive)
+apply (rule cprod_mono1)
+apply (rule gset_bd)
+apply (rule ordIso_imp_ordLeq)
+apply (rule ordIso_refl)
+apply (rule Card_order_cprod)
+done
+
+lemma Union_image_insert: "\<Union>f ` insert a B = f a \<union> \<Union>f ` B"
+by simp
+
+lemma Union_image_empty: "A \<union> \<Union>f ` {} = A"
+by simp
+
+definition collect where
+ "collect F x = (\<Union>f \<in> F. f x)"
+
+lemma collect_o: "collect F o g = collect ((\<lambda>f. f o g) ` F)"
+by (rule ext) (auto simp only: o_apply collect_def)
+
+lemma image_o_collect: "collect ((\<lambda>f. image g o f) ` F) = image g o collect F"
+by (rule ext) (auto simp add: collect_def)
+
+lemma conj_subset_def: "A \<subseteq> {x. P x \<and> Q x} = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
+by auto
+
+lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
+by auto
+
+lemma rev_bspec: "a \<in> A \<Longrightarrow> \<forall>z \<in> A. P z \<Longrightarrow> P a"
+by simp
+
+lemma Un_cong: "\<lbrakk>A = B; C = D\<rbrakk> \<Longrightarrow> A \<union> C = B \<union> D"
+by auto
+
+lemma UN_image_subset: "\<Union>f ` g x \<subseteq> X = (g x \<subseteq> {x. f x \<subseteq> X})"
+by auto
+
+lemma image_Collect_subsetI:
+ "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
+by auto
+
+lemma comp_set_bd_Union_o_collect: "|\<Union>\<Union>(\<lambda>f. f x) ` X| \<le>o hbd \<Longrightarrow> |(Union \<circ> collect X) x| \<le>o hbd"
+by (unfold o_apply collect_def SUP_def)
+
+lemma sum_case_comp_Inl:
+"sum_case f g \<circ> Inl = f"
+unfolding comp_def by simp
+
+lemma sum_case_expand_Inr: "f o Inl = g \<Longrightarrow> f x = sum_case g (f o Inr) x"
+by (auto split: sum.splits)
+
+lemma converse_mono:
+"R1 ^-1 \<subseteq> R2 ^-1 \<longleftrightarrow> R1 \<subseteq> R2"
+unfolding converse_def by auto
+
+lemma converse_shift:
+"R1 \<subseteq> R2 ^-1 \<Longrightarrow> R1 ^-1 \<subseteq> R2"
+unfolding converse_def by auto
+
+lemma converse_Times: "(A \<times> B) ^-1 = B \<times> A"
+by auto
+
+lemma equiv_triv1:
+assumes "equiv A R" and "(a, b) \<in> R" and "(a, c) \<in> R"
+shows "(b, c) \<in> R"
+using assms unfolding equiv_def sym_def trans_def by blast
+
+lemma equiv_triv2:
+assumes "equiv A R" and "(a, b) \<in> R" and "(b, c) \<in> R"
+shows "(a, c) \<in> R"
+using assms unfolding equiv_def trans_def by blast
+
+lemma equiv_proj:
+ assumes e: "equiv A R" and "z \<in> R"
+ shows "(proj R o fst) z = (proj R o snd) z"
+proof -
+ from assms(2) have z: "(fst z, snd z) \<in> R" by auto
+ have P: "\<And>x. (fst z, x) \<in> R \<Longrightarrow> (snd z, x) \<in> R" by (erule equiv_triv1[OF e z])
+ have "\<And>x. (snd z, x) \<in> R \<Longrightarrow> (fst z, x) \<in> R" by (erule equiv_triv2[OF e z])
+ with P show ?thesis unfolding proj_def[abs_def] by auto
+qed
+
+
+section{* Weak pullbacks: *}
+
+definition csquare where
+"csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
+
+definition wpull where
+"wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow>
+ (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow> (\<exists> a \<in> A. p1 a = b1 \<and> p2 a = b2))"
+
+lemma wpull_cong:
+"\<lbrakk>A' = A; B1' = B1; B2' = B2; wpull A B1 B2 f1 f2 p1 p2\<rbrakk> \<Longrightarrow> wpull A' B1' B2' f1 f2 p1 p2"
+by simp
+
+lemma wpull_id: "wpull UNIV B1 B2 id id id id"
+unfolding wpull_def by simp
+
+
+(* Weak pseudo-pullbacks *)
+
+definition wppull where
+"wppull A B1 B2 f1 f2 e1 e2 p1 p2 \<longleftrightarrow>
+ (\<forall> b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<longrightarrow>
+ (\<exists> a \<in> A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2))"
+
+
+(* The pullback of sets *)
+definition thePull where
+"thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}"
+
+lemma wpull_thePull:
+"wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd"
+unfolding wpull_def thePull_def by auto
+
+lemma wppull_thePull:
+assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
+shows
+"\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2.
+ j a' \<in> A \<and>
+ e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')"
+(is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')")
+proof(rule bchoice[of ?A' ?phi], default)
+ fix a' assume a': "a' \<in> ?A'"
+ hence "fst a' \<in> B1" unfolding thePull_def by auto
+ moreover
+ from a' have "snd a' \<in> B2" unfolding thePull_def by auto
+ moreover have "f1 (fst a') = f2 (snd a')"
+ using a' unfolding csquare_def thePull_def by auto
+ ultimately show "\<exists> ja'. ?phi a' ja'"
+ using assms unfolding wppull_def by auto
+qed
+
+lemma wpull_wppull:
+assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and
+1: "\<forall> a' \<in> A'. j a' \<in> A \<and> e1 (p1 (j a')) = e1 (p1' a') \<and> e2 (p2 (j a')) = e2 (p2' a')"
+shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2"
+unfolding wppull_def proof safe
+ fix b1 b2
+ assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2"
+ then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'"
+ using wp unfolding wpull_def by blast
+ show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2"
+ apply(rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto
+qed
+
+lemma wppull_id: "\<lbrakk>wpull UNIV UNIV UNIV f1 f2 p1 p2; e1 = id; e2 = id\<rbrakk> \<Longrightarrow>
+ wppull UNIV UNIV UNIV f1 f2 e1 e2 p1 p2"
+by (erule wpull_wppull) auto
+
+
+(* Operators: *)
+definition diag where "diag A \<equiv> {(a,a) | a. a \<in> A}"
+definition "Gr A f = {(a,f a) | a. a \<in> A}"
+definition image2 where "image2 A f g = {(f a, g a) | a. a \<in> A}"
+
+lemma diagI: "x \<in> A \<Longrightarrow> (x, x) \<in> diag A"
+unfolding diag_def by simp
+
+lemma diagE: "(a, b) \<in> diag A \<Longrightarrow> a = b"
+unfolding diag_def by simp
+
+lemma diagE': "x \<in> diag A \<Longrightarrow> fst x = snd x"
+unfolding diag_def by auto
+
+lemma diag_fst: "x \<in> diag A \<Longrightarrow> fst x \<in> A"
+unfolding diag_def by auto
+
+lemma diag_UNIV: "diag UNIV = Id"
+unfolding diag_def by auto
+
+lemma diag_converse: "diag A = (diag A) ^-1"
+unfolding diag_def by auto
+
+lemma diag_Comp: "diag A = diag A O diag A"
+unfolding diag_def by auto
+
+lemma diag_Gr: "diag A = Gr A id"
+unfolding diag_def Gr_def by simp
+
+lemma diag_UNIV_I: "x = y \<Longrightarrow> (x, y) \<in> diag UNIV"
+unfolding diag_def by auto
+
+lemma image2_eqI: "\<lbrakk>b = f x; c = g x; x \<in> A\<rbrakk> \<Longrightarrow> (b, c) \<in> image2 A f g"
+unfolding image2_def by auto
+
+lemma Id_def': "Id = {(a,b). a = b}"
+by auto
+
+lemma Id_alt: "Id = Gr UNIV id"
+unfolding Gr_def by auto
+
+lemma Id_subset: "Id \<subseteq> {(a, b). P a b \<or> a = b}"
+by auto
+
+lemma IdD: "(a, b) \<in> Id \<Longrightarrow> a = b"
+by auto
+
+lemma image2_Gr: "image2 A f g = (Gr A f)^-1 O (Gr A g)"
+unfolding image2_def Gr_def by auto
+
+lemma GrI: "\<lbrakk>x \<in> A; f x = fx\<rbrakk> \<Longrightarrow> (x, fx) \<in> Gr A f"
+unfolding Gr_def by simp
+
+lemma GrE: "(x, fx) \<in> Gr A f \<Longrightarrow> (x \<in> A \<Longrightarrow> f x = fx \<Longrightarrow> P) \<Longrightarrow> P"
+unfolding Gr_def by simp
+
+lemma GrD1: "(x, fx) \<in> Gr A f \<Longrightarrow> x \<in> A"
+unfolding Gr_def by simp
+
+lemma GrD2: "(x, fx) \<in> Gr A f \<Longrightarrow> f x = fx"
+unfolding Gr_def by simp
+
+lemma Gr_UNIV_id: "f = id \<Longrightarrow> (Gr UNIV f)^-1 O Gr UNIV f = Gr UNIV f"
+unfolding Gr_def by auto
+
+lemma Gr_fst_snd: "(Gr R fst)^-1 O Gr R snd = R"
+unfolding Gr_def by auto
+
+lemma Gr_mono: "A \<subseteq> B \<Longrightarrow> Gr A f \<subseteq> Gr B f"
+unfolding Gr_def by auto
+
+lemma subst_rel_def: "A = B \<Longrightarrow> (Gr A f)^-1 O Gr A g = (Gr B f)^-1 O Gr B g"
+by simp
+
+lemma abs_pred_def: "\<lbrakk>\<And>x y. (x, y) \<in> rel = pred x y\<rbrakk> \<Longrightarrow> rel = Collect (split pred)"
+by auto
+
+lemma Collect_split_cong: "Collect (split pred) = Collect (split pred') \<Longrightarrow> pred = pred'"
+by blast
+
+lemma pred_def_abs: "rel = Collect (split pred) \<Longrightarrow> pred = (\<lambda>x y. (x, y) \<in> rel)"
+by auto
+
+lemma wpull_Gr:
+"wpull (Gr A f) A (f ` A) f id fst snd"
+unfolding wpull_def Gr_def by auto
+
+lemma Gr_incl: "Gr A f \<subseteq> A <*> B \<longleftrightarrow> f ` A \<subseteq> B"
+unfolding Gr_def by auto
+
+lemma equiv_Image: "equiv A R \<Longrightarrow> (\<And>a b. (a, b) \<in> R \<Longrightarrow> a \<in> A \<and> b \<in> A \<and> R `` {a} = R `` {b})"
+unfolding equiv_def refl_on_def Image_def by (auto intro: transD symD)
+
+definition relImage where
+"relImage R f \<equiv> {(f a1, f a2) | a1 a2. (a1,a2) \<in> R}"
+
+definition relInvImage where
+"relInvImage A R f \<equiv> {(a1, a2) | a1 a2. a1 \<in> A \<and> a2 \<in> A \<and> (f a1, f a2) \<in> R}"
+
+lemma relImage_Gr:
+"\<lbrakk>R \<subseteq> A \<times> A\<rbrakk> \<Longrightarrow> relImage R f = (Gr A f)^-1 O R O Gr A f"
+unfolding relImage_def Gr_def relcomp_def by auto
+
+lemma relInvImage_Gr: "\<lbrakk>R \<subseteq> B \<times> B\<rbrakk> \<Longrightarrow> relInvImage A R f = Gr A f O R O (Gr A f)^-1"
+unfolding Gr_def relcomp_def image_def relInvImage_def by auto
+
+lemma relImage_mono:
+"R1 \<subseteq> R2 \<Longrightarrow> relImage R1 f \<subseteq> relImage R2 f"
+unfolding relImage_def by auto
+
+lemma relInvImage_mono:
+"R1 \<subseteq> R2 \<Longrightarrow> relInvImage A R1 f \<subseteq> relInvImage A R2 f"
+unfolding relInvImage_def by auto
+
+lemma relInvImage_diag:
+"(\<And>a1 a2. f a1 = f a2 \<longleftrightarrow> a1 = a2) \<Longrightarrow> relInvImage A (diag B) f \<subseteq> Id"
+unfolding relInvImage_def diag_def by auto
+
+lemma relInvImage_UNIV_relImage:
+"R \<subseteq> relInvImage UNIV (relImage R f) f"
+unfolding relInvImage_def relImage_def by auto
+
+lemma relImage_proj:
+assumes "equiv A R"
+shows "relImage R (proj R) \<subseteq> diag (A//R)"
+unfolding relImage_def diag_def apply safe
+using proj_iff[OF assms]
+by (metis assms equiv_Image proj_def proj_preserves)
+
+lemma relImage_relInvImage:
+assumes "R \<subseteq> f ` A <*> f ` A"
+shows "relImage (relInvImage A R f) f = R"
+using assms unfolding relImage_def relInvImage_def by fastforce
+
+
+(* Relation composition as a weak pseudo-pullback *)
+
+(* pick middle *)
+definition "pickM P Q a c = (SOME b. (a,b) \<in> P \<and> (b,c) \<in> Q)"
+
+lemma pickM:
+assumes "(a,c) \<in> P O Q"
+shows "(a, pickM P Q a c) \<in> P \<and> (pickM P Q a c, c) \<in> Q"
+unfolding pickM_def apply(rule someI_ex)
+using assms unfolding relcomp_def by auto
+
+definition fstO where "fstO P Q ac = (fst ac, pickM P Q (fst ac) (snd ac))"
+definition sndO where "sndO P Q ac = (pickM P Q (fst ac) (snd ac), snd ac)"
+
+lemma fstO_in: "ac \<in> P O Q \<Longrightarrow> fstO P Q ac \<in> P"
+by (metis assms fstO_def pickM surjective_pairing)
+
+lemma fst_fstO: "fst bc = (fst \<circ> fstO P Q) bc"
+unfolding comp_def fstO_def by simp
+
+lemma snd_sndO: "snd bc = (snd \<circ> sndO P Q) bc"
+unfolding comp_def sndO_def by simp
+
+lemma sndO_in: "ac \<in> P O Q \<Longrightarrow> sndO P Q ac \<in> Q"
+by (metis assms sndO_def pickM surjective_pairing)
+
+lemma csquare_fstO_sndO:
+"csquare (P O Q) snd fst (fstO P Q) (sndO P Q)"
+unfolding csquare_def fstO_def sndO_def using pickM by auto
+
+lemma wppull_fstO_sndO:
+shows "wppull (P O Q) P Q snd fst fst snd (fstO P Q) (sndO P Q)"
+using pickM unfolding wppull_def fstO_def sndO_def relcomp_def by auto
+
+lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)"
+by simp
+
+lemma snd_fst_flip: "snd xy = (fst o (%(x, y). (y, x))) xy"
+by (simp split: prod.split)
+
+lemma fst_snd_flip: "fst xy = (snd o (%(x, y). (y, x))) xy"
+by (simp split: prod.split)
+
+lemma flip_rel: "A \<subseteq> (R ^-1) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> R"
+by auto
+
+lemma fst_diag_id: "(fst \<circ> (%x. (x, x))) z = id z"
+by simp
+
+lemma snd_diag_id: "(snd \<circ> (%x. (x, x))) z = id z"
+by simp
+
+lemma fst_snd: "\<lbrakk>snd x = (y, z)\<rbrakk> \<Longrightarrow> fst (snd x) = y"
+by simp
+
+lemma snd_snd: "\<lbrakk>snd x = (y, z)\<rbrakk> \<Longrightarrow> snd (snd x) = z"
+by simp
+
+lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y"
+by simp
+
+lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z"
+by simp
+
+lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
+by auto
+
+lemma Collect_restrict': "{(x, y) | x y. phi x y \<and> P x y} \<subseteq> {(x, y) | x y. phi x y}"
+by auto
+
+lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
+by auto
+
+lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> rel.underS R j"
+unfolding rel.underS_def by simp
+
+lemma underS_E: "i \<in> rel.underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
+unfolding rel.underS_def by simp
+
+lemma underS_Field: "i \<in> rel.underS R j \<Longrightarrow> i \<in> Field R"
+unfolding rel.underS_def Field_def by auto
+
+lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
+unfolding Field_def by auto
+
+
+subsection {* Convolution product *}
+
+definition convol ("<_ , _>") where
+"<f , g> \<equiv> %a. (f a, g a)"
+
+lemma fst_convol:
+"fst o <f , g> = f"
+apply(rule ext)
+unfolding convol_def by simp
+
+lemma snd_convol:
+"snd o <f , g> = g"
+apply(rule ext)
+unfolding convol_def by simp
+
+lemma fst_convol': "fst (<f, g> x) = f x"
+using fst_convol unfolding convol_def by simp
+
+lemma snd_convol': "snd (<f, g> x) = g x"
+using snd_convol unfolding convol_def by simp
+
+lemma image_convolD: "\<lbrakk>(a, b) \<in> <f, g> ` X\<rbrakk> \<Longrightarrow> \<exists>x. x \<in> X \<and> a = f x \<and> b = g x"
+unfolding convol_def by auto
+
+lemma convol_expand_snd: "fst o f = g \<Longrightarrow> <g, snd o f> = f"
+unfolding convol_def by auto
+
+subsection{* Facts about functions *}
+
+lemma pointfreeE: "f o g = f' o g' \<Longrightarrow> f (g x) = f' (g' x)"
+unfolding o_def fun_eq_iff by simp
+
+lemma pointfree_idE: "f o g = id \<Longrightarrow> f (g x) = x"
+unfolding o_def fun_eq_iff by simp
+
+definition inver where
+ "inver g f A = (ALL a : A. g (f a) = a)"
+
+lemma bij_betw_iff_ex:
+ "bij_betw f A B = (EX g. g ` B = A \<and> inver g f A \<and> inver f g B)" (is "?L = ?R")
+proof (rule iffI)
+ assume ?L
+ hence f: "f ` A = B" and inj_f: "inj_on f A" unfolding bij_betw_def by auto
+ let ?phi = "% b a. a : A \<and> f a = b"
+ have "ALL b : B. EX a. ?phi b a" using f by blast
+ then obtain g where g: "ALL b : B. g b : A \<and> f (g b) = b"
+ using bchoice[of B ?phi] by blast
+ hence gg: "ALL b : f ` A. g b : A \<and> f (g b) = b" using f by blast
+ have gf: "inver g f A" unfolding inver_def by (metis gg imageI inj_f the_inv_into_f_f)
+ moreover have "g ` B \<le> A \<and> inver f g B" using g unfolding inver_def by blast
+ moreover have "A \<le> g ` B"
+ proof safe
+ fix a assume a: "a : A"
+ hence "f a : B" using f by auto
+ moreover have "a = g (f a)" using a gf unfolding inver_def by auto
+ ultimately show "a : g ` B" by blast
+ qed
+ ultimately show ?R by blast
+next
+ assume ?R
+ then obtain g where g: "g ` B = A \<and> inver g f A \<and> inver f g B" by blast
+ show ?L unfolding bij_betw_def
+ proof safe
+ show "inj_on f A" unfolding inj_on_def
+ proof safe
+ fix a1 a2 assume a: "a1 : A" "a2 : A" and "f a1 = f a2"
+ hence "g (f a1) = g (f a2)" by simp
+ thus "a1 = a2" using a g unfolding inver_def by simp
+ qed
+ next
+ fix a assume "a : A"
+ then obtain b where b: "b : B" and a: "a = g b" using g by blast
+ hence "b = f (g b)" using g unfolding inver_def by auto
+ thus "f a : B" unfolding a using b by simp
+ next
+ fix b assume "b : B"
+ hence "g b : A \<and> b = f (g b)" using g unfolding inver_def by auto
+ thus "b : f ` A" by auto
+ qed
+qed
+
+lemma bijI: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f"
+unfolding bij_def inj_on_def by auto blast
+
+lemma bij_betw_ex_weakE:
+ "\<lbrakk>bij_betw f A B\<rbrakk> \<Longrightarrow> \<exists>g. g ` B \<subseteq> A \<and> inver g f A \<and> inver f g B"
+by (auto simp only: bij_betw_iff_ex)
+
+lemma inver_surj: "\<lbrakk>g ` B \<subseteq> A; f ` A \<subseteq> B; inver g f A\<rbrakk> \<Longrightarrow> g ` B = A"
+unfolding inver_def by auto (rule rev_image_eqI, auto)
+
+lemma inver_mono: "\<lbrakk>A \<subseteq> B; inver f g B\<rbrakk> \<Longrightarrow> inver f g A"
+unfolding inver_def by auto
+
+lemma inver_pointfree: "inver f g A = (\<forall>a \<in> A. (f o g) a = a)"
+unfolding inver_def by simp
+
+lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
+unfolding bij_betw_def by auto
+
+lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
+unfolding bij_betw_def by auto
+
+lemma inverE: "\<lbrakk>inver f f' A; x \<in> A\<rbrakk> \<Longrightarrow> f (f' x) = x"
+unfolding inver_def by auto
+
+lemma bij_betw_inver1: "bij_betw f A B \<Longrightarrow> inver (inv_into A f) f A"
+unfolding bij_betw_def inver_def by auto
+
+lemma bij_betw_inver2: "bij_betw f A B \<Longrightarrow> inver f (inv_into A f) B"
+unfolding bij_betw_def inver_def by auto
+
+lemma bij_betwI: "\<lbrakk>bij_betw g B A; inver g f A; inver f g B\<rbrakk> \<Longrightarrow> bij_betw f A B"
+by (metis bij_betw_iff_ex bij_betw_imageE)
+
+lemma bij_betwI':
+ "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
+ \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
+ \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
+unfolding bij_betw_def inj_on_def by auto (metis rev_image_eqI)
+
+lemma o_bij:
+ assumes gf: "g o f = id" and fg: "f o g = id"
+ shows "bij f"
+unfolding bij_def inj_on_def surj_def proof safe
+ fix a1 a2 assume "f a1 = f a2"
+ hence "g ( f a1) = g (f a2)" by simp
+ thus "a1 = a2" using gf unfolding fun_eq_iff by simp
+next
+ fix b
+ have "b = f (g b)"
+ using fg unfolding fun_eq_iff by simp
+ thus "EX a. b = f a" by blast
+qed
+
+lemma surj_fun_eq:
+ assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
+ shows "g1 = g2"
+proof (rule ext)
+ fix y
+ from surj_on obtain x where "x \<in> X" and "y = f x" by blast
+ thus "g1 y = g2 y" using eq_on by simp
+qed
+
+lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
+unfolding wo_rel_def card_order_on_def by blast
+
+lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow>
+ \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
+unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
+
+lemma Card_order_trans:
+ "\<lbrakk>Card_order r; x \<noteq> y; (x, y) \<in> r; y \<noteq> z; (y, z) \<in> r\<rbrakk> \<Longrightarrow> x \<noteq> z \<and> (x, z) \<in> r"
+unfolding card_order_on_def well_order_on_def linear_order_on_def
+ partial_order_on_def preorder_on_def trans_def antisym_def by blast
+
+lemma Cinfinite_limit2:
+ assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
+ shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
+proof -
+ from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
+ unfolding card_order_on_def well_order_on_def linear_order_on_def
+ partial_order_on_def preorder_on_def by auto
+ obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
+ using Cinfinite_limit[OF x1 r] by blast
+ obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
+ using Cinfinite_limit[OF x2 r] by blast
+ show ?thesis
+ proof (cases "y1 = y2")
+ case True with y1 y2 show ?thesis by blast
+ next
+ case False
+ with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
+ unfolding total_on_def by auto
+ thus ?thesis
+ proof
+ assume *: "(y1, y2) \<in> r"
+ with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
+ with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
+ next
+ assume *: "(y2, y1) \<in> r"
+ with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
+ with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
+ qed
+ qed
+qed
+
+lemma Cinfinite_limit_finite: "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk>
+ \<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
+proof (induct X rule: finite_induct)
+ case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
+next
+ case (insert x X)
+ then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
+ then obtain z where z: "z \<in> Field r" "x \<noteq> z \<and> (x, z) \<in> r" "y \<noteq> z \<and> (y, z) \<in> r"
+ using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
+ show ?case by (metis Card_order_trans insert(5) insertE y(2) z)
+qed
+
+lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
+by auto
+
+(*helps resolution*)
+lemma well_order_induct_imp:
+ "wo_rel r \<Longrightarrow> (\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x) \<Longrightarrow>
+ x \<in> Field r \<longrightarrow> P x"
+by (erule wo_rel.well_order_induct)
+
+lemma meta_spec2:
+ assumes "(\<And>x y. PROP P x y)"
+ shows "PROP P x y"
+by (rule `(\<And>x y. PROP P x y)`)
+
+(*Extended List_Prefix*)
+
+definition prefCl where
+ "prefCl Kl = (\<forall> kl1 kl2. kl1 \<le> kl2 \<and> kl2 \<in> Kl \<longrightarrow> kl1 \<in> Kl)"
+definition PrefCl where
+ "PrefCl A n = (\<forall>kl kl'. kl \<in> A n \<and> kl' \<le> kl \<longrightarrow> (\<exists>m\<le>n. kl' \<in> A m))"
+
+lemma prefCl_UN:
+ "\<lbrakk>\<And>n. PrefCl A n\<rbrakk> \<Longrightarrow> prefCl (\<Union>n. A n)"
+unfolding prefCl_def PrefCl_def by fastforce
+
+definition Succ where "Succ Kl kl = {k . kl @ [k] \<in> Kl}"
+definition Shift where "Shift Kl k = {kl. k # kl \<in> Kl}"
+definition shift where "shift lab k = (\<lambda>kl. lab (k # kl))"
+
+lemmas sh_def = Shift_def shift_def
+
+lemma empty_Shift: "\<lbrakk>[] \<in> Kl; k \<in> Succ Kl []\<rbrakk> \<Longrightarrow> [] \<in> Shift Kl k"
+unfolding Shift_def Succ_def by simp
+
+lemma Shift_clists: "Kl \<subseteq> Field (clists r) \<Longrightarrow> Shift Kl k \<subseteq> Field (clists r)"
+unfolding Shift_def clists_def Field_card_of by auto
+
+lemma Shift_prefCl: "prefCl Kl \<Longrightarrow> prefCl (Shift Kl k)"
+unfolding prefCl_def Shift_def
+proof safe
+ fix kl1 kl2
+ assume "\<forall>kl1 kl2. kl1 \<le> kl2 \<and> kl2 \<in> Kl \<longrightarrow> kl1 \<in> Kl"
+ "kl1 \<le> kl2" "k # kl2 \<in> Kl"
+ thus "k # kl1 \<in> Kl" using Cons_prefix_Cons[of k kl1 k kl2] by blast
+qed
+
+lemma not_in_Shift: "kl \<notin> Shift Kl x \<Longrightarrow> x # kl \<notin> Kl"
+unfolding Shift_def by simp
+
+lemma prefCl_Succ: "\<lbrakk>prefCl Kl; k # kl \<in> Kl\<rbrakk> \<Longrightarrow> k \<in> Succ Kl []"
+unfolding Succ_def proof
+ assume "prefCl Kl" "k # kl \<in> Kl"
+ moreover have "k # [] \<le> k # kl" by auto
+ ultimately have "k # [] \<in> Kl" unfolding prefCl_def by blast
+ thus "[] @ [k] \<in> Kl" by simp
+qed
+
+lemma SuccD: "k \<in> Succ Kl kl \<Longrightarrow> kl @ [k] \<in> Kl"
+unfolding Succ_def by simp
+
+lemmas SuccE = SuccD[elim_format]
+
+lemma SuccI: "kl @ [k] \<in> Kl \<Longrightarrow> k \<in> Succ Kl kl"
+unfolding Succ_def by simp
+
+lemma ShiftD: "kl \<in> Shift Kl k \<Longrightarrow> k # kl \<in> Kl"
+unfolding Shift_def by simp
+
+lemma Succ_Shift: "Succ (Shift Kl k) kl = Succ Kl (k # kl)"
+unfolding Succ_def Shift_def by auto
+
+lemma ShiftI: "k # kl \<in> Kl \<Longrightarrow> kl \<in> Shift Kl k"
+unfolding Shift_def by simp
+
+lemma Func_cexp: "|Func A B| =o |B| ^c |A|"
+unfolding cexp_def Field_card_of by (simp only: card_of_refl)
+
+lemma clists_bound: "A \<in> Field (cpow (clists r)) - {{}} \<Longrightarrow> |A| \<le>o clists r"
+unfolding cpow_def clists_def Field_card_of by (auto simp: card_of_mono1)
+
+lemma cpow_clists_czero: "\<lbrakk>A \<in> Field (cpow (clists r)) - {{}}; |A| =o czero\<rbrakk> \<Longrightarrow> False"
+unfolding cpow_def clists_def
+by (auto simp add: card_of_ordIso_czero_iff_empty[symmetric])
+ (erule notE, erule ordIso_transitive, rule czero_ordIso)
+
+lemma incl_UNION_I:
+assumes "i \<in> I" and "A \<subseteq> F i"
+shows "A \<subseteq> UNION I F"
+using assms by auto
+
+lemma Nil_clists: "{[]} \<subseteq> Field (clists r)"
+unfolding clists_def Field_card_of by auto
+
+lemma Cons_clists:
+ "\<lbrakk>x \<in> Field r; xs \<in> Field (clists r)\<rbrakk> \<Longrightarrow> x # xs \<in> Field (clists r)"
+unfolding clists_def Field_card_of by auto
+
+lemma length_Cons: "length (x#xs) = Suc (length xs)"
+by simp
+
+lemma length_append_singleton: "length (xs @ [x]) = Suc (length xs)"
+by simp
+
+(*injection into the field of a cardinal*)
+definition "toCard_pred A r f \<equiv> inj_on f A \<and> f ` A \<subseteq> Field r \<and> Card_order r"
+definition "toCard A r \<equiv> SOME f. toCard_pred A r f"
+
+lemma ex_toCard_pred:
+"\<lbrakk>|A| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> \<exists> f. toCard_pred A r f"
+unfolding toCard_pred_def
+using card_of_ordLeq[of A "Field r"]
+ ordLeq_ordIso_trans[OF _ card_of_unique[of "Field r" r], of "|A|"]
+by blast
+
+lemma toCard_pred_toCard:
+ "\<lbrakk>|A| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> toCard_pred A r (toCard A r)"
+unfolding toCard_def using someI_ex[OF ex_toCard_pred] .
+
+lemma toCard_inj: "\<lbrakk>|A| \<le>o r; Card_order r; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow>
+ toCard A r x = toCard A r y \<longleftrightarrow> x = y"
+using toCard_pred_toCard unfolding inj_on_def toCard_pred_def by blast
+
+lemma toCard: "\<lbrakk>|A| \<le>o r; Card_order r; b \<in> A\<rbrakk> \<Longrightarrow> toCard A r b \<in> Field r"
+using toCard_pred_toCard unfolding toCard_pred_def by blast
+
+definition "fromCard A r k \<equiv> SOME b. b \<in> A \<and> toCard A r b = k"
+
+lemma fromCard_toCard:
+"\<lbrakk>|A| \<le>o r; Card_order r; b \<in> A\<rbrakk> \<Longrightarrow> fromCard A r (toCard A r b) = b"
+unfolding fromCard_def by (rule some_equality) (auto simp add: toCard_inj)
+
+(* pick according to the weak pullback *)
+definition pickWP_pred where
+"pickWP_pred A p1 p2 b1 b2 a \<equiv>
+ a \<in> A \<and> p1 a = b1 \<and> p2 a = b2"
+
+definition pickWP where
+"pickWP A p1 p2 b1 b2 \<equiv>
+ SOME a. pickWP_pred A p1 p2 b1 b2 a"
+
+lemma pickWP_pred:
+assumes "wpull A B1 B2 f1 f2 p1 p2" and
+"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
+shows "\<exists> a. pickWP_pred A p1 p2 b1 b2 a"
+using assms unfolding wpull_def pickWP_pred_def by blast
+
+lemma pickWP_pred_pickWP:
+assumes "wpull A B1 B2 f1 f2 p1 p2" and
+"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
+shows "pickWP_pred A p1 p2 b1 b2 (pickWP A p1 p2 b1 b2)"
+unfolding pickWP_def using assms by(rule someI_ex[OF pickWP_pred])
+
+lemma pickWP:
+assumes "wpull A B1 B2 f1 f2 p1 p2" and
+"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
+shows "pickWP A p1 p2 b1 b2 \<in> A"
+ "p1 (pickWP A p1 p2 b1 b2) = b1"
+ "p2 (pickWP A p1 p2 b1 b2) = b2"
+using assms pickWP_pred_pickWP unfolding pickWP_pred_def by fastforce+
+
+lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
+
+lemma Inl_Field_csum: "a \<in> Field r \<Longrightarrow> Inl a \<in> Field (r +c s)"
+unfolding Field_card_of csum_def by auto
+
+lemma Inr_Field_csum: "a \<in> Field s \<Longrightarrow> Inr a \<in> Field (r +c s)"
+unfolding Field_card_of csum_def by auto
+
+lemma nat_rec_0: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f 0 = f1"
+by auto
+
+lemma nat_rec_Suc: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f (Suc n) = f2 n (f n)"
+by auto
+
+lemma list_rec_Nil: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f [] = f1"
+by auto
+
+lemma list_rec_Cons: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f (x # xs) = f2 x xs (f xs)"
+by auto
+
+lemma sum_case_cong: "p = q \<Longrightarrow> sum_case f g p = sum_case f g q"
+by simp
+
+lemma sum_case_step:
+ "sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"
+ "sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"
+by auto
+
+lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+by (cases s) auto
+
+lemma obj_sum_base: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+by auto
+
+lemma obj_sum_step:
+ "\<lbrakk>\<forall>x. s = f (Inr (Inl x)) \<longrightarrow> P; \<forall>x. s = f (Inr (Inr x)) \<longrightarrow> P\<rbrakk> \<Longrightarrow> \<forall>x. s = f (Inr x) \<longrightarrow> P"
+by (metis obj_sumE)
+
+lemma not_arg_cong_Inr: "x \<noteq> y \<Longrightarrow> Inr x \<noteq> Inr y"
+by auto
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Basic_BNFs.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1529 @@
+(* Title: HOL/Codatatype/Basic_BNFs.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+Registration of various types as bounded natural functors.
+*)
+
+header {* Registration of Various Types as Bounded Natural Functors *}
+
+theory Basic_BNFs
+imports BNF_Def "~~/src/HOL/Quotient_Examples/FSet"
+ "~~/src/HOL/Library/Multiset" Countable_Set
+begin
+
+lemmas natLeq_card_order = natLeq_Card_order[unfolded Field_natLeq]
+
+lemma ctwo_card_order: "card_order ctwo"
+using Card_order_ctwo by (unfold ctwo_def Field_card_of)
+
+lemma natLeq_cinfinite: "cinfinite natLeq"
+unfolding cinfinite_def Field_natLeq by (rule nat_infinite)
+
+bnf_def ID = "id :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" ["\<lambda>x. {x}"] "\<lambda>_:: 'a. natLeq" ["id :: 'a \<Rightarrow> 'a"]
+apply auto
+apply (rule natLeq_card_order)
+apply (rule natLeq_cinfinite)
+apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])
+apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)
+apply (rule ordLeq_transitive)
+apply (rule ordLeq_cexp1[of natLeq])
+apply (rule Cinfinite_Cnotzero)
+apply (rule conjI)
+apply (rule natLeq_cinfinite)
+apply (rule natLeq_Card_order)
+apply (rule card_of_Card_order)
+apply (rule cexp_mono1)
+apply (rule ordLeq_csum1)
+apply (rule card_of_Card_order)
+apply (rule disjI2)
+apply (rule cone_ordLeq_cexp)
+apply (rule ordLeq_transitive)
+apply (rule cone_ordLeq_ctwo)
+apply (rule ordLeq_csum2)
+apply (rule Card_order_ctwo)
+apply (rule natLeq_Card_order)
+done
+
+lemma ID_pred[simp]: "ID_pred \<phi> = \<phi>"
+unfolding ID_pred_def ID_rel_def Gr_def fun_eq_iff by auto
+
+bnf_def DEADID = "id :: 'a \<Rightarrow> 'a" [] "\<lambda>_:: 'a. natLeq +c |UNIV :: 'a set|" ["SOME x :: 'a. True"]
+apply (auto simp add: wpull_id)
+apply (rule card_order_csum)
+apply (rule natLeq_card_order)
+apply (rule card_of_card_order_on)
+apply (rule cinfinite_csum)
+apply (rule disjI1)
+apply (rule natLeq_cinfinite)
+apply (rule ordLess_imp_ordLeq)
+apply (rule ordLess_ordLeq_trans)
+apply (rule ordLess_ctwo_cexp)
+apply (rule card_of_Card_order)
+apply (rule cexp_mono2'')
+apply (rule ordLeq_csum2)
+apply (rule card_of_Card_order)
+apply (rule ctwo_Cnotzero)
+by (rule card_of_Card_order)
+
+lemma DEADID_pred[simp]: "DEADID_pred = (op =)"
+unfolding DEADID_pred_def DEADID.rel_Id by simp
+
+ML {*
+
+signature BASIC_BNFS =
+sig
+ val ID_bnf: BNF_Def.BNF
+ val ID_rel_def: thm
+ val ID_pred_def: thm
+
+ val DEADID_bnf: BNF_Def.BNF
+end;
+
+structure Basic_BNFs : BASIC_BNFS =
+struct
+
+ val ID_bnf = the (BNF_Def.bnf_of @{context} "ID");
+ val DEADID_bnf = the (BNF_Def.bnf_of @{context} "DEADID");
+
+ val rel_def = BNF_Def.rel_def_of_bnf ID_bnf;
+ val ID_rel_def = rel_def RS sym;
+ val ID_pred_def =
+ Local_Defs.unfold @{context} [rel_def] (BNF_Def.pred_def_of_bnf ID_bnf) RS sym;
+
+end;
+*}
+
+definition sum_setl :: "'a + 'b \<Rightarrow> 'a set" where
+"sum_setl x = (case x of Inl z => {z} | _ => {})"
+
+definition sum_setr :: "'a + 'b \<Rightarrow> 'b set" where
+"sum_setr x = (case x of Inr z => {z} | _ => {})"
+
+lemmas sum_set_defs = sum_setl_def[abs_def] sum_setr_def[abs_def]
+
+bnf_def sum = sum_map [sum_setl, sum_setr] "\<lambda>_::'a + 'b. natLeq" [Inl, Inr]
+proof -
+ show "sum_map id id = id" by (rule sum_map.id)
+next
+ fix f1 f2 g1 g2
+ show "sum_map (g1 o f1) (g2 o f2) = sum_map g1 g2 o sum_map f1 f2"
+ by (rule sum_map.comp[symmetric])
+next
+ fix x f1 f2 g1 g2
+ assume a1: "\<And>z. z \<in> sum_setl x \<Longrightarrow> f1 z = g1 z" and
+ a2: "\<And>z. z \<in> sum_setr x \<Longrightarrow> f2 z = g2 z"
+ thus "sum_map f1 f2 x = sum_map g1 g2 x"
+ proof (cases x)
+ case Inl thus ?thesis using a1 by (clarsimp simp: sum_setl_def)
+ next
+ case Inr thus ?thesis using a2 by (clarsimp simp: sum_setr_def)
+ qed
+next
+ fix f1 f2
+ show "sum_setl o sum_map f1 f2 = image f1 o sum_setl"
+ by (rule ext, unfold o_apply) (simp add: sum_setl_def split: sum.split)
+next
+ fix f1 f2
+ show "sum_setr o sum_map f1 f2 = image f2 o sum_setr"
+ by (rule ext, unfold o_apply) (simp add: sum_setr_def split: sum.split)
+next
+ show "card_order natLeq" by (rule natLeq_card_order)
+next
+ show "cinfinite natLeq" by (rule natLeq_cinfinite)
+next
+ fix x
+ show "|sum_setl x| \<le>o natLeq"
+ apply (rule ordLess_imp_ordLeq)
+ apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
+ by (simp add: sum_setl_def split: sum.split)
+next
+ fix x
+ show "|sum_setr x| \<le>o natLeq"
+ apply (rule ordLess_imp_ordLeq)
+ apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
+ by (simp add: sum_setr_def split: sum.split)
+next
+ fix A1 :: "'a set" and A2 :: "'b set"
+ have in_alt: "{x. (case x of Inl z => {z} | _ => {}) \<subseteq> A1 \<and>
+ (case x of Inr z => {z} | _ => {}) \<subseteq> A2} = A1 <+> A2" (is "?L = ?R")
+ proof safe
+ fix x :: "'a + 'b"
+ assume "(case x of Inl z \<Rightarrow> {z} | _ \<Rightarrow> {}) \<subseteq> A1" "(case x of Inr z \<Rightarrow> {z} | _ \<Rightarrow> {}) \<subseteq> A2"
+ hence "x \<in> Inl ` A1 \<or> x \<in> Inr ` A2" by (cases x) simp+
+ thus "x \<in> A1 <+> A2" by blast
+ qed (auto split: sum.split)
+ show "|{x. sum_setl x \<subseteq> A1 \<and> sum_setr x \<subseteq> A2}| \<le>o
+ (( |A1| +c |A2| ) +c ctwo) ^c natLeq"
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule card_of_ordIso_subst)
+ apply (unfold sum_set_defs)
+ apply (rule in_alt)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule Plus_csum)
+ apply (rule ordLeq_transitive)
+ apply (rule ordLeq_csum1)
+ apply (rule Card_order_csum)
+ apply (rule ordLeq_cexp1)
+ apply (rule conjI)
+ using Field_natLeq UNIV_not_empty czeroE apply fast
+ apply (rule natLeq_Card_order)
+ by (rule Card_order_csum)
+next
+ fix A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22
+ assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"
+ hence
+ pull1: "\<And>b1 b2. \<lbrakk>b1 \<in> B11; b2 \<in> B21; f11 b1 = f21 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A1. p11 a = b1 \<and> p21 a = b2"
+ and pull2: "\<And>b1 b2. \<lbrakk>b1 \<in> B12; b2 \<in> B22; f12 b1 = f22 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A2. p12 a = b1 \<and> p22 a = b2"
+ unfolding wpull_def by blast+
+ show "wpull {x. sum_setl x \<subseteq> A1 \<and> sum_setr x \<subseteq> A2}
+ {x. sum_setl x \<subseteq> B11 \<and> sum_setr x \<subseteq> B12} {x. sum_setl x \<subseteq> B21 \<and> sum_setr x \<subseteq> B22}
+ (sum_map f11 f12) (sum_map f21 f22) (sum_map p11 p12) (sum_map p21 p22)"
+ (is "wpull ?in ?in1 ?in2 ?mapf1 ?mapf2 ?mapp1 ?mapp2")
+ proof (unfold wpull_def)
+ { fix B1 B2
+ assume *: "B1 \<in> ?in1" "B2 \<in> ?in2" "?mapf1 B1 = ?mapf2 B2"
+ have "\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2"
+ proof (cases B1)
+ case (Inl b1)
+ { fix b2 assume "B2 = Inr b2"
+ with Inl *(3) have False by simp
+ } then obtain b2 where Inl': "B2 = Inl b2" by (cases B2) (simp, blast)
+ with Inl * have "b1 \<in> B11" "b2 \<in> B21" "f11 b1 = f21 b2"
+ by (simp add: sum_setl_def)+
+ with pull1 obtain a where "a \<in> A1" "p11 a = b1" "p21 a = b2" by blast+
+ with Inl Inl' have "Inl a \<in> ?in" "?mapp1 (Inl a) = B1 \<and> ?mapp2 (Inl a) = B2"
+ by (simp add: sum_set_defs)+
+ thus ?thesis by blast
+ next
+ case (Inr b1)
+ { fix b2 assume "B2 = Inl b2"
+ with Inr *(3) have False by simp
+ } then obtain b2 where Inr': "B2 = Inr b2" by (cases B2) (simp, blast)
+ with Inr * have "b1 \<in> B12" "b2 \<in> B22" "f12 b1 = f22 b2"
+ by (simp add: sum_set_defs)+
+ with pull2 obtain a where "a \<in> A2" "p12 a = b1" "p22 a = b2" by blast+
+ with Inr Inr' have "Inr a \<in> ?in" "?mapp1 (Inr a) = B1 \<and> ?mapp2 (Inr a) = B2"
+ by (simp add: sum_set_defs)+
+ thus ?thesis by blast
+ qed
+ }
+ thus "\<forall>B1 B2. B1 \<in> ?in1 \<and> B2 \<in> ?in2 \<and> ?mapf1 B1 = ?mapf2 B2 \<longrightarrow>
+ (\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2)" by fastforce
+ qed
+qed (auto simp: sum_set_defs)
+
+lemma sum_pred[simp]:
+ "sum_pred \<phi> \<psi> x y =
+ (case x of Inl a1 \<Rightarrow> (case y of Inl a2 \<Rightarrow> \<phi> a1 a2 | Inr _ \<Rightarrow> False)
+ | Inr b1 \<Rightarrow> (case y of Inl _ \<Rightarrow> False | Inr b2 \<Rightarrow> \<psi> b1 b2))"
+unfolding sum_setl_def sum_setr_def sum_pred_def sum_rel_def Gr_def relcomp_unfold converse_unfold
+by (fastforce split: sum.splits)+
+
+lemma singleton_ordLeq_ctwo_natLeq: "|{x}| \<le>o ctwo *c natLeq"
+ apply (rule ordLeq_transitive)
+ apply (rule ordLeq_cprod2)
+ apply (rule ctwo_Cnotzero)
+ apply (auto simp: Field_card_of intro: card_of_card_order_on)
+ apply (rule cprod_mono2)
+ apply (rule ordLess_imp_ordLeq)
+ apply (unfold finite_iff_ordLess_natLeq[symmetric])
+ by simp
+
+definition fsts :: "'a \<times> 'b \<Rightarrow> 'a set" where
+"fsts x = {fst x}"
+
+definition snds :: "'a \<times> 'b \<Rightarrow> 'b set" where
+"snds x = {snd x}"
+
+lemmas prod_set_defs = fsts_def[abs_def] snds_def[abs_def]
+
+bnf_def prod = map_pair [fsts, snds] "\<lambda>_::'a \<times> 'b. ctwo *c natLeq" [Pair]
+proof (unfold prod_set_defs)
+ show "map_pair id id = id" by (rule map_pair.id)
+next
+ fix f1 f2 g1 g2
+ show "map_pair (g1 o f1) (g2 o f2) = map_pair g1 g2 o map_pair f1 f2"
+ by (rule map_pair.comp[symmetric])
+next
+ fix x f1 f2 g1 g2
+ assume "\<And>z. z \<in> {fst x} \<Longrightarrow> f1 z = g1 z" "\<And>z. z \<in> {snd x} \<Longrightarrow> f2 z = g2 z"
+ thus "map_pair f1 f2 x = map_pair g1 g2 x" by (cases x) simp
+next
+ fix f1 f2
+ show "(\<lambda>x. {fst x}) o map_pair f1 f2 = image f1 o (\<lambda>x. {fst x})"
+ by (rule ext, unfold o_apply) simp
+next
+ fix f1 f2
+ show "(\<lambda>x. {snd x}) o map_pair f1 f2 = image f2 o (\<lambda>x. {snd x})"
+ by (rule ext, unfold o_apply) simp
+next
+ show "card_order (ctwo *c natLeq)"
+ apply (rule card_order_cprod)
+ apply (rule ctwo_card_order)
+ by (rule natLeq_card_order)
+next
+ show "cinfinite (ctwo *c natLeq)"
+ apply (rule cinfinite_cprod2)
+ apply (rule ctwo_Cnotzero)
+ apply (rule conjI[OF _ natLeq_Card_order])
+ by (rule natLeq_cinfinite)
+next
+ fix x
+ show "|{fst x}| \<le>o ctwo *c natLeq"
+ by (rule singleton_ordLeq_ctwo_natLeq)
+next
+ fix x
+ show "|{snd x}| \<le>o ctwo *c natLeq"
+ by (rule singleton_ordLeq_ctwo_natLeq)
+next
+ fix A1 :: "'a set" and A2 :: "'b set"
+ have in_alt: "{x. {fst x} \<subseteq> A1 \<and> {snd x} \<subseteq> A2} = A1 \<times> A2" by auto
+ show "|{x. {fst x} \<subseteq> A1 \<and> {snd x} \<subseteq> A2}| \<le>o
+ ( ( |A1| +c |A2| ) +c ctwo) ^c (ctwo *c natLeq)"
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule card_of_ordIso_subst)
+ apply (rule in_alt)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule Times_cprod)
+ apply (rule ordLeq_transitive)
+ apply (rule cprod_csum_cexp)
+ apply (rule cexp_mono)
+ apply (rule ordLeq_csum1)
+ apply (rule Card_order_csum)
+ apply (rule ordLeq_cprod1)
+ apply (rule Card_order_ctwo)
+ apply (rule Cinfinite_Cnotzero)
+ apply (rule conjI[OF _ natLeq_Card_order])
+ apply (rule natLeq_cinfinite)
+ apply (rule disjI2)
+ apply (rule cone_ordLeq_cexp)
+ apply (rule ordLeq_transitive)
+ apply (rule cone_ordLeq_ctwo)
+ apply (rule ordLeq_csum2)
+ apply (rule Card_order_ctwo)
+ apply (rule notE)
+ apply (rule ctwo_not_czero)
+ apply assumption
+ by (rule Card_order_ctwo)
+next
+ fix A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22
+ assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"
+ thus "wpull {x. {fst x} \<subseteq> A1 \<and> {snd x} \<subseteq> A2}
+ {x. {fst x} \<subseteq> B11 \<and> {snd x} \<subseteq> B12} {x. {fst x} \<subseteq> B21 \<and> {snd x} \<subseteq> B22}
+ (map_pair f11 f12) (map_pair f21 f22) (map_pair p11 p12) (map_pair p21 p22)"
+ unfolding wpull_def by simp fast
+qed simp+
+
+lemma prod_pred[simp]:
+"prod_pred \<phi> \<psi> p1 p2 = (case p1 of (a1, b1) \<Rightarrow> case p2 of (a2, b2) \<Rightarrow> (\<phi> a1 a2 \<and> \<psi> b1 b2))"
+unfolding prod_set_defs prod_pred_def prod_rel_def Gr_def relcomp_unfold converse_unfold by auto
+(* TODO: pred characterization for each basic BNF *)
+
+(* Categorical version of pullback: *)
+lemma wpull_cat:
+assumes p: "wpull A B1 B2 f1 f2 p1 p2"
+and c: "f1 o q1 = f2 o q2"
+and r: "range q1 \<subseteq> B1" "range q2 \<subseteq> B2"
+obtains h where "range h \<subseteq> A \<and> q1 = p1 o h \<and> q2 = p2 o h"
+proof-
+ have *: "\<forall>d. \<exists>a \<in> A. p1 a = q1 d & p2 a = q2 d"
+ proof safe
+ fix d
+ have "f1 (q1 d) = f2 (q2 d)" using c unfolding comp_def[abs_def] by (rule fun_cong)
+ moreover
+ have "q1 d : B1" "q2 d : B2" using r unfolding image_def by auto
+ ultimately show "\<exists>a \<in> A. p1 a = q1 d \<and> p2 a = q2 d"
+ using p unfolding wpull_def by auto
+ qed
+ then obtain h where "!! d. h d \<in> A & p1 (h d) = q1 d & p2 (h d) = q2 d" by metis
+ thus ?thesis using that by fastforce
+qed
+
+lemma card_of_bounded_range:
+ "|{f :: 'd \<Rightarrow> 'a. range f \<subseteq> B}| \<le>o |Func (UNIV :: 'd set) B|" (is "|?LHS| \<le>o |?RHS|")
+proof -
+ let ?f = "\<lambda>f. %x. if f x \<in> B then Some (f x) else None"
+ have "inj_on ?f ?LHS" unfolding inj_on_def
+ proof (unfold fun_eq_iff, safe)
+ fix g :: "'d \<Rightarrow> 'a" and f :: "'d \<Rightarrow> 'a" and x
+ assume "range f \<subseteq> B" "range g \<subseteq> B" and eq: "\<forall>x. ?f f x = ?f g x"
+ hence "f x \<in> B" "g x \<in> B" by auto
+ with eq have "Some (f x) = Some (g x)" by metis
+ thus "f x = g x" by simp
+ qed
+ moreover have "?f ` ?LHS \<subseteq> ?RHS" unfolding Func_def by fastforce
+ ultimately show ?thesis using card_of_ordLeq by fast
+qed
+
+bnf_def "fun" = "op \<circ>" [range] "\<lambda>_:: 'a \<Rightarrow> 'b. natLeq +c |UNIV :: 'a set|"
+ ["%c x::'b::type. c::'a::type"]
+proof
+ fix f show "id \<circ> f = id f" by simp
+next
+ fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"
+ unfolding comp_def[abs_def] ..
+next
+ fix x f g
+ assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"
+ thus "f \<circ> x = g \<circ> x" by auto
+next
+ fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"
+ unfolding image_def comp_def[abs_def] by auto
+next
+ show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")
+ apply (rule card_order_csum)
+ apply (rule natLeq_card_order)
+ by (rule card_of_card_order_on)
+(* *)
+ show "cinfinite (natLeq +c ?U)"
+ apply (rule cinfinite_csum)
+ apply (rule disjI1)
+ by (rule natLeq_cinfinite)
+next
+ fix f :: "'d => 'a"
+ have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)
+ also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)
+ finally show "|range f| \<le>o natLeq +c ?U" .
+next
+ fix B :: "'a set"
+ have "|{f::'d => 'a. range f \<subseteq> B}| \<le>o |Func (UNIV :: 'd set) B|" by (rule card_of_bounded_range)
+ also have "|Func (UNIV :: 'd set) B| =o |B| ^c |UNIV :: 'd set|"
+ unfolding cexp_def Field_card_of by (rule card_of_refl)
+ also have "|B| ^c |UNIV :: 'd set| \<le>o
+ ( |B| +c ctwo) ^c (natLeq +c |UNIV :: 'd set| )"
+ apply (rule cexp_mono)
+ apply (rule ordLeq_csum1) apply (rule card_of_Card_order)
+ apply (rule ordLeq_csum2) apply (rule card_of_Card_order)
+ apply (rule disjI2) apply (rule cone_ordLeq_cexp)
+ apply (rule ordLeq_transitive) apply (rule cone_ordLeq_ctwo) apply (rule ordLeq_csum2)
+ apply (rule Card_order_ctwo)
+ apply (rule notE) apply (rule conjunct1) apply (rule Cnotzero_UNIV) apply blast
+ apply (rule card_of_Card_order)
+ done
+ finally
+ show "|{f::'d => 'a. range f \<subseteq> B}| \<le>o
+ ( |B| +c ctwo) ^c (natLeq +c |UNIV :: 'd set| )" .
+next
+ fix A B1 B2 f1 f2 p1 p2 assume p: "wpull A B1 B2 f1 f2 p1 p2"
+ show "wpull {h. range h \<subseteq> A} {g1. range g1 \<subseteq> B1} {g2. range g2 \<subseteq> B2}
+ (op \<circ> f1) (op \<circ> f2) (op \<circ> p1) (op \<circ> p2)"
+ unfolding wpull_def
+ proof safe
+ fix g1 g2 assume r: "range g1 \<subseteq> B1" "range g2 \<subseteq> B2"
+ and c: "f1 \<circ> g1 = f2 \<circ> g2"
+ show "\<exists>h \<in> {h. range h \<subseteq> A}. p1 \<circ> h = g1 \<and> p2 \<circ> h = g2"
+ using wpull_cat[OF p c r] by simp metis
+ qed
+qed auto
+
+lemma fun_pred[simp]: "fun_pred \<phi> f g = (\<forall>x. \<phi> (f x) (g x))"
+unfolding fun_rel_def fun_pred_def Gr_def relcomp_unfold converse_unfold
+by (auto intro!: exI dest!: in_mono)
+
+lemma card_of_list_in:
+ "|{xs. set xs \<subseteq> A}| \<le>o |Pfunc (UNIV :: nat set) A|" (is "|?LHS| \<le>o |?RHS|")
+proof -
+ let ?f = "%xs. %i. if i < length xs \<and> set xs \<subseteq> A then Some (nth xs i) else None"
+ have "inj_on ?f ?LHS" unfolding inj_on_def fun_eq_iff
+ proof safe
+ fix xs :: "'a list" and ys :: "'a list"
+ assume su: "set xs \<subseteq> A" "set ys \<subseteq> A" and eq: "\<forall>i. ?f xs i = ?f ys i"
+ hence *: "length xs = length ys"
+ by (metis linorder_cases option.simps(2) order_less_irrefl)
+ thus "xs = ys" by (rule nth_equalityI) (metis * eq su option.inject)
+ qed
+ moreover have "?f ` ?LHS \<subseteq> ?RHS" unfolding Pfunc_def by fastforce
+ ultimately show ?thesis using card_of_ordLeq by blast
+qed
+
+lemma list_in_empty: "A = {} \<Longrightarrow> {x. set x \<subseteq> A} = {[]}"
+by simp
+
+lemma card_of_Func: "|Func A B| =o |B| ^c |A|"
+unfolding cexp_def Field_card_of by (rule card_of_refl)
+
+lemma not_emp_czero_notIn_ordIso_Card_order:
+"A \<noteq> {} \<Longrightarrow> ( |A|, czero) \<notin> ordIso \<and> Card_order |A|"
+ apply (rule conjI)
+ apply (metis Field_card_of czeroE)
+ by (rule card_of_Card_order)
+
+lemma list_in_bd: "|{x. set x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq"
+proof -
+ fix A :: "'a set"
+ show "|{x. set x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq"
+ proof (cases "A = {}")
+ case False thus ?thesis
+ apply -
+ apply (rule ordLeq_transitive)
+ apply (rule card_of_list_in)
+ apply (rule ordLeq_transitive)
+ apply (erule card_of_Pfunc_Pow_Func)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule Times_cprod)
+ apply (rule cprod_cinfinite_bound)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule Pow_cexp_ctwo)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule cexp_cong2)
+ apply (rule card_of_nat)
+ apply (rule Card_order_ctwo)
+ apply (rule card_of_Card_order)
+ apply (rule natLeq_Card_order)
+ apply (rule disjI1)
+ apply (rule ctwo_Cnotzero)
+ apply (rule cexp_mono1)
+ apply (rule ordLeq_csum2)
+ apply (rule Card_order_ctwo)
+ apply (rule disjI1)
+ apply (rule ctwo_Cnotzero)
+ apply (rule natLeq_Card_order)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule card_of_Func)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule cexp_cong2)
+ apply (rule card_of_nat)
+ apply (rule card_of_Card_order)
+ apply (rule card_of_Card_order)
+ apply (rule natLeq_Card_order)
+ apply (rule disjI1)
+ apply (erule not_emp_czero_notIn_ordIso_Card_order)
+ apply (rule cexp_mono1)
+ apply (rule ordLeq_csum1)
+ apply (rule card_of_Card_order)
+ apply (rule disjI1)
+ apply (erule not_emp_czero_notIn_ordIso_Card_order)
+ apply (rule natLeq_Card_order)
+ apply (rule card_of_Card_order)
+ apply (rule card_of_Card_order)
+ apply (rule Cinfinite_cexp)
+ apply (rule ordLeq_csum2)
+ apply (rule Card_order_ctwo)
+ apply (rule conjI)
+ apply (rule natLeq_cinfinite)
+ by (rule natLeq_Card_order)
+ next
+ case True thus ?thesis
+ apply -
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule card_of_ordIso_subst)
+ apply (erule list_in_empty)
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule single_cone)
+ apply (rule cone_ordLeq_cexp)
+ apply (rule ordLeq_transitive)
+ apply (rule cone_ordLeq_ctwo)
+ apply (rule ordLeq_csum2)
+ by (rule Card_order_ctwo)
+ qed
+qed
+
+bnf_def list = map [set] "\<lambda>_::'a list. natLeq" ["[]"]
+proof -
+ show "map id = id" by (rule List.map.id)
+next
+ fix f g
+ show "map (g o f) = map g o map f" by (rule List.map.comp[symmetric])
+next
+ fix x f g
+ assume "\<And>z. z \<in> set x \<Longrightarrow> f z = g z"
+ thus "map f x = map g x" by simp
+next
+ fix f
+ show "set o map f = image f o set" by (rule ext, unfold o_apply, rule set_map)
+next
+ show "card_order natLeq" by (rule natLeq_card_order)
+next
+ show "cinfinite natLeq" by (rule natLeq_cinfinite)
+next
+ fix x
+ show "|set x| \<le>o natLeq"
+ apply (rule ordLess_imp_ordLeq)
+ apply (rule finite_ordLess_infinite[OF _ natLeq_Well_order])
+ unfolding Field_natLeq Field_card_of by (auto simp: card_of_well_order_on)
+next
+ fix A :: "'a set"
+ show "|{x. set x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" by (rule list_in_bd)
+next
+ fix A B1 B2 f1 f2 p1 p2
+ assume "wpull A B1 B2 f1 f2 p1 p2"
+ hence pull: "\<And>b1 b2. b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2 \<Longrightarrow> \<exists>a \<in> A. p1 a = b1 \<and> p2 a = b2"
+ unfolding wpull_def by auto
+ show "wpull {x. set x \<subseteq> A} {x. set x \<subseteq> B1} {x. set x \<subseteq> B2} (map f1) (map f2) (map p1) (map p2)"
+ (is "wpull ?A ?B1 ?B2 _ _ _ _")
+ proof (unfold wpull_def)
+ { fix as bs assume *: "as \<in> ?B1" "bs \<in> ?B2" "map f1 as = map f2 bs"
+ hence "length as = length bs" by (metis length_map)
+ hence "\<exists>zs \<in> ?A. map p1 zs = as \<and> map p2 zs = bs" using *
+ proof (induct as bs rule: list_induct2)
+ case (Cons a as b bs)
+ hence "a \<in> B1" "b \<in> B2" "f1 a = f2 b" by auto
+ with pull obtain z where "z \<in> A" "p1 z = a" "p2 z = b" by blast
+ moreover
+ from Cons obtain zs where "zs \<in> ?A" "map p1 zs = as" "map p2 zs = bs" by auto
+ ultimately have "z # zs \<in> ?A" "map p1 (z # zs) = a # as \<and> map p2 (z # zs) = b # bs" by auto
+ thus ?case by (rule_tac x = "z # zs" in bexI)
+ qed simp
+ }
+ thus "\<forall>as bs. as \<in> ?B1 \<and> bs \<in> ?B2 \<and> map f1 as = map f2 bs \<longrightarrow>
+ (\<exists>zs \<in> ?A. map p1 zs = as \<and> map p2 zs = bs)" by blast
+ qed
+qed auto
+
+bnf_def deadlist = "map id" [] "\<lambda>_::'a list. |lists (UNIV :: 'a set)|" ["[]"]
+by (auto simp add: cinfinite_def wpull_def infinite_UNIV_listI map.id
+ ordLeq_transitive ctwo_def card_of_card_order_on Field_card_of card_of_mono1 ordLeq_cexp2)
+
+(* Finite sets *)
+abbreviation afset where "afset \<equiv> abs_fset"
+abbreviation rfset where "rfset \<equiv> rep_fset"
+
+lemma fset_fset_member:
+"fset A = {a. a |\<in>| A}"
+unfolding fset_def fset_member_def by auto
+
+lemma afset_rfset:
+"afset (rfset x) = x"
+by (rule Quotient_fset[unfolded Quotient_def, THEN conjunct1, rule_format])
+
+lemma afset_rfset_id:
+"afset o rfset = id"
+unfolding comp_def afset_rfset id_def ..
+
+lemma rfset:
+"rfset A = rfset B \<longleftrightarrow> A = B"
+by (metis afset_rfset)
+
+lemma afset_set:
+"afset as = afset bs \<longleftrightarrow> set as = set bs"
+using Quotient_fset unfolding Quotient_def list_eq_def by auto
+
+lemma surj_afset:
+"\<exists> as. A = afset as"
+by (metis afset_rfset)
+
+lemma fset_def2:
+"fset = set o rfset"
+unfolding fset_def map_fun_def[abs_def] by simp
+
+lemma fset_def2_raw:
+"fset A = set (rfset A)"
+unfolding fset_def2 by simp
+
+lemma fset_comp_afset:
+"fset o afset = set"
+unfolding fset_def2 comp_def apply(rule ext)
+unfolding afset_set[symmetric] afset_rfset ..
+
+lemma fset_afset:
+"fset (afset as) = set as"
+unfolding fset_comp_afset[symmetric] by simp
+
+lemma set_rfset_afset:
+"set (rfset (afset as)) = set as"
+unfolding afset_set[symmetric] afset_rfset ..
+
+lemma map_fset_comp_afset:
+"(map_fset f) o afset = afset o (map f)"
+unfolding map_fset_def map_fun_def[abs_def] comp_def apply(rule ext)
+unfolding afset_set set_map set_rfset_afset id_apply ..
+
+lemma map_fset_afset:
+"(map_fset f) (afset as) = afset (map f as)"
+using map_fset_comp_afset unfolding comp_def fun_eq_iff by auto
+
+lemma fset_map_fset:
+"fset (map_fset f A) = (image f) (fset A)"
+apply(subst afset_rfset[symmetric, of A])
+unfolding map_fset_afset fset_afset set_map
+unfolding fset_def2_raw ..
+
+lemma map_fset_def2:
+"map_fset f = afset o (map f) o rfset"
+unfolding map_fset_def map_fun_def[abs_def] by simp
+
+lemma map_fset_def2_raw:
+"map_fset f A = afset (map f (rfset A))"
+unfolding map_fset_def2 by simp
+
+lemma finite_ex_fset:
+assumes "finite A"
+shows "\<exists> B. fset B = A"
+by (metis assms finite_list fset_afset)
+
+lemma wpull_image:
+assumes "wpull A B1 B2 f1 f2 p1 p2"
+shows "wpull (Pow A) (Pow B1) (Pow B2) (image f1) (image f2) (image p1) (image p2)"
+unfolding wpull_def Pow_def Bex_def mem_Collect_eq proof clarify
+ fix Y1 Y2 assume Y1: "Y1 \<subseteq> B1" and Y2: "Y2 \<subseteq> B2" and EQ: "f1 ` Y1 = f2 ` Y2"
+ def X \<equiv> "{a \<in> A. p1 a \<in> Y1 \<and> p2 a \<in> Y2}"
+ show "\<exists>X\<subseteq>A. p1 ` X = Y1 \<and> p2 ` X = Y2"
+ proof (rule exI[of _ X], intro conjI)
+ show "p1 ` X = Y1"
+ proof
+ show "Y1 \<subseteq> p1 ` X"
+ proof safe
+ fix y1 assume y1: "y1 \<in> Y1"
+ then obtain y2 where y2: "y2 \<in> Y2" and eq: "f1 y1 = f2 y2" using EQ by auto
+ then obtain x where "x \<in> A" and "p1 x = y1" and "p2 x = y2"
+ using assms y1 Y1 Y2 unfolding wpull_def by blast
+ thus "y1 \<in> p1 ` X" unfolding X_def using y1 y2 by auto
+ qed
+ qed(unfold X_def, auto)
+ show "p2 ` X = Y2"
+ proof
+ show "Y2 \<subseteq> p2 ` X"
+ proof safe
+ fix y2 assume y2: "y2 \<in> Y2"
+ then obtain y1 where y1: "y1 \<in> Y1" and eq: "f1 y1 = f2 y2" using EQ by force
+ then obtain x where "x \<in> A" and "p1 x = y1" and "p2 x = y2"
+ using assms y2 Y1 Y2 unfolding wpull_def by blast
+ thus "y2 \<in> p2 ` X" unfolding X_def using y1 y2 by auto
+ qed
+ qed(unfold X_def, auto)
+ qed(unfold X_def, auto)
+qed
+
+lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
+by (rule f_the_inv_into_f) (auto simp: inj_on_def fset_cong dest!: finite_ex_fset)
+
+bnf_def fset = map_fset [fset] "\<lambda>_::'a fset. natLeq" ["{||}"]
+proof -
+ show "map_fset id = id"
+ unfolding map_fset_def2 map_id o_id afset_rfset_id ..
+next
+ fix f g
+ show "map_fset (g o f) = map_fset g o map_fset f"
+ unfolding map_fset_def2 map.comp[symmetric] comp_def apply(rule ext)
+ unfolding afset_set set_map fset_def2_raw[symmetric] image_image[symmetric]
+ unfolding map_fset_afset[symmetric] map_fset_image afset_rfset
+ by (rule refl)
+next
+ fix x f g
+ assume "\<And>z. z \<in> fset x \<Longrightarrow> f z = g z"
+ hence "map f (rfset x) = map g (rfset x)"
+ apply(intro map_cong) unfolding fset_def2_raw by auto
+ thus "map_fset f x = map_fset g x" unfolding map_fset_def2_raw
+ by (rule arg_cong)
+next
+ fix f
+ show "fset o map_fset f = image f o fset"
+ unfolding comp_def fset_map_fset ..
+next
+ show "card_order natLeq" by (rule natLeq_card_order)
+next
+ show "cinfinite natLeq" by (rule natLeq_cinfinite)
+next
+ fix x
+ show "|fset x| \<le>o natLeq"
+ unfolding fset_def2_raw
+ apply (rule ordLess_imp_ordLeq)
+ apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
+ by (rule finite_set)
+next
+ fix A :: "'a set"
+ have "|{x. fset x \<subseteq> A}| \<le>o |afset ` {as. set as \<subseteq> A}|"
+ apply(rule card_of_mono1) unfolding fset_def2_raw apply auto
+ apply (rule image_eqI)
+ by (auto simp: afset_rfset)
+ also have "|afset ` {as. set as \<subseteq> A}| \<le>o |{as. set as \<subseteq> A}|" using card_of_image .
+ also have "|{as. set as \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" by (rule list_in_bd)
+ finally show "|{x. fset x \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" .
+next
+ fix A B1 B2 f1 f2 p1 p2
+ assume wp: "wpull A B1 B2 f1 f2 p1 p2"
+ hence "wpull (Pow A) (Pow B1) (Pow B2) (image f1) (image f2) (image p1) (image p2)"
+ by(rule wpull_image)
+ show "wpull {x. fset x \<subseteq> A} {x. fset x \<subseteq> B1} {x. fset x \<subseteq> B2}
+ (map_fset f1) (map_fset f2) (map_fset p1) (map_fset p2)"
+ unfolding wpull_def Pow_def Bex_def mem_Collect_eq proof clarify
+ fix y1 y2
+ assume Y1: "fset y1 \<subseteq> B1" and Y2: "fset y2 \<subseteq> B2"
+ assume "map_fset f1 y1 = map_fset f2 y2"
+ hence EQ: "f1 ` (fset y1) = f2 ` (fset y2)" unfolding map_fset_def2_raw
+ unfolding afset_set set_map fset_def2_raw .
+ with Y1 Y2 obtain X where X: "X \<subseteq> A"
+ and Y1: "p1 ` X = fset y1" and Y2: "p2 ` X = fset y2"
+ using wpull_image[OF wp] unfolding wpull_def Pow_def
+ unfolding Bex_def mem_Collect_eq apply -
+ apply(erule allE[of _ "fset y1"], erule allE[of _ "fset y2"]) by auto
+ have "\<forall> y1' \<in> fset y1. \<exists> x. x \<in> X \<and> y1' = p1 x" using Y1 by auto
+ then obtain q1 where q1: "\<forall> y1' \<in> fset y1. q1 y1' \<in> X \<and> y1' = p1 (q1 y1')" by metis
+ have "\<forall> y2' \<in> fset y2. \<exists> x. x \<in> X \<and> y2' = p2 x" using Y2 by auto
+ then obtain q2 where q2: "\<forall> y2' \<in> fset y2. q2 y2' \<in> X \<and> y2' = p2 (q2 y2')" by metis
+ def X' \<equiv> "q1 ` (fset y1) \<union> q2 ` (fset y2)"
+ have X': "X' \<subseteq> A" and Y1: "p1 ` X' = fset y1" and Y2: "p2 ` X' = fset y2"
+ using X Y1 Y2 q1 q2 unfolding X'_def by auto
+ have fX': "finite X'" unfolding X'_def by simp
+ then obtain x where X'eq: "X' = fset x" by (auto dest: finite_ex_fset)
+ show "\<exists>x. fset x \<subseteq> A \<and> map_fset p1 x = y1 \<and> map_fset p2 x = y2"
+ apply(intro exI[of _ "x"]) using X' Y1 Y2
+ unfolding X'eq map_fset_def2_raw fset_def2_raw set_map[symmetric]
+ afset_set[symmetric] afset_rfset by simp
+ qed
+qed auto
+
+lemma fset_pred[simp]: "fset_pred R a b \<longleftrightarrow>
+ ((\<forall>t \<in> fset a. (\<exists>u \<in> fset b. R t u)) \<and>
+ (\<forall>t \<in> fset b. (\<exists>u \<in> fset a. R u t)))" (is "?L = ?R")
+proof
+ assume ?L thus ?R unfolding fset_rel_def fset_pred_def
+ Gr_def relcomp_unfold converse_unfold
+ apply (simp add: subset_eq Ball_def)
+ apply (rule conjI)
+ apply (clarsimp, metis snd_conv)
+ by (clarsimp, metis fst_conv)
+next
+ assume ?R
+ def R' \<equiv> "the_inv fset (Collect (split R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?R'")
+ have "finite ?R'" by (intro finite_Int[OF disjI2] finite_cartesian_product) auto
+ hence *: "fset R' = ?R'" unfolding R'_def by (intro fset_to_fset)
+ show ?L unfolding fset_rel_def fset_pred_def Gr_def relcomp_unfold converse_unfold
+ proof (intro CollectI prod_caseI exI conjI)
+ from * show "(R', a) = (R', map_fset fst R')" using conjunct1[OF `?R`]
+ by (auto simp add: fset_cong[symmetric] image_def Int_def split: prod.splits)
+ from * show "(R', b) = (R', map_fset snd R')" using conjunct2[OF `?R`]
+ by (auto simp add: fset_cong[symmetric] image_def Int_def split: prod.splits)
+ qed (auto simp add: *)
+qed
+
+(* Countable sets *)
+
+lemma card_of_countable_sets_range:
+fixes A :: "'a set"
+shows "|{X. X \<subseteq> A \<and> countable X \<and> X \<noteq> {}}| \<le>o |{f::nat \<Rightarrow> 'a. range f \<subseteq> A}|"
+apply(rule card_of_ordLeqI[of fromNat]) using inj_on_fromNat
+unfolding inj_on_def by auto
+
+lemma card_of_countable_sets_Func:
+"|{X. X \<subseteq> A \<and> countable X \<and> X \<noteq> {}}| \<le>o |A| ^c natLeq"
+using card_of_countable_sets_range card_of_Func_UNIV[THEN ordIso_symmetric]
+unfolding cexp_def Field_natLeq Field_card_of
+by(rule ordLeq_ordIso_trans)
+
+lemma ordLeq_countable_subsets:
+"|A| \<le>o |{X. X \<subseteq> A \<and> countable X}|"
+apply(rule card_of_ordLeqI[of "\<lambda> a. {a}"]) unfolding inj_on_def by auto
+
+lemma finite_countable_subset:
+"finite {X. X \<subseteq> A \<and> countable X} \<longleftrightarrow> finite A"
+apply default
+ apply (erule contrapos_pp)
+ apply (rule card_of_ordLeq_infinite)
+ apply (rule ordLeq_countable_subsets)
+ apply assumption
+apply (rule finite_Collect_conjI)
+apply (rule disjI1)
+by (erule finite_Collect_subsets)
+
+lemma card_of_countable_sets:
+"|{X. X \<subseteq> A \<and> countable X}| \<le>o ( |A| +c ctwo) ^c natLeq"
+(is "|?L| \<le>o _")
+proof(cases "finite A")
+ let ?R = "Func (UNIV::nat set) (A <+> (UNIV::bool set))"
+ case True hence "finite ?L" by simp
+ moreover have "infinite ?R"
+ apply(rule infinite_Func[of _ "Inr True" "Inr False"]) by auto
+ ultimately show ?thesis unfolding cexp_def csum_def ctwo_def Field_natLeq Field_card_of
+ apply(intro ordLess_imp_ordLeq) by (rule finite_ordLess_infinite2)
+next
+ case False
+ hence "|{X. X \<subseteq> A \<and> countable X}| =o |{X. X \<subseteq> A \<and> countable X} - {{}}|"
+ by (intro card_of_infinite_diff_finitte finite.emptyI finite.insertI ordIso_symmetric)
+ (unfold finite_countable_subset)
+ also have "|{X. X \<subseteq> A \<and> countable X} - {{}}| \<le>o |A| ^c natLeq"
+ using card_of_countable_sets_Func[of A] unfolding set_diff_eq by auto
+ also have "|A| ^c natLeq \<le>o ( |A| +c ctwo) ^c natLeq"
+ apply(rule cexp_mono1_cone_ordLeq)
+ apply(rule ordLeq_csum1, rule card_of_Card_order)
+ apply (rule cone_ordLeq_cexp)
+ apply (rule cone_ordLeq_Cnotzero)
+ using csum_Cnotzero2 ctwo_Cnotzero apply blast
+ by (rule natLeq_Card_order)
+ finally show ?thesis .
+qed
+
+bnf_def cset = cIm [rcset] "\<lambda>_::'a cset. natLeq" ["cEmp"]
+proof -
+ show "cIm id = id" unfolding cIm_def[abs_def] id_def by auto
+next
+ fix f g show "cIm (g \<circ> f) = cIm g \<circ> cIm f"
+ unfolding cIm_def[abs_def] apply(rule ext) unfolding comp_def by auto
+next
+ fix C f g assume eq: "\<And>a. a \<in> rcset C \<Longrightarrow> f a = g a"
+ thus "cIm f C = cIm g C"
+ unfolding cIm_def[abs_def] unfolding image_def by auto
+next
+ fix f show "rcset \<circ> cIm f = op ` f \<circ> rcset" unfolding cIm_def[abs_def] by auto
+next
+ show "card_order natLeq" by (rule natLeq_card_order)
+next
+ show "cinfinite natLeq" by (rule natLeq_cinfinite)
+next
+ fix C show "|rcset C| \<le>o natLeq" using rcset unfolding countable_def .
+next
+ fix A :: "'a set"
+ have "|{Z. rcset Z \<subseteq> A}| \<le>o |acset ` {X. X \<subseteq> A \<and> countable X}|"
+ apply(rule card_of_mono1) unfolding Pow_def image_def
+ proof (rule Collect_mono, clarsimp)
+ fix x
+ assume "rcset x \<subseteq> A"
+ hence "rcset x \<subseteq> A \<and> countable (rcset x) \<and> x = acset (rcset x)"
+ using acset_rcset[of x] rcset[of x] by force
+ thus "\<exists>y \<subseteq> A. countable y \<and> x = acset y" by blast
+ qed
+ also have "|acset ` {X. X \<subseteq> A \<and> countable X}| \<le>o |{X. X \<subseteq> A \<and> countable X}|"
+ using card_of_image .
+ also have "|{X. X \<subseteq> A \<and> countable X}| \<le>o ( |A| +c ctwo) ^c natLeq"
+ using card_of_countable_sets .
+ finally show "|{Z. rcset Z \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" .
+next
+ fix A B1 B2 f1 f2 p1 p2
+ assume wp: "wpull A B1 B2 f1 f2 p1 p2"
+ show "wpull {x. rcset x \<subseteq> A} {x. rcset x \<subseteq> B1} {x. rcset x \<subseteq> B2}
+ (cIm f1) (cIm f2) (cIm p1) (cIm p2)"
+ unfolding wpull_def proof safe
+ fix y1 y2
+ assume Y1: "rcset y1 \<subseteq> B1" and Y2: "rcset y2 \<subseteq> B2"
+ assume "cIm f1 y1 = cIm f2 y2"
+ hence EQ: "f1 ` (rcset y1) = f2 ` (rcset y2)"
+ unfolding cIm_def by auto
+ with Y1 Y2 obtain X where X: "X \<subseteq> A"
+ and Y1: "p1 ` X = rcset y1" and Y2: "p2 ` X = rcset y2"
+ using wpull_image[OF wp] unfolding wpull_def Pow_def
+ unfolding Bex_def mem_Collect_eq apply -
+ apply(erule allE[of _ "rcset y1"], erule allE[of _ "rcset y2"]) by auto
+ have "\<forall> y1' \<in> rcset y1. \<exists> x. x \<in> X \<and> y1' = p1 x" using Y1 by auto
+ then obtain q1 where q1: "\<forall> y1' \<in> rcset y1. q1 y1' \<in> X \<and> y1' = p1 (q1 y1')" by metis
+ have "\<forall> y2' \<in> rcset y2. \<exists> x. x \<in> X \<and> y2' = p2 x" using Y2 by auto
+ then obtain q2 where q2: "\<forall> y2' \<in> rcset y2. q2 y2' \<in> X \<and> y2' = p2 (q2 y2')" by metis
+ def X' \<equiv> "q1 ` (rcset y1) \<union> q2 ` (rcset y2)"
+ have X': "X' \<subseteq> A" and Y1: "p1 ` X' = rcset y1" and Y2: "p2 ` X' = rcset y2"
+ using X Y1 Y2 q1 q2 unfolding X'_def by fast+
+ have fX': "countable X'" unfolding X'_def by simp
+ then obtain x where X'eq: "X' = rcset x" by (metis rcset_acset)
+ show "\<exists>x\<in>{x. rcset x \<subseteq> A}. cIm p1 x = y1 \<and> cIm p2 x = y2"
+ apply(intro bexI[of _ "x"]) using X' Y1 Y2 unfolding X'eq cIm_def by auto
+ qed
+qed (unfold cEmp_def, auto)
+
+
+(* Multisets *)
+
+lemma setsum_gt_0_iff:
+fixes f :: "'a \<Rightarrow> nat" assumes "finite A"
+shows "setsum f A > 0 \<longleftrightarrow> (\<exists> a \<in> A. f a > 0)"
+(is "?L \<longleftrightarrow> ?R")
+proof-
+ have "?L \<longleftrightarrow> \<not> setsum f A = 0" by fast
+ also have "... \<longleftrightarrow> (\<exists> a \<in> A. f a \<noteq> 0)" using assms by simp
+ also have "... \<longleftrightarrow> ?R" by simp
+ finally show ?thesis .
+qed
+
+(* *)
+definition mmap :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> nat) \<Rightarrow> 'b \<Rightarrow> nat" where
+"mmap h f b = setsum f {a. h a = b \<and> f a > 0}"
+
+lemma mmap_id: "mmap id = id"
+proof (rule ext)+
+ fix f a show "mmap id f a = id f a"
+ proof(cases "f a = 0")
+ case False
+ hence 1: "{aa. aa = a \<and> 0 < f aa} = {a}" by auto
+ show ?thesis by (simp add: mmap_def id_apply 1)
+ qed(unfold mmap_def, auto)
+qed
+
+lemma inj_on_setsum_inv:
+assumes f: "f \<in> multiset"
+and 1: "(0::nat) < setsum f {a. h a = b' \<and> 0 < f a}" (is "0 < setsum f ?A'")
+and 2: "{a. h a = b \<and> 0 < f a} = {a. h a = b' \<and> 0 < f a}" (is "?A = ?A'")
+shows "b = b'"
+proof-
+ have "finite ?A'" using f unfolding multiset_def by auto
+ hence "?A' \<noteq> {}" using 1 setsum_gt_0_iff by auto
+ thus ?thesis using 2 by auto
+qed
+
+lemma mmap_comp:
+fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
+assumes f: "f \<in> multiset"
+shows "mmap (h2 o h1) f = (mmap h2 o mmap h1) f"
+unfolding mmap_def[abs_def] comp_def proof(rule ext)+
+ fix c :: 'c
+ let ?A = "{a. h2 (h1 a) = c \<and> 0 < f a}"
+ let ?As = "\<lambda> b. {a. h1 a = b \<and> 0 < f a}"
+ let ?B = "{b. h2 b = c \<and> 0 < setsum f (?As b)}"
+ have 0: "{?As b | b. b \<in> ?B} = ?As ` ?B" by auto
+ have "\<And> b. finite (?As b)" using f unfolding multiset_def by simp
+ hence "?B = {b. h2 b = c \<and> ?As b \<noteq> {}}" using setsum_gt_0_iff by auto
+ hence A: "?A = \<Union> {?As b | b. b \<in> ?B}" by auto
+ have "setsum f ?A = setsum (setsum f) {?As b | b. b \<in> ?B}"
+ unfolding A apply(rule setsum_Union_disjoint)
+ using f unfolding multiset_def by auto
+ also have "... = setsum (setsum f) (?As ` ?B)" unfolding 0 ..
+ also have "... = setsum (setsum f o ?As) ?B" apply(rule setsum_reindex)
+ unfolding inj_on_def apply auto using inj_on_setsum_inv[OF f, of h1] by blast
+ also have "... = setsum (\<lambda> b. setsum f (?As b)) ?B" unfolding comp_def ..
+ finally show "setsum f ?A = setsum (\<lambda> b. setsum f (?As b)) ?B" .
+qed
+
+lemma mmap_comp1:
+fixes h1 :: "'a \<Rightarrow> 'b" and h2 :: "'b \<Rightarrow> 'c"
+assumes "f \<in> multiset"
+shows "mmap (\<lambda> a. h2 (h1 a)) f = mmap h2 (mmap h1 f)"
+using mmap_comp[OF assms] unfolding comp_def by auto
+
+lemma mmap:
+assumes "f \<in> multiset"
+shows "mmap h f \<in> multiset"
+using assms unfolding mmap_def[abs_def] multiset_def proof safe
+ assume fin: "finite {a. 0 < f a}" (is "finite ?A")
+ show "finite {b. 0 < setsum f {a. h a = b \<and> 0 < f a}}"
+ (is "finite {b. 0 < setsum f (?As b)}")
+ proof- let ?B = "{b. 0 < setsum f (?As b)}"
+ have "\<And> b. finite (?As b)" using assms unfolding multiset_def by simp
+ hence B: "?B = {b. ?As b \<noteq> {}}" using setsum_gt_0_iff by auto
+ hence "?B \<subseteq> h ` ?A" by auto
+ thus ?thesis using finite_surj[OF fin] by auto
+ qed
+qed
+
+lemma mmap_cong:
+assumes "\<And>a. a \<in># M \<Longrightarrow> f a = g a"
+shows "mmap f (count M) = mmap g (count M)"
+using assms unfolding mmap_def[abs_def] by (intro ext, intro setsum_cong) auto
+
+abbreviation supp where "supp f \<equiv> {a. f a > 0}"
+
+lemma mmap_image_comp:
+assumes f: "f \<in> multiset"
+shows "(supp o mmap h) f = (image h o supp) f"
+unfolding mmap_def[abs_def] comp_def proof-
+ have "\<And> b. finite {a. h a = b \<and> 0 < f a}" (is "\<And> b. finite (?As b)")
+ using f unfolding multiset_def by auto
+ thus "{b. 0 < setsum f (?As b)} = h ` {a. 0 < f a}"
+ using setsum_gt_0_iff by auto
+qed
+
+lemma mmap_image:
+assumes f: "f \<in> multiset"
+shows "supp (mmap h f) = h ` (supp f)"
+using mmap_image_comp[OF assms] unfolding comp_def .
+
+lemma set_of_Abs_multiset:
+assumes f: "f \<in> multiset"
+shows "set_of (Abs_multiset f) = supp f"
+using assms unfolding set_of_def by (auto simp: Abs_multiset_inverse)
+
+lemma supp_count:
+"supp (count M) = set_of M"
+using assms unfolding set_of_def by auto
+
+lemma multiset_of_surj:
+"multiset_of ` {as. set as \<subseteq> A} = {M. set_of M \<subseteq> A}"
+proof safe
+ fix M assume M: "set_of M \<subseteq> A"
+ obtain as where eq: "M = multiset_of as" using surj_multiset_of unfolding surj_def by auto
+ hence "set as \<subseteq> A" using M by auto
+ thus "M \<in> multiset_of ` {as. set as \<subseteq> A}" using eq by auto
+next
+ show "\<And>x xa xb. \<lbrakk>set xa \<subseteq> A; xb \<in> set_of (multiset_of xa)\<rbrakk> \<Longrightarrow> xb \<in> A"
+ by (erule set_mp) (unfold set_of_multiset_of)
+qed
+
+lemma card_of_set_of:
+"|{M. set_of M \<subseteq> A}| \<le>o |{as. set as \<subseteq> A}|"
+apply(rule card_of_ordLeqI2[of _ multiset_of]) using multiset_of_surj by auto
+
+lemma nat_sum_induct:
+assumes "\<And>n1 n2. (\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow> phi m1 m2) \<Longrightarrow> phi n1 n2"
+shows "phi (n1::nat) (n2::nat)"
+proof-
+ let ?chi = "\<lambda> n1n2 :: nat * nat. phi (fst n1n2) (snd n1n2)"
+ have "?chi (n1,n2)"
+ apply(induct rule: measure_induct[of "\<lambda> n1n2. fst n1n2 + snd n1n2" ?chi])
+ using assms by (metis fstI sndI)
+ thus ?thesis by simp
+qed
+
+lemma matrix_count:
+fixes ct1 ct2 :: "nat \<Rightarrow> nat"
+assumes "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
+shows
+"\<exists> ct. (\<forall> i1 \<le> n1. setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ct1 i1) \<and>
+ (\<forall> i2 \<le> n2. setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ct2 i2)"
+(is "?phi ct1 ct2 n1 n2")
+proof-
+ have "\<forall> ct1 ct2 :: nat \<Rightarrow> nat.
+ setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"
+ proof(induct rule: nat_sum_induct[of
+"\<lambda> n1 n2. \<forall> ct1 ct2 :: nat \<Rightarrow> nat.
+ setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2} \<longrightarrow> ?phi ct1 ct2 n1 n2"],
+ clarify)
+ fix n1 n2 :: nat and ct1 ct2 :: "nat \<Rightarrow> nat"
+ assume IH: "\<And> m1 m2. m1 + m2 < n1 + n2 \<Longrightarrow>
+ \<forall> dt1 dt2 :: nat \<Rightarrow> nat.
+ setsum dt1 {..<Suc m1} = setsum dt2 {..<Suc m2} \<longrightarrow> ?phi dt1 dt2 m1 m2"
+ and ss: "setsum ct1 {..<Suc n1} = setsum ct2 {..<Suc n2}"
+ show "?phi ct1 ct2 n1 n2"
+ proof(cases n1)
+ case 0 note n1 = 0
+ show ?thesis
+ proof(cases n2)
+ case 0 note n2 = 0
+ let ?ct = "\<lambda> i1 i2. ct2 0"
+ show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by simp
+ next
+ case (Suc m2) note n2 = Suc
+ let ?ct = "\<lambda> i1 i2. ct2 i2"
+ show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
+ qed
+ next
+ case (Suc m1) note n1 = Suc
+ show ?thesis
+ proof(cases n2)
+ case 0 note n2 = 0
+ let ?ct = "\<lambda> i1 i2. ct1 i1"
+ show ?thesis apply(rule exI[of _ ?ct]) using n1 n2 ss by auto
+ next
+ case (Suc m2) note n2 = Suc
+ show ?thesis
+ proof(cases "ct1 n1 \<le> ct2 n2")
+ case True
+ def dt2 \<equiv> "\<lambda> i2. if i2 = n2 then ct2 i2 - ct1 n1 else ct2 i2"
+ have "setsum ct1 {..<Suc m1} = setsum dt2 {..<Suc n2}"
+ unfolding dt2_def using ss n1 True by auto
+ hence "?phi ct1 dt2 m1 n2" using IH[of m1 n2] n1 by simp
+ then obtain dt where
+ 1: "\<And> i1. i1 \<le> m1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc n2} = ct1 i1" and
+ 2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc m1} = dt2 i2" by auto
+ let ?ct = "\<lambda> i1 i2. if i1 = n1 then (if i2 = n2 then ct1 n1 else 0)
+ else dt i1 i2"
+ show ?thesis apply(rule exI[of _ ?ct])
+ using n1 n2 1 2 True unfolding dt2_def by simp
+ next
+ case False
+ hence False: "ct2 n2 < ct1 n1" by simp
+ def dt1 \<equiv> "\<lambda> i1. if i1 = n1 then ct1 i1 - ct2 n2 else ct1 i1"
+ have "setsum dt1 {..<Suc n1} = setsum ct2 {..<Suc m2}"
+ unfolding dt1_def using ss n2 False by auto
+ hence "?phi dt1 ct2 n1 m2" using IH[of n1 m2] n2 by simp
+ then obtain dt where
+ 1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. dt i1 i2) {..<Suc m2} = dt1 i1" and
+ 2: "\<And> i2. i2 \<le> m2 \<Longrightarrow> setsum (\<lambda> i1. dt i1 i2) {..<Suc n1} = ct2 i2" by force
+ let ?ct = "\<lambda> i1 i2. if i2 = n2 then (if i1 = n1 then ct2 n2 else 0)
+ else dt i1 i2"
+ show ?thesis apply(rule exI[of _ ?ct])
+ using n1 n2 1 2 False unfolding dt1_def by simp
+ qed
+ qed
+ qed
+ qed
+ thus ?thesis using assms by auto
+qed
+
+definition
+"inj2 u B1 B2 \<equiv>
+ \<forall> b1 b1' b2 b2'. {b1,b1'} \<subseteq> B1 \<and> {b2,b2'} \<subseteq> B2 \<and> u b1 b2 = u b1' b2'
+ \<longrightarrow> b1 = b1' \<and> b2 = b2'"
+
+lemma matrix_count_finite:
+assumes B1: "B1 \<noteq> {}" "finite B1" and B2: "B2 \<noteq> {}" "finite B2" and u: "inj2 u B1 B2"
+and ss: "setsum N1 B1 = setsum N2 B2"
+shows "\<exists> M :: 'a \<Rightarrow> nat.
+ (\<forall> b1 \<in> B1. setsum (\<lambda> b2. M (u b1 b2)) B2 = N1 b1) \<and>
+ (\<forall> b2 \<in> B2. setsum (\<lambda> b1. M (u b1 b2)) B1 = N2 b2)"
+proof-
+ obtain n1 where "card B1 = Suc n1" using B1 by (metis card_insert finite.simps)
+ then obtain e1 where e1: "bij_betw e1 {..<Suc n1} B1"
+ using ex_bij_betw_finite_nat[OF B1(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
+ hence e1_inj: "inj_on e1 {..<Suc n1}" and e1_surj: "e1 ` {..<Suc n1} = B1"
+ unfolding bij_betw_def by auto
+ def f1 \<equiv> "inv_into {..<Suc n1} e1"
+ have f1: "bij_betw f1 B1 {..<Suc n1}"
+ and f1e1[simp]: "\<And> i1. i1 < Suc n1 \<Longrightarrow> f1 (e1 i1) = i1"
+ and e1f1[simp]: "\<And> b1. b1 \<in> B1 \<Longrightarrow> e1 (f1 b1) = b1" unfolding f1_def
+ apply (metis bij_betw_inv_into e1, metis bij_betw_inv_into_left e1 lessThan_iff)
+ by (metis e1_surj f_inv_into_f)
+ (* *)
+ obtain n2 where "card B2 = Suc n2" using B2 by (metis card_insert finite.simps)
+ then obtain e2 where e2: "bij_betw e2 {..<Suc n2} B2"
+ using ex_bij_betw_finite_nat[OF B2(2)] by (metis atLeast0LessThan bij_betw_the_inv_into)
+ hence e2_inj: "inj_on e2 {..<Suc n2}" and e2_surj: "e2 ` {..<Suc n2} = B2"
+ unfolding bij_betw_def by auto
+ def f2 \<equiv> "inv_into {..<Suc n2} e2"
+ have f2: "bij_betw f2 B2 {..<Suc n2}"
+ and f2e2[simp]: "\<And> i2. i2 < Suc n2 \<Longrightarrow> f2 (e2 i2) = i2"
+ and e2f2[simp]: "\<And> b2. b2 \<in> B2 \<Longrightarrow> e2 (f2 b2) = b2" unfolding f2_def
+ apply (metis bij_betw_inv_into e2, metis bij_betw_inv_into_left e2 lessThan_iff)
+ by (metis e2_surj f_inv_into_f)
+ (* *)
+ let ?ct1 = "N1 o e1" let ?ct2 = "N2 o e2"
+ have ss: "setsum ?ct1 {..<Suc n1} = setsum ?ct2 {..<Suc n2}"
+ unfolding setsum_reindex[OF e1_inj, symmetric] setsum_reindex[OF e2_inj, symmetric]
+ e1_surj e2_surj using ss .
+ obtain ct where
+ ct1: "\<And> i1. i1 \<le> n1 \<Longrightarrow> setsum (\<lambda> i2. ct i1 i2) {..<Suc n2} = ?ct1 i1" and
+ ct2: "\<And> i2. i2 \<le> n2 \<Longrightarrow> setsum (\<lambda> i1. ct i1 i2) {..<Suc n1} = ?ct2 i2"
+ using matrix_count[OF ss] by blast
+ (* *)
+ def A \<equiv> "{u b1 b2 | b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}"
+ have "\<forall> a \<in> A. \<exists> b1b2 \<in> B1 <*> B2. u (fst b1b2) (snd b1b2) = a"
+ unfolding A_def Ball_def mem_Collect_eq by auto
+ then obtain h1h2 where h12:
+ "\<And>a. a \<in> A \<Longrightarrow> u (fst (h1h2 a)) (snd (h1h2 a)) = a \<and> h1h2 a \<in> B1 <*> B2" by metis
+ def h1 \<equiv> "fst o h1h2" def h2 \<equiv> "snd o h1h2"
+ have h12[simp]: "\<And>a. a \<in> A \<Longrightarrow> u (h1 a) (h2 a) = a"
+ "\<And> a. a \<in> A \<Longrightarrow> h1 a \<in> B1" "\<And> a. a \<in> A \<Longrightarrow> h2 a \<in> B2"
+ using h12 unfolding h1_def h2_def by force+
+ {fix b1 b2 assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2"
+ hence inA: "u b1 b2 \<in> A" unfolding A_def by auto
+ hence "u b1 b2 = u (h1 (u b1 b2)) (h2 (u b1 b2))" by auto
+ moreover have "h1 (u b1 b2) \<in> B1" "h2 (u b1 b2) \<in> B2" using inA by auto
+ ultimately have "h1 (u b1 b2) = b1 \<and> h2 (u b1 b2) = b2"
+ using u b1 b2 unfolding inj2_def by fastforce
+ }
+ hence h1[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h1 (u b1 b2) = b1" and
+ h2[simp]: "\<And> b1 b2. \<lbrakk>b1 \<in> B1; b2 \<in> B2\<rbrakk> \<Longrightarrow> h2 (u b1 b2) = b2" by auto
+ def M \<equiv> "\<lambda> a. ct (f1 (h1 a)) (f2 (h2 a))"
+ show ?thesis
+ apply(rule exI[of _ M]) proof safe
+ fix b1 assume b1: "b1 \<in> B1"
+ hence f1b1: "f1 b1 \<le> n1" using f1 unfolding bij_betw_def
+ by (metis bij_betwE f1 lessThan_iff less_Suc_eq_le)
+ have "(\<Sum>b2\<in>B2. M (u b1 b2)) = (\<Sum>i2<Suc n2. ct (f1 b1) (f2 (e2 i2)))"
+ unfolding e2_surj[symmetric] setsum_reindex[OF e2_inj]
+ unfolding M_def comp_def apply(intro setsum_cong) apply force
+ by (metis e2_surj b1 h1 h2 imageI)
+ also have "... = N1 b1" using b1 ct1[OF f1b1] by simp
+ finally show "(\<Sum>b2\<in>B2. M (u b1 b2)) = N1 b1" .
+ next
+ fix b2 assume b2: "b2 \<in> B2"
+ hence f2b2: "f2 b2 \<le> n2" using f2 unfolding bij_betw_def
+ by (metis bij_betwE f2 lessThan_iff less_Suc_eq_le)
+ have "(\<Sum>b1\<in>B1. M (u b1 b2)) = (\<Sum>i1<Suc n1. ct (f1 (e1 i1)) (f2 b2))"
+ unfolding e1_surj[symmetric] setsum_reindex[OF e1_inj]
+ unfolding M_def comp_def apply(intro setsum_cong) apply force
+ by (metis e1_surj b2 h1 h2 imageI)
+ also have "... = N2 b2" using b2 ct2[OF f2b2] by simp
+ finally show "(\<Sum>b1\<in>B1. M (u b1 b2)) = N2 b2" .
+ qed
+qed
+
+lemma supp_vimage_mmap:
+assumes "M \<in> multiset"
+shows "supp M \<subseteq> f -` (supp (mmap f M))"
+using assms by (auto simp: mmap_image)
+
+lemma mmap_ge_0:
+assumes "M \<in> multiset"
+shows "0 < mmap f M b \<longleftrightarrow> (\<exists>a. 0 < M a \<and> f a = b)"
+proof-
+ have f: "finite {a. f a = b \<and> 0 < M a}" using assms unfolding multiset_def by auto
+ show ?thesis unfolding mmap_def setsum_gt_0_iff[OF f] by auto
+qed
+
+lemma finite_twosets:
+assumes "finite B1" and "finite B2"
+shows "finite {u b1 b2 |b1 b2. b1 \<in> B1 \<and> b2 \<in> B2}" (is "finite ?A")
+proof-
+ have A: "?A = (\<lambda> b1b2. u (fst b1b2) (snd b1b2)) ` (B1 <*> B2)" by force
+ show ?thesis unfolding A using finite_cartesian_product[OF assms] by auto
+qed
+
+lemma wp_mmap:
+fixes A :: "'a set" and B1 :: "'b1 set" and B2 :: "'b2 set"
+assumes wp: "wpull A B1 B2 f1 f2 p1 p2"
+shows
+"wpull {M. M \<in> multiset \<and> supp M \<subseteq> A}
+ {N1. N1 \<in> multiset \<and> supp N1 \<subseteq> B1} {N2. N2 \<in> multiset \<and> supp N2 \<subseteq> B2}
+ (mmap f1) (mmap f2) (mmap p1) (mmap p2)"
+unfolding wpull_def proof (safe, unfold Bex_def mem_Collect_eq)
+ fix N1 :: "'b1 \<Rightarrow> nat" and N2 :: "'b2 \<Rightarrow> nat"
+ assume mmap': "mmap f1 N1 = mmap f2 N2"
+ and N1[simp]: "N1 \<in> multiset" "supp N1 \<subseteq> B1"
+ and N2[simp]: "N2 \<in> multiset" "supp N2 \<subseteq> B2"
+ have mN1[simp]: "mmap f1 N1 \<in> multiset" using N1 by (auto simp: mmap)
+ have mN2[simp]: "mmap f2 N2 \<in> multiset" using N2 by (auto simp: mmap)
+ def P \<equiv> "mmap f1 N1"
+ have P1: "P = mmap f1 N1" and P2: "P = mmap f2 N2" unfolding P_def using mmap' by auto
+ note P = P1 P2
+ have P_mult[simp]: "P \<in> multiset" unfolding P_def using N1 by auto
+ have fin_N1[simp]: "finite (supp N1)" using N1(1) unfolding multiset_def by auto
+ have fin_N2[simp]: "finite (supp N2)" using N2(1) unfolding multiset_def by auto
+ have fin_P[simp]: "finite (supp P)" using P_mult unfolding multiset_def by auto
+ (* *)
+ def set1 \<equiv> "\<lambda> c. {b1 \<in> supp N1. f1 b1 = c}"
+ have set1[simp]: "\<And> c b1. b1 \<in> set1 c \<Longrightarrow> f1 b1 = c" unfolding set1_def by auto
+ have fin_set1: "\<And> c. c \<in> supp P \<Longrightarrow> finite (set1 c)"
+ using N1(1) unfolding set1_def multiset_def by auto
+ have set1_NE: "\<And> c. c \<in> supp P \<Longrightarrow> set1 c \<noteq> {}"
+ unfolding set1_def P1 mmap_ge_0[OF N1(1)] by auto
+ have supp_N1_set1: "supp N1 = (\<Union> c \<in> supp P. set1 c)"
+ using supp_vimage_mmap[OF N1(1), of f1] unfolding set1_def P1 by auto
+ hence set1_inclN1: "\<And>c. c \<in> supp P \<Longrightarrow> set1 c \<subseteq> supp N1" by auto
+ hence set1_incl: "\<And> c. c \<in> supp P \<Longrightarrow> set1 c \<subseteq> B1" using N1(2) by blast
+ have set1_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set1 c \<inter> set1 c' = {}"
+ unfolding set1_def by auto
+ have setsum_set1: "\<And> c. setsum N1 (set1 c) = P c"
+ unfolding P1 set1_def mmap_def apply(rule setsum_cong) by auto
+ (* *)
+ def set2 \<equiv> "\<lambda> c. {b2 \<in> supp N2. f2 b2 = c}"
+ have set2[simp]: "\<And> c b2. b2 \<in> set2 c \<Longrightarrow> f2 b2 = c" unfolding set2_def by auto
+ have fin_set2: "\<And> c. c \<in> supp P \<Longrightarrow> finite (set2 c)"
+ using N2(1) unfolding set2_def multiset_def by auto
+ have set2_NE: "\<And> c. c \<in> supp P \<Longrightarrow> set2 c \<noteq> {}"
+ unfolding set2_def P2 mmap_ge_0[OF N2(1)] by auto
+ have supp_N2_set2: "supp N2 = (\<Union> c \<in> supp P. set2 c)"
+ using supp_vimage_mmap[OF N2(1), of f2] unfolding set2_def P2 by auto
+ hence set2_inclN2: "\<And>c. c \<in> supp P \<Longrightarrow> set2 c \<subseteq> supp N2" by auto
+ hence set2_incl: "\<And> c. c \<in> supp P \<Longrightarrow> set2 c \<subseteq> B2" using N2(2) by blast
+ have set2_disj: "\<And> c c'. c \<noteq> c' \<Longrightarrow> set2 c \<inter> set2 c' = {}"
+ unfolding set2_def by auto
+ have setsum_set2: "\<And> c. setsum N2 (set2 c) = P c"
+ unfolding P2 set2_def mmap_def apply(rule setsum_cong) by auto
+ (* *)
+ have ss: "\<And> c. c \<in> supp P \<Longrightarrow> setsum N1 (set1 c) = setsum N2 (set2 c)"
+ unfolding setsum_set1 setsum_set2 ..
+ have "\<forall> c \<in> supp P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
+ \<exists> a \<in> A. p1 a = fst b1b2 \<and> p2 a = snd b1b2"
+ using wp set1_incl set2_incl unfolding wpull_def Ball_def mem_Collect_eq
+ by simp (metis set1 set2 set_rev_mp)
+ then obtain uu where uu:
+ "\<forall> c \<in> supp P. \<forall> b1b2 \<in> (set1 c) \<times> (set2 c).
+ uu c b1b2 \<in> A \<and> p1 (uu c b1b2) = fst b1b2 \<and> p2 (uu c b1b2) = snd b1b2" by metis
+ def u \<equiv> "\<lambda> c b1 b2. uu c (b1,b2)"
+ have u[simp]:
+ "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> A"
+ "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p1 (u c b1 b2) = b1"
+ "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> p2 (u c b1 b2) = b2"
+ using uu unfolding u_def by auto
+ {fix c assume c: "c \<in> supp P"
+ have "inj2 (u c) (set1 c) (set2 c)" unfolding inj2_def proof clarify
+ fix b1 b1' b2 b2'
+ assume "{b1, b1'} \<subseteq> set1 c" "{b2, b2'} \<subseteq> set2 c" and 0: "u c b1 b2 = u c b1' b2'"
+ hence "p1 (u c b1 b2) = b1 \<and> p2 (u c b1 b2) = b2 \<and>
+ p1 (u c b1' b2') = b1' \<and> p2 (u c b1' b2') = b2'"
+ using u(2)[OF c] u(3)[OF c] by simp metis
+ thus "b1 = b1' \<and> b2 = b2'" using 0 by auto
+ qed
+ } note inj = this
+ def sset \<equiv> "\<lambda> c. {u c b1 b2 | b1 b2. b1 \<in> set1 c \<and> b2 \<in> set2 c}"
+ have fin_sset[simp]: "\<And> c. c \<in> supp P \<Longrightarrow> finite (sset c)" unfolding sset_def
+ using fin_set1 fin_set2 finite_twosets by blast
+ have sset_A: "\<And> c. c \<in> supp P \<Longrightarrow> sset c \<subseteq> A" unfolding sset_def by auto
+ {fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
+ then obtain b1 b2 where b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
+ and a: "a = u c b1 b2" unfolding sset_def by auto
+ have "p1 a \<in> set1 c" and p2a: "p2 a \<in> set2 c"
+ using ac a b1 b2 c u(2) u(3) by simp+
+ hence "u c (p1 a) (p2 a) = a" unfolding a using b1 b2 inj[OF c]
+ unfolding inj2_def by (metis c u(2) u(3))
+ } note u_p12[simp] = this
+ {fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
+ hence "p1 a \<in> set1 c" unfolding sset_def by auto
+ }note p1[simp] = this
+ {fix c a assume c: "c \<in> supp P" and ac: "a \<in> sset c"
+ hence "p2 a \<in> set2 c" unfolding sset_def by auto
+ }note p2[simp] = this
+ (* *)
+ {fix c assume c: "c \<in> supp P"
+ hence "\<exists> M. (\<forall> b1 \<in> set1 c. setsum (\<lambda> b2. M (u c b1 b2)) (set2 c) = N1 b1) \<and>
+ (\<forall> b2 \<in> set2 c. setsum (\<lambda> b1. M (u c b1 b2)) (set1 c) = N2 b2)"
+ unfolding sset_def
+ using matrix_count_finite[OF set1_NE[OF c] fin_set1[OF c]
+ set2_NE[OF c] fin_set2[OF c] inj[OF c] ss[OF c]] by auto
+ }
+ then obtain Ms where
+ ss1: "\<And> c b1. \<lbrakk>c \<in> supp P; b1 \<in> set1 c\<rbrakk> \<Longrightarrow>
+ setsum (\<lambda> b2. Ms c (u c b1 b2)) (set2 c) = N1 b1" and
+ ss2: "\<And> c b2. \<lbrakk>c \<in> supp P; b2 \<in> set2 c\<rbrakk> \<Longrightarrow>
+ setsum (\<lambda> b1. Ms c (u c b1 b2)) (set1 c) = N2 b2"
+ by metis
+ def SET \<equiv> "\<Union> c \<in> supp P. sset c"
+ have fin_SET[simp]: "finite SET" unfolding SET_def apply(rule finite_UN_I) by auto
+ have SET_A: "SET \<subseteq> A" unfolding SET_def using sset_A by auto
+ have u_SET[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk> \<Longrightarrow> u c b1 b2 \<in> SET"
+ unfolding SET_def sset_def by blast
+ {fix c a assume c: "c \<in> supp P" and a: "a \<in> SET" and p1a: "p1 a \<in> set1 c"
+ then obtain c' where c': "c' \<in> supp P" and ac': "a \<in> sset c'"
+ unfolding SET_def by auto
+ hence "p1 a \<in> set1 c'" unfolding sset_def by auto
+ hence eq: "c = c'" using p1a c c' set1_disj by auto
+ hence "a \<in> sset c" using ac' by simp
+ } note p1_rev = this
+ {fix c a assume c: "c \<in> supp P" and a: "a \<in> SET" and p2a: "p2 a \<in> set2 c"
+ then obtain c' where c': "c' \<in> supp P" and ac': "a \<in> sset c'"
+ unfolding SET_def by auto
+ hence "p2 a \<in> set2 c'" unfolding sset_def by auto
+ hence eq: "c = c'" using p2a c c' set2_disj by auto
+ hence "a \<in> sset c" using ac' by simp
+ } note p2_rev = this
+ (* *)
+ have "\<forall> a \<in> SET. \<exists> c \<in> supp P. a \<in> sset c" unfolding SET_def by auto
+ then obtain h where h: "\<forall> a \<in> SET. h a \<in> supp P \<and> a \<in> sset (h a)" by metis
+ have h_u[simp]: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
+ \<Longrightarrow> h (u c b1 b2) = c"
+ by (metis h p2 set2 u(3) u_SET)
+ have h_u1: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
+ \<Longrightarrow> h (u c b1 b2) = f1 b1"
+ using h unfolding sset_def by auto
+ have h_u2: "\<And> c b1 b2. \<lbrakk>c \<in> supp P; b1 \<in> set1 c; b2 \<in> set2 c\<rbrakk>
+ \<Longrightarrow> h (u c b1 b2) = f2 b2"
+ using h unfolding sset_def by auto
+ def M \<equiv> "\<lambda> a. if a \<in> SET \<and> p1 a \<in> supp N1 \<and> p2 a \<in> supp N2 then Ms (h a) a else 0"
+ have sM: "supp M \<subseteq> SET" "supp M \<subseteq> p1 -` (supp N1)" "supp M \<subseteq> p2 -` (supp N2)"
+ unfolding M_def by auto
+ show "\<exists>M. (M \<in> multiset \<and> supp M \<subseteq> A) \<and> mmap p1 M = N1 \<and> mmap p2 M = N2"
+ proof(rule exI[of _ M], safe)
+ show "M \<in> multiset"
+ unfolding multiset_def using finite_subset[OF sM(1) fin_SET] by simp
+ next
+ fix a assume "0 < M a"
+ thus "a \<in> A" unfolding M_def using SET_A by (cases "a \<in> SET") auto
+ next
+ show "mmap p1 M = N1"
+ unfolding mmap_def[abs_def] proof(rule ext)
+ fix b1
+ let ?K = "{a. p1 a = b1 \<and> 0 < M a}"
+ show "setsum M ?K = N1 b1"
+ proof(cases "b1 \<in> supp N1")
+ case False
+ hence "?K = {}" using sM(2) by auto
+ thus ?thesis using False by auto
+ next
+ case True
+ def c \<equiv> "f1 b1"
+ have c: "c \<in> supp P" and b1: "b1 \<in> set1 c"
+ unfolding set1_def c_def P1 using True by (auto simp: mmap_image)
+ have "setsum M ?K = setsum M {a. p1 a = b1 \<and> a \<in> SET}"
+ apply(rule setsum_mono_zero_cong_left) unfolding M_def by auto
+ also have "... = setsum M ((\<lambda> b2. u c b1 b2) ` (set2 c))"
+ apply(rule setsum_cong) using c b1 proof safe
+ fix a assume p1a: "p1 a \<in> set1 c" and "0 < P c" and "a \<in> SET"
+ hence ac: "a \<in> sset c" using p1_rev by auto
+ hence "a = u c (p1 a) (p2 a)" using c by auto
+ moreover have "p2 a \<in> set2 c" using ac c by auto
+ ultimately show "a \<in> u c (p1 a) ` set2 c" by auto
+ next
+ fix b2 assume b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
+ hence "u c b1 b2 \<in> SET" using c by auto
+ qed auto
+ also have "... = setsum (\<lambda> b2. M (u c b1 b2)) (set2 c)"
+ unfolding comp_def[symmetric] apply(rule setsum_reindex)
+ using inj unfolding inj_on_def inj2_def using b1 c u(3) by blast
+ also have "... = N1 b1" unfolding ss1[OF c b1, symmetric]
+ apply(rule setsum_cong[OF refl]) unfolding M_def
+ using True h_u[OF c b1] set2_def u(2,3)[OF c b1] u_SET[OF c b1] by fastforce
+ finally show ?thesis .
+ qed
+ qed
+ next
+ show "mmap p2 M = N2"
+ unfolding mmap_def[abs_def] proof(rule ext)
+ fix b2
+ let ?K = "{a. p2 a = b2 \<and> 0 < M a}"
+ show "setsum M ?K = N2 b2"
+ proof(cases "b2 \<in> supp N2")
+ case False
+ hence "?K = {}" using sM(3) by auto
+ thus ?thesis using False by auto
+ next
+ case True
+ def c \<equiv> "f2 b2"
+ have c: "c \<in> supp P" and b2: "b2 \<in> set2 c"
+ unfolding set2_def c_def P2 using True by (auto simp: mmap_image)
+ have "setsum M ?K = setsum M {a. p2 a = b2 \<and> a \<in> SET}"
+ apply(rule setsum_mono_zero_cong_left) unfolding M_def by auto
+ also have "... = setsum M ((\<lambda> b1. u c b1 b2) ` (set1 c))"
+ apply(rule setsum_cong) using c b2 proof safe
+ fix a assume p2a: "p2 a \<in> set2 c" and "0 < P c" and "a \<in> SET"
+ hence ac: "a \<in> sset c" using p2_rev by auto
+ hence "a = u c (p1 a) (p2 a)" using c by auto
+ moreover have "p1 a \<in> set1 c" using ac c by auto
+ ultimately show "a \<in> (\<lambda>b1. u c b1 (p2 a)) ` set1 c" by auto
+ next
+ fix b2 assume b1: "b1 \<in> set1 c" and b2: "b2 \<in> set2 c"
+ hence "u c b1 b2 \<in> SET" using c by auto
+ qed auto
+ also have "... = setsum (M o (\<lambda> b1. u c b1 b2)) (set1 c)"
+ apply(rule setsum_reindex)
+ using inj unfolding inj_on_def inj2_def using b2 c u(2) by blast
+ also have "... = setsum (\<lambda> b1. M (u c b1 b2)) (set1 c)"
+ unfolding comp_def[symmetric] by simp
+ also have "... = N2 b2" unfolding ss2[OF c b2, symmetric]
+ apply(rule setsum_cong[OF refl]) unfolding M_def set2_def
+ using True h_u1[OF c _ b2] u(2,3)[OF c _ b2] u_SET[OF c _ b2]
+ unfolding set1_def by fastforce
+ finally show ?thesis .
+ qed
+ qed
+ qed
+qed
+
+definition mset_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
+"mset_map h = Abs_multiset \<circ> mmap h \<circ> count"
+
+bnf_def mset = mset_map [set_of] "\<lambda>_::'a multiset. natLeq" ["{#}"]
+unfolding mset_map_def
+proof -
+ show "Abs_multiset \<circ> mmap id \<circ> count = id" unfolding mmap_id by (auto simp: count_inverse)
+next
+ fix f g
+ show "Abs_multiset \<circ> mmap (g \<circ> f) \<circ> count =
+ Abs_multiset \<circ> mmap g \<circ> count \<circ> (Abs_multiset \<circ> mmap f \<circ> count)"
+ unfolding comp_def apply(rule ext)
+ by (auto simp: Abs_multiset_inverse count mmap_comp1 mmap)
+next
+ fix M f g assume eq: "\<And>a. a \<in> set_of M \<Longrightarrow> f a = g a"
+ thus "(Abs_multiset \<circ> mmap f \<circ> count) M = (Abs_multiset \<circ> mmap g \<circ> count) M" apply auto
+ unfolding cIm_def[abs_def] image_def
+ by (auto intro!: mmap_cong simp: Abs_multiset_inject count mmap)
+next
+ fix f show "set_of \<circ> (Abs_multiset \<circ> mmap f \<circ> count) = op ` f \<circ> set_of"
+ by (auto simp: count mmap mmap_image set_of_Abs_multiset supp_count)
+next
+ show "card_order natLeq" by (rule natLeq_card_order)
+next
+ show "cinfinite natLeq" by (rule natLeq_cinfinite)
+next
+ fix M show "|set_of M| \<le>o natLeq"
+ apply(rule ordLess_imp_ordLeq)
+ unfolding finite_iff_ordLess_natLeq[symmetric] using finite_set_of .
+next
+ fix A :: "'a set"
+ have "|{M. set_of M \<subseteq> A}| \<le>o |{as. set as \<subseteq> A}|" using card_of_set_of .
+ also have "|{as. set as \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq"
+ by (rule list_in_bd)
+ finally show "|{M. set_of M \<subseteq> A}| \<le>o ( |A| +c ctwo) ^c natLeq" .
+next
+ fix A B1 B2 f1 f2 p1 p2
+ let ?map = "\<lambda> f. Abs_multiset \<circ> mmap f \<circ> count"
+ assume wp: "wpull A B1 B2 f1 f2 p1 p2"
+ show "wpull {x. set_of x \<subseteq> A} {x. set_of x \<subseteq> B1} {x. set_of x \<subseteq> B2}
+ (?map f1) (?map f2) (?map p1) (?map p2)"
+ unfolding wpull_def proof safe
+ fix y1 y2
+ assume y1: "set_of y1 \<subseteq> B1" and y2: "set_of y2 \<subseteq> B2"
+ and m: "?map f1 y1 = ?map f2 y2"
+ def N1 \<equiv> "count y1" def N2 \<equiv> "count y2"
+ have "N1 \<in> multiset \<and> supp N1 \<subseteq> B1" and "N2 \<in> multiset \<and> supp N2 \<subseteq> B2"
+ and "mmap f1 N1 = mmap f2 N2"
+ using y1 y2 m unfolding N1_def N2_def
+ by (auto simp: Abs_multiset_inject count mmap)
+ then obtain M where M: "M \<in> multiset \<and> supp M \<subseteq> A"
+ and N1: "mmap p1 M = N1" and N2: "mmap p2 M = N2"
+ using wp_mmap[OF wp] unfolding wpull_def by auto
+ def x \<equiv> "Abs_multiset M"
+ show "\<exists>x\<in>{x. set_of x \<subseteq> A}. ?map p1 x = y1 \<and> ?map p2 x = y2"
+ apply(intro bexI[of _ x]) using M N1 N2 unfolding N1_def N2_def x_def
+ by (auto simp: count_inverse Abs_multiset_inverse)
+ qed
+qed (unfold set_of_empty, auto)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Codatatype.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,14 @@
+(* Title: HOL/Codatatype/Codatatype.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+The (co)datatype package.
+*)
+
+header {* The (Co)datatype Package *}
+
+theory Codatatype
+imports BNF_LFP BNF_GFP
+begin
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Countable_Set.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,328 @@
+(* Title: HOL/Codatatype/Countable_Set.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+(At most) countable sets.
+*)
+
+header {* (At Most) Countable Sets *}
+
+theory Countable_Set
+imports "../Ordinals_and_Cardinals_Base/Cardinal_Arithmetic"
+ "~~/src/HOL/Library/Countable"
+begin
+
+
+subsection{* Basics *}
+
+definition "countable A \<equiv> |A| \<le>o natLeq"
+
+lemma countable_card_of_nat:
+"countable A \<longleftrightarrow> |A| \<le>o |UNIV::nat set|"
+unfolding countable_def using card_of_nat
+using ordLeq_ordIso_trans ordIso_symmetric by blast
+
+lemma countable_ex_to_nat:
+fixes A :: "'a set"
+shows "countable A \<longleftrightarrow> (\<exists> f::'a\<Rightarrow>nat. inj_on f A)"
+unfolding countable_card_of_nat card_of_ordLeq[symmetric] by auto
+
+lemma countable_or_card_of:
+assumes "countable A"
+shows "(finite A \<and> |A| <o |UNIV::nat set| ) \<or>
+ (infinite A \<and> |A| =o |UNIV::nat set| )"
+apply (cases "finite A")
+ apply(metis finite_iff_cardOf_nat)
+ by (metis assms countable_card_of_nat infinite_iff_card_of_nat ordIso_iff_ordLeq)
+
+lemma countable_or:
+assumes "countable A"
+shows "(\<exists> f::'a\<Rightarrow>nat. finite A \<and> inj_on f A) \<or>
+ (\<exists> f::'a\<Rightarrow>nat. infinite A \<and> bij_betw f A UNIV)"
+using countable_or_card_of[OF assms]
+by (metis assms card_of_ordIso countable_ex_to_nat)
+
+lemma countable_cases_card_of[elim, consumes 1, case_names Fin Inf]:
+assumes "countable A"
+and "\<lbrakk>finite A; |A| <o |UNIV::nat set|\<rbrakk> \<Longrightarrow> phi"
+and "\<lbrakk>infinite A; |A| =o |UNIV::nat set|\<rbrakk> \<Longrightarrow> phi"
+shows phi
+using assms countable_or_card_of by blast
+
+lemma countable_cases[elim, consumes 1, case_names Fin Inf]:
+assumes "countable A"
+and "\<And> f::'a\<Rightarrow>nat. \<lbrakk>finite A; inj_on f A\<rbrakk> \<Longrightarrow> phi"
+and "\<And> f::'a\<Rightarrow>nat. \<lbrakk>infinite A; bij_betw f A UNIV\<rbrakk> \<Longrightarrow> phi"
+shows phi
+using assms countable_or by metis
+
+definition toNat_pred :: "'a set \<Rightarrow> ('a \<Rightarrow> nat) \<Rightarrow> bool"
+where
+"toNat_pred (A::'a set) f \<equiv>
+ (finite A \<and> inj_on f A) \<or> (infinite A \<and> bij_betw f A UNIV)"
+definition toNat where "toNat A \<equiv> SOME f. toNat_pred A f"
+
+lemma toNat_pred:
+assumes "countable A"
+shows "\<exists> f. toNat_pred A f"
+using assms countable_ex_to_nat toNat_pred_def by (cases rule: countable_cases) auto
+
+lemma toNat_pred_toNat:
+assumes "countable A"
+shows "toNat_pred A (toNat A)"
+unfolding toNat_def apply(rule someI_ex[of "toNat_pred A"])
+using toNat_pred[OF assms] .
+
+lemma bij_betw_toNat:
+assumes c: "countable A" and i: "infinite A"
+shows "bij_betw (toNat A) A (UNIV::nat set)"
+using toNat_pred_toNat[OF c] unfolding toNat_pred_def using i by auto
+
+lemma inj_on_toNat:
+assumes c: "countable A"
+shows "inj_on (toNat A) A"
+using c apply(cases rule: countable_cases)
+using bij_betw_toNat[OF c] toNat_pred_toNat[OF c]
+unfolding toNat_pred_def unfolding bij_betw_def by auto
+
+lemma toNat_inj[simp]:
+assumes c: "countable A" and a: "a \<in> A" and b: "b \<in> A"
+shows "toNat A a = toNat A b \<longleftrightarrow> a = b"
+using inj_on_toNat[OF c] using a b unfolding inj_on_def by auto
+
+lemma image_toNat:
+assumes c: "countable A" and i: "infinite A"
+shows "toNat A ` A = UNIV"
+using bij_betw_toNat[OF assms] unfolding bij_betw_def by simp
+
+lemma toNat_surj:
+assumes "countable A" and i: "infinite A"
+shows "\<exists> a. a \<in> A \<and> toNat A a = n"
+using image_toNat[OF assms]
+by (metis (no_types) image_iff iso_tuple_UNIV_I)
+
+definition
+"fromNat A n \<equiv>
+ if n \<in> toNat A ` A then inv_into A (toNat A) n
+ else (SOME a. a \<in> A)"
+
+lemma fromNat:
+assumes "A \<noteq> {}"
+shows "fromNat A n \<in> A"
+unfolding fromNat_def by (metis assms equals0I inv_into_into someI_ex)
+
+lemma toNat_fromNat[simp]:
+assumes "n \<in> toNat A ` A"
+shows "toNat A (fromNat A n) = n"
+by (metis assms f_inv_into_f fromNat_def)
+
+lemma infinite_toNat_fromNat[simp]:
+assumes c: "countable A" and i: "infinite A"
+shows "toNat A (fromNat A n) = n"
+apply(rule toNat_fromNat) using toNat_surj[OF assms]
+by (metis image_iff)
+
+lemma fromNat_toNat[simp]:
+assumes c: "countable A" and a: "a \<in> A"
+shows "fromNat A (toNat A a) = a"
+by (metis a c equals0D fromNat imageI toNat_fromNat toNat_inj)
+
+lemma fromNat_inj:
+assumes c: "countable A" and i: "infinite A"
+shows "fromNat A m = fromNat A n \<longleftrightarrow> m = n" (is "?L = ?R \<longleftrightarrow> ?K")
+proof-
+ have "?L = ?R \<longleftrightarrow> toNat A ?L = toNat A ?R"
+ unfolding toNat_inj[OF c fromNat[OF infinite_imp_nonempty[OF i]]
+ fromNat[OF infinite_imp_nonempty[OF i]]] ..
+ also have "... \<longleftrightarrow> ?K" using c i by simp
+ finally show ?thesis .
+qed
+
+lemma fromNat_surj:
+assumes c: "countable A" and a: "a \<in> A"
+shows "\<exists> n. fromNat A n = a"
+apply(rule exI[of _ "toNat A a"]) using assms by simp
+
+lemma fromNat_image_incl:
+assumes "A \<noteq> {}"
+shows "fromNat A ` UNIV \<subseteq> A"
+using fromNat[OF assms] by auto
+
+lemma incl_fromNat_image:
+assumes "countable A"
+shows "A \<subseteq> fromNat A ` UNIV"
+unfolding image_def using fromNat_surj[OF assms] by auto
+
+lemma fromNat_image[simp]:
+assumes "A \<noteq> {}" and "countable A"
+shows "fromNat A ` UNIV = A"
+by (metis assms equalityI fromNat_image_incl incl_fromNat_image)
+
+lemma fromNat_inject[simp]:
+assumes A: "A \<noteq> {}" "countable A" and B: "B \<noteq> {}" "countable B"
+shows "fromNat A = fromNat B \<longleftrightarrow> A = B"
+by (metis assms fromNat_image)
+
+lemma inj_on_fromNat:
+"inj_on fromNat ({A. A \<noteq> {} \<and> countable A})"
+unfolding inj_on_def by auto
+
+
+subsection {* Preservation under the set theoretic operations *}
+
+lemma contable_empty[simp,intro]:
+"countable {}"
+by (metis countable_ex_to_nat inj_on_empty)
+
+lemma incl_countable:
+assumes "A \<subseteq> B" and "countable B"
+shows "countable A"
+by (metis assms countable_ex_to_nat subset_inj_on)
+
+lemma countable_diff:
+assumes "countable A"
+shows "countable (A - B)"
+by (metis Diff_subset assms incl_countable)
+
+lemma finite_countable[simp]:
+assumes "finite A"
+shows "countable A"
+by (metis assms countable_ex_to_nat finite_imp_inj_to_nat_seg)
+
+lemma countable_singl[simp]:
+"countable {a}"
+by simp
+
+lemma countable_insert[simp]:
+"countable (insert a A) \<longleftrightarrow> countable A"
+proof
+ assume c: "countable A"
+ thus "countable (insert a A)"
+ apply (cases rule: countable_cases_card_of)
+ apply (metis finite_countable finite_insert)
+ unfolding countable_card_of_nat
+ by (metis infinite_card_of_insert ordIso_imp_ordLeq ordIso_transitive)
+qed(insert incl_countable, metis incl_countable subset_insertI)
+
+lemma contable_IntL[simp]:
+assumes "countable A"
+shows "countable (A \<inter> B)"
+by (metis Int_lower1 assms incl_countable)
+
+lemma contable_IntR[simp]:
+assumes "countable B"
+shows "countable (A \<inter> B)"
+by (metis assms contable_IntL inf.commute)
+
+lemma countable_UN[simp]:
+assumes cI: "countable I" and cA: "\<And> i. i \<in> I \<Longrightarrow> countable (A i)"
+shows "countable (\<Union> i \<in> I. A i)"
+using assms unfolding countable_card_of_nat
+apply(intro card_of_UNION_ordLeq_infinite) by auto
+
+lemma contable_Un[simp]:
+"countable (A \<union> B) \<longleftrightarrow> countable A \<and> countable B"
+proof safe
+ assume cA: "countable A" and cB: "countable B"
+ let ?I = "{0,Suc 0}" let ?As = "\<lambda> i. case i of 0 \<Rightarrow> A|Suc 0 \<Rightarrow> B"
+ have AB: "A \<union> B = (\<Union> i \<in> ?I. ?As i)" by simp
+ show "countable (A \<union> B)" unfolding AB apply(rule countable_UN)
+ using cA cB by auto
+qed (metis Un_upper1 incl_countable, metis Un_upper2 incl_countable)
+
+lemma countable_INT[simp]:
+assumes "i \<in> I" and "countable (A i)"
+shows "countable (\<Inter> i \<in> I. A i)"
+by (metis INF_insert assms contable_IntL insert_absorb)
+
+lemma countable_class[simp]:
+fixes A :: "('a::countable) set"
+shows "countable A"
+proof-
+ have "inj_on to_nat A" by (metis inj_on_to_nat)
+ thus ?thesis by (metis countable_ex_to_nat)
+qed
+
+lemma countable_image[simp]:
+assumes "countable A"
+shows "countable (f ` A)"
+using assms unfolding countable_card_of_nat
+by (metis card_of_image ordLeq_transitive)
+
+lemma countable_ordLeq:
+assumes "|A| \<le>o |B|" and "countable B"
+shows "countable A"
+using assms unfolding countable_card_of_nat by(rule ordLeq_transitive)
+
+lemma countable_ordLess:
+assumes AB: "|A| <o |B|" and B: "countable B"
+shows "countable A"
+using countable_ordLeq[OF ordLess_imp_ordLeq[OF AB] B] .
+
+lemma countable_vimage:
+assumes "B \<subseteq> range f" and "countable (f -` B)"
+shows "countable B"
+by (metis Int_absorb2 assms countable_image image_vimage_eq)
+
+lemma surj_countable_vimage:
+assumes s: "surj f" and c: "countable (f -` B)"
+shows "countable B"
+apply(rule countable_vimage[OF _ c]) using s by auto
+
+lemma countable_Collect[simp]:
+assumes "countable A"
+shows "countable {a \<in> A. \<phi> a}"
+by (metis Collect_conj_eq Int_absorb Int_commute Int_def assms contable_IntR)
+
+
+subsection{* The type of countable sets *}
+
+typedef (open) 'a cset = "{A :: 'a set. countable A}"
+apply(rule exI[of _ "{}"]) by simp
+
+abbreviation rcset where "rcset \<equiv> Rep_cset"
+abbreviation acset where "acset \<equiv> Abs_cset"
+
+lemmas acset_rcset = Rep_cset_inverse
+declare acset_rcset[simp]
+
+lemma acset_surj:
+"\<exists> A. countable A \<and> acset A = C"
+apply(cases rule: Abs_cset_cases[of C]) by auto
+
+lemma rcset_acset[simp]:
+assumes "countable A"
+shows "rcset (acset A) = A"
+using Abs_cset_inverse assms by auto
+
+lemma acset_inj[simp]:
+assumes "countable A" and "countable B"
+shows "acset A = acset B \<longleftrightarrow> A = B"
+using assms Abs_cset_inject by auto
+
+lemma rcset[simp]:
+"countable (rcset C)"
+using Rep_cset by simp
+
+lemma rcset_inj[simp]:
+"rcset C = rcset D \<longleftrightarrow> C = D"
+by (metis acset_rcset)
+
+lemma rcset_surj:
+assumes "countable A"
+shows "\<exists> C. rcset C = A"
+apply(cases rule: Rep_cset_cases[of A])
+using assms by auto
+
+definition "cIn a C \<equiv> (a \<in> rcset C)"
+definition "cEmp \<equiv> acset {}"
+definition "cIns a C \<equiv> acset (insert a (rcset C))"
+abbreviation cSingl where "cSingl a \<equiv> cIns a cEmp"
+definition "cUn C D \<equiv> acset (rcset C \<union> rcset D)"
+definition "cInt C D \<equiv> acset (rcset C \<inter> rcset D)"
+definition "cDif C D \<equiv> acset (rcset C - rcset D)"
+definition "cIm f C \<equiv> acset (f ` rcset C)"
+definition "cVim f D \<equiv> acset (f -` rcset D)"
+(* TODO eventually: nice setup for these operations, copied from the set setup *)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Equiv_Relations_More.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,161 @@
+(* Title: HOL/Codatatype/Equiv_Relations_More.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Some preliminaries on equivalence relations and quotients.
+*)
+
+header {* Some Preliminaries on Equivalence Relations and Quotients *}
+
+theory Equiv_Relations_More
+imports Equiv_Relations Hilbert_Choice
+begin
+
+
+(* Recall the following constants and lemmas:
+
+term Eps
+term "A//r"
+lemmas equiv_def
+lemmas refl_on_def
+ -- note that "reflexivity on" also assumes inclusion of the relation's field into r
+
+*)
+
+definition proj where "proj r x = r `` {x}"
+
+definition univ where "univ f X == f (Eps (%x. x \<in> X))"
+
+lemma proj_preserves:
+"x \<in> A \<Longrightarrow> proj r x \<in> A//r"
+unfolding proj_def by (rule quotientI)
+
+lemma proj_in_iff:
+assumes "equiv A r"
+shows "(proj r x \<in> A//r) = (x \<in> A)"
+apply(rule iffI, auto simp add: proj_preserves)
+unfolding proj_def quotient_def proof clarsimp
+ fix y assume y: "y \<in> A" and "r `` {x} = r `` {y}"
+ moreover have "y \<in> r `` {y}" using assms y unfolding equiv_def refl_on_def by blast
+ ultimately have "(x,y) \<in> r" by blast
+ thus "x \<in> A" using assms unfolding equiv_def refl_on_def by blast
+qed
+
+lemma proj_iff:
+"\<lbrakk>equiv A r; {x,y} \<subseteq> A\<rbrakk> \<Longrightarrow> (proj r x = proj r y) = ((x,y) \<in> r)"
+by (simp add: proj_def eq_equiv_class_iff)
+
+(*
+lemma in_proj: "\<lbrakk>equiv A r; x \<in> A\<rbrakk> \<Longrightarrow> x \<in> proj r x"
+unfolding proj_def equiv_def refl_on_def by blast
+*)
+
+lemma proj_image: "(proj r) ` A = A//r"
+unfolding proj_def[abs_def] quotient_def by blast
+
+lemma in_quotient_imp_non_empty:
+"\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<noteq> {}"
+unfolding quotient_def using equiv_class_self by fast
+
+lemma in_quotient_imp_in_rel:
+"\<lbrakk>equiv A r; X \<in> A//r; {x,y} \<subseteq> X\<rbrakk> \<Longrightarrow> (x,y) \<in> r"
+using quotient_eq_iff by fastforce
+
+lemma in_quotient_imp_closed:
+"\<lbrakk>equiv A r; X \<in> A//r; x \<in> X; (x,y) \<in> r\<rbrakk> \<Longrightarrow> y \<in> X"
+unfolding quotient_def equiv_def trans_def by blast
+
+lemma in_quotient_imp_subset:
+"\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<subseteq> A"
+using assms in_quotient_imp_in_rel equiv_type by fastforce
+
+lemma equiv_Eps_in:
+"\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> Eps (%x. x \<in> X) \<in> X"
+apply (rule someI2_ex)
+using in_quotient_imp_non_empty by blast
+
+lemma equiv_Eps_preserves:
+assumes ECH: "equiv A r" and X: "X \<in> A//r"
+shows "Eps (%x. x \<in> X) \<in> A"
+apply (rule in_mono[rule_format])
+ using assms apply (rule in_quotient_imp_subset)
+by (rule equiv_Eps_in) (rule assms)+
+
+lemma proj_Eps:
+assumes "equiv A r" and "X \<in> A//r"
+shows "proj r (Eps (%x. x \<in> X)) = X"
+unfolding proj_def proof auto
+ fix x assume x: "x \<in> X"
+ thus "(Eps (%x. x \<in> X), x) \<in> r" using assms equiv_Eps_in in_quotient_imp_in_rel by fast
+next
+ fix x assume "(Eps (%x. x \<in> X),x) \<in> r"
+ thus "x \<in> X" using in_quotient_imp_closed[OF assms equiv_Eps_in[OF assms]] by fast
+qed
+
+(*
+lemma Eps_proj:
+assumes "equiv A r" and "x \<in> A"
+shows "(Eps (%y. y \<in> proj r x), x) \<in> r"
+proof-
+ have 1: "proj r x \<in> A//r" using assms proj_preserves by fastforce
+ hence "Eps(%y. y \<in> proj r x) \<in> proj r x" using assms equiv_Eps_in by auto
+ moreover have "x \<in> proj r x" using assms in_proj by fastforce
+ ultimately show ?thesis using assms 1 in_quotient_imp_in_rel by fastforce
+qed
+
+lemma equiv_Eps_iff:
+assumes "equiv A r" and "{X,Y} \<subseteq> A//r"
+shows "((Eps (%x. x \<in> X),Eps (%y. y \<in> Y)) \<in> r) = (X = Y)"
+proof-
+ have "Eps (%x. x \<in> X) \<in> X \<and> Eps (%y. y \<in> Y) \<in> Y" using assms equiv_Eps_in by auto
+ thus ?thesis using assms quotient_eq_iff by fastforce
+qed
+
+lemma equiv_Eps_inj_on:
+assumes "equiv A r"
+shows "inj_on (%X. Eps (%x. x \<in> X)) (A//r)"
+unfolding inj_on_def proof clarify
+ fix X Y assume X: "X \<in> A//r" and Y: "Y \<in> A//r" and Eps: "Eps (%x. x \<in> X) = Eps (%y. y \<in> Y)"
+ hence "Eps (%x. x \<in> X) \<in> A" using assms equiv_Eps_preserves by auto
+ hence "(Eps (%x. x \<in> X), Eps (%y. y \<in> Y)) \<in> r"
+ using assms Eps unfolding quotient_def equiv_def refl_on_def by auto
+ thus "X= Y" using X Y assms equiv_Eps_iff by auto
+qed
+*)
+
+lemma univ_commute:
+assumes ECH: "equiv A r" and RES: "f respects r" and x: "x \<in> A"
+shows "(univ f) (proj r x) = f x"
+unfolding univ_def proof -
+ have prj: "proj r x \<in> A//r" using x proj_preserves by fast
+ hence "Eps (%y. y \<in> proj r x) \<in> A" using ECH equiv_Eps_preserves by fast
+ moreover have "proj r (Eps (%y. y \<in> proj r x)) = proj r x" using ECH prj proj_Eps by fast
+ ultimately have "(x, Eps (%y. y \<in> proj r x)) \<in> r" using x ECH proj_iff by fast
+ thus "f (Eps (%y. y \<in> proj r x)) = f x" using RES unfolding congruent_def by fastforce
+qed
+
+(*
+lemma univ_unique:
+assumes ECH: "equiv A r" and
+ RES: "f respects r" and COM: "\<forall> x \<in> A. G (proj r x) = f x"
+shows "\<forall> X \<in> A//r. G X = univ f X"
+proof
+ fix X assume "X \<in> A//r"
+ then obtain x where x: "x \<in> A" and X: "X = proj r x" using ECH proj_image[of r A] by blast
+ have "G X = f x" unfolding X using x COM by simp
+ thus "G X = univ f X" unfolding X using ECH RES x univ_commute by fastforce
+qed
+*)
+
+lemma univ_preserves:
+assumes ECH: "equiv A r" and RES: "f respects r" and
+ PRES: "\<forall> x \<in> A. f x \<in> B"
+shows "\<forall> X \<in> A//r. univ f X \<in> B"
+proof
+ fix X assume "X \<in> A//r"
+ then obtain x where x: "x \<in> A" and X: "X = proj r x" using ECH proj_image[of r A] by blast
+ hence "univ f X = f x" using assms univ_commute by fastforce
+ thus "univ f X \<in> B" using x PRES by simp
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/HFset.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,60 @@
+(* Title: Codatatype_Examples/HFset.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Hereditary sets.
+*)
+
+header {* Hereditary Sets *}
+
+theory HFset
+imports "../Codatatype/Codatatype"
+begin
+
+
+section {* Datatype definition *}
+
+bnf_data hfset: 'hfset = "'hfset fset"
+
+
+section {* Customization of terms *}
+
+subsection{* Constructors *}
+
+definition "Fold hs \<equiv> hfset_fld hs"
+
+lemma hfset_simps[simp]:
+"\<And>hs1 hs2. Fold hs1 = Fold hs2 \<longrightarrow> hs1 = hs2"
+unfolding Fold_def hfset.fld_inject by auto
+
+theorem hfset_cases[elim, case_names Fold]:
+assumes Fold: "\<And> hs. h = Fold hs \<Longrightarrow> phi"
+shows phi
+using Fold unfolding Fold_def
+by (cases rule: hfset.fld_exhaust[of h]) simp
+
+lemma hfset_induct[case_names Fold, induct type: hfset]:
+assumes Fold: "\<And> hs. (\<And> h. h |\<in>| hs \<Longrightarrow> phi h) \<Longrightarrow> phi (Fold hs)"
+shows "phi t"
+apply (induct rule: hfset.fld_induct)
+using Fold unfolding Fold_def fset_fset_member mem_Collect_eq ..
+
+(* alternative induction principle, using fset: *)
+lemma hfset_induct_fset[case_names Fold, induct type: hfset]:
+assumes Fold: "\<And> hs. (\<And> h. h \<in> fset hs \<Longrightarrow> phi h) \<Longrightarrow> phi (Fold hs)"
+shows "phi t"
+apply (induct rule: hfset_induct)
+using Fold by (metis notin_fset)
+
+subsection{* Recursion and iteration *}
+
+lemma hfset_rec:
+"hfset_rec R (Fold hs) = R (map_fset <id, hfset_rec R> hs)"
+using hfset.rec unfolding Fold_def .
+
+(* The iterator has a simpler form: *)
+lemma hfset_iter:
+"hfset_iter R (Fold hs) = R (map_fset (hfset_iter R) hs)"
+using hfset.iter unfolding Fold_def .
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Infinite_Derivation_Trees/Gram_Lang.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1366 @@
+(* Title: Gram_Lang.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Language of a grammar.
+*)
+
+header {* Language of a Grammar *}
+
+theory Gram_Lang
+imports Tree
+begin
+
+
+consts P :: "(N \<times> (T + N) set) set"
+axiomatization where
+ finite_N: "finite (UNIV::N set)"
+and finite_in_P: "\<And> n tns. (n,tns) \<in> P \<longrightarrow> finite tns"
+and used: "\<And> n. \<exists> tns. (n,tns) \<in> P"
+
+
+subsection{* Tree basics: frontier, interior, etc. *}
+
+lemma Tree_cong:
+assumes "root tr = root tr'" and "cont tr = cont tr'"
+shows "tr = tr'"
+by (metis Node_root_cont assms)
+
+inductive finiteT where
+Node: "\<lbrakk>finite as; (finiteT^#) as\<rbrakk> \<Longrightarrow> finiteT (Node a as)"
+monos lift_mono
+
+lemma finiteT_induct[consumes 1, case_names Node, induct pred: finiteT]:
+assumes 1: "finiteT tr"
+and IH: "\<And>as n. \<lbrakk>finite as; (\<phi>^#) as\<rbrakk> \<Longrightarrow> \<phi> (Node n as)"
+shows "\<phi> tr"
+using 1 apply(induct rule: finiteT.induct)
+apply(rule IH) apply assumption apply(elim mono_lift) by simp
+
+
+(* Frontier *)
+
+inductive inFr :: "N set \<Rightarrow> Tree \<Rightarrow> T \<Rightarrow> bool" where
+Base: "\<lbrakk>root tr \<in> ns; Inl t \<in> cont tr\<rbrakk> \<Longrightarrow> inFr ns tr t"
+|
+Ind: "\<lbrakk>root tr \<in> ns; Inr tr1 \<in> cont tr; inFr ns tr1 t\<rbrakk> \<Longrightarrow> inFr ns tr t"
+
+definition "Fr ns tr \<equiv> {t. inFr ns tr t}"
+
+lemma inFr_root_in: "inFr ns tr t \<Longrightarrow> root tr \<in> ns"
+by (metis inFr.simps)
+
+lemma inFr_mono:
+assumes "inFr ns tr t" and "ns \<subseteq> ns'"
+shows "inFr ns' tr t"
+using assms apply(induct arbitrary: ns' rule: inFr.induct)
+using Base Ind by (metis inFr.simps set_mp)+
+
+lemma inFr_Ind_minus:
+assumes "inFr ns1 tr1 t" and "Inr tr1 \<in> cont tr"
+shows "inFr (insert (root tr) ns1) tr t"
+using assms apply(induct rule: inFr.induct)
+ apply (metis inFr.simps insert_iff)
+ by (metis inFr.simps inFr_mono insertI1 subset_insertI)
+
+(* alternative definition *)
+inductive inFr2 :: "N set \<Rightarrow> Tree \<Rightarrow> T \<Rightarrow> bool" where
+Base: "\<lbrakk>root tr \<in> ns; Inl t \<in> cont tr\<rbrakk> \<Longrightarrow> inFr2 ns tr t"
+|
+Ind: "\<lbrakk>Inr tr1 \<in> cont tr; inFr2 ns1 tr1 t\<rbrakk>
+ \<Longrightarrow> inFr2 (insert (root tr) ns1) tr t"
+
+lemma inFr2_root_in: "inFr2 ns tr t \<Longrightarrow> root tr \<in> ns"
+apply(induct rule: inFr2.induct) by auto
+
+lemma inFr2_mono:
+assumes "inFr2 ns tr t" and "ns \<subseteq> ns'"
+shows "inFr2 ns' tr t"
+using assms apply(induct arbitrary: ns' rule: inFr2.induct)
+using Base Ind
+apply (metis subsetD) by (metis inFr2.simps insert_absorb insert_subset)
+
+lemma inFr2_Ind:
+assumes "inFr2 ns tr1 t" and "root tr \<in> ns" and "Inr tr1 \<in> cont tr"
+shows "inFr2 ns tr t"
+using assms apply(induct rule: inFr2.induct)
+ apply (metis inFr2.simps insert_absorb)
+ by (metis inFr2.simps insert_absorb)
+
+lemma inFr_inFr2:
+"inFr = inFr2"
+apply (rule ext)+ apply(safe)
+ apply(erule inFr.induct)
+ apply (metis (lifting) inFr2.Base)
+ apply (metis (lifting) inFr2_Ind)
+ apply(erule inFr2.induct)
+ apply (metis (lifting) inFr.Base)
+ apply (metis (lifting) inFr_Ind_minus)
+done
+
+lemma not_root_inFr:
+assumes "root tr \<notin> ns"
+shows "\<not> inFr ns tr t"
+by (metis assms inFr_root_in)
+
+theorem not_root_Fr:
+assumes "root tr \<notin> ns"
+shows "Fr ns tr = {}"
+using not_root_inFr[OF assms] unfolding Fr_def by auto
+
+
+(* Interior *)
+
+inductive inItr :: "N set \<Rightarrow> Tree \<Rightarrow> N \<Rightarrow> bool" where
+Base: "root tr \<in> ns \<Longrightarrow> inItr ns tr (root tr)"
+|
+Ind: "\<lbrakk>root tr \<in> ns; Inr tr1 \<in> cont tr; inItr ns tr1 n\<rbrakk> \<Longrightarrow> inItr ns tr n"
+
+definition "Itr ns tr \<equiv> {n. inItr ns tr n}"
+
+lemma inItr_root_in: "inItr ns tr n \<Longrightarrow> root tr \<in> ns"
+by (metis inItr.simps)
+
+lemma inItr_mono:
+assumes "inItr ns tr n" and "ns \<subseteq> ns'"
+shows "inItr ns' tr n"
+using assms apply(induct arbitrary: ns' rule: inItr.induct)
+using Base Ind by (metis inItr.simps set_mp)+
+
+
+(* The subtree relation *)
+
+inductive subtr where
+Refl: "root tr \<in> ns \<Longrightarrow> subtr ns tr tr"
+|
+Step: "\<lbrakk>root tr3 \<in> ns; subtr ns tr1 tr2; Inr tr2 \<in> cont tr3\<rbrakk> \<Longrightarrow> subtr ns tr1 tr3"
+
+lemma subtr_rootL_in:
+assumes "subtr ns tr1 tr2"
+shows "root tr1 \<in> ns"
+using assms apply(induct rule: subtr.induct) by auto
+
+lemma subtr_rootR_in:
+assumes "subtr ns tr1 tr2"
+shows "root tr2 \<in> ns"
+using assms apply(induct rule: subtr.induct) by auto
+
+lemmas subtr_roots_in = subtr_rootL_in subtr_rootR_in
+
+lemma subtr_mono:
+assumes "subtr ns tr1 tr2" and "ns \<subseteq> ns'"
+shows "subtr ns' tr1 tr2"
+using assms apply(induct arbitrary: ns' rule: subtr.induct)
+using Refl Step by (metis subtr.simps set_mp)+
+
+lemma subtr_trans_Un:
+assumes "subtr ns12 tr1 tr2" and "subtr ns23 tr2 tr3"
+shows "subtr (ns12 \<union> ns23) tr1 tr3"
+proof-
+ have "subtr ns23 tr2 tr3 \<Longrightarrow>
+ (\<forall> ns12 tr1. subtr ns12 tr1 tr2 \<longrightarrow> subtr (ns12 \<union> ns23) tr1 tr3)"
+ apply(induct rule: subtr.induct, safe)
+ apply (metis subtr_mono sup_commute sup_ge2)
+ by (metis (lifting) Step UnI2)
+ thus ?thesis using assms by auto
+qed
+
+lemma subtr_trans:
+assumes "subtr ns tr1 tr2" and "subtr ns tr2 tr3"
+shows "subtr ns tr1 tr3"
+using subtr_trans_Un[OF assms] by simp
+
+lemma subtr_StepL:
+assumes r: "root tr1 \<in> ns" and tr12: "Inr tr1 \<in> cont tr2" and s: "subtr ns tr2 tr3"
+shows "subtr ns tr1 tr3"
+apply(rule subtr_trans[OF _ s]) apply(rule Step[of tr2 ns tr1 tr1])
+by (metis assms subtr_rootL_in Refl)+
+
+(* alternative definition: *)
+inductive subtr2 where
+Refl: "root tr \<in> ns \<Longrightarrow> subtr2 ns tr tr"
+|
+Step: "\<lbrakk>root tr1 \<in> ns; Inr tr1 \<in> cont tr2; subtr2 ns tr2 tr3\<rbrakk> \<Longrightarrow> subtr2 ns tr1 tr3"
+
+lemma subtr2_rootL_in:
+assumes "subtr2 ns tr1 tr2"
+shows "root tr1 \<in> ns"
+using assms apply(induct rule: subtr2.induct) by auto
+
+lemma subtr2_rootR_in:
+assumes "subtr2 ns tr1 tr2"
+shows "root tr2 \<in> ns"
+using assms apply(induct rule: subtr2.induct) by auto
+
+lemmas subtr2_roots_in = subtr2_rootL_in subtr2_rootR_in
+
+lemma subtr2_mono:
+assumes "subtr2 ns tr1 tr2" and "ns \<subseteq> ns'"
+shows "subtr2 ns' tr1 tr2"
+using assms apply(induct arbitrary: ns' rule: subtr2.induct)
+using Refl Step by (metis subtr2.simps set_mp)+
+
+lemma subtr2_trans_Un:
+assumes "subtr2 ns12 tr1 tr2" and "subtr2 ns23 tr2 tr3"
+shows "subtr2 (ns12 \<union> ns23) tr1 tr3"
+proof-
+ have "subtr2 ns12 tr1 tr2 \<Longrightarrow>
+ (\<forall> ns23 tr3. subtr2 ns23 tr2 tr3 \<longrightarrow> subtr2 (ns12 \<union> ns23) tr1 tr3)"
+ apply(induct rule: subtr2.induct, safe)
+ apply (metis subtr2_mono sup_commute sup_ge2)
+ by (metis Un_iff subtr2.simps)
+ thus ?thesis using assms by auto
+qed
+
+lemma subtr2_trans:
+assumes "subtr2 ns tr1 tr2" and "subtr2 ns tr2 tr3"
+shows "subtr2 ns tr1 tr3"
+using subtr2_trans_Un[OF assms] by simp
+
+lemma subtr2_StepR:
+assumes r: "root tr3 \<in> ns" and tr23: "Inr tr2 \<in> cont tr3" and s: "subtr2 ns tr1 tr2"
+shows "subtr2 ns tr1 tr3"
+apply(rule subtr2_trans[OF s]) apply(rule Step[of _ _ tr3])
+by (metis assms subtr2_rootR_in Refl)+
+
+lemma subtr_subtr2:
+"subtr = subtr2"
+apply (rule ext)+ apply(safe)
+ apply(erule subtr.induct)
+ apply (metis (lifting) subtr2.Refl)
+ apply (metis (lifting) subtr2_StepR)
+ apply(erule subtr2.induct)
+ apply (metis (lifting) subtr.Refl)
+ apply (metis (lifting) subtr_StepL)
+done
+
+lemma subtr_inductL[consumes 1, case_names Refl Step]:
+assumes s: "subtr ns tr1 tr2" and Refl: "\<And>ns tr. \<phi> ns tr tr"
+and Step:
+"\<And>ns tr1 tr2 tr3.
+ \<lbrakk>root tr1 \<in> ns; Inr tr1 \<in> cont tr2; subtr ns tr2 tr3; \<phi> ns tr2 tr3\<rbrakk> \<Longrightarrow> \<phi> ns tr1 tr3"
+shows "\<phi> ns tr1 tr2"
+using s unfolding subtr_subtr2 apply(rule subtr2.induct)
+using Refl Step unfolding subtr_subtr2 by auto
+
+lemma subtr_UNIV_inductL[consumes 1, case_names Refl Step]:
+assumes s: "subtr UNIV tr1 tr2" and Refl: "\<And>tr. \<phi> tr tr"
+and Step:
+"\<And>tr1 tr2 tr3.
+ \<lbrakk>Inr tr1 \<in> cont tr2; subtr UNIV tr2 tr3; \<phi> tr2 tr3\<rbrakk> \<Longrightarrow> \<phi> tr1 tr3"
+shows "\<phi> tr1 tr2"
+using s apply(induct rule: subtr_inductL)
+apply(rule Refl) using Step subtr_mono by (metis subset_UNIV)
+
+(* Subtree versus frontier: *)
+lemma subtr_inFr:
+assumes "inFr ns tr t" and "subtr ns tr tr1"
+shows "inFr ns tr1 t"
+proof-
+ have "subtr ns tr tr1 \<Longrightarrow> (\<forall> t. inFr ns tr t \<longrightarrow> inFr ns tr1 t)"
+ apply(induct rule: subtr.induct, safe) by (metis inFr.Ind)
+ thus ?thesis using assms by auto
+qed
+
+corollary Fr_subtr:
+"Fr ns tr = \<Union> {Fr ns tr' | tr'. subtr ns tr' tr}"
+unfolding Fr_def proof safe
+ fix t assume t: "inFr ns tr t" hence "root tr \<in> ns" by (rule inFr_root_in)
+ thus "t \<in> \<Union>{{t. inFr ns tr' t} |tr'. subtr ns tr' tr}"
+ apply(intro UnionI[of "{t. inFr ns tr t}" _ t]) using t subtr.Refl by auto
+qed(metis subtr_inFr)
+
+lemma inFr_subtr:
+assumes "inFr ns tr t"
+shows "\<exists> tr'. subtr ns tr' tr \<and> Inl t \<in> cont tr'"
+using assms apply(induct rule: inFr.induct) apply safe
+ apply (metis subtr.Refl)
+ by (metis (lifting) subtr.Step)
+
+corollary Fr_subtr_cont:
+"Fr ns tr = \<Union> {Inl -` cont tr' | tr'. subtr ns tr' tr}"
+unfolding Fr_def
+apply safe
+apply (frule inFr_subtr)
+apply auto
+by (metis inFr.Base subtr_inFr subtr_rootL_in)
+
+(* Subtree versus interior: *)
+lemma subtr_inItr:
+assumes "inItr ns tr n" and "subtr ns tr tr1"
+shows "inItr ns tr1 n"
+proof-
+ have "subtr ns tr tr1 \<Longrightarrow> (\<forall> t. inItr ns tr n \<longrightarrow> inItr ns tr1 n)"
+ apply(induct rule: subtr.induct, safe) by (metis inItr.Ind)
+ thus ?thesis using assms by auto
+qed
+
+corollary Itr_subtr:
+"Itr ns tr = \<Union> {Itr ns tr' | tr'. subtr ns tr' tr}"
+unfolding Itr_def apply safe
+apply (metis (lifting, mono_tags) UnionI inItr_root_in mem_Collect_eq subtr.Refl)
+by (metis subtr_inItr)
+
+lemma inItr_subtr:
+assumes "inItr ns tr n"
+shows "\<exists> tr'. subtr ns tr' tr \<and> root tr' = n"
+using assms apply(induct rule: inItr.induct) apply safe
+ apply (metis subtr.Refl)
+ by (metis (lifting) subtr.Step)
+
+corollary Itr_subtr_cont:
+"Itr ns tr = {root tr' | tr'. subtr ns tr' tr}"
+unfolding Itr_def apply safe
+ apply (metis (lifting, mono_tags) UnionI inItr_subtr mem_Collect_eq vimageI2)
+ by (metis inItr.Base subtr_inItr subtr_rootL_in)
+
+
+subsection{* The immediate subtree function *}
+
+(* production of: *)
+abbreviation "prodOf tr \<equiv> (id \<oplus> root) ` (cont tr)"
+(* subtree of: *)
+definition "subtrOf tr n \<equiv> SOME tr'. Inr tr' \<in> cont tr \<and> root tr' = n"
+
+lemma subtrOf:
+assumes n: "Inr n \<in> prodOf tr"
+shows "Inr (subtrOf tr n) \<in> cont tr \<and> root (subtrOf tr n) = n"
+proof-
+ obtain tr' where "Inr tr' \<in> cont tr \<and> root tr' = n"
+ using n unfolding image_def by (metis (lifting) Inr_oplus_elim assms)
+ thus ?thesis unfolding subtrOf_def by(rule someI)
+qed
+
+lemmas Inr_subtrOf = subtrOf[THEN conjunct1]
+lemmas root_subtrOf[simp] = subtrOf[THEN conjunct2]
+
+lemma Inl_prodOf: "Inl -` (prodOf tr) = Inl -` (cont tr)"
+proof safe
+ fix t ttr assume "Inl t = (id \<oplus> root) ttr" and "ttr \<in> cont tr"
+ thus "t \<in> Inl -` cont tr" by(cases ttr, auto)
+next
+ fix t assume "Inl t \<in> cont tr" thus "t \<in> Inl -` prodOf tr"
+ by (metis (lifting) id_def image_iff sum_map.simps(1) vimageI2)
+qed
+
+lemma root_prodOf:
+assumes "Inr tr' \<in> cont tr"
+shows "Inr (root tr') \<in> prodOf tr"
+by (metis (lifting) assms image_iff sum_map.simps(2))
+
+
+subsection{* Derivation trees *}
+
+coinductive dtree where
+Tree: "\<lbrakk>(root tr, (id \<oplus> root) ` (cont tr)) \<in> P; inj_on root (Inr -` cont tr);
+ lift dtree (cont tr)\<rbrakk> \<Longrightarrow> dtree tr"
+monos lift_mono
+
+(* destruction rules: *)
+lemma dtree_P:
+assumes "dtree tr"
+shows "(root tr, (id \<oplus> root) ` (cont tr)) \<in> P"
+using assms unfolding dtree.simps by auto
+
+lemma dtree_inj_on:
+assumes "dtree tr"
+shows "inj_on root (Inr -` cont tr)"
+using assms unfolding dtree.simps by auto
+
+lemma dtree_inj[simp]:
+assumes "dtree tr" and "Inr tr1 \<in> cont tr" and "Inr tr2 \<in> cont tr"
+shows "root tr1 = root tr2 \<longleftrightarrow> tr1 = tr2"
+using assms dtree_inj_on unfolding inj_on_def by auto
+
+lemma dtree_lift:
+assumes "dtree tr"
+shows "lift dtree (cont tr)"
+using assms unfolding dtree.simps by auto
+
+
+(* coinduction:*)
+lemma dtree_coind[elim, consumes 1, case_names Hyp]:
+assumes phi: "\<phi> tr"
+and Hyp:
+"\<And> tr. \<phi> tr \<Longrightarrow>
+ (root tr, image (id \<oplus> root) (cont tr)) \<in> P \<and>
+ inj_on root (Inr -` cont tr) \<and>
+ lift (\<lambda> tr. \<phi> tr \<or> dtree tr) (cont tr)"
+shows "dtree tr"
+apply(rule dtree.coinduct[of \<phi> tr, OF phi])
+using Hyp by blast
+
+lemma dtree_raw_coind[elim, consumes 1, case_names Hyp]:
+assumes phi: "\<phi> tr"
+and Hyp:
+"\<And> tr. \<phi> tr \<Longrightarrow>
+ (root tr, image (id \<oplus> root) (cont tr)) \<in> P \<and>
+ inj_on root (Inr -` cont tr) \<and>
+ lift \<phi> (cont tr)"
+shows "dtree tr"
+using phi apply(induct rule: dtree_coind)
+using Hyp mono_lift
+by (metis (mono_tags) mono_lift)
+
+lemma dtree_subtr_inj_on:
+assumes d: "dtree tr1" and s: "subtr ns tr tr1"
+shows "inj_on root (Inr -` cont tr)"
+using s d apply(induct rule: subtr.induct)
+apply (metis (lifting) dtree_inj_on) by (metis dtree_lift lift_def)
+
+lemma dtree_subtr_P:
+assumes d: "dtree tr1" and s: "subtr ns tr tr1"
+shows "(root tr, (id \<oplus> root) ` cont tr) \<in> P"
+using s d apply(induct rule: subtr.induct)
+apply (metis (lifting) dtree_P) by (metis dtree_lift lift_def)
+
+lemma subtrOf_root[simp]:
+assumes tr: "dtree tr" and cont: "Inr tr' \<in> cont tr"
+shows "subtrOf tr (root tr') = tr'"
+proof-
+ have 0: "Inr (subtrOf tr (root tr')) \<in> cont tr" using Inr_subtrOf
+ by (metis (lifting) cont root_prodOf)
+ have "root (subtrOf tr (root tr')) = root tr'"
+ using root_subtrOf by (metis (lifting) cont root_prodOf)
+ thus ?thesis unfolding dtree_inj[OF tr 0 cont] .
+qed
+
+lemma surj_subtrOf:
+assumes "dtree tr" and 0: "Inr tr' \<in> cont tr"
+shows "\<exists> n. Inr n \<in> prodOf tr \<and> subtrOf tr n = tr'"
+apply(rule exI[of _ "root tr'"])
+using root_prodOf[OF 0] subtrOf_root[OF assms] by simp
+
+lemma dtree_subtr:
+assumes "dtree tr1" and "subtr ns tr tr1"
+shows "dtree tr"
+proof-
+ have "(\<exists> ns tr1. dtree tr1 \<and> subtr ns tr tr1) \<Longrightarrow> dtree tr"
+ proof (induct rule: dtree_raw_coind)
+ case (Hyp tr)
+ then obtain ns tr1 where tr1: "dtree tr1" and tr_tr1: "subtr ns tr tr1" by auto
+ show ?case unfolding lift_def proof safe
+ show "(root tr, (id \<oplus> root) ` cont tr) \<in> P" using dtree_subtr_P[OF tr1 tr_tr1] .
+ next
+ show "inj_on root (Inr -` cont tr)" using dtree_subtr_inj_on[OF tr1 tr_tr1] .
+ next
+ fix tr' assume tr': "Inr tr' \<in> cont tr"
+ have tr_tr1: "subtr (ns \<union> {root tr'}) tr tr1" using subtr_mono[OF tr_tr1] by auto
+ have "subtr (ns \<union> {root tr'}) tr' tr1" using subtr_StepL[OF _ tr' tr_tr1] by auto
+ thus "\<exists>ns' tr1. dtree tr1 \<and> subtr ns' tr' tr1" using tr1 by blast
+ qed
+ qed
+ thus ?thesis using assms by auto
+qed
+
+
+subsection{* Default trees *}
+
+(* Pick a left-hand side of a production for each nonterminal *)
+definition S where "S n \<equiv> SOME tns. (n,tns) \<in> P"
+
+lemma S_P: "(n, S n) \<in> P"
+using used unfolding S_def by(rule someI_ex)
+
+lemma finite_S: "finite (S n)"
+using S_P finite_in_P by auto
+
+
+(* The default tree of a nonterminal *)
+definition deftr :: "N \<Rightarrow> Tree" where
+"deftr \<equiv> coit id S"
+
+lemma deftr_simps[simp]:
+"root (deftr n) = n"
+"cont (deftr n) = image (id \<oplus> deftr) (S n)"
+using coit(1)[of id S n] coit(2)[of S n id, OF finite_S]
+unfolding deftr_def by simp_all
+
+lemmas root_deftr = deftr_simps(1)
+lemmas cont_deftr = deftr_simps(2)
+
+lemma root_o_deftr[simp]: "root o deftr = id"
+by (rule ext, auto)
+
+lemma dtree_deftr: "dtree (deftr n)"
+proof-
+ {fix tr assume "\<exists> n. tr = deftr n" hence "dtree tr"
+ apply(induct rule: dtree_raw_coind) apply safe
+ unfolding deftr_simps image_compose[symmetric] sum_map.comp id_o
+ root_o_deftr sum_map.id image_id id_apply apply(rule S_P)
+ unfolding inj_on_def lift_def by auto
+ }
+ thus ?thesis by auto
+qed
+
+
+subsection{* Hereditary substitution *}
+
+(* Auxiliary concept: The root-ommiting frontier: *)
+definition "inFrr ns tr t \<equiv> \<exists> tr'. Inr tr' \<in> cont tr \<and> inFr ns tr' t"
+definition "Frr ns tr \<equiv> {t. \<exists> tr'. Inr tr' \<in> cont tr \<and> t \<in> Fr ns tr'}"
+
+context
+fixes tr0 :: Tree
+begin
+
+definition "hsubst_r tr \<equiv> root tr"
+definition "hsubst_c tr \<equiv> if root tr = root tr0 then cont tr0 else cont tr"
+
+(* Hereditary substitution: *)
+definition hsubst :: "Tree \<Rightarrow> Tree" where
+"hsubst \<equiv> coit hsubst_r hsubst_c"
+
+lemma finite_hsubst_c: "finite (hsubst_c n)"
+unfolding hsubst_c_def by (metis finite_cont)
+
+lemma root_hsubst[simp]: "root (hsubst tr) = root tr"
+using coit(1)[of hsubst_r hsubst_c tr] unfolding hsubst_def hsubst_r_def by simp
+
+lemma root_o_subst[simp]: "root o hsubst = root"
+unfolding comp_def root_hsubst ..
+
+lemma cont_hsubst_eq[simp]:
+assumes "root tr = root tr0"
+shows "cont (hsubst tr) = (id \<oplus> hsubst) ` (cont tr0)"
+apply(subst id_o[symmetric, of id]) unfolding id_o
+using coit(2)[of hsubst_c tr hsubst_r, OF finite_hsubst_c]
+unfolding hsubst_def hsubst_c_def using assms by simp
+
+lemma hsubst_eq:
+assumes "root tr = root tr0"
+shows "hsubst tr = hsubst tr0"
+apply(rule Tree_cong) using assms cont_hsubst_eq by auto
+
+lemma cont_hsubst_neq[simp]:
+assumes "root tr \<noteq> root tr0"
+shows "cont (hsubst tr) = (id \<oplus> hsubst) ` (cont tr)"
+apply(subst id_o[symmetric, of id]) unfolding id_o
+using coit(2)[of hsubst_c tr hsubst_r, OF finite_hsubst_c]
+unfolding hsubst_def hsubst_c_def using assms by simp
+
+lemma Inl_cont_hsubst_eq[simp]:
+assumes "root tr = root tr0"
+shows "Inl -` cont (hsubst tr) = Inl -` (cont tr0)"
+unfolding cont_hsubst_eq[OF assms] by simp
+
+lemma Inr_cont_hsubst_eq[simp]:
+assumes "root tr = root tr0"
+shows "Inr -` cont (hsubst tr) = hsubst ` Inr -` cont tr0"
+unfolding cont_hsubst_eq[OF assms] by simp
+
+lemma Inl_cont_hsubst_neq[simp]:
+assumes "root tr \<noteq> root tr0"
+shows "Inl -` cont (hsubst tr) = Inl -` (cont tr)"
+unfolding cont_hsubst_neq[OF assms] by simp
+
+lemma Inr_cont_hsubst_neq[simp]:
+assumes "root tr \<noteq> root tr0"
+shows "Inr -` cont (hsubst tr) = hsubst ` Inr -` cont tr"
+unfolding cont_hsubst_neq[OF assms] by simp
+
+lemma dtree_hsubst:
+assumes tr0: "dtree tr0" and tr: "dtree tr"
+shows "dtree (hsubst tr)"
+proof-
+ {fix tr1 have "(\<exists> tr. dtree tr \<and> tr1 = hsubst tr) \<Longrightarrow> dtree tr1"
+ proof (induct rule: dtree_raw_coind)
+ case (Hyp tr1) then obtain tr
+ where dtr: "dtree tr" and tr1: "tr1 = hsubst tr" by auto
+ show ?case unfolding lift_def tr1 proof safe
+ show "(root (hsubst tr), prodOf (hsubst tr)) \<in> P"
+ unfolding tr1 apply(cases "root tr = root tr0")
+ using dtree_P[OF dtr] dtree_P[OF tr0]
+ by (auto simp add: image_compose[symmetric] sum_map.comp)
+ show "inj_on root (Inr -` cont (hsubst tr))"
+ apply(cases "root tr = root tr0") using dtree_inj_on[OF dtr] dtree_inj_on[OF tr0]
+ unfolding inj_on_def by (auto, blast)
+ fix tr' assume "Inr tr' \<in> cont (hsubst tr)"
+ thus "\<exists>tra. dtree tra \<and> tr' = hsubst tra"
+ apply(cases "root tr = root tr0", simp_all)
+ apply (metis dtree_lift lift_def tr0)
+ by (metis dtr dtree_lift lift_def)
+ qed
+ qed
+ }
+ thus ?thesis using assms by blast
+qed
+
+lemma Frr: "Frr ns tr = {t. inFrr ns tr t}"
+unfolding inFrr_def Frr_def Fr_def by auto
+
+lemma inFr_hsubst_imp:
+assumes "inFr ns (hsubst tr) t"
+shows "t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t \<or>
+ inFr (ns - {root tr0}) tr t"
+proof-
+ {fix tr1
+ have "inFr ns tr1 t \<Longrightarrow>
+ (\<And> tr. tr1 = hsubst tr \<Longrightarrow> (t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t \<or>
+ inFr (ns - {root tr0}) tr t))"
+ proof(induct rule: inFr.induct)
+ case (Base tr1 ns t tr)
+ hence rtr: "root tr1 \<in> ns" and t_tr1: "Inl t \<in> cont tr1" and tr1: "tr1 = hsubst tr"
+ by auto
+ show ?case
+ proof(cases "root tr1 = root tr0")
+ case True
+ hence "t \<in> Inl -` (cont tr0)" using t_tr1 unfolding tr1 by auto
+ thus ?thesis by simp
+ next
+ case False
+ hence "inFr (ns - {root tr0}) tr t" using t_tr1 unfolding tr1 apply simp
+ by (metis Base.prems Diff_iff root_hsubst inFr.Base rtr singletonE)
+ thus ?thesis by simp
+ qed
+ next
+ case (Ind tr1 ns tr1' t) note IH = Ind(4)
+ have rtr1: "root tr1 \<in> ns" and tr1'_tr1: "Inr tr1' \<in> cont tr1"
+ and t_tr1': "inFr ns tr1' t" and tr1: "tr1 = hsubst tr" using Ind by auto
+ have rtr1: "root tr1 = root tr" unfolding tr1 by simp
+ show ?case
+ proof(cases "root tr1 = root tr0")
+ case True
+ then obtain tr' where tr'_tr0: "Inr tr' \<in> cont tr0" and tr1': "tr1' = hsubst tr'"
+ using tr1'_tr1 unfolding tr1 by auto
+ show ?thesis using IH[OF tr1'] proof (elim disjE)
+ assume "inFr (ns - {root tr0}) tr' t"
+ thus ?thesis using tr'_tr0 unfolding inFrr_def by auto
+ qed auto
+ next
+ case False
+ then obtain tr' where tr'_tr: "Inr tr' \<in> cont tr" and tr1': "tr1' = hsubst tr'"
+ using tr1'_tr1 unfolding tr1 by auto
+ show ?thesis using IH[OF tr1'] proof (elim disjE)
+ assume "inFr (ns - {root tr0}) tr' t"
+ thus ?thesis using tr'_tr unfolding inFrr_def
+ by (metis Diff_iff False Ind(1) empty_iff inFr2_Ind inFr_inFr2 insert_iff rtr1)
+ qed auto
+ qed
+ qed
+ }
+ thus ?thesis using assms by auto
+qed
+
+lemma inFr_hsubst_notin:
+assumes "inFr ns tr t" and "root tr0 \<notin> ns"
+shows "inFr ns (hsubst tr) t"
+using assms apply(induct rule: inFr.induct)
+apply (metis Inl_cont_hsubst_neq inFr2.Base inFr_inFr2 root_hsubst vimageD vimageI2)
+by (metis (lifting) Inr_cont_hsubst_neq inFr.Ind rev_image_eqI root_hsubst vimageD vimageI2)
+
+lemma inFr_hsubst_minus:
+assumes "inFr (ns - {root tr0}) tr t"
+shows "inFr ns (hsubst tr) t"
+proof-
+ have 1: "inFr (ns - {root tr0}) (hsubst tr) t"
+ using inFr_hsubst_notin[OF assms] by simp
+ show ?thesis using inFr_mono[OF 1] by auto
+qed
+
+lemma inFr_self_hsubst:
+assumes "root tr0 \<in> ns"
+shows
+"inFr ns (hsubst tr0) t \<longleftrightarrow>
+ t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t"
+(is "?A \<longleftrightarrow> ?B \<or> ?C")
+apply(intro iffI)
+apply (metis inFr_hsubst_imp Diff_iff inFr_root_in insertI1) proof(elim disjE)
+ assume ?B thus ?A apply(intro inFr.Base) using assms by auto
+next
+ assume ?C then obtain tr where
+ tr_tr0: "Inr tr \<in> cont tr0" and t_tr: "inFr (ns - {root tr0}) tr t"
+ unfolding inFrr_def by auto
+ def tr1 \<equiv> "hsubst tr"
+ have 1: "inFr ns tr1 t" using t_tr unfolding tr1_def using inFr_hsubst_minus by auto
+ have "Inr tr1 \<in> cont (hsubst tr0)" unfolding tr1_def using tr_tr0 by auto
+ thus ?A using 1 inFr.Ind assms by (metis root_hsubst)
+qed
+
+theorem Fr_self_hsubst:
+assumes "root tr0 \<in> ns"
+shows "Fr ns (hsubst tr0) = Inl -` (cont tr0) \<union> Frr (ns - {root tr0}) tr0"
+using inFr_self_hsubst[OF assms] unfolding Frr Fr_def by auto
+
+end (* context *)
+
+
+subsection{* Regular trees *}
+
+hide_const regular
+
+definition "reg f tr \<equiv> \<forall> tr'. subtr UNIV tr' tr \<longrightarrow> tr' = f (root tr')"
+definition "regular tr \<equiv> \<exists> f. reg f tr"
+
+lemma reg_def2: "reg f tr \<longleftrightarrow> (\<forall> ns tr'. subtr ns tr' tr \<longrightarrow> tr' = f (root tr'))"
+unfolding reg_def using subtr_mono by (metis subset_UNIV)
+
+lemma regular_def2: "regular tr \<longleftrightarrow> (\<exists> f. reg f tr \<and> (\<forall> n. root (f n) = n))"
+unfolding regular_def proof safe
+ fix f assume f: "reg f tr"
+ def g \<equiv> "\<lambda> n. if inItr UNIV tr n then f n else deftr n"
+ show "\<exists>g. reg g tr \<and> (\<forall>n. root (g n) = n)"
+ apply(rule exI[of _ g])
+ using f deftr_simps(1) unfolding g_def reg_def apply safe
+ apply (metis (lifting) inItr.Base subtr_inItr subtr_rootL_in)
+ by (metis (full_types) inItr_subtr subtr_subtr2)
+qed auto
+
+lemma reg_root:
+assumes "reg f tr"
+shows "f (root tr) = tr"
+using assms unfolding reg_def
+by (metis (lifting) iso_tuple_UNIV_I subtr.Refl)
+
+
+lemma reg_Inr_cont:
+assumes "reg f tr" and "Inr tr' \<in> cont tr"
+shows "reg f tr'"
+by (metis (lifting) assms iso_tuple_UNIV_I reg_def subtr.Step)
+
+lemma reg_subtr:
+assumes "reg f tr" and "subtr ns tr' tr"
+shows "reg f tr'"
+using assms unfolding reg_def using subtr_trans[of UNIV tr] UNIV_I
+by (metis UNIV_eq_I UnCI Un_upper1 iso_tuple_UNIV_I subtr_mono subtr_trans)
+
+lemma regular_subtr:
+assumes r: "regular tr" and s: "subtr ns tr' tr"
+shows "regular tr'"
+using r reg_subtr[OF _ s] unfolding regular_def by auto
+
+lemma subtr_deftr:
+assumes "subtr ns tr' (deftr n)"
+shows "tr' = deftr (root tr')"
+proof-
+ {fix tr have "subtr ns tr' tr \<Longrightarrow> (\<forall> n. tr = deftr n \<longrightarrow> tr' = deftr (root tr'))"
+ apply (induct rule: subtr.induct)
+ proof(metis (lifting) deftr_simps(1), safe)
+ fix tr3 ns tr1 tr2 n
+ assume 1: "root (deftr n) \<in> ns" and 2: "subtr ns tr1 tr2"
+ and IH: "\<forall>n. tr2 = deftr n \<longrightarrow> tr1 = deftr (root tr1)"
+ and 3: "Inr tr2 \<in> cont (deftr n)"
+ have "tr2 \<in> deftr ` UNIV"
+ using 3 unfolding deftr_simps image_def
+ by (metis (lifting, full_types) 3 CollectI Inr_oplus_iff cont_deftr
+ iso_tuple_UNIV_I)
+ then obtain n where "tr2 = deftr n" by auto
+ thus "tr1 = deftr (root tr1)" using IH by auto
+ qed
+ }
+ thus ?thesis using assms by auto
+qed
+
+lemma reg_deftr: "reg deftr (deftr n)"
+unfolding reg_def using subtr_deftr by auto
+
+lemma dtree_subtrOf_Union:
+assumes "dtree tr"
+shows "\<Union>{K tr' |tr'. Inr tr' \<in> cont tr} =
+ \<Union>{K (subtrOf tr n) |n. Inr n \<in> prodOf tr}"
+unfolding Union_eq Bex_def mem_Collect_eq proof safe
+ fix x xa tr'
+ assume x: "x \<in> K tr'" and tr'_tr: "Inr tr' \<in> cont tr"
+ show "\<exists>X. (\<exists>n. X = K (subtrOf tr n) \<and> Inr n \<in> prodOf tr) \<and> x \<in> X"
+ apply(rule exI[of _ "K (subtrOf tr (root tr'))"]) apply(intro conjI)
+ apply(rule exI[of _ "root tr'"]) apply (metis (lifting) root_prodOf tr'_tr)
+ by (metis (lifting) assms subtrOf_root tr'_tr x)
+next
+ fix x X n ttr
+ assume x: "x \<in> K (subtrOf tr n)" and n: "Inr n = (id \<oplus> root) ttr" and ttr: "ttr \<in> cont tr"
+ show "\<exists>X. (\<exists>tr'. X = K tr' \<and> Inr tr' \<in> cont tr) \<and> x \<in> X"
+ apply(rule exI[of _ "K (subtrOf tr n)"]) apply(intro conjI)
+ apply(rule exI[of _ "subtrOf tr n"]) apply (metis imageI n subtrOf ttr)
+ using x .
+qed
+
+
+
+
+subsection {* Paths in a regular tree *}
+
+inductive path :: "(N \<Rightarrow> Tree) \<Rightarrow> N list \<Rightarrow> bool" for f where
+Base: "path f [n]"
+|
+Ind: "\<lbrakk>path f (n1 # nl); Inr (f n1) \<in> cont (f n)\<rbrakk>
+ \<Longrightarrow> path f (n # n1 # nl)"
+
+lemma path_NE:
+assumes "path f nl"
+shows "nl \<noteq> Nil"
+using assms apply(induct rule: path.induct) by auto
+
+lemma path_post:
+assumes f: "path f (n # nl)" and nl: "nl \<noteq> []"
+shows "path f nl"
+proof-
+ obtain n1 nl1 where nl: "nl = n1 # nl1" using nl by (cases nl, auto)
+ show ?thesis using assms unfolding nl using path.simps by (metis (lifting) list.inject)
+qed
+
+lemma path_post_concat:
+assumes "path f (nl1 @ nl2)" and "nl2 \<noteq> Nil"
+shows "path f nl2"
+using assms apply (induct nl1)
+apply (metis append_Nil) by (metis Nil_is_append_conv append_Cons path_post)
+
+lemma path_concat:
+assumes "path f nl1" and "path f ((last nl1) # nl2)"
+shows "path f (nl1 @ nl2)"
+using assms apply(induct rule: path.induct) apply simp
+by (metis append_Cons last.simps list.simps(3) path.Ind)
+
+lemma path_distinct:
+assumes "path f nl"
+shows "\<exists> nl'. path f nl' \<and> hd nl' = hd nl \<and> last nl' = last nl \<and>
+ set nl' \<subseteq> set nl \<and> distinct nl'"
+using assms proof(induct rule: length_induct)
+ case (1 nl) hence p_nl: "path f nl" by simp
+ then obtain n nl1 where nl: "nl = n # nl1" by (metis list.exhaust path_NE)
+ show ?case
+ proof(cases nl1)
+ case Nil
+ show ?thesis apply(rule exI[of _ nl]) using path.Base unfolding nl Nil by simp
+ next
+ case (Cons n1 nl2)
+ hence p1: "path f nl1" by (metis list.simps nl p_nl path_post)
+ show ?thesis
+ proof(cases "n \<in> set nl1")
+ case False
+ obtain nl1' where p1': "path f nl1'" and hd_nl1': "hd nl1' = hd nl1" and
+ l_nl1': "last nl1' = last nl1" and d_nl1': "distinct nl1'"
+ and s_nl1': "set nl1' \<subseteq> set nl1"
+ using 1(1)[THEN allE[of _ nl1]] p1 unfolding nl by auto
+ obtain nl2' where nl1': "nl1' = n1 # nl2'" using path_NE[OF p1'] hd_nl1'
+ unfolding Cons by(cases nl1', auto)
+ show ?thesis apply(intro exI[of _ "n # nl1'"]) unfolding nl proof safe
+ show "path f (n # nl1')" unfolding nl1'
+ apply(rule path.Ind, metis nl1' p1')
+ by (metis (lifting) Cons list.inject nl p1 p_nl path.simps path_NE)
+ qed(insert l_nl1' Cons nl1' s_nl1' d_nl1' False, auto)
+ next
+ case True
+ then obtain nl11 nl12 where nl1: "nl1 = nl11 @ n # nl12"
+ by (metis split_list)
+ have p12: "path f (n # nl12)"
+ apply(rule path_post_concat[of _ "n # nl11"]) using p_nl[unfolded nl nl1] by auto
+ obtain nl12' where p1': "path f nl12'" and hd_nl12': "hd nl12' = n" and
+ l_nl12': "last nl12' = last (n # nl12)" and d_nl12': "distinct nl12'"
+ and s_nl12': "set nl12' \<subseteq> {n} \<union> set nl12"
+ using 1(1)[THEN allE[of _ "n # nl12"]] p12 unfolding nl nl1 by auto
+ thus ?thesis apply(intro exI[of _ nl12']) unfolding nl nl1 by auto
+ qed
+ qed
+qed
+
+lemma path_subtr:
+assumes f: "\<And> n. root (f n) = n"
+and p: "path f nl"
+shows "subtr (set nl) (f (last nl)) (f (hd nl))"
+using p proof (induct rule: path.induct)
+ case (Ind n1 nl n) let ?ns1 = "insert n1 (set nl)"
+ have "path f (n1 # nl)"
+ and "subtr ?ns1 (f (last (n1 # nl))) (f n1)"
+ and fn1: "Inr (f n1) \<in> cont (f n)" using Ind by simp_all
+ hence fn1_flast: "subtr (insert n ?ns1) (f (last (n1 # nl))) (f n1)"
+ by (metis subset_insertI subtr_mono)
+ have 1: "last (n # n1 # nl) = last (n1 # nl)" by auto
+ have "subtr (insert n ?ns1) (f (last (n1 # nl))) (f n)"
+ using f subtr.Step[OF _ fn1_flast fn1] by auto
+ thus ?case unfolding 1 by simp
+qed (metis f hd.simps last_ConsL last_in_set not_Cons_self2 subtr.Refl)
+
+lemma reg_subtr_path_aux:
+assumes f: "reg f tr" and n: "subtr ns tr1 tr"
+shows "\<exists> nl. path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns"
+using n f proof(induct rule: subtr.induct)
+ case (Refl tr ns)
+ thus ?case
+ apply(intro exI[of _ "[root tr]"]) apply simp by (metis (lifting) path.Base reg_root)
+next
+ case (Step tr ns tr2 tr1)
+ hence rtr: "root tr \<in> ns" and tr1_tr: "Inr tr1 \<in> cont tr"
+ and tr2_tr1: "subtr ns tr2 tr1" and tr: "reg f tr" by auto
+ have tr1: "reg f tr1" using reg_subtr[OF tr] rtr tr1_tr
+ by (metis (lifting) Step.prems iso_tuple_UNIV_I reg_def subtr.Step)
+ obtain nl where nl: "path f nl" and f_nl: "f (hd nl) = tr1"
+ and last_nl: "f (last nl) = tr2" and set: "set nl \<subseteq> ns" using Step(3)[OF tr1] by auto
+ have 0: "path f (root tr # nl)" apply (subst path.simps)
+ using f_nl nl reg_root tr tr1_tr by (metis hd.simps neq_Nil_conv)
+ show ?case apply(rule exI[of _ "(root tr) # nl"])
+ using 0 reg_root tr last_nl nl path_NE rtr set by auto
+qed
+
+lemma reg_subtr_path:
+assumes f: "reg f tr" and n: "subtr ns tr1 tr"
+shows "\<exists> nl. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns"
+using reg_subtr_path_aux[OF assms] path_distinct[of f]
+by (metis (lifting) order_trans)
+
+lemma subtr_iff_path:
+assumes r: "reg f tr" and f: "\<And> n. root (f n) = n"
+shows "subtr ns tr1 tr \<longleftrightarrow>
+ (\<exists> nl. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns)"
+proof safe
+ fix nl assume p: "path f nl" and nl: "set nl \<subseteq> ns"
+ have "subtr (set nl) (f (last nl)) (f (hd nl))"
+ apply(rule path_subtr) using p f by simp_all
+ thus "subtr ns (f (last nl)) (f (hd nl))"
+ using subtr_mono nl by auto
+qed(insert reg_subtr_path[OF r], auto)
+
+lemma inFr_iff_path:
+assumes r: "reg f tr" and f: "\<And> n. root (f n) = n"
+shows
+"inFr ns tr t \<longleftrightarrow>
+ (\<exists> nl tr1. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and>
+ set nl \<subseteq> ns \<and> Inl t \<in> cont tr1)"
+apply safe
+apply (metis (no_types) inFr_subtr r reg_subtr_path)
+by (metis f inFr.Base path_subtr subtr_inFr subtr_mono subtr_rootL_in)
+
+
+
+subsection{* The regular cut of a tree *}
+
+context fixes tr0 :: Tree
+begin
+
+(* Picking a subtree of a certain root: *)
+definition "pick n \<equiv> SOME tr. subtr UNIV tr tr0 \<and> root tr = n"
+
+lemma pick:
+assumes "inItr UNIV tr0 n"
+shows "subtr UNIV (pick n) tr0 \<and> root (pick n) = n"
+proof-
+ have "\<exists> tr. subtr UNIV tr tr0 \<and> root tr = n"
+ using assms by (metis (lifting) inItr_subtr)
+ thus ?thesis unfolding pick_def by(rule someI_ex)
+qed
+
+lemmas subtr_pick = pick[THEN conjunct1]
+lemmas root_pick = pick[THEN conjunct2]
+
+lemma dtree_pick:
+assumes tr0: "dtree tr0" and n: "inItr UNIV tr0 n"
+shows "dtree (pick n)"
+using dtree_subtr[OF tr0 subtr_pick[OF n]] .
+
+definition "regOf_r n \<equiv> root (pick n)"
+definition "regOf_c n \<equiv> (id \<oplus> root) ` cont (pick n)"
+
+(* The regular tree of a function: *)
+definition regOf :: "N \<Rightarrow> Tree" where
+"regOf \<equiv> coit regOf_r regOf_c"
+
+lemma finite_regOf_c: "finite (regOf_c n)"
+unfolding regOf_c_def by (metis finite_cont finite_imageI)
+
+lemma root_regOf_pick: "root (regOf n) = root (pick n)"
+using coit(1)[of regOf_r regOf_c n] unfolding regOf_def regOf_r_def by simp
+
+lemma root_regOf[simp]:
+assumes "inItr UNIV tr0 n"
+shows "root (regOf n) = n"
+unfolding root_regOf_pick root_pick[OF assms] ..
+
+lemma cont_regOf[simp]:
+"cont (regOf n) = (id \<oplus> (regOf o root)) ` cont (pick n)"
+apply(subst id_o[symmetric, of id]) unfolding sum_map.comp[symmetric]
+unfolding image_compose unfolding regOf_c_def[symmetric]
+using coit(2)[of regOf_c n regOf_r, OF finite_regOf_c]
+unfolding regOf_def ..
+
+lemma Inl_cont_regOf[simp]:
+"Inl -` (cont (regOf n)) = Inl -` (cont (pick n))"
+unfolding cont_regOf by simp
+
+lemma Inr_cont_regOf:
+"Inr -` (cont (regOf n)) = (regOf \<circ> root) ` (Inr -` cont (pick n))"
+unfolding cont_regOf by simp
+
+lemma subtr_regOf:
+assumes n: "inItr UNIV tr0 n" and "subtr UNIV tr1 (regOf n)"
+shows "\<exists> n1. inItr UNIV tr0 n1 \<and> tr1 = regOf n1"
+proof-
+ {fix tr ns assume "subtr UNIV tr1 tr"
+ hence "tr = regOf n \<longrightarrow> (\<exists> n1. inItr UNIV tr0 n1 \<and> tr1 = regOf n1)"
+ proof (induct rule: subtr_UNIV_inductL)
+ case (Step tr2 tr1 tr)
+ show ?case proof
+ assume "tr = regOf n"
+ then obtain n1 where tr2: "Inr tr2 \<in> cont tr1"
+ and tr1_tr: "subtr UNIV tr1 tr" and n1: "inItr UNIV tr0 n1" and tr1: "tr1 = regOf n1"
+ using Step by auto
+ obtain tr2' where tr2: "tr2 = regOf (root tr2')"
+ and tr2': "Inr tr2' \<in> cont (pick n1)"
+ using tr2 Inr_cont_regOf[of n1]
+ unfolding tr1 image_def o_def using vimage_eq by auto
+ have "inItr UNIV tr0 (root tr2')"
+ using inItr.Base inItr.Ind n1 pick subtr_inItr tr2' by (metis iso_tuple_UNIV_I)
+ thus "\<exists>n2. inItr UNIV tr0 n2 \<and> tr2 = regOf n2" using tr2 by blast
+ qed
+ qed(insert n, auto)
+ }
+ thus ?thesis using assms by auto
+qed
+
+lemma root_regOf_root:
+assumes n: "inItr UNIV tr0 n" and t_tr: "t_tr \<in> cont (pick n)"
+shows "(id \<oplus> (root \<circ> regOf \<circ> root)) t_tr = (id \<oplus> root) t_tr"
+using assms apply(cases t_tr)
+ apply (metis (lifting) sum_map.simps(1))
+ using pick regOf_def regOf_r_def coit(1)
+ inItr.Base o_apply subtr_StepL subtr_inItr sum_map.simps(2)
+ by (metis UNIV_I)
+
+lemma regOf_P:
+assumes tr0: "dtree tr0" and n: "inItr UNIV tr0 n"
+shows "(n, (id \<oplus> root) ` cont (regOf n)) \<in> P" (is "?L \<in> P")
+proof-
+ have "?L = (n, (id \<oplus> root) ` cont (pick n))"
+ unfolding cont_regOf image_compose[symmetric] sum_map.comp id_o o_assoc
+ unfolding Pair_eq apply(rule conjI[OF refl]) apply(rule image_cong[OF refl])
+ by(rule root_regOf_root[OF n])
+ moreover have "... \<in> P" by (metis (lifting) dtree_pick root_pick dtree_P n tr0)
+ ultimately show ?thesis by simp
+qed
+
+lemma dtree_regOf:
+assumes tr0: "dtree tr0" and "inItr UNIV tr0 n"
+shows "dtree (regOf n)"
+proof-
+ {fix tr have "\<exists> n. inItr UNIV tr0 n \<and> tr = regOf n \<Longrightarrow> dtree tr"
+ proof (induct rule: dtree_raw_coind)
+ case (Hyp tr)
+ then obtain n where n: "inItr UNIV tr0 n" and tr: "tr = regOf n" by auto
+ show ?case unfolding lift_def apply safe
+ apply (metis (lifting) regOf_P root_regOf n tr tr0)
+ unfolding tr Inr_cont_regOf unfolding inj_on_def apply clarsimp using root_regOf
+ apply (metis UNIV_I inItr.Base n pick subtr2.simps subtr_inItr subtr_subtr2)
+ by (metis n subtr.Refl subtr_StepL subtr_regOf tr UNIV_I)
+ qed
+ }
+ thus ?thesis using assms by blast
+qed
+
+(* The regular cut of a tree: *)
+definition "rcut \<equiv> regOf (root tr0)"
+
+theorem reg_rcut: "reg regOf rcut"
+unfolding reg_def rcut_def
+by (metis inItr.Base root_regOf subtr_regOf UNIV_I)
+
+lemma rcut_reg:
+assumes "reg regOf tr0"
+shows "rcut = tr0"
+using assms unfolding rcut_def reg_def by (metis subtr.Refl UNIV_I)
+
+theorem rcut_eq: "rcut = tr0 \<longleftrightarrow> reg regOf tr0"
+using reg_rcut rcut_reg by metis
+
+theorem regular_rcut: "regular rcut"
+using reg_rcut unfolding regular_def by blast
+
+theorem Fr_rcut: "Fr UNIV rcut \<subseteq> Fr UNIV tr0"
+proof safe
+ fix t assume "t \<in> Fr UNIV rcut"
+ then obtain tr where t: "Inl t \<in> cont tr" and tr: "subtr UNIV tr (regOf (root tr0))"
+ using Fr_subtr[of UNIV "regOf (root tr0)"] unfolding rcut_def
+ by (metis (full_types) Fr_def inFr_subtr mem_Collect_eq)
+ obtain n where n: "inItr UNIV tr0 n" and tr: "tr = regOf n" using tr
+ by (metis (lifting) inItr.Base subtr_regOf UNIV_I)
+ have "Inl t \<in> cont (pick n)" using t using Inl_cont_regOf[of n] unfolding tr
+ by (metis (lifting) vimageD vimageI2)
+ moreover have "subtr UNIV (pick n) tr0" using subtr_pick[OF n] ..
+ ultimately show "t \<in> Fr UNIV tr0" unfolding Fr_subtr_cont by auto
+qed
+
+theorem dtree_rcut:
+assumes "dtree tr0"
+shows "dtree rcut"
+unfolding rcut_def using dtree_regOf[OF assms inItr.Base] by simp
+
+theorem root_rcut[simp]: "root rcut = root tr0"
+unfolding rcut_def
+by (metis (lifting) root_regOf inItr.Base reg_def reg_root subtr_rootR_in)
+
+end (* context *)
+
+
+subsection{* Recursive description of the regular tree frontiers *}
+
+lemma regular_inFr:
+assumes r: "regular tr" and In: "root tr \<in> ns"
+and t: "inFr ns tr t"
+shows "t \<in> Inl -` (cont tr) \<or>
+ (\<exists> tr'. Inr tr' \<in> cont tr \<and> inFr (ns - {root tr}) tr' t)"
+(is "?L \<or> ?R")
+proof-
+ obtain f where r: "reg f tr" and f: "\<And>n. root (f n) = n"
+ using r unfolding regular_def2 by auto
+ obtain nl tr1 where d_nl: "distinct nl" and p: "path f nl" and hd_nl: "f (hd nl) = tr"
+ and l_nl: "f (last nl) = tr1" and s_nl: "set nl \<subseteq> ns" and t_tr1: "Inl t \<in> cont tr1"
+ using t unfolding inFr_iff_path[OF r f] by auto
+ obtain n nl1 where nl: "nl = n # nl1" by (metis (lifting) p path.simps)
+ hence f_n: "f n = tr" using hd_nl by simp
+ have n_nl1: "n \<notin> set nl1" using d_nl unfolding nl by auto
+ show ?thesis
+ proof(cases nl1)
+ case Nil hence "tr = tr1" using f_n l_nl unfolding nl by simp
+ hence ?L using t_tr1 by simp thus ?thesis by simp
+ next
+ case (Cons n1 nl2) note nl1 = Cons
+ have 1: "last nl1 = last nl" "hd nl1 = n1" unfolding nl nl1 by simp_all
+ have p1: "path f nl1" and n1_tr: "Inr (f n1) \<in> cont tr"
+ using path.simps[of f nl] p f_n unfolding nl nl1 by auto
+ have r1: "reg f (f n1)" using reg_Inr_cont[OF r n1_tr] .
+ have 0: "inFr (set nl1) (f n1) t" unfolding inFr_iff_path[OF r1 f]
+ apply(intro exI[of _ nl1], intro exI[of _ tr1])
+ using d_nl unfolding 1 l_nl unfolding nl using p1 t_tr1 by auto
+ have root_tr: "root tr = n" by (metis f f_n)
+ have "inFr (ns - {root tr}) (f n1) t" apply(rule inFr_mono[OF 0])
+ using s_nl unfolding root_tr unfolding nl using n_nl1 by auto
+ thus ?thesis using n1_tr by auto
+ qed
+qed
+
+theorem regular_Fr:
+assumes r: "regular tr" and In: "root tr \<in> ns"
+shows "Fr ns tr =
+ Inl -` (cont tr) \<union>
+ \<Union> {Fr (ns - {root tr}) tr' | tr'. Inr tr' \<in> cont tr}"
+unfolding Fr_def
+using In inFr.Base regular_inFr[OF assms] apply safe
+apply (simp, metis (full_types) UnionI mem_Collect_eq)
+apply simp
+by (simp, metis (lifting) inFr_Ind_minus insert_Diff)
+
+
+subsection{* The generated languages *}
+
+(* The (possibly inifinite tree) generated language *)
+definition "L ns n \<equiv> {Fr ns tr | tr. dtree tr \<and> root tr = n}"
+
+(* The regular-tree generated language *)
+definition "Lr ns n \<equiv> {Fr ns tr | tr. dtree tr \<and> root tr = n \<and> regular tr}"
+
+theorem L_rec_notin:
+assumes "n \<notin> ns"
+shows "L ns n = {{}}"
+using assms unfolding L_def apply safe
+ using not_root_Fr apply force
+ apply(rule exI[of _ "deftr n"])
+ by (metis (no_types) dtree_deftr not_root_Fr root_deftr)
+
+theorem Lr_rec_notin:
+assumes "n \<notin> ns"
+shows "Lr ns n = {{}}"
+using assms unfolding Lr_def apply safe
+ using not_root_Fr apply force
+ apply(rule exI[of _ "deftr n"])
+ by (metis (no_types) regular_def dtree_deftr not_root_Fr reg_deftr root_deftr)
+
+lemma dtree_subtrOf:
+assumes "dtree tr" and "Inr n \<in> prodOf tr"
+shows "dtree (subtrOf tr n)"
+by (metis assms dtree_lift lift_def subtrOf)
+
+theorem Lr_rec_in:
+assumes n: "n \<in> ns"
+shows "Lr ns n \<subseteq>
+{Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
+ (n,tns) \<in> P \<and>
+ (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> Lr (ns - {n}) n')}"
+(is "Lr ns n \<subseteq> {?F tns K | tns K. (n,tns) \<in> P \<and> ?\<phi> tns K}")
+proof safe
+ fix ts assume "ts \<in> Lr ns n"
+ then obtain tr where dtr: "dtree tr" and r: "root tr = n" and tr: "regular tr"
+ and ts: "ts = Fr ns tr" unfolding Lr_def by auto
+ def tns \<equiv> "(id \<oplus> root) ` (cont tr)"
+ def K \<equiv> "\<lambda> n'. Fr (ns - {n}) (subtrOf tr n')"
+ show "\<exists>tns K. ts = ?F tns K \<and> (n, tns) \<in> P \<and> ?\<phi> tns K"
+ apply(rule exI[of _ tns], rule exI[of _ K]) proof(intro conjI allI impI)
+ show "ts = Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns}"
+ unfolding ts regular_Fr[OF tr n[unfolded r[symmetric]]]
+ unfolding tns_def K_def r[symmetric]
+ unfolding Inl_prodOf dtree_subtrOf_Union[OF dtr] ..
+ show "(n, tns) \<in> P" unfolding tns_def r[symmetric] using dtree_P[OF dtr] .
+ fix n' assume "Inr n' \<in> tns" thus "K n' \<in> Lr (ns - {n}) n'"
+ unfolding K_def Lr_def mem_Collect_eq apply(intro exI[of _ "subtrOf tr n'"])
+ using dtr tr apply(intro conjI refl) unfolding tns_def
+ apply(erule dtree_subtrOf[OF dtr])
+ apply (metis subtrOf)
+ by (metis Inr_subtrOf UNIV_I regular_subtr subtr.simps)
+ qed
+qed
+
+lemma hsubst_aux:
+fixes n ftr tns
+assumes n: "n \<in> ns" and tns: "finite tns" and
+1: "\<And> n'. Inr n' \<in> tns \<Longrightarrow> dtree (ftr n')"
+defines "tr \<equiv> Node n ((id \<oplus> ftr) ` tns)" defines "tr' \<equiv> hsubst tr tr"
+shows "Fr ns tr' = Inl -` tns \<union> \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
+(is "_ = ?B") proof-
+ have rtr: "root tr = n" and ctr: "cont tr = (id \<oplus> ftr) ` tns"
+ unfolding tr_def using tns by auto
+ have Frr: "Frr (ns - {n}) tr = \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
+ unfolding Frr_def ctr by auto
+ have "Fr ns tr' = Inl -` (cont tr) \<union> Frr (ns - {n}) tr"
+ using Fr_self_hsubst[OF n[unfolded rtr[symmetric]]] unfolding tr'_def rtr ..
+ also have "... = ?B" unfolding ctr Frr by simp
+ finally show ?thesis .
+qed
+
+theorem L_rec_in:
+assumes n: "n \<in> ns"
+shows "
+{Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
+ (n,tns) \<in> P \<and>
+ (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> L (ns - {n}) n')}
+ \<subseteq> L ns n"
+proof safe
+ fix tns K
+ assume P: "(n, tns) \<in> P" and 0: "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> L (ns - {n}) n'"
+ {fix n' assume "Inr n' \<in> tns"
+ hence "K n' \<in> L (ns - {n}) n'" using 0 by auto
+ hence "\<exists> tr'. K n' = Fr (ns - {n}) tr' \<and> dtree tr' \<and> root tr' = n'"
+ unfolding L_def mem_Collect_eq by auto
+ }
+ then obtain ftr where 0: "\<And> n'. Inr n' \<in> tns \<Longrightarrow>
+ K n' = Fr (ns - {n}) (ftr n') \<and> dtree (ftr n') \<and> root (ftr n') = n'"
+ by metis
+ def tr \<equiv> "Node n ((id \<oplus> ftr) ` tns)" def tr' \<equiv> "hsubst tr tr"
+ have rtr: "root tr = n" and ctr: "cont tr = (id \<oplus> ftr) ` tns"
+ unfolding tr_def by (simp, metis P cont_Node finite_imageI finite_in_P)
+ have prtr: "prodOf tr = tns" apply(rule Inl_Inr_image_cong)
+ unfolding ctr apply simp apply simp apply safe
+ using 0 unfolding image_def apply force apply simp by (metis 0 vimageI2)
+ have 1: "{K n' |n'. Inr n' \<in> tns} = {Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
+ using 0 by auto
+ have dtr: "dtree tr" apply(rule dtree.Tree)
+ apply (metis (lifting) P prtr rtr)
+ unfolding inj_on_def ctr lift_def using 0 by auto
+ hence dtr': "dtree tr'" unfolding tr'_def by (metis dtree_hsubst)
+ have tns: "finite tns" using finite_in_P P by simp
+ have "Inl -` tns \<union> \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns} \<in> L ns n"
+ unfolding L_def mem_Collect_eq apply(intro exI[of _ tr'] conjI)
+ using dtr' 0 hsubst_aux[OF assms tns, of ftr] unfolding tr_def tr'_def by auto
+ thus "Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} \<in> L ns n" unfolding 1 .
+qed
+
+lemma card_N: "(n::N) \<in> ns \<Longrightarrow> card (ns - {n}) < card ns"
+by (metis finite_N Diff_UNIV Diff_infinite_finite card_Diff1_less finite.emptyI)
+
+function LL where
+"LL ns n =
+ (if n \<notin> ns then {{}} else
+ {Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
+ (n,tns) \<in> P \<and>
+ (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> LL (ns - {n}) n')})"
+by(pat_completeness, auto)
+termination apply(relation "inv_image (measure card) fst")
+using card_N by auto
+
+declare LL.simps[code] (* TODO: Does code generation for LL work? *)
+declare LL.simps[simp del]
+
+theorem Lr_LL: "Lr ns n \<subseteq> LL ns n"
+proof (induct ns arbitrary: n rule: measure_induct[of card])
+ case (1 ns n) show ?case proof(cases "n \<in> ns")
+ case False thus ?thesis unfolding Lr_rec_notin[OF False] by (simp add: LL.simps)
+ next
+ case True show ?thesis apply(rule subset_trans)
+ using Lr_rec_in[OF True] apply assumption
+ unfolding LL.simps[of ns n] using True 1 card_N proof clarsimp
+ fix tns K
+ assume "n \<in> ns" hence c: "card (ns - {n}) < card ns" using card_N by blast
+ assume "(n, tns) \<in> P"
+ and "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> Lr (ns - {n}) n'"
+ thus "\<exists>tnsa Ka.
+ Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} =
+ Inl -` tnsa \<union> \<Union>{Ka n' |n'. Inr n' \<in> tnsa} \<and>
+ (n, tnsa) \<in> P \<and> (\<forall>n'. Inr n' \<in> tnsa \<longrightarrow> Ka n' \<in> LL (ns - {n}) n')"
+ apply(intro exI[of _ tns] exI[of _ K]) using c 1 by auto
+ qed
+ qed
+qed
+
+theorem LL_L: "LL ns n \<subseteq> L ns n"
+proof (induct ns arbitrary: n rule: measure_induct[of card])
+ case (1 ns n) show ?case proof(cases "n \<in> ns")
+ case False thus ?thesis unfolding L_rec_notin[OF False] by (simp add: LL.simps)
+ next
+ case True show ?thesis apply(rule subset_trans)
+ prefer 2 using L_rec_in[OF True] apply assumption
+ unfolding LL.simps[of ns n] using True 1 card_N proof clarsimp
+ fix tns K
+ assume "n \<in> ns" hence c: "card (ns - {n}) < card ns" using card_N by blast
+ assume "(n, tns) \<in> P"
+ and "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> LL (ns - {n}) n'"
+ thus "\<exists>tnsa Ka.
+ Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} =
+ Inl -` tnsa \<union> \<Union>{Ka n' |n'. Inr n' \<in> tnsa} \<and>
+ (n, tnsa) \<in> P \<and> (\<forall>n'. Inr n' \<in> tnsa \<longrightarrow> Ka n' \<in> L (ns - {n}) n')"
+ apply(intro exI[of _ tns] exI[of _ K]) using c 1 by auto
+ qed
+ qed
+qed
+
+(* The subsumpsion relation between languages *)
+definition "subs L1 L2 \<equiv> \<forall> ts2 \<in> L2. \<exists> ts1 \<in> L1. ts1 \<subseteq> ts2"
+
+lemma incl_subs[simp]: "L2 \<subseteq> L1 \<Longrightarrow> subs L1 L2"
+unfolding subs_def by auto
+
+lemma subs_refl[simp]: "subs L1 L1" unfolding subs_def by auto
+
+lemma subs_trans: "\<lbrakk>subs L1 L2; subs L2 L3\<rbrakk> \<Longrightarrow> subs L1 L3"
+unfolding subs_def by (metis subset_trans)
+
+(* Language equivalence *)
+definition "leqv L1 L2 \<equiv> subs L1 L2 \<and> subs L2 L1"
+
+lemma subs_leqv[simp]: "leqv L1 L2 \<Longrightarrow> subs L1 L2"
+unfolding leqv_def by auto
+
+lemma subs_leqv_sym[simp]: "leqv L1 L2 \<Longrightarrow> subs L2 L1"
+unfolding leqv_def by auto
+
+lemma leqv_refl[simp]: "leqv L1 L1" unfolding leqv_def by auto
+
+lemma leqv_trans:
+assumes 12: "leqv L1 L2" and 23: "leqv L2 L3"
+shows "leqv L1 L3"
+using assms unfolding leqv_def by (metis (lifting) subs_trans)
+
+lemma leqv_sym: "leqv L1 L2 \<Longrightarrow> leqv L2 L1"
+unfolding leqv_def by auto
+
+lemma leqv_Sym: "leqv L1 L2 \<longleftrightarrow> leqv L2 L1"
+unfolding leqv_def by auto
+
+lemma Lr_incl_L: "Lr ns ts \<subseteq> L ns ts"
+unfolding Lr_def L_def by auto
+
+lemma Lr_subs_L: "subs (Lr UNIV ts) (L UNIV ts)"
+unfolding subs_def proof safe
+ fix ts2 assume "ts2 \<in> L UNIV ts"
+ then obtain tr where ts2: "ts2 = Fr UNIV tr" and dtr: "dtree tr" and rtr: "root tr = ts"
+ unfolding L_def by auto
+ thus "\<exists>ts1\<in>Lr UNIV ts. ts1 \<subseteq> ts2"
+ apply(intro bexI[of _ "Fr UNIV (rcut tr)"])
+ unfolding Lr_def L_def using Fr_rcut dtree_rcut root_rcut regular_rcut by auto
+qed
+
+theorem Lr_leqv_L: "leqv (Lr UNIV ts) (L UNIV ts)"
+using Lr_subs_L unfolding leqv_def by (metis (lifting) Lr_incl_L incl_subs)
+
+theorem LL_leqv_L: "leqv (LL UNIV ts) (L UNIV ts)"
+by (metis (lifting) LL_L Lr_LL Lr_subs_L incl_subs leqv_def subs_trans)
+
+theorem LL_leqv_Lr: "leqv (LL UNIV ts) (Lr UNIV ts)"
+using Lr_leqv_L LL_leqv_L by (metis leqv_Sym leqv_trans)
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Infinite_Derivation_Trees/Parallel.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,143 @@
+theory Parallel
+imports Tree
+begin
+
+
+consts Nplus :: "N \<Rightarrow> N \<Rightarrow> N" (infixl "+" 60)
+
+axiomatization where
+ Nplus_comm: "(a::N) + b = b + (a::N)"
+and Nplus_assoc: "((a::N) + b) + c = a + (b + c)"
+
+
+
+section{* Parallel composition *}
+
+fun par_r where "par_r (tr1,tr2) = root tr1 + root tr2"
+fun par_c where
+"par_c (tr1,tr2) =
+ Inl ` (Inl -` (cont tr1 \<union> cont tr2)) \<union>
+ Inr ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
+
+declare par_r.simps[simp del] declare par_c.simps[simp del]
+
+definition par :: "Tree \<times> Tree \<Rightarrow> Tree" where
+"par \<equiv> coit par_r par_c"
+
+abbreviation par_abbr (infixr "\<parallel>" 80) where "tr1 \<parallel> tr2 \<equiv> par (tr1, tr2)"
+
+lemma finite_par_c: "finite (par_c (tr1, tr2))"
+unfolding par_c.simps apply(rule finite_UnI)
+ apply (metis finite_Un finite_cont finite_imageI finite_vimageI inj_Inl)
+ apply(intro finite_imageI finite_cartesian_product finite_vimageI)
+ using finite_cont by auto
+
+lemma root_par: "root (tr1 \<parallel> tr2) = root tr1 + root tr2"
+using coit(1)[of par_r par_c "(tr1,tr2)"] unfolding par_def par_r.simps by simp
+
+lemma cont_par:
+"cont (tr1 \<parallel> tr2) = (id \<oplus> par) ` par_c (tr1,tr2)"
+using coit(2)[of par_c "(tr1,tr2)" par_r, OF finite_par_c]
+unfolding par_def ..
+
+lemma Inl_cont_par[simp]:
+"Inl -` (cont (tr1 \<parallel> tr2)) = Inl -` (cont tr1 \<union> cont tr2)"
+unfolding cont_par par_c.simps by auto
+
+lemma Inr_cont_par[simp]:
+"Inr -` (cont (tr1 \<parallel> tr2)) = par ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
+unfolding cont_par par_c.simps by auto
+
+lemma Inl_in_cont_par:
+"Inl t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (Inl t \<in> cont tr1 \<or> Inl t \<in> cont tr2)"
+using Inl_cont_par[of tr1 tr2] unfolding vimage_def by auto
+
+lemma Inr_in_cont_par:
+"Inr t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (t \<in> par ` (Inr -` cont tr1 \<times> Inr -` cont tr2))"
+using Inr_cont_par[of tr1 tr2] unfolding vimage_def by auto
+
+
+section{* =-coinductive proofs *}
+
+(* Detailed proofs of commutativity and associativity: *)
+theorem par_com: "tr1 \<parallel> tr2 = tr2 \<parallel> tr1"
+proof-
+ let ?\<phi> = "\<lambda> trA trB. \<exists> tr1 tr2. trA = tr1 \<parallel> tr2 \<and> trB = tr2 \<parallel> tr1"
+ {fix trA trB
+ assume "?\<phi> trA trB" hence "trA = trB"
+ proof (induct rule: Tree_coind, safe)
+ fix tr1 tr2
+ show "root (tr1 \<parallel> tr2) = root (tr2 \<parallel> tr1)"
+ unfolding root_par by (rule Nplus_comm)
+ next
+ fix tr1 tr2 :: Tree
+ let ?trA = "tr1 \<parallel> tr2" let ?trB = "tr2 \<parallel> tr1"
+ show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
+ unfolding lift2_def proof(intro conjI allI impI)
+ fix n show "Inl n \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> Inl n \<in> cont (tr2 \<parallel> tr1)"
+ unfolding Inl_in_cont_par by auto
+ next
+ fix trA' assume "Inr trA' \<in> cont ?trA"
+ then obtain tr1' tr2' where "trA' = tr1' \<parallel> tr2'"
+ and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+ unfolding Inr_in_cont_par by auto
+ thus "\<exists> trB'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
+ apply(intro exI[of _ "tr2' \<parallel> tr1'"]) unfolding Inr_in_cont_par by auto
+ next
+ fix trB' assume "Inr trB' \<in> cont ?trB"
+ then obtain tr1' tr2' where "trB' = tr2' \<parallel> tr1'"
+ and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+ unfolding Inr_in_cont_par by auto
+ thus "\<exists> trA'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
+ apply(intro exI[of _ "tr1' \<parallel> tr2'"]) unfolding Inr_in_cont_par by auto
+ qed
+ qed
+ }
+ thus ?thesis by blast
+qed
+
+theorem par_assoc: "(tr1 \<parallel> tr2) \<parallel> tr3 = tr1 \<parallel> (tr2 \<parallel> tr3)"
+proof-
+ let ?\<phi> =
+ "\<lambda> trA trB. \<exists> tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and>
+ trB = tr1 \<parallel> (tr2 \<parallel> tr3)"
+ {fix trA trB
+ assume "?\<phi> trA trB" hence "trA = trB"
+ proof (induct rule: Tree_coind, safe)
+ fix tr1 tr2 tr3
+ show "root ((tr1 \<parallel> tr2) \<parallel> tr3) = root (tr1 \<parallel> (tr2 \<parallel> tr3))"
+ unfolding root_par by (rule Nplus_assoc)
+ next
+ fix tr1 tr2 tr3
+ let ?trA = "(tr1 \<parallel> tr2) \<parallel> tr3" let ?trB = "tr1 \<parallel> (tr2 \<parallel> tr3)"
+ show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
+ unfolding lift2_def proof(intro conjI allI impI)
+ fix n show "Inl n \<in> (cont ?trA) \<longleftrightarrow> Inl n \<in> (cont ?trB)"
+ unfolding Inl_in_cont_par by simp
+ next
+ fix trA' assume "Inr trA' \<in> cont ?trA"
+ then obtain tr1' tr2' tr3' where "trA' = (tr1' \<parallel> tr2') \<parallel> tr3'"
+ and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+ and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
+ thus "\<exists> trB'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
+ apply(intro exI[of _ "tr1' \<parallel> (tr2' \<parallel> tr3')"])
+ unfolding Inr_in_cont_par by auto
+ next
+ fix trB' assume "Inr trB' \<in> cont ?trB"
+ then obtain tr1' tr2' tr3' where "trB' = tr1' \<parallel> (tr2' \<parallel> tr3')"
+ and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+ and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
+ thus "\<exists> trA'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
+ apply(intro exI[of _ "(tr1' \<parallel> tr2') \<parallel> tr3'"])
+ unfolding Inr_in_cont_par by auto
+ qed
+ qed
+ }
+ thus ?thesis by blast
+qed
+
+
+
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Infinite_Derivation_Trees/Prelim.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,66 @@
+(* Title: Gram_Tree.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Preliminaries
+*)
+
+
+theory Prelim
+imports "../../Codatatype/Codatatype"
+begin
+
+declare fset_to_fset[simp]
+
+lemma fst_snd_convol_o[simp]: "<fst o s, snd o s> = s"
+apply(rule ext) by (simp add: convol_def)
+
+abbreviation sm_abbrev (infix "\<oplus>" 60)
+where "f \<oplus> g \<equiv> Sum_Type.sum_map f g"
+
+lemma sum_map_InlD: "(f \<oplus> g) z = Inl x \<Longrightarrow> \<exists>y. z = Inl y \<and> f y = x"
+by (cases z) auto
+
+lemma sum_map_InrD: "(f \<oplus> g) z = Inr x \<Longrightarrow> \<exists>y. z = Inr y \<and> g y = x"
+by (cases z) auto
+
+abbreviation sum_case_abbrev ("[[_,_]]" 800)
+where "[[f,g]] \<equiv> Sum_Type.sum_case f g"
+
+lemma inj_Inl[simp]: "inj Inl" unfolding inj_on_def by auto
+lemma inj_Inr[simp]: "inj Inr" unfolding inj_on_def by auto
+
+lemma Inl_oplus_elim:
+assumes "Inl tr \<in> (id \<oplus> f) ` tns"
+shows "Inl tr \<in> tns"
+using assms apply clarify by (case_tac x, auto)
+
+lemma Inl_oplus_iff[simp]: "Inl tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> Inl tr \<in> tns"
+using Inl_oplus_elim
+by (metis id_def image_iff sum_map.simps(1))
+
+lemma Inl_m_oplus[simp]: "Inl -` (id \<oplus> f) ` tns = Inl -` tns"
+using Inl_oplus_iff unfolding vimage_def by auto
+
+lemma Inr_oplus_elim:
+assumes "Inr tr \<in> (id \<oplus> f) ` tns"
+shows "\<exists> n. Inr n \<in> tns \<and> f n = tr"
+using assms apply clarify by (case_tac x, auto)
+
+lemma Inr_oplus_iff[simp]:
+"Inr tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> (\<exists> n. Inr n \<in> tns \<and> f n = tr)"
+apply (rule iffI)
+ apply (metis Inr_oplus_elim)
+by (metis image_iff sum_map.simps(2))
+
+lemma Inr_m_oplus[simp]: "Inr -` (id \<oplus> f) ` tns = f ` (Inr -` tns)"
+using Inr_oplus_iff unfolding vimage_def by auto
+
+lemma Inl_Inr_image_cong:
+assumes "Inl -` A = Inl -` B" and "Inr -` A = Inr -` B"
+shows "A = B"
+apply safe using assms apply(case_tac x, auto) by(case_tac x, auto)
+
+
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Infinite_Derivation_Trees/Tree.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,326 @@
+(* Title: Gram_Tree.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Trees with nonterminal internal nodes and terminal leafs.
+*)
+
+
+header {* Trees with nonterminal internal nodes and terminal leafs *}
+
+
+theory Tree
+imports Prelim
+begin
+
+typedecl N typedecl T
+
+bnf_codata Tree: 'Tree = "N \<times> (T + 'Tree) fset"
+
+
+section {* Sugar notations for Tree *}
+
+subsection{* Setup for map, set, pred *}
+
+(* These should be eventually inferred from compositionality *)
+
+lemma TreeBNF_map:
+"TreeBNF_map f (n,as) = (n, map_fset (id \<oplus> f) as)"
+unfolding TreeBNF_map_def id_apply
+sum_map_def by simp
+
+lemma TreeBNF_map':
+"TreeBNF_map f n_as = (fst n_as, map_fset (id \<oplus> f) (snd n_as))"
+using TreeBNF_map by(cases n_as, simp)
+
+
+definition
+"llift2 \<phi> as1 as2 \<longleftrightarrow>
+ (\<forall> n. Inl n \<in> fset as1 \<longleftrightarrow> Inl n \<in> fset as2) \<and>
+ (\<forall> tr1. Inr tr1 \<in> fset as1 \<longrightarrow> (\<exists> tr2. Inr tr2 \<in> fset as2 \<and> \<phi> tr1 tr2)) \<and>
+ (\<forall> tr2. Inr tr2 \<in> fset as2 \<longrightarrow> (\<exists> tr1. Inr tr1 \<in> fset as1 \<and> \<phi> tr1 tr2))"
+
+lemma TreeBNF_pred: "TreeBNF_pred \<phi> (n1,as1) (n2,as2) \<longleftrightarrow> n1 = n2 \<and> llift2 \<phi> as1 as2"
+unfolding llift2_def TreeBNF.pred_unfold
+apply (auto split: sum.splits)
+apply (metis sumE)
+apply (metis sumE)
+apply (metis sumE)
+apply (metis sumE)
+apply (metis sumE sum.simps(1,2,4))
+apply (metis sumE sum.simps(1,2,4))
+done
+
+
+subsection{* Constructors *}
+
+definition NNode :: "N \<Rightarrow> (T + Tree)fset \<Rightarrow> Tree"
+where "NNode n as \<equiv> Tree_fld (n,as)"
+
+lemmas ctor_defs = NNode_def
+
+
+subsection {* Pre-selectors *}
+
+(* These are mere auxiliaries *)
+
+definition "asNNode tr \<equiv> SOME n_as. NNode (fst n_as) (snd n_as) = tr"
+lemmas pre_sel_defs = asNNode_def
+
+
+subsection {* Selectors *}
+
+(* One for each pair (constructor, constructor argument) *)
+
+(* For NNode: *)
+definition root :: "Tree \<Rightarrow> N" where "root tr = fst (asNNode tr)"
+definition ccont :: "Tree \<Rightarrow> (T + Tree)fset" where "ccont tr = snd (asNNode tr)"
+
+lemmas sel_defs = root_def ccont_def
+
+
+subsection {* Basic properties *}
+
+(* Constructors versus selectors *)
+lemma NNode_surj: "\<exists> n as. NNode n as = tr"
+unfolding NNode_def
+by (metis Tree.fld_unf pair_collapse)
+
+lemma NNode_asNNode:
+"NNode (fst (asNNode tr)) (snd (asNNode tr)) = tr"
+proof-
+ obtain n as where "NNode n as = tr" using NNode_surj[of tr] by blast
+ hence "NNode (fst (n,as)) (snd (n,as)) = tr" by simp
+ thus ?thesis unfolding asNNode_def by(rule someI)
+qed
+
+theorem NNode_root_ccont[simp]:
+"NNode (root tr) (ccont tr) = tr"
+using NNode_asNNode unfolding root_def ccont_def .
+
+(* Constructors *)
+theorem TTree_simps[simp]:
+"NNode n as = NNode n' as' \<longleftrightarrow> n = n' \<and> as = as'"
+unfolding ctor_defs Tree.fld_inject by auto
+
+theorem TTree_cases[elim, case_names NNode Choice]:
+assumes NNode: "\<And> n as. tr = NNode n as \<Longrightarrow> phi"
+shows phi
+proof(cases rule: Tree.fld_exhaust[of tr])
+ fix x assume "tr = Tree_fld x"
+ thus ?thesis
+ apply(cases x)
+ using NNode unfolding ctor_defs apply blast
+ done
+qed
+
+(* Constructors versus selectors *)
+theorem TTree_sel_ctor[simp]:
+"root (NNode n as) = n"
+"ccont (NNode n as) = as"
+unfolding root_def ccont_def
+by (metis (no_types) NNode_asNNode TTree_simps)+
+
+
+subsection{* Coinduction *}
+
+theorem TTree_coind_Node[elim, consumes 1, case_names NNode, induct pred: "HOL.eq"]:
+assumes phi: "\<phi> tr1 tr2" and
+NNode: "\<And> n1 n2 as1 as2.
+ \<lbrakk>\<phi> (NNode n1 as1) (NNode n2 as2)\<rbrakk> \<Longrightarrow>
+ n1 = n2 \<and> llift2 \<phi> as1 as2"
+shows "tr1 = tr2"
+apply(rule mp[OF Tree.pred_coinduct[of \<phi> tr1 tr2] phi]) proof clarify
+ fix tr1 tr2 assume \<phi>: "\<phi> tr1 tr2"
+ show "TreeBNF_pred \<phi> (Tree_unf tr1) (Tree_unf tr2)"
+ apply(cases rule: Tree.fld_exhaust[of tr1], cases rule: Tree.fld_exhaust[of tr2])
+ apply (simp add: Tree.unf_fld)
+ apply(case_tac x, case_tac xa, simp)
+ unfolding TreeBNF_pred apply(rule NNode) using \<phi> unfolding NNode_def by simp
+qed
+
+theorem TTree_coind[elim, consumes 1, case_names LLift]:
+assumes phi: "\<phi> tr1 tr2" and
+LLift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow>
+ root tr1 = root tr2 \<and> llift2 \<phi> (ccont tr1) (ccont tr2)"
+shows "tr1 = tr2"
+using phi apply(induct rule: TTree_coind_Node)
+using LLift by (metis TTree_sel_ctor)
+
+
+
+subsection {* Coiteration *}
+
+(* Preliminaries: *)
+declare Tree.unf_fld[simp]
+declare Tree.fld_unf[simp]
+
+lemma Tree_unf_NNode[simp]:
+"Tree_unf (NNode n as) = (n,as)"
+unfolding NNode_def Tree.unf_fld ..
+
+lemma Tree_unf_root_ccont:
+"Tree_unf tr = (root tr, ccont tr)"
+unfolding root_def ccont_def
+by (metis (lifting) NNode_asNNode Tree_unf_NNode)
+
+(* Coiteration *)
+definition TTree_coit ::
+"('b \<Rightarrow> N) \<Rightarrow> ('b \<Rightarrow> (T + 'b) fset) \<Rightarrow> 'b \<Rightarrow> Tree"
+where "TTree_coit rt ct \<equiv> Tree_coiter <rt,ct>"
+
+lemma Tree_coit_coit:
+"Tree_coiter s = TTree_coit (fst o s) (snd o s)"
+apply(rule ext)
+unfolding TTree_coit_def by simp
+
+theorem TTree_coit:
+"root (TTree_coit rt ct b) = rt b"
+"ccont (TTree_coit rt ct b) = map_fset (id \<oplus> TTree_coit rt ct) (ct b)"
+using Tree.coiter[of "<rt,ct>" b] unfolding Tree_coit_coit fst_convol snd_convol
+unfolding TreeBNF_map' fst_convol' snd_convol'
+unfolding Tree_unf_root_ccont by simp_all
+
+(* Corecursion, stronger than coitation *)
+definition TTree_corec ::
+"('b \<Rightarrow> N) \<Rightarrow> ('b \<Rightarrow> (T + (Tree + 'b)) fset) \<Rightarrow> 'b \<Rightarrow> Tree"
+where "TTree_corec rt ct \<equiv> Tree_corec <rt,ct>"
+
+lemma Tree_corec_corec:
+"Tree_corec s = TTree_corec (fst o s) (snd o s)"
+apply(rule ext)
+unfolding TTree_corec_def by simp
+
+theorem TTree_corec:
+"root (TTree_corec rt ct b) = rt b"
+"ccont (TTree_corec rt ct b) = map_fset (id \<oplus> ([[id, TTree_corec rt ct]]) ) (ct b)"
+using Tree.corec[of "<rt,ct>" b] unfolding Tree_corec_corec fst_convol snd_convol
+unfolding TreeBNF_map' fst_convol' snd_convol'
+unfolding Tree_unf_root_ccont by simp_all
+
+
+subsection{* The characteristic theorems transported from fset to set *}
+
+definition "Node n as \<equiv> NNode n (the_inv fset as)"
+definition "cont \<equiv> fset o ccont"
+definition "coit rt ct \<equiv> TTree_coit rt (the_inv fset o ct)"
+definition "corec rt ct \<equiv> TTree_corec rt (the_inv fset o ct)"
+
+definition lift ("_ ^#" 200) where
+"lift \<phi> as \<longleftrightarrow> (\<forall> tr. Inr tr \<in> as \<longrightarrow> \<phi> tr)"
+
+definition lift2 ("_ ^#2" 200) where
+"lift2 \<phi> as1 as2 \<longleftrightarrow>
+ (\<forall> n. Inl n \<in> as1 \<longleftrightarrow> Inl n \<in> as2) \<and>
+ (\<forall> tr1. Inr tr1 \<in> as1 \<longrightarrow> (\<exists> tr2. Inr tr2 \<in> as2 \<and> \<phi> tr1 tr2)) \<and>
+ (\<forall> tr2. Inr tr2 \<in> as2 \<longrightarrow> (\<exists> tr1. Inr tr1 \<in> as1 \<and> \<phi> tr1 tr2))"
+
+definition liftS ("_ ^#s" 200) where
+"liftS trs = {as. Inr -` as \<subseteq> trs}"
+
+lemma lift2_llift2:
+"\<lbrakk>finite as1; finite as2\<rbrakk> \<Longrightarrow>
+ lift2 \<phi> as1 as2 \<longleftrightarrow> llift2 \<phi> (the_inv fset as1) (the_inv fset as2)"
+unfolding lift2_def llift2_def by auto
+
+lemma llift2_lift2:
+"llift2 \<phi> as1 as2 \<longleftrightarrow> lift2 \<phi> (fset as1) (fset as2)"
+using lift2_llift2 by (metis finite_fset fset_cong fset_to_fset)
+
+lemma mono_lift:
+assumes "(\<phi>^#) as"
+and "\<And> tr. \<phi> tr \<Longrightarrow> \<phi>' tr"
+shows "(\<phi>'^#) as"
+using assms unfolding lift_def[abs_def] by blast
+
+lemma mono_liftS:
+assumes "trs1 \<subseteq> trs2 "
+shows "(trs1 ^#s) \<subseteq> (trs2 ^#s)"
+using assms unfolding liftS_def[abs_def] by blast
+
+lemma lift_mono:
+assumes "\<phi> \<le> \<phi>'"
+shows "(\<phi>^#) \<le> (\<phi>'^#)"
+using assms unfolding lift_def[abs_def] by blast
+
+lemma mono_lift2:
+assumes "(\<phi>^#2) as1 as2"
+and "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow> \<phi>' tr1 tr2"
+shows "(\<phi>'^#2) as1 as2"
+using assms unfolding lift2_def[abs_def] by blast
+
+lemma lift2_mono:
+assumes "\<phi> \<le> \<phi>'"
+shows "(\<phi>^#2) \<le> (\<phi>'^#2)"
+using assms unfolding lift2_def[abs_def] by blast
+
+lemma finite_cont[simp]: "finite (cont tr)"
+unfolding cont_def by auto
+
+theorem Node_root_cont[simp]:
+"Node (root tr) (cont tr) = tr"
+using NNode_root_ccont unfolding Node_def cont_def
+by (metis cont_def finite_cont fset_cong fset_to_fset o_def)
+
+theorem Tree_simps[simp]:
+assumes "finite as" and "finite as'"
+shows "Node n as = Node n' as' \<longleftrightarrow> n = n' \<and> as = as'"
+using assms TTree_simps unfolding Node_def
+by (metis fset_to_fset)
+
+theorem Tree_cases[elim, case_names Node Choice]:
+assumes Node: "\<And> n as. \<lbrakk>finite as; tr = Node n as\<rbrakk> \<Longrightarrow> phi"
+shows phi
+apply(cases rule: TTree_cases[of tr])
+using Node unfolding Node_def
+by (metis Node Node_root_cont finite_cont)
+
+theorem Tree_sel_ctor[simp]:
+"root (Node n as) = n"
+"finite as \<Longrightarrow> cont (Node n as) = as"
+unfolding Node_def cont_def by auto
+
+theorems root_Node = Tree_sel_ctor(1)
+theorems cont_Node = Tree_sel_ctor(2)
+
+theorem Tree_coind_Node[elim, consumes 1, case_names Node]:
+assumes phi: "\<phi> tr1 tr2" and
+Node:
+"\<And> n1 n2 as1 as2.
+ \<lbrakk>finite as1; finite as2; \<phi> (Node n1 as1) (Node n2 as2)\<rbrakk>
+ \<Longrightarrow> n1 = n2 \<and> (\<phi>^#2) as1 as2"
+shows "tr1 = tr2"
+using phi apply(induct rule: TTree_coind_Node)
+unfolding llift2_lift2 apply(rule Node)
+unfolding Node_def
+apply (metis finite_fset)
+apply (metis finite_fset)
+by (metis finite_fset fset_cong fset_to_fset)
+
+theorem Tree_coind[elim, consumes 1, case_names Lift, induct pred: "HOL.eq"]:
+assumes phi: "\<phi> tr1 tr2" and
+Lift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow>
+ root tr1 = root tr2 \<and> (\<phi>^#2) (cont tr1) (cont tr2)"
+shows "tr1 = tr2"
+using phi apply(induct rule: TTree_coind)
+unfolding llift2_lift2 apply(rule Lift[unfolded cont_def comp_def]) .
+
+theorem coit:
+"root (coit rt ct b) = rt b"
+"finite (ct b) \<Longrightarrow> cont (coit rt ct b) = image (id \<oplus> coit rt ct) (ct b)"
+using TTree_coit[of rt "the_inv fset \<circ> ct" b] unfolding coit_def
+apply - apply metis
+unfolding cont_def comp_def
+by (metis (no_types) fset_to_fset map_fset_image)
+
+
+theorem corec:
+"root (corec rt ct b) = rt b"
+"finite (ct b) \<Longrightarrow> cont (corec rt ct b) = image (id \<oplus> ([[id, corec rt ct]])) (ct b)"
+using TTree_corec[of rt "the_inv fset \<circ> ct" b] unfolding corec_def
+apply - apply metis
+unfolding cont_def comp_def
+by (metis (no_types) fset_to_fset map_fset_image)
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Lambda_Term.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,259 @@
+(* Title: Codatatype_Examples/Lambda_Term.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Lambda-terms.
+*)
+
+header {* Lambda-Terms *}
+
+theory Lambda_Term
+imports "../Codatatype/Codatatype"
+begin
+
+
+section {* Datatype definition *}
+
+bnf_data trm: 'trm = "'a + 'trm \<times> 'trm + 'a \<times> 'trm + ('a \<times> 'trm) fset \<times> 'trm"
+
+
+section {* Customization of terms *}
+
+subsection{* Set and map *}
+
+lemma trmBNF_set2_Lt: "trmBNF_set2 (Inr (Inr (Inr (xts, t)))) = snd ` (fset xts) \<union> {t}"
+unfolding trmBNF_set2_def sum_set_defs prod_set_defs collect_def[abs_def]
+by auto
+
+lemma trmBNF_set2_Var: "\<And>x. trmBNF_set2 (Inl x) = {}"
+and trmBNF_set2_App:
+"\<And>t1 t2. trmBNF_set2 (Inr (Inl t1t2)) = {fst t1t2, snd t1t2}"
+and trmBNF_set2_Lam:
+"\<And>x t. trmBNF_set2 (Inr (Inr (Inl (x, t)))) = {t}"
+unfolding trmBNF_set2_def sum_set_defs prod_set_defs collect_def[abs_def]
+by auto
+
+lemma trmBNF_map:
+"\<And> a1. trmBNF_map f1 f2 (Inl a1) = Inl (f1 a1)"
+"\<And> a2 b2. trmBNF_map f1 f2 (Inr (Inl (a2,b2))) = Inr (Inl (f2 a2, f2 b2))"
+"\<And> a1 a2. trmBNF_map f1 f2 (Inr (Inr (Inl (a1,a2)))) = Inr (Inr (Inl (f1 a1, f2 a2)))"
+"\<And> a1a2s a2.
+ trmBNF_map f1 f2 (Inr (Inr (Inr (a1a2s, a2)))) =
+ Inr (Inr (Inr (map_fset (\<lambda> (a1', a2'). (f1 a1', f2 a2')) a1a2s, f2 a2)))"
+unfolding trmBNF_map_def collect_def[abs_def] map_pair_def by auto
+
+
+subsection{* Constructors *}
+
+definition "Var x \<equiv> trm_fld (Inl x)"
+definition "App t1 t2 \<equiv> trm_fld (Inr (Inl (t1,t2)))"
+definition "Lam x t \<equiv> trm_fld (Inr (Inr (Inl (x,t))))"
+definition "Lt xts t \<equiv> trm_fld (Inr (Inr (Inr (xts,t))))"
+
+lemmas ctor_defs = Var_def App_def Lam_def Lt_def
+
+theorem trm_simps[simp]:
+"\<And>x y. Var x = Var y \<longleftrightarrow> x = y"
+"\<And>t1 t2 t1' t2'. App t1 t2 = App t1' t2' \<longleftrightarrow> t1 = t1' \<and> t2 = t2'"
+"\<And>x x' t t'. Lam x t = Lam x' t' \<longleftrightarrow> x = x' \<and> t = t'"
+"\<And> xts xts' t t'. Lt xts t = Lt xts' t' \<longleftrightarrow> xts = xts' \<and> t = t'"
+(* *)
+"\<And> x t1 t2. Var x \<noteq> App t1 t2" "\<And>x y t. Var x \<noteq> Lam y t" "\<And> x xts t. Var x \<noteq> Lt xts t"
+"\<And> t1 t2 x t. App t1 t2 \<noteq> Lam x t" "\<And> t1 t2 xts t. App t1 t2 \<noteq> Lt xts t"
+"\<And>x t xts t1. Lam x t \<noteq> Lt xts t1"
+unfolding ctor_defs trm.fld_inject by auto
+
+theorem trm_cases[elim, case_names Var App Lam Lt]:
+assumes Var: "\<And> x. t = Var x \<Longrightarrow> phi"
+and App: "\<And> t1 t2. t = App t1 t2 \<Longrightarrow> phi"
+and Lam: "\<And> x t1. t = Lam x t1 \<Longrightarrow> phi"
+and Lt: "\<And> xts t1. t = Lt xts t1 \<Longrightarrow> phi"
+shows phi
+proof(cases rule: trm.fld_exhaust[of t])
+ fix x assume "t = trm_fld x"
+ thus ?thesis
+ apply(cases x) using Var unfolding ctor_defs apply blast
+ apply(case_tac b) using App unfolding ctor_defs apply(case_tac a, blast)
+ apply(case_tac ba) using Lam unfolding ctor_defs apply(case_tac a, blast)
+ apply(case_tac bb) using Lt unfolding ctor_defs by blast
+qed
+
+lemma trm_induct[case_names Var App Lam Lt, induct type: trm]:
+assumes Var: "\<And> (x::'a). phi (Var x)"
+and App: "\<And> t1 t2. \<lbrakk>phi t1; phi t2\<rbrakk> \<Longrightarrow> phi (App t1 t2)"
+and Lam: "\<And> x t. phi t \<Longrightarrow> phi (Lam x t)"
+and Lt: "\<And> xts t. \<lbrakk>\<And> x1 t1. (x1,t1) |\<in>| xts \<Longrightarrow> phi t1; phi t\<rbrakk> \<Longrightarrow> phi (Lt xts t)"
+shows "phi t"
+proof(induct rule: trm.fld_induct)
+ fix u :: "'a + 'a trm \<times> 'a trm + 'a \<times> 'a trm + ('a \<times> 'a trm) fset \<times> 'a trm"
+ assume IH: "\<And>t. t \<in> trmBNF_set2 u \<Longrightarrow> phi t"
+ show "phi (trm_fld u)"
+ proof(cases u)
+ case (Inl x)
+ show ?thesis using Var unfolding Var_def Inl .
+ next
+ case (Inr uu) note Inr1 = Inr
+ show ?thesis
+ proof(cases uu)
+ case (Inl t1t2)
+ obtain t1 t2 where t1t2: "t1t2 = (t1,t2)" by (cases t1t2, blast)
+ show ?thesis unfolding Inr1 Inl t1t2 App_def[symmetric] apply(rule App)
+ using IH unfolding Inr1 Inl trmBNF_set2_App t1t2 fst_conv snd_conv by blast+
+ next
+ case (Inr uuu) note Inr2 = Inr
+ show ?thesis
+ proof(cases uuu)
+ case (Inl xt)
+ obtain x t where xt: "xt = (x,t)" by (cases xt, blast)
+ show ?thesis unfolding Inr1 Inr2 Inl xt Lam_def[symmetric] apply(rule Lam)
+ using IH unfolding Inr1 Inr2 Inl trmBNF_set2_Lam xt by blast
+ next
+ case (Inr xts_t)
+ obtain xts t where xts_t: "xts_t = (xts,t)" by (cases xts_t, blast)
+ show ?thesis unfolding Inr1 Inr2 Inr xts_t Lt_def[symmetric] apply(rule Lt) using IH
+ unfolding Inr1 Inr2 Inr trmBNF_set2_Lt xts_t fset_fset_member image_def by auto
+ qed
+ qed
+ qed
+qed
+
+
+subsection{* Recursion and iteration *}
+
+definition
+"sumJoin4 f1 f2 f3 f4 \<equiv>
+\<lambda> k. (case k of
+ Inl x1 \<Rightarrow> f1 x1
+|Inr k1 \<Rightarrow> (case k1 of
+ Inl ((s2,a2),(t2,b2)) \<Rightarrow> f2 s2 a2 t2 b2
+|Inr k2 \<Rightarrow> (case k2 of Inl (x3,(t3,b3)) \<Rightarrow> f3 x3 t3 b3
+|Inr (xts,(t4,b4)) \<Rightarrow> f4 xts t4 b4)))"
+
+lemma sumJoin4_simps[simp]:
+"\<And>x. sumJoin4 var app lam lt (Inl x) = var x"
+"\<And> t1 a1 t2 a2. sumJoin4 var app lam lt (Inr (Inl ((t1,a1),(t2,a2)))) = app t1 a1 t2 a2"
+"\<And> x t a. sumJoin4 var app lam lt (Inr (Inr (Inl (x,(t,a))))) = lam x t a"
+"\<And> xtas t a. sumJoin4 var app lam lt (Inr (Inr (Inr (xtas,(t,a))))) = lt xtas t a"
+unfolding sumJoin4_def by auto
+
+definition "trmrec var app lam lt \<equiv> trm_rec (sumJoin4 var app lam lt)"
+
+lemma trmrec_Var[simp]:
+"trmrec var app lam lt (Var x) = var x"
+unfolding trmrec_def Var_def trm.rec trmBNF_map(1) by simp
+
+lemma trmrec_App[simp]:
+"trmrec var app lam lt (App t1 t2) =
+ app t1 (trmrec var app lam lt t1) t2 (trmrec var app lam lt t2)"
+unfolding trmrec_def App_def trm.rec trmBNF_map(2) convol_def by simp
+
+lemma trmrec_Lam[simp]:
+"trmrec var app lam lt (Lam x t) = lam x t (trmrec var app lam lt t)"
+unfolding trmrec_def Lam_def trm.rec trmBNF_map(3) convol_def by simp
+
+lemma trmrec_Lt[simp]:
+"trmrec var app lam lt (Lt xts t) =
+ lt (map_fset (\<lambda> (x,t). (x,t,trmrec var app lam lt t)) xts) t (trmrec var app lam lt t)"
+unfolding trmrec_def Lt_def trm.rec trmBNF_map(4) convol_def by simp
+
+definition
+"sumJoinI4 f1 f2 f3 f4 \<equiv>
+\<lambda> k. (case k of
+ Inl x1 \<Rightarrow> f1 x1
+|Inr k1 \<Rightarrow> (case k1 of
+ Inl (a2,b2) \<Rightarrow> f2 a2 b2
+|Inr k2 \<Rightarrow> (case k2 of Inl (x3,b3) \<Rightarrow> f3 x3 b3
+|Inr (xts,b4) \<Rightarrow> f4 xts b4)))"
+
+lemma sumJoinI4_simps[simp]:
+"\<And>x. sumJoinI4 var app lam lt (Inl x) = var x"
+"\<And> a1 a2. sumJoinI4 var app lam lt (Inr (Inl (a1,a2))) = app a1 a2"
+"\<And> x a. sumJoinI4 var app lam lt (Inr (Inr (Inl (x,a)))) = lam x a"
+"\<And> xtas a. sumJoinI4 var app lam lt (Inr (Inr (Inr (xtas,a)))) = lt xtas a"
+unfolding sumJoinI4_def by auto
+
+(* The iterator has a simpler, hence more manageable type. *)
+definition "trmiter var app lam lt \<equiv> trm_iter (sumJoinI4 var app lam lt)"
+
+lemma trmiter_Var[simp]:
+"trmiter var app lam lt (Var x) = var x"
+unfolding trmiter_def Var_def trm.iter trmBNF_map(1) by simp
+
+lemma trmiter_App[simp]:
+"trmiter var app lam lt (App t1 t2) =
+ app (trmiter var app lam lt t1) (trmiter var app lam lt t2)"
+unfolding trmiter_def App_def trm.iter trmBNF_map(2) by simp
+
+lemma trmiter_Lam[simp]:
+"trmiter var app lam lt (Lam x t) = lam x (trmiter var app lam lt t)"
+unfolding trmiter_def Lam_def trm.iter trmBNF_map(3) by simp
+
+lemma trmiter_Lt[simp]:
+"trmiter var app lam lt (Lt xts t) =
+ lt (map_fset (\<lambda> (x,t). (x,trmiter var app lam lt t)) xts) (trmiter var app lam lt t)"
+unfolding trmiter_def Lt_def trm.iter trmBNF_map(4) by simp
+
+
+subsection{* Example: The set of all variables varsOf and free variables fvarsOf of a term: *}
+
+definition "varsOf = trmiter
+(\<lambda> x. {x})
+(\<lambda> X1 X2. X1 \<union> X2)
+(\<lambda> x X. X \<union> {x})
+(\<lambda> xXs Y. Y \<union> (\<Union> { {x} \<union> X | x X. (x,X) |\<in>| xXs}))"
+
+lemma varsOf_simps[simp]:
+"varsOf (Var x) = {x}"
+"varsOf (App t1 t2) = varsOf t1 \<union> varsOf t2"
+"varsOf (Lam x t) = varsOf t \<union> {x}"
+"varsOf (Lt xts t) =
+ varsOf t \<union> (\<Union> { {x} \<union> X | x X. (x,X) |\<in>| map_fset (\<lambda> (x,t1). (x,varsOf t1)) xts})"
+unfolding varsOf_def by simp_all
+
+definition "fvarsOf = trmiter
+(\<lambda> x. {x})
+(\<lambda> X1 X2. X1 \<union> X2)
+(\<lambda> x X. X - {x})
+(\<lambda> xtXs Y. Y - {x | x X. (x,X) |\<in>| xtXs} \<union> (\<Union> {X | x X. (x,X) |\<in>| xtXs}))"
+
+lemma fvarsOf_simps[simp]:
+"fvarsOf (Var x) = {x}"
+"fvarsOf (App t1 t2) = fvarsOf t1 \<union> fvarsOf t2"
+"fvarsOf (Lam x t) = fvarsOf t - {x}"
+"fvarsOf (Lt xts t) =
+ fvarsOf t
+ - {x | x X. (x,X) |\<in>| map_fset (\<lambda> (x,t1). (x,fvarsOf t1)) xts}
+ \<union> (\<Union> {X | x X. (x,X) |\<in>| map_fset (\<lambda> (x,t1). (x,fvarsOf t1)) xts})"
+unfolding fvarsOf_def by simp_all
+
+lemma diff_Un_incl_triv: "\<lbrakk>A \<subseteq> D; C \<subseteq> E\<rbrakk> \<Longrightarrow> A - B \<union> C \<subseteq> D \<union> E" by blast
+
+lemma in_map_fset_iff:
+"(x, X) |\<in>| map_fset (\<lambda>(x, t1). (x, f t1)) xts \<longleftrightarrow>
+ (\<exists> t1. (x,t1) |\<in>| xts \<and> X = f t1)"
+unfolding map_fset_def2_raw in_fset fset_afset unfolding fset_def2_raw by auto
+
+lemma fvarsOf_varsOf: "fvarsOf t \<subseteq> varsOf t"
+proof induct
+ case (Lt xts t)
+ thus ?case unfolding fvarsOf_simps varsOf_simps
+ proof (elim diff_Un_incl_triv)
+ show
+ "\<Union>{X | x X. (x, X) |\<in>| map_fset (\<lambda>(x, t1). (x, fvarsOf t1)) xts}
+ \<subseteq> \<Union>{{x} \<union> X |x X. (x, X) |\<in>| map_fset (\<lambda>(x, t1). (x, varsOf t1)) xts}"
+ (is "_ \<subseteq> \<Union> ?L")
+ proof(rule Sup_mono, safe)
+ fix a x X
+ assume "(x, X) |\<in>| map_fset (\<lambda>(x, t1). (x, fvarsOf t1)) xts"
+ then obtain t1 where x_t1: "(x,t1) |\<in>| xts" and X: "X = fvarsOf t1"
+ unfolding in_map_fset_iff by auto
+ let ?Y = "varsOf t1"
+ have x_Y: "(x,?Y) |\<in>| map_fset (\<lambda>(x, t1). (x, varsOf t1)) xts"
+ using x_t1 unfolding in_map_fset_iff by auto
+ show "\<exists> Y \<in> ?L. X \<subseteq> Y" unfolding X using Lt(1) x_Y x_t1 by auto
+ qed
+ qed
+qed auto
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/ListF.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,171 @@
+(* Title: Codatatype_Examples/ListF.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Finite lists.
+*)
+
+header {* Finite Lists *}
+
+theory ListF
+imports "../Codatatype/Codatatype"
+begin
+
+bnf_data listF: 'list = "unit + 'a \<times> 'list"
+
+definition "NilF = listF_fld (Inl ())"
+definition "Conss a as \<equiv> listF_fld (Inr (a, as))"
+
+lemma listF_map_NilF[simp]: "listF_map f NilF = NilF"
+unfolding listF_map_def listFBNF_map_def NilF_def listF.iter by simp
+
+lemma listF_map_Conss[simp]:
+ "listF_map f (Conss x xs) = Conss (f x) (listF_map f xs)"
+unfolding listF_map_def listFBNF_map_def Conss_def listF.iter by simp
+
+lemma listF_set_NilF[simp]: "listF_set NilF = {}"
+unfolding listF_set_def NilF_def listF.iter listFBNF_set1_def listFBNF_set2_def
+ sum_set_defs listFBNF_map_def collect_def[abs_def] by simp
+
+lemma listF_set_Conss[simp]: "listF_set (Conss x xs) = {x} \<union> listF_set xs"
+unfolding listF_set_def Conss_def listF.iter listFBNF_set1_def listFBNF_set2_def
+ sum_set_defs prod_set_defs listFBNF_map_def collect_def[abs_def] by simp
+
+lemma iter_sum_case_NilF: "listF_iter (sum_case f g) NilF = f ()"
+unfolding NilF_def listF.iter listFBNF_map_def by simp
+
+
+lemma iter_sum_case_Conss:
+ "listF_iter (sum_case f g) (Conss y ys) = g (y, listF_iter (sum_case f g) ys)"
+unfolding Conss_def listF.iter listFBNF_map_def by simp
+
+(* familiar induction principle *)
+lemma listF_induct:
+ fixes xs :: "'a listF"
+ assumes IB: "P NilF" and IH: "\<And>x xs. P xs \<Longrightarrow> P (Conss x xs)"
+ shows "P xs"
+proof (rule listF.fld_induct)
+ fix xs :: "unit + 'a \<times> 'a listF"
+ assume raw_IH: "\<And>a. a \<in> listFBNF_set2 xs \<Longrightarrow> P a"
+ show "P (listF_fld xs)"
+ proof (cases xs)
+ case (Inl a) with IB show ?thesis unfolding NilF_def by simp
+ next
+ case (Inr b)
+ then obtain y ys where yys: "listF_fld xs = Conss y ys"
+ unfolding Conss_def listF.fld_inject by (blast intro: prod.exhaust)
+ hence "ys \<in> listFBNF_set2 xs"
+ unfolding listFBNF_set2_def Conss_def listF.fld_inject sum_set_defs prod_set_defs
+ collect_def[abs_def] by simp
+ with raw_IH have "P ys" by blast
+ with IH have "P (Conss y ys)" by blast
+ with yys show ?thesis by simp
+ qed
+qed
+
+rep_datatype NilF Conss
+by (blast intro: listF_induct) (auto simp add: NilF_def Conss_def listF.fld_inject)
+
+definition Singll ("[[_]]") where
+ [simp]: "Singll a \<equiv> Conss a NilF"
+
+definition appendd (infixr "@@" 65) where
+ "appendd \<equiv> listF_iter (sum_case (\<lambda> _. id) (\<lambda> (a,f) bs. Conss a (f bs)))"
+
+definition "lrev \<equiv> listF_iter (sum_case (\<lambda> _. NilF) (\<lambda> (b,bs). bs @@ [[b]]))"
+
+lemma lrev_NilF[simp]: "lrev NilF = NilF"
+unfolding lrev_def by (simp add: iter_sum_case_NilF)
+
+lemma lrev_Conss[simp]: "lrev (Conss y ys) = lrev ys @@ [[y]]"
+unfolding lrev_def by (simp add: iter_sum_case_Conss)
+
+lemma NilF_appendd[simp]: "NilF @@ ys = ys"
+unfolding appendd_def by (simp add: iter_sum_case_NilF)
+
+lemma Conss_append[simp]: "Conss x xs @@ ys = Conss x (xs @@ ys)"
+unfolding appendd_def by (simp add: iter_sum_case_Conss)
+
+lemma appendd_NilF[simp]: "xs @@ NilF = xs"
+by (rule listF_induct) auto
+
+lemma appendd_assoc[simp]: "(xs @@ ys) @@ zs = xs @@ ys @@ zs"
+by (rule listF_induct) auto
+
+lemma lrev_appendd[simp]: "lrev (xs @@ ys) = lrev ys @@ lrev xs"
+by (rule listF_induct[of _ xs]) auto
+
+lemma listF_map_appendd[simp]:
+ "listF_map f (xs @@ ys) = listF_map f xs @@ listF_map f ys"
+by (rule listF_induct[of _ xs]) auto
+
+lemma lrev_listF_map[simp]: "lrev (listF_map f xs) = listF_map f (lrev xs)"
+by (rule listF_induct[of _ xs]) auto
+
+lemma lrev_lrev[simp]: "lrev (lrev as) = as"
+by (rule listF_induct) auto
+
+fun lengthh where
+ "lengthh NilF = 0"
+| "lengthh (Conss x xs) = Suc (lengthh xs)"
+
+fun nthh where
+ "nthh (Conss x xs) 0 = x"
+| "nthh (Conss x xs) (Suc n) = nthh xs n"
+| "nthh xs i = undefined"
+
+lemma lengthh_listF_map[simp]: "lengthh (listF_map f xs) = lengthh xs"
+by (rule listF_induct[of _ xs]) auto
+
+lemma nthh_listF_map[simp]:
+ "i < lengthh xs \<Longrightarrow> nthh (listF_map f xs) i = f (nthh xs i)"
+by (induct rule: nthh.induct) auto
+
+lemma nthh_listF_set[simp]: "i < lengthh xs \<Longrightarrow> nthh xs i \<in> listF_set xs"
+by (induct rule: nthh.induct) auto
+
+lemma NilF_iff[iff]: "(lengthh xs = 0) = (xs = NilF)"
+by (induct xs) auto
+
+lemma Conss_iff[iff]:
+ "(lengthh xs = Suc n) = (\<exists>y ys. xs = Conss y ys \<and> lengthh ys = n)"
+by (induct xs) auto
+
+lemma Conss_iff'[iff]:
+ "(Suc n = lengthh xs) = (\<exists>y ys. xs = Conss y ys \<and> lengthh ys = n)"
+by (induct xs) (simp, simp, blast)
+
+lemma listF_induct2: "\<lbrakk>lengthh xs = lengthh ys; P NilF NilF;
+ \<And>x xs y ys. P xs ys \<Longrightarrow> P (Conss x xs) (Conss y ys)\<rbrakk> \<Longrightarrow> P xs ys"
+by (induct xs arbitrary: ys rule: listF_induct) auto
+
+fun zipp where
+ "zipp NilF NilF = NilF"
+| "zipp (Conss x xs) (Conss y ys) = Conss (x, y) (zipp xs ys)"
+| "zipp xs ys = undefined"
+
+lemma listF_map_fst_zip[simp]:
+ "lengthh xs = lengthh ys \<Longrightarrow> listF_map fst (zipp xs ys) = xs"
+by (erule listF_induct2) auto
+
+lemma listF_map_snd_zip[simp]:
+ "lengthh xs = lengthh ys \<Longrightarrow> listF_map snd (zipp xs ys) = ys"
+by (erule listF_induct2) auto
+
+lemma lengthh_zip[simp]:
+ "lengthh xs = lengthh ys \<Longrightarrow> lengthh (zipp xs ys) = lengthh xs"
+by (erule listF_induct2) auto
+
+lemma nthh_zip[simp]:
+ assumes *: "lengthh xs = lengthh ys"
+ shows "i < lengthh xs \<Longrightarrow> nthh (zipp xs ys) i = (nthh xs i, nthh ys i)"
+proof (induct arbitrary: i rule: listF_induct2[OF *])
+ case (2 x xs y ys) thus ?case by (induct i) auto
+qed simp
+
+lemma list_set_nthh[simp]:
+ "(x \<in> listF_set xs) \<Longrightarrow> (\<exists>i < lengthh xs. nthh xs i = x)"
+by (induct xs) (auto, induct rule: nthh.induct, auto)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Misc_Codata.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,88 @@
+(* Title: Codatatype_Examples/Misc_Data.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Miscellaneous codatatype declarations.
+*)
+
+header {* Miscellaneous Codatatype Declarations *}
+
+theory Misc_Codata
+imports "../Codatatype/Codatatype"
+begin
+
+ML {* quick_and_dirty := false *}
+
+ML {* PolyML.fullGC (); *}
+
+bnf_codata simple: 'a = "unit + unit + unit + unit"
+
+bnf_codata stream: 's = "'a \<times> 's"
+
+bnf_codata llist: 'llist = "unit + 'a \<times> 'llist"
+
+bnf_codata some_passive: 'a = "'a + 'b + 'c + 'd + 'e"
+
+(*
+ ('a, 'b1, 'b2) F1 = 'a * 'b1 + 'a * 'b2
+ ('a, 'b1, 'b2) F2 = unit + 'b1 * 'b2
+*)
+
+bnf_codata F1: 'b1 = "'a \<times> 'b1 + 'a \<times> 'b2"
+and F2: 'b2 = "unit + 'b1 * 'b2"
+
+bnf_codata EXPR: 'E = "'T + 'T \<times> 'E"
+and TERM: 'T = "'F + 'F \<times> 'T"
+and FACTOR: 'F = "'a + 'b + 'E"
+
+bnf_codata llambda:
+ 'trm = "string +
+ 'trm \<times> 'trm +
+ string \<times> 'trm +
+ (string \<times> 'trm) fset \<times> 'trm"
+
+bnf_codata par_llambda:
+ 'trm = "'a +
+ 'trm \<times> 'trm +
+ 'a \<times> 'trm +
+ ('a \<times> 'trm) fset \<times> 'trm"
+
+(*
+ 'a tree = Empty | Node of 'a * 'a forest ('b = unit + 'a * 'c)
+ 'a forest = Nil | Cons of 'a tree * 'a forest ('c = unit + 'b * 'c)
+*)
+
+bnf_codata tree: 'tree = "unit + 'a \<times> 'forest"
+and forest: 'forest = "unit + 'tree \<times> 'forest"
+
+bnf_codata CPS: 'a = "'b + 'b \<Rightarrow> 'a"
+
+bnf_codata fun_rhs: 'a = "'b1 \<Rightarrow> 'b2 \<Rightarrow> 'b3 \<Rightarrow> 'b4 \<Rightarrow> 'b5 \<Rightarrow> 'b6 \<Rightarrow> 'b7 \<Rightarrow> 'b8 \<Rightarrow> 'b9 \<Rightarrow> 'a"
+
+bnf_codata fun_rhs': 'a = "'b1 \<Rightarrow> 'b2 \<Rightarrow> 'b3 \<Rightarrow> 'b4 \<Rightarrow> 'b5 \<Rightarrow> 'b6 \<Rightarrow> 'b7 \<Rightarrow> 'b8 \<Rightarrow> 'b9 \<Rightarrow> 'b10 \<Rightarrow>
+ 'b11 \<Rightarrow> 'b12 \<Rightarrow> 'b13 \<Rightarrow> 'b14 \<Rightarrow> 'b15 \<Rightarrow> 'b16 \<Rightarrow> 'b17 \<Rightarrow> 'b18 \<Rightarrow> 'b19 \<Rightarrow> 'b20 \<Rightarrow> 'a"
+
+bnf_codata some_killing: 'a = "'b \<Rightarrow> 'd \<Rightarrow> ('a + 'c)"
+and in_here: 'c = "'d \<times> 'b + 'e"
+
+bnf_codata some_killing': 'a = "'b \<Rightarrow> 'd \<Rightarrow> ('a + 'c)"
+and in_here': 'c = "'d + 'e"
+
+bnf_codata some_killing'': 'a = "'b \<Rightarrow> 'c"
+and in_here'': 'c = "'d \<times> 'b + 'e"
+
+bnf_codata less_killing: 'a = "'b \<Rightarrow> 'c"
+
+(* SLOW, MEMORY-HUNGRY
+bnf_codata K1': 'K1 = "'K2 + 'a list"
+and K2': 'K2 = "'K3 + 'c fset"
+and K3': 'K3 = "'K3 + 'K4 + 'K4 \<times> 'K5"
+and K4': 'K4 = "'K5 + 'a list list list"
+and K5': 'K5 = "'K6"
+and K6': 'K6 = "'K7"
+and K7': 'K7 = "'K8"
+and K8': 'K8 = "'K1 list"
+*)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Misc_Data.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,161 @@
+(* Title: Codatatype_Examples/Misc_Data.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Miscellaneous datatype declarations.
+*)
+
+header {* Miscellaneous Datatype Declarations *}
+
+theory Misc_Data
+imports "../Codatatype/Codatatype"
+begin
+
+ML {* quick_and_dirty := false *}
+
+ML {* PolyML.fullGC (); *}
+
+bnf_data simple: 'a = "unit + unit + unit + unit"
+
+bnf_data mylist: 'list = "unit + 'a \<times> 'list"
+
+bnf_data some_passive: 'a = "'a + 'b + 'c + 'd + 'e"
+
+bnf_data lambda:
+ 'trm = "string +
+ 'trm \<times> 'trm +
+ string \<times> 'trm +
+ (string \<times> 'trm) fset \<times> 'trm"
+
+bnf_data par_lambda:
+ 'trm = "'a +
+ 'trm \<times> 'trm +
+ 'a \<times> 'trm +
+ ('a \<times> 'trm) fset \<times> 'trm"
+
+(*
+ ('a, 'b1, 'b2) F1 = 'a * 'b1 + 'a * 'b2
+ ('a, 'b1, 'b2) F2 = unit + 'b1 * 'b2
+*)
+
+bnf_data F1: 'b1 = "'a \<times> 'b1 + 'a \<times> 'b2"
+and F2: 'b2 = "unit + 'b1 * 'b2"
+
+(*
+ 'a tree = Empty | Node of 'a * 'a forest ('b = unit + 'a * 'c)
+ 'a forest = Nil | Cons of 'a tree * 'a forest ('c = unit + 'b * 'c)
+*)
+
+bnf_data tree: 'tree = "unit + 'a \<times> 'forest"
+and forest: 'forest = "unit + 'tree \<times> 'forest"
+
+(*
+ 'a tree = Empty | Node of 'a branch * 'a branch ('b = unit + 'c * 'c)
+' a branch = Branch of 'a * 'a tree ('c = 'a * 'b)
+*)
+
+bnf_data tree': 'tree = "unit + 'branch \<times> 'branch"
+and branch: 'branch = "'a \<times> 'tree"
+
+(*
+ exp = Sum of term * exp ('c = 'd + 'd * 'c)
+ term = Prod of factor * term ('d = 'e + 'e * 'd)
+ factor = C 'a | V 'b | Paren exp ('e = 'a + 'b + 'c)
+*)
+
+bnf_data EXPR: 'E = "'T + 'T \<times> 'E"
+and TERM: 'T = "'F + 'F \<times> 'T"
+and FACTOR: 'F = "'a + 'b + 'E"
+
+bnf_data some_killing: 'a = "'b \<Rightarrow> 'd \<Rightarrow> ('a + 'c)"
+and in_here: 'c = "'d \<times> 'b + 'e"
+
+bnf_data nofail1: 'a = "'a \<times> 'b + 'b"
+bnf_data nofail2: 'a = "('a \<times> 'b \<times> 'a \<times> 'b) list"
+bnf_data nofail3: 'a = "'b \<times> ('a \<times> 'b \<times> 'a \<times> 'b) fset"
+bnf_data nofail4: 'a = "('a \<times> ('a \<times> 'b \<times> 'a \<times> 'b) fset) list"
+
+(*
+bnf_data fail: 'a = "'a \<times> 'b \<times> 'a \<times> 'b list"
+bnf_data fail: 'a = "'a \<times> 'b \<times> 'a \<times> 'b"
+bnf_data fail: 'a = "'a \<times> 'b + 'a"
+bnf_data fail: 'a = "'a \<times> 'b"
+*)
+
+bnf_data L1: 'L1 = "'L2 list"
+and L2: 'L2 = "'L1 fset + 'L2"
+
+bnf_data K1: 'K1 = "'K2"
+and K2: 'K2 = "'K3"
+and K3: 'K3 = "'K1 list"
+
+bnf_data t1: 't1 = "'t3 + 't2"
+and t2: 't2 = "'t1"
+and t3: 't3 = "unit"
+
+bnf_data t1': 't1 = "'t2 + 't3"
+and t2': 't2 = "'t1"
+and t3': 't3 = "unit"
+
+(*
+bnf_data fail1: 'L1 = "'L2"
+and fail2: 'L2 = "'L3"
+and fail2: 'L3 = "'L1"
+
+bnf_data fail1: 'L1 = "'L2 list \<times> 'L2"
+and fail2: 'L2 = "'L2 fset \<times> 'L3"
+and fail2: 'L3 = "'L1"
+
+bnf_data fail1: 'L1 = "'L2 list \<times> 'L2"
+and fail2: 'L2 = "'L1 fset \<times> 'L1"
+*)
+(* SLOW
+bnf_data K1': 'K1 = "'K2 + 'a list"
+and K2': 'K2 = "'K3 + 'c fset"
+and K3': 'K3 = "'K3 + 'K4 + 'K4 \<times> 'K5"
+and K4': 'K4 = "'K5 + 'a list list list"
+and K5': 'K5 = "'K6"
+and K6': 'K6 = "'K7"
+and K7': 'K7 = "'K8"
+and K8': 'K8 = "'K1 list"
+
+(*time comparison*)
+datatype ('a, 'c) D1 = A1 "('a, 'c) D2" | B1 "'a list"
+ and ('a, 'c) D2 = A2 "('a, 'c) D3" | B2 "'c list"
+ and ('a, 'c) D3 = A3 "('a, 'c) D3" | B3 "('a, 'c) D4" | C3 "('a, 'c) D4" "('a, 'c) D5"
+ and ('a, 'c) D4 = A4 "('a, 'c) D5" | B4 "'a list list list"
+ and ('a, 'c) D5 = A5 "('a, 'c) D6"
+ and ('a, 'c) D6 = A6 "('a, 'c) D7"
+ and ('a, 'c) D7 = A7 "('a, 'c) D8"
+ and ('a, 'c) D8 = A8 "('a, 'c) D1 list"
+*)
+
+(* fail:
+bnf_data t1: 't1 = "'t2 * 't3 + 't2 * 't4"
+and t2: 't2 = "unit"
+and t3: 't3 = 't4
+and t4: 't4 = 't1
+*)
+
+bnf_data k1: 'k1 = "'k2 * 'k3 + 'k2 * 'k4"
+and k2: 'k2 = unit
+and k3: 'k3 = 'k4
+and k4: 'k4 = unit
+
+bnf_data tt1: 'tt1 = "'tt3 * 'tt2 + 'tt2 * 'tt4"
+and tt2: 'tt2 = unit
+and tt3: 'tt3 = 'tt1
+and tt4: 'tt4 = unit
+(* SLOW
+bnf_data s1: 's1 = "'s2 * 's3 * 's4 + 's3 + 's2 * 's6 + 's4 * 's2 + 's2 * 's2"
+and s2: 's2 = "'s7 * 's5 + 's5 * 's4 * 's6"
+and s3: 's3 = "'s1 * 's7 * 's2 + 's3 * 's3 + 's4 * 's5"
+and s4: 's4 = 's5
+and s5: 's5 = unit
+and s6: 's6 = "'s6 + 's1 * 's2 + 's6"
+and s7: 's7 = "'s8 + 's5"
+and s8: 's8 = nat
+*)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Process.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,742 @@
+(* Title: Codatatype_Examples/Process.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Processes.
+*)
+
+header {* Processes *}
+
+theory Process
+imports "../Codatatype/Codatatype"
+begin
+
+bnf_codata process: 'p = "'a * 'p + 'p * 'p"
+(* codatatype
+ 'a process = Action (prefOf :: 'a) (contOf :: 'a process) |
+ Choice (ch1Of :: 'a process) (ch2Of :: 'a process)
+*)
+
+(* Read: prefix of, continuation of, choice 1 of, choice 2 of *)
+
+section {* Customization *}
+
+subsection{* Setup for map, set, pred *}
+
+(* These should be eventually inferred from compositionality *)
+
+lemma processBNF_map[simp]:
+"processBNF_map fa fp (Inl (a ,p)) = Inl (fa a, fp p)"
+"processBNF_map fa fp (Inr (p1, p2)) = Inr (fp p1, fp p2)"
+unfolding processBNF_map_def by auto
+
+lemma processBNF_pred[simp]:
+"processBNF_pred (op =) \<phi> (Inl (a,p)) (Inl (a',p')) \<longleftrightarrow> a = a' \<and> \<phi> p p'"
+"processBNF_pred (op =) \<phi> (Inr (p,q)) (Inr (p',q')) \<longleftrightarrow> \<phi> p p' \<and> \<phi> q q'"
+"\<not> processBNF_pred (op =) \<phi> (Inl ap) (Inr q1q2)"
+"\<not> processBNF_pred (op =) \<phi> (Inr q1q2) (Inl ap)"
+by (auto simp: diag_def processBNF.pred_unfold)
+
+
+subsection{* Constructors *}
+
+definition Action :: "'a \<Rightarrow> 'a process \<Rightarrow> 'a process"
+where "Action a p \<equiv> process_fld (Inl (a,p))"
+
+definition Choice :: "'a process \<Rightarrow> 'a process \<Rightarrow> 'a process"
+where "Choice p1 p2 \<equiv> process_fld (Inr (p1,p2))"
+
+lemmas ctor_defs = Action_def Choice_def
+
+
+subsection {* Discriminators *}
+
+(* One discriminator for each constructor. By the constructor exhaustiveness,
+one of them is of course redundant, so for n constructors we only need n-1
+discriminators. However, keeping n discriminators seems more uniform. *)
+
+definition isAction :: "'a process \<Rightarrow> bool"
+where "isAction q \<equiv> \<exists> a p. q = Action a p"
+
+definition isChoice :: "'a process \<Rightarrow> bool"
+where "isChoice q \<equiv> \<exists> p1 p2. q = Choice p1 p2"
+
+lemmas discr_defs = isAction_def isChoice_def
+
+
+subsection {* Pre-selectors *}
+
+(* These are mere auxiliaries *)
+
+definition "asAction q \<equiv> SOME ap. q = Action (fst ap) (snd ap)"
+definition "asChoice q \<equiv> SOME p1p2. q = Choice (fst p1p2) (snd p1p2)"
+lemmas pre_sel_defs = asAction_def asChoice_def
+
+
+subsection {* Selectors *}
+
+(* One for each pair (constructor, constructor argument) *)
+
+(* For Action: *)
+definition prefOf :: "'a process \<Rightarrow> 'a" where "prefOf q = fst (asAction q)"
+definition contOf :: "'a process \<Rightarrow> 'a process" where "contOf q = snd (asAction q)"
+
+(* For Choice: *)
+definition ch1Of :: "'a process \<Rightarrow> 'a process" where "ch1Of q = fst (asChoice q)"
+definition ch2Of :: "'a process \<Rightarrow> 'a process" where "ch2Of q = snd (asChoice q)"
+
+lemmas sel_defs = prefOf_def contOf_def ch1Of_def ch2Of_def
+
+
+subsection {* Basic properties *}
+
+(* Selectors versus discriminators *)
+lemma isAction_asAction:
+"isAction q \<longleftrightarrow> q = Action (fst (asAction q)) (snd (asAction q))"
+(is "?L \<longleftrightarrow> ?R")
+proof
+ assume ?L
+ then obtain a p where q: "q = Action a p" unfolding isAction_def by auto
+ show ?R unfolding asAction_def q by (rule someI[of _ "(a,p)"]) simp
+qed(unfold isAction_def, auto)
+
+theorem isAction_prefOf_contOf:
+"isAction q \<longleftrightarrow> q = Action (prefOf q) (contOf q)"
+using isAction_asAction unfolding prefOf_def contOf_def .
+
+lemma isChoice_asChoice:
+"isChoice q \<longleftrightarrow> q = Choice (fst (asChoice q)) (snd (asChoice q))"
+(is "?L \<longleftrightarrow> ?R")
+proof
+ assume ?L
+ then obtain p1 p2 where q: "q = Choice p1 p2" unfolding isChoice_def by auto
+ show ?R unfolding asChoice_def q by (rule someI[of _ "(p1,p2)"]) simp
+qed(unfold isChoice_def, auto)
+
+theorem isChoice_ch1Of_ch2Of:
+"isChoice q \<longleftrightarrow> q = Choice (ch1Of q) (ch2Of q)"
+using isChoice_asChoice unfolding ch1Of_def ch2Of_def .
+
+(* Constructors *)
+theorem process_simps[simp]:
+"Action a p = Action a' p' \<longleftrightarrow> a = a' \<and> p = p'"
+"Choice p1 p2 = Choice p1' p2' \<longleftrightarrow> p1 = p1' \<and> p2 = p2'"
+(* *)
+"Action a p \<noteq> Choice p1 p2"
+"Choice p1 p2 \<noteq> Action a p"
+unfolding ctor_defs process.fld_inject by auto
+
+theorem process_cases[elim, case_names Action Choice]:
+assumes Action: "\<And> a p. q = Action a p \<Longrightarrow> phi"
+and Choice: "\<And> p1 p2. q = Choice p1 p2 \<Longrightarrow> phi"
+shows phi
+proof(cases rule: process.fld_exhaust[of q])
+ fix x assume "q = process_fld x"
+ thus ?thesis
+ apply(cases x)
+ apply(case_tac a) using Action unfolding ctor_defs apply blast
+ apply(case_tac b) using Choice unfolding ctor_defs apply blast
+ done
+qed
+
+(* Constructors versus discriminators *)
+theorem isAction_isChoice:
+"isAction p \<or> isChoice p"
+unfolding isAction_def isChoice_def by (cases rule: process_cases) auto
+
+theorem isAction_Action[simp]: "isAction (Action a p)"
+unfolding isAction_def by auto
+
+theorem isAction_Choice[simp]: "\<not> isAction (Choice p1 p2)"
+unfolding isAction_def by auto
+
+theorem isChoice_Choice[simp]: "isChoice (Choice p1 p2)"
+unfolding isChoice_def by auto
+
+theorem isChoice_Action[simp]: "\<not> isChoice (Action a p)"
+unfolding isChoice_def by auto
+
+theorem not_isAction_isChoice: "\<not> (isAction p \<and> isChoice p)"
+by (cases rule: process_cases[of p]) auto
+
+(* Constructors versus selectors *)
+theorem dest_ctor[simp]:
+"prefOf (Action a p) = a"
+"contOf (Action a p) = p"
+"ch1Of (Choice p1 p2) = p1"
+"ch2Of (Choice p1 p2) = p2"
+using isAction_Action[of a p]
+ isChoice_Choice[of p1 p2]
+unfolding isAction_prefOf_contOf
+ isChoice_ch1Of_ch2Of by auto
+
+theorem ctor_dtor[simp]:
+"\<And> p. isAction p \<Longrightarrow> Action (prefOf p) (contOf p) = p"
+"\<And> p. isChoice p \<Longrightarrow> Choice (ch1Of p) (ch2Of p) = p"
+unfolding isAction_def isChoice_def by auto
+
+
+subsection{* Coinduction *}
+
+theorem process_coind[elim, consumes 1, case_names iss Action Choice, induct pred: "HOL.eq"]:
+assumes phi: "\<phi> p p'" and
+iss: "\<And>p p'. \<phi> p p' \<Longrightarrow> (isAction p \<longleftrightarrow> isAction p') \<and> (isChoice p \<longleftrightarrow> isChoice p')" and
+Act: "\<And> a a' p p'. \<phi> (Action a p) (Action a' p') \<Longrightarrow> a = a' \<and> \<phi> p p'" and
+Ch: "\<And> p q p' q'. \<phi> (Choice p q) (Choice p' q') \<Longrightarrow> \<phi> p p' \<and> \<phi> q q'"
+shows "p = p'"
+proof(intro mp[OF process.pred_coinduct, of \<phi>, OF _ phi], clarify)
+ fix p p' assume \<phi>: "\<phi> p p'"
+ show "processBNF_pred (op =) \<phi> (process_unf p) (process_unf p')"
+ proof(cases rule: process_cases[of p])
+ case (Action a q) note p = Action
+ hence "isAction p'" using iss[OF \<phi>] by (cases rule: process_cases[of p'], auto)
+ then obtain a' q' where p': "p' = Action a' q'" by (cases rule: process_cases[of p'], auto)
+ have 0: "a = a' \<and> \<phi> q q'" using Act[OF \<phi>[unfolded p p']] .
+ have unf: "process_unf p = Inl (a,q)" "process_unf p' = Inl (a',q')"
+ unfolding p p' Action_def process.unf_fld by simp_all
+ show ?thesis using 0 unfolding unf by simp
+ next
+ case (Choice p1 p2) note p = Choice
+ hence "isChoice p'" using iss[OF \<phi>] by (cases rule: process_cases[of p'], auto)
+ then obtain p1' p2' where p': "p' = Choice p1' p2'"
+ by (cases rule: process_cases[of p'], auto)
+ have 0: "\<phi> p1 p1' \<and> \<phi> p2 p2'" using Ch[OF \<phi>[unfolded p p']] .
+ have unf: "process_unf p = Inr (p1,p2)" "process_unf p' = Inr (p1',p2')"
+ unfolding p p' Choice_def process.unf_fld by simp_all
+ show ?thesis using 0 unfolding unf by simp
+ qed
+qed
+
+(* Stronger coinduction, up to equality: *)
+theorem process_coind_upto[elim, consumes 1, case_names iss Action Choice]:
+assumes phi: "\<phi> p p'" and
+iss: "\<And>p p'. \<phi> p p' \<Longrightarrow> (isAction p \<longleftrightarrow> isAction p') \<and> (isChoice p \<longleftrightarrow> isChoice p')" and
+Act: "\<And> a a' p p'. \<phi> (Action a p) (Action a' p') \<Longrightarrow> a = a' \<and> (\<phi> p p' \<or> p = p')" and
+Ch: "\<And> p q p' q'. \<phi> (Choice p q) (Choice p' q') \<Longrightarrow> (\<phi> p p' \<or> p = p') \<and> (\<phi> q q' \<or> q = q')"
+shows "p = p'"
+proof(intro mp[OF process.pred_coinduct_upto, of \<phi>, OF _ phi], clarify)
+ fix p p' assume \<phi>: "\<phi> p p'"
+ show "processBNF_pred (op =) (\<lambda>a b. \<phi> a b \<or> a = b) (process_unf p) (process_unf p')"
+ proof(cases rule: process_cases[of p])
+ case (Action a q) note p = Action
+ hence "isAction p'" using iss[OF \<phi>] by (cases rule: process_cases[of p'], auto)
+ then obtain a' q' where p': "p' = Action a' q'" by (cases rule: process_cases[of p'], auto)
+ have 0: "a = a' \<and> (\<phi> q q' \<or> q = q')" using Act[OF \<phi>[unfolded p p']] .
+ have unf: "process_unf p = Inl (a,q)" "process_unf p' = Inl (a',q')"
+ unfolding p p' Action_def process.unf_fld by simp_all
+ show ?thesis using 0 unfolding unf by simp
+ next
+ case (Choice p1 p2) note p = Choice
+ hence "isChoice p'" using iss[OF \<phi>] by (cases rule: process_cases[of p'], auto)
+ then obtain p1' p2' where p': "p' = Choice p1' p2'"
+ by (cases rule: process_cases[of p'], auto)
+ have 0: "(\<phi> p1 p1' \<or> p1 = p1') \<and> (\<phi> p2 p2' \<or> p2 = p2')" using Ch[OF \<phi>[unfolded p p']] .
+ have unf: "process_unf p = Inr (p1,p2)" "process_unf p' = Inr (p1',p2')"
+ unfolding p p' Choice_def process.unf_fld by simp_all
+ show ?thesis using 0 unfolding unf by simp
+ qed
+qed
+
+
+subsection {* Coiteration and corecursion *}
+
+(* Preliminaries: *)
+definition
+"join22 isA pr co isC c1 c2 \<equiv>
+ \<lambda> P. if isA P then Inl (pr P, co P)
+ else if isC P then Inr (c1 P, c2 P)
+ else undefined"
+
+declare process.unf_fld[simp]
+declare process.fld_unf[simp]
+
+lemma unf_Action[simp]:
+"process_unf (Action a p) = Inl (a,p)"
+unfolding Action_def process.unf_fld ..
+
+lemma unf_Choice[simp]:
+"process_unf (Choice p1 p2) = Inr (p1,p2)"
+unfolding Choice_def process.unf_fld ..
+
+lemma isAction_unf:
+assumes "isAction p"
+shows "process_unf p = Inl (prefOf p, contOf p)"
+using assms unfolding isAction_def by auto
+
+lemma isChoice_unf:
+assumes "isChoice p"
+shows "process_unf p = Inr (ch1Of p, ch2Of p)"
+using assms unfolding isChoice_def by auto
+
+lemma unf_join22:
+"process_unf p = join22 isAction prefOf contOf isChoice ch1Of ch2Of p"
+unfolding join22_def
+using isAction_unf isChoice_unf not_isAction_isChoice isAction_isChoice by auto
+
+lemma isA_join22:
+assumes "isA P"
+shows "join22 isA pr co isC c1 c2 P = Inl (pr P, co P)"
+using assms unfolding join22_def by auto
+
+lemma isC_join22:
+assumes "\<not> isA P" and "isC P"
+shows "join22 isA pr co isC c1 c2 P = Inr (c1 P, c2 P)"
+using assms unfolding join22_def by auto
+
+(* Coiteration *)
+definition pcoiter ::
+"('b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'b)
+ \<Rightarrow>
+ ('b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b)
+ \<Rightarrow>
+ 'b \<Rightarrow> 'a process"
+where "pcoiter isA pr co isC c1 c2 \<equiv> process_coiter (join22 isA pr co isC c1 c2)"
+
+lemma unf_prefOf:
+assumes "process_unf q = Inl (a,p)"
+shows "prefOf q = a"
+using assms by (cases rule: process_cases[of q]) auto
+
+lemma unf_contOf:
+assumes "process_unf q = Inl (a,p)"
+shows "contOf q = p"
+using assms by (cases rule: process_cases[of q]) auto
+
+lemma unf_ch1Of:
+assumes "process_unf q = Inr (p1,p2)"
+shows "ch1Of q = p1"
+using assms by (cases rule: process_cases[of q]) auto
+
+lemma unf_ch2Of:
+assumes "process_unf q = Inr (p1,p2)"
+shows "ch2Of q = p2"
+using assms by (cases rule: process_cases[of q]) auto
+
+theorem pcoiter:
+"\<And>P. isA P \<Longrightarrow>
+ pcoiter isA pr co isC c1 c2 P =
+ Action (pr P)
+ (pcoiter isA pr co isC c1 c2 (co P))"
+"\<And>P. \<lbrakk>\<not> isA P; isC P\<rbrakk> \<Longrightarrow>
+ pcoiter isA pr co isC c1 c2 P =
+ Choice (pcoiter isA pr co isC c1 c2 (c1 P))
+ (pcoiter isA pr co isC c1 c2 (c2 P))"
+proof-
+ fix P
+ let ?f = "pcoiter isA pr co isC c1 c2" let ?s = "join22 isA pr co isC c1 c2"
+ assume isA: "isA P"
+ have unf: "process_unf (process_coiter ?s P) = Inl (pr P, ?f (co P))"
+ using process.coiter[of ?s P]
+ unfolding isA_join22[of isA P "pr" co isC c1 c2, OF isA]
+ processBNF_map id_apply pcoiter_def .
+ thus "?f P = Action (pr P) (?f (co P))"
+ unfolding pcoiter_def Action_def using process.fld_unf by metis
+next
+ fix P
+ let ?f = "pcoiter isA pr co isC c1 c2" let ?s = "join22 isA pr co isC c1 c2"
+ assume isA: "\<not> isA P" and isC: "isC P"
+ have unf: "process_unf (process_coiter ?s P) = Inr (?f (c1 P), ?f (c2 P))"
+ using process.coiter[of ?s P]
+ unfolding isC_join22[of isA P isC "pr" co c1 c2, OF isA isC]
+ processBNF_map id_apply pcoiter_def .
+ thus "?f P = Choice (?f (c1 P)) (?f (c2 P))"
+ unfolding pcoiter_def Choice_def using process.fld_unf by metis
+qed
+
+(* Corecursion, more general than coiteration (often unnecessarily more general): *)
+definition pcorec ::
+"('b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a process + 'b)
+ \<Rightarrow>
+ ('b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a process + 'b) \<Rightarrow> ('b \<Rightarrow> 'a process + 'b)
+ \<Rightarrow>
+ 'b \<Rightarrow> 'a process"
+where
+"pcorec isA pr co isC c1 c2 \<equiv> process_corec (join22 isA pr co isC c1 c2)"
+
+theorem pcorec_Action:
+assumes isA: "isA P"
+shows
+"case co P of
+ Inl p \<Rightarrow> pcorec isA pr co isC c1 c2 P = Action (pr P) p
+ |Inr Q \<Rightarrow> pcorec isA pr co isC c1 c2 P =
+ Action (pr P)
+ (pcorec isA pr co isC c1 c2 Q)"
+proof-
+ let ?f = "pcorec isA pr co isC c1 c2" let ?s = "join22 isA pr co isC c1 c2"
+ show ?thesis
+ proof(cases "co P")
+ case (Inl p)
+ have "process_unf (process_corec ?s P) = Inl (pr P, p)"
+ using process.corec[of ?s P]
+ unfolding isA_join22[of isA P "pr" co isC c1 c2, OF isA]
+ processBNF_map id_apply pcorec_def Inl by simp
+ thus ?thesis unfolding Inl pcorec_def Action_def using process.fld_unf by (simp, metis)
+ next
+ case (Inr Q)
+ have "process_unf (process_corec ?s P) = Inl (pr P, ?f Q)"
+ using process.corec[of ?s P]
+ unfolding isA_join22[of isA P "pr" co isC c1 c2, OF isA]
+ processBNF_map id_apply pcorec_def Inr by simp
+ thus ?thesis unfolding Inr pcorec_def Action_def using process.fld_unf by (simp, metis)
+ qed
+qed
+
+theorem pcorec_Choice:
+assumes isA: "\<not> isA P" and isC: "isC P"
+shows
+"case (c1 P,c2 P) of
+ (Inl p1, Inl p2) \<Rightarrow> pcorec isA pr co isC c1 c2 P =
+ Choice p1 p2
+ |(Inl p1, Inr Q2) \<Rightarrow> pcorec isA pr co isC c1 c2 P =
+ Choice p1
+ (pcorec isA pr co isC c1 c2 Q2)
+ |(Inr Q1, Inl p2) \<Rightarrow> pcorec isA pr co isC c1 c2 P =
+ Choice (pcorec isA pr co isC c1 c2 Q1)
+ p2
+ |(Inr Q1, Inr Q2) \<Rightarrow> pcorec isA pr co isC c1 c2 P =
+ Choice (pcorec isA pr co isC c1 c2 Q1)
+ (pcorec isA pr co isC c1 c2 Q2)"
+proof-
+ let ?f = "pcorec isA pr co isC c1 c2" let ?s = "join22 isA pr co isC c1 c2"
+ show ?thesis
+ proof(cases "c1 P")
+ case (Inl p1) note c1 = Inl
+ show ?thesis
+ proof(cases "c2 P")
+ case (Inl p2) note c2 = Inl
+ have "process_unf (process_corec ?s P) = Inr (p1, p2)"
+ using process.corec[of ?s P]
+ unfolding isC_join22[of isA P isC "pr" co c1 c2, OF isA isC]
+ processBNF_map id_apply pcorec_def c1 c2 by simp
+ thus ?thesis unfolding c1 c2 pcorec_def Choice_def using process.fld_unf by (simp, metis)
+ next
+ case (Inr Q2) note c2 = Inr
+ have "process_unf (process_corec ?s P) = Inr (p1, ?f Q2)"
+ using process.corec[of ?s P]
+ unfolding isC_join22[of isA P isC "pr" co c1 c2, OF isA isC]
+ processBNF_map id_apply pcorec_def c1 c2 by simp
+ thus ?thesis unfolding c1 c2 pcorec_def Choice_def using process.fld_unf by (simp, metis)
+ qed
+ next
+ case (Inr Q1) note c1 = Inr
+ show ?thesis
+ proof(cases "c2 P")
+ case (Inl p2) note c2 = Inl
+ have "process_unf (process_corec ?s P) = Inr (?f Q1, p2)"
+ using process.corec[of ?s P]
+ unfolding isC_join22[of isA P isC "pr" co c1 c2, OF isA isC]
+ processBNF_map id_apply pcorec_def c1 c2 by simp
+ thus ?thesis unfolding c1 c2 pcorec_def Choice_def using process.fld_unf by (simp, metis)
+ next
+ case (Inr Q2) note c2 = Inr
+ have "process_unf (process_corec ?s P) = Inr (?f Q1, ?f Q2)"
+ using process.corec[of ?s P]
+ unfolding isC_join22[of isA P isC "pr" co c1 c2, OF isA isC]
+ processBNF_map id_apply pcorec_def c1 c2 by simp
+ thus ?thesis unfolding c1 c2 pcorec_def Choice_def using process.fld_unf by (simp, metis)
+ qed
+ qed
+qed
+
+theorems pcorec = pcorec_Action pcorec_Choice
+
+
+section{* Coinductive definition of the notion of trace *}
+
+(* Say we have a type of streams: *)
+typedecl 'a stream
+consts Ccons :: "'a \<Rightarrow> 'a stream \<Rightarrow> 'a stream"
+
+(* Use the existing coinductive package (distinct from our
+new codatatype package, but highly compatible with it): *)
+
+coinductive trace where
+"trace p as \<Longrightarrow> trace (Action a p) (Ccons a as)"
+|
+"trace p as \<or> trace q as \<Longrightarrow> trace (Choice p q) as"
+
+
+section{* Examples of corecursive definitions: *}
+
+subsection{* Single-guard fixpoint definition *}
+
+definition
+"BX \<equiv>
+ pcoiter
+ (\<lambda> P. True)
+ (\<lambda> P. ''a'')
+ (\<lambda> P. P)
+ undefined
+ undefined
+ undefined
+ ()"
+
+lemma BX: "BX = Action ''a'' BX"
+unfolding BX_def
+using pcoiter(1)[of "\<lambda> P. True" "()" "\<lambda> P. ''a''" "\<lambda> P. P"] by simp
+
+
+subsection{* Multi-guard fixpoint definitions, simulated with auxiliary arguments *}
+
+datatype x_y_ax = x | y | ax
+
+definition "isA \<equiv> \<lambda> K. case K of x \<Rightarrow> False |y \<Rightarrow> True |ax \<Rightarrow> True"
+definition "pr \<equiv> \<lambda> K. case K of x \<Rightarrow> undefined |y \<Rightarrow> ''b'' |ax \<Rightarrow> ''a''"
+definition "co \<equiv> \<lambda> K. case K of x \<Rightarrow> undefined |y \<Rightarrow> x |ax \<Rightarrow> x"
+lemmas Action_defs = isA_def pr_def co_def
+
+definition "isC \<equiv> \<lambda> K. case K of x \<Rightarrow> True |y \<Rightarrow> False |ax \<Rightarrow> False"
+definition "c1 \<equiv> \<lambda> K. case K of x \<Rightarrow> ax |y \<Rightarrow> undefined |ax \<Rightarrow> undefined"
+definition "c2 \<equiv> \<lambda> K. case K of x \<Rightarrow> y |y \<Rightarrow> undefined |ax \<Rightarrow> undefined"
+lemmas Choice_defs = isC_def c1_def c2_def
+
+definition "F \<equiv> pcoiter isA pr co isC c1 c2"
+definition "X = F x" definition "Y = F y" definition "AX = F ax"
+
+lemma X_Y_AX: "X = Choice AX Y" "Y = Action ''b'' X" "AX = Action ''a'' X"
+unfolding X_def Y_def AX_def F_def
+using pcoiter(2)[of isA x isC "pr" co c1 c2]
+ pcoiter(1)[of isA y "pr" co isC c1 c2]
+ pcoiter(1)[of isA ax "pr" co isC c1 c2]
+unfolding Action_defs Choice_defs by simp_all
+
+(* end product: *)
+lemma X_AX:
+"X = Choice AX (Action ''b'' X)"
+"AX = Action ''a'' X"
+using X_Y_AX by simp_all
+
+
+
+section{* Case study: Multi-guard fixpoint definitions, without auxiliary arguments *}
+
+hide_const x y ax X Y AX
+
+(* Process terms *)
+datatype ('a,'pvar) process_term =
+ VAR 'pvar |
+ PROC "'a process" |
+ ACT 'a "('a,'pvar) process_term" | CH "('a,'pvar) process_term" "('a,'pvar) process_term"
+
+(* below, sys represents a system of equations *)
+fun isACT where
+"isACT sys (VAR X) =
+ (case sys X of ACT a T \<Rightarrow> True |PROC p \<Rightarrow> isAction p |_ \<Rightarrow> False)"
+|
+"isACT sys (PROC p) = isAction p"
+|
+"isACT sys (ACT a T) = True"
+|
+"isACT sys (CH T1 T2) = False"
+
+fun PREF where
+"PREF sys (VAR X) =
+ (case sys X of ACT a T \<Rightarrow> a | PROC p \<Rightarrow> prefOf p)"
+|
+"PREF sys (PROC p) = prefOf p"
+|
+"PREF sys (ACT a T) = a"
+
+fun CONT where
+"CONT sys (VAR X) =
+ (case sys X of ACT a T \<Rightarrow> T | PROC p \<Rightarrow> PROC (contOf p))"
+|
+"CONT sys (PROC p) = PROC (contOf p)"
+|
+"CONT sys (ACT a T) = T"
+
+fun isCH where
+"isCH sys (VAR X) =
+ (case sys X of CH T1 T2 \<Rightarrow> True |PROC p \<Rightarrow> isChoice p |_ \<Rightarrow> False)"
+|
+"isCH sys (PROC p) = isChoice p"
+|
+"isCH sys (ACT a T) = False"
+|
+"isCH sys (CH T1 T2) = True"
+
+fun CH1 where
+"CH1 sys (VAR X) =
+ (case sys X of CH T1 T2 \<Rightarrow> T1 |PROC p \<Rightarrow> PROC (ch1Of p))"
+|
+"CH1 sys (PROC p) = PROC (ch1Of p)"
+|
+"CH1 sys (CH T1 T2) = T1"
+
+fun CH2 where
+"CH2 sys (VAR X) =
+ (case sys X of CH T1 T2 \<Rightarrow> T2 |PROC p \<Rightarrow> PROC (ch2Of p))"
+|
+"CH2 sys (PROC p) = PROC (ch2Of p)"
+|
+"CH2 sys (CH T1 T2) = T2"
+
+definition "guarded sys \<equiv> \<forall> X Y. sys X \<noteq> VAR Y"
+
+lemma guarded_isACT_isCH:
+assumes g: "guarded sys"
+shows "isACT sys T \<or> isCH sys T"
+proof(induct T)
+ case (VAR X)
+ thus ?case
+ using g isAction_isChoice unfolding guarded_def by (cases "sys X", auto)
+qed(insert isAction_isChoice assms, unfold guarded_def, auto)
+
+definition
+"solution sys \<equiv>
+ pcoiter
+ (isACT sys)
+ (PREF sys)
+ (CONT sys)
+ (isCH sys)
+ (CH1 sys)
+ (CH2 sys)"
+
+lemma solution_Action:
+assumes "isACT sys T"
+shows "solution sys T = Action (PREF sys T) (solution sys (CONT sys T))"
+unfolding solution_def
+using pcoiter(1)[of "isACT sys" T "PREF sys" "CONT sys"
+ "isCH sys" "CH1 sys" "CH2 sys"] assms by simp
+
+lemma solution_Choice:
+assumes "\<not> isACT sys T" "isCH sys T"
+shows "solution sys T = Choice (solution sys (CH1 sys T)) (solution sys (CH2 sys T))"
+unfolding solution_def
+using pcoiter(2)[of "isACT sys" T "isCH sys" "PREF sys" "CONT sys"
+ "CH1 sys" "CH2 sys"] assms by simp
+
+lemma isACT_VAR:
+assumes g: "guarded sys"
+shows "isACT sys (VAR X) \<longleftrightarrow> isACT sys (sys X)"
+using g unfolding guarded_def by (cases "sys X") auto
+
+lemma isCH_VAR:
+assumes g: "guarded sys"
+shows "isCH sys (VAR X) \<longleftrightarrow> isCH sys (sys X)"
+using g unfolding guarded_def by (cases "sys X") auto
+
+lemma solution_VAR:
+assumes g: "guarded sys"
+shows "solution sys (VAR X) = solution sys (sys X)"
+proof(cases "isACT sys (VAR X)")
+ case True
+ hence T: "isACT sys (sys X)" unfolding isACT_VAR[OF g] .
+ show ?thesis
+ unfolding solution_Action[OF T] using solution_Action[of sys "VAR X"] True g
+ unfolding guarded_def by (cases "sys X", auto)
+next
+ case False note FFalse = False
+ hence TT: "\<not> isACT sys (sys X)" unfolding isACT_VAR[OF g] .
+ show ?thesis
+ proof(cases "isCH sys (VAR X)")
+ case True
+ hence T: "isCH sys (sys X)" unfolding isCH_VAR[OF g] .
+ show ?thesis
+ unfolding solution_Choice[OF TT T] using solution_Choice[of sys "VAR X"] FFalse True g
+ unfolding guarded_def by (cases "sys X", auto)
+ next
+ case False
+ hence False using FFalse guarded_isACT_isCH[OF g, of "VAR X"] by simp
+ thus ?thesis by simp
+ qed
+qed
+
+lemma solution_PROC[simp]:
+"solution sys (PROC p) = p"
+proof-
+ {fix q assume "q = solution sys (PROC p)"
+ hence "p = q"
+ proof(induct rule: process_coind)
+ case (iss p p')
+ from isAction_isChoice[of p] show ?case
+ proof
+ assume p: "isAction p"
+ hence 0: "isACT sys (PROC p)" by simp
+ thus ?thesis using iss not_isAction_isChoice
+ unfolding solution_Action[OF 0] by auto
+ next
+ assume "isChoice p"
+ hence 0: "isCH sys (PROC p)" and p: "\<not> isAction p"
+ using not_isAction_isChoice by auto
+ hence 1: "\<not> isACT sys (PROC p)" by simp
+ show ?thesis using 0 iss not_isAction_isChoice
+ unfolding solution_Choice[OF 1 0] by auto
+ qed
+ next
+ case (Action a a' p p')
+ hence 0: "isACT sys (PROC (Action a p))" by simp
+ show ?case using Action unfolding solution_Action[OF 0] by simp
+ next
+ case (Choice p q p' q')
+ hence 0: "isCH sys (PROC (Choice p q))" by simp
+ hence 1: "\<not> isACT sys (PROC (Choice p q))" using not_isAction_isChoice by auto
+ show ?case using Choice unfolding solution_Choice[OF 1 0] by simp
+ qed
+ }
+ thus ?thesis by metis
+qed
+
+lemma solution_ACT[simp]:
+"solution sys (ACT a T) = Action a (solution sys T)"
+by (metis CONT.simps(3) PREF.simps(3) isACT.simps(3) solution_Action)
+
+lemma solution_CH[simp]:
+"solution sys (CH T1 T2) = Choice (solution sys T1) (solution sys T2)"
+by (metis CH1.simps(3) CH2.simps(3) isACT.simps(4) isCH.simps(4) solution_Choice)
+
+
+(* Example: *)
+
+fun sys where
+"sys 0 = CH (VAR (Suc 0)) (ACT ''b'' (VAR 0))"
+|
+"sys (Suc 0) = ACT ''a'' (VAR 0)"
+| (* dummy guarded term for variables outside the system: *)
+"sys X = ACT ''a'' (VAR 0)"
+
+lemma guarded_sys:
+"guarded sys"
+unfolding guarded_def proof (intro allI)
+ fix X Y show "sys X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto)
+qed
+
+(* the actual processes: *)
+definition "x \<equiv> solution sys (VAR 0)"
+definition "ax \<equiv> solution sys (VAR (Suc 0))"
+
+(* end product: *)
+lemma x_ax:
+"x = Choice ax (Action ''b'' x)"
+"ax = Action ''a'' x"
+unfolding x_def ax_def by (subst solution_VAR[OF guarded_sys], simp)+
+
+
+(* Thanks to the inclusion of processes as process terms, one can
+also consider parametrized systems of equations---here, x is a (semantic)
+process parameter: *)
+
+fun sys' where
+"sys' 0 = CH (PROC x) (ACT ''b'' (VAR 0))"
+|
+"sys' (Suc 0) = CH (ACT ''a'' (VAR 0)) (PROC x)"
+| (* dummy guarded term : *)
+"sys' X = ACT ''a'' (VAR 0)"
+
+lemma guarded_sys':
+"guarded sys'"
+unfolding guarded_def proof (intro allI)
+ fix X Y show "sys' X \<noteq> VAR Y" by (cases X, simp, case_tac nat, auto)
+qed
+
+(* the actual processes: *)
+definition "y \<equiv> solution sys' (VAR 0)"
+definition "ay \<equiv> solution sys' (VAR (Suc 0))"
+
+(* end product: *)
+lemma y_ay:
+"y = Choice x (Action ''b'' y)"
+"ay = Choice (Action ''a'' y) x"
+unfolding y_def ay_def by (subst solution_VAR[OF guarded_sys'], simp)+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/Stream.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,159 @@
+(* Title: Codatatype_Examples/Stream.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Infinite streams.
+*)
+
+header {* Infinite Streams *}
+
+theory Stream
+imports TreeFI
+begin
+
+bnf_codata stream: 's = "'a \<times> 's"
+
+(* selectors for streams *)
+definition "hdd as \<equiv> fst (stream_unf as)"
+definition "tll as \<equiv> snd (stream_unf as)"
+
+lemma coiter_pair_fun_hdd[simp]: "hdd (stream_coiter (f \<odot> g) t) = f t"
+unfolding hdd_def pair_fun_def stream.coiter by simp
+
+lemma coiter_pair_fun_tll[simp]: "tll (stream_coiter (f \<odot> g) t) =
+ stream_coiter (f \<odot> g) (g t)"
+unfolding tll_def pair_fun_def stream.coiter by simp
+
+(* infinite trees: *)
+coinductive infiniteTr where
+"\<lbrakk>tr' \<in> listF_set (sub tr); infiniteTr tr'\<rbrakk> \<Longrightarrow> infiniteTr tr"
+
+lemma infiniteTr_coind_upto[consumes 1, case_names sub]:
+assumes *: "phi tr" and
+**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr' \<or> infiniteTr tr'"
+shows "infiniteTr tr"
+using assms by (elim infiniteTr.coinduct) blast
+
+lemma infiniteTr_coind[consumes 1, case_names sub, induct pred: infiniteTr]:
+assumes *: "phi tr" and
+**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr'"
+shows "infiniteTr tr"
+using assms by (elim infiniteTr.coinduct) blast
+
+lemma infiniteTr_sub[simp]:
+"infiniteTr tr \<Longrightarrow> (\<exists> tr' \<in> listF_set (sub tr). infiniteTr tr')"
+by (erule infiniteTr.cases) blast
+
+definition "konigPath \<equiv> stream_coiter
+ (lab \<odot> (\<lambda>tr. SOME tr'. tr' \<in> listF_set (sub tr) \<and> infiniteTr tr'))"
+
+lemma hdd_simps1[simp]: "hdd (konigPath t) = lab t"
+unfolding konigPath_def by simp
+
+lemma tll_simps2[simp]: "tll (konigPath t) =
+ konigPath (SOME tr. tr \<in> listF_set (sub t) \<and> infiniteTr tr)"
+unfolding konigPath_def by simp
+
+(* proper paths in trees: *)
+coinductive properPath where
+"\<lbrakk>hdd as = lab tr; tr' \<in> listF_set (sub tr); properPath (tll as) tr'\<rbrakk> \<Longrightarrow>
+ properPath as tr"
+
+lemma properPath_coind_upto[consumes 1, case_names hdd_lab sub]:
+assumes *: "phi as tr" and
+**: "\<And> as tr. phi as tr \<Longrightarrow> hdd as = lab tr" and
+***: "\<And> as tr.
+ phi as tr \<Longrightarrow>
+ \<exists> tr' \<in> listF_set (sub tr). phi (tll as) tr' \<or> properPath (tll as) tr'"
+shows "properPath as tr"
+using assms by (elim properPath.coinduct) blast
+
+lemma properPath_coind[consumes 1, case_names hdd_lab sub, induct pred: properPath]:
+assumes *: "phi as tr" and
+**: "\<And> as tr. phi as tr \<Longrightarrow> hdd as = lab tr" and
+***: "\<And> as tr.
+ phi as tr \<Longrightarrow>
+ \<exists> tr' \<in> listF_set (sub tr). phi (tll as) tr'"
+shows "properPath as tr"
+using properPath_coind_upto[of phi, OF * **] *** by blast
+
+lemma properPath_hdd_lab:
+"properPath as tr \<Longrightarrow> hdd as = lab tr"
+by (erule properPath.cases) blast
+
+lemma properPath_sub:
+"properPath as tr \<Longrightarrow>
+ \<exists> tr' \<in> listF_set (sub tr). phi (tll as) tr' \<or> properPath (tll as) tr'"
+by (erule properPath.cases) blast
+
+(* prove the following by coinduction *)
+theorem Konig:
+ assumes "infiniteTr tr"
+ shows "properPath (konigPath tr) tr"
+proof-
+ {fix as
+ assume "infiniteTr tr \<and> as = konigPath tr" hence "properPath as tr"
+ proof (induct rule: properPath_coind, safe)
+ fix t
+ let ?t = "SOME t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'"
+ assume "infiniteTr t"
+ hence "\<exists>t' \<in> listF_set (sub t). infiniteTr t'" by simp
+ hence "\<exists>t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'" by blast
+ hence "?t \<in> listF_set (sub t) \<and> infiniteTr ?t" by (elim someI_ex)
+ moreover have "tll (konigPath t) = konigPath ?t" by simp
+ ultimately show "\<exists>t' \<in> listF_set (sub t).
+ infiniteTr t' \<and> tll (konigPath t) = konigPath t'" by blast
+ qed simp
+ }
+ thus ?thesis using assms by blast
+qed
+
+(* some more stream theorems *)
+
+lemma stream_map[simp]: "stream_map f = stream_coiter (f o hdd \<odot> tll)"
+unfolding stream_map_def pair_fun_def hdd_def[abs_def] tll_def[abs_def]
+ map_pair_def o_def prod_case_beta by simp
+
+lemma streamBNF_pred[simp]: "streamBNF_pred \<phi>1 \<phi>2 a b = (\<phi>1 (fst a) (fst b) \<and> \<phi>2 (snd a) (snd b))"
+by (auto simp: streamBNF.pred_unfold)
+
+lemmas stream_coind = mp[OF stream.pred_coinduct, unfolded streamBNF_pred[abs_def],
+ folded hdd_def tll_def]
+
+definition plus :: "nat stream \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<oplus>" 66) where
+ [simp]: "plus xs ys =
+ stream_coiter ((%(xs, ys). hdd xs + hdd ys) \<odot> (%(xs, ys). (tll xs, tll ys))) (xs, ys)"
+
+definition scalar :: "nat \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<cdot>" 68) where
+ [simp]: "scalar n = stream_map (\<lambda>x. n * x)"
+
+definition ones :: "nat stream" where [simp]: "ones = stream_coiter ((%x. 1) \<odot> id) ()"
+definition twos :: "nat stream" where [simp]: "twos = stream_coiter ((%x. 2) \<odot> id) ()"
+definition ns :: "nat \<Rightarrow> nat stream" where [simp]: "ns n = scalar n ones"
+
+lemma "ones \<oplus> ones = twos"
+by (intro stream_coind[where phi="%x1 x2. \<exists>x. x1 = ones \<oplus> ones \<and> x2 = twos"])
+ auto
+
+lemma "n \<cdot> twos = ns (2 * n)"
+by (intro stream_coind[where phi="%x1 x2. \<exists>n. x1 = n \<cdot> twos \<and> x2 = ns (2 * n)"])
+ force+
+
+lemma prod_scalar: "(n * m) \<cdot> xs = n \<cdot> m \<cdot> xs"
+by (intro stream_coind[where phi="%x1 x2. \<exists>n m xs. x1 = (n * m) \<cdot> xs \<and> x2 = n \<cdot> m \<cdot> xs"])
+ force+
+
+lemma scalar_plus: "n \<cdot> (xs \<oplus> ys) = n \<cdot> xs \<oplus> n \<cdot> ys"
+by (intro stream_coind[where phi="%x1 x2. \<exists>n xs ys. x1 = n \<cdot> (xs \<oplus> ys) \<and> x2 = n \<cdot> xs \<oplus> n \<cdot> ys"])
+ (force simp: add_mult_distrib2)+
+
+lemma plus_comm: "xs \<oplus> ys = ys \<oplus> xs"
+by (intro stream_coind[where phi="%x1 x2. \<exists>xs ys. x1 = xs \<oplus> ys \<and> x2 = ys \<oplus> xs"])
+ force+
+
+lemma plus_assoc: "(xs \<oplus> ys) \<oplus> zs = xs \<oplus> ys \<oplus> zs"
+by (intro stream_coind[where phi="%x1 x2. \<exists>xs ys zs. x1 = (xs \<oplus> ys) \<oplus> zs \<and> x2 = xs \<oplus> ys \<oplus> zs"])
+ force+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/TreeFI.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,81 @@
+(* Title: Codatatype_Examples/TreeFI.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Finitely branching possibly infinite trees.
+*)
+
+header {* Finitely Branching Possibly Infinite Trees *}
+
+theory TreeFI
+imports ListF
+begin
+
+bnf_codata treeFI: 'tree = "'a \<times> 'tree listF"
+
+lemma treeFIBNF_listF_set[simp]: "treeFIBNF_set2 (i, xs) = listF_set xs"
+unfolding treeFIBNF_set2_def collect_def[abs_def] prod_set_defs
+by (auto simp add: listF.set_natural')
+
+(* selectors for trees *)
+definition "lab tr \<equiv> fst (treeFI_unf tr)"
+definition "sub tr \<equiv> snd (treeFI_unf tr)"
+
+lemma unf[simp]: "treeFI_unf tr = (lab tr, sub tr)"
+unfolding lab_def sub_def by simp
+
+definition pair_fun (infixr "\<odot>" 50) where
+ "f \<odot> g \<equiv> \<lambda>x. (f x, g x)"
+
+lemma coiter_pair_fun_lab: "lab (treeFI_coiter (f \<odot> g) t) = f t"
+unfolding lab_def pair_fun_def treeFI.coiter treeFIBNF_map_def by simp
+
+lemma coiter_pair_fun_sub: "sub (treeFI_coiter (f \<odot> g) t) = listF_map (treeFI_coiter (f \<odot> g)) (g t)"
+unfolding sub_def pair_fun_def treeFI.coiter treeFIBNF_map_def by simp
+
+(* Tree reverse:*)
+definition "trev \<equiv> treeFI_coiter (lab \<odot> lrev o sub)"
+
+lemma trev_simps1[simp]: "lab (trev t) = lab t"
+unfolding trev_def by (simp add: coiter_pair_fun_lab)
+
+lemma trev_simps2[simp]: "sub (trev t) = listF_map trev (lrev (sub t))"
+unfolding trev_def by (simp add: coiter_pair_fun_sub)
+
+lemma treeFI_coinduct:
+assumes *: "phi x y"
+and step: "\<And>a b. phi a b \<Longrightarrow>
+ lab a = lab b \<and>
+ lengthh (sub a) = lengthh (sub b) \<and>
+ (\<forall>i < lengthh (sub a). phi (nthh (sub a) i) (nthh (sub b) i))"
+shows "x = y"
+proof (rule mp[OF treeFI.unf_coinduct, of phi, OF _ *])
+ fix a b :: "'a treeFI"
+ let ?zs = "zipp (sub a) (sub b)"
+ let ?z = "(lab a, ?zs)"
+ assume "phi a b"
+ with step have step': "lab a = lab b" "lengthh (sub a) = lengthh (sub b)"
+ "\<forall>i < lengthh (sub a). phi (nthh (sub a) i) (nthh (sub b) i)" by auto
+ hence "treeFIBNF_map id fst ?z = treeFI_unf a" "treeFIBNF_map id snd ?z = treeFI_unf b"
+ unfolding treeFIBNF_map_def by auto
+ moreover have "\<forall>(x, y) \<in> treeFIBNF_set2 ?z. phi x y"
+ proof safe
+ fix z1 z2
+ assume "(z1, z2) \<in> treeFIBNF_set2 ?z"
+ hence "(z1, z2) \<in> listF_set ?zs" by auto
+ hence "\<exists>i < lengthh ?zs. nthh ?zs i = (z1, z2)" by auto
+ with step'(2) obtain i where "i < lengthh (sub a)"
+ "nthh (sub a) i = z1" "nthh (sub b) i = z2" by auto
+ with step'(3) show "phi z1 z2" by auto
+ qed
+ ultimately show "\<exists>z.
+ (treeFIBNF_map id fst z = treeFI_unf a \<and>
+ treeFIBNF_map id snd z = treeFI_unf b) \<and>
+ (\<forall>x y. (x, y) \<in> treeFIBNF_set2 z \<longrightarrow> phi x y)" by blast
+qed
+
+lemma trev_trev: "trev (trev tr) = tr"
+by (rule treeFI_coinduct[of "%a b. trev (trev b) = a"]) auto
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Examples/TreeFsetI.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,56 @@
+(* Title: Codatatype_Examples/TreeFsetI.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Finitely branching possibly infinite trees, with sets of children.
+*)
+
+header {* Finitely Branching Possibly Infinite Trees, with Sets of Children *}
+
+theory TreeFsetI
+imports "../Codatatype/Codatatype"
+begin
+
+definition pair_fun (infixr "\<odot>" 50) where
+ "f \<odot> g \<equiv> \<lambda>x. (f x, g x)"
+
+bnf_codata treeFsetI: 't = "'a \<times> 't fset"
+
+(* selectors for trees *)
+definition "lab t \<equiv> fst (treeFsetI_unf t)"
+definition "sub t \<equiv> snd (treeFsetI_unf t)"
+
+lemma unf[simp]: "treeFsetI_unf t = (lab t, sub t)"
+unfolding lab_def sub_def by simp
+
+lemma coiter_pair_fun_lab: "lab (treeFsetI_coiter (f \<odot> g) t) = f t"
+unfolding lab_def pair_fun_def treeFsetI.coiter treeFsetIBNF_map_def by simp
+
+lemma coiter_pair_fun_sub: "sub (treeFsetI_coiter (f \<odot> g) t) = map_fset (treeFsetI_coiter (f \<odot> g)) (g t)"
+unfolding sub_def pair_fun_def treeFsetI.coiter treeFsetIBNF_map_def by simp
+
+(* tree map (contrived example): *)
+definition "tmap f \<equiv> treeFsetI_coiter (f o lab \<odot> sub)"
+
+lemma tmap_simps1[simp]: "lab (tmap f t) = f (lab t)"
+unfolding tmap_def by (simp add: coiter_pair_fun_lab)
+
+lemma trev_simps2[simp]: "sub (tmap f t) = map_fset (tmap f) (sub t)"
+unfolding tmap_def by (simp add: coiter_pair_fun_sub)
+
+lemma treeFsetIBNF_pred[simp]: "treeFsetIBNF_pred R1 R2 a b = (R1 (fst a) (fst b) \<and>
+ (\<forall>t \<in> fset (snd a). (\<exists>u \<in> fset (snd b). R2 t u)) \<and>
+ (\<forall>t \<in> fset (snd b). (\<exists>u \<in> fset (snd a). R2 u t)))"
+apply (cases a)
+apply (cases b)
+apply (simp add: treeFsetIBNF.pred_unfold)
+done
+
+lemmas treeFsetI_coind = mp[OF treeFsetI.pred_coinduct]
+
+lemma "tmap (f o g) x = tmap f (tmap g x)"
+by (intro treeFsetI_coind[where phi="%x1 x2. \<exists>x. x1 = tmap (f o g) x \<and> x2 = tmap f (tmap g x)"])
+ force+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/README.html Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,58 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
+
+<html>
+
+<head>
+ <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
+ <title>Codatatype Package</title>
+</head>
+
+<body>
+
+<h3><i>Codatatype</i>: A (co)datatype package based on bounded natural functors
+(BNFs)</h3>
+
+<p>
+The <i>Codatatype</i> package provides a fully modular framework for
+constructing inductive and coinductive datatypes in HOL, with support for mixed
+mutual and nested (co)recursion. Mixed (co)recursion enables type definitions
+involving both datatypes and codatatypes, such as the type of finitely branching
+trees of possibly infinite depth. The framework draws heavily from category
+theory.
+
+<p>
+The package is described in the following paper:
+
+<ul>
+ <li><a href="http://www21.in.tum.de/~traytel/papers/codatatypes/index.html">Foundational, Compositional (Co)datatypes for Higher-Order Logic—Category Theory Applied to Theorem Proving</a>, <br>
+ Dmitriy Traytel, Andrei Popescu, and Jasmin Christian Blanchette.<br>
+ <i>Logic in Computer Science (LICS 2012)</i>, 2012.
+</ul>
+
+<p>
+The main entry point for applications is <tt>Codatatype.thy</tt>.
+The <tt>Examples</tt> directory contains various examples of (co)datatypes,
+including the examples from the paper.
+
+<p>
+The key notion underlying the package is that of a <i>bounded natural functor</i>
+(<i>BNF</i>)—an enriched type constructor satisfying specific properties
+preserved by interesting categorical operations (composition, least fixed point,
+and greatest fixed point). The <tt>Basic_BNFs.thy</tt> file registers
+various basic types, notably for sums, products, function spaces, finite sets,
+multisets, and countable sets. Custom BNFs can be registered as well.
+
+<p>
+<b>Warning:</b> The package is under development. Future versions are expected
+to support multiple constructors and selectors per (co)datatype (instead of a
+single <i>fld</i> or <i>unf</i> constant) and provide a nicer syntax for
+(co)datatype and (co)recursive function definitions. Please contact
+any of
+<a href="mailto:traytel@in.tum.de">the</a>
+<a href="mailto:popescua@in.tum.de">above</a>
+<a href="mailto:blanchette@in.tum.de">authors</a>
+if you have questions or comments.
+
+</body>
+
+</html>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_comp.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,834 @@
+(* Title: HOL/Codatatype/Tools/bnf_comp.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+Composition of bounded natural functors.
+*)
+
+signature BNF_COMP =
+sig
+ type unfold_thms
+ val empty_unfold: unfold_thms
+ val map_unfolds_of: unfold_thms -> thm list
+ val set_unfoldss_of: unfold_thms -> thm list list
+ val rel_unfolds_of: unfold_thms -> thm list
+ val pred_unfolds_of: unfold_thms -> thm list
+
+ val default_comp_sort: (string * sort) list list -> (string * sort) list
+ val bnf_of_typ: BNF_Def.const_policy -> binding -> (binding -> binding) ->
+ ((string * sort) list list -> (string * sort) list) -> typ -> unfold_thms * Proof.context ->
+ (BNF_Def.BNF * (typ list * typ list)) * (unfold_thms * Proof.context)
+ val bnf_of_typ_cmd: binding * string -> Proof.context -> Proof.context
+ val seal_bnf: unfold_thms -> binding -> typ list -> BNF_Def.BNF -> Proof.context ->
+ BNF_Def.BNF * local_theory
+ val normalize_bnfs: (int -> binding -> binding) -> ''a list list -> ''a list ->
+ (''a list list -> ''a list) -> BNF_Def.BNF list -> unfold_thms -> Proof.context ->
+ (int list list * ''a list) * (BNF_Def.BNF list * (unfold_thms * Proof.context))
+end;
+
+structure BNF_Comp : BNF_COMP =
+struct
+
+open BNF_Def
+open BNF_Util
+open BNF_Tactics
+open BNF_Comp_Tactics
+
+type unfold_thms = {
+ map_unfolds: thm list,
+ set_unfoldss: thm list list,
+ rel_unfolds: thm list,
+ pred_unfolds: thm list
+};
+
+fun add_to_thms thms NONE = thms
+ | add_to_thms thms (SOME new) = if Thm.is_reflexive new then thms else insert Thm.eq_thm new thms;
+fun adds_to_thms thms NONE = thms
+ | adds_to_thms thms (SOME news) = insert (eq_set Thm.eq_thm) (filter_refl news) thms;
+
+fun mk_unfold_thms maps setss rels preds =
+ {map_unfolds = maps, set_unfoldss = setss, rel_unfolds = rels, pred_unfolds = preds};
+
+val empty_unfold = mk_unfold_thms [] [] [] [];
+
+fun add_to_unfold_opt map_opt sets_opt rel_opt pred_opt
+ {map_unfolds = maps, set_unfoldss = setss, rel_unfolds = rels, pred_unfolds = preds} = {
+ map_unfolds = add_to_thms maps map_opt,
+ set_unfoldss = adds_to_thms setss sets_opt,
+ rel_unfolds = add_to_thms rels rel_opt,
+ pred_unfolds = add_to_thms preds pred_opt};
+
+fun add_to_unfold map sets rel pred =
+ add_to_unfold_opt (SOME map) (SOME sets) (SOME rel) (SOME pred);
+
+val map_unfolds_of = #map_unfolds;
+val set_unfoldss_of = #set_unfoldss;
+val rel_unfolds_of = #rel_unfolds;
+val pred_unfolds_of = #pred_unfolds;
+
+val bdTN = "bdT";
+
+val compN = "comp_"
+fun mk_killN n = "kill" ^ string_of_int n ^ "_";
+fun mk_liftN n = "lift" ^ string_of_int n ^ "_";
+fun mk_permuteN src dest =
+ "permute_" ^ implode (map string_of_int src) ^ "_" ^ implode (map string_of_int dest) ^ "_";
+
+val no_thm = refl;
+val Collect_split_box_equals = box_equals RS @{thm Collect_split_cong};
+val abs_pred_sym = sym RS @{thm abs_pred_def};
+val abs_pred_sym_pred_abs = abs_pred_sym RS @{thm pred_def_abs};
+
+(*copied from Envir.expand_term_free*)
+fun expand_term_const defs =
+ let
+ val eqs = map ((fn ((x, U), u) => (x, (U, u))) o apfst dest_Const) defs;
+ val get = fn Const (x, _) => AList.lookup (op =) eqs x | _ => NONE;
+ in Envir.expand_term get end;
+
+fun clean_compose_bnf const_policy qualify b outer inners (unfold, lthy) =
+ let
+ val olive = live_of_bnf outer;
+ val onwits = nwits_of_bnf outer;
+ val odead = dead_of_bnf outer;
+ val inner = hd inners;
+ val ilive = live_of_bnf inner;
+ val ideads = map dead_of_bnf inners;
+ val inwitss = map nwits_of_bnf inners;
+
+ (* TODO: check olive = length inners > 0,
+ forall inner from inners. ilive = live,
+ forall inner from inners. idead = dead *)
+
+ val (oDs, lthy1) = apfst (map TFree)
+ (Variable.invent_types (replicate odead HOLogic.typeS) lthy);
+ val (Dss, lthy2) = apfst (map (map TFree))
+ (fold_map Variable.invent_types (map (fn n => replicate n HOLogic.typeS) ideads) lthy1);
+ val (Ass, lthy3) = apfst (replicate ilive o map TFree)
+ (Variable.invent_types (replicate ilive HOLogic.typeS) lthy2);
+ val As = if ilive > 0 then hd Ass else [];
+ val Ass_repl = replicate olive As;
+ val (Bs, _(*lthy4*)) = apfst (map TFree)
+ (Variable.invent_types (replicate ilive HOLogic.typeS) lthy3);
+ val Bss_repl = replicate olive Bs;
+
+ val (((fs', Asets), xs), _(*names_lthy*)) = lthy
+ |> apfst snd o mk_Frees' "f" (map2 (curry (op -->)) As Bs)
+ ||>> mk_Frees "A" (map (HOLogic.mk_setT) As)
+ ||>> mk_Frees "x" As;
+
+ val CAs = map3 mk_T_of_bnf Dss Ass_repl inners;
+ val CCA = mk_T_of_bnf oDs CAs outer;
+ val CBs = map3 mk_T_of_bnf Dss Bss_repl inners;
+ val outer_sets = mk_sets_of_bnf (replicate olive oDs) (replicate olive CAs) outer;
+ val inner_setss = map3 mk_sets_of_bnf (map (replicate ilive) Dss) (replicate olive Ass) inners;
+ val inner_bds = map3 mk_bd_of_bnf Dss Ass_repl inners;
+ val outer_bd = mk_bd_of_bnf oDs CAs outer;
+
+ (*%f1 ... fn. outer.map (inner_1.map f1 ... fn) ... (inner_m.map f1 ... fn)*)
+ val comp_map = fold_rev Term.abs fs'
+ (Term.list_comb (mk_map_of_bnf oDs CAs CBs outer,
+ map2 (fn Ds => (fn f => Term.list_comb (f, map Bound ((ilive - 1) downto 0))) o
+ mk_map_of_bnf Ds As Bs) Dss inners));
+
+ (*Union o collect {outer.set_1 ... outer.set_m} o outer.map inner_1.set_i ... inner_m.set_i*)
+ (*Union o collect {image inner_1.set_i o outer.set_1 ... image inner_m.set_i o outer.set_m}*)
+ fun mk_comp_set i =
+ let
+ val (setTs, T) = `(replicate olive o HOLogic.mk_setT) (nth As i);
+ val outer_set = mk_collect
+ (mk_sets_of_bnf (replicate olive oDs) (replicate olive setTs) outer)
+ (mk_T_of_bnf oDs setTs outer --> HOLogic.mk_setT T);
+ val inner_sets = map (fn sets => nth sets i) inner_setss;
+ val outer_map = mk_map_of_bnf oDs CAs setTs outer;
+ val map_inner_sets = Term.list_comb (outer_map, inner_sets);
+ val collect_image = mk_collect
+ (map2 (fn f => fn set => HOLogic.mk_comp (mk_image f, set)) inner_sets outer_sets)
+ (CCA --> HOLogic.mk_setT T);
+ in
+ (Library.foldl1 HOLogic.mk_comp [mk_Union T, outer_set, map_inner_sets],
+ HOLogic.mk_comp (mk_Union T, collect_image))
+ end;
+
+ val (comp_sets, comp_sets_alt) = map_split mk_comp_set (0 upto ilive - 1);
+
+ (*(inner_1.bd +c ... +c inner_m.bd) *c outer.bd*)
+ val comp_bd = Term.absdummy CCA (mk_cprod
+ (Library.foldr1 (uncurry mk_csum) inner_bds) outer_bd);
+
+ fun comp_map_id_tac {context = ctxt, ...} =
+ let
+ (*order the theorems by reverse size to prevent bad interaction with nonconfluent rewrite
+ rules*)
+ val thms = (map map_id_of_bnf inners
+ |> map (`(Term.size_of_term o Thm.prop_of))
+ |> sort (rev_order o int_ord o pairself fst)
+ |> map snd) @ [map_id_of_bnf outer];
+ in
+ (EVERY' (map (fn thm => subst_tac ctxt [thm]) thms) THEN' rtac refl) 1
+ end;
+
+ fun comp_map_comp_tac _ =
+ mk_comp_map_comp_tac (map_comp_of_bnf outer) (map_cong_of_bnf outer)
+ (map map_comp_of_bnf inners);
+
+ fun mk_single_comp_set_natural_tac i _ =
+ mk_comp_set_natural_tac (map_comp_of_bnf outer) (map_cong_of_bnf outer)
+ (collect_set_natural_of_bnf outer)
+ (map ((fn thms => nth thms i) o set_natural_of_bnf) inners);
+
+ val comp_set_natural_tacs = map mk_single_comp_set_natural_tac (0 upto ilive - 1);
+
+ fun comp_bd_card_order_tac _ =
+ mk_comp_bd_card_order_tac (map bd_card_order_of_bnf inners) (bd_card_order_of_bnf outer);
+
+ fun comp_bd_cinfinite_tac _ =
+ mk_comp_bd_cinfinite_tac (bd_cinfinite_of_bnf inner) (bd_cinfinite_of_bnf outer);
+
+ val comp_set_alt_thms =
+ if ! quick_and_dirty then
+ replicate ilive no_thm
+ else
+ map (fn goal => Skip_Proof.prove lthy [] [] goal
+ (fn {context, ...} => (mk_comp_set_alt_tac context (collect_set_natural_of_bnf outer))))
+ (map2 (curry (HOLogic.mk_Trueprop o HOLogic.mk_eq)) comp_sets comp_sets_alt);
+
+ fun comp_map_cong_tac _ =
+ mk_comp_map_cong_tac comp_set_alt_thms (map_cong_of_bnf outer) (map map_cong_of_bnf inners);
+
+ val comp_set_bd_tacs =
+ if ! quick_and_dirty then
+ replicate (length comp_set_alt_thms) (K all_tac)
+ else
+ let
+ val outer_set_bds = set_bd_of_bnf outer;
+ val inner_set_bdss = map set_bd_of_bnf inners;
+ val inner_bd_Card_orders = map bd_Card_order_of_bnf inners;
+ fun comp_single_set_bd_thm i j =
+ @{thm comp_single_set_bd} OF [nth inner_bd_Card_orders j, nth (nth inner_set_bdss j) i,
+ nth outer_set_bds j]
+ val single_set_bd_thmss =
+ map ((fn f => map f (0 upto olive - 1)) o comp_single_set_bd_thm) (0 upto ilive - 1);
+ in
+ map2 (fn comp_set_alt => fn single_set_bds => fn {context, ...} =>
+ mk_comp_set_bd_tac context comp_set_alt single_set_bds)
+ comp_set_alt_thms single_set_bd_thmss
+ end;
+
+ val comp_in_alt_thm =
+ if ! quick_and_dirty then
+ no_thm
+ else
+ let
+ val comp_in = mk_in Asets comp_sets CCA;
+ val comp_in_alt = mk_in (map2 (mk_in Asets) inner_setss CAs) outer_sets CCA;
+ val goal =
+ fold_rev Logic.all Asets
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (comp_in, comp_in_alt)));
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (fn {context, ...} => mk_comp_in_alt_tac context comp_set_alt_thms)
+ end;
+
+ fun comp_in_bd_tac _ =
+ mk_comp_in_bd_tac comp_in_alt_thm (map in_bd_of_bnf inners) (in_bd_of_bnf outer)
+ (map bd_Cinfinite_of_bnf inners) (bd_Card_order_of_bnf outer);
+
+ fun comp_map_wpull_tac _ =
+ mk_map_wpull_tac comp_in_alt_thm (map map_wpull_of_bnf inners) (map_wpull_of_bnf outer);
+
+ val tacs = [comp_map_id_tac, comp_map_comp_tac, comp_map_cong_tac] @ comp_set_natural_tacs @
+ [comp_bd_card_order_tac, comp_bd_cinfinite_tac] @ comp_set_bd_tacs @
+ [comp_in_bd_tac, comp_map_wpull_tac];
+
+ val outer_wits = mk_wits_of_bnf (replicate onwits oDs) (replicate onwits CAs) outer;
+
+ val inner_witss = map (map (fn (I, wit) => Term.list_comb (wit, map (nth xs) I)))
+ (map3 (fn Ds => fn n => mk_wits_of_bnf (replicate n Ds) (replicate n As))
+ Dss inwitss inners);
+
+ val inner_witsss = map (map (nth inner_witss) o fst) outer_wits;
+
+ val comp_wits = (inner_witsss, (map (single o snd) outer_wits))
+ |-> map2 (fold (map_product (fn iwit => fn owit => owit $ iwit)))
+ |> flat
+ |> map (`(fn t => Term.add_frees t []))
+ |> minimize_wits
+ |> map (fn (frees, t) => fold absfree frees t);
+
+ fun wit_tac {context = ctxt, ...} =
+ mk_comp_wit_tac ctxt (wit_thms_of_bnf outer) (collect_set_natural_of_bnf outer)
+ (maps wit_thms_of_bnf inners);
+
+ val (bnf', lthy') =
+ add_bnf const_policy (K Derive_Some_Facts) qualify tacs wit_tac (SOME (oDs @ flat Dss))
+ ((((b, comp_map), comp_sets), comp_bd), comp_wits) lthy;
+
+ val outer_rel_Gr = rel_Gr_of_bnf outer RS sym;
+ val outer_rel_cong = rel_cong_of_bnf outer;
+
+ val comp_rel_unfold_thm =
+ trans OF [rel_def_of_bnf bnf',
+ trans OF [comp_in_alt_thm RS @{thm subst_rel_def},
+ trans OF [@{thm arg_cong2[of _ _ _ _ relcomp]} OF
+ [trans OF [outer_rel_Gr RS @{thm arg_cong[of _ _ converse]},
+ rel_converse_of_bnf outer RS sym], outer_rel_Gr],
+ trans OF [rel_O_of_bnf outer RS sym, outer_rel_cong OF
+ (map (fn bnf => rel_def_of_bnf bnf RS sym) inners)]]]];
+
+ val comp_pred_unfold_thm = Collect_split_box_equals OF [comp_rel_unfold_thm,
+ pred_def_of_bnf bnf' RS abs_pred_sym,
+ trans OF [outer_rel_cong OF (map (fn bnf => pred_def_of_bnf bnf RS abs_pred_sym) inners),
+ pred_def_of_bnf outer RS abs_pred_sym]];
+
+ val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
+ comp_rel_unfold_thm comp_pred_unfold_thm unfold;
+ in
+ (bnf', (unfold', lthy'))
+ end;
+
+fun clean_compose_bnf_cmd (outer, inners) lthy =
+ let
+ val outer = the (bnf_of lthy outer)
+ val inners = map (the o bnf_of lthy) inners
+ val b = name_of_bnf outer
+ |> Binding.prefix_name compN
+ |> Binding.suffix_name ("_" ^ implode (map (Binding.name_of o name_of_bnf) inners));
+ in
+ (snd o snd) (clean_compose_bnf Dont_Inline I b outer inners
+ (empty_unfold, lthy))
+ end;
+
+(* Killing live variables *)
+
+fun killN_bnf qualify n bnf (unfold, lthy) = if n = 0 then (bnf, (unfold, lthy)) else
+ let
+ val b = Binding.prefix_name (mk_killN n) (name_of_bnf bnf);
+ val live = live_of_bnf bnf;
+ val dead = dead_of_bnf bnf;
+ val nwits = nwits_of_bnf bnf;
+
+ (* TODO: check 0 < n <= live *)
+
+ val (Ds, lthy1) = apfst (map TFree)
+ (Variable.invent_types (replicate dead HOLogic.typeS) lthy);
+ val ((killedAs, As), lthy2) = apfst (`(take n) o map TFree)
+ (Variable.invent_types (replicate live HOLogic.typeS) lthy1);
+ val (Bs, _(*lthy3*)) = apfst (append killedAs o map TFree)
+ (Variable.invent_types (replicate (live - n) HOLogic.typeS) lthy2);
+
+ val ((Asets, lives), _(*names_lthy*)) = lthy
+ |> mk_Frees "A" (map (HOLogic.mk_setT) (drop n As))
+ ||>> mk_Frees "x" (drop n As);
+ val xs = map (fn T => HOLogic.choice_const T $ absdummy T @{term True}) killedAs @ lives;
+
+ val T = mk_T_of_bnf Ds As bnf;
+
+ (*bnf.map id ... id*)
+ val killN_map = Term.list_comb (mk_map_of_bnf Ds As Bs bnf, map HOLogic.id_const killedAs);
+
+ val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;
+ val killN_sets = drop n bnf_sets;
+
+ (*(|UNIV :: A1 set| +c ... +c |UNIV :: An set|) *c bnf.bd*)
+ val bnf_bd = mk_bd_of_bnf Ds As bnf;
+ val killN_bd = mk_cprod
+ (Library.foldr1 (uncurry mk_csum) (map (mk_card_of o HOLogic.mk_UNIV) killedAs)) bnf_bd;
+
+ fun killN_map_id_tac _ = rtac (map_id_of_bnf bnf) 1;
+ fun killN_map_comp_tac {context, ...} =
+ Local_Defs.unfold_tac context ((map_comp_of_bnf bnf RS sym) :: @{thms o_assoc id_o o_id}) THEN
+ rtac refl 1;
+ fun killN_map_cong_tac {context, ...} =
+ mk_killN_map_cong_tac context n (live - n) (map_cong_of_bnf bnf);
+ val killN_set_natural_tacs =
+ map (fn thm => fn _ => rtac thm 1) (drop n (set_natural_of_bnf bnf));
+ fun killN_bd_card_order_tac _ = mk_killN_bd_card_order_tac n (bd_card_order_of_bnf bnf);
+ fun killN_bd_cinfinite_tac _ = mk_killN_bd_cinfinite_tac (bd_Cinfinite_of_bnf bnf);
+ val killN_set_bd_tacs =
+ map (fn thm => fn _ => mk_killN_set_bd_tac (bd_Card_order_of_bnf bnf) thm)
+ (drop n (set_bd_of_bnf bnf));
+
+ val killN_in_alt_thm =
+ if ! quick_and_dirty then
+ no_thm
+ else
+ let
+ val killN_in = mk_in Asets killN_sets T;
+ val killN_in_alt = mk_in (map HOLogic.mk_UNIV killedAs @ Asets) bnf_sets T;
+ val goal =
+ fold_rev Logic.all Asets (HOLogic.mk_Trueprop (HOLogic.mk_eq (killN_in, killN_in_alt)));
+ in
+ Skip_Proof.prove lthy [] [] goal (K killN_in_alt_tac)
+ end;
+
+ fun killN_in_bd_tac _ =
+ mk_killN_in_bd_tac n (live > n) killN_in_alt_thm (in_bd_of_bnf bnf)
+ (bd_Card_order_of_bnf bnf) (bd_Cinfinite_of_bnf bnf) (bd_Cnotzero_of_bnf bnf);
+ fun killN_map_wpull_tac _ =
+ mk_map_wpull_tac killN_in_alt_thm [] (map_wpull_of_bnf bnf);
+
+ val tacs = [killN_map_id_tac, killN_map_comp_tac, killN_map_cong_tac] @ killN_set_natural_tacs @
+ [killN_bd_card_order_tac, killN_bd_cinfinite_tac] @ killN_set_bd_tacs @
+ [killN_in_bd_tac, killN_map_wpull_tac];
+
+ val wits = mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf;
+
+ val killN_wits = map (fn t => fold absfree (Term.add_frees t []) t)
+ (map (fn (I, wit) => Term.list_comb (wit, map (nth xs) I)) wits);
+
+ fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);
+
+ val (bnf', lthy') =
+ add_bnf Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME (killedAs @ Ds))
+ ((((b, killN_map), killN_sets), Term.absdummy T killN_bd), killN_wits) lthy;
+
+ val rel_Gr = rel_Gr_of_bnf bnf RS sym;
+
+ val killN_rel_unfold_thm =
+ trans OF [rel_def_of_bnf bnf',
+ trans OF [killN_in_alt_thm RS @{thm subst_rel_def},
+ trans OF [@{thm arg_cong2[of _ _ _ _ relcomp]} OF
+ [trans OF [rel_Gr RS @{thm arg_cong[of _ _ converse]}, rel_converse_of_bnf bnf RS sym],
+ rel_Gr],
+ trans OF [rel_O_of_bnf bnf RS sym, rel_cong_of_bnf bnf OF
+ (replicate n @{thm trans[OF Gr_UNIV_id[OF refl] Id_alt[symmetric]]} @
+ replicate (live - n) @{thm Gr_fst_snd})]]]];
+
+ val killN_pred_unfold_thm = Collect_split_box_equals OF
+ [Local_Defs.unfold lthy' @{thms Id_def'} killN_rel_unfold_thm,
+ pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];
+
+ val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
+ killN_rel_unfold_thm killN_pred_unfold_thm unfold;
+ in
+ (bnf', (unfold', lthy'))
+ end;
+
+fun killN_bnf_cmd (n, raw_bnf) lthy =
+ (snd o snd) (killN_bnf I n (the (bnf_of lthy raw_bnf)) (empty_unfold, lthy));
+
+(* Adding dummy live variables *)
+
+fun liftN_bnf qualify n bnf (unfold, lthy) = if n = 0 then (bnf, (unfold, lthy)) else
+ let
+ val b = Binding.prefix_name (mk_liftN n) (name_of_bnf bnf);
+ val live = live_of_bnf bnf;
+ val dead = dead_of_bnf bnf;
+ val nwits = nwits_of_bnf bnf;
+
+ (* TODO: check 0 < n *)
+
+ val (Ds, lthy1) = apfst (map TFree)
+ (Variable.invent_types (replicate dead HOLogic.typeS) lthy);
+ val ((newAs, As), lthy2) = apfst (chop n o map TFree)
+ (Variable.invent_types (replicate (n + live) HOLogic.typeS) lthy1);
+ val ((newBs, Bs), _(*lthy3*)) = apfst (chop n o map TFree)
+ (Variable.invent_types (replicate (n + live) HOLogic.typeS) lthy2);
+
+ val (Asets, _(*names_lthy*)) = lthy
+ |> mk_Frees "A" (map (HOLogic.mk_setT) (newAs @ As));
+
+ val T = mk_T_of_bnf Ds As bnf;
+
+ (*%f1 ... fn. bnf.map*)
+ val liftN_map =
+ fold_rev Term.absdummy (map2 (curry (op -->)) newAs newBs) (mk_map_of_bnf Ds As Bs bnf);
+
+ val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;
+ val liftN_sets = map (fn A => absdummy T (HOLogic.mk_set A [])) newAs @ bnf_sets;
+
+ val liftN_bd = mk_bd_of_bnf Ds As bnf;
+
+ fun liftN_map_id_tac _ = rtac (map_id_of_bnf bnf) 1;
+ fun liftN_map_comp_tac {context, ...} =
+ Local_Defs.unfold_tac context ((map_comp_of_bnf bnf RS sym) :: @{thms o_assoc id_o o_id}) THEN
+ rtac refl 1;
+ fun liftN_map_cong_tac {context, ...} =
+ rtac (map_cong_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac context 1);
+ val liftN_set_natural_tacs =
+ if ! quick_and_dirty then
+ replicate (n + live) (K all_tac)
+ else
+ replicate n (K empty_natural_tac) @
+ map (fn thm => fn _ => rtac thm 1) (set_natural_of_bnf bnf);
+ fun liftN_bd_card_order_tac _ = rtac (bd_card_order_of_bnf bnf) 1;
+ fun liftN_bd_cinfinite_tac _ = rtac (bd_cinfinite_of_bnf bnf) 1;
+ val liftN_set_bd_tacs =
+ if ! quick_and_dirty then
+ replicate (n + live) (K all_tac)
+ else
+ replicate n (K (mk_liftN_set_bd_tac (bd_Card_order_of_bnf bnf))) @
+ (map (fn thm => fn _ => rtac thm 1) (set_bd_of_bnf bnf));
+
+ val liftN_in_alt_thm =
+ if ! quick_and_dirty then
+ no_thm
+ else
+ let
+ val liftN_in = mk_in Asets liftN_sets T;
+ val liftN_in_alt = mk_in (drop n Asets) bnf_sets T;
+ val goal =
+ fold_rev Logic.all Asets (HOLogic.mk_Trueprop (HOLogic.mk_eq (liftN_in, liftN_in_alt)))
+ in
+ Skip_Proof.prove lthy [] [] goal (K liftN_in_alt_tac)
+ end;
+
+ fun liftN_in_bd_tac _ =
+ mk_liftN_in_bd_tac n liftN_in_alt_thm (in_bd_of_bnf bnf) (bd_Card_order_of_bnf bnf);
+ fun liftN_map_wpull_tac _ =
+ mk_map_wpull_tac liftN_in_alt_thm [] (map_wpull_of_bnf bnf);
+
+ val tacs = [liftN_map_id_tac, liftN_map_comp_tac, liftN_map_cong_tac] @ liftN_set_natural_tacs @
+ [liftN_bd_card_order_tac, liftN_bd_cinfinite_tac] @ liftN_set_bd_tacs @
+ [liftN_in_bd_tac, liftN_map_wpull_tac];
+
+ val liftN_wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
+
+ fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);
+
+ val (bnf', lthy') =
+ add_bnf Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME Ds)
+ ((((b, liftN_map), liftN_sets), Term.absdummy T liftN_bd), liftN_wits) lthy;
+
+ val liftN_rel_unfold_thm =
+ trans OF [rel_def_of_bnf bnf',
+ trans OF [liftN_in_alt_thm RS @{thm subst_rel_def}, rel_def_of_bnf bnf RS sym]];
+
+ val liftN_pred_unfold_thm = Collect_split_box_equals OF [liftN_rel_unfold_thm,
+ pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];
+
+ val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
+ liftN_rel_unfold_thm liftN_pred_unfold_thm unfold;
+ in
+ (bnf', (unfold', lthy'))
+ end;
+
+fun liftN_bnf_cmd (n, raw_bnf) lthy =
+ (snd o snd) (liftN_bnf I n (the (bnf_of lthy raw_bnf)) (empty_unfold, lthy));
+
+(* Changing the order of live variables *)
+
+fun permute_bnf qualify src dest bnf (unfold, lthy) = if src = dest then (bnf, (unfold, lthy)) else
+ let
+ val b = Binding.prefix_name (mk_permuteN src dest) (name_of_bnf bnf);
+ val live = live_of_bnf bnf;
+ val dead = dead_of_bnf bnf;
+ val nwits = nwits_of_bnf bnf;
+ fun permute xs = mk_permute src dest xs;
+ fun permute_rev xs = mk_permute dest src xs;
+
+ val (Ds, lthy1) = apfst (map TFree)
+ (Variable.invent_types (replicate dead HOLogic.typeS) lthy);
+ val (As, lthy2) = apfst (map TFree)
+ (Variable.invent_types (replicate live HOLogic.typeS) lthy1);
+ val (Bs, _(*lthy3*)) = apfst (map TFree)
+ (Variable.invent_types (replicate live HOLogic.typeS) lthy2);
+
+ val (Asets, _(*names_lthy*)) = lthy
+ |> mk_Frees "A" (map (HOLogic.mk_setT) (permute As));
+
+ val T = mk_T_of_bnf Ds As bnf;
+
+ (*%f(1) ... f(n). bnf.map f\<sigma>(1) ... f\<sigma>(n)*)
+ val permute_map = fold_rev Term.absdummy (permute (map2 (curry op -->) As Bs))
+ (Term.list_comb (mk_map_of_bnf Ds As Bs bnf,
+ permute_rev (map Bound ((live - 1) downto 0))));
+
+ val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;
+ val permute_sets = permute bnf_sets;
+
+ val permute_bd = mk_bd_of_bnf Ds As bnf;
+
+ fun permute_map_id_tac _ = rtac (map_id_of_bnf bnf) 1;
+ fun permute_map_comp_tac _ = rtac (map_comp_of_bnf bnf) 1;
+ fun permute_map_cong_tac {context, ...} =
+ rtac (map_cong_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac context 1);
+ val permute_set_natural_tacs =
+ permute (map (fn thm => fn _ => rtac thm 1) (set_natural_of_bnf bnf));
+ fun permute_bd_card_order_tac _ = rtac (bd_card_order_of_bnf bnf) 1;
+ fun permute_bd_cinfinite_tac _ = rtac (bd_cinfinite_of_bnf bnf) 1;
+ val permute_set_bd_tacs = permute (map (fn thm => fn _ => rtac thm 1) (set_bd_of_bnf bnf));
+
+ val permute_in_alt_thm =
+ if ! quick_and_dirty then
+ no_thm
+ else
+ let
+ val permute_in = mk_in Asets permute_sets T;
+ val permute_in_alt = mk_in (permute_rev Asets) bnf_sets T;
+ val goal =
+ fold_rev Logic.all Asets
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (permute_in, permute_in_alt)));
+ in
+ Skip_Proof.prove lthy [] [] goal (K (mk_permute_in_alt_tac src dest))
+ end;
+
+ fun permute_in_bd_tac _ =
+ mk_permute_in_bd_tac src dest permute_in_alt_thm (in_bd_of_bnf bnf)
+ (bd_Card_order_of_bnf bnf);
+ fun permute_map_wpull_tac _ =
+ mk_map_wpull_tac permute_in_alt_thm [] (map_wpull_of_bnf bnf);
+
+ val tacs = [permute_map_id_tac, permute_map_comp_tac, permute_map_cong_tac] @
+ permute_set_natural_tacs @ [permute_bd_card_order_tac, permute_bd_cinfinite_tac] @
+ permute_set_bd_tacs @ [permute_in_bd_tac, permute_map_wpull_tac];
+
+ val permute_wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
+
+ fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);
+
+ val (bnf', lthy') =
+ add_bnf Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME Ds)
+ ((((b, permute_map), permute_sets), Term.absdummy T permute_bd), permute_wits) lthy;
+
+ val permute_rel_unfold_thm =
+ trans OF [rel_def_of_bnf bnf',
+ trans OF [permute_in_alt_thm RS @{thm subst_rel_def}, rel_def_of_bnf bnf RS sym]];
+
+ val permute_pred_unfold_thm = Collect_split_box_equals OF [permute_rel_unfold_thm,
+ pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];
+
+ val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
+ permute_rel_unfold_thm permute_pred_unfold_thm unfold;
+ in
+ (bnf', (unfold', lthy'))
+ end;
+
+fun permute_bnf_cmd ((src, dest), raw_bnf) lthy =
+ (snd o snd) (permute_bnf I src dest (the (bnf_of lthy raw_bnf))
+ (empty_unfold, lthy));
+
+(* Hide the type of the bound (optimization) and unfold the definitions (nicer to the user) *)
+
+fun seal_bnf unfold b Ds bnf lthy =
+ let
+ val live = live_of_bnf bnf;
+ val nwits = nwits_of_bnf bnf;
+
+ val (As, lthy1) = apfst (map TFree)
+ (Variable.invent_types (replicate live HOLogic.typeS) (fold Variable.declare_typ Ds lthy));
+ val (Bs, _) = apfst (map TFree)
+ (Variable.invent_types (replicate live HOLogic.typeS) lthy1);
+
+ val map_unfolds = filter_refl (map_unfolds_of unfold);
+ val set_unfoldss = map filter_refl (set_unfoldss_of unfold);
+
+ val expand_maps = fold expand_term_const (map (single o Logic.dest_equals o Thm.prop_of)
+ map_unfolds);
+ val expand_sets = fold expand_term_const (map (map (Logic.dest_equals o Thm.prop_of))
+ set_unfoldss);
+ val unfold_maps = fold (Local_Defs.unfold lthy o single) map_unfolds;
+ val unfold_sets = fold (Local_Defs.unfold lthy) set_unfoldss;
+ val unfold_defs = unfold_sets o unfold_maps;
+ val bnf_map = expand_maps (mk_map_of_bnf Ds As Bs bnf);
+ val bnf_sets = map (expand_maps o expand_sets)
+ (mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf);
+ val bnf_bd = mk_bd_of_bnf Ds As bnf;
+ val T = mk_T_of_bnf Ds As bnf;
+
+ (*bd should only depend on dead type variables!*)
+ val bd_repT = fst (dest_relT (fastype_of bnf_bd));
+ val bdT_bind = Binding.suffix_name ("_" ^ bdTN) b;
+ val params = fold Term.add_tfreesT Ds [];
+
+ val ((bdT_name, (bdT_glob_info, bdT_loc_info)), (lthy', lthy)) =
+ lthy
+ |> Typedef.add_typedef true NONE (bdT_bind, params, NoSyn)
+ (HOLogic.mk_UNIV bd_repT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1)
+ ||> `Local_Theory.restore;
+
+ val phi = Proof_Context.export_morphism lthy lthy';
+
+ val bnf_bd' = mk_dir_image bnf_bd
+ (Const (#Abs_name bdT_glob_info, bd_repT --> Type (bdT_name, map TFree params)))
+
+ val set_def = Morphism.thm phi (the (#set_def bdT_loc_info));
+ val Abs_inject = Morphism.thm phi (#Abs_inject bdT_loc_info);
+ val Abs_cases = Morphism.thm phi (#Abs_cases bdT_loc_info);
+
+ val Abs_bdT_inj = mk_Abs_inj_thm set_def Abs_inject;
+ val Abs_bdT_bij = mk_Abs_bij_thm lthy' set_def Abs_inject Abs_cases;
+
+ val bd_ordIso = @{thm dir_image} OF [Abs_bdT_inj, bd_Card_order_of_bnf bnf];
+ val bd_card_order =
+ @{thm card_order_dir_image} OF [Abs_bdT_bij, bd_card_order_of_bnf bnf];
+ val bd_cinfinite =
+ (@{thm Cinfinite_cong} OF [bd_ordIso, bd_Cinfinite_of_bnf bnf]) RS conjunct1;
+
+ val set_bds =
+ map (fn thm => @{thm ordLeq_ordIso_trans} OF [thm, bd_ordIso]) (set_bd_of_bnf bnf);
+ val in_bd =
+ @{thm ordLeq_ordIso_trans} OF [in_bd_of_bnf bnf,
+ @{thm cexp_cong2_Cnotzero} OF [bd_ordIso, if live = 0 then
+ @{thm ctwo_Cnotzero} else @{thm ctwo_Cnotzero} RS @{thm csum_Cnotzero2},
+ bd_Card_order_of_bnf bnf]];
+
+ fun mk_tac thm {context = ctxt, prems = _} = (rtac (unfold_defs thm) THEN'
+ SOLVE o REPEAT_DETERM o (atac ORELSE' Goal.assume_rule_tac ctxt)) 1;
+ val tacs =
+ map mk_tac ([map_id_of_bnf bnf, map_comp_of_bnf bnf, map_cong_of_bnf bnf] @
+ set_natural_of_bnf bnf) @
+ map K [rtac bd_card_order 1, rtac bd_cinfinite 1] @
+ map mk_tac (set_bds @ [in_bd, map_wpull_of_bnf bnf]);
+
+ val bnf_wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
+
+ fun wit_tac _ = mk_simple_wit_tac (map unfold_defs (wit_thms_of_bnf bnf));
+
+ val (bnf', lthy') = add_bnf Hardly_Inline (K Derive_All_Facts) I tacs wit_tac NONE
+ ((((b, bnf_map), bnf_sets), Term.absdummy T bnf_bd'), bnf_wits) lthy;
+
+ val defs' = filter_refl (map_def_of_bnf bnf' :: set_defs_of_bnf bnf');
+ val unfold_defs' = unfold_defs o Local_Defs.unfold lthy' defs';
+
+ val rel_def = unfold_defs' (rel_def_of_bnf bnf');
+ val rel_unfold = Local_Defs.unfold lthy'
+ (map unfold_defs (filter_refl (rel_unfolds_of unfold))) rel_def;
+
+ val unfold_defs'' =
+ unfold_defs' o (Local_Defs.unfold lthy' (filter_refl [rel_def_of_bnf bnf']));
+
+ val pred_def = unfold_defs'' (pred_def_of_bnf bnf' RS abs_pred_sym_pred_abs);
+ val pred_unfold = Local_Defs.unfold lthy'
+ (map unfold_defs (filter_refl (pred_unfolds_of unfold))) pred_def;
+
+ fun note thmN thms =
+ snd o Local_Theory.note
+ ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms);
+ in
+ (bnf', lthy'
+ |> note rel_unfoldN [rel_unfold]
+ |> note pred_unfoldN [pred_unfold])
+ end;
+
+(* Composition pipeline *)
+
+fun permute_and_kill qualify n src dest bnf =
+ bnf
+ |> permute_bnf qualify src dest
+ #> uncurry (killN_bnf qualify n);
+
+fun lift_and_permute qualify n src dest bnf =
+ bnf
+ |> liftN_bnf qualify n
+ #> uncurry (permute_bnf qualify src dest);
+
+fun normalize_bnfs qualify Ass Ds sort bnfs unfold lthy =
+ let
+ val before_kill_src = map (fn As => 0 upto (length As - 1)) Ass;
+ val kill_poss = map (find_indices Ds) Ass;
+ val live_poss = map2 (subtract (op =)) kill_poss before_kill_src;
+ val before_kill_dest = map2 append kill_poss live_poss;
+ val kill_ns = map length kill_poss;
+ val (inners', (unfold', lthy')) =
+ fold_map5 (fn i => permute_and_kill (qualify i))
+ (if length bnfs = 1 then [0] else (1 upto length bnfs))
+ kill_ns before_kill_src before_kill_dest bnfs (unfold, lthy);
+
+ val Ass' = map2 (map o nth) Ass live_poss;
+ val As = sort Ass';
+ val after_lift_dest = replicate (length Ass') (0 upto (length As - 1));
+ val old_poss = map (map (fn x => find_index (fn y => x = y) As)) Ass';
+ val new_poss = map2 (subtract (op =)) old_poss after_lift_dest;
+ val after_lift_src = map2 append new_poss old_poss;
+ val lift_ns = map (fn xs => length As - length xs) Ass';
+ in
+ ((kill_poss, As), fold_map5 (fn i => lift_and_permute (qualify i))
+ (if length bnfs = 1 then [0] else (1 upto length bnfs))
+ lift_ns after_lift_src after_lift_dest inners' (unfold', lthy'))
+ end;
+
+fun default_comp_sort Ass =
+ Library.sort (Term_Ord.typ_ord o pairself TFree) (fold (fold (insert (op =))) Ass []);
+
+fun compose_bnf const_policy qualify' b sort outer inners oDs Dss tfreess (unfold, lthy) =
+ let
+ val name = Binding.name_of b;
+ fun qualify i bind =
+ let val namei = if i > 0 then name ^ string_of_int i else name;
+ in
+ if member (op =) (#2 (Binding.dest bind)) (namei, true) then qualify' bind
+ else qualify' (Binding.prefix_name namei bind)
+ end;
+
+ val Ass = map (map dest_TFree) tfreess;
+ val Ds = fold (fold Term.add_tfreesT) (oDs :: Dss) [];
+
+ val ((kill_poss, As), (inners', (unfold', lthy'))) =
+ normalize_bnfs qualify Ass Ds sort inners unfold lthy;
+
+ val Ds = oDs @ flat (map3 (append oo map o nth) tfreess kill_poss Dss);
+ val As = map TFree As;
+ in
+ apfst (rpair (Ds, As)) (clean_compose_bnf const_policy I b outer inners' (unfold', lthy'))
+ end;
+
+fun compose_bnf_cmd (((((b, outer), inners), oDs), Dss), Ass) lthy = (snd o snd)
+ (compose_bnf Dont_Inline I b default_comp_sort (the (bnf_of lthy outer))
+ (map (the o bnf_of lthy) inners)
+ (map (Syntax.read_typ lthy) oDs) (map (map (Syntax.read_typ lthy)) Dss)
+ (map (map (Syntax.read_typ lthy)) Ass) (empty_unfold, lthy));
+
+fun bnf_of_typ _ _ _ _ (T as TFree _) (unfold, lthy) =
+ ((Basic_BNFs.ID_bnf, ([], [T])), (add_to_unfold_opt NONE NONE
+ (SOME Basic_BNFs.ID_rel_def) (SOME Basic_BNFs.ID_pred_def) unfold, lthy))
+ | bnf_of_typ _ _ _ _ (TVar _) _ = error "Unexpected schematic variable"
+ | bnf_of_typ const_policy b qualify' sort (T as Type (C, Ts)) (unfold, lthy) =
+ let val tfrees = Term.add_tfreesT T [];
+ in
+ if null tfrees then
+ ((Basic_BNFs.DEADID_bnf, ([T], [])), (unfold, lthy))
+ else if forall (can Term.dest_TFree) Ts andalso length Ts = length tfrees then let
+ val bnf = the (bnf_of lthy (Long_Name.base_name C));
+ val T' = T_of_bnf bnf;
+ val deads = deads_of_bnf bnf;
+ val lives = lives_of_bnf bnf;
+ val tvars' = Term.add_tvarsT T' [];
+ val deads_lives =
+ pairself (map (Term.typ_subst_TVars (map fst tvars' ~~ map TFree tfrees)))
+ (deads, lives);
+ val rel_def = rel_def_of_bnf bnf;
+ val unfold' = add_to_unfold_opt NONE NONE (SOME (rel_def RS sym))
+ (SOME (Local_Defs.unfold lthy [rel_def] (pred_def_of_bnf bnf) RS sym)) unfold;
+ in ((bnf, deads_lives), (unfold', lthy)) end
+ else
+ let
+ (* FIXME: we should allow several BNFs with the same base name *)
+ val Tname = Long_Name.base_name C;
+ val name = Binding.name_of b ^ "_" ^ Tname;
+ fun qualify i bind =
+ let val namei = if i > 0 then name ^ string_of_int i else name;
+ in
+ if member (op =) (#2 (Binding.dest bind)) (namei, true) then qualify' bind
+ else qualify' (Binding.prefix_name namei bind)
+ end;
+ val outer = the (bnf_of lthy Tname);
+ val odead = dead_of_bnf outer;
+ val olive = live_of_bnf outer;
+ val oDs_pos = find_indices [TFree ("dead", [])]
+ (snd (dest_Type
+ (mk_T_of_bnf (replicate odead (TFree ("dead", []))) (replicate olive dummyT) outer)));
+ val oDs = map (nth Ts) oDs_pos;
+ val Ts' = map (nth Ts) (subtract (op =) oDs_pos (0 upto length Ts - 1));
+ val ((inners, (Dss, Ass)), (unfold', lthy')) =
+ apfst (apsnd split_list o split_list)
+ (fold_map2 (fn i =>
+ bnf_of_typ Smart_Inline (Binding.name (name ^ string_of_int i)) (qualify i) sort)
+ (if length Ts' = 1 then [0] else (1 upto length Ts'))
+ Ts' (unfold, lthy));
+ in
+ compose_bnf const_policy (qualify 0) b sort outer inners oDs Dss Ass (unfold', lthy')
+ end
+ end;
+
+fun bnf_of_typ_cmd (b, rawT) lthy = (snd o snd)
+ (bnf_of_typ Dont_Inline b I default_comp_sort (Syntax.read_typ lthy rawT)
+ (empty_unfold, lthy));
+
+val _ =
+ Outer_Syntax.local_theory @{command_spec "bnf_of_typ"} "parse a type as composition of BNFs"
+ (((Parse.binding --| Parse.$$$ "=") -- Parse.typ) >> bnf_of_typ_cmd);
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_comp_tactics.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,394 @@
+(* Title: HOL/Codatatype/Tools/bnf_comp_tactics.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+Tactics for composition of bounded natural functors.
+*)
+
+signature BNF_COMP_TACTICS =
+sig
+ val mk_comp_bd_card_order_tac: thm list -> thm -> tactic
+ val mk_comp_bd_cinfinite_tac: thm -> thm -> tactic
+ val mk_comp_in_alt_tac: Proof.context -> thm list -> tactic
+ val mk_comp_in_bd_tac: thm -> thm list -> thm -> thm list -> thm -> tactic
+ val mk_comp_map_comp_tac: thm -> thm -> thm list -> tactic
+ val mk_comp_map_cong_tac: thm list -> thm -> thm list -> tactic
+ val mk_comp_set_alt_tac: Proof.context -> thm -> tactic
+ val mk_comp_set_bd_tac: Proof.context -> thm -> thm list -> tactic
+ val mk_comp_set_natural_tac: thm -> thm -> thm -> thm list -> tactic
+ val mk_comp_wit_tac: Proof.context -> thm list -> thm -> thm list -> tactic
+
+ val mk_killN_bd_card_order_tac: int -> thm -> tactic
+ val mk_killN_bd_cinfinite_tac: thm -> tactic
+ val killN_in_alt_tac: tactic
+ val mk_killN_in_bd_tac: int -> bool -> thm -> thm -> thm -> thm -> thm -> tactic
+ val mk_killN_map_cong_tac: Proof.context -> int -> int -> thm -> tactic
+ val mk_killN_set_bd_tac: thm -> thm -> tactic
+
+ val empty_natural_tac: tactic
+ val liftN_in_alt_tac: tactic
+ val mk_liftN_in_bd_tac: int -> thm -> thm -> thm -> tactic
+ val mk_liftN_set_bd_tac: thm -> tactic
+
+ val mk_permute_in_alt_tac: ''a list -> ''a list -> tactic
+ val mk_permute_in_bd_tac: ''a list -> ''a list -> thm -> thm -> thm -> tactic
+end;
+
+structure BNF_Comp_Tactics : BNF_COMP_TACTICS =
+struct
+
+open BNF_Util
+open BNF_Tactics
+
+val arg_cong_Union = @{thm arg_cong[of _ _ Union]};
+val o_eq_dest_lhs = @{thm o_eq_dest_lhs};
+val set_mp = @{thm set_mp};
+val trans_image_cong_o_apply = @{thm trans[OF image_cong[OF o_apply refl]]};
+val trans_o_apply = @{thm trans[OF o_apply]};
+
+
+
+(* Composition *)
+
+fun mk_comp_set_alt_tac ctxt collect_set_natural =
+ Local_Defs.unfold_tac ctxt @{thms sym[OF o_assoc]} THEN
+ Local_Defs.unfold_tac ctxt [collect_set_natural RS sym] THEN
+ rtac refl 1;
+
+fun mk_comp_map_comp_tac Gmap_comp Gmap_cong map_comps =
+ EVERY' ([rtac ext, rtac sym, rtac trans_o_apply,
+ rtac (Gmap_comp RS sym RS o_eq_dest_lhs RS trans), rtac Gmap_cong] @
+ map (fn thm => rtac (thm RS sym RS fun_cong)) map_comps) 1;
+
+fun mk_comp_set_natural_tac Gmap_comp Gmap_cong Gset_natural set_naturals =
+ EVERY' ([rtac ext] @
+ replicate 3 (rtac trans_o_apply) @
+ [rtac (arg_cong_Union RS trans),
+ rtac (@{thm arg_cong2[of _ _ _ _ collect, OF refl]} RS trans),
+ rtac (Gmap_comp RS sym RS o_eq_dest_lhs RS trans),
+ rtac Gmap_cong] @
+ map (fn thm => rtac (thm RS fun_cong)) set_naturals @
+ [rtac (Gset_natural RS o_eq_dest_lhs), rtac sym, rtac trans_o_apply,
+ rtac trans_image_cong_o_apply, rtac trans_image_cong_o_apply,
+ rtac (@{thm image_cong} OF [Gset_natural RS o_eq_dest_lhs RS arg_cong_Union, refl] RS trans),
+ rtac @{thm trans[OF pointfreeE[OF Union_natural[symmetric]]]}, rtac arg_cong_Union,
+ rtac @{thm trans[OF o_eq_dest_lhs[OF image_o_collect[symmetric]]]},
+ rtac @{thm fun_cong[OF arg_cong[of _ _ collect]]}] @
+ [REPEAT_DETERM_N (length set_naturals) o EVERY' [rtac @{thm trans[OF image_insert]},
+ rtac @{thm arg_cong2[of _ _ _ _ insert]}, rtac ext, rtac trans_o_apply,
+ rtac @{thm trans[OF image_cong[OF o_apply refl]]}, rtac @{thm trans[OF image_image]},
+ rtac @{thm sym[OF trans[OF o_apply]]}, rtac @{thm image_cong[OF refl o_apply]}],
+ rtac @{thm image_empty}]) 1;
+
+fun mk_comp_map_cong_tac comp_set_alts map_cong map_congs =
+ let
+ val n = length comp_set_alts;
+ in
+ (if n = 0 then rtac refl 1
+ else rtac map_cong 1 THEN
+ EVERY' (map_index (fn (i, map_cong) =>
+ rtac map_cong THEN' EVERY' (map_index (fn (k, set_alt) =>
+ EVERY' [select_prem_tac n (dtac @{thm meta_spec}) (k + 1), etac @{thm meta_mp},
+ rtac (equalityD2 RS set_mp), rtac (set_alt RS fun_cong RS trans),
+ rtac trans_o_apply, rtac (@{thm collect_def} RS arg_cong_Union),
+ rtac @{thm UnionI}, rtac @{thm UN_I}, REPEAT_DETERM_N i o rtac @{thm insertI2},
+ rtac @{thm insertI1}, rtac (o_apply RS equalityD2 RS set_mp),
+ etac @{thm imageI}, atac])
+ comp_set_alts))
+ map_congs) 1)
+ end;
+
+fun mk_comp_bd_card_order_tac Fbd_card_orders Gbd_card_order =
+ let
+ val (card_orders, last_card_order) = split_last Fbd_card_orders;
+ fun gen_before thm = rtac @{thm card_order_csum} THEN' rtac thm;
+ in
+ (rtac @{thm card_order_cprod} THEN'
+ WRAP' gen_before (K (K all_tac)) card_orders (rtac last_card_order) THEN'
+ rtac Gbd_card_order) 1
+ end;
+
+fun mk_comp_bd_cinfinite_tac Fbd_cinfinite Gbd_cinfinite =
+ (rtac @{thm cinfinite_cprod} THEN'
+ ((K (TRY ((rtac @{thm cinfinite_csum} THEN' rtac disjI1) 1)) THEN'
+ ((rtac @{thm cinfinite_csum} THEN' rtac disjI1 THEN' rtac Fbd_cinfinite) ORELSE'
+ rtac Fbd_cinfinite)) ORELSE'
+ rtac Fbd_cinfinite) THEN'
+ rtac Gbd_cinfinite) 1;
+
+fun mk_comp_set_bd_tac ctxt comp_set_alt Gset_Fset_bds =
+ let
+ val (bds, last_bd) = split_last Gset_Fset_bds;
+ fun gen_before bd =
+ rtac ctrans THEN' rtac @{thm Un_csum} THEN'
+ rtac ctrans THEN' rtac @{thm csum_mono} THEN'
+ rtac bd;
+ fun gen_after _ = rtac @{thm ordIso_imp_ordLeq} THEN' rtac @{thm cprod_csum_distrib1};
+ in
+ Local_Defs.unfold_tac ctxt [comp_set_alt] THEN
+ rtac @{thm comp_set_bd_Union_o_collect} 1 THEN
+ Local_Defs.unfold_tac ctxt @{thms Union_image_insert Union_image_empty Union_Un_distrib
+ o_apply} THEN
+ (rtac ctrans THEN'
+ WRAP' gen_before gen_after bds (rtac last_bd) THEN'
+ rtac @{thm ordIso_imp_ordLeq} THEN'
+ rtac @{thm cprod_com}) 1
+ end;
+
+val comp_in_alt_thms = @{thms o_apply collect_def SUP_def image_insert image_empty Union_insert
+ Union_empty Un_empty_right Union_Un_distrib Un_subset_iff conj_subset_def UN_image_subset
+ conj_assoc};
+
+fun mk_comp_in_alt_tac ctxt comp_set_alts =
+ Local_Defs.unfold_tac ctxt (comp_set_alts @ comp_in_alt_thms) THEN
+ Local_Defs.unfold_tac ctxt @{thms set_eq_subset} THEN
+ rtac conjI 1 THEN
+ REPEAT_DETERM (
+ rtac @{thm subsetI} 1 THEN
+ Local_Defs.unfold_tac ctxt @{thms mem_Collect_eq Ball_def} THEN
+ (REPEAT_DETERM (CHANGED (etac conjE 1)) THEN
+ REPEAT_DETERM (CHANGED ((
+ (rtac conjI THEN' (atac ORELSE' rtac @{thm subset_UNIV})) ORELSE'
+ atac ORELSE'
+ (rtac @{thm subset_UNIV})) 1)) ORELSE rtac @{thm subset_UNIV} 1));
+
+fun mk_comp_in_bd_tac comp_in_alt Fin_bds Gin_bd Fbd_Cinfs Gbd_Card_order =
+ let
+ val (bds, last_bd) = split_last Fin_bds;
+ val (Cinfs, _) = split_last Fbd_Cinfs;
+ fun gen_before (bd, _) = rtac ctrans THEN' rtac @{thm csum_mono} THEN' rtac bd;
+ fun gen_after (_, (bd_Cinf, next_bd_Cinf)) =
+ TRY o (rtac @{thm csum_cexp} THEN'
+ rtac bd_Cinf THEN'
+ (TRY o (rtac @{thm Cinfinite_csum} THEN' rtac disjI1) THEN' rtac next_bd_Cinf ORELSE'
+ rtac next_bd_Cinf) THEN'
+ ((rtac @{thm Card_order_csum} THEN' rtac @{thm ordLeq_csum2}) ORELSE'
+ (rtac @{thm Card_order_ctwo} THEN' rtac @{thm ordLeq_refl})) THEN'
+ rtac @{thm Card_order_ctwo});
+ in
+ (rtac @{thm ordIso_ordLeq_trans} THEN'
+ rtac @{thm card_of_ordIso_subst} THEN'
+ rtac comp_in_alt THEN'
+ rtac ctrans THEN'
+ rtac Gin_bd THEN'
+ rtac @{thm ordLeq_ordIso_trans} THEN'
+ rtac @{thm cexp_mono1} THEN'
+ rtac @{thm ordLeq_ordIso_trans} THEN'
+ rtac @{thm csum_mono1} THEN'
+ WRAP' gen_before gen_after (bds ~~ (Cinfs ~~ tl Fbd_Cinfs)) (rtac last_bd) THEN'
+ rtac @{thm csum_absorb1} THEN'
+ rtac @{thm Cinfinite_cexp} THEN'
+ (rtac @{thm ordLeq_csum2} ORELSE' rtac @{thm ordLeq_refl}) THEN'
+ rtac @{thm Card_order_ctwo} THEN'
+ (TRY o (rtac @{thm Cinfinite_csum} THEN' rtac disjI1) THEN' rtac (hd Fbd_Cinfs) ORELSE'
+ rtac (hd Fbd_Cinfs)) THEN'
+ rtac @{thm ctwo_ordLeq_Cinfinite} THEN'
+ rtac @{thm Cinfinite_cexp} THEN'
+ (rtac @{thm ordLeq_csum2} ORELSE' rtac @{thm ordLeq_refl}) THEN'
+ rtac @{thm Card_order_ctwo} THEN'
+ (TRY o (rtac @{thm Cinfinite_csum} THEN' rtac disjI1) THEN' rtac (hd Fbd_Cinfs) ORELSE'
+ rtac (hd Fbd_Cinfs)) THEN'
+ rtac disjI1 THEN'
+ TRY o rtac @{thm csum_Cnotzero2} THEN'
+ rtac @{thm ctwo_Cnotzero} THEN'
+ rtac Gbd_Card_order THEN'
+ rtac @{thm cexp_cprod} THEN'
+ TRY o rtac @{thm csum_Cnotzero2} THEN'
+ rtac @{thm ctwo_Cnotzero}) 1
+ end;
+
+val comp_wit_thms = @{thms Union_empty_conv o_apply collect_def SUP_def
+ Union_image_insert Union_image_empty};
+
+fun mk_comp_wit_tac ctxt Gwit_thms collect_set_natural Fwit_thms =
+ ALLGOALS (dtac @{thm in_Union_o_assoc}) THEN
+ Local_Defs.unfold_tac ctxt (collect_set_natural :: comp_wit_thms) THEN
+ REPEAT_DETERM (
+ atac 1 ORELSE
+ REPEAT_DETERM (eresolve_tac @{thms UnionE UnE imageE} 1) THEN
+ (TRY o dresolve_tac Gwit_thms THEN'
+ (etac FalseE ORELSE'
+ hyp_subst_tac THEN'
+ dresolve_tac Fwit_thms THEN'
+ (etac FalseE ORELSE' atac))) 1);
+
+
+
+(* Kill operation *)
+
+fun mk_killN_map_cong_tac ctxt n m map_cong =
+ (rtac map_cong THEN' EVERY' (replicate n (rtac refl)) THEN'
+ EVERY' (replicate m (Goal.assume_rule_tac ctxt))) 1;
+
+fun mk_killN_bd_card_order_tac n bd_card_order =
+ (rtac @{thm card_order_cprod} THEN'
+ K (REPEAT_DETERM_N (n - 1)
+ ((rtac @{thm card_order_csum} THEN'
+ rtac @{thm card_of_card_order_on}) 1)) THEN'
+ rtac @{thm card_of_card_order_on} THEN'
+ rtac bd_card_order) 1;
+
+fun mk_killN_bd_cinfinite_tac bd_Cinfinite =
+ (rtac @{thm cinfinite_cprod2} THEN'
+ TRY o rtac @{thm csum_Cnotzero1} THEN'
+ rtac @{thm Cnotzero_UNIV} THEN'
+ rtac bd_Cinfinite) 1;
+
+fun mk_killN_set_bd_tac bd_Card_order set_bd =
+ (rtac ctrans THEN'
+ rtac set_bd THEN'
+ rtac @{thm ordLeq_cprod2} THEN'
+ TRY o rtac @{thm csum_Cnotzero1} THEN'
+ rtac @{thm Cnotzero_UNIV} THEN'
+ rtac bd_Card_order) 1
+
+val killN_in_alt_tac =
+ ((rtac @{thm Collect_cong} THEN' rtac @{thm iffI}) 1 THEN
+ REPEAT_DETERM (CHANGED (etac conjE 1)) THEN
+ REPEAT_DETERM (CHANGED ((etac conjI ORELSE'
+ rtac conjI THEN' rtac @{thm subset_UNIV}) 1)) THEN
+ (rtac @{thm subset_UNIV} ORELSE' atac) 1 THEN
+ REPEAT_DETERM (CHANGED (etac conjE 1)) THEN
+ REPEAT_DETERM (CHANGED ((etac conjI ORELSE' atac) 1))) ORELSE
+ ((rtac @{thm UNIV_eq_I} THEN' rtac CollectI) 1 THEN
+ REPEAT_DETERM (TRY (rtac conjI 1) THEN rtac @{thm subset_UNIV} 1));
+
+fun mk_killN_in_bd_tac n nontrivial_killN_in in_alt in_bd bd_Card_order bd_Cinfinite bd_Cnotzero =
+ (rtac @{thm ordIso_ordLeq_trans} THEN'
+ rtac @{thm card_of_ordIso_subst} THEN'
+ rtac in_alt THEN'
+ rtac ctrans THEN'
+ rtac in_bd THEN'
+ rtac @{thm ordIso_ordLeq_trans} THEN'
+ rtac @{thm cexp_cong1}) 1 THEN
+ (if nontrivial_killN_in then
+ rtac @{thm ordIso_transitive} 1 THEN
+ REPEAT_DETERM_N (n - 1)
+ ((rtac @{thm csum_cong1} THEN'
+ rtac @{thm ordIso_symmetric} THEN'
+ rtac @{thm csum_assoc} THEN'
+ rtac @{thm ordIso_transitive}) 1) THEN
+ (rtac @{thm ordIso_refl} THEN'
+ rtac @{thm Card_order_csum} THEN'
+ rtac @{thm ordIso_transitive} THEN'
+ rtac @{thm csum_assoc} THEN'
+ rtac @{thm ordIso_transitive} THEN'
+ rtac @{thm csum_cong1} THEN'
+ K (mk_flatten_assoc_tac
+ (rtac @{thm ordIso_refl} THEN'
+ FIRST' [rtac @{thm card_of_Card_order}, rtac @{thm Card_order_csum}])
+ @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_cong}) THEN'
+ rtac @{thm ordIso_refl} THEN'
+ (rtac @{thm card_of_Card_order} ORELSE' rtac @{thm Card_order_csum})) 1
+ else all_tac) THEN
+ (rtac @{thm csum_com} THEN'
+ rtac bd_Card_order THEN'
+ rtac disjI1 THEN'
+ rtac @{thm csum_Cnotzero2} THEN'
+ rtac @{thm ctwo_Cnotzero} THEN'
+ rtac disjI1 THEN'
+ rtac @{thm csum_Cnotzero2} THEN'
+ TRY o rtac @{thm csum_Cnotzero1} THEN'
+ rtac @{thm Cnotzero_UNIV} THEN'
+ rtac @{thm ordLeq_ordIso_trans} THEN'
+ rtac @{thm cexp_mono1} THEN'
+ rtac ctrans THEN'
+ rtac @{thm csum_mono2} THEN'
+ rtac @{thm ordLeq_cprod1} THEN'
+ (rtac @{thm card_of_Card_order} ORELSE' rtac @{thm Card_order_csum}) THEN'
+ rtac bd_Cnotzero THEN'
+ rtac @{thm csum_cexp'} THEN'
+ rtac @{thm Cinfinite_cprod2} THEN'
+ TRY o rtac @{thm csum_Cnotzero1} THEN'
+ rtac @{thm Cnotzero_UNIV} THEN'
+ rtac bd_Cinfinite THEN'
+ ((rtac @{thm Card_order_ctwo} THEN' rtac @{thm ordLeq_refl}) ORELSE'
+ (rtac @{thm Card_order_csum} THEN' rtac @{thm ordLeq_csum2})) THEN'
+ rtac @{thm Card_order_ctwo} THEN'
+ rtac disjI1 THEN'
+ rtac @{thm csum_Cnotzero2} THEN'
+ TRY o rtac @{thm csum_Cnotzero1} THEN'
+ rtac @{thm Cnotzero_UNIV} THEN'
+ rtac bd_Card_order THEN'
+ rtac @{thm cexp_cprod_ordLeq} THEN'
+ TRY o rtac @{thm csum_Cnotzero2} THEN'
+ rtac @{thm ctwo_Cnotzero} THEN'
+ rtac @{thm Cinfinite_cprod2} THEN'
+ TRY o rtac @{thm csum_Cnotzero1} THEN'
+ rtac @{thm Cnotzero_UNIV} THEN'
+ rtac bd_Cinfinite THEN'
+ rtac bd_Cnotzero THEN'
+ rtac @{thm ordLeq_cprod2} THEN'
+ TRY o rtac @{thm csum_Cnotzero1} THEN'
+ rtac @{thm Cnotzero_UNIV} THEN'
+ rtac bd_Card_order) 1;
+
+
+
+(* Lift operation *)
+
+val empty_natural_tac = rtac @{thm empty_natural} 1;
+
+fun mk_liftN_set_bd_tac bd_Card_order = (rtac @{thm Card_order_empty} THEN' rtac bd_Card_order) 1;
+
+val liftN_in_alt_tac =
+ ((rtac @{thm Collect_cong} THEN' rtac @{thm iffI}) 1 THEN
+ REPEAT_DETERM (CHANGED (etac conjE 1)) THEN
+ REPEAT_DETERM (CHANGED ((etac conjI ORELSE' atac) 1)) THEN
+ REPEAT_DETERM (CHANGED (etac conjE 1)) THEN
+ REPEAT_DETERM (CHANGED ((etac conjI ORELSE'
+ rtac conjI THEN' rtac @{thm empty_subsetI}) 1)) THEN
+ (rtac @{thm empty_subsetI} ORELSE' atac) 1) ORELSE
+ ((rtac sym THEN' rtac @{thm UNIV_eq_I} THEN' rtac CollectI) 1 THEN
+ REPEAT_DETERM (TRY (rtac conjI 1) THEN rtac @{thm empty_subsetI} 1));
+
+fun mk_liftN_in_bd_tac n in_alt in_bd bd_Card_order =
+ (rtac @{thm ordIso_ordLeq_trans} THEN'
+ rtac @{thm card_of_ordIso_subst} THEN'
+ rtac in_alt THEN'
+ rtac ctrans THEN'
+ rtac in_bd THEN'
+ rtac @{thm cexp_mono1}) 1 THEN
+ ((rtac @{thm csum_mono1} 1 THEN
+ REPEAT_DETERM_N (n - 1)
+ ((rtac ctrans THEN'
+ rtac @{thm ordLeq_csum2} THEN'
+ (rtac @{thm Card_order_csum} ORELSE' rtac @{thm card_of_Card_order})) 1) THEN
+ (rtac @{thm ordLeq_csum2} THEN'
+ (rtac @{thm Card_order_csum} ORELSE' rtac @{thm card_of_Card_order})) 1) ORELSE
+ (rtac @{thm ordLeq_csum2} THEN' rtac @{thm Card_order_ctwo}) 1) THEN
+ (rtac disjI1 THEN' TRY o rtac @{thm csum_Cnotzero2} THEN' rtac @{thm ctwo_Cnotzero}
+ THEN' rtac bd_Card_order) 1;
+
+
+
+(* Permute operation *)
+
+fun mk_permute_in_alt_tac src dest =
+ (rtac @{thm Collect_cong} THEN'
+ mk_rotate_eq_tac (rtac refl) trans @{thm conj_assoc} @{thm conj_commute} @{thm conj_cong}
+ dest src) 1;
+
+fun mk_permute_in_bd_tac src dest in_alt in_bd bd_Card_order =
+ (rtac @{thm ordIso_ordLeq_trans} THEN'
+ rtac @{thm card_of_ordIso_subst} THEN'
+ rtac in_alt THEN'
+ rtac @{thm ordLeq_ordIso_trans} THEN'
+ rtac in_bd THEN'
+ rtac @{thm cexp_cong1} THEN'
+ rtac @{thm csum_cong1} THEN'
+ mk_rotate_eq_tac
+ (rtac @{thm ordIso_refl} THEN'
+ FIRST' [rtac @{thm card_of_Card_order}, rtac @{thm Card_order_csum}])
+ @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong}
+ src dest THEN'
+ rtac bd_Card_order THEN'
+ rtac disjI1 THEN'
+ TRY o rtac @{thm csum_Cnotzero2} THEN'
+ rtac @{thm ctwo_Cnotzero} THEN'
+ rtac disjI1 THEN'
+ TRY o rtac @{thm csum_Cnotzero2} THEN'
+ rtac @{thm ctwo_Cnotzero}) 1;
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_def.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1217 @@
+(* Title: HOL/Codatatype/Tools/bnf_def.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+Definition of bounded natural functors.
+*)
+
+signature BNF_DEF =
+sig
+ type BNF
+ type nonemptiness_witness = {I: int list, wit: term, prop: thm list}
+
+ val bnf_of: Proof.context -> string -> BNF option
+ val name_of_bnf: BNF -> binding
+ val T_of_bnf: BNF -> typ
+ val live_of_bnf: BNF -> int
+ val lives_of_bnf: BNF -> typ list
+ val dead_of_bnf: BNF -> int
+ val deads_of_bnf: BNF -> typ list
+ val nwits_of_bnf: BNF -> int
+
+ val mapN: string
+ val setN: string
+ val relN: string
+ val predN: string
+ val mk_setN: int -> string
+ val rel_unfoldN: string
+ val pred_unfoldN: string
+
+ val mk_T_of_bnf: typ list -> typ list -> BNF -> typ
+ val mk_bd_of_bnf: typ list -> typ list -> BNF -> term
+ val mk_map_of_bnf: typ list -> typ list -> typ list -> BNF -> term
+ val mk_pred_of_bnf: typ list -> typ list -> typ list -> BNF -> term
+ val mk_rel_of_bnf: typ list -> typ list -> typ list -> BNF -> term
+ val mk_sets_of_bnf: typ list list -> typ list list -> BNF -> term list
+ val mk_wits_of_bnf: typ list list -> typ list list -> BNF -> (int list * term) list
+
+ val bd_Card_order_of_bnf: BNF -> thm
+ val bd_Cinfinite_of_bnf: BNF -> thm
+ val bd_Cnotzero_of_bnf: BNF -> thm
+ val bd_card_order_of_bnf: BNF -> thm
+ val bd_cinfinite_of_bnf: BNF -> thm
+ val collect_set_natural_of_bnf: BNF -> thm
+ val in_bd_of_bnf: BNF -> thm
+ val in_cong_of_bnf: BNF -> thm
+ val in_mono_of_bnf: BNF -> thm
+ val in_rel_of_bnf: BNF -> thm
+ val map_comp'_of_bnf: BNF -> thm
+ val map_comp_of_bnf: BNF -> thm
+ val map_cong_of_bnf: BNF -> thm
+ val map_def_of_bnf: BNF -> thm
+ val map_id'_of_bnf: BNF -> thm
+ val map_id_of_bnf: BNF -> thm
+ val map_wppull_of_bnf: BNF -> thm
+ val map_wpull_of_bnf: BNF -> thm
+ val pred_def_of_bnf: BNF -> thm
+ val rel_Gr_of_bnf: BNF -> thm
+ val rel_Id_of_bnf: BNF -> thm
+ val rel_O_of_bnf: BNF -> thm
+ val rel_cong_of_bnf: BNF -> thm
+ val rel_converse_of_bnf: BNF -> thm
+ val rel_def_of_bnf: BNF -> thm
+ val rel_mono_of_bnf: BNF -> thm
+ val set_bd_of_bnf: BNF -> thm list
+ val set_defs_of_bnf: BNF -> thm list
+ val set_natural'_of_bnf: BNF -> thm list
+ val set_natural_of_bnf: BNF -> thm list
+ val sets_of_bnf: BNF -> term list
+ val wit_thms_of_bnf: BNF -> thm list
+ val wit_thmss_of_bnf: BNF -> thm list list
+
+ val mk_witness: int list * term -> thm list -> nonemptiness_witness
+ val minimize_wits: (''a list * term) list -> (''a list * term) list
+ val wits_of_bnf: BNF -> nonemptiness_witness list
+
+ datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline
+ datatype fact_policy =
+ Derive_Some_Facts | Derive_All_Facts | Derive_All_Facts_Note_Most | Note_All_Facts_and_Axioms
+ val bnf_note_all: bool Config.T
+ val user_policy: Proof.context -> fact_policy
+
+ val print_bnfs: Proof.context -> unit
+ val add_bnf: const_policy -> (Proof.context -> fact_policy) -> (binding -> binding) ->
+ ({prems: thm list, context: Proof.context} -> tactic) list ->
+ ({prems: thm list, context: Proof.context} -> tactic) -> typ list option ->
+ (((binding * term) * term list) * term) * term list -> local_theory ->
+ BNF * local_theory
+
+ val filter_refl: thm list -> thm list
+ val add_bnf_cmd: (((binding * string) * string list) * string) * string list -> local_theory ->
+ Proof.state
+end;
+
+structure BNF_Def : BNF_DEF =
+struct
+
+open BNF_Util
+open BNF_Tactics
+
+type axioms = {
+ map_id: thm,
+ map_comp: thm,
+ map_cong: thm,
+ set_natural: thm list,
+ bd_card_order: thm,
+ bd_cinfinite: thm,
+ set_bd: thm list,
+ in_bd: thm,
+ map_wpull: thm
+};
+
+fun mk_axioms' ((((((((id, comp), cong), nat), c_o), cinf), set_bd), in_bd), wpull) =
+ {map_id = id, map_comp = comp, map_cong = cong, set_natural = nat, bd_card_order = c_o,
+ bd_cinfinite = cinf, set_bd = set_bd, in_bd = in_bd, map_wpull = wpull};
+
+fun dest_cons [] = raise Empty
+ | dest_cons (x :: xs) = (x, xs);
+
+fun mk_axioms n thms = thms
+ |> map the_single
+ |> dest_cons
+ ||>> dest_cons
+ ||>> dest_cons
+ ||>> chop n
+ ||>> dest_cons
+ ||>> dest_cons
+ ||>> chop n
+ ||>> dest_cons
+ ||> the_single
+ |> mk_axioms';
+
+fun dest_axioms {map_id, map_comp, map_cong, set_natural,
+ bd_card_order, bd_cinfinite, set_bd, in_bd, map_wpull} =
+ [map_id, map_comp, map_cong] @ set_natural @ [bd_card_order, bd_cinfinite] @
+ set_bd @ [in_bd, map_wpull];
+
+fun map_axioms f
+ {map_id = map_id, map_comp = map_comp, map_cong = map_cong, set_natural = set_natural,
+ bd_card_order = bd_card_order, bd_cinfinite = bd_cinfinite,
+ set_bd = set_bd, in_bd = in_bd, map_wpull = map_wpull} =
+ {map_id = f map_id,
+ map_comp = f map_comp,
+ map_cong = f map_cong,
+ set_natural = map f set_natural,
+ bd_card_order = f bd_card_order,
+ bd_cinfinite = f bd_cinfinite,
+ set_bd = map f set_bd,
+ in_bd = f in_bd,
+ map_wpull = f map_wpull};
+
+val morph_axioms = map_axioms o Morphism.thm;
+
+type defs = {
+ map_def: thm,
+ set_defs: thm list,
+ rel_def: thm,
+ pred_def: thm
+}
+
+fun mk_defs map sets rel pred = {map_def = map, set_defs = sets, rel_def = rel, pred_def = pred};
+
+fun map_defs f {map_def = map, set_defs = sets, rel_def = rel, pred_def = pred} =
+ {map_def = f map, set_defs = List.map f sets, rel_def = f rel, pred_def = f pred};
+
+val morph_defs = map_defs o Morphism.thm;
+
+type facts = {
+ bd_Card_order: thm,
+ bd_Cinfinite: thm,
+ bd_Cnotzero: thm,
+ collect_set_natural: thm lazy,
+ in_cong: thm lazy,
+ in_mono: thm lazy,
+ in_rel: thm lazy,
+ map_comp': thm lazy,
+ map_id': thm lazy,
+ map_wppull: thm lazy,
+ rel_cong: thm lazy,
+ rel_mono: thm lazy,
+ rel_Id: thm lazy,
+ rel_Gr: thm lazy,
+ rel_converse: thm lazy,
+ rel_O: thm lazy,
+ set_natural': thm lazy list
+};
+
+fun mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero
+ collect_set_natural in_cong in_mono in_rel map_comp' map_id' map_wppull
+ rel_cong rel_mono rel_Id rel_Gr rel_converse rel_O set_natural' = {
+ bd_Card_order = bd_Card_order,
+ bd_Cinfinite = bd_Cinfinite,
+ bd_Cnotzero = bd_Cnotzero,
+ collect_set_natural = collect_set_natural,
+ in_cong = in_cong,
+ in_mono = in_mono,
+ in_rel = in_rel,
+ map_comp' = map_comp',
+ map_id' = map_id',
+ map_wppull = map_wppull,
+ rel_cong = rel_cong,
+ rel_mono = rel_mono,
+ rel_Id = rel_Id,
+ rel_Gr = rel_Gr,
+ rel_converse = rel_converse,
+ rel_O = rel_O,
+ set_natural' = set_natural'};
+
+fun map_facts f {
+ bd_Card_order,
+ bd_Cinfinite,
+ bd_Cnotzero,
+ collect_set_natural,
+ in_cong,
+ in_mono,
+ in_rel,
+ map_comp',
+ map_id',
+ map_wppull,
+ rel_cong,
+ rel_mono,
+ rel_Id,
+ rel_Gr,
+ rel_converse,
+ rel_O,
+ set_natural'} =
+ {bd_Card_order = f bd_Card_order,
+ bd_Cinfinite = f bd_Cinfinite,
+ bd_Cnotzero = f bd_Cnotzero,
+ collect_set_natural = Lazy.map f collect_set_natural,
+ in_cong = Lazy.map f in_cong,
+ in_mono = Lazy.map f in_mono,
+ in_rel = Lazy.map f in_rel,
+ map_comp' = Lazy.map f map_comp',
+ map_id' = Lazy.map f map_id',
+ map_wppull = Lazy.map f map_wppull,
+ rel_cong = Lazy.map f rel_cong,
+ rel_mono = Lazy.map f rel_mono,
+ rel_Id = Lazy.map f rel_Id,
+ rel_Gr = Lazy.map f rel_Gr,
+ rel_converse = Lazy.map f rel_converse,
+ rel_O = Lazy.map f rel_O,
+ set_natural' = map (Lazy.map f) set_natural'};
+
+val morph_facts = map_facts o Morphism.thm;
+
+type nonemptiness_witness = {
+ I: int list,
+ wit: term,
+ prop: thm list
+};
+
+fun mk_witness (I, wit) prop = {I = I, wit = wit, prop = prop};
+fun map_witness f g {I, wit, prop} = {I = I, wit = f wit, prop = map g prop};
+fun morph_witness phi = map_witness (Morphism.term phi) (Morphism.thm phi);
+
+datatype BNF = BNF of {
+ name: binding,
+ T: typ,
+ live: int,
+ lives: typ list, (*source type variables of map, only for composition*)
+ lives': typ list, (*target type variables of map, only for composition*)
+ dead: int,
+ deads: typ list, (*only for composition*)
+ map: term,
+ sets: term list,
+ bd: term,
+ axioms: axioms,
+ defs: defs,
+ facts: facts,
+ nwits: int,
+ wits: nonemptiness_witness list,
+ rel: term,
+ pred: term
+};
+
+(* getters *)
+
+fun rep_bnf (BNF bnf) = bnf;
+val name_of_bnf = #name o rep_bnf;
+val T_of_bnf = #T o rep_bnf;
+fun mk_T_of_bnf Ds Ts bnf =
+ let val bnf_rep = rep_bnf bnf
+ in Term.typ_subst_atomic ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)) (#T bnf_rep) end;
+val live_of_bnf = #live o rep_bnf;
+val lives_of_bnf = #lives o rep_bnf;
+val dead_of_bnf = #dead o rep_bnf;
+val deads_of_bnf = #deads o rep_bnf;
+val axioms_of_bnf = #axioms o rep_bnf;
+val facts_of_bnf = #facts o rep_bnf;
+val nwits_of_bnf = #nwits o rep_bnf;
+val wits_of_bnf = #wits o rep_bnf;
+
+(*terms*)
+val map_of_bnf = #map o rep_bnf;
+val sets_of_bnf = #sets o rep_bnf;
+fun mk_map_of_bnf Ds Ts Us bnf =
+ let val bnf_rep = rep_bnf bnf;
+ in
+ Term.subst_atomic_types
+ ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts) @ (#lives' bnf_rep ~~ Us)) (#map bnf_rep)
+ end;
+fun mk_sets_of_bnf Dss Tss bnf =
+ let val bnf_rep = rep_bnf bnf;
+ in
+ map2 (fn (Ds, Ts) => Term.subst_atomic_types
+ ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts))) (Dss ~~ Tss) (#sets bnf_rep)
+ end;
+val bd_of_bnf = #bd o rep_bnf;
+fun mk_bd_of_bnf Ds Ts bnf =
+ let val bnf_rep = rep_bnf bnf;
+ in Term.subst_atomic_types ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)) (#bd bnf_rep) end;
+fun mk_wits_of_bnf Dss Tss bnf =
+ let
+ val bnf_rep = rep_bnf bnf;
+ val wits = map (fn x => (#I x, #wit x)) (#wits bnf_rep);
+ in
+ map2 (fn (Ds, Ts) => apsnd (Term.subst_atomic_types
+ ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)))) (Dss ~~ Tss) wits
+ end;
+val rel_of_bnf = #rel o rep_bnf;
+fun mk_rel_of_bnf Ds Ts Us bnf =
+ let val bnf_rep = rep_bnf bnf;
+ in
+ Term.subst_atomic_types
+ ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts) @ (#lives' bnf_rep ~~ Us)) (#rel bnf_rep)
+ end;
+val pred_of_bnf = #pred o rep_bnf;
+fun mk_pred_of_bnf Ds Ts Us bnf =
+ let val bnf_rep = rep_bnf bnf;
+ in
+ Term.subst_atomic_types
+ ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts) @ (#lives' bnf_rep ~~ Us)) (#pred bnf_rep)
+ end;
+
+(*thms*)
+val bd_card_order_of_bnf = #bd_card_order o #axioms o rep_bnf;
+val bd_cinfinite_of_bnf = #bd_cinfinite o #axioms o rep_bnf;
+val bd_Card_order_of_bnf = #bd_Card_order o #facts o rep_bnf;
+val bd_Cinfinite_of_bnf = #bd_Cinfinite o #facts o rep_bnf;
+val bd_Cnotzero_of_bnf = #bd_Cnotzero o #facts o rep_bnf;
+val collect_set_natural_of_bnf = Lazy.force o #collect_set_natural o #facts o rep_bnf;
+val in_bd_of_bnf = #in_bd o #axioms o rep_bnf;
+val in_cong_of_bnf = Lazy.force o #in_cong o #facts o rep_bnf;
+val in_mono_of_bnf = Lazy.force o #in_mono o #facts o rep_bnf;
+val in_rel_of_bnf = Lazy.force o #in_rel o #facts o rep_bnf;
+val map_def_of_bnf = #map_def o #defs o rep_bnf;
+val map_id_of_bnf = #map_id o #axioms o rep_bnf;
+val map_id'_of_bnf = Lazy.force o #map_id' o #facts o rep_bnf;
+val map_comp_of_bnf = #map_comp o #axioms o rep_bnf;
+val map_comp'_of_bnf = Lazy.force o #map_comp' o #facts o rep_bnf;
+val map_cong_of_bnf = #map_cong o #axioms o rep_bnf;
+val map_wppull_of_bnf = Lazy.force o #map_wppull o #facts o rep_bnf;
+val map_wpull_of_bnf = #map_wpull o #axioms o rep_bnf;
+val pred_def_of_bnf = #pred_def o #defs o rep_bnf;
+val rel_cong_of_bnf = Lazy.force o #rel_cong o #facts o rep_bnf;
+val rel_mono_of_bnf = Lazy.force o #rel_mono o #facts o rep_bnf;
+val rel_def_of_bnf = #rel_def o #defs o rep_bnf;
+val rel_Id_of_bnf = Lazy.force o #rel_Id o #facts o rep_bnf;
+val rel_Gr_of_bnf = Lazy.force o #rel_Gr o #facts o rep_bnf;
+val rel_converse_of_bnf = Lazy.force o #rel_converse o #facts o rep_bnf;
+val rel_O_of_bnf = Lazy.force o #rel_O o #facts o rep_bnf;
+val set_bd_of_bnf = #set_bd o #axioms o rep_bnf;
+val set_defs_of_bnf = #set_defs o #defs o rep_bnf;
+val set_natural_of_bnf = #set_natural o #axioms o rep_bnf;
+val set_natural'_of_bnf = map Lazy.force o #set_natural' o #facts o rep_bnf;
+val wit_thms_of_bnf = maps #prop o wits_of_bnf;
+val wit_thmss_of_bnf = map #prop o wits_of_bnf;
+
+fun mk_bnf name T live lives lives' dead deads map sets bd axioms defs facts wits rel pred =
+ BNF {name = name, T = T,
+ live = live, lives = lives, lives' = lives', dead = dead, deads = deads,
+ map = map, sets = sets, bd = bd,
+ axioms = axioms, defs = defs, facts = facts,
+ nwits = length wits, wits = wits, rel = rel, pred = pred};
+
+fun morph_bnf phi (BNF {name = name, T = T, live = live, lives = lives, lives' = lives',
+ dead = dead, deads = deads, map = map, sets = sets, bd = bd,
+ axioms = axioms, defs = defs, facts = facts,
+ nwits = nwits, wits = wits, rel = rel, pred = pred}) =
+ BNF {name = Morphism.binding phi name, T = Morphism.typ phi T,
+ live = live, lives = List.map (Morphism.typ phi) lives,
+ lives' = List.map (Morphism.typ phi) lives',
+ dead = dead, deads = List.map (Morphism.typ phi) deads,
+ map = Morphism.term phi map, sets = List.map (Morphism.term phi) sets,
+ bd = Morphism.term phi bd,
+ axioms = morph_axioms phi axioms,
+ defs = morph_defs phi defs,
+ facts = morph_facts phi facts,
+ nwits = nwits,
+ wits = List.map (morph_witness phi) wits,
+ rel = Morphism.term phi rel, pred = Morphism.term phi pred};
+
+fun eq_bnf (BNF {T = T1, live = live1, dead = dead1, ...},
+ BNF {T = T2, live = live2, dead = dead2, ...}) =
+ Type.could_unify (T1, T2) andalso live1 = live2 andalso dead1 = dead2;
+
+structure Data = Generic_Data
+(
+ type T = BNF Symtab.table;
+ val empty = Symtab.empty;
+ val extend = I;
+ val merge = Symtab.merge (eq_bnf);
+);
+
+val bnf_of = Symtab.lookup o Data.get o Context.Proof;
+
+
+
+(* Utilities *)
+
+fun normalize_set insts instA set =
+ let
+ val (T, T') = dest_funT (fastype_of set);
+ val A = fst (Term.dest_TVar (HOLogic.dest_setT T'));
+ val params = Term.add_tvar_namesT T [];
+ in Term.subst_TVars ((A :: params) ~~ (instA :: insts)) set end;
+
+fun normalize_rel ctxt instTs instA instB rel =
+ let
+ val thy = Proof_Context.theory_of ctxt;
+ val tyenv =
+ Sign.typ_match thy (fastype_of rel, Library.foldr (op -->) (instTs, mk_relT (instA, instB)))
+ Vartab.empty;
+ in Envir.subst_term (tyenv, Vartab.empty) rel end;
+
+fun normalize_pred ctxt instTs instA instB pred =
+ let
+ val thy = Proof_Context.theory_of ctxt;
+ val tyenv =
+ Sign.typ_match thy (fastype_of pred,
+ Library.foldr (op -->) (instTs, instA --> instB --> HOLogic.boolT)) Vartab.empty;
+ in Envir.subst_term (tyenv, Vartab.empty) pred end;
+
+fun normalize_wit insts CA As wit =
+ let
+ fun strip_param (Ts, T as Type (@{type_name fun}, [T1, T2])) =
+ if Type.raw_instance (CA, T) then (Ts, T) else strip_param (T1 :: Ts, T2)
+ | strip_param x = x;
+ val (Ts, T) = strip_param ([], fastype_of wit);
+ val subst = Term.add_tvar_namesT T [] ~~ insts;
+ fun find y = find_index (fn x => x = y) As;
+ in
+ (map (find o Term.typ_subst_TVars subst) (rev Ts), Term.subst_TVars subst wit)
+ end;
+
+fun minimize_wits wits =
+ let
+ fun minimize done [] = done
+ | minimize done ((I, wit : term) :: todo) =
+ if exists (fn (J, _) => subset (op =) (J, I)) (done @ todo)
+ then minimize done todo
+ else minimize ((I, wit) :: done) todo;
+ in minimize [] wits end;
+
+fun unfold_defs_tac lthy defs mk_tac context = Local_Defs.unfold_tac lthy defs THEN mk_tac context;
+
+
+
+(* Names *)
+
+fun nonzero_string_of_int 0 = ""
+ | nonzero_string_of_int n = string_of_int n;
+
+val mapN = "map";
+val setN = "set";
+fun mk_setN i = setN ^ nonzero_string_of_int i;
+val bdN = "bd";
+val witN = "wit";
+fun mk_witN i = witN ^ nonzero_string_of_int i;
+val relN = "rel";
+val predN = "pred";
+val rel_unfoldN = relN ^ "_unfold";
+val pred_unfoldN = predN ^ "_unfold";
+
+val bd_card_orderN = "bd_card_order";
+val bd_cinfiniteN = "bd_cinfinite";
+val bd_Card_orderN = "bd_Card_order";
+val bd_CinfiniteN = "bd_Cinfinite";
+val bd_CnotzeroN = "bd_Cnotzero";
+val collect_set_naturalN = "collect_set_natural";
+val in_bdN = "in_bd";
+val in_congN = "in_cong";
+val in_monoN = "in_mono";
+val in_relN = "in_rel";
+val map_idN = "map_id";
+val map_id'N = "map_id'";
+val map_compN = "map_comp";
+val map_comp'N = "map_comp'";
+val map_congN = "map_cong";
+val map_wppullN = "map_wppull";
+val map_wpullN = "map_wpull";
+val rel_congN = "rel_cong";
+val rel_IdN = "rel_Id";
+val rel_GrN = "rel_Gr";
+val rel_converseN = "rel_converse";
+val rel_ON = "rel_comp";
+val set_naturalN = "set_natural";
+val set_natural'N = "set_natural'";
+val set_bdN = "set_bd";
+
+datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline;
+
+datatype fact_policy =
+ Derive_Some_Facts | Derive_All_Facts | Derive_All_Facts_Note_Most | Note_All_Facts_and_Axioms;
+
+val bnf_note_all = Attrib.setup_config_bool @{binding bnf_note_all} (K false);
+
+fun user_policy ctxt =
+ if Config.get ctxt bnf_note_all then Note_All_Facts_and_Axioms
+ else Derive_All_Facts_Note_Most;
+
+val smart_max_inline_size = 25; (*FUDGE*)
+
+val no_def = Drule.reflexive_thm;
+val no_fact = refl;
+
+fun is_reflexive th =
+ let val t = Thm.prop_of th;
+ in
+ op aconv (Logic.dest_equals t)
+ handle TERM _ => op aconv (HOLogic.dest_eq (HOLogic.dest_Trueprop t))
+ handle TERM _ => false
+ end;
+
+val filter_refl = filter_out is_reflexive;
+
+
+
+(* Define new BNFs *)
+
+fun prepare_bnf const_policy mk_fact_policy qualify prep_term Ds_opt
+ ((((raw_b, raw_map), raw_sets), raw_bd_Abs), raw_wits) no_defs_lthy =
+ let
+ val fact_policy = mk_fact_policy no_defs_lthy;
+ val b = qualify raw_b;
+ val live = length raw_sets;
+ val nwits = length raw_wits;
+
+ val map_rhs = prep_term no_defs_lthy raw_map;
+ val set_rhss = map (prep_term no_defs_lthy) raw_sets;
+ val (bd_rhsT, bd_rhs) = (case prep_term no_defs_lthy raw_bd_Abs of
+ Abs (_, T, t) => (T, t)
+ | _ => error "Bad bound constant");
+ val wit_rhss = map (prep_term no_defs_lthy) raw_wits;
+
+ val map_bind_def = (fn () => Binding.suffix_name ("_" ^ mapN) b, map_rhs);
+ val set_binds_defs =
+ let
+ val bs = if live = 1 then [fn () => Binding.suffix_name ("_" ^ setN) b]
+ else map (fn i => fn () => Binding.suffix_name ("_" ^ mk_setN i) b) (1 upto live)
+ in map2 pair bs set_rhss end;
+ val bd_bind_def = (fn () => Binding.suffix_name ("_" ^ bdN) b, bd_rhs);
+ val wit_binds_defs =
+ let
+ val bs = if nwits = 1 then [fn () => Binding.suffix_name ("_" ^ witN) b]
+ else map (fn i => fn () => Binding.suffix_name ("_" ^ mk_witN i) b) (1 upto nwits);
+ in map2 pair bs wit_rhss end;
+
+ fun maybe_define needed_for_fp (b, rhs) lthy =
+ let
+ val inline =
+ (not needed_for_fp orelse fact_policy = Derive_Some_Facts) andalso
+ (case const_policy of
+ Dont_Inline => false
+ | Hardly_Inline => Term.is_Free rhs orelse Term.is_Const rhs
+ | Smart_Inline => Term.size_of_term rhs <= smart_max_inline_size
+ | Do_Inline => true)
+ in
+ if inline then
+ ((rhs, no_def), lthy)
+ else
+ let val b = b () in
+ apfst (apsnd snd) (Local_Theory.define ((b, NoSyn), ((Thm.def_binding b, []), rhs))
+ lthy)
+ end
+ end;
+ fun maybe_restore lthy0 lthy = lthy |> not (pointer_eq (lthy0, lthy)) ? Local_Theory.restore;
+
+ val (((((bnf_map_term, raw_map_def),
+ (bnf_set_terms, raw_set_defs)),
+ (bnf_bd_term, raw_bd_def)),
+ (bnf_wit_terms, raw_wit_defs)), (lthy', lthy)) =
+ no_defs_lthy
+ |> maybe_define false map_bind_def
+ ||>> apfst split_list o fold_map (maybe_define false) set_binds_defs
+ ||>> maybe_define false bd_bind_def
+ ||>> apfst split_list o fold_map (maybe_define false) wit_binds_defs
+ ||> `(maybe_restore no_defs_lthy);
+
+ (*transforms defined frees into consts (and more)*)
+ val phi = Proof_Context.export_morphism lthy lthy';
+
+ val bnf_map_def = Morphism.thm phi raw_map_def;
+ val bnf_set_defs = map (Morphism.thm phi) raw_set_defs;
+ val bnf_bd_def = Morphism.thm phi raw_bd_def;
+ val bnf_wit_defs = map (Morphism.thm phi) raw_wit_defs;
+
+ val one_step_defs = filter_refl (bnf_map_def :: bnf_bd_def :: bnf_set_defs @ bnf_wit_defs);
+
+ val _ = case map_filter (try dest_Free)
+ (bnf_map_term :: bnf_set_terms @ [bnf_bd_term] @ bnf_wit_terms) of
+ [] => ()
+ | frees => Proof_Display.print_consts true lthy (K false) frees;
+
+ val bnf_map = Morphism.term phi bnf_map_term;
+
+ fun iter_split ((Ts, T1), T2) = if length Ts < live then error "Bad map function"
+ else if length Ts = live then ((Ts, T1), T2)
+ else iter_split (split_last Ts, T1 --> T2);
+
+ (*TODO: handle errors*)
+ (*simple shape analysis of a map function*)
+ val (((alphas, betas), CA), _) =
+ apfst (apfst (map_split dest_funT))
+ (iter_split (apfst split_last (strip_type (fastype_of bnf_map))));
+
+ val CA_params = map TVar (Term.add_tvarsT CA []);
+
+ val bnf_sets = map2 (normalize_set CA_params) alphas (map (Morphism.term phi) bnf_set_terms);
+ val bdT = Morphism.typ phi bd_rhsT;
+ val bnf_bd =
+ Term.subst_TVars (Term.add_tvar_namesT bdT [] ~~ CA_params) (Morphism.term phi bnf_bd_term);
+ val bnf_wits = map (normalize_wit CA_params CA alphas o Morphism.term phi) bnf_wit_terms;
+
+ (*TODO: assert Ds = (TVars of bnf_map) \ (alphas @ betas) as sets*)
+ val deads = (case Ds_opt of
+ NONE => subtract (op =) (alphas @ betas) (map TVar (Term.add_tvars bnf_map []))
+ | SOME Ds => map (Morphism.typ phi) Ds);
+ val dead = length deads;
+
+ (*FIXME: check DUP here, not in after_qed*)
+ val key = Name_Space.full_name Name_Space.default_naming b;
+
+ (*TODO: further checks of type of bnf_map*)
+ (*TODO: check types of bnf_sets*)
+ (*TODO: check type of bnf_bd*)
+
+ val ((((((((((As', Bs'), Cs), Ds), B1Ts), B2Ts), domTs), ranTs), ranTs'), ranTs''),
+ (Ts, T)) = lthy'
+ |> mk_TFrees live
+ ||>> mk_TFrees live
+ ||>> mk_TFrees live
+ ||>> mk_TFrees dead
+ ||>> mk_TFrees live
+ ||>> mk_TFrees live
+ ||>> mk_TFrees live
+ ||>> mk_TFrees live
+ ||>> mk_TFrees live
+ ||>> mk_TFrees live
+ ||> fst o mk_TFrees 1
+ ||> the_single
+ ||> `(replicate live);
+
+ fun mk_bnf_map As' Bs' =
+ Term.subst_atomic_types ((deads ~~ Ds) @ (alphas ~~ As') @ (betas ~~ Bs')) bnf_map;
+ fun mk_bnf_t As' t =
+ Term.subst_atomic_types ((deads ~~ Ds) @ (alphas ~~ As')) t;
+ fun mk_bnf_T As' T =
+ Term.typ_subst_atomic ((deads ~~ Ds) @ (alphas ~~ As')) T;
+
+ val (setRTs, RTs) = map_split (`HOLogic.mk_setT o HOLogic.mk_prodT) (As' ~~ Bs');
+ val setRTsAsCs = map (HOLogic.mk_setT o HOLogic.mk_prodT) (As' ~~ Cs);
+ val setRTsBsCs = map (HOLogic.mk_setT o HOLogic.mk_prodT) (Bs' ~~ Cs);
+ val setRT's = map (HOLogic.mk_setT o HOLogic.mk_prodT) (Bs' ~~ As');
+ val self_setRTs = map (HOLogic.mk_setT o HOLogic.mk_prodT) (As' ~~ As');
+ val QTs = map2 (fn T => fn U => T --> U --> HOLogic.boolT) As' Bs';
+
+ val bnf_map_AsAs = mk_bnf_map As' As';
+ val bnf_map_AsBs = mk_bnf_map As' Bs';
+ val bnf_map_AsCs = mk_bnf_map As' Cs;
+ val bnf_map_BsCs = mk_bnf_map Bs' Cs;
+ val bnf_sets_As = map (mk_bnf_t As') bnf_sets;
+ val bnf_sets_Bs = map (mk_bnf_t Bs') bnf_sets;
+ val bnf_bd_As = mk_bnf_t As' bnf_bd;
+ val bnf_wit_As = map (apsnd (mk_bnf_t As')) bnf_wits;
+ val CA' = mk_bnf_T As' CA;
+ val CB' = mk_bnf_T Bs' CA;
+ val CC' = mk_bnf_T Cs CA;
+ val CRs' = mk_bnf_T RTs CA;
+
+ val ((((((((((((((((((((((((fs, fs_copy), gs), hs), (x, x')), (y, y')), (z, z')), zs), As),
+ As_copy), Xs), B1s), B2s), f1s), f2s), e1s), e2s), p1s), p2s), bs),
+ (Rs, Rs')), Rs_copy), Ss), (Qs, Qs')), _) = lthy'
+ |> mk_Frees "f" (map2 (curry (op -->)) As' Bs')
+ ||>> mk_Frees "f" (map2 (curry (op -->)) As' Bs')
+ ||>> mk_Frees "g" (map2 (curry (op -->)) Bs' Cs)
+ ||>> mk_Frees "h" (map2 (curry (op -->)) As' Ts)
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "x") CA'
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "y") CB'
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "z") CRs'
+ ||>> mk_Frees "z" As'
+ ||>> mk_Frees "A" (map HOLogic.mk_setT As')
+ ||>> mk_Frees "A" (map HOLogic.mk_setT As')
+ ||>> mk_Frees "A" (map HOLogic.mk_setT domTs)
+ ||>> mk_Frees "B1" (map HOLogic.mk_setT B1Ts)
+ ||>> mk_Frees "B2" (map HOLogic.mk_setT B2Ts)
+ ||>> mk_Frees "f1" (map2 (curry (op -->)) B1Ts ranTs)
+ ||>> mk_Frees "f2" (map2 (curry (op -->)) B2Ts ranTs)
+ ||>> mk_Frees "e1" (map2 (curry (op -->)) B1Ts ranTs')
+ ||>> mk_Frees "e2" (map2 (curry (op -->)) B2Ts ranTs'')
+ ||>> mk_Frees "p1" (map2 (curry (op -->)) domTs B1Ts)
+ ||>> mk_Frees "p2" (map2 (curry (op -->)) domTs B2Ts)
+ ||>> mk_Frees "b" As'
+ ||>> mk_Frees' "R" setRTs
+ ||>> mk_Frees "R" setRTs
+ ||>> mk_Frees "S" setRTsBsCs
+ ||>> mk_Frees' "Q" QTs;
+
+ val goal_map_id =
+ let
+ val bnf_map_app_id = Term.list_comb
+ (bnf_map_AsAs, map HOLogic.id_const As');
+ in
+ HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (bnf_map_app_id, HOLogic.id_const CA'))
+ end;
+
+ val goal_map_comp =
+ let
+ val bnf_map_app_comp = Term.list_comb
+ (bnf_map_AsCs, map2 (curry HOLogic.mk_comp) gs fs);
+ val comp_bnf_map_app = HOLogic.mk_comp
+ (Term.list_comb (bnf_map_BsCs, gs),
+ Term.list_comb (bnf_map_AsBs, fs));
+ in
+ fold_rev Logic.all (fs @ gs)
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (bnf_map_app_comp, comp_bnf_map_app)))
+ end;
+
+ val goal_map_cong =
+ let
+ fun mk_prem z set f f_copy =
+ Logic.all z (Logic.mk_implies
+ (HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set $ x)),
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (f $ z, f_copy $ z))));
+ val prems = map4 mk_prem zs bnf_sets_As fs fs_copy;
+ val eq = HOLogic.mk_eq (Term.list_comb (bnf_map_AsBs, fs) $ x,
+ Term.list_comb (bnf_map_AsBs, fs_copy) $ x);
+ in
+ fold_rev Logic.all (x :: fs @ fs_copy)
+ (Logic.list_implies (prems, HOLogic.mk_Trueprop eq))
+ end;
+
+ val goal_set_naturals =
+ let
+ fun mk_goal setA setB f =
+ let
+ val set_comp_map =
+ HOLogic.mk_comp (setB, Term.list_comb (bnf_map_AsBs, fs));
+ val image_comp_set = HOLogic.mk_comp (mk_image f, setA);
+ in
+ fold_rev Logic.all fs
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (set_comp_map, image_comp_set)))
+ end;
+ in
+ map3 mk_goal bnf_sets_As bnf_sets_Bs fs
+ end;
+
+ val goal_card_order_bd = HOLogic.mk_Trueprop (mk_card_order bnf_bd_As);
+
+ val goal_cinfinite_bd = HOLogic.mk_Trueprop (mk_cinfinite bnf_bd_As);
+
+ val goal_set_bds =
+ let
+ fun mk_goal set =
+ Logic.all x (HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of (set $ x)) bnf_bd_As));
+ in
+ map mk_goal bnf_sets_As
+ end;
+
+ val goal_in_bd =
+ let
+ val bd = mk_cexp
+ (if live = 0 then ctwo
+ else mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo)
+ bnf_bd_As;
+ in
+ fold_rev Logic.all As
+ (HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of (mk_in As bnf_sets_As CA')) bd))
+ end;
+
+ val goal_map_wpull =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ (map8 mk_wpull Xs B1s B2s f1s f2s (replicate live NONE) p1s p2s);
+ val CX = mk_bnf_T domTs CA;
+ val CB1 = mk_bnf_T B1Ts CA;
+ val CB2 = mk_bnf_T B2Ts CA;
+ val bnf_sets_CX = map2 (normalize_set (map (mk_bnf_T domTs) CA_params)) domTs bnf_sets;
+ val bnf_sets_CB1 = map2 (normalize_set (map (mk_bnf_T B1Ts) CA_params)) B1Ts bnf_sets;
+ val bnf_sets_CB2 = map2 (normalize_set (map (mk_bnf_T B2Ts) CA_params)) B2Ts bnf_sets;
+ val bnf_map_app_f1 = Term.list_comb (mk_bnf_map B1Ts ranTs, f1s);
+ val bnf_map_app_f2 = Term.list_comb (mk_bnf_map B2Ts ranTs, f2s);
+ val bnf_map_app_p1 = Term.list_comb (mk_bnf_map domTs B1Ts, p1s);
+ val bnf_map_app_p2 = Term.list_comb (mk_bnf_map domTs B2Ts, p2s);
+
+ val map_wpull = mk_wpull (mk_in Xs bnf_sets_CX CX)
+ (mk_in B1s bnf_sets_CB1 CB1) (mk_in B2s bnf_sets_CB2 CB2)
+ bnf_map_app_f1 bnf_map_app_f2 NONE bnf_map_app_p1 bnf_map_app_p2;
+ in
+ fold_rev Logic.all (Xs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)
+ (Logic.list_implies (prems, HOLogic.mk_Trueprop map_wpull))
+ end;
+
+ val goals =
+ [goal_map_id, goal_map_comp, goal_map_cong] @ goal_set_naturals @
+ [goal_card_order_bd, goal_cinfinite_bd] @ goal_set_bds @
+ [goal_in_bd, goal_map_wpull];
+
+ fun mk_wit_goals (I, wit) =
+ let
+ val xs = map (nth bs) I;
+ fun wit_goal i =
+ let
+ val z = nth zs i;
+ val set_wit = nth bnf_sets_As i $ Term.list_comb (wit, xs);
+ val concl = HOLogic.mk_Trueprop
+ (if member (op =) I i then HOLogic.mk_eq (z, nth bs i)
+ else @{term False});
+ in
+ fold_rev Logic.all (z :: xs)
+ (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set_wit)), concl))
+ end;
+ in
+ map wit_goal (0 upto live - 1)
+ end;
+
+ val wit_goalss = map mk_wit_goals bnf_wit_As;
+
+ fun after_qed thms lthy =
+ let
+ val (axioms, wit_thms) = apfst (mk_axioms live) (chop (length goals) thms);
+
+ val bd_Card_order =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_Card_order (bnf_bd_As)))
+ (fn _ => mk_Card_order_tac (#bd_card_order axioms));
+
+ val bd_Cinfinite = @{thm conjI} OF [#bd_cinfinite axioms, bd_Card_order];
+ val bd_Cnotzero = bd_Cinfinite RS @{thm Cinfinite_Cnotzero};
+
+ fun mk_lazy f = if fact_policy <> Derive_Some_Facts then Lazy.value (f ()) else Lazy.lazy f;
+
+ fun mk_collect_set_natural () =
+ let
+ val defT = mk_bnf_T Ts CA --> HOLogic.mk_setT T;
+ val collect_map = HOLogic.mk_comp
+ (mk_collect (map (mk_bnf_t Ts) bnf_sets) defT,
+ Term.list_comb (mk_bnf_map As' Ts, hs));
+ val image_collect = mk_collect
+ (map2 (fn h => fn set => HOLogic.mk_comp (mk_image h, set)) hs bnf_sets_As)
+ defT;
+ (*collect {set1 ... setm} o map f1 ... fm = collect {f1` o set1 ... fm` o setm}*)
+ val goal =
+ fold_rev Logic.all hs
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (collect_map, image_collect)));
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (fn {context = ctxt, ...} => mk_collect_set_natural_tac ctxt (#set_natural axioms))
+ end;
+
+ val collect_set_natural = mk_lazy mk_collect_set_natural;
+
+ fun mk_in_mono () =
+ let
+ val prems_mono = map2 (HOLogic.mk_Trueprop oo mk_subset) As As_copy;
+ val goal_in_mono =
+ fold_rev Logic.all (As @ As_copy)
+ (Logic.list_implies (prems_mono, HOLogic.mk_Trueprop
+ (mk_subset (mk_in As bnf_sets_As CA') (mk_in As_copy bnf_sets_As CA'))));
+ in
+ Skip_Proof.prove lthy [] [] goal_in_mono (K (mk_in_mono_tac live))
+ end;
+
+ val in_mono = mk_lazy mk_in_mono;
+
+ fun mk_in_cong () =
+ let
+ val prems_cong = map2 (HOLogic.mk_Trueprop oo curry HOLogic.mk_eq) As As_copy;
+ val goal_in_cong =
+ fold_rev Logic.all (As @ As_copy)
+ (Logic.list_implies (prems_cong, HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (mk_in As bnf_sets_As CA', mk_in As_copy bnf_sets_As CA'))));
+ in
+ Skip_Proof.prove lthy [] [] goal_in_cong (K ((TRY o hyp_subst_tac THEN' rtac refl) 1))
+ end;
+
+ val in_cong = mk_lazy mk_in_cong;
+
+ val map_id' = mk_lazy (fn () => mk_id' (#map_id axioms));
+ val map_comp' = mk_lazy (fn () => mk_comp' (#map_comp axioms));
+
+ val set_natural' =
+ map (fn thm => mk_lazy (fn () => mk_set_natural' thm)) (#set_natural axioms);
+
+ (* relator *)
+
+ (*%R1 .. Rn. Gr (in R1 .. Rn) (map fst .. fst)^-1 O Gr (in R1 .. Rn) (map snd .. snd)*)
+ val rel_rhs =
+ let
+ val map1 = Term.list_comb (mk_bnf_map RTs As', map fst_const RTs);
+ val map2 = Term.list_comb (mk_bnf_map RTs Bs', map snd_const RTs);
+ val bnf_in = mk_in Rs (map (mk_bnf_t RTs) bnf_sets) CRs';
+ in
+ fold_rev Term.absfree Rs'
+ (mk_rel_comp (mk_converse (mk_Gr bnf_in map1), mk_Gr bnf_in map2))
+ end;
+ val rel_bind_def = (fn () => Binding.suffix_name ("_" ^ relN) b, rel_rhs);
+
+ val ((bnf_rel_term, raw_rel_def), (lthy, lthy_old)) =
+ lthy
+ |> maybe_define true rel_bind_def
+ ||> `(maybe_restore lthy);
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val bnf_rel = Morphism.term phi bnf_rel_term;
+
+ fun mk_bnf_rel setRTs CA' CB' = normalize_rel lthy setRTs CA' CB' bnf_rel;
+
+ val relAsBs = mk_bnf_rel setRTs CA' CB';
+ val bnf_rel_def = Morphism.thm phi raw_rel_def;
+ val rel_def_unabs =
+ if fact_policy <> Derive_Some_Facts then
+ mk_unabs_def live (bnf_rel_def RS meta_eq_to_obj_eq)
+ else
+ no_fact;
+
+ val pred_rhs = fold absfree (y' :: x' :: rev Qs') (HOLogic.mk_mem (HOLogic.mk_prod (x, y),
+ Term.list_comb (relAsBs, map3 (fn Q => fn T => fn U =>
+ HOLogic.Collect_const (HOLogic.mk_prodT (T, U)) $ HOLogic.mk_split Q)
+ Qs As' Bs')));
+ val pred_bind_def = (fn () => Binding.suffix_name ("_" ^ predN) b, pred_rhs);
+
+ val ((bnf_pred_term, raw_pred_def), (lthy, lthy_old)) =
+ lthy
+ |> maybe_define true pred_bind_def
+ ||> `(maybe_restore lthy);
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val bnf_pred = Morphism.term phi bnf_pred_term;
+
+ fun mk_bnf_pred QTs CA' CB' = normalize_pred lthy QTs CA' CB' bnf_pred;
+
+ val pred = mk_bnf_pred QTs CA' CB';
+ val bnf_pred_def = Morphism.thm phi raw_pred_def;
+ val pred_def_unabs =
+ if fact_policy <> Derive_Some_Facts then
+ mk_unabs_def (live + 2) (bnf_pred_def RS meta_eq_to_obj_eq)
+ else
+ no_fact;
+
+ fun mk_map_wppull () =
+ let
+ val prems = if live = 0 then [] else
+ [HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map8 mk_wpull Xs B1s B2s f1s f2s (map SOME (e1s ~~ e2s)) p1s p2s))];
+ val CX = mk_bnf_T domTs CA;
+ val CB1 = mk_bnf_T B1Ts CA;
+ val CB2 = mk_bnf_T B2Ts CA;
+ val bnf_sets_CX =
+ map2 (normalize_set (map (mk_bnf_T domTs) CA_params)) domTs bnf_sets;
+ val bnf_sets_CB1 =
+ map2 (normalize_set (map (mk_bnf_T B1Ts) CA_params)) B1Ts bnf_sets;
+ val bnf_sets_CB2 =
+ map2 (normalize_set (map (mk_bnf_T B2Ts) CA_params)) B2Ts bnf_sets;
+ val bnf_map_app_f1 = Term.list_comb (mk_bnf_map B1Ts ranTs, f1s);
+ val bnf_map_app_f2 = Term.list_comb (mk_bnf_map B2Ts ranTs, f2s);
+ val bnf_map_app_e1 = Term.list_comb (mk_bnf_map B1Ts ranTs', e1s);
+ val bnf_map_app_e2 = Term.list_comb (mk_bnf_map B2Ts ranTs'', e2s);
+ val bnf_map_app_p1 = Term.list_comb (mk_bnf_map domTs B1Ts, p1s);
+ val bnf_map_app_p2 = Term.list_comb (mk_bnf_map domTs B2Ts, p2s);
+
+ val concl = mk_wpull (mk_in Xs bnf_sets_CX CX)
+ (mk_in B1s bnf_sets_CB1 CB1) (mk_in B2s bnf_sets_CB2 CB2)
+ bnf_map_app_f1 bnf_map_app_f2 (SOME (bnf_map_app_e1, bnf_map_app_e2))
+ bnf_map_app_p1 bnf_map_app_p2;
+
+ val goal =
+ fold_rev Logic.all (Xs @ B1s @ B2s @ f1s @ f2s @ e1s @ e2s @ p1s @ p2s)
+ (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (fn _ => mk_map_wppull_tac (#map_id axioms) (#map_cong axioms)
+ (#map_wpull axioms) (Lazy.force map_comp') (map Lazy.force set_natural'))
+ end;
+
+ val map_wppull = mk_lazy mk_map_wppull;
+
+ fun mk_rel_Gr () =
+ let
+ val lhs = Term.list_comb (relAsBs, map2 mk_Gr As fs);
+ val rhs = mk_Gr (mk_in As bnf_sets_As CA') (Term.list_comb (bnf_map_AsBs, fs));
+ val goal = fold_rev Logic.all (As @ fs)
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)));
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (mk_rel_Gr_tac bnf_rel_def (#map_id axioms) (#map_cong axioms)
+ (#map_wpull axioms) (Lazy.force in_cong) (Lazy.force map_id')
+ (Lazy.force map_comp') (map Lazy.force set_natural'))
+ end;
+
+ val rel_Gr = mk_lazy mk_rel_Gr;
+
+ fun mk_rel_prems f = map2 (HOLogic.mk_Trueprop oo f) Rs Rs_copy
+ fun mk_rel_concl f = HOLogic.mk_Trueprop
+ (f (Term.list_comb (relAsBs, Rs), Term.list_comb (relAsBs, Rs_copy)));
+
+ fun mk_rel_mono () =
+ let
+ val mono_prems = mk_rel_prems mk_subset;
+ val mono_concl = mk_rel_concl (uncurry mk_subset);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Rs @ Rs_copy) (Logic.list_implies (mono_prems, mono_concl)))
+ (mk_rel_mono_tac bnf_rel_def (Lazy.force in_mono))
+ end;
+
+ fun mk_rel_cong () =
+ let
+ val cong_prems = mk_rel_prems (curry HOLogic.mk_eq);
+ val cong_concl = mk_rel_concl HOLogic.mk_eq;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Rs @ Rs_copy) (Logic.list_implies (cong_prems, cong_concl)))
+ (fn _ => (TRY o hyp_subst_tac THEN' rtac refl) 1)
+ end;
+
+ val rel_mono = mk_lazy mk_rel_mono;
+ val rel_cong = mk_lazy mk_rel_cong;
+
+ fun mk_rel_Id () =
+ let val relAsAs = mk_bnf_rel self_setRTs CA' CA' in
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (Term.list_comb (relAsAs, map Id_const As'), Id_const CA')))
+ (mk_rel_Id_tac live (Lazy.force rel_Gr) (#map_id axioms))
+ end;
+
+ val rel_Id = mk_lazy mk_rel_Id;
+
+ fun mk_rel_converse () =
+ let
+ val relBsAs = mk_bnf_rel setRT's CB' CA';
+ val lhs = Term.list_comb (relBsAs, map mk_converse Rs);
+ val rhs = mk_converse (Term.list_comb (relAsBs, Rs));
+ val le_goal = fold_rev Logic.all Rs (HOLogic.mk_Trueprop (mk_subset lhs rhs));
+ val le_thm = Skip_Proof.prove lthy [] [] le_goal
+ (mk_rel_converse_le_tac bnf_rel_def (Lazy.force rel_Id) (#map_cong axioms)
+ (Lazy.force map_comp') (map Lazy.force set_natural'))
+ val goal = fold_rev Logic.all Rs (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)));
+ in
+ Skip_Proof.prove lthy [] [] goal (fn _ => mk_rel_converse_tac le_thm)
+ end;
+
+ val rel_converse = mk_lazy mk_rel_converse;
+
+ fun mk_rel_O () =
+ let
+ val relAsCs = mk_bnf_rel setRTsAsCs CA' CC';
+ val relBsCs = mk_bnf_rel setRTsBsCs CB' CC';
+ val lhs = Term.list_comb (relAsCs, map2 (curry mk_rel_comp) Rs Ss);
+ val rhs = mk_rel_comp (Term.list_comb (relAsBs, Rs), Term.list_comb (relBsCs, Ss));
+ val goal =
+ fold_rev Logic.all (Rs @ Ss) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)));
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (mk_rel_O_tac bnf_rel_def (Lazy.force rel_Id) (#map_cong axioms)
+ (Lazy.force map_wppull) (Lazy.force map_comp') (map Lazy.force set_natural'))
+ end;
+
+ val rel_O = mk_lazy mk_rel_O;
+
+ fun mk_in_rel () =
+ let
+ val bnf_in = mk_in Rs (map (mk_bnf_t RTs) bnf_sets) CRs';
+ val map1 = Term.list_comb (mk_bnf_map RTs As', map fst_const RTs);
+ val map2 = Term.list_comb (mk_bnf_map RTs Bs', map snd_const RTs);
+ val map_fst_eq = HOLogic.mk_eq (map1 $ z, x);
+ val map_snd_eq = HOLogic.mk_eq (map2 $ z, y);
+ val lhs = HOLogic.mk_mem (HOLogic.mk_prod (x, y), Term.list_comb (relAsBs, Rs));
+ val rhs =
+ HOLogic.mk_exists (fst z', snd z', HOLogic.mk_conj (HOLogic.mk_mem (z, bnf_in),
+ HOLogic.mk_conj (map_fst_eq, map_snd_eq)));
+ val goal =
+ fold_rev Logic.all (x :: y :: Rs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)));
+ in
+ Skip_Proof.prove lthy [] [] goal (mk_in_rel_tac bnf_rel_def (length bnf_sets))
+ end;
+
+ val in_rel = mk_lazy mk_in_rel;
+
+ val defs = mk_defs bnf_map_def bnf_set_defs rel_def_unabs pred_def_unabs;
+
+ val facts = mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_natural
+ in_cong in_mono in_rel map_comp' map_id' map_wppull
+ rel_cong rel_mono rel_Id rel_Gr rel_converse rel_O set_natural';
+
+ val wits = map2 mk_witness bnf_wits wit_thms;
+
+ val bnf_rel = Term.subst_atomic_types
+ ((Ds ~~ deads) @ (As' ~~ alphas) @ (Bs' ~~ betas)) relAsBs;
+ val bnf_pred = Term.subst_atomic_types
+ ((Ds ~~ deads) @ (As' ~~ alphas) @ (Bs' ~~ betas)) pred;
+
+ val bnf = mk_bnf b CA live alphas betas dead deads bnf_map bnf_sets bnf_bd axioms defs facts
+ wits bnf_rel bnf_pred;
+
+ fun note thmN thms =
+ snd o Local_Theory.note
+ ((qualify (Binding.qualify true (Binding.name_of b) (Binding.name thmN)), []), thms);
+
+ fun lazy_note thmN thms = note thmN (map Lazy.force thms);
+ in
+ (bnf, lthy
+ |> (if fact_policy = Note_All_Facts_and_Axioms then
+ let
+ val witNs = if length wits = 1 then [witN] else map mk_witN (1 upto length wits);
+ in
+ note bd_card_orderN [#bd_card_order axioms]
+ #> note bd_cinfiniteN [#bd_cinfinite axioms]
+ #> note bd_Card_orderN [#bd_Card_order facts]
+ #> note bd_CinfiniteN [#bd_Cinfinite facts]
+ #> note bd_CnotzeroN [#bd_Cnotzero facts]
+ #> lazy_note collect_set_naturalN [#collect_set_natural facts]
+ #> note in_bdN [#in_bd axioms]
+ #> lazy_note in_monoN [#in_mono facts]
+ #> lazy_note in_relN [#in_rel facts]
+ #> note map_compN [#map_comp axioms]
+ #> note map_idN [#map_id axioms]
+ #> note map_wpullN [#map_wpull axioms]
+ #> note set_naturalN (#set_natural axioms)
+ #> note set_bdN (#set_bd axioms)
+ #> fold2 note witNs wit_thms
+ end
+ else
+ I)
+ |> (if fact_policy = Note_All_Facts_and_Axioms orelse
+ fact_policy = Derive_All_Facts_Note_Most then
+ lazy_note rel_IdN [#rel_Id facts]
+ #> lazy_note rel_GrN [#rel_Gr facts]
+ #> lazy_note rel_converseN [#rel_converse facts]
+ #> lazy_note rel_ON [#rel_O facts]
+ #> lazy_note map_id'N [#map_id' facts]
+ #> lazy_note map_comp'N [#map_comp' facts]
+ #> note map_congN [#map_cong axioms]
+ #> lazy_note set_natural'N (#set_natural' facts)
+ #> Local_Theory.declaration {syntax = false, pervasive = true}
+ (fn phi => Data.map (Symtab.update_new (key, morph_bnf phi bnf)))
+ else
+ I))
+ end;
+ in
+ (goals, wit_goalss, after_qed, lthy', one_step_defs)
+ end;
+
+fun add_bnf const_policy fact_policy qualify tacs wit_tac Ds =
+ (fn (goals, wit_goalss, after_qed, lthy, defs) =>
+ let
+ val wits_tac = K (TRYALL Goal.conjunction_tac) THEN' unfold_defs_tac lthy defs wit_tac;
+ val wit_goals = wit_goalss |> map Logic.mk_conjunction_balanced;
+ val wit_goal = Logic.mk_conjunction_balanced wit_goals;
+ val wit_thms =
+ Skip_Proof.prove lthy [] [] wit_goal wits_tac
+ |> Conjunction.elim_balanced (length wit_goals)
+ |> map2 (Conjunction.elim_balanced o length) wit_goalss
+ |> map (map (Thm.forall_elim_vars 0));
+ in
+ (map2 (Skip_Proof.prove lthy [] []) goals (map (unfold_defs_tac lthy defs) tacs))
+ |> (fn thms => after_qed (map single thms @ wit_thms) lthy)
+ end) oo prepare_bnf const_policy fact_policy qualify
+ (fn ctxt => singleton (Type_Infer_Context.infer_types ctxt)) Ds;
+
+val add_bnf_cmd = (fn (goals, wit_goals, after_qed, lthy, defs) =>
+ Proof.unfolding ([[(defs, [])]])
+ (Proof.theorem NONE (snd oo after_qed)
+ (map (single o rpair []) goals @ map (map (rpair [])) wit_goals) lthy)) oo
+ prepare_bnf Do_Inline user_policy I Syntax.read_term NONE;
+
+fun print_bnfs ctxt =
+ let
+ fun pretty_set sets i = Pretty.block
+ [Pretty.str (mk_setN (i + 1) ^ ":"), Pretty.brk 1,
+ Pretty.quote (Syntax.pretty_term ctxt (nth sets i))];
+
+ fun pretty_bnf (key, BNF {T = T, map = map, sets = sets, bd = bd,
+ live = live, lives = lives, dead = dead, deads = deads, ...}) =
+ Pretty.big_list
+ (Pretty.string_of (Pretty.block [Pretty.str key, Pretty.str ":", Pretty.brk 1,
+ Pretty.quote (Syntax.pretty_typ ctxt T)]))
+ ([Pretty.block [Pretty.str "live:", Pretty.brk 1, Pretty.str (string_of_int live),
+ Pretty.brk 3, Pretty.list "[" "]" (List.map (Syntax.pretty_typ ctxt) lives)],
+ Pretty.block [Pretty.str "dead:", Pretty.brk 1, Pretty.str (string_of_int dead),
+ Pretty.brk 3, Pretty.list "[" "]" (List.map (Syntax.pretty_typ ctxt) deads)],
+ Pretty.block [Pretty.str (mapN ^ ":"), Pretty.brk 1,
+ Pretty.quote (Syntax.pretty_term ctxt map)]] @
+ List.map (pretty_set sets) (0 upto length sets - 1) @
+ [Pretty.block [Pretty.str (bdN ^ ":"), Pretty.brk 1,
+ Pretty.quote (Syntax.pretty_term ctxt bd)]]);
+ in
+ Pretty.big_list "BNFs:" (map pretty_bnf (Symtab.dest (Data.get (Context.Proof ctxt))))
+ |> Pretty.writeln
+ end;
+
+val _ =
+ Outer_Syntax.improper_command @{command_spec "print_bnfs"} "print all BNFs"
+ (Scan.succeed (Toplevel.keep (print_bnfs o Toplevel.context_of)));
+
+val _ =
+ Outer_Syntax.local_theory_to_proof @{command_spec "bnf_def"} "define a BNF for an existing type"
+ (((Parse.binding --| Parse.$$$ "=") -- Parse.term --
+ (Parse.$$$ "[" |-- Parse.list Parse.term --| Parse.$$$ "]") -- Parse.term --
+ (Parse.$$$ "[" |-- Parse.list Parse.term --| Parse.$$$ "]")) >> add_bnf_cmd);
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_fp_util.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,279 @@
+(* Title: HOL/Codatatype/Tools/bnf_fp_util.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Shared library for the datatype and the codatatype construction.
+*)
+
+signature BNF_FP_UTIL =
+sig
+ val time: Timer.real_timer -> string -> Timer.real_timer
+
+ val IITN: string
+ val LevN: string
+ val algN: string
+ val behN: string
+ val bisN: string
+ val carTN: string
+ val coN: string
+ val coinductN: string
+ val coiterN: string
+ val coiter_uniqueN: string
+ val corecN: string
+ val fldN: string
+ val fld_coiterN: string
+ val fld_exhaustN: string
+ val fld_induct2N: string
+ val fld_inductN: string
+ val fld_injectN: string
+ val fld_unfN: string
+ val hsetN: string
+ val hset_recN: string
+ val inductN: string
+ val isNodeN: string
+ val iterN: string
+ val iter_uniqueN: string
+ val lsbisN: string
+ val map_simpsN: string
+ val map_uniqueN: string
+ val min_algN: string
+ val morN: string
+ val pred_coinductN: string
+ val pred_coinduct_uptoN: string
+ val recN: string
+ val rel_coinductN: string
+ val rel_coinduct_uptoN: string
+ val rvN: string
+ val set_inclN: string
+ val set_set_inclN: string
+ val strTN: string
+ val str_initN: string
+ val sum_bdN: string
+ val sum_bdTN: string
+ val unfN: string
+ val unf_coinductN: string
+ val unf_coinduct_uptoN: string
+ val unf_exhaustN: string
+ val unf_fldN: string
+ val unf_injectN: string
+ val uniqueN: string
+ val uptoN: string
+
+ val mk_exhaustN: string -> string
+ val mk_injectN: string -> string
+ val mk_nchotomyN: string -> string
+ val mk_set_simpsN: int -> string
+ val mk_set_minimalN: int -> string
+ val mk_set_inductN: int -> string
+
+ val typedef: bool -> binding option -> binding * (string * sort) list * mixfix -> term ->
+ (binding * binding) option -> tactic -> local_theory -> (string * Typedef.info) * local_theory
+
+ val split_conj_thm: thm -> thm list
+ val split_conj_prems: int -> thm -> thm
+
+ val list_all_free: term list -> term -> term
+ val list_exists_free: term list -> term -> term
+
+ val mk_Field: term -> term
+ val mk_union: term * term -> term
+
+ val mk_tactics: 'a -> 'a -> 'a -> 'a list -> 'a -> 'a -> 'a list -> 'a -> 'a -> 'a list
+
+ val fixpoint: ('a * 'a -> bool) -> ('a list -> 'a list) -> 'a list -> 'a list
+ val interleave: 'a list -> 'a list -> 'a list
+
+ val certifyT: Proof.context -> typ -> ctyp
+ val certify: Proof.context -> term -> cterm
+
+ val fp_bnf: (binding list -> typ list list -> BNF_Def.BNF list ->
+ Proof.context -> Proof.context) ->
+ binding list -> ((string * sort) * typ) list -> Proof.context -> Proof.context
+ val fp_bnf_cmd: (binding list -> typ list list -> BNF_Def.BNF list ->
+ Proof.context -> Proof.context) ->
+ binding list * (string list * string list) -> Proof.context -> Proof.context
+end;
+
+structure BNF_FP_Util : BNF_FP_UTIL =
+struct
+
+open BNF_Comp
+open BNF_Def
+open BNF_Util
+
+val timing = true;
+fun time timer msg = (if timing
+ then warning (msg ^ ": " ^ ATP_Util.string_from_time (Timer.checkRealTimer timer))
+ else (); Timer.startRealTimer ());
+
+(*TODO: is this really different from Typedef.add_typedef_global?*)
+fun typedef def opt_name typ set opt_morphs tac lthy =
+ let
+ val ((name, info), (lthy, lthy_old)) =
+ lthy
+ |> Typedef.add_typedef def opt_name typ set opt_morphs tac
+ ||> `Local_Theory.restore;
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ in
+ ((name, Typedef.transform_info phi info), lthy)
+ end;
+
+val coN = "co"
+val algN = "alg"
+val IITN = "IITN"
+val iterN = "iter"
+val coiterN = coN ^ iterN
+val uniqueN = "_unique"
+val iter_uniqueN = iterN ^ uniqueN
+val coiter_uniqueN = coiterN ^ uniqueN
+val fldN = "fld"
+val unfN = "unf"
+val fld_coiterN = fldN ^ "_" ^ coiterN
+val map_simpsN = mapN ^ "_simps"
+val map_uniqueN = mapN ^ uniqueN
+val min_algN = "min_alg"
+val morN = "mor"
+val bisN = "bis"
+val lsbisN = "lsbis"
+val sum_bdTN = "sbdT"
+val sum_bdN = "sbd"
+val carTN = "carT"
+val strTN = "strT"
+val isNodeN = "isNode"
+val LevN = "Lev"
+val rvN = "recover"
+val behN = "beh"
+fun mk_set_simpsN i = mk_setN i ^ "_simps"
+fun mk_set_minimalN i = mk_setN i ^ "_minimal"
+fun mk_set_inductN i = mk_setN i ^ "_induct"
+
+val str_initN = "str_init"
+val recN = "rec"
+val corecN = coN ^ recN
+
+val fld_unfN = fldN ^ "_" ^ unfN
+val unf_fldN = unfN ^ "_" ^ fldN
+fun mk_nchotomyN s = s ^ "_nchotomy"
+fun mk_injectN s = s ^ "_inject"
+fun mk_exhaustN s = s ^ "_exhaust"
+val fld_injectN = mk_injectN fldN
+val fld_exhaustN = mk_exhaustN fldN
+val unf_injectN = mk_injectN unfN
+val unf_exhaustN = mk_exhaustN unfN
+val inductN = "induct"
+val coinductN = coN ^ inductN
+val fld_inductN = fldN ^ "_" ^ inductN
+val fld_induct2N = fld_inductN ^ "2"
+val unf_coinductN = unfN ^ "_" ^ coinductN
+val rel_coinductN = relN ^ "_" ^ coinductN
+val pred_coinductN = predN ^ "_" ^ coinductN
+val uptoN = "upto"
+val unf_coinduct_uptoN = unf_coinductN ^ "_" ^ uptoN
+val rel_coinduct_uptoN = rel_coinductN ^ "_" ^ uptoN
+val pred_coinduct_uptoN = pred_coinductN ^ "_" ^ uptoN
+val hsetN = "Hset"
+val hset_recN = hsetN ^ "_rec"
+val set_inclN = "set_incl"
+val set_set_inclN = "set_set_incl"
+
+fun mk_Field r =
+ let val T = fst (dest_relT (fastype_of r));
+ in Const (@{const_name Field}, mk_relT (T, T) --> HOLogic.mk_setT T) $ r end;
+
+val mk_union = HOLogic.mk_binop @{const_name sup};
+
+(*dangerous; use with monotonic, converging functions only!*)
+fun fixpoint eq f X = if subset eq (f X, X) then X else fixpoint eq f (f X);
+
+val list_exists_free =
+ fold_rev (fn free => fn P =>
+ let val (x, T) = Term.dest_Free free;
+ in HOLogic.exists_const T $ Term.absfree (x, T) P end);
+
+val list_all_free =
+ fold_rev (fn free => fn P =>
+ let val (x, T) = Term.dest_Free free;
+ in HOLogic.all_const T $ Term.absfree (x, T) P end);
+
+(*stolen from ~~/src/HOL/Tools/SMT/smt_utils.ML*)
+fun certify ctxt = Thm.cterm_of (Proof_Context.theory_of ctxt);
+fun certifyT ctxt = Thm.ctyp_of (Proof_Context.theory_of ctxt);
+
+(* stolen from "~~/src/HOL/Tools/Datatype/datatype_aux.ML" *)
+fun split_conj_thm th =
+ ((th RS conjunct1)::(split_conj_thm (th RS conjunct2))) handle THM _ => [th];
+
+fun split_conj_prems limit th =
+ let
+ fun split n i th =
+ if i = n then th else split n (i + 1) (conjI RSN (i, th)) handle THM _ => th;
+ in split limit 1 th end;
+
+fun mk_tactics mid mcomp mcong snat bdco bdinf sbd inbd wpull =
+ [mid, mcomp, mcong] @ snat @ [bdco, bdinf] @ sbd @ [inbd, wpull];
+
+fun interleave xs ys = flat (map2 (fn x => fn y => [x, y]) xs ys);
+
+fun fp_sort lhss Ass = Library.sort (Term_Ord.typ_ord o pairself TFree)
+ (subtract (op =) lhss (fold (fold (insert (op =))) Ass [])) @ lhss;
+
+fun mk_fp_bnf timer construct bs sort bnfs deads lives unfold lthy =
+ let
+ (* TODO: assert that none of the deads is a lhs *)
+
+ val name = fold_rev (fn b => fn s => Binding.name_of b ^ s) bs "";
+ fun qualify i bind =
+ let val namei = if i > 0 then name ^ string_of_int i else name;
+ in
+ if member (op =) (#2 (Binding.dest bind)) (namei, true) then bind
+ else Binding.prefix_name namei bind
+ end;
+
+ val Ass = map (map dest_TFree) lives;
+ val Ds = fold (fold Term.add_tfreesT) deads [];
+
+ val timer = time (timer "Construction of BNFs");
+
+ val ((kill_poss, _), (bnfs', (unfold', lthy'))) =
+ normalize_bnfs qualify Ass Ds sort bnfs unfold lthy;
+
+ val Dss = map3 (append oo map o nth) lives kill_poss deads;
+
+ val (bnfs'', lthy'') =
+ fold_map3 (seal_bnf unfold') (map (Binding.suffix_name "BNF") bs) Dss bnfs' lthy';
+
+ val timer = time (timer "Normalization & sealing of BNFs");
+
+ val res = construct bs Dss bnfs'' lthy'';
+
+ val timer = time (timer "FP construction in total");
+ in
+ res
+ end;
+
+fun fp_bnf construct bs eqs lthy =
+ let
+ val timer = time (Timer.startRealTimer ());
+ val (lhss, rhss) = split_list eqs;
+ val sort = fp_sort lhss;
+ val ((bnfs, (Dss, Ass)), (unfold, lthy')) = apfst (apsnd split_list o split_list)
+ (fold_map2 (fn b => bnf_of_typ Smart_Inline (Binding.suffix_name "RAW" b) I sort) bs rhss
+ (empty_unfold, lthy));
+ in
+ mk_fp_bnf timer construct bs sort bnfs Dss Ass unfold lthy'
+ end;
+
+fun fp_bnf_cmd construct (bs, (raw_lhss, raw_bnfs)) lthy =
+ let
+ val timer = time (Timer.startRealTimer ());
+ val lhss = map (dest_TFree o Syntax.read_typ lthy) raw_lhss;
+ val sort = fp_sort lhss;
+ val ((bnfs, (Dss, Ass)), (unfold, lthy')) = apfst (apsnd split_list o split_list)
+ (fold_map2 (fn b => fn rawT =>
+ (bnf_of_typ Smart_Inline (Binding.suffix_name "RAW" b) I sort (Syntax.read_typ lthy rawT)))
+ bs raw_bnfs (empty_unfold, lthy));
+ in
+ mk_fp_bnf timer construct bs sort bnfs Dss Ass unfold lthy'
+ end;
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_gfp.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,2784 @@
+(* Title: HOL/Codatatype/Tools/bnf_gfp.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+Codatatype construction.
+*)
+
+signature BNF_GFP =
+sig
+ val bnf_gfp: binding list -> typ list list -> BNF_Def.BNF list -> Proof.context -> Proof.context
+end;
+
+structure BNF_GFP : BNF_GFP =
+struct
+
+open BNF_Def
+open BNF_Util
+open BNF_Tactics
+open BNF_FP_Util
+open BNF_GFP_Util
+open BNF_GFP_Tactics
+
+(*all bnfs have the same lives*)
+fun bnf_gfp bs Dss_insts bnfs lthy =
+ let
+ val timer = time (Timer.startRealTimer ());
+
+ val live = live_of_bnf (hd bnfs);
+ val n = length bnfs; (*active*)
+ val ks = 1 upto n;
+ val m = live - n (*passive, if 0 don't generate a new bnf*);
+ val ls = 1 upto m;
+ val b = Binding.name (fold_rev (fn b => fn s => Binding.name_of b ^ s) bs "");
+
+ fun note thmN thms = snd o Local_Theory.note
+ ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms);
+ fun notes thmN thmss = fold2 (fn b => fn thms => snd o Local_Theory.note
+ ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms)) bs thmss;
+
+ (* TODO: check if m, n etc are sane *)
+
+ val Dss = map (fn Ds => map TFree (fold Term.add_tfreesT Ds [])) Dss_insts;
+ val deads = distinct (op =) (flat Dss);
+ val names_lthy = fold Variable.declare_typ deads lthy;
+
+ (* tvars *)
+ val ((((((((passiveAs, activeAs), allAs)), (passiveBs, activeBs)),
+ (passiveCs, activeCs)), passiveXs), passiveYs), idxT) = names_lthy
+ |> mk_TFrees live
+ |> apfst (`(chop m))
+ ||> mk_TFrees live
+ ||>> apfst (chop m)
+ ||> mk_TFrees live
+ ||>> apfst (chop m)
+ ||>> mk_TFrees m
+ ||>> mk_TFrees m
+ ||> fst o mk_TFrees 1
+ ||> the_single;
+
+ val Ass = replicate n allAs;
+ val allBs = passiveAs @ activeBs;
+ val Bss = replicate n allBs;
+ val allCs = passiveAs @ activeCs;
+ val allCs' = passiveBs @ activeCs;
+ val Css' = replicate n allCs';
+
+ (* typs *)
+ fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs;
+ val (params, params') = `(map dest_TFree) (deads @ passiveAs);
+ val FTsAs = mk_FTs allAs;
+ val FTsBs = mk_FTs allBs;
+ val FTsCs = mk_FTs allCs;
+ val ATs = map HOLogic.mk_setT passiveAs;
+ val BTs = map HOLogic.mk_setT activeAs;
+ val B'Ts = map HOLogic.mk_setT activeBs;
+ val B''Ts = map HOLogic.mk_setT activeCs;
+ val sTs = map2 (fn T => fn U => T --> U) activeAs FTsAs;
+ val s'Ts = map2 (fn T => fn U => T --> U) activeBs FTsBs;
+ val s''Ts = map2 (fn T => fn U => T --> U) activeCs FTsCs;
+ val fTs = map2 (fn T => fn U => T --> U) activeAs activeBs;
+ val all_fTs = map2 (fn T => fn U => T --> U) allAs allBs;
+ val self_fTs = map (fn T => T --> T) activeAs;
+ val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs;
+ val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs';
+ val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs;
+ val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs;
+ val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs;
+ val setsRTs = map HOLogic.mk_setT sRTs;
+ val setRTs = map HOLogic.mk_setT RTs;
+ val all_sbisT = HOLogic.mk_tupleT setsRTs;
+ val setR'Ts = map HOLogic.mk_setT R'Ts;
+ val FRTs = mk_FTs (passiveAs @ RTs);
+ val sumBsAs = map2 (curry mk_sumT) activeBs activeAs;
+ val sumFTs = mk_FTs (passiveAs @ sumBsAs);
+ val sum_sTs = map2 (fn T => fn U => T --> U) activeAs sumFTs;
+
+ (* terms *)
+ val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;
+ val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;
+ val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;
+ val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;
+ val map_Inls = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ sumBsAs)) bnfs;
+ val map_Inls_rev = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ sumBsAs)) Bss bnfs;
+ val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;
+ val map_snds = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;
+ fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)
+ (map (replicate live) (replicate n Ts)) bnfs;
+ val setssAs = mk_setss allAs;
+ val setssAs' = transpose setssAs;
+ val bis_setss = mk_setss (passiveAs @ RTs);
+ val relsAsBs = map4 mk_rel_of_bnf Dss Ass Bss bnfs;
+ val bds = map3 mk_bd_of_bnf Dss Ass bnfs;
+ val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
+ val sum_bdT = fst (dest_relT (fastype_of sum_bd));
+ val witss = map wits_of_bnf bnfs;
+
+ val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;
+ val Zeros = map (fn empty =>
+ HOLogic.mk_tuple (map (fn U => absdummy U empty) activeAs)) emptys;
+ val hrecTs = map fastype_of Zeros;
+ val hsetTs = map (fn hrecT => Library.foldr (op -->) (sTs, HOLogic.natT --> hrecT)) hrecTs;
+
+ val (((((((((((((((((((((((((((((((((((zs, zs'), zs_copy), zs_copy2),
+ z's), As), As_copy), Bs), Bs_copy), B's), B''s), ss), sum_ss), s's), s''s), fs), fs_copy),
+ self_fs), all_fs), gs), all_gs), xFs), xFs_copy), RFs), (Rtuple, Rtuple')), (hrecs, hrecs')),
+ (nat, nat')), Rs), Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), Kss),
+ names_lthy) = lthy
+ |> mk_Frees' "b" activeAs
+ ||>> mk_Frees "b" activeAs
+ ||>> mk_Frees "b" activeAs
+ ||>> mk_Frees "b" activeBs
+ ||>> mk_Frees "A" ATs
+ ||>> mk_Frees "A" ATs
+ ||>> mk_Frees "B" BTs
+ ||>> mk_Frees "B" BTs
+ ||>> mk_Frees "B'" B'Ts
+ ||>> mk_Frees "B''" B''Ts
+ ||>> mk_Frees "s" sTs
+ ||>> mk_Frees "sums" sum_sTs
+ ||>> mk_Frees "s'" s'Ts
+ ||>> mk_Frees "s''" s''Ts
+ ||>> mk_Frees "f" fTs
+ ||>> mk_Frees "f" fTs
+ ||>> mk_Frees "f" self_fTs
+ ||>> mk_Frees "f" all_fTs
+ ||>> mk_Frees "g" gTs
+ ||>> mk_Frees "g" all_gTs
+ ||>> mk_Frees "x" FTsAs
+ ||>> mk_Frees "x" FTsAs
+ ||>> mk_Frees "x" FRTs
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Rtuple") all_sbisT
+ ||>> mk_Frees' "rec" hrecTs
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT
+ ||>> mk_Frees "R" setRTs
+ ||>> mk_Frees "R" setRTs
+ ||>> mk_Frees "R'" setR'Ts
+ ||>> mk_Frees "R" setsRTs
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") idxT
+ ||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT)
+ ||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs)
+ ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) activeAs) ATs);
+
+ val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
+ val passive_diags = map mk_diag As;
+ val active_UNIVs = map HOLogic.mk_UNIV activeAs;
+ val sum_UNIVs = map HOLogic.mk_UNIV sumBsAs;
+ val passive_ids = map HOLogic.id_const passiveAs;
+ val active_ids = map HOLogic.id_const activeAs;
+ val Inls = map2 Inl_const activeBs activeAs;
+ val fsts = map fst_const RTs;
+ val snds = map snd_const RTs;
+
+ (* thms *)
+ val bd_card_orders = map bd_card_order_of_bnf bnfs;
+ val bd_card_order = hd bd_card_orders
+ val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
+ val bd_Card_order = hd bd_Card_orders;
+ val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs;
+ val bd_Cinfinite = hd bd_Cinfinites;
+ val bd_Cnotzeros = map bd_Cnotzero_of_bnf bnfs;
+ val bd_Cnotzero = hd bd_Cnotzeros;
+ val in_bds = map in_bd_of_bnf bnfs;
+ val in_monos = map in_mono_of_bnf bnfs;
+ val map_comps = map map_comp_of_bnf bnfs;
+ val map_comp's = map map_comp'_of_bnf bnfs;
+ val map_congs = map map_cong_of_bnf bnfs;
+ val map_id's = map map_id'_of_bnf bnfs;
+ val pred_defs = map pred_def_of_bnf bnfs;
+ val rel_congs = map rel_cong_of_bnf bnfs;
+ val rel_converses = map rel_converse_of_bnf bnfs;
+ val rel_defs = map rel_def_of_bnf bnfs;
+ val rel_Grs = map rel_Gr_of_bnf bnfs;
+ val rel_Ids = map rel_Id_of_bnf bnfs;
+ val rel_monos = map rel_mono_of_bnf bnfs;
+ val rel_Os = map rel_O_of_bnf bnfs;
+ val map_wpulls = map map_wpull_of_bnf bnfs;
+ val set_bdss = map set_bd_of_bnf bnfs;
+ val set_natural'ss = map set_natural'_of_bnf bnfs;
+
+ val timer = time (timer "Extracted terms & thms");
+
+ (* derived thms *)
+
+ (*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x)=
+ map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*)
+ fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp =
+ let
+ val lhs = Term.list_comb (mapBsCs, all_gs) $
+ (Term.list_comb (mapAsBs, passive_ids @ fs) $ x);
+ val rhs =
+ Term.list_comb (mapAsCs, take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (x :: fs @ all_gs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
+ (K (mk_map_comp_id_tac map_comp))
+ end;
+
+ val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comp's;
+
+ (*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==>
+ map id ... id f(m+1) ... f(m+n) x = x*)
+ fun mk_map_congL x mapAsAs sets map_cong map_id' =
+ let
+ fun mk_prem set f z z' =
+ HOLogic.mk_Trueprop
+ (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));
+ val prems = map4 mk_prem (drop m sets) self_fs zs zs';
+ val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x))
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))
+ (K (mk_map_congL_tac m map_cong map_id'))
+ end;
+
+ val map_congL_thms = map5 mk_map_congL xFs mapsAsAs setssAs map_congs map_id's;
+ val in_mono'_thms = map (fn thm =>
+ (thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos;
+ val in_cong'_thms = map (fn bnf => in_cong_of_bnf bnf OF (replicate m refl)) bnfs;
+
+ val map_arg_cong_thms =
+ let
+ val prems = map2 (fn x => fn y =>
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))) xFs xFs_copy;
+ val maps = map (fn map => Term.list_comb (map, all_fs)) mapsAsBs;
+ val concls = map3 (fn x => fn y => fn map =>
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (map $ x, map $ y))) xFs xFs_copy maps;
+ val goals =
+ map4 (fn prem => fn concl => fn x => fn y =>
+ fold_rev Logic.all (x :: y :: all_fs) (Logic.mk_implies (prem, concl)))
+ prems concls xFs xFs_copy;
+ in
+ map (fn goal => Skip_Proof.prove lthy [] [] goal
+ (K ((hyp_subst_tac THEN' rtac refl) 1))) goals
+ end;
+
+ val timer = time (timer "Derived simple theorems");
+
+ (* coalgebra *)
+
+ val coalg_bind = Binding.suffix_name ("_" ^ coN ^ algN) b;
+ val coalg_name = Binding.name_of coalg_bind;
+ val coalg_def_bind = (Thm.def_binding coalg_bind, []);
+
+ (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in A1 .. Am B1 ... Bn)*)
+ val coalg_spec =
+ let
+ val coalgT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);
+
+ val ins = map3 mk_in (replicate n (As @ Bs)) setssAs FTsAs;
+ fun mk_coalg_conjunct B s X z z' =
+ mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));
+
+ val lhs = Term.list_comb (Free (coalg_name, coalgT), As @ Bs @ ss);
+ val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_coalg_conjunct Bs ss ins zs zs')
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (coalg_bind, NONE, NoSyn), (coalg_def_bind, coalg_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));
+ val coalg_def = Morphism.thm phi coalg_def_free;
+
+ fun mk_coalg As Bs ss =
+ let
+ val args = As @ Bs @ ss;
+ val Ts = map fastype_of args;
+ val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);
+ in
+ Term.list_comb (Const (coalg, coalgT), args)
+ end;
+
+ val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
+
+ val coalg_in_thms = map (fn i =>
+ coalg_def RS @{thm subst[of _ _ "%x. x"]} RS mk_conjunctN n i RS bspec) ks
+
+ val coalg_set_thmss =
+ let
+ val coalg_prem = HOLogic.mk_Trueprop (mk_coalg As Bs ss);
+ fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
+ fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) B);
+ val prems = map2 mk_prem zs Bs;
+ val conclss = map3 (fn s => fn x => fn sets => map2 (mk_concl s x) (As @ Bs) sets)
+ ss zs setssAs;
+ val goalss = map3 (fn x => fn prem => fn concls => map (fn concl =>
+ fold_rev Logic.all (x :: As @ Bs @ ss)
+ (Logic.list_implies (coalg_prem :: [prem], concl))) concls) zs prems conclss;
+ in
+ map (fn goals => map (fn goal => Skip_Proof.prove lthy [] [] goal
+ (K (mk_coalg_set_tac coalg_def))) goals) goalss
+ end;
+
+ val coalg_set_thmss' = transpose coalg_set_thmss;
+
+ fun mk_tcoalg ATs BTs = mk_coalg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);
+
+ val tcoalg_thm =
+ let
+ val goal = fold_rev Logic.all ss
+ (HOLogic.mk_Trueprop (mk_tcoalg passiveAs activeAs ss))
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (K (stac coalg_def 1 THEN CONJ_WRAP
+ (K (EVERY' [rtac ballI, rtac CollectI,
+ CONJ_WRAP' (K (EVERY' [rtac @{thm subset_UNIV}])) allAs] 1)) ss))
+ end;
+
+ val timer = time (timer "Coalgebra definition & thms");
+
+ (* morphism *)
+
+ val mor_bind = Binding.suffix_name ("_" ^ morN) b;
+ val mor_name = Binding.name_of mor_bind;
+ val mor_def_bind = (Thm.def_binding mor_bind, []);
+
+ (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)
+ (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.
+ Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)
+ val mor_spec =
+ let
+ val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);
+
+ fun mk_fbetw f B1 B2 z z' =
+ mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
+ fun mk_mor B mapAsBs f s s' z z' =
+ mk_Ball B (Term.absfree z' (HOLogic.mk_eq
+ (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));
+ val lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);
+ val rhs = HOLogic.mk_conj
+ (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),
+ Library.foldr1 HOLogic.mk_conj (map7 mk_mor Bs mapsAsBs fs ss s's zs zs'))
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (mor_bind, NONE, NoSyn), (mor_def_bind, mor_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
+ val mor_def = Morphism.thm phi mor_def_free;
+
+ fun mk_mor Bs1 ss1 Bs2 ss2 fs =
+ let
+ val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
+ val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
+ val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
+ in
+ Term.list_comb (Const (mor, morT), args)
+ end;
+
+ val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
+
+ val (mor_image_thms, morE_thms) =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
+ fun mk_image_goal f B1 B2 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)
+ (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_subset (mk_image f $ B1) B2)));
+ val image_goals = map3 mk_image_goal fs Bs B's;
+ fun mk_elim_goal B mapAsBs f s s' x =
+ fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)
+ (Logic.list_implies ([prem, HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B))],
+ HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x)))));
+ val elim_goals = map6 mk_elim_goal Bs mapsAsBs fs ss s's zs;
+ fun prove goal =
+ Skip_Proof.prove lthy [] [] goal (K (mk_mor_elim_tac mor_def));
+ in
+ (map prove image_goals, map prove elim_goals)
+ end;
+
+ val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;
+
+ val mor_incl_thm =
+ let
+ val prems = map2 (HOLogic.mk_Trueprop oo mk_subset) Bs Bs_copy;
+ val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))
+ (K (mk_mor_incl_tac mor_def map_id's))
+ end;
+
+ val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);
+
+ val mor_comp_thm =
+ let
+ val prems =
+ [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
+ HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
+ val concl =
+ HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
+ (Logic.list_implies (prems, concl)))
+ (K (mk_mor_comp_tac mor_def mor_image'_thms morE_thms map_comp_id_thms))
+ end;
+
+ val mor_cong_thm =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
+ val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)
+ (Logic.list_implies (prems, concl)))
+ (K ((hyp_subst_tac THEN' atac) 1))
+ end;
+
+ val mor_UNIV_thm =
+ let
+ fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
+ (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),
+ HOLogic.mk_comp (s', f));
+ val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
+ val rhs = Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct mapsAsBs fs ss s's);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (ss @ s's @ fs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
+ (K (mk_mor_UNIV_tac morE_thms mor_def))
+ end;
+
+ val mor_str_thm =
+ let
+ val maps = map2 (fn Ds => fn bnf => Term.list_comb
+ (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all ss (HOLogic.mk_Trueprop
+ (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss)))
+ (K (mk_mor_str_tac ks mor_UNIV_thm))
+ end;
+
+ val mor_sum_case_thm =
+ let
+ val maps = map3 (fn s => fn sum_s => fn map =>
+ mk_sum_case (HOLogic.mk_comp (Term.list_comb (map, passive_ids @ Inls), s)) sum_s)
+ s's sum_ss map_Inls;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (s's @ sum_ss) (HOLogic.mk_Trueprop
+ (mk_mor (map HOLogic.mk_UNIV activeBs) s's sum_UNIVs maps Inls)))
+ (K (mk_mor_sum_case_tac ks mor_UNIV_thm))
+ end;
+
+ val timer = time (timer "Morphism definition & thms");
+
+ fun hset_rec_bind j = Binding.suffix_name ("_" ^ hset_recN ^ (if m = 1 then "" else
+ string_of_int j)) b;
+ val hset_rec_name = Binding.name_of o hset_rec_bind;
+ val hset_rec_def_bind = rpair [] o Thm.def_binding o hset_rec_bind;
+
+ fun hset_rec_spec j Zero hsetT hrec hrec' =
+ let
+ fun mk_Suc s setsAs z z' =
+ let
+ val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m setsAs);
+ fun mk_UN set k = mk_UNION (set $ (s $ z)) (mk_nthN n hrec k);
+ in
+ Term.absfree z'
+ (mk_union (set $ (s $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))
+ end;
+
+ val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'
+ (HOLogic.mk_tuple (map4 mk_Suc ss setssAs zs zs')));
+
+ val lhs = Term.list_comb (Free (hset_rec_name j, hsetT), ss);
+ val rhs = mk_nat_rec Zero Suc;
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((hset_rec_frees, (_, hset_rec_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map5 (fn j => fn Zero => fn hsetT => fn hrec => fn hrec' => Specification.definition
+ (SOME (hset_rec_bind j, NONE, NoSyn),
+ (hset_rec_def_bind j, hset_rec_spec j Zero hsetT hrec hrec')))
+ ls Zeros hsetTs hrecs hrecs'
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val hset_rec_defs = map (Morphism.thm phi) hset_rec_def_frees;
+ val hset_recs = map (fst o Term.dest_Const o Morphism.term phi) hset_rec_frees;
+
+ fun mk_hset_rec ss nat i j T =
+ let
+ val args = ss @ [nat];
+ val Ts = map fastype_of ss;
+ val bTs = map domain_type Ts;
+ val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) bTs)
+ val hset_recT = Library.foldr (op -->) (Ts, HOLogic.natT --> hrecT);
+ in
+ mk_nthN n (Term.list_comb (Const (nth hset_recs (j - 1), hset_recT), args)) i
+ end;
+
+ val hset_rec_0ss = mk_rec_simps n @{thm nat_rec_0} hset_rec_defs;
+ val hset_rec_Sucss = mk_rec_simps n @{thm nat_rec_Suc} hset_rec_defs;
+ val hset_rec_0ss' = transpose hset_rec_0ss;
+ val hset_rec_Sucss' = transpose hset_rec_Sucss;
+
+ fun hset_bind i j = Binding.suffix_name ("_" ^ hsetN ^
+ (if m = 1 then "" else string_of_int j)) (nth bs (i - 1));
+ val hset_name = Binding.name_of oo hset_bind;
+ val hset_def_bind = rpair [] o Thm.def_binding oo hset_bind;
+
+ fun hset_spec i j =
+ let
+ val U = nth activeAs (i - 1);
+ val z = nth zs (i - 1);
+ val T = nth passiveAs (j - 1);
+ val setT = HOLogic.mk_setT T;
+ val hsetT = Library.foldr (op -->) (sTs, U --> setT);
+
+ val lhs = Term.list_comb (Free (hset_name i j, hsetT), ss @ [z]);
+ val rhs = mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
+ (Term.absfree nat' (mk_hset_rec ss nat i j T $ z));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((hset_frees, (_, hset_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map (fn i => fold_map (fn j => Specification.definition
+ (SOME (hset_bind i j, NONE, NoSyn), (hset_def_bind i j, hset_spec i j))) ls) ks
+ |>> map (apsnd split_list o split_list)
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val hset_defss = map (map (Morphism.thm phi)) hset_def_frees;
+ val hset_defss' = transpose hset_defss;
+ val hset_namess = map (map (fst o Term.dest_Const o Morphism.term phi)) hset_frees;
+
+ fun mk_hset ss i j T =
+ let
+ val Ts = map fastype_of ss;
+ val bTs = map domain_type Ts;
+ val hsetT = Library.foldr (op -->) (Ts, nth bTs (i - 1) --> HOLogic.mk_setT T);
+ in
+ Term.list_comb (Const (nth (nth hset_namess (i - 1)) (j - 1), hsetT), ss)
+ end;
+
+ val hsetssAs = map (fn i => map2 (mk_hset ss i) ls passiveAs) ks;
+
+ val (set_incl_hset_thmss, set_hset_incl_hset_thmsss) =
+ let
+ fun mk_set_incl_hset s x set hset = fold_rev Logic.all (x :: ss)
+ (HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (hset $ x)));
+
+ fun mk_set_hset_incl_hset s x y set hset1 hset2 =
+ fold_rev Logic.all (x :: y :: ss)
+ (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x, set $ (s $ y))),
+ HOLogic.mk_Trueprop (mk_subset (hset1 $ x) (hset2 $ y))));
+
+ val set_incl_hset_goalss =
+ map4 (fn s => fn x => fn sets => fn hsets =>
+ map2 (mk_set_incl_hset s x) (take m sets) hsets)
+ ss zs setssAs hsetssAs;
+
+ (*xk : F(i)set(m+k) (si yi) ==> F(k)_hset(j) s1 ... sn xk <= F(i)_hset(j) s1 ... sn yi*)
+ val set_hset_incl_hset_goalsss =
+ map4 (fn si => fn yi => fn sets => fn hsetsi =>
+ map3 (fn xk => fn set => fn hsetsk =>
+ map2 (mk_set_hset_incl_hset si xk yi set) hsetsk hsetsi)
+ zs_copy (drop m sets) hsetssAs)
+ ss zs setssAs hsetssAs;
+ in
+ (map3 (fn goals => fn defs => fn rec_Sucs =>
+ map3 (fn goal => fn def => fn rec_Suc =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_set_incl_hset_tac def rec_Suc)))
+ goals defs rec_Sucs)
+ set_incl_hset_goalss hset_defss hset_rec_Sucss,
+ map3 (fn goalss => fn defsi => fn rec_Sucs =>
+ map3 (fn k => fn goals => fn defsk =>
+ map4 (fn goal => fn defk => fn defi => fn rec_Suc =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_set_hset_incl_hset_tac n [defk, defi] rec_Suc k)))
+ goals defsk defsi rec_Sucs)
+ ks goalss hset_defss)
+ set_hset_incl_hset_goalsss hset_defss hset_rec_Sucss)
+ end;
+
+ val set_incl_hset_thmss' = transpose set_incl_hset_thmss;
+ val set_hset_incl_hset_thmsss' = transpose (map transpose set_hset_incl_hset_thmsss);
+ val set_hset_incl_hset_thmsss'' = map transpose set_hset_incl_hset_thmsss';
+ val set_hset_thmss = map (map (fn thm => thm RS @{thm set_mp})) set_incl_hset_thmss;
+ val set_hset_hset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))
+ set_hset_incl_hset_thmsss;
+ val set_hset_thmss' = transpose set_hset_thmss;
+ val set_hset_hset_thmsss' = transpose (map transpose set_hset_hset_thmsss);
+
+ val set_incl_hin_thmss =
+ let
+ fun mk_set_incl_hin s x hsets1 set hsets2 T =
+ fold_rev Logic.all (x :: ss @ As)
+ (Logic.list_implies
+ (map2 (fn hset => fn A => HOLogic.mk_Trueprop (mk_subset (hset $ x) A)) hsets1 As,
+ HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (mk_in As hsets2 T))));
+
+ val set_incl_hin_goalss =
+ map4 (fn s => fn x => fn sets => fn hsets =>
+ map3 (mk_set_incl_hin s x hsets) (drop m sets) hsetssAs activeAs)
+ ss zs setssAs hsetssAs;
+ in
+ map2 (map2 (fn goal => fn thms =>
+ Skip_Proof.prove lthy [] [] goal (K (mk_set_incl_hin_tac thms))))
+ set_incl_hin_goalss set_hset_incl_hset_thmsss
+ end;
+
+ val hset_minimal_thms =
+ let
+ fun mk_passive_prem set s x K =
+ Logic.all x (HOLogic.mk_Trueprop (mk_subset (set $ (s $ x)) (K $ x)));
+
+ fun mk_active_prem s x1 K1 set x2 K2 =
+ fold_rev Logic.all [x1, x2]
+ (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (x2, set $ (s $ x1))),
+ HOLogic.mk_Trueprop (mk_subset (K2 $ x2) (K1 $ x1))));
+
+ val premss = map2 (fn j => fn Ks =>
+ map4 mk_passive_prem (map (fn xs => nth xs (j - 1)) setssAs) ss zs Ks @
+ flat (map4 (fn sets => fn s => fn x1 => fn K1 =>
+ map3 (mk_active_prem s x1 K1) (drop m sets) zs_copy Ks) setssAs ss zs Ks))
+ ls Kss;
+
+ val hset_rec_minimal_thms =
+ let
+ fun mk_conjunct j T i K x = mk_subset (mk_hset_rec ss nat i j T $ x) (K $ x);
+ fun mk_concl j T Ks = list_all_free zs
+ (Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs));
+ val concls = map3 mk_concl ls passiveAs Kss;
+
+ val goals = map2 (fn prems => fn concl =>
+ Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls
+
+ val ctss =
+ map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
+ in
+ map4 (fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_hset_rec_minimal_tac m cts hset_rec_0s hset_rec_Sucs)))
+ goals ctss hset_rec_0ss' hset_rec_Sucss'
+ end;
+
+ fun mk_conjunct j T i K x = mk_subset (mk_hset ss i j T $ x) (K $ x);
+ fun mk_concl j T Ks = Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T) ks Ks zs);
+ val concls = map3 mk_concl ls passiveAs Kss;
+
+ val goals = map3 (fn Ks => fn prems => fn concl =>
+ fold_rev Logic.all (Ks @ ss @ zs)
+ (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))) Kss premss concls;
+ in
+ map3 (fn goal => fn hset_defs => fn hset_rec_minimal =>
+ Skip_Proof.prove lthy [] [] goal
+ (mk_hset_minimal_tac n hset_defs hset_rec_minimal))
+ goals hset_defss' hset_rec_minimal_thms
+ end;
+
+ val mor_hset_thmss =
+ let
+ val mor_hset_rec_thms =
+ let
+ fun mk_conjunct j T i f x B =
+ HOLogic.mk_imp (HOLogic.mk_mem (x, B), HOLogic.mk_eq
+ (mk_hset_rec s's nat i j T $ (f $ x), mk_hset_rec ss nat i j T $ x));
+
+ fun mk_concl j T = list_all_free zs
+ (Library.foldr1 HOLogic.mk_conj (map4 (mk_conjunct j T) ks fs zs Bs));
+ val concls = map2 mk_concl ls passiveAs;
+
+ val ctss =
+ map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
+
+ val goals = map (fn concl =>
+ Logic.list_implies ([coalg_prem, mor_prem], HOLogic.mk_Trueprop concl)) concls;
+ in
+ map5 (fn j => fn goal => fn cts => fn hset_rec_0s => fn hset_rec_Sucs =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (K (mk_mor_hset_rec_tac m n cts j hset_rec_0s hset_rec_Sucs
+ morE_thms set_natural'ss coalg_set_thmss))))
+ ls goals ctss hset_rec_0ss' hset_rec_Sucss'
+ end;
+
+ val mor_hset_rec_thmss = map (fn thm => map (fn i =>
+ mk_specN n thm RS mk_conjunctN n i RS mp) ks) mor_hset_rec_thms;
+
+ fun mk_prem x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (x, B));
+
+ fun mk_concl j T i f x = HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (mk_hset s's i j T $ (f $ x), mk_hset ss i j T $ x));
+
+ val goalss = map2 (fn j => fn T => map4 (fn i => fn f => fn x => fn B =>
+ fold_rev Logic.all (x :: As @ Bs @ ss @ B's @ s's @ fs)
+ (Logic.list_implies ([coalg_prem, mor_prem,
+ mk_prem x B], mk_concl j T i f x))) ks fs zs Bs) ls passiveAs;
+ in
+ map3 (map3 (fn goal => fn hset_def => fn mor_hset_rec =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_mor_hset_tac hset_def mor_hset_rec))))
+ goalss hset_defss' mor_hset_rec_thmss
+ end;
+
+ val timer = time (timer "Hereditary sets");
+
+ (* bisimulation *)
+
+ val bis_bind = Binding.suffix_name ("_" ^ bisN) b;
+ val bis_name = Binding.name_of bis_bind;
+ val bis_def_bind = (Thm.def_binding bis_bind, []);
+
+ fun mk_bis_le_conjunct R B1 B2 = mk_subset R (mk_Times (B1, B2));
+ val bis_le = Library.foldr1 HOLogic.mk_conj (map3 mk_bis_le_conjunct Rs Bs B's)
+
+ val bis_spec =
+ let
+ val bisT = Library.foldr (op -->) (ATs @ BTs @ sTs @ B'Ts @ s'Ts @ setRTs, HOLogic.boolT);
+
+ val fst_args = passive_ids @ fsts;
+ val snd_args = passive_ids @ snds;
+ fun mk_bis R s s' b1 b2 RF map1 map2 sets =
+ list_all_free [b1, b2] (HOLogic.mk_imp
+ (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
+ mk_Bex (mk_in (As @ Rs) sets (snd (dest_Free RF))) (Term.absfree (dest_Free RF)
+ (HOLogic.mk_conj
+ (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),
+ HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));
+
+ val lhs = Term.list_comb (Free (bis_name, bisT), As @ Bs @ ss @ B's @ s's @ Rs);
+ val rhs = HOLogic.mk_conj
+ (bis_le, Library.foldr1 HOLogic.mk_conj
+ (map9 mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (bis_bind, NONE, NoSyn), (bis_def_bind, bis_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val bis = fst (Term.dest_Const (Morphism.term phi bis_free));
+ val bis_def = Morphism.thm phi bis_def_free;
+
+ fun mk_bis As Bs1 ss1 Bs2 ss2 Rs =
+ let
+ val args = As @ Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;
+ val Ts = map fastype_of args;
+ val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);
+ in
+ Term.list_comb (Const (bis, bisT), args)
+ end;
+
+ val bis_cong_thm =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ (mk_bis As Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)
+ val concl = HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs_copy);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs @ Rs_copy)
+ (Logic.list_implies (prems, concl)))
+ (K ((hyp_subst_tac THEN' atac) 1))
+ end;
+
+ val bis_rel_thm =
+ let
+ fun mk_conjunct R s s' b1 b2 rel =
+ list_all_free [b1, b2] (HOLogic.mk_imp
+ (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
+ HOLogic.mk_mem (HOLogic.mk_prod (s $ b1, s' $ b2),
+ Term.list_comb (rel, passive_diags @ Rs))));
+
+ val rhs = HOLogic.mk_conj
+ (bis_le, Library.foldr1 HOLogic.mk_conj
+ (map6 mk_conjunct Rs ss s's zs z's relsAsBs))
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_bis As Bs ss B's s's Rs, rhs))))
+ (K (mk_bis_rel_tac m bis_def rel_defs map_comp's map_congs set_natural'ss))
+ end;
+
+ val bis_converse_thm =
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
+ (Logic.mk_implies
+ (HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
+ HOLogic.mk_Trueprop (mk_bis As B's s's Bs ss (map mk_converse Rs)))))
+ (K (mk_bis_converse_tac m bis_rel_thm rel_congs rel_converses));
+
+ val bis_O_thm =
+ let
+ val prems =
+ [HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's Rs),
+ HOLogic.mk_Trueprop (mk_bis As B's s's B''s s''s R's)];
+ val concl =
+ HOLogic.mk_Trueprop (mk_bis As Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ B''s @ s''s @ Rs @ R's)
+ (Logic.list_implies (prems, concl)))
+ (K (mk_bis_O_tac m bis_rel_thm rel_congs rel_Os))
+ end;
+
+ val bis_Gr_thm =
+ let
+ val concl =
+ HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map2 mk_Gr Bs fs));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ fs)
+ (Logic.list_implies ([coalg_prem, mor_prem], concl)))
+ (mk_bis_Gr_tac bis_rel_thm rel_Grs mor_image_thms morE_thms coalg_in_thms)
+ end;
+
+ val bis_image2_thm = bis_cong_thm OF
+ ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::
+ replicate n @{thm image2_Gr});
+
+ val bis_diag_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::
+ replicate n @{thm diag_Gr});
+
+ val bis_Union_thm =
+ let
+ val prem =
+ HOLogic.mk_Trueprop (mk_Ball Idx
+ (Term.absfree idx' (mk_bis As Bs ss B's s's (map (fn R => R $ idx) Ris))));
+ val concl =
+ HOLogic.mk_Trueprop (mk_bis As Bs ss B's s's (map (mk_UNION Idx) Ris));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Idx :: As @ Bs @ ss @ B's @ s's @ Ris)
+ (Logic.mk_implies (prem, concl)))
+ (mk_bis_Union_tac bis_def in_mono'_thms)
+ end;
+
+ (* self-bisimulation *)
+
+ fun mk_sbis As Bs ss Rs = mk_bis As Bs ss Bs ss Rs;
+
+ val sbis_prem = HOLogic.mk_Trueprop (mk_sbis As Bs ss sRs);
+
+ (* largest self-bisimulation *)
+
+ fun lsbis_bind i = Binding.suffix_name ("_" ^ lsbisN ^ (if n = 1 then "" else
+ string_of_int i)) b;
+ val lsbis_name = Binding.name_of o lsbis_bind;
+ val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;
+
+ val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs
+ (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis As Bs ss sRs)));
+
+ fun lsbis_spec i RT =
+ let
+ fun mk_lsbisT RT =
+ Library.foldr (op -->) (map fastype_of (As @ Bs @ ss), RT);
+ val lhs = Term.list_comb (Free (lsbis_name i, mk_lsbisT RT), As @ Bs @ ss);
+ val rhs = mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map2 (fn i => fn RT => Specification.definition
+ (SOME (lsbis_bind i, NONE, NoSyn), (lsbis_def_bind i, lsbis_spec i RT))) ks setsRTs
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val lsbis_defs = map (Morphism.thm phi) lsbis_def_frees;
+ val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;
+
+ fun mk_lsbis As Bs ss i =
+ let
+ val args = As @ Bs @ ss;
+ val Ts = map fastype_of args;
+ val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));
+ val lsbisT = Library.foldr (op -->) (Ts, RT);
+ in
+ Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)
+ end;
+
+ val sbis_lsbis_thm =
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss)
+ (HOLogic.mk_Trueprop (mk_sbis As Bs ss (map (mk_lsbis As Bs ss) ks))))
+ (K (mk_sbis_lsbis_tac lsbis_defs bis_Union_thm bis_cong_thm));
+
+ val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS
+ (bis_def RS @{thm subst[of _ _ "%x. x"]} RS conjunct1 RS mk_conjunctN n i)) ks;
+ val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS
+ (bis_def RS @{thm subst[of _ _ "%x. x"]} RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;
+
+ val incl_lsbis_thms =
+ let
+ fun mk_concl i R = HOLogic.mk_Trueprop (mk_subset R (mk_lsbis As Bs ss i));
+ val goals = map2 (fn i => fn R => fold_rev Logic.all (As @ Bs @ ss @ sRs)
+ (Logic.mk_implies (sbis_prem, mk_concl i R))) ks sRs;
+ in
+ map3 (fn goal => fn i => fn def => Skip_Proof.prove lthy [] [] goal
+ (K (mk_incl_lsbis_tac n i def))) goals ks lsbis_defs
+ end;
+
+ val equiv_lsbis_thms =
+ let
+ fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis As Bs ss i));
+ val goals = map2 (fn i => fn B => fold_rev Logic.all (As @ Bs @ ss)
+ (Logic.mk_implies (coalg_prem, mk_concl i B))) ks Bs;
+ in
+ map3 (fn goal => fn l_incl => fn incl_l =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_equiv_lsbis_tac sbis_lsbis_thm l_incl incl_l
+ bis_diag_thm bis_converse_thm bis_O_thm)))
+ goals lsbis_incl_thms incl_lsbis_thms
+ end;
+
+ val timer = time (timer "Bisimulations");
+
+ (* bounds *)
+
+ val (lthy, sbd, sbdT,
+ sbd_card_order, sbd_Cinfinite, sbd_Cnotzero, sbd_Card_order, set_sbdss, in_sbds) =
+ if n = 1
+ then (lthy, sum_bd, sum_bdT,
+ bd_card_order, bd_Cinfinite, bd_Cnotzero, bd_Card_order, set_bdss, in_bds)
+ else
+ let
+ val sbdT_bind = Binding.suffix_name ("_" ^ sum_bdTN) b;
+
+ val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =
+ typedef true NONE (sbdT_bind, params, NoSyn)
+ (HOLogic.mk_UNIV sum_bdT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
+
+ val sbdT = Type (sbdT_name, params');
+ val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);
+
+ val sbd_bind = Binding.suffix_name ("_" ^ sum_bdN) b;
+ val sbd_name = Binding.name_of sbd_bind;
+ val sbd_def_bind = (Thm.def_binding sbd_bind, []);
+
+ val sbd_spec = HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (Free (sbd_name, mk_relT (`I sbdT)), mk_dir_image sum_bd Abs_sbdT));
+
+ val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (sbd_bind, NONE, NoSyn), (sbd_def_bind, sbd_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val sbd_def = Morphism.thm phi sbd_def_free;
+ val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));
+
+ val sbdT_set_def = the (#set_def sbdT_loc_info);
+ val sbdT_Abs_inject = #Abs_inject sbdT_loc_info;
+ val sbdT_Abs_cases = #Abs_cases sbdT_loc_info;
+
+ val Abs_sbdT_inj = mk_Abs_inj_thm sbdT_set_def sbdT_Abs_inject;
+ val Abs_sbdT_bij = mk_Abs_bij_thm lthy sbdT_set_def sbdT_Abs_inject sbdT_Abs_cases;
+
+ fun mk_sum_Cinfinite [thm] = thm
+ | mk_sum_Cinfinite (thm :: thms) =
+ @{thm Cinfinite_csum_strong} OF [thm, mk_sum_Cinfinite thms];
+
+ val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
+ val sum_Card_order = sum_Cinfinite RS conjunct2;
+
+ fun mk_sum_card_order [thm] = thm
+ | mk_sum_card_order (thm :: thms) =
+ @{thm card_order_csum} OF [thm, mk_sum_card_order thms];
+
+ val sum_card_order = mk_sum_card_order bd_card_orders;
+
+ val sbd_ordIso = Local_Defs.fold lthy [sbd_def]
+ (@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order]);
+ val sbd_card_order = Local_Defs.fold lthy [sbd_def]
+ (@{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]);
+ val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];
+ val sbd_Cnotzero = sbd_Cinfinite RS @{thm Cinfinite_Cnotzero};
+ val sbd_Card_order = sbd_Cinfinite RS conjunct2;
+
+ fun mk_set_sbd i bd_Card_order bds =
+ map (fn thm => @{thm ordLeq_ordIso_trans} OF
+ [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;
+ val set_sbdss = map3 mk_set_sbd ks bd_Card_orders set_bdss;
+
+ fun mk_in_sbd i Co Cnz bd =
+ Cnz RS ((@{thm ordLeq_ordIso_trans} OF
+ [(Co RS mk_ordLeq_csum n i (Co RS @{thm ordLeq_refl})), sbd_ordIso]) RS
+ (bd RS @{thm ordLeq_transitive[OF _
+ cexp_mono2_Cnotzero[OF _ csum_Cnotzero2[OF ctwo_Cnotzero]]]}));
+ val in_sbds = map4 mk_in_sbd ks bd_Card_orders bd_Cnotzeros in_bds;
+ in
+ (lthy, sbd, sbdT,
+ sbd_card_order, sbd_Cinfinite, sbd_Cnotzero, sbd_Card_order, set_sbdss, in_sbds)
+ end;
+
+ fun mk_sbd_sbd 1 = sbd_Card_order RS @{thm ordIso_refl}
+ | mk_sbd_sbd n = @{thm csum_absorb1} OF
+ [sbd_Cinfinite, mk_sbd_sbd (n - 1) RS @{thm ordIso_imp_ordLeq}];
+
+ val sbd_sbd_thm = mk_sbd_sbd n;
+
+ val sbdTs = replicate n sbdT;
+ val sum_sbd = Library.foldr1 (uncurry mk_csum) (replicate n sbd);
+ val sum_sbdT = mk_sumTN sbdTs;
+ val sum_sbd_listT = HOLogic.listT sum_sbdT;
+ val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;
+ val bdTs = passiveAs @ replicate n sbdT;
+ val to_sbd_maps = map4 mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;
+ val bdFTs = mk_FTs bdTs;
+ val sbdFT = mk_sumTN bdFTs;
+ val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);
+ val treeQT = HOLogic.mk_setT treeT;
+ val treeTs = passiveAs @ replicate n treeT;
+ val treeQTs = passiveAs @ replicate n treeQT;
+ val treeFTs = mk_FTs treeTs;
+ val tree_maps = map4 mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;
+ val final_maps = map4 mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;
+ val tree_setss = mk_setss treeTs;
+ val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);
+
+ val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];
+ val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);
+ val Lev_recT = fastype_of Zero;
+ val LevT = Library.foldr (op -->) (sTs, HOLogic.natT --> Lev_recT);
+
+ val Nil = HOLogic.mk_tuple (map3 (fn i => fn z => fn z'=>
+ Term.absfree z' (mk_InN activeAs z i)) ks zs zs');
+ val rv_recT = fastype_of Nil;
+ val rvT = Library.foldr (op -->) (sTs, sum_sbd_listT --> rv_recT);
+
+ val (((((((((((sumx, sumx'), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')),
+ (lab, lab')), (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')),
+ names_lthy) = names_lthy
+ |> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT
+ ||>> mk_Frees' "k" sbdTs
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT
+ ||>> mk_Frees "x" bdFTs
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT;
+
+ val (k, k') = (hd kks, hd kks')
+
+ val timer = time (timer "Bounds");
+
+ (* tree coalgebra *)
+
+ fun isNode_bind i = Binding.suffix_name ("_" ^ isNodeN ^ (if n = 1 then "" else
+ string_of_int i)) b;
+ val isNode_name = Binding.name_of o isNode_bind;
+ val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;
+
+ val isNodeT =
+ Library.foldr (op -->) (map fastype_of (As @ [Kl, lab, kl]), HOLogic.boolT);
+
+ val Succs = map3 (fn i => fn k => fn k' =>
+ HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))
+ ks kks kks';
+
+ fun isNode_spec sets x i =
+ let
+ val (passive_sets, active_sets) = chop m (map (fn set => set $ x) sets);
+ val lhs = Term.list_comb (Free (isNode_name i, isNodeT), As @ [Kl, lab, kl]);
+ val rhs = list_exists_free [x]
+ (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::
+ map2 mk_subset passive_sets As @ map2 (curry HOLogic.mk_eq) active_sets Succs));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn x => fn sets => Specification.definition
+ (SOME (isNode_bind i, NONE, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))
+ ks xs isNode_setss
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val isNode_defs = map (Morphism.thm phi) isNode_def_frees;
+ val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;
+
+ fun mk_isNode As kl i =
+ Term.list_comb (Const (nth isNodes (i - 1), isNodeT), As @ [Kl, lab, kl]);
+
+ val isTree =
+ let
+ val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);
+ val Field = mk_subset Kl (mk_Field (mk_clists sum_sbd));
+ val prefCl = mk_prefCl Kl;
+
+ val tree = mk_Ball Kl (Term.absfree kl'
+ (HOLogic.mk_conj
+ (Library.foldr1 HOLogic.mk_disj (map (mk_isNode As kl) ks),
+ Library.foldr1 HOLogic.mk_conj (map4 (fn Succ => fn i => fn k => fn k' =>
+ mk_Ball Succ (Term.absfree k' (mk_isNode As
+ (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))
+ Succs ks kks kks'))));
+
+ val undef = list_all_free [kl] (HOLogic.mk_imp
+ (HOLogic.mk_not (HOLogic.mk_mem (kl, Kl)),
+ HOLogic.mk_eq (lab $ kl, mk_undefined sbdFT)));
+ in
+ Library.foldr1 HOLogic.mk_conj [empty, Field, prefCl, tree, undef]
+ end;
+
+ fun carT_bind i = Binding.suffix_name ("_" ^ carTN ^ (if n = 1 then "" else
+ string_of_int i)) b;
+ val carT_name = Binding.name_of o carT_bind;
+ val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;
+
+ fun carT_spec i =
+ let
+ val carTT = Library.foldr (op -->) (ATs, HOLogic.mk_setT treeT);
+
+ val lhs = Term.list_comb (Free (carT_name i, carTT), As);
+ val rhs = HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
+ (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),
+ HOLogic.mk_conj (isTree, mk_isNode As (HOLogic.mk_list sum_sbdT []) i))));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map (fn i => Specification.definition
+ (SOME (carT_bind i, NONE, NoSyn), (carT_def_bind i, carT_spec i))) ks
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val carT_defs = map (Morphism.thm phi) carT_def_frees;
+ val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;
+
+ fun mk_carT As i = Term.list_comb
+ (Const (nth carTs (i - 1),
+ Library.foldr (op -->) (map fastype_of As, HOLogic.mk_setT treeT)), As);
+
+ fun strT_bind i = Binding.suffix_name ("_" ^ strTN ^ (if n = 1 then "" else
+ string_of_int i)) b;
+ val strT_name = Binding.name_of o strT_bind;
+ val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;
+
+ fun strT_spec mapFT FT i =
+ let
+ val strTT = treeT --> FT;
+
+ fun mk_f i k k' =
+ let val in_k = mk_InN sbdTs k i;
+ in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;
+
+ val f = Term.list_comb (mapFT, passive_ids @ map3 mk_f ks kks kks');
+ val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));
+ val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);
+ val lhs = Free (strT_name i, strTT);
+ val rhs = HOLogic.mk_split (Term.absfree Kl' (Term.absfree lab'
+ (mk_sum_caseN fs $ (lab $ HOLogic.mk_list sum_sbdT []))));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn mapFT => fn FT => Specification.definition
+ (SOME (strT_bind i, NONE, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))
+ ks tree_maps treeFTs
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val strT_defs = map ((fn def => trans OF [def RS fun_cong, @{thm prod.cases}]) o
+ Morphism.thm phi) strT_def_frees;
+ val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;
+
+ fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);
+
+ val carTAs = map (mk_carT As) ks;
+ val carTAs_copy = map (mk_carT As_copy) ks;
+ val strTAs = map2 mk_strT treeFTs ks;
+ val hset_strTss = map (fn i => map2 (mk_hset strTAs i) ls passiveAs) ks;
+
+ val coalgT_thm =
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_coalg As carTAs strTAs)))
+ (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_natural'ss);
+
+ val card_of_carT_thms =
+ let
+ val lhs = mk_card_of
+ (HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
+ (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)), isTree))));
+ val rhs = mk_cexp
+ (if m = 0 then ctwo else
+ (mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo))
+ (mk_cexp sbd sbd);
+ val card_of_carT =
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_ordLeq lhs rhs)))
+ (K (mk_card_of_carT_tac m isNode_defs sbd_sbd_thm
+ sbd_card_order sbd_Card_order sbd_Cinfinite sbd_Cnotzero in_sbds))
+ in
+ map (fn def => @{thm ordLeq_transitive[OF
+ card_of_mono1[OF ord_eq_le_trans[OF _ Collect_restrict']]]} OF [def, card_of_carT])
+ carT_defs
+ end;
+
+ val carT_set_thmss =
+ let
+ val Kl_lab = HOLogic.mk_prod (Kl, lab);
+ fun mk_goal carT strT set k i =
+ fold_rev Logic.all (sumx :: Kl :: lab :: k :: kl :: As)
+ (Logic.list_implies (map HOLogic.mk_Trueprop
+ [HOLogic.mk_mem (Kl_lab, carT), HOLogic.mk_mem (mk_Cons sumx kl, Kl),
+ HOLogic.mk_eq (sumx, mk_InN sbdTs k i)],
+ HOLogic.mk_Trueprop (HOLogic.mk_mem
+ (HOLogic.mk_prod (mk_Shift Kl sumx, mk_shift lab sumx),
+ set $ (strT $ Kl_lab)))));
+
+ val goalss = map3 (fn carT => fn strT => fn sets =>
+ map3 (mk_goal carT strT) (drop m sets) kks ks) carTAs strTAs tree_setss;
+ in
+ map6 (fn i => fn goals =>
+ fn carT_def => fn strT_def => fn isNode_def => fn set_naturals =>
+ map2 (fn goal => fn set_natural =>
+ Skip_Proof.prove lthy [] [] goal
+ (mk_carT_set_tac n i carT_def strT_def isNode_def set_natural))
+ goals (drop m set_naturals))
+ ks goalss carT_defs strT_defs isNode_defs set_natural'ss
+ end;
+
+ val carT_set_thmss' = transpose carT_set_thmss;
+
+ val isNode_hset_thmss =
+ let
+ val Kl_lab = HOLogic.mk_prod (Kl, lab);
+ fun mk_Kl_lab carT = HOLogic.mk_mem (Kl_lab, carT);
+
+ val strT_hset_thmsss =
+ let
+ val strT_hset_thms =
+ let
+ fun mk_lab_kl i x = HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i);
+
+ fun mk_inner_conjunct j T i x set i' carT =
+ HOLogic.mk_imp (HOLogic.mk_conj (mk_Kl_lab carT, mk_lab_kl i x),
+ mk_subset (set $ x) (mk_hset strTAs i' j T $ Kl_lab));
+
+ fun mk_conjunct j T i x set =
+ Library.foldr1 HOLogic.mk_conj (map2 (mk_inner_conjunct j T i x set) ks carTAs);
+
+ fun mk_concl j T = list_all_free (Kl :: lab :: xs @ As)
+ (HOLogic.mk_imp (HOLogic.mk_mem (kl, Kl),
+ Library.foldr1 HOLogic.mk_conj (map3 (mk_conjunct j T)
+ ks xs (map (fn xs => nth xs (j - 1)) isNode_setss))));
+ val concls = map2 mk_concl ls passiveAs;
+
+ val cTs = [SOME (certifyT lthy sum_sbdT)];
+ val arg_cong_cTs = map (SOME o certifyT lthy) treeFTs;
+ val ctss =
+ map (fn phi => map (SOME o certify lthy) [Term.absfree kl' phi, kl]) concls;
+
+ val goals = map HOLogic.mk_Trueprop concls;
+ in
+ map5 (fn j => fn goal => fn cts => fn set_incl_hsets => fn set_hset_incl_hsetss =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (K (mk_strT_hset_tac n m j arg_cong_cTs cTs cts
+ carT_defs strT_defs isNode_defs
+ set_incl_hsets set_hset_incl_hsetss coalg_set_thmss' carT_set_thmss'
+ coalgT_thm set_natural'ss))))
+ ls goals ctss set_incl_hset_thmss' set_hset_incl_hset_thmsss''
+ end;
+
+ val strT_hset'_thms = map (fn thm => mk_specN (2 + n + m) thm RS mp) strT_hset_thms;
+ in
+ map (fn thm => map (fn i => map (fn i' =>
+ thm RS mk_conjunctN n i RS mk_conjunctN n i' RS mp) ks) ks) strT_hset'_thms
+ end;
+
+ val carT_prems = map (fn carT =>
+ HOLogic.mk_Trueprop (HOLogic.mk_mem (Kl_lab, carT))) carTAs_copy;
+ val prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, Kl));
+ val in_prems = map (fn hsets =>
+ HOLogic.mk_Trueprop (HOLogic.mk_mem (Kl_lab, mk_in As hsets treeT))) hset_strTss;
+ val isNode_premss = replicate n (map (HOLogic.mk_Trueprop o mk_isNode As_copy kl) ks);
+ val conclss = replicate n (map (HOLogic.mk_Trueprop o mk_isNode As kl) ks);
+ in
+ map5 (fn carT_prem => fn isNode_prems => fn in_prem => fn concls => fn strT_hset_thmss =>
+ map4 (fn isNode_prem => fn concl => fn isNode_def => fn strT_hset_thms =>
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Kl :: lab :: kl :: As @ As_copy)
+ (Logic.list_implies ([carT_prem, prem, isNode_prem, in_prem], concl)))
+ (mk_isNode_hset_tac n isNode_def strT_hset_thms))
+ isNode_prems concls isNode_defs
+ (if m = 0 then replicate n [] else transpose strT_hset_thmss))
+ carT_prems isNode_premss in_prems conclss
+ (if m = 0 then replicate n [] else transpose (map transpose strT_hset_thmsss))
+ end;
+
+ val timer = time (timer "Tree coalgebra");
+
+ fun mk_to_sbd s x i i' =
+ mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
+ fun mk_from_sbd s x i i' =
+ mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
+
+ fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>
+ thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;
+
+ val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};
+ val to_sbd_thmss = mk_to_sbd_thmss @{thm toCard};
+ val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};
+
+ val Lev_bind = Binding.suffix_name ("_" ^ LevN) b;
+ val Lev_name = Binding.name_of Lev_bind;
+ val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);
+
+ val Lev_spec =
+ let
+ fun mk_Suc i s setsAs a a' =
+ let
+ val sets = drop m setsAs;
+ fun mk_set i' set b =
+ let
+ val Cons = HOLogic.mk_eq (kl_copy,
+ mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)
+ val b_set = HOLogic.mk_mem (b, set $ (s $ a));
+ val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);
+ in
+ HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]
+ (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))
+ end;
+ in
+ Term.absfree a' (Library.foldl1 mk_union (map3 mk_set ks sets zs_copy))
+ end;
+
+ val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'
+ (HOLogic.mk_tuple (map5 mk_Suc ks ss setssAs zs zs')));
+
+ val lhs = Term.list_comb (Free (Lev_name, LevT), ss);
+ val rhs = mk_nat_rec Zero Suc;
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (Lev_bind, NONE, NoSyn), (Lev_def_bind, Lev_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val Lev_def = Morphism.thm phi Lev_def_free;
+ val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));
+
+ fun mk_Lev ss nat i =
+ let
+ val Ts = map fastype_of ss;
+ val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->
+ HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));
+ in
+ mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i
+ end;
+
+ val Lev_0s = flat (mk_rec_simps n @{thm nat_rec_0} [Lev_def]);
+ val Lev_Sucs = flat (mk_rec_simps n @{thm nat_rec_Suc} [Lev_def]);
+
+ val rv_bind = Binding.suffix_name ("_" ^ rvN) b;
+ val rv_name = Binding.name_of rv_bind;
+ val rv_def_bind = rpair [] (Thm.def_binding rv_bind);
+
+ val rv_spec =
+ let
+ fun mk_Cons i s b b' =
+ let
+ fun mk_case i' =
+ Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));
+ in
+ Term.absfree b' (mk_sum_caseN (map mk_case ks) $ sumx)
+ end;
+
+ val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'
+ (HOLogic.mk_tuple (map4 mk_Cons ks ss zs zs'))));
+
+ val lhs = Term.list_comb (Free (rv_name, rvT), ss);
+ val rhs = mk_list_rec Nil Cons;
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (rv_bind, NONE, NoSyn), (rv_def_bind, rv_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val rv_def = Morphism.thm phi rv_def_free;
+ val rv = fst (Term.dest_Const (Morphism.term phi rv_free));
+
+ fun mk_rv ss kl i =
+ let
+ val Ts = map fastype_of ss;
+ val As = map domain_type Ts;
+ val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->
+ HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));
+ in
+ mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i
+ end;
+
+ val rv_Nils = flat (mk_rec_simps n @{thm list_rec_Nil} [rv_def]);
+ val rv_Conss = flat (mk_rec_simps n @{thm list_rec_Cons} [rv_def]);
+
+ fun beh_bind i = Binding.suffix_name ("_" ^ behN ^ (if n = 1 then "" else
+ string_of_int i)) b;
+ val beh_name = Binding.name_of o beh_bind;
+ val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;
+
+ fun beh_spec i z =
+ let
+ val mk_behT = Library.foldr (op -->) (map fastype_of (ss @ [z]), treeT);
+
+ fun mk_case i to_sbd_map s k k' =
+ Term.absfree k' (mk_InN bdFTs
+ (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);
+
+ val Lab = Term.absfree kl' (mk_If
+ (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))
+ (mk_sum_caseN (map5 mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z))
+ (mk_undefined sbdFT));
+
+ val lhs = Term.list_comb (Free (beh_name i, mk_behT), ss) $ z;
+ val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
+ (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map2 (fn i => fn z => Specification.definition
+ (SOME (beh_bind i, NONE, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+
+ val beh_defs = map (Morphism.thm phi) beh_def_frees;
+ val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;
+
+ fun mk_beh ss i =
+ let
+ val Ts = map fastype_of ss;
+ val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);
+ in
+ Term.list_comb (Const (nth behs (i - 1), behT), ss)
+ end;
+
+ val Lev_sbd_thms =
+ let
+ fun mk_conjunct i z = mk_subset (mk_Lev ss nat i $ z) (mk_Field (mk_clists sum_sbd));
+ val goal = list_all_free zs
+ (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
+
+ val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
+
+ val Lev_sbd = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (K (mk_Lev_sbd_tac cts Lev_0s Lev_Sucs to_sbd_thmss)));
+
+ val Lev_sbd' = mk_specN n Lev_sbd;
+ in
+ map (fn i => Lev_sbd' RS mk_conjunctN n i) ks
+ end;
+
+ val (length_Lev_thms, length_Lev'_thms) =
+ let
+ fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
+ HOLogic.mk_eq (mk_size kl, nat));
+ val goal = list_all_free (kl :: zs)
+ (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
+
+ val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
+
+ val length_Lev = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (K (mk_length_Lev_tac cts Lev_0s Lev_Sucs)));
+
+ val length_Lev' = mk_specN (n + 1) length_Lev;
+ val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;
+
+ fun mk_goal i z = fold_rev Logic.all (z :: kl :: nat :: ss) (Logic.mk_implies
+ (HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z)),
+ HOLogic.mk_Trueprop (HOLogic.mk_mem (kl, mk_Lev ss (mk_size kl) i $ z))));
+ val goals = map2 mk_goal ks zs;
+
+ val length_Levs' = map2 (fn goal => fn length_Lev =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_length_Lev'_tac length_Lev))) goals length_Levs;
+ in
+ (length_Levs, length_Levs')
+ end;
+
+ val prefCl_Lev_thms =
+ let
+ fun mk_conjunct i z = HOLogic.mk_imp
+ (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), mk_subset kl_copy kl),
+ HOLogic.mk_mem (kl_copy, mk_Lev ss (mk_size kl_copy) i $ z));
+ val goal = list_all_free (kl :: kl_copy :: zs)
+ (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
+
+ val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
+
+ val prefCl_Lev = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (K (mk_prefCl_Lev_tac cts Lev_0s Lev_Sucs)));
+
+ val prefCl_Lev' = mk_specN (n + 2) prefCl_Lev;
+ in
+ map (fn i => prefCl_Lev' RS mk_conjunctN n i RS mp) ks
+ end;
+
+ val rv_last_thmss =
+ let
+ fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]
+ (HOLogic.mk_eq
+ (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,
+ mk_InN activeAs z_copy i'));
+ val goal = list_all_free (k :: zs)
+ (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>
+ Library.foldr1 HOLogic.mk_conj
+ (map2 (mk_conjunct i z) ks zs_copy)) ks zs));
+
+ val cTs = [SOME (certifyT lthy sum_sbdT)];
+ val cts = map (SOME o certify lthy) [Term.absfree kl' goal, kl];
+
+ val rv_last = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (K (mk_rv_last_tac cTs cts rv_Nils rv_Conss)));
+
+ val rv_last' = mk_specN (n + 1) rv_last;
+ in
+ map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks
+ end;
+
+ val set_rv_Lev_thmsss = if m = 0 then replicate n (replicate n []) else
+ let
+ fun mk_case s sets z z_free = Term.absfree z_free (Library.foldr1 HOLogic.mk_conj
+ (map2 (fn set => fn A => mk_subset (set $ (s $ z)) A) (take m sets) As));
+
+ fun mk_conjunct i z B = HOLogic.mk_imp
+ (HOLogic.mk_conj (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), HOLogic.mk_mem (z, B)),
+ mk_sum_caseN (map4 mk_case ss setssAs zs zs') $ (mk_rv ss kl i $ z));
+
+ val goal = list_all_free (kl :: zs)
+ (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct ks zs Bs));
+
+ val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
+
+ val set_rv_Lev = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] []
+ (Logic.mk_implies (coalg_prem, HOLogic.mk_Trueprop goal))
+ (K (mk_set_rv_Lev_tac m cts Lev_0s Lev_Sucs rv_Nils rv_Conss
+ coalg_set_thmss from_to_sbd_thmss)));
+
+ val set_rv_Lev' = mk_specN (n + 1) set_rv_Lev;
+ in
+ map (fn i => map (fn i' =>
+ split_conj_thm (if n = 1 then set_rv_Lev' RS mk_conjunctN n i RS mp
+ else set_rv_Lev' RS mk_conjunctN n i RS mp RSN
+ (2, @{thm sum_case_cong} RS @{thm subst[of _ _ "%x. x"]}) RS
+ (mk_sum_casesN n i' RS @{thm subst[of _ _ "%x. x"]}))) ks) ks
+ end;
+
+ val set_Lev_thmsss =
+ let
+ fun mk_conjunct i z =
+ let
+ fun mk_conjunct' i' sets s z' =
+ let
+ fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp
+ (HOLogic.mk_mem (z'', set $ (s $ z')),
+ HOLogic.mk_mem (mk_append (kl,
+ HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),
+ mk_Lev ss (HOLogic.mk_Suc nat) i $ z));
+ in
+ HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),
+ (Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct'' ks (drop m sets) zs_copy2)))
+ end;
+ in
+ HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
+ Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct' ks setssAs ss zs_copy))
+ end;
+
+ val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)
+ (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
+
+ val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
+
+ val set_Lev = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (K (mk_set_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss)));
+
+ val set_Lev' = mk_specN (3 * n + 1) set_Lev;
+ in
+ map (fn i => map (fn i' => map (fn i'' => set_Lev' RS
+ mk_conjunctN n i RS mp RS
+ mk_conjunctN n i' RS mp RS
+ mk_conjunctN n i'' RS mp) ks) ks) ks
+ end;
+
+ val set_image_Lev_thmsss =
+ let
+ fun mk_conjunct i z =
+ let
+ fun mk_conjunct' i' sets =
+ let
+ fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp
+ (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),
+ HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));
+ in
+ HOLogic.mk_imp (HOLogic.mk_mem
+ (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),
+ mk_Lev ss (HOLogic.mk_Suc nat) i $ z),
+ (Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct'' ks sets ss zs_copy)))
+ end;
+ in
+ HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
+ Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))
+ end;
+
+ val goal = list_all_free (kl :: k :: zs @ zs_copy)
+ (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
+
+ val cts = map (SOME o certify lthy) [Term.absfree nat' goal, nat];
+
+ val set_image_Lev = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (K (mk_set_image_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss
+ from_to_sbd_thmss to_sbd_inj_thmss)));
+
+ val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;
+ in
+ map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS
+ mk_conjunctN n i RS mp RS
+ mk_conjunctN n i'' RS mp RS
+ mk_conjunctN n i' RS mp) ks) ks) ks
+ end;
+
+ val mor_beh_thm =
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss) (Logic.mk_implies (coalg_prem,
+ HOLogic.mk_Trueprop (mk_mor Bs ss carTAs strTAs (map (mk_beh ss) ks)))))
+ (mk_mor_beh_tac m mor_def mor_cong_thm
+ beh_defs carT_defs strT_defs isNode_defs
+ to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbd_thms
+ length_Lev_thms length_Lev'_thms prefCl_Lev_thms rv_last_thmss
+ set_rv_Lev_thmsss set_Lev_thmsss set_image_Lev_thmsss
+ set_natural'ss coalg_set_thmss map_comp_id_thms map_congs map_arg_cong_thms);
+
+ val timer = time (timer "Behavioral morphism");
+
+ fun mk_LSBIS As i = mk_lsbis As (map (mk_carT As) ks) strTAs i;
+ fun mk_car_final As i =
+ mk_quotient (mk_carT As i) (mk_LSBIS As i);
+ fun mk_str_final As i =
+ mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),
+ passive_ids @ map (mk_proj o mk_LSBIS As) ks), nth strTAs (i - 1)));
+
+ val car_finalAs = map (mk_car_final As) ks;
+ val str_finalAs = map (mk_str_final As) ks;
+ val car_finals = map (mk_car_final passive_UNIVs) ks;
+ val str_finals = map (mk_str_final passive_UNIVs) ks;
+
+ val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;
+ val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;
+
+ val congruent_str_final_thms =
+ let
+ fun mk_goal R final_map strT =
+ fold_rev Logic.all As (HOLogic.mk_Trueprop
+ (mk_congruent R (HOLogic.mk_comp
+ (Term.list_comb (final_map, passive_ids @ map (mk_proj o mk_LSBIS As) ks), strT))));
+
+ val goals = map3 mk_goal (map (mk_LSBIS As) ks) final_maps strTAs;
+ in
+ map4 (fn goal => fn lsbisE => fn map_comp_id => fn map_cong =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_congruent_str_final_tac m lsbisE map_comp_id map_cong equiv_LSBIS_thms)))
+ goals lsbisE_thms map_comp_id_thms map_congs
+ end;
+
+ val coalg_final_thm = Skip_Proof.prove lthy [] [] (fold_rev Logic.all As
+ (HOLogic.mk_Trueprop (mk_coalg As car_finalAs str_finalAs)))
+ (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms
+ set_natural'ss coalgT_set_thmss));
+
+ val mor_T_final_thm = Skip_Proof.prove lthy [] [] (fold_rev Logic.all As
+ (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finalAs str_finalAs
+ (map (mk_proj o mk_LSBIS As) ks))))
+ (K (mk_mor_T_final_tac mor_def congruent_str_final_thms equiv_LSBIS_thms));
+
+ val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];
+ val in_car_final_thms = map (fn mor_image' => mor_image' OF
+ [tcoalg_thm RS mor_final_thm, UNIV_I]) mor_image'_thms;
+
+ val timer = time (timer "Final coalgebra");
+
+ val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
+ lthy
+ |> fold_map3 (fn b => fn car_final => fn in_car_final =>
+ typedef false NONE (b, params, NoSyn) car_final NONE
+ (EVERY' [rtac exI, rtac in_car_final] 1)) bs car_finals in_car_final_thms
+ |>> apsnd split_list o split_list;
+
+ val Ts = map (fn name => Type (name, params')) T_names;
+ fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
+ val Ts' = mk_Ts passiveBs;
+ val Ts'' = mk_Ts passiveCs;
+ val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;
+ val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;
+
+ val Reps = map #Rep T_loc_infos;
+ val Rep_injects = map #Rep_inject T_loc_infos;
+ val Rep_inverses = map #Rep_inverse T_loc_infos;
+ val Abs_inverses = map #Abs_inverse T_loc_infos;
+
+ val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
+
+ val UNIVs = map HOLogic.mk_UNIV Ts;
+ val FTs = mk_FTs (passiveAs @ Ts);
+ val FTs' = mk_FTs (passiveBs @ Ts);
+ val prodTs = map (HOLogic.mk_prodT o `I) Ts;
+ val prodFTs = mk_FTs (passiveAs @ prodTs);
+ val FTs_setss = mk_setss (passiveAs @ Ts);
+ val FTs'_setss = mk_setss (passiveBs @ Ts);
+ val prodFT_setss = mk_setss (passiveAs @ prodTs);
+ val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;
+ val map_FT_nths = map2 (fn Ds =>
+ mk_map_of_bnf Ds (passiveAs @ prodTs) (passiveAs @ Ts)) Dss bnfs;
+ val fstsTs = map fst_const prodTs;
+ val sndsTs = map snd_const prodTs;
+ val unfTs = map2 (curry (op -->)) Ts FTs;
+ val fldTs = map2 (curry (op -->)) FTs Ts;
+ val coiter_fTs = map2 (curry op -->) activeAs Ts;
+ val corec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) sum_sTs;
+ val corec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls;
+ val corec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_Inls_rev;
+ val corec_Inls = map (Term.subst_atomic_types (activeBs ~~ Ts)) Inls;
+
+ val (((((((((((((Jzs, Jzs'), (Jz's, Jz's')), Jzs_copy), Jzs1), Jzs2), Jpairs),
+ FJzs), TRs), coiter_fs), coiter_fs_copy), corec_ss), phis), names_lthy) = names_lthy
+ |> mk_Frees' "z" Ts
+ ||>> mk_Frees' "z" Ts'
+ ||>> mk_Frees "z" Ts
+ ||>> mk_Frees "z1" Ts
+ ||>> mk_Frees "z2" Ts
+ ||>> mk_Frees "j" (map2 (curry HOLogic.mk_prodT) Ts Ts')
+ ||>> mk_Frees "x" prodFTs
+ ||>> mk_Frees "R" (map (mk_relT o `I) Ts)
+ ||>> mk_Frees "f" coiter_fTs
+ ||>> mk_Frees "g" coiter_fTs
+ ||>> mk_Frees "s" corec_sTs
+ ||>> mk_Frees "phi" (map (fn T => T --> T --> HOLogic.boolT) Ts);
+
+ fun unf_bind i = Binding.suffix_name ("_" ^ unfN) (nth bs (i - 1));
+ val unf_name = Binding.name_of o unf_bind;
+ val unf_def_bind = rpair [] o Thm.def_binding o unf_bind;
+
+ fun unf_spec i rep str map_FT unfT Jz Jz' =
+ let
+ val lhs = Free (unf_name i, unfT);
+ val rhs = Term.absfree Jz'
+ (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $
+ (str $ (rep $ Jz)));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((unf_frees, (_, unf_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map7 (fn i => fn rep => fn str => fn map => fn unfT => fn Jz => fn Jz' =>
+ Specification.definition
+ (SOME (unf_bind i, NONE, NoSyn), (unf_def_bind i, unf_spec i rep str map unfT Jz Jz')))
+ ks Rep_Ts str_finals map_FTs unfTs Jzs Jzs'
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ fun mk_unfs passive =
+ map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (deads @ passive)) o
+ Morphism.term phi) unf_frees;
+ val unfs = mk_unfs passiveAs;
+ val unf's = mk_unfs passiveBs;
+ val unf_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) unf_def_frees;
+
+ val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;
+ val (mor_Rep_thm, mor_Abs_thm) =
+ let
+ val mor_Rep =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor UNIVs unfs car_finals str_finals Rep_Ts))
+ (mk_mor_Rep_tac m (mor_def :: unf_defs) Reps Abs_inverses coalg_final_set_thmss
+ map_comp_id_thms map_congL_thms);
+
+ val mor_Abs =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs unfs Abs_Ts))
+ (mk_mor_Abs_tac (mor_def :: unf_defs) Abs_inverses);
+ in
+ (mor_Rep, mor_Abs)
+ end;
+
+ val timer = time (timer "unf definitions & thms");
+
+ fun coiter_bind i = Binding.suffix_name ("_" ^ coN ^ iterN) (nth bs (i - 1));
+ val coiter_name = Binding.name_of o coiter_bind;
+ val coiter_def_bind = rpair [] o Thm.def_binding o coiter_bind;
+
+ fun coiter_spec i T AT abs f z z' =
+ let
+ val coiterT = Library.foldr (op -->) (sTs, AT --> T);
+
+ val lhs = Term.list_comb (Free (coiter_name i, coiterT), ss);
+ val rhs = Term.absfree z' (abs $ (f $ z));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((coiter_frees, (_, coiter_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map7 (fn i => fn T => fn AT => fn abs => fn f => fn z => fn z' =>
+ Specification.definition
+ (SOME (coiter_bind i, NONE, NoSyn), (coiter_def_bind i, coiter_spec i T AT abs f z z')))
+ ks Ts activeAs Abs_Ts (map (fn i => HOLogic.mk_comp
+ (mk_proj (mk_LSBIS passive_UNIVs i), mk_beh ss i)) ks) zs zs'
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val coiters = map (fst o dest_Const o Morphism.term phi) coiter_frees;
+ fun mk_coiter Ts ss i = Term.list_comb (Const (nth coiters (i - 1), Library.foldr (op -->)
+ (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
+ val coiter_defs = map ((fn thm => thm RS fun_cong) o Morphism.thm phi) coiter_def_frees;
+
+ val mor_coiter_thm =
+ let
+ val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;
+ val morEs' = map (fn thm =>
+ (thm OF [tcoalg_thm RS mor_final_thm, UNIV_I]) RS sym) morE_thms;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all ss
+ (HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs unfs (map (mk_coiter Ts ss) ks))))
+ (K (mk_mor_coiter_tac m mor_UNIV_thm unf_defs coiter_defs Abs_inverses' morEs'
+ map_comp_id_thms map_congs))
+ end;
+ val coiter_thms = map (fn thm => (thm OF [mor_coiter_thm, UNIV_I]) RS sym) morE_thms;
+
+ val (raw_coind_thms, raw_coind_thm) =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_sbis passive_UNIVs UNIVs unfs TRs);
+ val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map2 (fn R => fn T => mk_subset R (Id_const T)) TRs Ts));
+ val goal = fold_rev Logic.all TRs (Logic.mk_implies (prem, concl));
+ in
+ `split_conj_thm (Skip_Proof.prove lthy [] [] goal
+ (K (mk_raw_coind_tac bis_def bis_cong_thm bis_O_thm bis_converse_thm bis_Gr_thm
+ tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm
+ lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects)))
+ end;
+
+ val unique_mor_thms =
+ let
+ val prems = [HOLogic.mk_Trueprop (mk_coalg passive_UNIVs Bs ss), HOLogic.mk_Trueprop
+ (HOLogic.mk_conj (mk_mor Bs ss UNIVs unfs coiter_fs,
+ mk_mor Bs ss UNIVs unfs coiter_fs_copy))];
+ fun mk_fun_eq B f g z = HOLogic.mk_imp
+ (HOLogic.mk_mem (z, B), HOLogic.mk_eq (f $ z, g $ z));
+ val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map4 mk_fun_eq Bs coiter_fs coiter_fs_copy zs));
+
+ val unique_mor = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ coiter_fs @ coiter_fs_copy @ zs)
+ (Logic.list_implies (prems, unique)))
+ (K (mk_unique_mor_tac raw_coind_thms bis_image2_thm));
+ in
+ map (fn thm => conjI RSN (2, thm RS mp)) (split_conj_thm unique_mor)
+ end;
+
+ val (coiter_unique_mor_thms, coiter_unique_mor_thm) =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs unfs coiter_fs);
+ fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_coiter Ts ss i);
+ val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map2 mk_fun_eq coiter_fs ks));
+
+ val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);
+ val mor_thm = mor_comp_thm OF [tcoalg_thm RS mor_final_thm, mor_Abs_thm];
+
+ val unique_mor = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (ss @ coiter_fs) (Logic.mk_implies (prem, unique)))
+ (K (mk_coiter_unique_mor_tac raw_coind_thms bis_thm mor_thm coiter_defs));
+ in
+ `split_conj_thm unique_mor
+ end;
+
+ val (coiter_unique_thms, coiter_unique_thm) = `split_conj_thm (split_conj_prems n
+ (mor_UNIV_thm RS @{thm ssubst[of _ _ "%x. x"]} RS coiter_unique_mor_thm));
+
+ val coiter_unf_thms = map (fn thm => mor_id_thm RS thm RS sym) coiter_unique_mor_thms;
+
+ val coiter_o_unf_thms =
+ let
+ val mor = mor_comp_thm OF [mor_str_thm, mor_coiter_thm];
+ in
+ map2 (fn unique => fn coiter_fld =>
+ trans OF [mor RS unique, coiter_fld]) coiter_unique_mor_thms coiter_unf_thms
+ end;
+
+ val timer = time (timer "coiter definitions & thms");
+
+ val map_unfs = map2 (fn Ds => fn bnf =>
+ Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
+ map HOLogic.id_const passiveAs @ unfs)) Dss bnfs;
+
+ fun fld_bind i = Binding.suffix_name ("_" ^ fldN) (nth bs (i - 1));
+ val fld_name = Binding.name_of o fld_bind;
+ val fld_def_bind = rpair [] o Thm.def_binding o fld_bind;
+
+ fun fld_spec i fldT =
+ let
+ val lhs = Free (fld_name i, fldT);
+ val rhs = mk_coiter Ts map_unfs i;
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((fld_frees, (_, fld_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map2 (fn i => fn fldT =>
+ Specification.definition
+ (SOME (fld_bind i, NONE, NoSyn), (fld_def_bind i, fld_spec i fldT))) ks fldTs
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ fun mk_flds params =
+ map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
+ fld_frees;
+ val flds = mk_flds params';
+ val fld_defs = map (Morphism.thm phi) fld_def_frees;
+
+ val fld_o_unf_thms = map2 (Local_Defs.fold lthy o single) fld_defs coiter_o_unf_thms;
+
+ val unf_o_fld_thms =
+ let
+ fun mk_goal unf fld FT =
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf, fld), HOLogic.id_const FT));
+ val goals = map3 mk_goal unfs flds FTs;
+ in
+ map5 (fn goal => fn fld_def => fn coiter => fn map_comp_id => fn map_congL =>
+ Skip_Proof.prove lthy [] [] goal
+ (mk_unf_o_fld_tac fld_def coiter map_comp_id map_congL coiter_o_unf_thms))
+ goals fld_defs coiter_thms map_comp_id_thms map_congL_thms
+ end;
+
+ val unf_fld_thms = map (fn thm => thm RS @{thm pointfree_idE}) unf_o_fld_thms;
+ val fld_unf_thms = map (fn thm => thm RS @{thm pointfree_idE}) fld_o_unf_thms;
+
+ val bij_unf_thms =
+ map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) fld_o_unf_thms unf_o_fld_thms;
+ val inj_unf_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_unf_thms;
+ val surj_unf_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_unf_thms;
+ val unf_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_unf_thms;
+ val unf_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_unf_thms;
+ val unf_exhaust_thms = map (fn thm => thm RS @{thm exE}) unf_nchotomy_thms;
+
+ val bij_fld_thms =
+ map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) unf_o_fld_thms fld_o_unf_thms;
+ val inj_fld_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_fld_thms;
+ val surj_fld_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_fld_thms;
+ val fld_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_fld_thms;
+ val fld_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_fld_thms;
+ val fld_exhaust_thms = map (fn thm => thm RS @{thm exE}) fld_nchotomy_thms;
+
+ val fld_coiter_thms = map3 (fn unf_inject => fn coiter => fn unf_fld =>
+ iffD1 OF [unf_inject, trans OF [coiter, unf_fld RS sym]])
+ unf_inject_thms coiter_thms unf_fld_thms;
+
+ val timer = time (timer "fld definitions & thms");
+
+ val corec_Inl_sum_thms =
+ let
+ val mor = mor_comp_thm OF [mor_sum_case_thm, mor_coiter_thm];
+ in
+ map2 (fn unique => fn coiter_unf =>
+ trans OF [mor RS unique, coiter_unf]) coiter_unique_mor_thms coiter_unf_thms
+ end;
+
+ fun corec_bind i = Binding.suffix_name ("_" ^ coN ^ recN) (nth bs (i - 1));
+ val corec_name = Binding.name_of o corec_bind;
+ val corec_def_bind = rpair [] o Thm.def_binding o corec_bind;
+
+ fun corec_spec i T AT =
+ let
+ val corecT = Library.foldr (op -->) (corec_sTs, AT --> T);
+ val maps = map3 (fn unf => fn sum_s => fn map => mk_sum_case
+ (HOLogic.mk_comp (Term.list_comb (map, passive_ids @ corec_Inls), unf)) sum_s)
+ unfs corec_ss corec_maps;
+
+ val lhs = Term.list_comb (Free (corec_name i, corecT), corec_ss);
+ val rhs = HOLogic.mk_comp (mk_coiter Ts maps i, Inr_const T AT);
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((corec_frees, (_, corec_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn T => fn AT =>
+ Specification.definition
+ (SOME (corec_bind i, NONE, NoSyn), (corec_def_bind i, corec_spec i T AT)))
+ ks Ts activeAs
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val corecs = map (fst o dest_Const o Morphism.term phi) corec_frees;
+ fun mk_corec ss i = Term.list_comb (Const (nth corecs (i - 1), Library.foldr (op -->)
+ (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
+ val corec_defs = map (Morphism.thm phi) corec_def_frees;
+
+ val sum_cases =
+ map2 (fn T => fn i => mk_sum_case (HOLogic.id_const T) (mk_corec corec_ss i)) Ts ks;
+ val corec_thms =
+ let
+ fun mk_goal i corec_s corec_map unf z =
+ let
+ val lhs = unf $ (mk_corec corec_ss i $ z);
+ val rhs = Term.list_comb (corec_map, passive_ids @ sum_cases) $ (corec_s $ z);
+ in
+ fold_rev Logic.all (z :: corec_ss) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
+ end;
+ val goals = map5 mk_goal ks corec_ss corec_maps_rev unfs zs;
+ in
+ map3 (fn goal => fn coiter => fn map_cong =>
+ Skip_Proof.prove lthy [] [] goal
+ (mk_corec_tac m corec_defs coiter map_cong corec_Inl_sum_thms))
+ goals coiter_thms map_congs
+ end;
+
+ val timer = time (timer "corec definitions & thms");
+
+ val (unf_coinduct_thm, coinduct_params, rel_coinduct_thm, pred_coinduct_thm,
+ unf_coinduct_upto_thm, rel_coinduct_upto_thm, pred_coinduct_upto_thm) =
+ let
+ val zs = Jzs1 @ Jzs2;
+ val frees = phis @ zs;
+
+ fun mk_Ids Id = if Id then map Id_const passiveAs else map mk_diag passive_UNIVs;
+
+ fun mk_phi upto_eq phi z1 z2 = if upto_eq
+ then Term.absfree (dest_Free z1) (Term.absfree (dest_Free z2)
+ (HOLogic.mk_disj (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2))))
+ else phi;
+
+ fun phi_rels upto_eq = map4 (fn phi => fn T => fn z1 => fn z2 =>
+ HOLogic.Collect_const (HOLogic.mk_prodT (T, T)) $
+ HOLogic.mk_split (mk_phi upto_eq phi z1 z2)) phis Ts Jzs1 Jzs2;
+
+ val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;
+
+ fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));
+ val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map3 mk_concl phis Jzs1 Jzs2));
+
+ fun mk_rel_prem upto_eq phi unf rel Jz Jz_copy =
+ let
+ val concl = HOLogic.mk_mem (HOLogic.mk_tuple [unf $ Jz, unf $ Jz_copy],
+ Term.list_comb (rel, mk_Ids upto_eq @ phi_rels upto_eq));
+ in
+ HOLogic.mk_Trueprop
+ (list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl)))
+ end;
+
+ val rel_prems = map5 (mk_rel_prem false) phis unfs rels Jzs Jzs_copy;
+ val rel_upto_prems = map5 (mk_rel_prem true) phis unfs rels Jzs Jzs_copy;
+
+ val rel_coinduct_goal = fold_rev Logic.all frees (Logic.list_implies (rel_prems, concl));
+ val coinduct_params = rev (Term.add_tfrees rel_coinduct_goal []);
+
+ val rel_coinduct = Local_Defs.unfold lthy @{thms diag_UNIV}
+ (Skip_Proof.prove lthy [] [] rel_coinduct_goal
+ (K (mk_rel_coinduct_tac ks raw_coind_thm bis_rel_thm)));
+
+ fun mk_unf_prem upto_eq phi unf map_nth sets Jz Jz_copy FJz =
+ let
+ val xs = [Jz, Jz_copy];
+
+ fun mk_map_conjunct nths x =
+ HOLogic.mk_eq (Term.list_comb (map_nth, passive_ids @ nths) $ FJz, unf $ x);
+
+ fun mk_set_conjunct set phi z1 z2 =
+ list_all_free [z1, z2]
+ (HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (z1, z2), set $ FJz),
+ mk_phi upto_eq phi z1 z2 $ z1 $ z2));
+
+ val concl = list_exists_free [FJz] (HOLogic.mk_conj
+ (Library.foldr1 HOLogic.mk_conj (map2 mk_map_conjunct [fstsTs, sndsTs] xs),
+ Library.foldr1 HOLogic.mk_conj
+ (map4 mk_set_conjunct (drop m sets) phis Jzs1 Jzs2)));
+ in
+ fold_rev Logic.all xs (Logic.mk_implies
+ (HOLogic.mk_Trueprop (Term.list_comb (phi, xs)), HOLogic.mk_Trueprop concl))
+ end;
+
+ fun mk_unf_prems upto_eq =
+ map7 (mk_unf_prem upto_eq) phis unfs map_FT_nths prodFT_setss Jzs Jzs_copy FJzs
+
+ val unf_prems = mk_unf_prems false;
+ val unf_upto_prems = mk_unf_prems true;
+
+ val unf_coinduct_goal = fold_rev Logic.all frees (Logic.list_implies (unf_prems, concl));
+ val unf_coinduct = Skip_Proof.prove lthy [] [] unf_coinduct_goal
+ (K (mk_unf_coinduct_tac m ks raw_coind_thm bis_def));
+
+ val cTs = map (SOME o certifyT lthy o TFree) coinduct_params;
+ val cts = map3 (SOME o certify lthy ooo mk_phi true) phis Jzs1 Jzs2;
+
+ val rel_coinduct_upto = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all zs (Logic.list_implies (rel_upto_prems, concl)))
+ (K (mk_rel_coinduct_upto_tac m cTs cts rel_coinduct rel_monos rel_Ids)));
+
+ val unf_coinduct_upto = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all zs (Logic.list_implies (unf_upto_prems, concl)))
+ (K (mk_unf_coinduct_upto_tac ks cTs cts unf_coinduct bis_def
+ (tcoalg_thm RS bis_diag_thm))));
+
+ val pred_coinduct = rel_coinduct
+ |> Local_Defs.unfold lthy @{thms Id_def'}
+ |> Local_Defs.fold lthy pred_defs;
+ val pred_coinduct_upto = rel_coinduct_upto
+ |> Local_Defs.unfold lthy @{thms Id_def'}
+ |> Local_Defs.fold lthy pred_defs;
+ in
+ (unf_coinduct, rev (Term.add_tfrees unf_coinduct_goal []), rel_coinduct, pred_coinduct,
+ unf_coinduct_upto, rel_coinduct_upto, pred_coinduct_upto)
+ end;
+
+ val timer = time (timer "coinduction");
+
+ (*register new codatatypes as BNFs*)
+ val lthy = if m = 0 then lthy else
+ let
+ val fTs = map2 (curry op -->) passiveAs passiveBs;
+ val gTs = map2 (curry op -->) passiveBs passiveCs;
+ val f1Ts = map2 (curry op -->) passiveAs passiveYs;
+ val f2Ts = map2 (curry op -->) passiveBs passiveYs;
+ val p1Ts = map2 (curry op -->) passiveXs passiveAs;
+ val p2Ts = map2 (curry op -->) passiveXs passiveBs;
+ val pTs = map2 (curry op -->) passiveXs passiveCs;
+ val uTs = map2 (curry op -->) Ts Ts';
+ val JRTs = map2 (curry mk_relT) passiveAs passiveBs;
+ val JphiTs = map2 (fn T => fn U => T --> U --> HOLogic.boolT) passiveAs passiveBs;
+ val prodTs = map2 (curry HOLogic.mk_prodT) Ts Ts';
+ val B1Ts = map HOLogic.mk_setT passiveAs;
+ val B2Ts = map HOLogic.mk_setT passiveBs;
+ val AXTs = map HOLogic.mk_setT passiveXs;
+ val XTs = mk_Ts passiveXs;
+ val YTs = mk_Ts passiveYs;
+
+ val ((((((((((((((((((((fs, fs'), (fs_copy, fs'_copy)), (gs, gs')), us),
+ (Jys, Jys')), (Jys_copy, Jys'_copy)), set_induct_phiss), JRs), Jphis),
+ B1s), B2s), AXs), Xs), f1s), f2s), p1s), p2s), ps), (ys, ys')), names_lthy) = names_lthy
+ |> mk_Frees' "f" fTs
+ ||>> mk_Frees' "f" fTs
+ ||>> mk_Frees' "g" gTs
+ ||>> mk_Frees "u" uTs
+ ||>> mk_Frees' "b" Ts'
+ ||>> mk_Frees' "b" Ts'
+ ||>> mk_Freess "phi" (map (fn T => map (fn U => T --> U --> HOLogic.boolT) Ts) passiveAs)
+ ||>> mk_Frees "R" JRTs
+ ||>> mk_Frees "phi" JphiTs
+ ||>> mk_Frees "B1" B1Ts
+ ||>> mk_Frees "B2" B2Ts
+ ||>> mk_Frees "A" AXTs
+ ||>> mk_Frees "x" XTs
+ ||>> mk_Frees "f1" f1Ts
+ ||>> mk_Frees "f2" f2Ts
+ ||>> mk_Frees "p1" p1Ts
+ ||>> mk_Frees "p2" p2Ts
+ ||>> mk_Frees "p" pTs
+ ||>> mk_Frees' "y" passiveAs;
+
+ val map_FTFT's = map2 (fn Ds =>
+ mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+
+ fun mk_maps ATs BTs Ts mk_T =
+ map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs;
+ fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts);
+ fun mk_map mk_const mk_T Ts fs Ts' unfs mk_maps =
+ mk_coiter Ts' (map2 (fn unf => fn Fmap =>
+ HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, unf)) unfs (mk_maps Ts mk_T));
+ val mk_map_id = mk_map HOLogic.id_const I;
+ val mk_mapsAB = mk_maps passiveAs passiveBs;
+ val mk_mapsBC = mk_maps passiveBs passiveCs;
+ val mk_mapsAC = mk_maps passiveAs passiveCs;
+ val mk_mapsAY = mk_maps passiveAs passiveYs;
+ val mk_mapsBY = mk_maps passiveBs passiveYs;
+ val mk_mapsXA = mk_maps passiveXs passiveAs;
+ val mk_mapsXB = mk_maps passiveXs passiveBs;
+ val mk_mapsXC = mk_maps passiveXs passiveCs;
+ val fs_maps = map (mk_map_id Ts fs Ts' unfs mk_mapsAB) ks;
+ val fs_copy_maps = map (mk_map_id Ts fs_copy Ts' unfs mk_mapsAB) ks;
+ val gs_maps = map (mk_map_id Ts' gs Ts'' unf's mk_mapsBC) ks;
+ val fgs_maps =
+ map (mk_map_id Ts (map2 (curry HOLogic.mk_comp) gs fs) Ts'' unfs mk_mapsAC) ks;
+ val Xunfs = mk_unfs passiveXs;
+ val UNIV's = map HOLogic.mk_UNIV Ts';
+ val CUNIVs = map HOLogic.mk_UNIV passiveCs;
+ val UNIV''s = map HOLogic.mk_UNIV Ts'';
+ val fstsTsTs' = map fst_const prodTs;
+ val sndsTsTs' = map snd_const prodTs;
+ val unf''s = mk_unfs passiveCs;
+ val f1s_maps = map (mk_map_id Ts f1s YTs unfs mk_mapsAY) ks;
+ val f2s_maps = map (mk_map_id Ts' f2s YTs unf's mk_mapsBY) ks;
+ val pid_maps = map (mk_map_id XTs ps Ts'' Xunfs mk_mapsXC) ks;
+ val pfst_Fmaps =
+ map (mk_Fmap fst_const p1s prodTs) (mk_mapsXA prodTs (fst o HOLogic.dest_prodT));
+ val psnd_Fmaps =
+ map (mk_Fmap snd_const p2s prodTs) (mk_mapsXB prodTs (snd o HOLogic.dest_prodT));
+ val p1id_Fmaps = map (mk_Fmap HOLogic.id_const p1s prodTs) (mk_mapsXA prodTs I);
+ val p2id_Fmaps = map (mk_Fmap HOLogic.id_const p2s prodTs) (mk_mapsXB prodTs I);
+ val pid_Fmaps = map (mk_Fmap HOLogic.id_const ps prodTs) (mk_mapsXC prodTs I);
+
+ val (map_simp_thms, map_thms) =
+ let
+ fun mk_goal fs_map map unf unf' = fold_rev Logic.all fs
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf', fs_map),
+ HOLogic.mk_comp (Term.list_comb (map, fs @ fs_maps), unf))));
+ val goals = map4 mk_goal fs_maps map_FTFT's unfs unf's;
+ val cTs = map (SOME o certifyT lthy) FTs';
+ val maps = map5 (fn goal => fn cT => fn coiter => fn map_comp' => fn map_cong =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_map_tac m n cT coiter map_comp' map_cong)))
+ goals cTs coiter_thms map_comp's map_congs;
+ in
+ map_split (fn thm => (thm RS @{thm pointfreeE}, thm)) maps
+ end;
+
+ val map_comp_thms =
+ let
+ val goal = fold_rev Logic.all (fs @ gs)
+ (HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map3 (fn fmap => fn gmap => fn fgmap =>
+ HOLogic.mk_eq (HOLogic.mk_comp (gmap, fmap), fgmap))
+ fs_maps gs_maps fgs_maps)))
+ in
+ split_conj_thm (Skip_Proof.prove lthy [] [] goal
+ (K (mk_map_comp_tac m n map_thms map_comps map_congs coiter_unique_thm)))
+ end;
+
+ val (map_unique_thms, map_unique_thm) =
+ let
+ fun mk_prem u map unf unf' =
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf', u),
+ HOLogic.mk_comp (Term.list_comb (map, fs @ us), unf)));
+ val prems = map4 mk_prem us map_FTFT's unfs unf's;
+ val goal =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map2 (curry HOLogic.mk_eq) us fs_maps));
+ val unique = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (us @ fs) (Logic.list_implies (prems, goal)))
+ (mk_map_unique_tac coiter_unique_thm map_comps);
+ in
+ `split_conj_thm unique
+ end;
+
+ val timer = time (timer "map functions for the new codatatypes");
+
+ val bd = mk_ccexp sbd sbd;
+
+ val timer = time (timer "bounds for the new codatatypes");
+
+ fun mk_set_Ts T = passiveAs @ replicate n (HOLogic.mk_setT T);
+ val setsss = map (mk_setss o mk_set_Ts) passiveAs;
+ val map_setss = map (fn T => map2 (fn Ds =>
+ mk_map_of_bnf Ds (passiveAs @ Ts) (mk_set_Ts T)) Dss bnfs) passiveAs;
+
+ val setss_by_bnf = map (fn i => map2 (mk_hset unfs i) ls passiveAs) ks;
+ val setss_by_bnf' = map (fn i => map2 (mk_hset unf's i) ls passiveBs) ks;
+ val setss_by_range = transpose setss_by_bnf;
+
+ val set_simp_thmss =
+ let
+ fun mk_simp_goal relate pas_set act_sets sets unf z set =
+ relate (set $ z, mk_union (pas_set $ (unf $ z),
+ Library.foldl1 mk_union
+ (map2 (fn X => mk_UNION (X $ (unf $ z))) act_sets sets)));
+ fun mk_goals eq =
+ map2 (fn i => fn sets =>
+ map4 (fn Fsets =>
+ mk_simp_goal eq (nth Fsets (i - 1)) (drop m Fsets) sets)
+ FTs_setss unfs Jzs sets)
+ ls setss_by_range;
+
+ val le_goals = map
+ (fold_rev Logic.all Jzs o HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj)
+ (mk_goals (uncurry mk_subset));
+ val set_le_thmss = map split_conj_thm
+ (map4 (fn goal => fn hset_minimal => fn set_hsets => fn set_hset_hsetss =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_set_le_tac n hset_minimal set_hsets set_hset_hsetss)))
+ le_goals hset_minimal_thms set_hset_thmss' set_hset_hset_thmsss');
+
+ val simp_goalss = map (map2 (fn z => fn goal =>
+ Logic.all z (HOLogic.mk_Trueprop goal)) Jzs)
+ (mk_goals HOLogic.mk_eq);
+ in
+ map4 (map4 (fn goal => fn set_le => fn set_incl_hset => fn set_hset_incl_hsets =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_set_simp_tac n set_le set_incl_hset set_hset_incl_hsets))))
+ simp_goalss set_le_thmss set_incl_hset_thmss' set_hset_incl_hset_thmsss'
+ end;
+
+ val timer = time (timer "set functions for the new codatatypes");
+
+ val colss = map2 (fn j => fn T =>
+ map (fn i => mk_hset_rec unfs nat i j T) ks) ls passiveAs;
+ val colss' = map2 (fn j => fn T =>
+ map (fn i => mk_hset_rec unf's nat i j T) ks) ls passiveBs;
+ val Xcolss = map2 (fn j => fn T =>
+ map (fn i => mk_hset_rec Xunfs nat i j T) ks) ls passiveXs;
+
+ val col_natural_thmss =
+ let
+ fun mk_col_natural f map z col col' =
+ HOLogic.mk_eq (mk_image f $ (col $ z), col' $ (map $ z));
+
+ fun mk_goal f cols cols' = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
+ (map4 (mk_col_natural f) fs_maps Jzs cols cols'));
+
+ val goals = map3 mk_goal fs colss colss';
+
+ val ctss =
+ map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) goals;
+
+ val thms = map4 (fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (mk_col_natural_tac cts rec_0s rec_Sucs map_simp_thms set_natural'ss)))
+ goals ctss hset_rec_0ss' hset_rec_Sucss';
+ in
+ map (split_conj_thm o mk_specN n) thms
+ end;
+
+ val col_bd_thmss =
+ let
+ fun mk_col_bd z col = mk_ordLeq (mk_card_of (col $ z)) sbd;
+
+ fun mk_goal cols = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
+ (map2 mk_col_bd Jzs cols));
+
+ val goals = map mk_goal colss;
+
+ val ctss =
+ map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) goals;
+
+ val thms = map5 (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] (HOLogic.mk_Trueprop goal)
+ (K (mk_col_bd_tac m j cts rec_0s rec_Sucs
+ sbd_Card_order sbd_Cinfinite set_sbdss))))
+ ls goals ctss hset_rec_0ss' hset_rec_Sucss';
+ in
+ map (split_conj_thm o mk_specN n) thms
+ end;
+
+ val map_cong_thms =
+ let
+ val cTs = map (SOME o certifyT lthy o
+ Term.typ_subst_atomic (passiveAs ~~ passiveBs) o TFree) coinduct_params;
+
+ fun mk_prem z set f g y y' =
+ mk_Ball (set $ z) (Term.absfree y' (HOLogic.mk_eq (f $ y, g $ y)));
+
+ fun mk_prems sets z =
+ Library.foldr1 HOLogic.mk_conj (map5 (mk_prem z) sets fs fs_copy ys ys')
+
+ fun mk_map_cong sets z fmap gmap =
+ HOLogic.mk_imp (mk_prems sets z, HOLogic.mk_eq (fmap $ z, gmap $ z));
+
+ fun mk_coind_body sets (x, T) z fmap gmap y y_copy =
+ HOLogic.mk_conj
+ (HOLogic.mk_mem (z, HOLogic.mk_Collect (x, T, mk_prems sets z)),
+ HOLogic.mk_conj (HOLogic.mk_eq (y, fmap $ z),
+ HOLogic.mk_eq (y_copy, gmap $ z)))
+
+ fun mk_cphi sets (z' as (x, T)) z fmap gmap y' y y'_copy y_copy =
+ HOLogic.mk_exists (x, T, mk_coind_body sets z' z fmap gmap y y_copy)
+ |> Term.absfree y'_copy
+ |> Term.absfree y'
+ |> certify lthy;
+
+ val cphis =
+ map9 mk_cphi setss_by_bnf Jzs' Jzs fs_maps fs_copy_maps Jys' Jys Jys'_copy Jys_copy;
+
+ val coinduct = Drule.instantiate' cTs (map SOME cphis) unf_coinduct_thm;
+
+ val goal =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map4 mk_map_cong setss_by_bnf Jzs fs_maps fs_copy_maps));
+
+ val thm = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (K (mk_mcong_tac m (rtac coinduct) map_comp's map_simp_thms map_congs set_natural'ss
+ set_hset_thmss set_hset_hset_thmsss)))
+ in
+ split_conj_thm thm
+ end;
+
+ val B1_ins = map2 (mk_in B1s) setss_by_bnf Ts;
+ val B2_ins = map2 (mk_in B2s) setss_by_bnf' Ts';
+ val thePulls = map4 mk_thePull B1_ins B2_ins f1s_maps f2s_maps;
+ val thePullTs = passiveXs @ map2 (curry HOLogic.mk_prodT) Ts Ts';
+ val thePull_ins = map2 (mk_in (AXs @ thePulls)) (mk_setss thePullTs) (mk_FTs thePullTs);
+ val pickFs = map5 mk_pickWP thePull_ins pfst_Fmaps psnd_Fmaps
+ (map2 (curry (op $)) unfs Jzs) (map2 (curry (op $)) unf's Jz's);
+ val pickF_ss = map3 (fn pickF => fn z => fn z' =>
+ HOLogic.mk_split (Term.absfree z (Term.absfree z' pickF))) pickFs Jzs' Jz's';
+ val picks = map (mk_coiter XTs pickF_ss) ks;
+
+ val wpull_prem = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map8 mk_wpull AXs B1s B2s f1s f2s (replicate m NONE) p1s p2s));
+
+ val map_eq_thms = map2 (fn simp => fn diff => box_equals OF [diff RS iffD2, simp, simp])
+ map_simp_thms unf_inject_thms;
+ val map_wpull_thms = map (fn thm => thm OF
+ (replicate m asm_rl @ replicate n @{thm wpull_thePull})) map_wpulls;
+ val pickWP_assms_tacs =
+ map3 mk_pickWP_assms_tac set_incl_hset_thmss set_incl_hin_thmss map_eq_thms;
+
+ val coalg_thePull_thm =
+ let
+ val coalg = HOLogic.mk_Trueprop
+ (mk_coalg CUNIVs thePulls (map2 (curry HOLogic.mk_comp) pid_Fmaps pickF_ss));
+ val goal = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s @ ps)
+ (Logic.mk_implies (wpull_prem, coalg));
+ in
+ Skip_Proof.prove lthy [] [] goal (mk_coalg_thePull_tac m coalg_def map_wpull_thms
+ set_natural'ss pickWP_assms_tacs)
+ end;
+
+ val (mor_thePull_fst_thm, mor_thePull_snd_thm, mor_thePull_pick_thm) =
+ let
+ val mor_fst = HOLogic.mk_Trueprop
+ (mk_mor thePulls (map2 (curry HOLogic.mk_comp) p1id_Fmaps pickF_ss)
+ UNIVs unfs fstsTsTs');
+ val mor_snd = HOLogic.mk_Trueprop
+ (mk_mor thePulls (map2 (curry HOLogic.mk_comp) p2id_Fmaps pickF_ss)
+ UNIV's unf's sndsTsTs');
+ val mor_pick = HOLogic.mk_Trueprop
+ (mk_mor thePulls (map2 (curry HOLogic.mk_comp) pid_Fmaps pickF_ss)
+ UNIV''s unf''s (map2 (curry HOLogic.mk_comp) pid_maps picks));
+
+ val goal_fst = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)
+ (Logic.mk_implies (wpull_prem, mor_fst));
+ val goal_snd = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)
+ (Logic.mk_implies (wpull_prem, mor_snd));
+ val goal_pick = fold_rev Logic.all (AXs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s @ ps)
+ (Logic.mk_implies (wpull_prem, mor_pick));
+ in
+ (Skip_Proof.prove lthy [] [] goal_fst (mk_mor_thePull_fst_tac m mor_def map_wpull_thms
+ map_comp's pickWP_assms_tacs),
+ Skip_Proof.prove lthy [] [] goal_snd (mk_mor_thePull_snd_tac m mor_def map_wpull_thms
+ map_comp's pickWP_assms_tacs),
+ Skip_Proof.prove lthy [] [] goal_pick (mk_mor_thePull_pick_tac mor_def coiter_thms
+ map_comp's))
+ end;
+
+ val pick_col_thmss =
+ let
+ fun mk_conjunct AX Jpair pick thePull col =
+ HOLogic.mk_imp (HOLogic.mk_mem (Jpair, thePull), mk_subset (col $ (pick $ Jpair)) AX);
+
+ fun mk_concl AX cols =
+ list_all_free Jpairs (Library.foldr1 HOLogic.mk_conj
+ (map4 (mk_conjunct AX) Jpairs picks thePulls cols));
+
+ val concls = map2 mk_concl AXs Xcolss;
+
+ val ctss =
+ map (fn phi => map (SOME o certify lthy) [Term.absfree nat' phi, nat]) concls;
+
+ val goals =
+ map (fn concl => Logic.mk_implies (wpull_prem, HOLogic.mk_Trueprop concl)) concls;
+
+ val thms = map5 (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
+ singleton (Proof_Context.export names_lthy lthy) (Skip_Proof.prove lthy [] [] goal
+ (mk_pick_col_tac m j cts rec_0s rec_Sucs coiter_thms set_natural'ss map_wpull_thms
+ pickWP_assms_tacs)))
+ ls goals ctss hset_rec_0ss' hset_rec_Sucss';
+ in
+ map (map (fn thm => thm RS mp) o split_conj_thm o mk_specN n) thms
+ end;
+
+ val timer = time (timer "helpers for BNF properties");
+
+ val map_id_tacs = map2 (K oo mk_map_id_tac map_thms) coiter_unique_thms coiter_unf_thms;
+ val map_comp_tacs = map (fn thm => K (rtac (thm RS sym) 1)) map_comp_thms;
+ val map_cong_tacs = map (mk_map_cong_tac m) map_cong_thms;
+ val set_nat_tacss =
+ map2 (map2 (K oo mk_set_natural_tac)) hset_defss (transpose col_natural_thmss);
+
+ val bd_co_tacs = replicate n (K (mk_bd_card_order_tac sbd_card_order));
+ val bd_cinf_tacs = replicate n (K (mk_bd_cinfinite_tac sbd_Cinfinite));
+
+ val set_bd_tacss =
+ map2 (map2 (K oo mk_set_bd_tac sbd_Cinfinite)) hset_defss (transpose col_bd_thmss);
+
+ val in_bd_tacs = map7 (fn i => fn isNode_hsets => fn carT_def =>
+ fn card_of_carT => fn mor_image => fn Rep_inverse => fn mor_hsets =>
+ K (mk_in_bd_tac (nth isNode_hsets (i - 1)) isNode_hsets carT_def
+ card_of_carT mor_image Rep_inverse mor_hsets
+ sbd_Cnotzero sbd_Card_order mor_Rep_thm coalgT_thm mor_T_final_thm tcoalg_thm))
+ ks isNode_hset_thmss carT_defs card_of_carT_thms
+ mor_image'_thms Rep_inverses (transpose mor_hset_thmss);
+
+ val map_wpull_tacs =
+ map3 (K ooo mk_wpull_tac m coalg_thePull_thm mor_thePull_fst_thm mor_thePull_snd_thm
+ mor_thePull_pick_thm) unique_mor_thms (transpose pick_col_thmss) hset_defss;
+
+ val tacss = map9 mk_tactics map_id_tacs map_comp_tacs map_cong_tacs set_nat_tacss bd_co_tacs
+ bd_cinf_tacs set_bd_tacss in_bd_tacs map_wpull_tacs;
+
+ val fld_witss =
+ let
+ val witss = map2 (fn Ds => fn bnf => mk_wits_of_bnf
+ (replicate (nwits_of_bnf bnf) Ds)
+ (replicate (nwits_of_bnf bnf) (passiveAs @ Ts)) bnf) Dss bnfs;
+ fun close_wit (I, wit) = fold_rev Term.absfree (map (nth ys') I) wit;
+ fun wit_apply (arg_I, arg_wit) (fun_I, fun_wit) =
+ (union (op =) arg_I fun_I, fun_wit $ arg_wit);
+
+ fun gen_arg support i =
+ if i < m then [([i], nth ys i)]
+ else maps (mk_wit support (nth flds (i - m)) (i - m)) (nth support (i - m))
+ and mk_wit support fld i (I, wit) =
+ let val args = map (gen_arg (nth_map i (remove (op =) (I, wit)) support)) I;
+ in
+ (args, [([], wit)])
+ |-> fold (map_product wit_apply)
+ |> map (apsnd (fn t => fld $ t))
+ |> minimize_wits
+ end;
+ in
+ map3 (fn fld => fn i => map close_wit o minimize_wits o maps (mk_wit witss fld i))
+ flds (0 upto n - 1) witss
+ end;
+
+ val wit_tac = mk_wit_tac n unf_fld_thms (flat set_simp_thmss) (maps wit_thms_of_bnf bnfs);
+
+ val (Jbnfs, lthy) =
+ fold_map6 (fn tacs => fn b => fn map => fn sets => fn T => fn wits =>
+ add_bnf Dont_Inline user_policy I tacs wit_tac (SOME deads)
+ ((((b, fold_rev Term.absfree fs' map), sets), absdummy T bd), wits))
+ tacss bs fs_maps setss_by_bnf Ts fld_witss lthy;
+
+ val fold_maps = Local_Defs.fold lthy (map (fn bnf =>
+ mk_unabs_def m (map_def_of_bnf bnf RS @{thm meta_eq_to_obj_eq})) Jbnfs);
+
+ val fold_sets = Local_Defs.fold lthy (maps (fn bnf =>
+ map (fn thm => thm RS @{thm meta_eq_to_obj_eq}) (set_defs_of_bnf bnf)) Jbnfs);
+
+ val timer = time (timer "registered new codatatypes as BNFs");
+
+ val (set_incl_thmss, set_set_incl_thmsss, set_induct_thms) =
+ let
+ fun tinst_of unf =
+ map (SOME o certify lthy) (unf :: remove (op =) unf unfs);
+ fun tinst_of' unf = case tinst_of unf of t :: ts => t :: NONE :: ts;
+ val Tinst = map (pairself (certifyT lthy))
+ (map Logic.varifyT_global (deads @ allAs) ~~ (deads @ passiveAs @ Ts));
+ val set_incl_thmss =
+ map2 (fn unf => map (singleton (Proof_Context.export names_lthy lthy) o
+ fold_sets o Drule.instantiate' [] (tinst_of' unf) o
+ Thm.instantiate (Tinst, []) o Drule.zero_var_indexes))
+ unfs set_incl_hset_thmss;
+
+ val tinst = interleave (map (SOME o certify lthy) unfs) (replicate n NONE)
+ val set_minimal_thms =
+ map (fold_sets o Drule.instantiate' [] tinst o Thm.instantiate (Tinst, []) o
+ Drule.zero_var_indexes)
+ hset_minimal_thms;
+
+ val set_set_incl_thmsss =
+ map2 (fn unf => map (map (singleton (Proof_Context.export names_lthy lthy) o
+ fold_sets o Drule.instantiate' [] (NONE :: tinst_of' unf) o
+ Thm.instantiate (Tinst, []) o Drule.zero_var_indexes)))
+ unfs set_hset_incl_hset_thmsss;
+
+ val set_set_incl_thmsss' = transpose (map transpose set_set_incl_thmsss);
+
+ val incls =
+ maps (map (fn thm => thm RS @{thm subset_Collect_iff})) set_incl_thmss @
+ @{thms subset_Collect_iff[OF subset_refl]};
+
+ fun mk_induct_tinst phis jsets y y' =
+ map4 (fn phi => fn jset => fn Jz => fn Jz' =>
+ SOME (certify lthy (Term.absfree Jz' (HOLogic.mk_Collect (fst y', snd y',
+ HOLogic.mk_conj (HOLogic.mk_mem (y, jset $ Jz), phi $ y $ Jz))))))
+ phis jsets Jzs Jzs';
+ val set_induct_thms =
+ map6 (fn set_minimal => fn set_set_inclss => fn jsets => fn y => fn y' => fn phis =>
+ ((set_minimal
+ |> Drule.instantiate' [] (mk_induct_tinst phis jsets y y')
+ |> fold_sets |> Local_Defs.unfold lthy incls) OF
+ (replicate n ballI @
+ maps (map (fn thm => thm RS @{thm subset_CollectI})) set_set_inclss))
+ |> singleton (Proof_Context.export names_lthy lthy)
+ |> rule_by_tactic lthy (ALLGOALS (TRY o etac asm_rl)))
+ set_minimal_thms set_set_incl_thmsss' setss_by_range ys ys' set_induct_phiss
+ in
+ (set_incl_thmss, set_set_incl_thmsss, set_induct_thms)
+ end;
+
+ val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+ val Jrels = map (mk_rel_of_bnf deads passiveAs passiveBs) Jbnfs;
+ val preds = map2 (fn Ds => mk_pred_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+ val Jpreds = map (mk_pred_of_bnf deads passiveAs passiveBs) Jbnfs;
+
+ val JrelRs = map (fn Jrel => Term.list_comb (Jrel, JRs)) Jrels;
+ val relRs = map (fn rel => Term.list_comb (rel, JRs @ JrelRs)) rels;
+ val Jpredphis = map (fn Jrel => Term.list_comb (Jrel, Jphis)) Jpreds;
+ val predphis = map (fn rel => Term.list_comb (rel, Jphis @ Jpredphis)) preds;
+
+ val in_rels = map in_rel_of_bnf bnfs;
+ val in_Jrels = map in_rel_of_bnf Jbnfs;
+ val Jpred_defs =
+ map (Drule.abs_def o (fn thm => thm RS @{thm eq_reflection}) o pred_def_of_bnf) Jbnfs;
+
+ val folded_map_simp_thms = map fold_maps map_simp_thms;
+ val folded_set_simp_thmss = map (map fold_sets) set_simp_thmss;
+ val folded_set_simp_thmss' = transpose folded_set_simp_thmss;
+
+ val Jrel_unfold_thms =
+ let
+ fun mk_goal Jz Jz' unf unf' JrelR relR = fold_rev Logic.all (Jz :: Jz' :: JRs)
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (HOLogic.mk_mem (HOLogic.mk_prod (Jz, Jz'), JrelR),
+ HOLogic.mk_mem (HOLogic.mk_prod (unf $ Jz, unf' $ Jz'), relR))));
+ val goals = map6 mk_goal Jzs Jz's unfs unf's JrelRs relRs;
+ in
+ map12 (fn i => fn goal => fn in_rel => fn map_comp => fn map_cong =>
+ fn map_simp => fn set_simps => fn unf_inject => fn unf_fld =>
+ fn set_naturals => fn set_incls => fn set_set_inclss =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_rel_unfold_tac in_Jrels i in_rel map_comp map_cong map_simp set_simps
+ unf_inject unf_fld set_naturals set_incls set_set_inclss)))
+ ks goals in_rels map_comp's map_congs folded_map_simp_thms folded_set_simp_thmss'
+ unf_inject_thms unf_fld_thms set_natural'ss set_incl_thmss set_set_incl_thmsss
+ end;
+
+ val Jpred_unfold_thms =
+ let
+ fun mk_goal Jz Jz' unf unf' Jpredphi predphi = fold_rev Logic.all (Jz :: Jz' :: Jphis)
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (Jpredphi $ Jz $ Jz', predphi $ (unf $ Jz) $ (unf' $ Jz'))));
+ val goals = map6 mk_goal Jzs Jz's unfs unf's Jpredphis predphis;
+ in
+ map3 (fn goal => fn pred_def => fn Jrel_unfold =>
+ Skip_Proof.prove lthy [] [] goal (mk_pred_unfold_tac pred_def Jpred_defs Jrel_unfold))
+ goals pred_defs Jrel_unfold_thms
+ end;
+
+ val timer = time (timer "additional properties");
+
+ val ls' = if m = 1 then [0] else ls;
+ in
+ lthy
+ |> note map_uniqueN [fold_maps map_unique_thm]
+ |> notes map_simpsN (map single folded_map_simp_thms)
+ |> fold2 (fn i => notes (mk_set_simpsN i) o map single) ls' folded_set_simp_thmss
+ |> notes set_inclN set_incl_thmss
+ |> notes set_set_inclN (map flat set_set_incl_thmsss) (* nicer names? *)
+ |> fold2 (fn i => note (mk_set_inductN i) o single) ls' set_induct_thms
+ |> notes rel_unfoldN (map single Jrel_unfold_thms)
+ |> notes pred_unfoldN (map single Jpred_unfold_thms)
+ end;
+ in
+ lthy
+ |> notes coiterN (map single coiter_thms)
+ |> notes coiter_uniqueN (map single coiter_unique_thms)
+ |> notes corecN (map single corec_thms)
+ |> notes unf_fldN (map single unf_fld_thms)
+ |> notes fld_unfN (map single fld_unf_thms)
+ |> notes unf_injectN (map single unf_inject_thms)
+ |> notes unf_exhaustN (map single unf_exhaust_thms)
+ |> notes fld_injectN (map single fld_inject_thms)
+ |> notes fld_exhaustN (map single fld_exhaust_thms)
+ |> notes fld_coiterN (map single fld_coiter_thms)
+ |> note unf_coinductN [unf_coinduct_thm]
+ |> note rel_coinductN [rel_coinduct_thm]
+ |> note pred_coinductN [pred_coinduct_thm]
+ |> note unf_coinduct_uptoN [unf_coinduct_upto_thm]
+ |> note rel_coinduct_uptoN [rel_coinduct_upto_thm]
+ |> note pred_coinduct_uptoN [pred_coinduct_upto_thm]
+ end;
+
+val _ =
+ Outer_Syntax.local_theory @{command_spec "bnf_codata"} "greatest fixed points for BNF equations"
+ (Parse.and_list1
+ ((Parse.binding --| Parse.$$$ ":") -- (Parse.typ --| Parse.$$$ "=" -- Parse.typ)) >>
+ (fp_bnf_cmd bnf_gfp o apsnd split_list o split_list));
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_gfp_tactics.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1536 @@
+(* Title: HOL/Codatatype/Tools/bnf_gfp_tactics.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+Tactics for the codatatype construction.
+*)
+
+signature BNF_GFP_TACTICS =
+sig
+ val mk_Lev_sbd_tac: cterm option list -> thm list -> thm list -> thm list list -> tactic
+ val mk_bd_card_order_tac: thm -> tactic
+ val mk_bd_cinfinite_tac: thm -> tactic
+ val mk_bis_Gr_tac: thm -> thm list -> thm list -> thm list -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_bis_O_tac: int -> thm -> thm list -> thm list -> tactic
+ val mk_bis_Union_tac: thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_bis_converse_tac: int -> thm -> thm list -> thm list -> tactic
+ val mk_bis_rel_tac: int -> thm -> thm list -> thm list -> thm list -> thm list list -> tactic
+ val mk_carT_set_tac: int -> int -> thm -> thm -> thm -> thm ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_card_of_carT_tac: int -> thm list -> thm -> thm -> thm -> thm -> thm -> thm list -> tactic
+ val mk_coalgT_tac: int -> thm list -> thm list -> thm list list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_coalg_final_tac: int -> thm -> thm list -> thm list -> thm list list -> thm list list ->
+ tactic
+ val mk_coalg_set_tac: thm -> tactic
+ val mk_coalg_thePull_tac: int -> thm -> thm list -> thm list list -> (int -> tactic) list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_coiter_unique_mor_tac: thm list -> thm -> thm -> thm list -> tactic
+ val mk_col_bd_tac: int -> int -> cterm option list -> thm list -> thm list -> thm -> thm ->
+ thm list list -> tactic
+ val mk_col_natural_tac: cterm option list -> thm list -> thm list -> thm list -> thm list list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_congruent_str_final_tac: int -> thm -> thm -> thm -> thm list -> tactic
+ val mk_corec_tac: int -> thm list -> thm -> thm -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_equiv_lsbis_tac: thm -> thm -> thm -> thm -> thm -> thm -> tactic
+ val mk_hset_minimal_tac: int -> thm list -> thm -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_hset_rec_minimal_tac: int -> cterm option list -> thm list -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_in_bd_tac: thm -> thm list -> thm -> thm -> thm -> thm -> thm list -> thm -> thm -> thm ->
+ thm -> thm -> thm -> tactic
+ val mk_incl_lsbis_tac: int -> int -> thm -> tactic
+ val mk_isNode_hset_tac: int -> thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_length_Lev'_tac: thm -> tactic
+ val mk_length_Lev_tac: cterm option list -> thm list -> thm list -> tactic
+ val mk_map_comp_tac: int -> int -> thm list -> thm list -> thm list -> thm -> tactic
+ val mk_mcong_tac: int -> (int -> tactic) -> thm list -> thm list -> thm list -> thm list list ->
+ thm list list -> thm list list list -> tactic
+ val mk_map_id_tac: thm list -> thm -> thm -> tactic
+ val mk_map_tac: int -> int -> ctyp option -> thm -> thm -> thm -> tactic
+ val mk_map_unique_tac: thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_mor_Abs_tac: thm list -> thm list -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_mor_Rep_tac: int -> thm list -> thm list -> thm list -> thm list list -> thm list ->
+ thm list -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_mor_T_final_tac: thm -> thm list -> thm list -> tactic
+ val mk_mor_UNIV_tac: thm list -> thm -> tactic
+ val mk_mor_beh_tac: int -> thm -> thm -> thm list -> thm list -> thm list -> thm list ->
+ thm list list -> thm list list -> thm list -> thm list -> thm list -> thm list -> thm list ->
+ thm list -> thm list -> thm list -> thm list list -> thm list list list -> thm list list list ->
+ thm list list list -> thm list list -> thm list list -> thm list -> thm list -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_mor_coiter_tac: int -> thm -> thm list -> thm list -> thm list -> thm list -> thm list ->
+ thm list -> tactic
+ val mk_mor_comp_tac: thm -> thm list -> thm list -> thm list -> tactic
+ val mk_mor_elim_tac: thm -> tactic
+ val mk_mor_hset_rec_tac: int -> int -> cterm option list -> int -> thm list -> thm list ->
+ thm list -> thm list list -> thm list list -> tactic
+ val mk_mor_hset_tac: thm -> thm -> tactic
+ val mk_mor_incl_tac: thm -> thm list -> tactic
+ val mk_mor_str_tac: 'a list -> thm -> tactic
+ val mk_mor_sum_case_tac: 'a list -> thm -> tactic
+ val mk_mor_thePull_fst_tac: int -> thm -> thm list -> thm list -> (int -> tactic) list ->
+ {prems: thm list, context: Proof.context} -> tactic
+ val mk_mor_thePull_snd_tac: int -> thm -> thm list -> thm list -> (int -> tactic) list ->
+ {prems: thm list, context: Proof.context} -> tactic
+ val mk_mor_thePull_pick_tac: thm -> thm list -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_prefCl_Lev_tac: cterm option list -> thm list -> thm list -> tactic
+ val mk_pickWP_assms_tac: thm list -> thm list -> thm -> (int -> tactic)
+ val mk_pick_col_tac: int -> int -> cterm option list -> thm list -> thm list -> thm list ->
+ thm list list -> thm list -> (int -> tactic) list -> {prems: 'a, context: Proof.context} ->
+ tactic
+ val mk_raw_coind_tac: thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm list ->
+ thm list -> thm list -> thm -> thm list -> tactic
+ val mk_rel_coinduct_tac: 'a list -> thm -> thm -> tactic
+ val mk_rel_coinduct_upto_tac: int -> ctyp option list -> cterm option list -> thm -> thm list ->
+ thm list -> tactic
+ val mk_rel_unfold_tac: thm list -> int -> thm -> thm -> thm -> thm -> thm list -> thm -> thm ->
+ thm list -> thm list -> thm list list -> tactic
+ val mk_rv_last_tac: ctyp option list -> cterm option list -> thm list -> thm list -> tactic
+ val mk_sbis_lsbis_tac: thm list -> thm -> thm -> tactic
+ val mk_set_Lev_tac: cterm option list -> thm list -> thm list -> thm list -> thm list ->
+ thm list list -> tactic
+ val mk_set_bd_tac: thm -> thm -> thm -> tactic
+ val mk_set_hset_incl_hset_tac: int -> thm list -> thm -> int -> tactic
+ val mk_set_image_Lev_tac: cterm option list -> thm list -> thm list -> thm list -> thm list ->
+ thm list list -> thm list list -> tactic
+ val mk_set_incl_hin_tac: thm list -> tactic
+ val mk_set_incl_hset_tac: thm -> thm -> tactic
+ val mk_set_le_tac: int -> thm -> thm list -> thm list list -> tactic
+ val mk_set_natural_tac: thm -> thm -> tactic
+ val mk_set_rv_Lev_tac: int -> cterm option list -> thm list -> thm list -> thm list -> thm list ->
+ thm list list -> thm list list -> tactic
+ val mk_set_simp_tac: int -> thm -> thm -> thm list -> tactic
+ val mk_strT_hset_tac: int -> int -> int -> ctyp option list -> ctyp option list ->
+ cterm option list -> thm list -> thm list -> thm list -> thm list -> thm list list ->
+ thm list list -> thm list list -> thm -> thm list list -> tactic
+ val mk_unf_coinduct_tac: int -> int list -> thm -> thm -> tactic
+ val mk_unf_coinduct_upto_tac: int list -> ctyp option list -> cterm option list -> thm -> thm ->
+ thm -> tactic
+ val mk_unf_o_fld_tac: thm -> thm -> thm -> thm -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_unique_mor_tac: thm list -> thm -> tactic
+ val mk_wit_tac: int -> thm list -> thm list -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_wpull_tac: int -> thm -> thm -> thm -> thm -> thm -> thm list -> thm list -> tactic
+end;
+
+structure BNF_GFP_Tactics : BNF_GFP_TACTICS =
+struct
+
+open BNF_Tactics
+open BNF_Util
+open BNF_GFP_Util
+
+fun mk_coalg_set_tac coalg_def =
+ dtac (coalg_def RS @{thm subst[of _ _ "\<lambda>x. x"]}) 1 THEN
+ REPEAT_DETERM (etac conjE 1) THEN
+ EVERY' [dtac @{thm rev_bspec}, atac] 1 THEN
+ REPEAT_DETERM (eresolve_tac [CollectE, conjE] 1) THEN atac 1;
+
+fun mk_mor_elim_tac mor_def =
+ (dtac (subst OF [mor_def]) THEN'
+ REPEAT o etac conjE THEN'
+ TRY o rtac @{thm image_subsetI} THEN'
+ etac bspec THEN'
+ atac) 1;
+
+fun mk_mor_incl_tac mor_def map_id's =
+ (stac mor_def THEN'
+ rtac conjI THEN'
+ CONJ_WRAP' (K (EVERY' [rtac ballI, etac @{thm set_mp}, stac @{thm id_apply}, atac]))
+ map_id's THEN'
+ CONJ_WRAP' (fn thm =>
+ (EVERY' [rtac ballI, rtac (thm RS trans), rtac sym, rtac (@{thm id_apply} RS arg_cong)]))
+ map_id's) 1;
+
+fun mk_mor_comp_tac mor_def mor_images morEs map_comp_ids =
+ let
+ fun fbetw_tac image = EVERY' [rtac ballI, stac o_apply, etac image, etac image, atac];
+ fun mor_tac ((mor_image, morE), map_comp_id) =
+ EVERY' [rtac ballI, stac o_apply, rtac trans, rtac (map_comp_id RS sym), rtac trans,
+ etac (morE RS arg_cong), atac, etac morE, etac mor_image, atac];
+ in
+ (stac mor_def THEN' rtac conjI THEN'
+ CONJ_WRAP' fbetw_tac mor_images THEN'
+ CONJ_WRAP' mor_tac ((mor_images ~~ morEs) ~~ map_comp_ids)) 1
+ end;
+
+fun mk_mor_UNIV_tac morEs mor_def =
+ let
+ val n = length morEs;
+ fun mor_tac morE = EVERY' [rtac ext, rtac trans, rtac o_apply, rtac trans, etac morE,
+ rtac UNIV_I, rtac sym, rtac o_apply];
+ in
+ EVERY' [rtac iffI, CONJ_WRAP' mor_tac morEs,
+ stac mor_def, rtac conjI, CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) morEs,
+ CONJ_WRAP' (fn i =>
+ EVERY' [dtac (mk_conjunctN n i), rtac ballI, etac @{thm pointfreeE}]) (1 upto n)] 1
+ end;
+
+fun mk_mor_str_tac ks mor_UNIV =
+ (stac mor_UNIV THEN' CONJ_WRAP' (K (rtac refl)) ks) 1;
+
+fun mk_mor_sum_case_tac ks mor_UNIV =
+ (stac mor_UNIV THEN' CONJ_WRAP' (K (rtac @{thm sum_case_comp_Inl[symmetric]})) ks) 1;
+
+fun mk_set_incl_hset_tac def rec_Suc =
+ EVERY' (stac def ::
+ map rtac [@{thm incl_UNION_I}, UNIV_I, @{thm ord_le_eq_trans}, @{thm Un_upper1},
+ sym, rec_Suc]) 1;
+
+fun mk_set_hset_incl_hset_tac n defs rec_Suc i =
+ EVERY' (map (TRY oo stac) defs @
+ map rtac [@{thm UN_least}, subsetI, @{thm UN_I}, UNIV_I, @{thm set_mp}, equalityD2, rec_Suc,
+ UnI2, mk_UnN n i] @
+ [etac @{thm UN_I}, atac]) 1;
+
+fun mk_set_incl_hin_tac incls =
+ if null incls then rtac @{thm subset_UNIV} 1
+ else EVERY' [rtac subsetI, rtac CollectI,
+ CONJ_WRAP' (fn incl => EVERY' [rtac subset_trans, etac incl, atac]) incls] 1;
+
+fun mk_hset_rec_minimal_tac m cts rec_0s rec_Sucs {context = ctxt, prems = _} =
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn thm => EVERY'
+ [rtac @{thm ord_eq_le_trans}, rtac thm, rtac @{thm empty_subsetI}]) rec_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn rec_Suc => EVERY'
+ [rtac @{thm ord_eq_le_trans}, rtac rec_Suc,
+ if m = 0 then K all_tac
+ else (rtac @{thm Un_least} THEN' Goal.assume_rule_tac ctxt),
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
+ (K (EVERY' [rtac @{thm UN_least}, REPEAT_DETERM o eresolve_tac [allE, conjE],
+ rtac subset_trans, atac, Goal.assume_rule_tac ctxt])) rec_0s])
+ rec_Sucs] 1;
+
+fun mk_hset_minimal_tac n hset_defs hset_rec_minimal {context = ctxt, prems = _} =
+ (CONJ_WRAP' (fn def => (EVERY' [rtac @{thm ord_eq_le_trans}, rtac def,
+ rtac @{thm UN_least}, rtac rev_mp, rtac hset_rec_minimal,
+ EVERY' (replicate ((n + 1) * n) (Goal.assume_rule_tac ctxt)), rtac impI,
+ REPEAT_DETERM o eresolve_tac [allE, conjE], atac])) hset_defs) 1
+
+fun mk_mor_hset_rec_tac m n cts j rec_0s rec_Sucs morEs set_naturalss coalg_setss =
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn thm => EVERY' (map rtac [impI, thm RS trans, thm RS sym])) rec_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP'
+ (fn (rec_Suc, (morE, ((passive_set_naturals, active_set_naturals), coalg_sets))) =>
+ EVERY' [rtac impI, rtac (rec_Suc RS trans), rtac (rec_Suc RS trans RS sym),
+ if m = 0 then K all_tac
+ else EVERY' [rtac @{thm Un_cong}, rtac box_equals,
+ rtac (nth passive_set_naturals (j - 1) RS sym),
+ rtac @{thm trans[OF fun_cong[OF image_id] id_apply]}, etac (morE RS arg_cong), atac],
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
+ (fn (i, (set_natural, coalg_set)) =>
+ EVERY' [rtac sym, rtac trans, rtac (refl RSN (2, @{thm UN_cong})),
+ etac (morE RS sym RS arg_cong RS trans), atac, rtac set_natural,
+ rtac (@{thm UN_simps(10)} RS trans), rtac (refl RS @{thm UN_cong}),
+ ftac coalg_set, atac, dtac @{thm set_mp}, atac, rtac mp, rtac (mk_conjunctN n i),
+ REPEAT_DETERM o etac allE, atac, atac])
+ (rev ((1 upto n) ~~ (active_set_naturals ~~ coalg_sets)))])
+ (rec_Sucs ~~ (morEs ~~ (map (chop m) set_naturalss ~~ map (drop m) coalg_setss)))] 1;
+
+fun mk_mor_hset_tac hset_def mor_hset_rec =
+ EVERY' [rtac (hset_def RS trans), rtac (refl RS @{thm UN_cong} RS trans), etac mor_hset_rec,
+ atac, atac, rtac (hset_def RS sym)] 1
+
+fun mk_bis_rel_tac m bis_def rel_defs map_comps map_congs set_naturalss =
+ let
+ val n = length rel_defs;
+ val thms = ((1 upto n) ~~ map_comps ~~ map_congs ~~ set_naturalss ~~ rel_defs);
+
+ fun mk_if_tac ((((i, map_comp), map_cong), set_naturals), rel_def) =
+ EVERY' [rtac allI, rtac allI, rtac impI, dtac (mk_conjunctN n i),
+ etac allE, etac allE, etac impE, atac, etac bexE, etac conjE,
+ rtac (rel_def RS equalityD2 RS @{thm set_mp}),
+ rtac @{thm relcompI}, rtac @{thm converseI},
+ EVERY' (map (fn thm =>
+ EVERY' [rtac @{thm GrI}, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ rtac CollectI,
+ CONJ_WRAP' (fn (i, thm) =>
+ if i <= m
+ then EVERY' [rtac @{thm ord_eq_le_trans}, rtac thm, rtac subset_trans,
+ etac @{thm image_mono}, rtac @{thm image_subsetI}, etac @{thm diagI}]
+ else EVERY' [rtac @{thm ord_eq_le_trans}, rtac trans, rtac thm,
+ rtac @{thm trans[OF fun_cong[OF image_id] id_apply]}, atac])
+ (1 upto (m + n) ~~ set_naturals),
+ rtac trans, rtac trans, rtac map_comp, rtac map_cong, REPEAT_DETERM_N m o rtac thm,
+ REPEAT_DETERM_N n o rtac (@{thm o_id} RS fun_cong), atac])
+ @{thms fst_diag_id snd_diag_id})];
+
+ fun mk_only_if_tac ((((i, map_comp), map_cong), set_naturals), rel_def) =
+ EVERY' [dtac (mk_conjunctN n i), rtac allI, rtac allI, rtac impI,
+ etac allE, etac allE, etac impE, atac,
+ dtac (rel_def RS equalityD1 RS @{thm set_mp}),
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm relcompE}, @{thm converseE}],
+ REPEAT_DETERM o eresolve_tac [@{thm GrE}, exE, conjE],
+ REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ REPEAT_DETERM o etac conjE,
+ hyp_subst_tac,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ rtac bexI, rtac conjI, rtac trans, rtac map_comp,
+ REPEAT_DETERM_N m o stac @{thm id_o},
+ REPEAT_DETERM_N n o stac @{thm o_id},
+ etac sym, rtac trans, rtac map_comp,
+ REPEAT_DETERM_N m o stac @{thm id_o},
+ REPEAT_DETERM_N n o stac @{thm o_id},
+ rtac trans, rtac map_cong,
+ REPEAT_DETERM_N m o EVERY' [rtac @{thm diagE'}, etac @{thm set_mp}, atac],
+ REPEAT_DETERM_N n o rtac refl,
+ etac sym, rtac CollectI,
+ CONJ_WRAP' (fn (i, thm) =>
+ if i <= m
+ then EVERY' [rtac @{thm ord_eq_le_trans}, rtac thm, rtac @{thm image_subsetI},
+ rtac @{thm diag_fst}, etac @{thm set_mp}, atac]
+ else EVERY' [rtac @{thm ord_eq_le_trans}, rtac trans, rtac thm,
+ rtac @{thm trans[OF fun_cong[OF image_id] id_apply]}, atac])
+ (1 upto (m + n) ~~ set_naturals)];
+ in
+ EVERY' [rtac (bis_def RS trans),
+ rtac iffI, etac conjE, etac conjI, CONJ_WRAP' mk_if_tac thms,
+ etac conjE, etac conjI, CONJ_WRAP' mk_only_if_tac thms] 1
+ end;
+
+fun mk_bis_converse_tac m bis_rel rel_congs rel_converses =
+ EVERY' [stac bis_rel, dtac (bis_rel RS @{thm subst[of _ _ "%x. x"]}),
+ REPEAT_DETERM o etac conjE, rtac conjI,
+ CONJ_WRAP' (K (EVERY' [rtac @{thm converse_shift}, etac subset_trans,
+ rtac equalityD2, rtac @{thm converse_Times}])) rel_congs,
+ CONJ_WRAP' (fn (rel_cong, rel_converse) =>
+ EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm set_mp[OF equalityD2]},
+ rtac (rel_cong RS trans),
+ REPEAT_DETERM_N m o rtac @{thm diag_converse},
+ REPEAT_DETERM_N (length rel_congs) o rtac refl,
+ rtac rel_converse,
+ REPEAT_DETERM o etac allE,
+ rtac @{thm converseI}, etac mp, etac @{thm converseD}]) (rel_congs ~~ rel_converses)] 1;
+
+fun mk_bis_O_tac m bis_rel rel_congs rel_Os =
+ EVERY' [stac bis_rel, REPEAT_DETERM o dtac (bis_rel RS @{thm subst[of _ _ "%x. x"]}),
+ REPEAT_DETERM o etac conjE, rtac conjI,
+ CONJ_WRAP' (K (EVERY' [etac @{thm relcomp_subset_Sigma}, atac])) rel_congs,
+ CONJ_WRAP' (fn (rel_cong, rel_O) =>
+ EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm set_mp[OF equalityD2]},
+ rtac (rel_cong RS trans),
+ REPEAT_DETERM_N m o rtac @{thm diag_Comp},
+ REPEAT_DETERM_N (length rel_congs) o rtac refl,
+ rtac rel_O,
+ etac @{thm relcompE},
+ REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ etac conjE, hyp_subst_tac,
+ REPEAT_DETERM o etac allE, rtac @{thm relcompI},
+ etac mp, atac, etac mp, atac]) (rel_congs ~~ rel_Os)] 1;
+
+fun mk_bis_Gr_tac bis_rel rel_Grs mor_images morEs coalg_ins
+ {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt (bis_rel :: @{thm diag_Gr} :: rel_Grs) THEN
+ EVERY' [rtac conjI,
+ CONJ_WRAP' (fn thm => rtac (@{thm Gr_incl} RS ssubst) THEN' etac thm) mor_images,
+ CONJ_WRAP' (fn (coalg_in, morE) =>
+ EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm GrI}, etac coalg_in,
+ etac @{thm GrD1}, etac (morE RS trans), etac @{thm GrD1},
+ etac (@{thm GrD2} RS arg_cong)]) (coalg_ins ~~ morEs)] 1;
+
+fun mk_bis_Union_tac bis_def in_monos {context = ctxt, prems = _} =
+ let
+ val n = length in_monos;
+ val ks = 1 upto n;
+ in
+ Local_Defs.unfold_tac ctxt [bis_def] THEN
+ EVERY' [rtac conjI,
+ CONJ_WRAP' (fn i =>
+ EVERY' [rtac @{thm UN_least}, dtac bspec, atac,
+ dtac conjunct1, etac (mk_conjunctN n i)]) ks,
+ CONJ_WRAP' (fn (i, in_mono) =>
+ EVERY' [rtac allI, rtac allI, rtac impI, etac @{thm UN_E}, dtac bspec, atac,
+ dtac conjunct2, dtac (mk_conjunctN n i), etac allE, etac allE, dtac mp,
+ atac, etac bexE, rtac bexI, atac, rtac in_mono,
+ REPEAT_DETERM_N n o etac @{thm incl_UNION_I[OF _ subset_refl]},
+ atac]) (ks ~~ in_monos)] 1
+ end;
+
+fun mk_sbis_lsbis_tac lsbis_defs bis_Union bis_cong =
+ let
+ val n = length lsbis_defs;
+ in
+ EVERY' [rtac (Thm.permute_prems 0 1 bis_cong), EVERY' (map rtac lsbis_defs),
+ rtac bis_Union, rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE, exE],
+ hyp_subst_tac, etac bis_cong, EVERY' (map (rtac o mk_nth_conv n) (1 upto n))] 1
+ end;
+
+fun mk_incl_lsbis_tac n i lsbis_def =
+ EVERY' [rtac @{thm xt1(3)}, rtac lsbis_def, rtac @{thm incl_UNION_I}, rtac CollectI,
+ REPEAT_DETERM_N n o rtac exI, rtac conjI, rtac refl, atac, rtac equalityD2,
+ rtac (mk_nth_conv n i)] 1;
+
+fun mk_equiv_lsbis_tac sbis_lsbis lsbis_incl incl_lsbis bis_diag bis_converse bis_O =
+ EVERY' [rtac @{thm ssubst[of _ _ "%x. x", OF equiv_def]},
+
+ rtac conjI, rtac @{thm ssubst[of _ _ "%x. x", OF refl_on_def]},
+ rtac conjI, rtac lsbis_incl, rtac ballI, rtac @{thm set_mp},
+ rtac incl_lsbis, rtac bis_diag, atac, etac @{thm diagI},
+
+ rtac conjI, rtac @{thm ssubst[of _ _ "%x. x", OF sym_def]},
+ rtac allI, rtac allI, rtac impI, rtac @{thm set_mp},
+ rtac incl_lsbis, rtac bis_converse, rtac sbis_lsbis, etac @{thm converseI},
+
+ rtac @{thm ssubst[of _ _ "%x. x", OF trans_def]},
+ rtac allI, rtac allI, rtac allI, rtac impI, rtac impI, rtac @{thm set_mp},
+ rtac incl_lsbis, rtac bis_O, rtac sbis_lsbis, rtac sbis_lsbis,
+ etac @{thm relcompI}, atac] 1;
+
+fun mk_coalgT_tac m defs strT_defs set_naturalss {context = ctxt, prems = _} =
+ let
+ val n = length strT_defs;
+ val ks = 1 upto n;
+ fun coalg_tac (i, ((passive_sets, active_sets), def)) =
+ EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
+ hyp_subst_tac, rtac (def RS trans RS @{thm ssubst_mem}), etac (arg_cong RS trans),
+ rtac (mk_sum_casesN n i), rtac CollectI,
+ EVERY' (map (fn thm => EVERY' [rtac conjI, rtac (thm RS @{thm ord_eq_le_trans}),
+ etac ((trans OF [@{thm image_id} RS fun_cong, @{thm id_apply}]) RS
+ @{thm ord_eq_le_trans})]) passive_sets),
+ CONJ_WRAP' (fn (i, thm) => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
+ rtac @{thm image_subsetI}, rtac CollectI, rtac exI, rtac exI, rtac conjI, rtac refl,
+ rtac conjI,
+ rtac conjI, etac @{thm empty_Shift}, dtac @{thm set_rev_mp},
+ etac equalityD1, etac CollectD,
+ rtac conjI, etac @{thm Shift_clists},
+ rtac conjI, etac @{thm Shift_prefCl},
+ rtac conjI, rtac ballI,
+ rtac conjI, dtac @{thm iffD1[OF ball_conj_distrib]}, dtac conjunct1,
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt @{thms Succ_Shift shift_def}),
+ etac bspec, etac @{thm ShiftD},
+ CONJ_WRAP' (fn i => EVERY' [rtac ballI, etac CollectE, dtac @{thm ShiftD},
+ dtac bspec, etac thin_rl, atac, dtac conjunct2, dtac (mk_conjunctN n i),
+ dtac bspec, rtac CollectI, etac @{thm set_mp[OF equalityD1[OF Succ_Shift]]},
+ REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI,
+ rtac conjI, rtac (@{thm shift_def} RS fun_cong RS trans),
+ rtac (@{thm append_Cons} RS sym RS arg_cong RS trans), atac,
+ REPEAT_DETERM_N m o (rtac conjI THEN' atac),
+ CONJ_WRAP' (K (EVERY' [etac trans, rtac @{thm Collect_cong},
+ rtac @{thm eqset_imp_iff}, rtac sym, rtac trans, rtac @{thm Succ_Shift},
+ rtac (@{thm append_Cons} RS sym RS arg_cong)])) ks]) ks,
+ rtac allI, rtac impI, REPEAT_DETERM o eresolve_tac [allE, impE],
+ etac @{thm not_in_Shift}, rtac trans, rtac (@{thm shift_def} RS fun_cong), atac,
+ dtac bspec, atac, dtac conjunct2, dtac (mk_conjunctN n i), dtac bspec,
+ etac @{thm set_mp[OF equalityD1]}, atac,
+ REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI,
+ rtac conjI, rtac (@{thm shift_def} RS fun_cong RS trans),
+ etac (@{thm append_Nil} RS sym RS arg_cong RS trans),
+ REPEAT_DETERM_N m o (rtac conjI THEN' atac),
+ CONJ_WRAP' (K (EVERY' [etac trans, rtac @{thm Collect_cong},
+ rtac @{thm eqset_imp_iff}, rtac sym, rtac trans, rtac @{thm Succ_Shift},
+ rtac (@{thm append_Nil} RS sym RS arg_cong)])) ks]) (ks ~~ active_sets)];
+ in
+ Local_Defs.unfold_tac ctxt defs THEN
+ CONJ_WRAP' coalg_tac (ks ~~ (map (chop m) set_naturalss ~~ strT_defs)) 1
+ end;
+
+fun mk_card_of_carT_tac m isNode_defs sbd_sbd
+ sbd_card_order sbd_Card_order sbd_Cinfinite sbd_Cnotzero in_sbds =
+ let
+ val n = length isNode_defs;
+ in
+ EVERY' [rtac (Thm.permute_prems 0 1 ctrans),
+ rtac @{thm card_of_Sigma_ordLeq_Cinfinite}, rtac @{thm Cinfinite_cexp},
+ if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
+ rtac @{thm Card_order_ctwo}, rtac @{thm Cinfinite_cexp},
+ rtac @{thm ctwo_ordLeq_Cinfinite}, rtac sbd_Cinfinite, rtac sbd_Cinfinite,
+ rtac ctrans, rtac @{thm card_of_diff},
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm card_of_Field_ordIso},
+ rtac @{thm Card_order_cpow}, rtac @{thm ordIso_ordLeq_trans},
+ rtac @{thm cpow_cexp_ctwo}, rtac ctrans, rtac @{thm cexp_mono1_Cnotzero},
+ if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
+ rtac @{thm Card_order_ctwo}, rtac @{thm ctwo_Cnotzero}, rtac @{thm Card_order_clists},
+ rtac @{thm cexp_mono2_Cnotzero}, rtac @{thm ordIso_ordLeq_trans},
+ rtac @{thm clists_Cinfinite},
+ if n = 1 then rtac sbd_Cinfinite else rtac (sbd_Cinfinite RS @{thm Cinfinite_csum1}),
+ rtac @{thm ordIso_ordLeq_trans}, rtac sbd_sbd, rtac @{thm infinite_ordLeq_cexp},
+ rtac sbd_Cinfinite,
+ if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
+ rtac @{thm Cnotzero_clists},
+ rtac ballI, rtac @{thm ordIso_ordLeq_trans}, rtac @{thm card_of_Func_Ffunc},
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm Func_cexp},
+ rtac ctrans, rtac @{thm cexp_mono},
+ rtac @{thm ordLeq_ordIso_trans},
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1
+ (sbd_Cinfinite RS @{thm Cinfinite_cexp[OF ordLeq_csum2[OF Card_order_ctwo]]}
+ RSN (3, @{thm Un_Cinfinite_bound}))))
+ (fn thm => EVERY' [rtac ctrans, rtac @{thm card_of_image}, rtac thm]) (rev in_sbds),
+ rtac @{thm cexp_cong1_Cnotzero}, rtac @{thm csum_cong1},
+ REPEAT_DETERM_N m o rtac @{thm csum_cong2},
+ CONJ_WRAP_GEN' (rtac @{thm csum_cong})
+ (K (rtac (sbd_Card_order RS @{thm card_of_Field_ordIso}))) in_sbds,
+ rtac sbd_Card_order,
+ rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
+ rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
+ rtac @{thm ordLeq_ordIso_trans}, etac @{thm clists_bound},
+ rtac @{thm clists_Cinfinite}, TRY o rtac @{thm Cinfinite_csum1}, rtac sbd_Cinfinite,
+ rtac disjI2, rtac @{thm cone_ordLeq_cexp}, rtac @{thm cone_ordLeq_cexp},
+ rtac ctrans, rtac @{thm cone_ordLeq_ctwo}, rtac @{thm ordLeq_csum2},
+ rtac @{thm Card_order_ctwo}, rtac FalseE, etac @{thm cpow_clists_czero}, atac,
+ rtac @{thm card_of_Card_order},
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm cexp_cprod},
+ rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm cexp_cong2_Cnotzero},
+ rtac @{thm ordIso_transitive}, rtac @{thm cprod_cong2}, rtac sbd_sbd,
+ rtac @{thm cprod_infinite}, rtac sbd_Cinfinite,
+ rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac @{thm Card_order_cprod},
+ rtac ctrans, rtac @{thm cexp_mono1_Cnotzero},
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm ordIso_transitive}, rtac @{thm csum_cong1},
+ rtac @{thm ordIso_transitive},
+ REPEAT_DETERM_N m o rtac @{thm csum_cong2},
+ rtac sbd_sbd,
+ BNF_Tactics.mk_rotate_eq_tac (rtac @{thm ordIso_refl} THEN'
+ FIRST' [rtac @{thm card_of_Card_order},
+ rtac @{thm Card_order_csum}, rtac sbd_Card_order])
+ @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong}
+ (1 upto m + 1) (m + 1 :: (1 upto m)),
+ if m = 0 then K all_tac else EVERY' [rtac @{thm ordIso_transitive}, rtac @{thm csum_assoc}],
+ rtac @{thm csum_com}, rtac @{thm csum_cexp'}, rtac sbd_Cinfinite,
+ if m = 0 then rtac @{thm Card_order_ctwo} else rtac @{thm Card_order_csum},
+ if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
+ rtac @{thm Card_order_ctwo},
+ rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac sbd_Card_order,
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm cexp_cprod_ordLeq},
+ if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
+ rtac sbd_Cinfinite, rtac sbd_Cnotzero, rtac @{thm ordLeq_refl}, rtac sbd_Card_order,
+ rtac @{thm cexp_mono2_Cnotzero}, rtac @{thm infinite_ordLeq_cexp},
+ rtac sbd_Cinfinite,
+ if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
+ rtac sbd_Cnotzero,
+ rtac @{thm card_of_mono1}, rtac subsetI,
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm prod_caseE}], hyp_subst_tac,
+ rtac @{thm SigmaI}, rtac @{thm DiffI}, rtac @{thm set_mp}, rtac equalityD2,
+ rtac (@{thm cpow_def} RS arg_cong RS trans), rtac (@{thm Pow_def} RS arg_cong RS trans),
+ rtac @{thm Field_card_of}, rtac CollectI, atac, rtac notI, etac @{thm singletonE},
+ hyp_subst_tac, etac @{thm emptyE}, rtac (@{thm Ffunc_def} RS equalityD2 RS @{thm set_mp}),
+ rtac CollectI, rtac conjI, rtac ballI, dtac bspec, etac thin_rl, atac, dtac conjunct1,
+ CONJ_WRAP_GEN' (etac disjE) (fn (i, def) => EVERY'
+ [rtac (mk_UnN n i), dtac (def RS @{thm subst[of _ _ "%x. x"]}),
+ REPEAT_DETERM o eresolve_tac [exE, conjE], rtac @{thm image_eqI}, atac, rtac CollectI,
+ REPEAT_DETERM_N m o (rtac conjI THEN' atac),
+ CONJ_WRAP' (K (EVERY' [etac @{thm ord_eq_le_trans}, rtac subset_trans,
+ rtac @{thm subset_UNIV}, rtac equalityD2, rtac @{thm Field_card_order},
+ rtac sbd_card_order])) isNode_defs]) (1 upto n ~~ isNode_defs),
+ atac] 1
+ end;
+
+fun mk_carT_set_tac n i carT_def strT_def isNode_def set_natural {context = ctxt, prems = _}=
+ EVERY' [dtac (carT_def RS equalityD1 RS @{thm set_mp}),
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
+ dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ etac conjE, hyp_subst_tac,
+ dtac (isNode_def RS @{thm subst[of _ _ "%x. x"]}),
+ REPEAT_DETERM o eresolve_tac [exE, conjE],
+ rtac (equalityD2 RS @{thm set_mp}),
+ rtac (strT_def RS arg_cong RS trans),
+ etac (arg_cong RS trans),
+ fo_rtac (mk_sum_casesN n i RS arg_cong RS trans) ctxt,
+ rtac set_natural, rtac imageI, etac (equalityD2 RS @{thm set_mp}), rtac CollectI,
+ etac @{thm prefCl_Succ}, atac] 1;
+
+fun mk_strT_hset_tac n m j arg_cong_cTs cTs cts carT_defs strT_defs isNode_defs
+ set_incl_hsets set_hset_incl_hsetss coalg_setss carT_setss coalgT set_naturalss =
+ let
+ val set_naturals = map (fn xs => nth xs (j - 1)) set_naturalss;
+ val ks = 1 upto n;
+ fun base_tac (i, (cT, (strT_def, (set_incl_hset, set_natural)))) =
+ CONJ_WRAP' (fn (i', (carT_def, isNode_def)) => rtac impI THEN' etac conjE THEN'
+ (if i = i'
+ then EVERY' [rtac @{thm xt1(4)}, rtac set_incl_hset,
+ rtac (strT_def RS arg_cong RS trans), etac (arg_cong RS trans),
+ rtac (Thm.permute_prems 0 1 (set_natural RS box_equals)),
+ rtac (trans OF [@{thm image_id} RS fun_cong, @{thm id_apply}]),
+ rtac (mk_sum_casesN n i RS (Drule.instantiate' [cT] [] arg_cong) RS sym)]
+ else EVERY' [dtac (carT_def RS equalityD1 RS @{thm set_mp}),
+ REPEAT_DETERM o eresolve_tac [CollectE, exE], etac conjE,
+ dtac conjunct2, dtac @{thm Pair_eq[THEN subst, of "%x. x"]}, etac conjE,
+ hyp_subst_tac, dtac (isNode_def RS @{thm subst[of _ _ "%x. x"]}),
+ REPEAT_DETERM o eresolve_tac [exE, conjE],
+ rtac (mk_InN_not_InM i i' RS notE), etac (sym RS trans), atac]))
+ (ks ~~ (carT_defs ~~ isNode_defs));
+ fun step_tac (i, (coalg_sets, (carT_sets, set_hset_incl_hsets))) =
+ dtac (mk_conjunctN n i) THEN'
+ CONJ_WRAP' (fn (coalg_set, (carT_set, set_hset_incl_hset)) =>
+ EVERY' [rtac impI, etac conjE, etac impE, rtac conjI,
+ rtac (coalgT RS coalg_set RS @{thm set_mp}), atac, etac carT_set, atac, atac,
+ etac (@{thm shift_def} RS fun_cong RS trans), etac subset_trans,
+ rtac set_hset_incl_hset, etac carT_set, atac, atac])
+ (coalg_sets ~~ (carT_sets ~~ set_hset_incl_hsets));
+ in
+ EVERY' [rtac (Drule.instantiate' cTs cts @{thm list.induct}),
+ REPEAT_DETERM o rtac allI, rtac impI,
+ CONJ_WRAP' base_tac
+ (ks ~~ (arg_cong_cTs ~~ (strT_defs ~~ (set_incl_hsets ~~ set_naturals)))),
+ REPEAT_DETERM o rtac allI, rtac impI,
+ REPEAT_DETERM o eresolve_tac [allE, impE], etac @{thm ShiftI},
+ CONJ_WRAP' (fn i => dtac (mk_conjunctN n i) THEN' rtac (mk_sumEN n) THEN'
+ CONJ_WRAP_GEN' (K all_tac) step_tac
+ (ks ~~ (drop m coalg_setss ~~ (carT_setss ~~ set_hset_incl_hsetss)))) ks] 1
+ end;
+
+fun mk_isNode_hset_tac n isNode_def strT_hsets {context = ctxt, prems = _} =
+ let
+ val m = length strT_hsets;
+ in
+ if m = 0 then atac 1
+ else (Local_Defs.unfold_tac ctxt [isNode_def] THEN
+ EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
+ rtac exI, rtac conjI, atac,
+ CONJ_WRAP' (fn (thm, i) => if i > m then atac
+ else EVERY' [rtac (thm RS subset_trans), atac, rtac conjI, atac, atac, atac])
+ (strT_hsets @ (replicate n mp) ~~ (1 upto (m + n)))] 1)
+ end;
+
+fun mk_Lev_sbd_tac cts Lev_0s Lev_Sucs to_sbdss =
+ let
+ val n = length Lev_0s;
+ val ks = 1 upto n;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn Lev_0 =>
+ EVERY' (map rtac [@{thm ord_eq_le_trans}, Lev_0, @{thm Nil_clists}])) Lev_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn (Lev_Suc, to_sbds) =>
+ EVERY' [rtac @{thm ord_eq_le_trans}, rtac Lev_Suc,
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
+ (fn (i, to_sbd) => EVERY' [rtac subsetI,
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ rtac @{thm Cons_clists}, rtac (mk_InN_Field n i), etac to_sbd,
+ etac @{thm set_rev_mp}, REPEAT_DETERM o etac allE,
+ etac (mk_conjunctN n i)])
+ (rev (ks ~~ to_sbds))])
+ (Lev_Sucs ~~ to_sbdss)] 1
+ end;
+
+fun mk_length_Lev_tac cts Lev_0s Lev_Sucs =
+ let
+ val n = length Lev_0s;
+ val ks = n downto 1;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn Lev_0 =>
+ EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS @{thm set_mp}),
+ etac @{thm singletonE}, etac ssubst, rtac @{thm list.size(3)}]) Lev_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn Lev_Suc =>
+ EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}),
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
+ (fn i => EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ rtac trans, rtac @{thm length_Cons}, rtac @{thm arg_cong[of _ _ Suc]},
+ REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i), etac mp, atac]) ks])
+ Lev_Sucs] 1
+ end;
+
+fun mk_length_Lev'_tac length_Lev =
+ EVERY' [ftac length_Lev, etac ssubst, atac] 1;
+
+fun mk_prefCl_Lev_tac cts Lev_0s Lev_Sucs =
+ let
+ val n = length Lev_0s;
+ val ks = n downto 1;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn Lev_0 =>
+ EVERY' [rtac impI, etac conjE, dtac (Lev_0 RS equalityD1 RS @{thm set_mp}),
+ etac @{thm singletonE}, hyp_subst_tac, dtac @{thm prefix_Nil[THEN subst, of "%x. x"]},
+ hyp_subst_tac, rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF list.size(3)]]]]},
+ rtac Lev_0, rtac @{thm singletonI}]) Lev_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn (Lev_0, Lev_Suc) =>
+ EVERY' [rtac impI, etac conjE, dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}),
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
+ (fn i => EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ dtac @{thm prefix_Cons[THEN subst, of "%x. x"]}, etac disjE, hyp_subst_tac,
+ rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF list.size(3)]]]]},
+ rtac Lev_0, rtac @{thm singletonI},
+ REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac,
+ rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF length_Cons]]]]},
+ rtac Lev_Suc, rtac (mk_UnN n i), rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI,
+ rtac refl, etac conjI, REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i),
+ etac mp, etac conjI, atac]) ks])
+ (Lev_0s ~~ Lev_Sucs)] 1
+ end;
+
+fun mk_rv_last_tac cTs cts rv_Nils rv_Conss =
+ let
+ val n = length rv_Nils;
+ val ks = 1 upto n;
+ in
+ EVERY' [rtac (Drule.instantiate' cTs cts @{thm list.induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn rv_Cons =>
+ CONJ_WRAP' (fn (i, rv_Nil) => (EVERY' [rtac exI,
+ rtac (@{thm append_Nil} RS arg_cong RS trans),
+ rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS arg_cong RS trans), rtac rv_Nil]))
+ (ks ~~ rv_Nils))
+ rv_Conss,
+ REPEAT_DETERM o rtac allI, rtac (mk_sumEN n),
+ EVERY' (map (fn i =>
+ CONJ_WRAP' (fn rv_Cons => EVERY' [REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i),
+ CONJ_WRAP' (fn i' => EVERY' [dtac (mk_conjunctN n i'), etac exE, rtac exI,
+ rtac (@{thm append_Cons} RS arg_cong RS trans),
+ rtac (rv_Cons RS trans), etac (@{thm sum_case_cong} RS arg_cong RS trans),
+ rtac (mk_sum_casesN n i RS arg_cong RS trans), atac])
+ ks])
+ rv_Conss)
+ ks)] 1
+ end;
+
+fun mk_set_rv_Lev_tac m cts Lev_0s Lev_Sucs rv_Nils rv_Conss coalg_setss from_to_sbdss =
+ let
+ val n = length Lev_0s;
+ val ks = 1 upto n;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn (i, ((Lev_0, rv_Nil), coalg_sets)) =>
+ EVERY' [rtac impI, REPEAT_DETERM o etac conjE,
+ dtac (Lev_0 RS equalityD1 RS @{thm set_mp}), etac @{thm singletonE}, etac ssubst,
+ rtac (rv_Nil RS arg_cong RS @{thm ssubst[of _ _ "%x. x"]}),
+ rtac (mk_sum_casesN n i RS @{thm ssubst[of _ _ "%x. x"]}),
+ CONJ_WRAP' (fn thm => etac thm THEN' atac) (take m coalg_sets)])
+ (ks ~~ ((Lev_0s ~~ rv_Nils) ~~ coalg_setss)),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), (from_to_sbds, coalg_sets)) =>
+ EVERY' [rtac impI, etac conjE, dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}),
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
+ (fn (i, (from_to_sbd, coalg_set)) =>
+ EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ rtac (rv_Cons RS arg_cong RS @{thm ssubst[of _ _ "%x. x"]}),
+ rtac (mk_sum_casesN n i RS arg_cong RS trans RS @{thm ssubst[of _ _ "%x. x"]}),
+ etac (from_to_sbd RS arg_cong), REPEAT_DETERM o etac allE,
+ dtac (mk_conjunctN n i), etac mp, etac conjI, etac @{thm set_rev_mp},
+ etac coalg_set, atac])
+ (rev (ks ~~ (from_to_sbds ~~ drop m coalg_sets)))])
+ ((Lev_Sucs ~~ rv_Conss) ~~ (from_to_sbdss ~~ coalg_setss))] 1
+ end;
+
+fun mk_set_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbdss =
+ let
+ val n = length Lev_0s;
+ val ks = 1 upto n;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn ((i, (Lev_0, Lev_Suc)), rv_Nil) =>
+ EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS @{thm set_mp}),
+ etac @{thm singletonE}, hyp_subst_tac,
+ CONJ_WRAP' (fn i' => rtac impI THEN' dtac (sym RS trans) THEN' rtac rv_Nil THEN'
+ (if i = i'
+ then EVERY' [dtac (mk_InN_inject n i), hyp_subst_tac,
+ CONJ_WRAP' (fn (i'', Lev_0'') =>
+ EVERY' [rtac impI, rtac @{thm ssubst_mem[OF append_Nil]},
+ rtac (Lev_Suc RS equalityD2 RS @{thm set_mp}), rtac (mk_UnN n i''),
+ rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl,
+ etac conjI, rtac (Lev_0'' RS equalityD2 RS @{thm set_mp}),
+ rtac @{thm singletonI}])
+ (ks ~~ Lev_0s)]
+ else etac (mk_InN_not_InM i' i RS notE)))
+ ks])
+ ((ks ~~ (Lev_0s ~~ Lev_Sucs)) ~~ rv_Nils),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), from_to_sbds) =>
+ EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}),
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
+ (fn (i, from_to_sbd) =>
+ EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ CONJ_WRAP' (fn i' => rtac impI THEN'
+ CONJ_WRAP' (fn i'' =>
+ EVERY' [rtac impI, rtac (Lev_Suc RS equalityD2 RS @{thm set_mp}),
+ rtac @{thm ssubst_mem[OF append_Cons]}, rtac (mk_UnN n i),
+ rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl,
+ rtac conjI, atac, dtac (sym RS trans RS sym),
+ rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS trans),
+ etac (from_to_sbd RS arg_cong), REPEAT_DETERM o etac allE,
+ dtac (mk_conjunctN n i), dtac mp, atac,
+ dtac (mk_conjunctN n i'), dtac mp, atac,
+ dtac (mk_conjunctN n i''), etac mp, atac])
+ ks)
+ ks])
+ (rev (ks ~~ from_to_sbds))])
+ ((Lev_Sucs ~~ rv_Conss) ~~ from_to_sbdss)] 1
+ end;
+
+fun mk_set_image_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbdss to_sbd_injss =
+ let
+ val n = length Lev_0s;
+ val ks = 1 upto n;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn ((i, (Lev_0, Lev_Suc)), rv_Nil) =>
+ EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS @{thm set_mp}),
+ etac @{thm singletonE}, hyp_subst_tac,
+ CONJ_WRAP' (fn i' => rtac impI THEN'
+ CONJ_WRAP' (fn i'' => rtac impI THEN' dtac (sym RS trans) THEN' rtac rv_Nil THEN'
+ (if i = i''
+ then EVERY' [dtac @{thm ssubst_mem[OF sym[OF append_Nil]]},
+ dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}), dtac (mk_InN_inject n i),
+ hyp_subst_tac,
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
+ (fn k => REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN'
+ dtac @{thm list.inject[THEN subst, of "%x. x"]} THEN' etac conjE THEN'
+ (if k = i'
+ then EVERY' [dtac (mk_InN_inject n k), hyp_subst_tac, etac imageI]
+ else etac (mk_InN_not_InM i' k RS notE)))
+ (rev ks)]
+ else etac (mk_InN_not_InM i'' i RS notE)))
+ ks)
+ ks])
+ ((ks ~~ (Lev_0s ~~ Lev_Sucs)) ~~ rv_Nils),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), (from_to_sbds, to_sbd_injs)) =>
+ EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}),
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
+ (fn (i, (from_to_sbd, to_sbd_inj)) =>
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN' hyp_subst_tac THEN'
+ CONJ_WRAP' (fn i' => rtac impI THEN'
+ dtac @{thm ssubst_mem[OF sym[OF append_Cons]]} THEN'
+ dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}) THEN'
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn k =>
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN'
+ dtac @{thm list.inject[THEN subst, of "%x. x"]} THEN' etac conjE THEN'
+ (if k = i
+ then EVERY' [dtac (mk_InN_inject n i),
+ dtac (Thm.permute_prems 0 2 (to_sbd_inj RS @{thm subst[of _ _ "%x. x"]})),
+ atac, atac, hyp_subst_tac] THEN'
+ CONJ_WRAP' (fn i'' =>
+ EVERY' [rtac impI, dtac (sym RS trans),
+ rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS arg_cong RS trans),
+ etac (from_to_sbd RS arg_cong),
+ REPEAT_DETERM o etac allE,
+ dtac (mk_conjunctN n i), dtac mp, atac,
+ dtac (mk_conjunctN n i'), dtac mp, atac,
+ dtac (mk_conjunctN n i''), etac mp, etac sym])
+ ks
+ else etac (mk_InN_not_InM i k RS notE)))
+ (rev ks))
+ ks)
+ (rev (ks ~~ (from_to_sbds ~~ to_sbd_injs)))])
+ ((Lev_Sucs ~~ rv_Conss) ~~ (from_to_sbdss ~~ to_sbd_injss))] 1
+ end;
+
+fun mk_mor_beh_tac m mor_def mor_cong beh_defs carT_defs strT_defs isNode_defs
+ to_sbd_injss from_to_sbdss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbds length_Levs length_Lev's
+ prefCl_Levs rv_lastss set_rv_Levsss set_Levsss set_image_Levsss set_naturalss coalg_setss
+ map_comp_ids map_congs map_arg_congs {context = ctxt, prems = _} =
+ let
+ val n = length beh_defs;
+ val ks = 1 upto n;
+
+ fun fbetw_tac (i, (carT_def, (isNode_def, (Lev_0, (rv_Nil, (Lev_sbd,
+ ((length_Lev, length_Lev'), (prefCl_Lev, (rv_lasts, (set_naturals,
+ (coalg_sets, (set_rv_Levss, (set_Levss, set_image_Levss))))))))))))) =
+ EVERY' [rtac ballI, rtac (carT_def RS equalityD2 RS @{thm set_mp}),
+ rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, rtac conjI,
+ rtac conjI,
+ rtac @{thm UN_I}, rtac UNIV_I, rtac (Lev_0 RS equalityD2 RS @{thm set_mp}),
+ rtac @{thm singletonI},
+ rtac conjI,
+ rtac @{thm UN_least}, rtac Lev_sbd,
+ rtac conjI,
+ rtac @{thm prefCl_UN}, rtac ssubst, rtac @{thm PrefCl_def}, REPEAT_DETERM o rtac allI,
+ rtac impI, etac conjE, rtac exI, rtac conjI, rtac @{thm ord_le_eq_trans},
+ etac @{thm prefix_length_le}, etac length_Lev, rtac prefCl_Lev, etac conjI, atac,
+ rtac conjI,
+ rtac ballI, etac @{thm UN_E}, rtac conjI,
+ if n = 1 then K all_tac else rtac (mk_sumEN n),
+ EVERY' (map6 (fn i => fn isNode_def => fn set_naturals =>
+ fn set_rv_Levs => fn set_Levs => fn set_image_Levs =>
+ EVERY' [rtac (mk_disjIN n i), rtac (isNode_def RS ssubst),
+ rtac exI, rtac conjI,
+ (if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
+ else rtac (@{thm if_P} RS arg_cong RS trans) THEN' etac length_Lev' THEN'
+ etac (@{thm sum_case_cong} RS trans) THEN' rtac (mk_sum_casesN n i)),
+ EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
+ EVERY' [rtac conjI, rtac @{thm ord_eq_le_trans}, rtac (set_natural RS trans),
+ rtac @{thm trans[OF fun_cong[OF image_id] id_apply]},
+ etac set_rv_Lev, TRY o atac, etac conjI, atac])
+ (take m set_naturals) set_rv_Levs),
+ CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
+ EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
+ rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
+ if n = 1 then rtac refl else atac, atac, rtac subsetI,
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
+ rtac set_image_Lev, atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
+ etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
+ if n = 1 then rtac refl else atac])
+ (drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
+ ks isNode_defs set_naturalss set_rv_Levss set_Levss set_image_Levss),
+ CONJ_WRAP' (fn (i, (rv_last, (isNode_def, (set_naturals,
+ (set_rv_Levs, (set_Levs, set_image_Levs)))))) =>
+ EVERY' [rtac ballI,
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
+ rtac (rev_mp OF [rv_last, impI]), etac exE, rtac (isNode_def RS ssubst),
+ rtac exI, rtac conjI,
+ (if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
+ else rtac (@{thm if_P} RS trans) THEN' etac length_Lev' THEN'
+ etac (@{thm sum_case_cong} RS trans) THEN' rtac (mk_sum_casesN n i)),
+ EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
+ EVERY' [rtac conjI, rtac @{thm ord_eq_le_trans}, rtac (set_natural RS trans),
+ rtac @{thm trans[OF fun_cong[OF image_id] id_apply]},
+ etac set_rv_Lev, TRY o atac, etac conjI, atac])
+ (take m set_naturals) set_rv_Levs),
+ CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
+ EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
+ rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
+ if n = 1 then rtac refl else atac, atac, rtac subsetI,
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
+ REPEAT_DETERM_N 4 o etac thin_rl,
+ rtac set_image_Lev,
+ atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
+ etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
+ if n = 1 then rtac refl else atac])
+ (drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
+ (ks ~~ (rv_lasts ~~ (isNode_defs ~~ (set_naturalss ~~
+ (set_rv_Levss ~~ (set_Levss ~~ set_image_Levss)))))),
+ (**)
+ rtac allI, rtac impI, rtac @{thm if_not_P}, rtac notI,
+ etac notE, etac @{thm UN_I[OF UNIV_I]},
+ (*root isNode*)
+ rtac (isNode_def RS ssubst), rtac exI, rtac conjI, rtac (@{thm if_P} RS trans),
+ rtac length_Lev', rtac (Lev_0 RS equalityD2 RS @{thm set_mp}), rtac @{thm singletonI},
+ CONVERSION (Conv.top_conv
+ (K (Conv.try_conv (Conv.rewr_conv (rv_Nil RS eq_reflection)))) ctxt),
+ if n = 1 then rtac refl else rtac (mk_sum_casesN n i),
+ EVERY' (map2 (fn set_natural => fn coalg_set =>
+ EVERY' [rtac conjI, rtac @{thm ord_eq_le_trans}, rtac (set_natural RS trans),
+ rtac @{thm trans[OF fun_cong[OF image_id] id_apply]}, etac coalg_set, atac])
+ (take m set_naturals) (take m coalg_sets)),
+ CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
+ EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
+ rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, rtac set_Lev,
+ rtac (Lev_0 RS equalityD2 RS @{thm set_mp}), rtac @{thm singletonI}, rtac rv_Nil,
+ atac, rtac subsetI,
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
+ rtac set_image_Lev, rtac (Lev_0 RS equalityD2 RS @{thm set_mp}),
+ rtac @{thm singletonI}, dtac length_Lev',
+ etac @{thm set_mp[OF equalityD1[OF arg_cong[OF
+ trans[OF length_append_singleton arg_cong[of _ _ Suc, OF list.size(3)]]]]]},
+ rtac rv_Nil])
+ (drop m set_naturals ~~ (nth set_Levss (i - 1) ~~ nth set_image_Levss (i - 1)))];
+
+ fun mor_tac (i, (strT_def, (((Lev_0, Lev_Suc), (rv_Nil, rv_Cons)),
+ ((map_comp_id, (map_cong, map_arg_cong)), (length_Lev', (from_to_sbds, to_sbd_injs)))))) =
+ EVERY' [rtac ballI, rtac sym, rtac trans, rtac strT_def,
+ rtac (@{thm if_P} RS
+ (if n = 1 then map_arg_cong else @{thm sum_case_cong}) RS trans),
+ rtac (@{thm list.size(3)} RS arg_cong RS trans RS equalityD2 RS @{thm set_mp}),
+ rtac Lev_0, rtac @{thm singletonI},
+ CONVERSION (Conv.top_conv
+ (K (Conv.try_conv (Conv.rewr_conv (rv_Nil RS eq_reflection)))) ctxt),
+ if n = 1 then K all_tac
+ else (rtac (@{thm sum_case_cong} RS trans) THEN'
+ rtac (mk_sum_casesN n i) THEN' rtac (mk_sum_casesN n i RS trans)),
+ rtac (map_comp_id RS trans), rtac (map_cong OF replicate m refl),
+ EVERY' (map3 (fn i' => fn to_sbd_inj => fn from_to_sbd =>
+ DETERM o EVERY' [rtac trans, rtac o_apply, rtac ssubst, rtac @{thm Pair_eq}, rtac conjI,
+ rtac trans, rtac @{thm Shift_def},
+ rtac equalityI, rtac subsetI, etac thin_rl, etac thin_rl,
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm UN_E}], dtac length_Lev', dtac asm_rl,
+ etac thin_rl, dtac @{thm set_rev_mp[OF _ equalityD1]},
+ rtac (@{thm length_Cons} RS arg_cong RS trans), rtac Lev_Suc,
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn i'' =>
+ EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
+ dtac @{thm list.inject[THEN subst, of "%x. x"]}, etac conjE,
+ if i' = i'' then EVERY' [dtac (mk_InN_inject n i'),
+ dtac (Thm.permute_prems 0 2 (to_sbd_inj RS @{thm subst[of _ _ "%x. x"]})),
+ atac, atac, hyp_subst_tac, etac @{thm UN_I[OF UNIV_I]}]
+ else etac (mk_InN_not_InM i' i'' RS notE)])
+ (rev ks),
+ rtac @{thm UN_least}, rtac subsetI, rtac CollectI, rtac @{thm UN_I[OF UNIV_I]},
+ rtac (Lev_Suc RS equalityD2 RS @{thm set_mp}), rtac (mk_UnN n i'), rtac CollectI,
+ REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, etac conjI, atac,
+ rtac trans, rtac @{thm shift_def}, rtac ssubst, rtac @{thm fun_eq_iff}, rtac allI,
+ rtac @{thm if_cong}, rtac (@{thm length_Cons} RS arg_cong RS trans), rtac iffI,
+ dtac asm_rl, dtac asm_rl, dtac asm_rl,
+ dtac (Lev_Suc RS equalityD1 RS @{thm set_mp}),
+ CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn i'' =>
+ EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
+ dtac @{thm list.inject[THEN subst, of "%x. x"]}, etac conjE,
+ if i' = i'' then EVERY' [dtac (mk_InN_inject n i'),
+ dtac (Thm.permute_prems 0 2 (to_sbd_inj RS @{thm subst[of _ _ "%x. x"]})),
+ atac, atac, hyp_subst_tac, atac]
+ else etac (mk_InN_not_InM i' i'' RS notE)])
+ (rev ks),
+ rtac (Lev_Suc RS equalityD2 RS @{thm set_mp}), rtac (mk_UnN n i'), rtac CollectI,
+ REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, etac conjI, atac,
+ CONVERSION (Conv.top_conv
+ (K (Conv.try_conv (Conv.rewr_conv (rv_Cons RS eq_reflection)))) ctxt),
+ if n = 1 then K all_tac
+ else rtac @{thm sum_case_cong} THEN' rtac (mk_sum_casesN n i' RS trans),
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt [from_to_sbd]), rtac refl,
+ rtac refl])
+ ks to_sbd_injs from_to_sbds)];
+ in
+ (rtac mor_cong THEN'
+ EVERY' (map (fn thm => rtac (thm RS ext)) beh_defs) THEN'
+ stac mor_def THEN' rtac conjI THEN'
+ CONJ_WRAP' fbetw_tac
+ (ks ~~ (carT_defs ~~ (isNode_defs ~~ (Lev_0s ~~ (rv_Nils ~~ (Lev_sbds ~~
+ ((length_Levs ~~ length_Lev's) ~~ (prefCl_Levs ~~ (rv_lastss ~~
+ (set_naturalss ~~ (coalg_setss ~~
+ (set_rv_Levsss ~~ (set_Levsss ~~ set_image_Levsss))))))))))))) THEN'
+ CONJ_WRAP' mor_tac
+ (ks ~~ (strT_defs ~~ (((Lev_0s ~~ Lev_Sucs) ~~ (rv_Nils ~~ rv_Conss)) ~~
+ ((map_comp_ids ~~ (map_congs ~~ map_arg_congs)) ~~
+ (length_Lev's ~~ (from_to_sbdss ~~ to_sbd_injss))))))) 1
+ end;
+
+fun mk_congruent_str_final_tac m lsbisE map_comp_id map_cong equiv_LSBISs =
+ EVERY' [rtac @{thm congruentI}, dtac lsbisE,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE, bexE], rtac (o_apply RS trans),
+ etac (sym RS arg_cong RS trans), rtac (map_comp_id RS trans),
+ rtac (map_cong RS trans), REPEAT_DETERM_N m o rtac refl,
+ EVERY' (map (fn equiv_LSBIS =>
+ EVERY' [rtac @{thm equiv_proj}, rtac equiv_LSBIS, etac @{thm set_mp}, atac])
+ equiv_LSBISs), rtac sym, rtac (o_apply RS trans),
+ etac (sym RS arg_cong RS trans), rtac map_comp_id] 1;
+
+fun mk_coalg_final_tac m coalg_def congruent_str_finals equiv_LSBISs set_naturalss coalgT_setss =
+ EVERY' [stac coalg_def,
+ CONJ_WRAP' (fn ((set_naturals, coalgT_sets), (equiv_LSBIS, congruent_str_final)) =>
+ EVERY' [rtac @{thm univ_preserves}, rtac equiv_LSBIS, rtac congruent_str_final,
+ rtac ballI, rtac @{thm ssubst_mem}, rtac o_apply, rtac CollectI,
+ EVERY' (map2 (fn set_natural => fn coalgT_set =>
+ EVERY' [rtac conjI, rtac (set_natural RS @{thm ord_eq_le_trans}),
+ rtac @{thm ord_eq_le_trans[OF trans[OF fun_cong[OF image_id] id_apply]]},
+ etac coalgT_set])
+ (take m set_naturals) (take m coalgT_sets)),
+ CONJ_WRAP' (fn (equiv_LSBIS, (set_natural, coalgT_set)) =>
+ EVERY' [rtac (set_natural RS @{thm ord_eq_le_trans}),
+ rtac @{thm image_subsetI}, rtac ssubst, rtac @{thm proj_in_iff},
+ rtac equiv_LSBIS, etac @{thm set_rev_mp}, etac coalgT_set])
+ (equiv_LSBISs ~~ drop m (set_naturals ~~ coalgT_sets))])
+ ((set_naturalss ~~ coalgT_setss) ~~ (equiv_LSBISs ~~ congruent_str_finals))] 1;
+
+fun mk_mor_T_final_tac mor_def congruent_str_finals equiv_LSBISs =
+ EVERY' [stac mor_def, rtac conjI,
+ CONJ_WRAP' (fn equiv_LSBIS =>
+ EVERY' [rtac ballI, rtac ssubst, rtac @{thm proj_in_iff}, rtac equiv_LSBIS, atac])
+ equiv_LSBISs,
+ CONJ_WRAP' (fn (equiv_LSBIS, congruent_str_final) =>
+ EVERY' [rtac ballI, rtac sym, rtac trans, rtac @{thm univ_commute}, rtac equiv_LSBIS,
+ rtac congruent_str_final, atac, rtac o_apply])
+ (equiv_LSBISs ~~ congruent_str_finals)] 1;
+
+fun mk_mor_Rep_tac m defs Reps Abs_inverses coalg_final_setss map_comp_ids map_congLs
+ {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt defs THEN
+ EVERY' [rtac conjI,
+ CONJ_WRAP' (fn thm => rtac ballI THEN' rtac thm) Reps,
+ CONJ_WRAP' (fn (Rep, ((map_comp_id, map_congL), coalg_final_sets)) =>
+ EVERY' [rtac ballI, rtac (map_comp_id RS trans), rtac map_congL,
+ EVERY' (map2 (fn Abs_inverse => fn coalg_final_set =>
+ EVERY' [rtac ballI, rtac (o_apply RS trans), rtac Abs_inverse,
+ etac @{thm set_rev_mp}, rtac coalg_final_set, rtac Rep])
+ Abs_inverses (drop m coalg_final_sets))])
+ (Reps ~~ ((map_comp_ids ~~ map_congLs) ~~ coalg_final_setss))] 1;
+
+fun mk_mor_Abs_tac defs Abs_inverses {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt defs THEN
+ EVERY' [rtac conjI,
+ CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) Abs_inverses,
+ CONJ_WRAP' (fn thm => rtac ballI THEN' etac (thm RS arg_cong RS sym)) Abs_inverses] 1;
+
+fun mk_mor_coiter_tac m mor_UNIV unf_defs coiter_defs Abs_inverses morEs map_comp_ids map_congs =
+ EVERY' [rtac @{thm ssubst[of _ _ "%x. x"]}, rtac mor_UNIV,
+ CONJ_WRAP' (fn ((Abs_inverse, morE), ((unf_def, coiter_def), (map_comp_id, map_cong))) =>
+ EVERY' [rtac ext, rtac (o_apply RS trans RS sym), rtac (unf_def RS trans),
+ rtac (coiter_def RS arg_cong RS trans), rtac (Abs_inverse RS arg_cong RS trans),
+ rtac (morE RS arg_cong RS trans), rtac (map_comp_id RS trans),
+ rtac (o_apply RS trans RS sym), rtac map_cong,
+ REPEAT_DETERM_N m o rtac refl,
+ EVERY' (map (fn thm => rtac (thm RS trans) THEN' rtac (o_apply RS sym)) coiter_defs)])
+ ((Abs_inverses ~~ morEs) ~~ ((unf_defs ~~ coiter_defs) ~~ (map_comp_ids ~~ map_congs)))] 1;
+
+fun mk_raw_coind_tac bis_def bis_cong bis_O bis_converse bis_Gr tcoalg coalgT mor_T_final
+ sbis_lsbis lsbis_incls incl_lsbiss equiv_LSBISs mor_Rep Rep_injects =
+ let
+ val n = length Rep_injects;
+ in
+ EVERY' [rtac rev_mp, ftac (bis_def RS @{thm subst[of _ _ "%x. x"]}),
+ REPEAT_DETERM o etac conjE, rtac bis_cong, rtac bis_O, rtac bis_converse,
+ rtac bis_Gr, rtac tcoalg, rtac mor_Rep, rtac bis_O, atac, rtac bis_Gr, rtac tcoalg,
+ rtac mor_Rep, REPEAT_DETERM_N n o etac @{thm relImage_Gr},
+ rtac impI, rtac rev_mp, rtac bis_cong, rtac bis_O, rtac bis_Gr, rtac coalgT,
+ rtac mor_T_final, rtac bis_O, rtac sbis_lsbis, rtac bis_converse, rtac bis_Gr, rtac coalgT,
+ rtac mor_T_final, EVERY' (map (fn thm => rtac (thm RS @{thm relInvImage_Gr})) lsbis_incls),
+ rtac impI,
+ CONJ_WRAP' (fn (Rep_inject, (equiv_LSBIS , (incl_lsbis, lsbis_incl))) =>
+ EVERY' [rtac subset_trans, rtac @{thm relInvImage_UNIV_relImage}, rtac subset_trans,
+ rtac @{thm relInvImage_mono}, rtac subset_trans, etac incl_lsbis,
+ rtac @{thm ord_eq_le_trans}, rtac @{thm sym[OF relImage_relInvImage]},
+ rtac @{thm xt1(3)}, rtac @{thm Sigma_cong},
+ rtac @{thm proj_image}, rtac @{thm proj_image}, rtac lsbis_incl,
+ rtac subset_trans, rtac @{thm relImage_mono}, rtac incl_lsbis, atac,
+ rtac @{thm relImage_proj}, rtac equiv_LSBIS, rtac @{thm relInvImage_diag},
+ rtac Rep_inject])
+ (Rep_injects ~~ (equiv_LSBISs ~~ (incl_lsbiss ~~ lsbis_incls)))] 1
+ end;
+
+fun mk_unique_mor_tac raw_coinds bis =
+ CONJ_WRAP' (fn raw_coind =>
+ EVERY' [rtac impI, rtac (bis RS raw_coind RS @{thm set_mp} RS @{thm IdD}), atac,
+ etac conjunct1, atac, etac conjunct2, rtac @{thm image2_eqI}, rtac refl, rtac refl, atac])
+ raw_coinds 1;
+
+fun mk_coiter_unique_mor_tac raw_coinds bis mor coiter_defs =
+ CONJ_WRAP' (fn (raw_coind, coiter_def) =>
+ EVERY' [rtac ext, etac (bis RS raw_coind RS @{thm set_mp} RS @{thm IdD}), rtac mor,
+ rtac @{thm image2_eqI}, rtac refl, rtac (coiter_def RS arg_cong RS trans),
+ rtac (o_apply RS sym), rtac UNIV_I]) (raw_coinds ~~ coiter_defs) 1;
+
+fun mk_unf_o_fld_tac fld_def coiter map_comp_id map_congL coiter_o_unfs
+ {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt [fld_def] THEN EVERY' [rtac ext, rtac trans, rtac o_apply,
+ rtac trans, rtac coiter, rtac trans, rtac map_comp_id, rtac trans, rtac map_congL,
+ EVERY' (map (fn thm =>
+ rtac ballI THEN' rtac (trans OF [thm RS fun_cong, @{thm id_apply}])) coiter_o_unfs),
+ rtac sym, rtac @{thm id_apply}] 1;
+
+fun mk_corec_tac m corec_defs coiter map_cong corec_Inls {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt corec_defs THEN EVERY' [rtac trans, rtac (o_apply RS arg_cong),
+ rtac trans, rtac coiter, fo_rtac (@{thm sum.cases(2)} RS arg_cong RS trans) ctxt, rtac map_cong,
+ REPEAT_DETERM_N m o rtac refl,
+ EVERY' (map (fn thm => rtac @{thm sum_case_expand_Inr} THEN' rtac thm) corec_Inls)] 1;
+
+fun mk_rel_coinduct_tac ks raw_coind bis_rel =
+ EVERY' [rtac rev_mp, rtac raw_coind, rtac ssubst, rtac bis_rel, rtac conjI,
+ CONJ_WRAP' (K (rtac @{thm ord_le_eq_trans[OF subset_UNIV UNIV_Times_UNIV[THEN sym]]})) ks,
+ CONJ_WRAP' (K (EVERY' [rtac allI, rtac allI, rtac impI,
+ REPEAT_DETERM o etac allE, etac mp, etac CollectE, etac @{thm splitD}])) ks,
+ rtac impI, REPEAT_DETERM o etac conjE,
+ CONJ_WRAP' (K (EVERY' [rtac impI, rtac @{thm IdD}, etac @{thm set_mp},
+ rtac CollectI, etac @{thm prod_caseI}])) ks] 1;
+
+fun mk_rel_coinduct_upto_tac m cTs cts rel_coinduct rel_monos rel_Ids =
+ EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts rel_coinduct),
+ EVERY' (map2 (fn rel_mono => fn rel_Id =>
+ EVERY' [REPEAT_DETERM o resolve_tac [allI, impI], REPEAT_DETERM o etac allE,
+ etac disjE, etac mp, atac, hyp_subst_tac, rtac (rel_mono RS @{thm set_mp}),
+ REPEAT_DETERM_N m o rtac @{thm subset_refl},
+ REPEAT_DETERM_N (length rel_monos) o rtac @{thm Id_subset},
+ rtac (rel_Id RS equalityD2 RS @{thm set_mp}), rtac @{thm IdI}])
+ rel_monos rel_Ids),
+ rtac impI, REPEAT_DETERM o etac conjE,
+ CONJ_WRAP' (K (rtac impI THEN' etac mp THEN' etac disjI1)) rel_Ids] 1;
+
+fun mk_unf_coinduct_tac m ks raw_coind bis_def =
+ let
+ val n = length ks;
+ in
+ EVERY' [rtac rev_mp, rtac raw_coind, rtac ssubst, rtac bis_def, rtac conjI,
+ CONJ_WRAP' (K (rtac @{thm ord_le_eq_trans[OF subset_UNIV UNIV_Times_UNIV[THEN sym]]})) ks,
+ CONJ_WRAP' (fn i => EVERY' [select_prem_tac n (dtac asm_rl) i, REPEAT_DETERM o rtac allI,
+ rtac impI, REPEAT_DETERM o dtac @{thm meta_spec}, etac CollectE, etac @{thm meta_impE},
+ atac, etac exE, etac conjE, etac conjE, rtac bexI, rtac conjI,
+ etac @{thm fst_conv[THEN subst]}, etac @{thm snd_conv[THEN subst]},
+ rtac CollectI, REPEAT_DETERM_N m o (rtac conjI THEN' rtac @{thm subset_UNIV}),
+ CONJ_WRAP' (fn i' => EVERY' [rtac subsetI, rtac CollectI, dtac (mk_conjunctN n i'),
+ REPEAT_DETERM o etac allE, etac mp, rtac @{thm ssubst_mem[OF pair_collapse]}, atac])
+ ks])
+ ks,
+ rtac impI,
+ CONJ_WRAP' (fn i => EVERY' [rtac impI, dtac (mk_conjunctN n i),
+ rtac @{thm subst[OF pair_in_Id_conv]}, etac @{thm set_mp},
+ rtac CollectI, etac (refl RSN (2, @{thm subst_Pair}))]) ks] 1
+ end;
+
+fun mk_unf_coinduct_upto_tac ks cTs cts unf_coinduct bis_def bis_diag =
+ EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts unf_coinduct),
+ EVERY' (map (fn i =>
+ EVERY' [etac disjE, REPEAT_DETERM o dtac @{thm meta_spec}, etac @{thm meta_mp},
+ atac, rtac rev_mp, rtac subst, rtac bis_def, rtac bis_diag,
+ rtac impI, dtac conjunct2, dtac (mk_conjunctN (length ks) i), REPEAT_DETERM o etac allE,
+ etac impE, etac @{thm diag_UNIV_I}, REPEAT_DETERM o eresolve_tac [bexE, conjE, CollectE],
+ rtac exI, rtac conjI, etac conjI, atac,
+ CONJ_WRAP' (K (EVERY' [REPEAT_DETERM o resolve_tac [allI, impI],
+ rtac disjI2, rtac @{thm diagE}, etac @{thm set_mp}, atac])) ks])
+ ks),
+ rtac impI, REPEAT_DETERM o etac conjE,
+ CONJ_WRAP' (K (rtac impI THEN' etac mp THEN' etac disjI1)) ks] 1;
+
+fun mk_map_tac m n cT coiter map_comp' map_cong =
+ EVERY' [rtac ext, rtac (o_apply RS trans RS sym), rtac (o_apply RS trans RS sym),
+ rtac (coiter RS trans), rtac (Thm.permute_prems 0 1 (map_comp' RS box_equals)), rtac map_cong,
+ REPEAT_DETERM_N m o rtac (@{thm id_o} RS fun_cong),
+ REPEAT_DETERM_N n o rtac (@{thm o_id} RS fun_cong),
+ rtac (o_apply RS (Drule.instantiate' [cT] [] arg_cong) RS sym)] 1;
+
+fun mk_set_le_tac n hset_minimal set_hsets set_hset_hsetss =
+ EVERY' [rtac hset_minimal,
+ REPEAT_DETERM_N n o rtac @{thm Un_upper1},
+ REPEAT_DETERM_N n o
+ EVERY' (map3 (fn i => fn set_hset => fn set_hset_hsets =>
+ EVERY' [rtac subsetI, rtac @{thm UnI2}, rtac (mk_UnN n i), etac @{thm UN_I},
+ etac UnE, etac set_hset, REPEAT_DETERM_N (n - 1) o etac UnE,
+ EVERY' (map (fn thm => EVERY' [etac @{thm UN_E}, etac thm, atac]) set_hset_hsets)])
+ (1 upto n) set_hsets set_hset_hsetss)] 1;
+
+fun mk_set_simp_tac n set_le set_incl_hset set_hset_incl_hsets =
+ EVERY' [rtac equalityI, rtac set_le, rtac @{thm Un_least}, rtac set_incl_hset,
+ REPEAT_DETERM_N (n - 1) o rtac @{thm Un_least},
+ EVERY' (map (fn thm => rtac @{thm UN_least} THEN' etac thm) set_hset_incl_hsets)] 1;
+
+fun mk_map_id_tac maps coiter_unique coiter_unf =
+ EVERY' [rtac (coiter_unique RS trans), EVERY' (map (fn thm => rtac (thm RS sym)) maps),
+ rtac coiter_unf] 1;
+
+fun mk_map_comp_tac m n maps map_comps map_congs coiter_unique =
+ EVERY' [rtac coiter_unique,
+ EVERY' (map3 (fn map_thm => fn map_comp => fn map_cong =>
+ EVERY' (map rtac
+ ([@{thm o_assoc} RS trans,
+ @{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_comp RS sym, refl] RS trans,
+ @{thm o_assoc} RS trans RS sym,
+ @{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_thm, refl] RS trans,
+ @{thm o_assoc} RS sym RS trans, map_thm RS arg_cong RS trans, @{thm o_assoc} RS trans,
+ @{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_comp RS sym, refl] RS trans,
+ ext, o_apply RS trans, o_apply RS trans RS sym, map_cong] @
+ replicate m (@{thm id_o} RS fun_cong) @
+ replicate n (@{thm o_id} RS fun_cong))))
+ maps map_comps map_congs)] 1;
+
+val o_apply_trans_sym = o_apply RS trans RS sym;
+val fst_convol_fun_cong_sym = @{thm fst_convol} RS fun_cong RS sym;
+val snd_convol_fun_cong_sym = @{thm snd_convol} RS fun_cong RS sym;
+
+fun mk_mcong_tac m coinduct_tac map_comp's map_simps map_congs set_naturalss set_hsetss
+ set_hset_hsetsss =
+ let
+ val n = length map_comp's;
+ val ks = 1 upto n;
+ in
+ EVERY' ([rtac rev_mp,
+ coinduct_tac] @
+ maps (fn (((((map_comp'_trans, map_simps_trans), map_cong), set_naturals), set_hsets),
+ set_hset_hsetss) =>
+ [REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac, rtac exI, rtac conjI, rtac conjI,
+ rtac map_comp'_trans, rtac sym, rtac map_simps_trans, rtac map_cong,
+ REPEAT_DETERM_N m o (rtac o_apply_trans_sym THEN' rtac @{thm id_apply}),
+ REPEAT_DETERM_N n o rtac fst_convol_fun_cong_sym,
+ rtac map_comp'_trans, rtac sym, rtac map_simps_trans, rtac map_cong,
+ EVERY' (maps (fn set_hset =>
+ [rtac o_apply_trans_sym, rtac (@{thm id_apply} RS trans), etac CollectE,
+ REPEAT_DETERM o etac conjE, etac bspec, etac set_hset]) set_hsets),
+ REPEAT_DETERM_N n o rtac snd_convol_fun_cong_sym,
+ CONJ_WRAP' (fn (set_natural, set_hset_hsets) =>
+ EVERY' [REPEAT_DETERM o rtac allI, rtac impI, rtac @{thm image_convolD},
+ etac @{thm set_rev_mp}, rtac @{thm ord_eq_le_trans}, rtac set_natural,
+ rtac @{thm image_mono}, rtac subsetI, rtac CollectI, etac CollectE,
+ REPEAT_DETERM o etac conjE,
+ CONJ_WRAP' (fn set_hset_hset =>
+ EVERY' [rtac ballI, etac bspec, etac set_hset_hset, atac]) set_hset_hsets])
+ (drop m set_naturals ~~ set_hset_hsetss)])
+ (map (fn th => th RS trans) map_comp's ~~ map (fn th => th RS trans) map_simps ~~
+ map_congs ~~ set_naturalss ~~ set_hsetss ~~ set_hset_hsetsss) @
+ [rtac impI,
+ CONJ_WRAP' (fn k =>
+ EVERY' [rtac impI, dtac (mk_conjunctN n k), etac mp, rtac exI, rtac conjI, etac CollectI,
+ rtac conjI, rtac refl, rtac refl]) ks]) 1
+ end
+
+fun mk_map_unique_tac coiter_unique map_comps {context = ctxt, prems = _} =
+ rtac coiter_unique 1 THEN
+ Local_Defs.unfold_tac ctxt (map (fn thm => thm RS sym) map_comps @ @{thms o_assoc id_o o_id}) THEN
+ ALLGOALS (etac sym);
+
+fun mk_col_natural_tac cts rec_0s rec_Sucs map_simps set_naturalss
+ {context = ctxt, prems = _} =
+ let
+ val n = length map_simps;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI, SELECT_GOAL (Local_Defs.unfold_tac ctxt rec_0s),
+ CONJ_WRAP' (K (rtac @{thm image_empty})) rec_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn (rec_Suc, (map_simp, set_nats)) => EVERY'
+ [SELECT_GOAL (Local_Defs.unfold_tac ctxt
+ (rec_Suc :: map_simp :: set_nats @ @{thms image_Un image_UN UN_simps(10)})),
+ rtac @{thm Un_cong}, rtac refl,
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
+ (fn i => EVERY' [rtac @{thm UN_cong[OF refl]},
+ REPEAT_DETERM o etac allE, etac (mk_conjunctN n i)]) (n downto 1)])
+ (rec_Sucs ~~ (map_simps ~~ set_naturalss))] 1
+ end;
+
+fun mk_set_natural_tac hset_def col_natural =
+ EVERY' (map rtac [ext, (o_apply RS trans), (hset_def RS trans), sym,
+ (o_apply RS trans), (@{thm image_cong} OF [hset_def, refl] RS trans),
+ (@{thm image_UN} RS trans), (refl RS @{thm UN_cong}), col_natural]) 1;
+
+fun mk_col_bd_tac m j cts rec_0s rec_Sucs sbd_Card_order sbd_Cinfinite set_sbdss =
+ let
+ val n = length rec_0s;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn rec_0 => EVERY' (map rtac [@{thm ordIso_ordLeq_trans},
+ @{thm card_of_ordIso_subst}, rec_0, @{thm Card_order_empty}, sbd_Card_order])) rec_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn (rec_Suc, set_sbds) => EVERY'
+ [rtac @{thm ordIso_ordLeq_trans}, rtac @{thm card_of_ordIso_subst}, rtac rec_Suc,
+ rtac (sbd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})), rtac (nth set_sbds (j - 1)),
+ REPEAT_DETERM_N (n - 1) o rtac (sbd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})),
+ EVERY' (map2 (fn i => fn set_sbd => EVERY' [rtac @{thm UNION_Cinfinite_bound},
+ rtac set_sbd, rtac ballI, REPEAT_DETERM o etac allE,
+ etac (mk_conjunctN n i), rtac sbd_Cinfinite]) (1 upto n) (drop m set_sbds))])
+ (rec_Sucs ~~ set_sbdss)] 1
+ end;
+
+fun mk_set_bd_tac sbd_Cinfinite hset_def col_bd =
+ EVERY' (map rtac [@{thm ordIso_ordLeq_trans}, @{thm card_of_ordIso_subst}, hset_def,
+ ctrans, @{thm UNION_Cinfinite_bound}, @{thm ordIso_ordLeq_trans}, @{thm card_of_nat},
+ @{thm natLeq_ordLeq_cinfinite}, sbd_Cinfinite, ballI, col_bd, sbd_Cinfinite,
+ ctrans, @{thm infinite_ordLeq_cexp}, sbd_Cinfinite, @{thm cexp_ordLeq_ccexp}]) 1;
+
+fun mk_in_bd_tac isNode_hset isNode_hsets carT_def card_of_carT mor_image Rep_inverse mor_hsets
+ sbd_Cnotzero sbd_Card_order mor_Rep coalgT mor_T_final tcoalg =
+ let
+ val n = length isNode_hsets;
+ val in_hin_tac = rtac CollectI THEN'
+ CONJ_WRAP' (fn mor_hset => EVERY' (map etac
+ [mor_hset OF [coalgT, mor_T_final] RS sym RS @{thm ord_eq_le_trans},
+ arg_cong RS sym RS @{thm ord_eq_le_trans},
+ mor_hset OF [tcoalg, mor_Rep, UNIV_I] RS @{thm ord_eq_le_trans}])) mor_hsets;
+ in
+ EVERY' [rtac (Thm.permute_prems 0 1 @{thm ordLeq_transitive}), rtac ctrans,
+ rtac @{thm card_of_image}, rtac @{thm ordIso_ordLeq_trans},
+ rtac @{thm card_of_ordIso_subst}, rtac @{thm sym[OF proj_image]}, rtac ctrans,
+ rtac @{thm card_of_image}, rtac ctrans, rtac card_of_carT, rtac @{thm cexp_mono2_Cnotzero},
+ rtac @{thm cexp_ordLeq_ccexp}, rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
+ rtac @{thm Cnotzero_cexp}, rtac sbd_Cnotzero, rtac sbd_Card_order,
+ rtac @{thm card_of_mono1}, rtac subsetI, rtac @{thm image_eqI}, rtac sym,
+ rtac Rep_inverse, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ rtac @{thm set_mp}, rtac equalityD2, rtac @{thm sym[OF proj_image]}, rtac imageE,
+ rtac @{thm set_rev_mp}, rtac mor_image, rtac mor_Rep, rtac UNIV_I, rtac equalityD2,
+ rtac @{thm proj_image}, rtac @{thm image_eqI}, atac,
+ ftac (carT_def RS equalityD1 RS @{thm set_mp}),
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ rtac (carT_def RS equalityD2 RS @{thm set_mp}), rtac CollectI, REPEAT_DETERM o rtac exI,
+ rtac conjI, rtac refl, rtac conjI, etac conjI, etac conjI, etac conjI, rtac conjI,
+ rtac ballI, dtac bspec, atac, REPEAT_DETERM o etac conjE, rtac conjI,
+ CONJ_WRAP_GEN' (etac disjE) (fn (i, isNode_hset) =>
+ EVERY' [rtac (mk_disjIN n i), rtac isNode_hset, atac, atac, atac, in_hin_tac])
+ (1 upto n ~~ isNode_hsets),
+ CONJ_WRAP' (fn isNode_hset =>
+ EVERY' [rtac ballI, rtac isNode_hset, atac, ftac CollectD, etac @{thm SuccD},
+ etac bspec, atac, in_hin_tac])
+ isNode_hsets,
+ atac, rtac isNode_hset, atac, atac, atac, in_hin_tac] 1
+ end;
+
+fun mk_bd_card_order_tac sbd_card_order =
+ EVERY' (map rtac [@{thm card_order_ccexp}, sbd_card_order, sbd_card_order]) 1;
+
+fun mk_bd_cinfinite_tac sbd_Cinfinite =
+ EVERY' (map rtac [@{thm cinfinite_ccexp}, @{thm ctwo_ordLeq_Cinfinite},
+ sbd_Cinfinite, sbd_Cinfinite]) 1;
+
+fun mk_pickWP_assms_tac set_incl_hsets set_incl_hins map_eq =
+ let
+ val m = length set_incl_hsets;
+ in
+ EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
+ EVERY' (map (fn thm => rtac conjI THEN' etac (thm RS @{thm subset_trans})) set_incl_hsets),
+ CONJ_WRAP' (fn thm => rtac thm THEN' REPEAT_DETERM_N m o atac) set_incl_hins,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
+ EVERY' (map (fn thm => rtac conjI THEN' etac (thm RS @{thm subset_trans})) set_incl_hsets),
+ CONJ_WRAP' (fn thm => rtac thm THEN' REPEAT_DETERM_N m o atac) set_incl_hins,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE], etac map_eq]
+ end;
+
+fun mk_coalg_thePull_tac m coalg_def map_wpulls set_naturalss pickWP_assms_tacs
+ {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt [coalg_def] THEN
+ CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, set_naturals)) =>
+ EVERY' [rtac ballI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
+ hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
+ EVERY' (map (etac o mk_conjunctN m) (1 upto m)),
+ pickWP_assms_tac,
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt @{thms o_apply prod.cases}), rtac impI,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ rtac CollectI,
+ REPEAT_DETERM_N m o (rtac conjI THEN' rtac @{thm subset_UNIV}),
+ CONJ_WRAP' (fn set_natural =>
+ EVERY' [rtac @{thm ord_eq_le_trans}, rtac trans, rtac set_natural,
+ rtac @{thm trans[OF fun_cong[OF image_id] id_apply]}, atac])
+ (drop m set_naturals)])
+ (map_wpulls ~~ (pickWP_assms_tacs ~~ set_naturalss)) 1;
+
+fun mk_mor_thePull_nth_tac conv pick m mor_def map_wpulls map_comps pickWP_assms_tacs
+ {context = ctxt, prems = _} =
+ let
+ val n = length map_comps;
+ in
+ Local_Defs.unfold_tac ctxt [mor_def] THEN
+ EVERY' [rtac conjI,
+ CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) (1 upto n),
+ CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, map_comp)) =>
+ EVERY' [rtac ballI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}, conjE],
+ hyp_subst_tac,
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt @{thms o_apply prod.cases}),
+ rtac (map_comp RS trans),
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt (conv :: @{thms o_id id_o})),
+ rtac (map_wpull RS pick), REPEAT_DETERM_N m o atac,
+ pickWP_assms_tac])
+ (map_wpulls ~~ (pickWP_assms_tacs ~~ map_comps))] 1
+ end;
+
+val mk_mor_thePull_fst_tac = mk_mor_thePull_nth_tac @{thm fst_conv} @{thm pickWP(2)};
+val mk_mor_thePull_snd_tac = mk_mor_thePull_nth_tac @{thm snd_conv} @{thm pickWP(3)};
+
+fun mk_mor_thePull_pick_tac mor_def coiters map_comps {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt [mor_def, @{thm thePull_def}] THEN rtac conjI 1 THEN
+ CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) coiters 1 THEN
+ CONJ_WRAP' (fn (coiter, map_comp) =>
+ EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}, conjE],
+ hyp_subst_tac,
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt (coiter :: map_comp :: @{thms comp_def id_def})),
+ rtac refl])
+ (coiters ~~ map_comps) 1;
+
+fun mk_pick_col_tac m j cts rec_0s rec_Sucs coiters set_naturalss map_wpulls pickWP_assms_tacs
+ {context = ctxt, prems = _} =
+ let
+ val n = length rec_0s;
+ val ks = n downto 1;
+ in
+ EVERY' [rtac (Drule.instantiate' [] cts @{thm nat_induct}),
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn thm => EVERY'
+ [rtac impI, rtac @{thm ord_eq_le_trans}, rtac thm, rtac @{thm empty_subsetI}]) rec_0s,
+ REPEAT_DETERM o rtac allI,
+ CONJ_WRAP' (fn (rec_Suc, ((coiter, set_naturals), (map_wpull, pickWP_assms_tac))) =>
+ EVERY' [rtac impI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
+ hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
+ EVERY' (map (etac o mk_conjunctN m) (1 upto m)),
+ pickWP_assms_tac,
+ rtac impI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ rtac @{thm ord_eq_le_trans}, rtac rec_Suc,
+ rtac @{thm Un_least},
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt [coiter, nth set_naturals (j - 1),
+ @{thm prod.cases}]),
+ etac @{thm ord_eq_le_trans[OF trans [OF fun_cong[OF image_id] id_apply]]},
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least})) (fn (i, set_natural) =>
+ EVERY' [rtac @{thm UN_least},
+ SELECT_GOAL (Local_Defs.unfold_tac ctxt [coiter, set_natural, @{thm prod.cases}]),
+ etac imageE, hyp_subst_tac, REPEAT_DETERM o etac allE,
+ dtac (mk_conjunctN n i), etac mp, etac @{thm set_mp}, atac])
+ (ks ~~ rev (drop m set_naturals))])
+ (rec_Sucs ~~ ((coiters ~~ set_naturalss) ~~ (map_wpulls ~~ pickWP_assms_tacs)))] 1
+ end;
+
+fun mk_wpull_tac m coalg_thePull mor_thePull_fst mor_thePull_snd mor_thePull_pick
+ mor_unique pick_cols hset_defs =
+ EVERY' [rtac @{thm ssubst[of _ _ "%x. x", OF wpull_def]}, REPEAT_DETERM o rtac allI, rtac impI,
+ REPEAT_DETERM o etac conjE, rtac bexI, rtac conjI,
+ rtac box_equals, rtac mor_unique,
+ rtac coalg_thePull, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
+ rtac mor_thePull_pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
+ rtac mor_thePull_fst, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
+ rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
+ rtac @{thm prod_caseI}, etac conjI, etac conjI, atac, rtac o_apply, rtac @{thm fst_conv},
+ rtac box_equals, rtac mor_unique,
+ rtac coalg_thePull, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
+ rtac mor_thePull_pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
+ rtac mor_thePull_snd, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
+ rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
+ rtac @{thm prod_caseI}, etac conjI, etac conjI, atac, rtac o_apply, rtac @{thm snd_conv},
+ rtac CollectI,
+ CONJ_WRAP' (fn (pick, def) =>
+ EVERY' [rtac (def RS @{thm ord_eq_le_trans}), rtac @{thm UN_least},
+ rtac pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
+ rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
+ rtac @{thm prod_caseI}, etac conjI, etac conjI, atac])
+ (pick_cols ~~ hset_defs)] 1;
+
+fun mk_wit_tac n unf_flds set_simp wit {context = ctxt, prems = _} =
+ REPEAT_DETERM (atac 1 ORELSE
+ EVERY' [dtac @{thm set_rev_mp}, rtac equalityD1, resolve_tac set_simp,
+ K (Local_Defs.unfold_tac ctxt unf_flds),
+ REPEAT_DETERM_N n o etac UnE,
+ REPEAT_DETERM o
+ (TRY o REPEAT_DETERM o etac UnE THEN' TRY o etac @{thm UN_E} THEN'
+ (eresolve_tac wit ORELSE'
+ (dresolve_tac wit THEN'
+ (etac FalseE ORELSE'
+ EVERY' [hyp_subst_tac, dtac @{thm set_rev_mp}, rtac equalityD1, resolve_tac set_simp,
+ K (Local_Defs.unfold_tac ctxt unf_flds), REPEAT_DETERM_N n o etac UnE]))))] 1);
+
+fun mk_rel_unfold_tac in_Jrels i in_rel map_comp map_cong map_simp set_simps unf_inject unf_fld
+ set_naturals set_incls set_set_inclss =
+ let
+ val m = length set_incls;
+ val n = length set_set_inclss;
+ val (passive_set_naturals, active_set_naturals) = chop m set_naturals;
+ val in_Jrel = nth in_Jrels (i - 1);
+ val if_tac =
+ EVERY' [dtac (in_Jrel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
+ rtac (in_rel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
+ EVERY' (map2 (fn set_natural => fn set_incl =>
+ EVERY' [rtac conjI, rtac @{thm ord_eq_le_trans}, rtac set_natural,
+ rtac @{thm ord_eq_le_trans}, rtac @{thm trans[OF fun_cong[OF image_id] id_apply]},
+ etac (set_incl RS @{thm subset_trans})])
+ passive_set_naturals set_incls),
+ CONJ_WRAP' (fn (in_Jrel, (set_natural, set_set_incls)) =>
+ EVERY' [rtac @{thm ord_eq_le_trans}, rtac set_natural, rtac @{thm image_subsetI},
+ rtac (in_Jrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
+ CONJ_WRAP' (fn thm => etac (thm RS @{thm subset_trans}) THEN' atac) set_set_incls,
+ rtac conjI, rtac refl, rtac refl])
+ (in_Jrels ~~ (active_set_naturals ~~ set_set_inclss)),
+ CONJ_WRAP' (fn conv =>
+ EVERY' [rtac trans, rtac map_comp, rtac trans, rtac map_cong,
+ REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
+ REPEAT_DETERM_N n o EVERY' (map rtac [trans, o_apply, conv]),
+ rtac trans, rtac sym, rtac map_simp, rtac (unf_inject RS iffD2), atac])
+ @{thms fst_conv snd_conv}];
+ val only_if_tac =
+ EVERY' [dtac (in_rel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
+ rtac (in_Jrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
+ CONJ_WRAP' (fn (set_simp, passive_set_natural) =>
+ EVERY' [rtac @{thm ord_eq_le_trans}, rtac set_simp, rtac @{thm Un_least},
+ rtac @{thm ord_eq_le_trans}, rtac box_equals, rtac passive_set_natural,
+ rtac (unf_fld RS sym RS arg_cong), rtac @{thm trans[OF fun_cong[OF image_id] id_apply]},
+ atac,
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
+ (fn (active_set_natural, in_Jrel) => EVERY' [rtac @{thm ord_eq_le_trans},
+ rtac @{thm UN_cong[OF _ refl]}, rtac @{thm box_equals[OF _ _ refl]},
+ rtac active_set_natural, rtac (unf_fld RS sym RS arg_cong), rtac @{thm UN_least},
+ dtac @{thm set_rev_mp}, etac @{thm image_mono}, etac imageE,
+ dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Jrel RS iffD1),
+ dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE,
+ dtac (Thm.permute_prems 0 1 @{thm ssubst_mem}), atac,
+ hyp_subst_tac, REPEAT_DETERM o eresolve_tac [CollectE, conjE], atac])
+ (rev (active_set_naturals ~~ in_Jrels))])
+ (set_simps ~~ passive_set_naturals),
+ rtac conjI,
+ REPEAT_DETERM_N 2 o EVERY'[rtac (unf_inject RS iffD1), rtac trans, rtac map_simp,
+ rtac box_equals, rtac map_comp, rtac (unf_fld RS sym RS arg_cong), rtac trans,
+ rtac map_cong, REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
+ EVERY' (map (fn in_Jrel => EVERY' [rtac trans, rtac o_apply, dtac @{thm set_rev_mp}, atac,
+ dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Jrel RS iffD1), dtac @{thm someI_ex},
+ REPEAT_DETERM o etac conjE, atac]) in_Jrels),
+ atac]]
+ in
+ EVERY' [rtac iffI, if_tac, only_if_tac] 1
+ end;
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_gfp_util.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,261 @@
+(* Title: HOL/Codatatype/Tools/bnf_gfp_util.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Library for the codatatype construction.
+*)
+
+signature BNF_GFP_UTIL =
+sig
+ val mk_rec_simps: int -> thm -> thm list -> thm list list
+
+ val dest_listT: typ -> typ
+
+ val mk_Cons: term -> term -> term
+ val mk_If: term -> term -> term -> term
+ val mk_Shift: term -> term -> term
+ val mk_Succ: term -> term -> term
+ val mk_Times: term * term -> term
+ val mk_append: term * term -> term
+ val mk_congruent: term -> term -> term
+ val mk_clists: term -> term
+ val mk_diag: term -> term
+ val mk_equiv: term -> term -> term
+ val mk_fromCard: term -> term -> term
+ val mk_list_rec: term -> term -> term
+ val mk_nat_rec: term -> term -> term
+ val mk_pickWP: term -> term -> term -> term -> term -> term
+ val mk_prefCl: term -> term
+ val mk_proj: term -> term
+ val mk_quotient: term -> term -> term
+ val mk_shift: term -> term -> term
+ val mk_size: term -> term
+ val mk_thePull: term -> term -> term -> term -> term
+ val mk_toCard: term -> term -> term
+ val mk_undefined: typ -> term
+ val mk_univ: term -> term
+
+ val Inl_const: typ -> typ -> term
+ val Inr_const: typ -> typ -> term
+
+ val mk_Inl: term -> typ -> term
+ val mk_Inr: term -> typ -> term
+
+ val mk_InN: typ list -> term -> int -> term
+
+ val mk_sumTN: typ list -> typ
+ val mk_sum_case: term -> term -> term
+ val mk_sum_caseN: term list -> term
+
+ val mk_disjIN: int -> int -> thm
+ val mk_specN: int -> thm -> thm
+ val mk_sum_casesN: int -> int -> thm
+ val mk_sumEN: int -> thm
+
+ val mk_InN_Field: int -> int -> thm
+ val mk_InN_inject: int -> int -> thm
+ val mk_InN_not_InM: int -> int -> thm
+end;
+
+structure BNF_GFP_Util : BNF_GFP_UTIL =
+struct
+
+open BNF_Util
+
+val mk_append = HOLogic.mk_binop @{const_name append};
+
+fun mk_equiv B R =
+ Const (@{const_name equiv}, fastype_of B --> fastype_of R --> HOLogic.boolT) $ B $ R;
+
+fun mk_Sigma (A, B) =
+ let
+ val AT = fastype_of A;
+ val BT = fastype_of B;
+ val ABT = mk_relT (HOLogic.dest_setT AT, HOLogic.dest_setT (range_type BT));
+ in Const (@{const_name Sigma}, AT --> BT --> ABT) $ A $ B end;
+
+fun mk_diag A =
+ let
+ val AT = fastype_of A;
+ val AAT = mk_relT (HOLogic.dest_setT AT, HOLogic.dest_setT AT);
+ in Const (@{const_name diag}, AT --> AAT) $ A end;
+
+fun mk_Times (A, B) =
+ let val AT = HOLogic.dest_setT (fastype_of A);
+ in mk_Sigma (A, Term.absdummy AT B) end;
+
+fun dest_listT (Type (@{type_name list}, [T])) = T
+ | dest_listT T = raise TYPE ("dest_setT: set type expected", [T], []);
+
+fun mk_Succ Kl kl =
+ let val T = fastype_of kl;
+ in
+ Const (@{const_name Succ},
+ HOLogic.mk_setT T --> T --> HOLogic.mk_setT (dest_listT T)) $ Kl $ kl
+ end;
+
+fun mk_Shift Kl k =
+ let val T = fastype_of Kl;
+ in
+ Const (@{const_name Shift}, T --> dest_listT (HOLogic.dest_setT T) --> T) $ Kl $ k
+ end;
+
+fun mk_shift lab k =
+ let val T = fastype_of lab;
+ in
+ Const (@{const_name shift}, T --> dest_listT (Term.domain_type T) --> T) $ lab $ k
+ end;
+
+fun mk_prefCl A =
+ Const (@{const_name prefCl}, fastype_of A --> HOLogic.boolT) $ A;
+
+fun mk_clists r =
+ let val T = fastype_of r;
+ in Const (@{const_name clists}, T --> mk_relT (`I (HOLogic.listT (fst (dest_relT T))))) $ r end;
+
+fun mk_toCard A r =
+ let
+ val AT = fastype_of A;
+ val rT = fastype_of r;
+ in
+ Const (@{const_name toCard},
+ AT --> rT --> HOLogic.dest_setT AT --> fst (dest_relT rT)) $ A $ r
+ end;
+
+fun mk_fromCard A r =
+ let
+ val AT = fastype_of A;
+ val rT = fastype_of r;
+ in
+ Const (@{const_name fromCard},
+ AT --> rT --> fst (dest_relT rT) --> HOLogic.dest_setT AT) $ A $ r
+ end;
+
+fun mk_Cons x xs =
+ let val T = fastype_of xs;
+ in Const (@{const_name Cons}, dest_listT T --> T --> T) $ x $ xs end;
+
+fun mk_size t = HOLogic.size_const (fastype_of t) $ t;
+
+fun mk_If p t f =
+ let val T = fastype_of t;
+ in Const (@{const_name If}, HOLogic.boolT --> T --> T --> T) $ p $ t $ f end;
+
+fun mk_quotient A R =
+ let val T = fastype_of A;
+ in Const (@{const_name quotient}, T --> fastype_of R --> HOLogic.mk_setT T) $ A $ R end;
+
+fun mk_proj R =
+ let val ((AT, BT), T) = `dest_relT (fastype_of R);
+ in Const (@{const_name proj}, T --> AT --> HOLogic.mk_setT BT) $ R end;
+
+fun mk_univ f =
+ let val ((AT, BT), T) = `dest_funT (fastype_of f);
+ in Const (@{const_name univ}, T --> HOLogic.mk_setT AT --> BT) $ f end;
+
+fun mk_congruent R f =
+ Const (@{const_name congruent}, fastype_of R --> fastype_of f --> HOLogic.boolT) $ R $ f;
+
+fun mk_undefined T = Const (@{const_name undefined}, T);
+
+fun mk_nat_rec Zero Suc =
+ let val T = fastype_of Zero;
+ in Const (@{const_name nat_rec}, T --> fastype_of Suc --> HOLogic.natT --> T) $ Zero $ Suc end;
+
+fun mk_list_rec Nil Cons =
+ let
+ val T = fastype_of Nil;
+ val (U, consT) = `(Term.domain_type) (fastype_of Cons);
+ in
+ Const (@{const_name list_rec}, T --> consT --> HOLogic.listT U --> T) $ Nil $ Cons
+ end;
+
+fun mk_thePull B1 B2 f1 f2 =
+ let
+ val fT1 = fastype_of f1;
+ val fT2 = fastype_of f2;
+ val BT1 = domain_type fT1;
+ val BT2 = domain_type fT2;
+ in
+ Const (@{const_name thePull}, HOLogic.mk_setT BT1 --> HOLogic.mk_setT BT2 --> fT1 --> fT2 -->
+ mk_relT (BT1, BT2)) $ B1 $ B2 $ f1 $ f2
+ end;
+
+fun mk_pickWP A f1 f2 b1 b2 =
+ let
+ val fT1 = fastype_of f1;
+ val fT2 = fastype_of f2;
+ val AT = domain_type fT1;
+ val BT1 = range_type fT1;
+ val BT2 = range_type fT2;
+ in
+ Const (@{const_name pickWP}, HOLogic.mk_setT AT --> fT1 --> fT2 --> BT1 --> BT2 --> AT) $
+ A $ f1 $ f2 $ b1 $ b2
+ end;
+
+fun mk_disjIN 1 1 = @{thm TrueE[OF TrueI]}
+ | mk_disjIN _ 1 = disjI1
+ | mk_disjIN 2 2 = disjI2
+ | mk_disjIN n m = (mk_disjIN (n - 1) (m - 1)) RS disjI2;
+
+fun mk_sumTN Ts = Library.foldr1 (mk_sumT) Ts;
+
+fun Inl_const LT RT = Const (@{const_name Inl}, LT --> mk_sumT (LT, RT));
+fun mk_Inl t RT = Inl_const (fastype_of t) RT $ t;
+
+fun Inr_const LT RT = Const (@{const_name Inr}, RT --> mk_sumT (LT, RT));
+fun mk_Inr t LT = Inr_const LT (fastype_of t) $ t;
+
+fun mk_InN [_] t 1 = t
+ | mk_InN (_ :: Ts) t 1 = mk_Inl t (mk_sumTN Ts)
+ | mk_InN (LT :: Ts) t m = mk_Inr (mk_InN Ts t (m - 1)) LT
+ | mk_InN Ts t _ = raise (TYPE ("mk_InN", Ts, [t]));
+
+fun mk_sum_case f g =
+ let
+ val fT = fastype_of f;
+ val gT = fastype_of g;
+ in
+ Const (@{const_name sum_case},
+ fT --> gT --> mk_sumT (domain_type fT, domain_type gT) --> range_type fT) $ f $ g
+ end;
+
+fun mk_sum_caseN [f] = f
+ | mk_sum_caseN (f :: fs) = mk_sum_case f (mk_sum_caseN fs);
+
+fun mk_InN_not_InM 1 _ = @{thm Inl_not_Inr}
+ | mk_InN_not_InM n m =
+ if n > m then mk_InN_not_InM m n RS @{thm not_sym}
+ else mk_InN_not_InM (n - 1) (m - 1) RS @{thm not_arg_cong_Inr};
+
+fun mk_InN_Field 1 1 = @{thm TrueE[OF TrueI]}
+ | mk_InN_Field _ 1 = @{thm Inl_Field_csum}
+ | mk_InN_Field 2 2 = @{thm Inr_Field_csum}
+ | mk_InN_Field n m = mk_InN_Field (n - 1) (m - 1) RS @{thm Inr_Field_csum};
+
+fun mk_InN_inject 1 _ = @{thm TrueE[OF TrueI]}
+ | mk_InN_inject _ 1 = @{thm Inl_inject}
+ | mk_InN_inject 2 2 = @{thm Inr_inject}
+ | mk_InN_inject n m = @{thm Inr_inject} RS mk_InN_inject (n - 1) (m - 1);
+
+fun mk_specN 0 thm = thm
+ | mk_specN n thm = mk_specN (n - 1) (thm RS spec);
+
+fun mk_sum_casesN 1 1 = @{thm refl}
+ | mk_sum_casesN _ 1 = @{thm sum.cases(1)}
+ | mk_sum_casesN 2 2 = @{thm sum.cases(2)}
+ | mk_sum_casesN n m = trans OF [@{thm sum_case_step(2)}, mk_sum_casesN (n - 1) (m - 1)];
+
+local
+ fun mk_sumEN' 1 = @{thm obj_sum_step}
+ | mk_sumEN' n = mk_sumEN' (n - 1) RSN (2, @{thm obj_sum_step});
+in
+ fun mk_sumEN 1 = @{thm obj_sum_base}
+ | mk_sumEN 2 = @{thm sumE}
+ | mk_sumEN n = (mk_sumEN' (n - 2) RSN (2, @{thm obj_sumE})) OF replicate n (impI RS allI);
+end;
+
+fun mk_rec_simps n rec_thm defs = map (fn i =>
+ map (fn def => def RS rec_thm RS mk_nthI n i RS fun_cong) defs) (1 upto n);
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_lfp.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1763 @@
+(* Title: HOL/Codatatype/Tools/bnf_lfp.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Datatype construction.
+*)
+
+signature BNF_LFP =
+sig
+ val bnf_lfp: binding list -> typ list list -> BNF_Def.BNF list -> Proof.context -> Proof.context
+end;
+
+structure BNF_LFP : BNF_LFP =
+struct
+
+open BNF_Def
+open BNF_Util
+open BNF_Tactics
+open BNF_FP_Util
+open BNF_LFP_Util
+open BNF_LFP_Tactics
+
+(*all bnfs have the same lives*)
+fun bnf_lfp bs Dss_insts bnfs lthy =
+ let
+ val timer = time (Timer.startRealTimer ());
+ val live = live_of_bnf (hd bnfs)
+ val n = length bnfs; (*active*)
+ val ks = 1 upto n
+ val m = live - n (*passive, if 0 don't generate a new bnf*)
+ val b = Binding.name (fold_rev (fn b => fn s => Binding.name_of b ^ s) bs "");
+
+ fun note thmN thms = snd o Local_Theory.note
+ ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms);
+ fun notes thmN thmss = fold2 (fn b => fn thms => snd o Local_Theory.note
+ ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms)) bs thmss;
+
+ (* TODO: check if m, n etc are sane *)
+
+ val Dss = map (fn Ds => map TFree (fold Term.add_tfreesT Ds [])) Dss_insts;
+ val deads = distinct (op =) (flat Dss);
+ val names_lthy = fold Variable.declare_typ deads lthy;
+
+ (* tvars *)
+ val (((((((passiveAs, activeAs), allAs)), (passiveBs, activeBs)),
+ activeCs), passiveXs), passiveYs) = names_lthy
+ |> mk_TFrees live
+ |> apfst (`(chop m))
+ ||> mk_TFrees live
+ ||>> apfst (chop m)
+ ||>> mk_TFrees n
+ ||>> mk_TFrees m
+ ||> fst o mk_TFrees m;
+
+ val Ass = replicate n allAs;
+ val allBs = passiveAs @ activeBs;
+ val Bss = replicate n allBs;
+ val allCs = passiveAs @ activeCs;
+ val allCs' = passiveBs @ activeCs;
+ val Css' = replicate n allCs';
+
+ (* typs *)
+ fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs;
+ val (params, params') = `(map dest_TFree) (deads @ passiveAs);
+ val FTsAs = mk_FTs allAs;
+ val FTsBs = mk_FTs allBs;
+ val FTsCs = mk_FTs allCs;
+ val ATs = map HOLogic.mk_setT passiveAs;
+ val BTs = map HOLogic.mk_setT activeAs;
+ val B'Ts = map HOLogic.mk_setT activeBs;
+ val B''Ts = map HOLogic.mk_setT activeCs;
+ val sTs = map2 (curry (op -->)) FTsAs activeAs;
+ val s'Ts = map2 (curry (op -->)) FTsBs activeBs;
+ val s''Ts = map2 (curry (op -->)) FTsCs activeCs;
+ val fTs = map2 (curry (op -->)) activeAs activeBs;
+ val inv_fTs = map2 (curry (op -->)) activeBs activeAs;
+ val self_fTs = map2 (curry (op -->)) activeAs activeAs;
+ val gTs = map2 (curry (op -->)) activeBs activeCs;
+ val all_gTs = map2 (curry (op -->)) allBs allCs';
+ val prodBsAs = map2 (curry HOLogic.mk_prodT) activeBs activeAs;
+ val prodFTs = mk_FTs (passiveAs @ prodBsAs);
+ val prod_sTs = map2 (curry (op -->)) prodFTs activeAs;
+
+ (* terms *)
+ val mapsAsAs = map4 mk_map_of_bnf Dss Ass Ass bnfs;
+ val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;
+ val mapsBsAs = map4 mk_map_of_bnf Dss Bss Ass bnfs;
+ val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;
+ val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;
+ val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ prodBsAs)) Bss bnfs;
+ val map_fsts_rev = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ prodBsAs)) bnfs;
+ fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)
+ (map (replicate live) (replicate n Ts)) bnfs;
+ val setssAs = mk_setss allAs;
+ val bds = map3 mk_bd_of_bnf Dss Ass bnfs;
+ val witss = map wits_of_bnf bnfs;
+
+ val (((((((((((((((((((zs, zs'), As), Bs), Bs_copy), B's), B''s), ss), prod_ss), s's), s''s),
+ fs), fs_copy), inv_fs), self_fs), gs), all_gs), (xFs, xFs')), (yFs, yFs')),
+ names_lthy) = lthy
+ |> mk_Frees' "z" activeAs
+ ||>> mk_Frees "A" ATs
+ ||>> mk_Frees "B" BTs
+ ||>> mk_Frees "B" BTs
+ ||>> mk_Frees "B'" B'Ts
+ ||>> mk_Frees "B''" B''Ts
+ ||>> mk_Frees "s" sTs
+ ||>> mk_Frees "prods" prod_sTs
+ ||>> mk_Frees "s'" s'Ts
+ ||>> mk_Frees "s''" s''Ts
+ ||>> mk_Frees "f" fTs
+ ||>> mk_Frees "f" fTs
+ ||>> mk_Frees "f" inv_fTs
+ ||>> mk_Frees "f" self_fTs
+ ||>> mk_Frees "g" gTs
+ ||>> mk_Frees "g" all_gTs
+ ||>> mk_Frees' "x" FTsAs
+ ||>> mk_Frees' "y" FTsBs;
+
+ val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
+ val active_UNIVs = map HOLogic.mk_UNIV activeAs;
+ val prod_UNIVs = map HOLogic.mk_UNIV prodBsAs;
+ val passive_ids = map HOLogic.id_const passiveAs;
+ val active_ids = map HOLogic.id_const activeAs;
+ val fsts = map fst_const prodBsAs;
+
+ (* thms *)
+ val bd_card_orders = map bd_card_order_of_bnf bnfs;
+ val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
+ val bd_Card_order = hd bd_Card_orders;
+ val bd_Cinfinite = bd_Cinfinite_of_bnf (hd bnfs);
+ val bd_Cnotzeros = map bd_Cnotzero_of_bnf bnfs;
+ val bd_Cnotzero = hd bd_Cnotzeros;
+ val in_bds = map in_bd_of_bnf bnfs;
+ val map_comp's = map map_comp'_of_bnf bnfs;
+ val map_congs = map map_cong_of_bnf bnfs;
+ val map_ids = map map_id_of_bnf bnfs;
+ val map_id's = map map_id'_of_bnf bnfs;
+ val map_wpulls = map map_wpull_of_bnf bnfs;
+ val set_bdss = map set_bd_of_bnf bnfs;
+ val set_natural'ss = map set_natural'_of_bnf bnfs;
+
+ val timer = time (timer "Extracted terms & thms");
+
+ (* nonemptiness check *)
+ fun new_wit X wit = subset (op =) (#I wit, (0 upto m - 1) @ map snd X);
+
+ fun enrich X = map_filter (fn i =>
+ (case find_first (fn (_, i') => i = i') X of
+ NONE =>
+ (case find_index (new_wit X) (nth witss (i - m)) of
+ ~1 => NONE
+ | j => SOME (j, i))
+ | SOME ji => SOME ji)) (m upto m + n - 1);
+ val reachable = fixpoint (op =) enrich [];
+ val _ = if map snd reachable = (m upto m + n - 1) then ()
+ else error "The datatype could not be generated because nonemptiness could not be proved";
+
+ val wit_thms =
+ flat (map2 (fn bnf => fn (j, _) => nth (wit_thmss_of_bnf bnf) j) bnfs reachable);
+
+ val timer = time (timer "Checked nonemptiness");
+
+ (* derived thms *)
+
+ (*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x)=
+ map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*)
+ fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp =
+ let
+ val lhs = Term.list_comb (mapBsCs, all_gs) $
+ (Term.list_comb (mapAsBs, passive_ids @ fs) $ x);
+ val rhs = Term.list_comb (mapAsCs,
+ take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (x :: fs @ all_gs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
+ (K (mk_map_comp_id_tac map_comp))
+ end;
+
+ val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comp's;
+
+ (*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==>
+ map id ... id f(m+1) ... f(m+n) x = x*)
+ fun mk_map_congL x mapAsAs sets map_cong map_id' =
+ let
+ fun mk_prem set f z z' = HOLogic.mk_Trueprop
+ (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));
+ val prems = map4 mk_prem (drop m sets) self_fs zs zs';
+ val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x))
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))
+ (K (mk_map_congL_tac m map_cong map_id'))
+ end;
+
+ val map_congL_thms = map5 mk_map_congL xFs mapsAsAs setssAs map_congs map_id's;
+ val in_mono'_thms = map (fn bnf => in_mono_of_bnf bnf OF (replicate m subset_refl)) bnfs
+ val in_cong'_thms = map (fn bnf => in_cong_of_bnf bnf OF (replicate m refl)) bnfs
+
+ val timer = time (timer "Derived simple theorems");
+
+ (* algebra *)
+
+ val alg_bind = Binding.suffix_name ("_" ^ algN) b;
+ val alg_name = Binding.name_of alg_bind;
+ val alg_def_bind = (Thm.def_binding alg_bind, []);
+
+ (*forall i = 1 ... n: (\<forall>x \<in> Fi_in A1 .. Am B1 ... Bn. si x \<in> Bi)*)
+ val alg_spec =
+ let
+ val algT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);
+
+ val ins = map3 mk_in (replicate n (As @ Bs)) setssAs FTsAs;
+ fun mk_alg_conjunct B s X x x' =
+ mk_Ball X (Term.absfree x' (HOLogic.mk_mem (s $ x, B)));
+
+ val lhs = Term.list_comb (Free (alg_name, algT), As @ Bs @ ss);
+ val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_alg_conjunct Bs ss ins xFs xFs')
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((alg_free, (_, alg_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (alg_bind, NONE, NoSyn), (alg_def_bind, alg_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val alg = fst (Term.dest_Const (Morphism.term phi alg_free));
+ val alg_def = Morphism.thm phi alg_def_free;
+
+ fun mk_alg As Bs ss =
+ let
+ val args = As @ Bs @ ss;
+ val Ts = map fastype_of args;
+ val algT = Library.foldr (op -->) (Ts, HOLogic.boolT);
+ in
+ Term.list_comb (Const (alg, algT), args)
+ end;
+
+ val alg_set_thms =
+ let
+ val alg_prem = HOLogic.mk_Trueprop (mk_alg As Bs ss);
+ fun mk_prem x set B = HOLogic.mk_Trueprop (mk_subset (set $ x) B);
+ fun mk_concl s x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (s $ x, B));
+ val premss = map2 ((fn x => fn sets => map2 (mk_prem x) sets (As @ Bs))) xFs setssAs;
+ val concls = map3 mk_concl ss xFs Bs;
+ val goals = map3 (fn x => fn prems => fn concl =>
+ fold_rev Logic.all (x :: As @ Bs @ ss)
+ (Logic.list_implies (alg_prem :: prems, concl))) xFs premss concls;
+ in
+ map (fn goal => Skip_Proof.prove lthy [] [] goal
+ (K (mk_alg_set_tac alg_def))) goals
+ end;
+
+ fun mk_talg ATs BTs = mk_alg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);
+
+ val talg_thm =
+ let
+ val goal = fold_rev Logic.all ss
+ (HOLogic.mk_Trueprop (mk_talg passiveAs activeAs ss))
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (K (stac alg_def 1 THEN CONJ_WRAP (K (EVERY' [rtac ballI, rtac UNIV_I] 1)) ss))
+ end;
+
+ val timer = time (timer "Algebra definition & thms");
+
+ val alg_not_empty_thms =
+ let
+ val alg_prem =
+ HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val concls = map (HOLogic.mk_Trueprop o mk_not_empty) Bs;
+ val goals =
+ map (fn concl =>
+ fold_rev Logic.all (Bs @ ss) (Logic.mk_implies (alg_prem, concl))) concls;
+ in
+ map2 (fn goal => fn alg_set =>
+ Skip_Proof.prove lthy [] []
+ goal (K (mk_alg_not_empty_tac alg_set alg_set_thms wit_thms)))
+ goals alg_set_thms
+ end;
+
+ val timer = time (timer "Proved nonemptiness");
+
+ (* morphism *)
+
+ val mor_bind = Binding.suffix_name ("_" ^ morN) b;
+ val mor_name = Binding.name_of mor_bind;
+ val mor_def_bind = (Thm.def_binding mor_bind, []);
+
+ (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. f x \<in> B'i)*)
+ (*mor) forall i = 1 ... n: (\<forall>x \<in> Fi_in UNIV ... UNIV B1 ... Bn.
+ f (s1 x) = s1' (Fi_map id ... id f1 ... fn x))*)
+ val mor_spec =
+ let
+ val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);
+
+ fun mk_fbetw f B1 B2 z z' =
+ mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
+ fun mk_mor sets mapAsBs f s s' T x x' =
+ mk_Ball (mk_in (passive_UNIVs @ Bs) sets T)
+ (Term.absfree x' (HOLogic.mk_eq (f $ (s $ x), s' $
+ (Term.list_comb (mapAsBs, passive_ids @ fs) $ x))));
+ val lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);
+ val rhs = HOLogic.mk_conj
+ (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),
+ Library.foldr1 HOLogic.mk_conj
+ (map8 mk_mor setssAs mapsAsBs fs ss s's FTsAs xFs xFs'))
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (mor_bind, NONE, NoSyn), (mor_def_bind, mor_spec))
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
+ val mor_def = Morphism.thm phi mor_def_free;
+
+ fun mk_mor Bs1 ss1 Bs2 ss2 fs =
+ let
+ val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
+ val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
+ val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
+ in
+ Term.list_comb (Const (mor, morT), args)
+ end;
+
+ val (mor_image_thms, morE_thms) =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
+ fun mk_image_goal f B1 B2 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)
+ (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_subset (mk_image f $ B1) B2)));
+ val image_goals = map3 mk_image_goal fs Bs B's;
+ fun mk_elim_prem sets x T = HOLogic.mk_Trueprop
+ (HOLogic.mk_mem (x, mk_in (passive_UNIVs @ Bs) sets T));
+ fun mk_elim_goal sets mapAsBs f s s' x T =
+ fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)
+ (Logic.list_implies ([prem, mk_elim_prem sets x T],
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (f $ (s $ x),
+ s' $ Term.list_comb (mapAsBs, passive_ids @ fs @ [x])))));
+ val elim_goals = map7 mk_elim_goal setssAs mapsAsBs fs ss s's xFs FTsAs;
+ fun prove goal =
+ Skip_Proof.prove lthy [] [] goal (K (mk_mor_elim_tac mor_def));
+ in
+ (map prove image_goals, map prove elim_goals)
+ end;
+
+ val mor_incl_thm =
+ let
+ val prems = map2 (HOLogic.mk_Trueprop oo mk_subset) Bs Bs_copy;
+ val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))
+ (K (mk_mor_incl_tac mor_def map_id's))
+ end;
+
+ val mor_comp_thm =
+ let
+ val prems =
+ [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
+ HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
+ val concl =
+ HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
+ (Logic.list_implies (prems, concl)))
+ (K (mk_mor_comp_tac mor_def set_natural'ss map_comp_id_thms))
+ end;
+
+ val mor_inv_thm =
+ let
+ fun mk_inv_prem f inv_f B B' = HOLogic.mk_conj (mk_subset (mk_image inv_f $ B') B,
+ HOLogic.mk_conj (mk_inver inv_f f B, mk_inver f inv_f B'));
+ val prems = map HOLogic.mk_Trueprop
+ ([mk_mor Bs ss B's s's fs,
+ mk_alg passive_UNIVs Bs ss,
+ mk_alg passive_UNIVs B's s's] @
+ map4 mk_inv_prem fs inv_fs Bs B's);
+ val concl = HOLogic.mk_Trueprop (mk_mor B's s's Bs ss inv_fs);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ inv_fs)
+ (Logic.list_implies (prems, concl)))
+ (K (mk_mor_inv_tac alg_def mor_def
+ set_natural'ss morE_thms map_comp_id_thms map_congL_thms))
+ end;
+
+ val mor_cong_thm =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
+ val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)
+ (Logic.list_implies (prems, concl)))
+ (K ((hyp_subst_tac THEN' atac) 1))
+ end;
+
+ val mor_str_thm =
+ let
+ val maps = map2 (fn Ds => fn bnf => Term.list_comb
+ (mk_map_of_bnf Ds (passiveAs @ FTsAs) allAs bnf, passive_ids @ ss)) Dss bnfs;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all ss (HOLogic.mk_Trueprop
+ (mk_mor (map HOLogic.mk_UNIV FTsAs) maps active_UNIVs ss ss)))
+ (K (mk_mor_str_tac ks mor_def))
+ end;
+
+ val mor_convol_thm =
+ let
+ val maps = map3 (fn s => fn prod_s => fn map =>
+ mk_convol (HOLogic.mk_comp (s, Term.list_comb (map, passive_ids @ fsts)), prod_s))
+ s's prod_ss map_fsts;
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (s's @ prod_ss) (HOLogic.mk_Trueprop
+ (mk_mor prod_UNIVs maps (map HOLogic.mk_UNIV activeBs) s's fsts)))
+ (K (mk_mor_convol_tac ks mor_def))
+ end;
+
+ val mor_UNIV_thm =
+ let
+ fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
+ (HOLogic.mk_comp (f, s),
+ HOLogic.mk_comp (s', Term.list_comb (mapAsBs, passive_ids @ fs)));
+ val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
+ val rhs = Library.foldr1 HOLogic.mk_conj (map4 mk_conjunct mapsAsBs fs ss s's);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (ss @ s's @ fs) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))))
+ (K (mk_mor_UNIV_tac m morE_thms mor_def))
+ end;
+
+ val timer = time (timer "Morphism definition & thms");
+
+ (* isomorphism *)
+
+ (*mor Bs1 ss1 Bs2 ss2 fs \<and> (\<exists>gs. mor Bs2 ss2 Bs1 ss1 fs \<and>
+ forall i = 1 ... n. (inver gs[i] fs[i] Bs1[i] \<and> inver fs[i] gs[i] Bs2[i]))*)
+ fun mk_iso Bs1 ss1 Bs2 ss2 fs gs =
+ let
+ val ex_inv_mor = list_exists_free gs
+ (HOLogic.mk_conj (mk_mor Bs2 ss2 Bs1 ss1 gs,
+ Library.foldr1 HOLogic.mk_conj (map2 (curry HOLogic.mk_conj)
+ (map3 mk_inver gs fs Bs1) (map3 mk_inver fs gs Bs2))));
+ in
+ HOLogic.mk_conj (mk_mor Bs1 ss1 Bs2 ss2 fs, ex_inv_mor)
+ end;
+
+ val iso_alt_thm =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ [mk_alg passive_UNIVs Bs ss,
+ mk_alg passive_UNIVs B's s's]
+ val concl = HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_iso Bs ss B's s's fs inv_fs,
+ HOLogic.mk_conj (mk_mor Bs ss B's s's fs,
+ Library.foldr1 HOLogic.mk_conj (map3 mk_bij_betw fs Bs B's))));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs) (Logic.list_implies (prems, concl)))
+ (K (mk_iso_alt_tac mor_image_thms mor_inv_thm))
+ end;
+
+ val timer = time (timer "Isomorphism definition & thms");
+
+ (* algebra copies *)
+
+ val (copy_alg_thm, ex_copy_alg_thm) =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ (mk_alg passive_UNIVs Bs ss :: map3 mk_bij_betw inv_fs B's Bs);
+ val inver_prems = map HOLogic.mk_Trueprop
+ (map3 mk_inver inv_fs fs Bs @ map3 mk_inver fs inv_fs B's);
+ val all_prems = prems @ inver_prems;
+ fun mk_s f s mapT y y' = Term.absfree y' (f $ (s $
+ (Term.list_comb (mapT, passive_ids @ inv_fs) $ y)));
+
+ val alg = HOLogic.mk_Trueprop
+ (mk_alg passive_UNIVs B's (map5 mk_s fs ss mapsBsAs yFs yFs'));
+ val copy_str_thm = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
+ (Logic.list_implies (all_prems, alg)))
+ (K (mk_copy_str_tac set_natural'ss alg_def alg_set_thms));
+
+ val iso = HOLogic.mk_Trueprop
+ (mk_iso B's (map5 mk_s fs ss mapsBsAs yFs yFs') Bs ss inv_fs fs_copy);
+ val copy_alg_thm = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
+ (Logic.list_implies (all_prems, iso)))
+ (K (mk_copy_alg_tac set_natural'ss alg_set_thms mor_def iso_alt_thm copy_str_thm));
+
+ val ex = HOLogic.mk_Trueprop
+ (list_exists_free s's
+ (HOLogic.mk_conj (mk_alg passive_UNIVs B's s's,
+ mk_iso B's s's Bs ss inv_fs fs_copy)));
+ val ex_copy_alg_thm = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
+ (Logic.list_implies (prems, ex)))
+ (K (mk_ex_copy_alg_tac n copy_str_thm copy_alg_thm));
+ in
+ (copy_alg_thm, ex_copy_alg_thm)
+ end;
+
+ val timer = time (timer "Copy thms");
+
+
+ (* bounds *)
+
+ val sum_Card_order = if n = 1 then bd_Card_order else @{thm Card_order_csum};
+ val sum_Cnotzero = if n = 1 then bd_Cnotzero else bd_Cnotzero RS @{thm csum_Cnotzero1};
+ val sum_Cinfinite = if n = 1 then bd_Cinfinite else bd_Cinfinite RS @{thm Cinfinite_csum1};
+ fun mk_set_bd_sums i bd_Card_order bds =
+ if n = 1 then bds
+ else map (fn thm => bd_Card_order RS mk_ordLeq_csum n i thm) bds;
+ val set_bd_sumss = map3 mk_set_bd_sums ks bd_Card_orders set_bdss;
+
+ fun mk_in_bd_sum i Co Cnz bd =
+ if n = 1 then bd
+ else Cnz RS ((Co RS mk_ordLeq_csum n i (Co RS @{thm ordLeq_refl})) RS
+ (bd RS @{thm ordLeq_transitive[OF _
+ cexp_mono2_Cnotzero[OF _ csum_Cnotzero2[OF ctwo_Cnotzero]]]}));
+ val in_bd_sums = map4 mk_in_bd_sum ks bd_Card_orders bd_Cnotzeros in_bds;
+
+ val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
+ val suc_bd = mk_cardSuc sum_bd;
+ val field_suc_bd = mk_Field suc_bd;
+ val suc_bdT = fst (dest_relT (fastype_of suc_bd));
+ fun mk_Asuc_bd [] = mk_cexp ctwo suc_bd
+ | mk_Asuc_bd As =
+ mk_cexp (mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo) suc_bd;
+
+ val suc_bd_Card_order = if n = 1 then bd_Card_order RS @{thm cardSuc_Card_order}
+ else @{thm cardSuc_Card_order[OF Card_order_csum]};
+ val suc_bd_Cinfinite = if n = 1 then bd_Cinfinite RS @{thm Cinfinite_cardSuc}
+ else bd_Cinfinite RS @{thm Cinfinite_cardSuc[OF Cinfinite_csum1]};
+ val suc_bd_Cnotzero = suc_bd_Cinfinite RS @{thm Cinfinite_Cnotzero};
+ val suc_bd_worel = suc_bd_Card_order RS @{thm Card_order_wo_rel}
+ val basis_Asuc = if m = 0 then @{thm ordLeq_refl[OF Card_order_ctwo]}
+ else @{thm ordLeq_csum2[OF Card_order_ctwo]};
+ val Asuc_bd_Cinfinite = suc_bd_Cinfinite RS (basis_Asuc RS @{thm Cinfinite_cexp});
+ val Asuc_bd_Cnotzero = Asuc_bd_Cinfinite RS @{thm Cinfinite_Cnotzero};
+
+ val suc_bd_Asuc_bd = @{thm ordLess_ordLeq_trans[OF
+ ordLess_ctwo_cexp
+ cexp_mono1_Cnotzero[OF _ ctwo_Cnotzero]]} OF
+ [suc_bd_Card_order, basis_Asuc, suc_bd_Card_order];
+
+ val Asuc_bdT = fst (dest_relT (fastype_of (mk_Asuc_bd As)));
+ val II_BTs = replicate n (HOLogic.mk_setT Asuc_bdT);
+ val II_sTs = map2 (fn Ds => fn bnf =>
+ mk_T_of_bnf Ds (passiveAs @ replicate n Asuc_bdT) bnf --> Asuc_bdT) Dss bnfs;
+
+ val (((((((idxs, Asi_name), (idx, idx')), (jdx, jdx')), II_Bs), II_ss), Asuc_fs),
+ names_lthy) = names_lthy
+ |> mk_Frees "i" (replicate n suc_bdT)
+ ||>> (fn ctxt => apfst the_single (mk_fresh_names ctxt 1 "Asi"))
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") suc_bdT
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "j") suc_bdT
+ ||>> mk_Frees "IIB" II_BTs
+ ||>> mk_Frees "IIs" II_sTs
+ ||>> mk_Frees "f" (map (fn T => Asuc_bdT --> T) activeAs);
+
+ val suc_bd_limit_thm =
+ let
+ val prem = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map (fn idx => HOLogic.mk_mem (idx, field_suc_bd)) idxs));
+ fun mk_conjunct idx = HOLogic.mk_conj (mk_not_eq idx jdx,
+ HOLogic.mk_mem (HOLogic.mk_prod (idx, jdx), suc_bd));
+ val concl = HOLogic.mk_Trueprop (mk_Bex field_suc_bd
+ (Term.absfree jdx' (Library.foldr1 HOLogic.mk_conj (map mk_conjunct idxs))));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all idxs (Logic.list_implies ([prem], concl)))
+ (K (mk_bd_limit_tac n suc_bd_Cinfinite))
+ end;
+
+ val timer = time (timer "Bounds");
+
+
+ (* minimal algebra *)
+
+ fun mk_minG Asi i k = mk_UNION (mk_underS suc_bd $ i)
+ (Term.absfree jdx' (mk_nthN n (Asi $ jdx) k));
+
+ fun mk_minH_component As Asi i sets Ts s k =
+ HOLogic.mk_binop @{const_name "sup"}
+ (mk_minG Asi i k, mk_image s $ mk_in (As @ map (mk_minG Asi i) ks) sets Ts);
+
+ fun mk_min_algs As ss =
+ let
+ val BTs = map (range_type o fastype_of) ss;
+ val Ts = map (HOLogic.dest_setT o fastype_of) As @ BTs;
+ val (Asi, Asi') = `Free (Asi_name, suc_bdT -->
+ Library.foldr1 HOLogic.mk_prodT (map HOLogic.mk_setT BTs));
+ in
+ mk_worec suc_bd (Term.absfree Asi' (Term.absfree idx' (HOLogic.mk_tuple
+ (map4 (mk_minH_component As Asi idx) (mk_setss Ts) (mk_FTs Ts) ss ks))))
+ end;
+
+ val (min_algs_thms, min_algs_mono_thms, card_of_min_algs_thm, least_min_algs_thm) =
+ let
+ val i_field = HOLogic.mk_mem (idx, field_suc_bd);
+ val min_algs = mk_min_algs As ss;
+ val min_algss = map (fn k => mk_nthN n (min_algs $ idx) k) ks;
+
+ val concl = HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (min_algs $ idx, HOLogic.mk_tuple
+ (map4 (mk_minH_component As min_algs idx) setssAs FTsAs ss ks)));
+ val goal = fold_rev Logic.all (idx :: As @ ss)
+ (Logic.mk_implies (HOLogic.mk_Trueprop i_field, concl));
+
+ val min_algs_thm = Skip_Proof.prove lthy [] [] goal
+ (K (mk_min_algs_tac suc_bd_worel in_cong'_thms));
+
+ val min_algs_thms = map (fn k => min_algs_thm RS mk_nthI n k) ks;
+
+ fun mk_mono_goal min_alg =
+ fold_rev Logic.all (As @ ss) (HOLogic.mk_Trueprop (mk_relChain suc_bd
+ (Term.absfree idx' min_alg)));
+
+ val monos = map2 (fn goal => fn min_algs =>
+ Skip_Proof.prove lthy [] [] goal (K (mk_min_algs_mono_tac min_algs)))
+ (map mk_mono_goal min_algss) min_algs_thms;
+
+ val Asuc_bd = mk_Asuc_bd As;
+
+ fun mk_card_conjunct min_alg = mk_ordLeq (mk_card_of min_alg) Asuc_bd;
+ val card_conjunction = Library.foldr1 HOLogic.mk_conj (map mk_card_conjunct min_algss);
+ val card_cT = certifyT lthy suc_bdT;
+ val card_ct = certify lthy (Term.absfree idx' card_conjunction);
+
+ val card_of = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (HOLogic.mk_imp (i_field, card_conjunction)))
+ (K (mk_min_algs_card_of_tac card_cT card_ct
+ m suc_bd_worel min_algs_thms in_bd_sums
+ sum_Card_order sum_Cnotzero suc_bd_Card_order suc_bd_Cinfinite suc_bd_Cnotzero
+ suc_bd_Asuc_bd Asuc_bd_Cinfinite Asuc_bd_Cnotzero)));
+
+ val least_prem = HOLogic.mk_Trueprop (mk_alg As Bs ss);
+ val least_conjunction = Library.foldr1 HOLogic.mk_conj (map2 mk_subset min_algss Bs);
+ val least_cT = certifyT lthy suc_bdT;
+ val least_ct = certify lthy (Term.absfree idx' least_conjunction);
+
+ val least = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] []
+ (Logic.mk_implies (least_prem,
+ HOLogic.mk_Trueprop (HOLogic.mk_imp (i_field, least_conjunction))))
+ (K (mk_min_algs_least_tac least_cT least_ct
+ suc_bd_worel min_algs_thms alg_set_thms)));
+ in
+ (min_algs_thms, monos, card_of, least)
+ end;
+
+ val timer = time (timer "min_algs definition & thms");
+
+ fun min_alg_bind i = Binding.suffix_name
+ ("_" ^ min_algN ^ (if n = 1 then "" else string_of_int i)) b;
+ val min_alg_name = Binding.name_of o min_alg_bind;
+ val min_alg_def_bind = rpair [] o Thm.def_binding o min_alg_bind;
+
+ fun min_alg_spec i =
+ let
+ val min_algT =
+ Library.foldr (op -->) (ATs @ sTs, HOLogic.mk_setT (nth activeAs (i - 1)));
+
+ val lhs = Term.list_comb (Free (min_alg_name i, min_algT), As @ ss);
+ val rhs = mk_UNION (field_suc_bd)
+ (Term.absfree idx' (mk_nthN n (mk_min_algs As ss $ idx) i));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((min_alg_frees, (_, min_alg_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map (fn i => Specification.definition
+ (SOME (min_alg_bind i, NONE, NoSyn), (min_alg_def_bind i, min_alg_spec i))) ks
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val min_algs = map (fst o Term.dest_Const o Morphism.term phi) min_alg_frees;
+ val min_alg_defs = map (Morphism.thm phi) min_alg_def_frees;
+
+ fun mk_min_alg As ss i =
+ let
+ val T = HOLogic.mk_setT (range_type (fastype_of (nth ss (i - 1))))
+ val args = As @ ss;
+ val Ts = map fastype_of args;
+ val min_algT = Library.foldr (op -->) (Ts, T);
+ in
+ Term.list_comb (Const (nth min_algs (i - 1), min_algT), args)
+ end;
+
+ val (alg_min_alg_thm, card_of_min_alg_thms, least_min_alg_thms, mor_incl_min_alg_thm) =
+ let
+ val min_algs = map (mk_min_alg As ss) ks;
+
+ val goal = fold_rev Logic.all (As @ ss) (HOLogic.mk_Trueprop (mk_alg As min_algs ss));
+ val alg_min_alg = Skip_Proof.prove lthy [] [] goal
+ (K (mk_alg_min_alg_tac m alg_def min_alg_defs suc_bd_limit_thm sum_Cinfinite
+ set_bd_sumss min_algs_thms min_algs_mono_thms));
+
+ val Asuc_bd = mk_Asuc_bd As;
+ fun mk_card_of_thm min_alg def = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ ss)
+ (HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of min_alg) Asuc_bd)))
+ (K (mk_card_of_min_alg_tac def card_of_min_algs_thm
+ suc_bd_Card_order suc_bd_Asuc_bd Asuc_bd_Cinfinite));
+
+ val least_prem = HOLogic.mk_Trueprop (mk_alg As Bs ss);
+ fun mk_least_thm min_alg B def = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss)
+ (Logic.mk_implies (least_prem, HOLogic.mk_Trueprop (mk_subset min_alg B))))
+ (K (mk_least_min_alg_tac def least_min_algs_thm));
+
+ val leasts = map3 mk_least_thm min_algs Bs min_alg_defs;
+
+ val incl_prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val incl_min_algs = map (mk_min_alg passive_UNIVs ss) ks;
+ val incl = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss)
+ (Logic.mk_implies (incl_prem,
+ HOLogic.mk_Trueprop (mk_mor incl_min_algs ss Bs ss active_ids))))
+ (K (EVERY' (rtac mor_incl_thm :: map etac leasts) 1));
+ in
+ (alg_min_alg,
+ map2 mk_card_of_thm min_algs min_alg_defs,
+ leasts, incl)
+ end;
+
+ val timer = time (timer "Minimal algebra definition & thms");
+
+ val II_repT = HOLogic.mk_prodT (HOLogic.mk_tupleT II_BTs, HOLogic.mk_tupleT II_sTs);
+ val IIT_bind = Binding.suffix_name ("_" ^ IITN) b;
+
+ val ((IIT_name, (IIT_glob_info, IIT_loc_info)), lthy) =
+ typedef true NONE (IIT_bind, params, NoSyn)
+ (HOLogic.mk_UNIV II_repT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
+
+ val IIT = Type (IIT_name, params');
+ val Abs_IIT = Const (#Abs_name IIT_glob_info, II_repT --> IIT);
+ val Rep_IIT = Const (#Rep_name IIT_glob_info, IIT --> II_repT);
+ val Abs_IIT_inverse_thm =
+ mk_Abs_inverse_thm (the (#set_def IIT_loc_info)) (#Abs_inverse IIT_loc_info);
+
+ val initT = IIT --> Asuc_bdT;
+ val active_initTs = replicate n initT;
+ val init_FTs = map2 (fn Ds => mk_T_of_bnf Ds (passiveAs @ active_initTs)) Dss bnfs;
+ val init_fTs = map (fn T => initT --> T) activeAs;
+
+ val (((((((iidx, iidx'), init_xs), (init_xFs, init_xFs')),
+ init_fs), init_fs_copy), init_phis), names_lthy) = names_lthy
+ |> yield_singleton (apfst (op ~~) oo mk_Frees' "i") IIT
+ ||>> mk_Frees "ix" active_initTs
+ ||>> mk_Frees' "x" init_FTs
+ ||>> mk_Frees "f" init_fTs
+ ||>> mk_Frees "f" init_fTs
+ ||>> mk_Frees "phi" (replicate n (initT --> HOLogic.boolT));
+
+ val II = HOLogic.mk_Collect (fst iidx', IIT, list_exists_free (II_Bs @ II_ss)
+ (HOLogic.mk_conj (HOLogic.mk_eq (iidx,
+ Abs_IIT $ (HOLogic.mk_prod (HOLogic.mk_tuple II_Bs, HOLogic.mk_tuple II_ss))),
+ mk_alg passive_UNIVs II_Bs II_ss)));
+
+ val select_Bs = map (mk_nthN n (HOLogic.mk_fst (Rep_IIT $ iidx))) ks;
+ val select_ss = map (mk_nthN n (HOLogic.mk_snd (Rep_IIT $ iidx))) ks;
+
+ fun str_init_bind i = Binding.suffix_name ("_" ^ str_initN ^ (if n = 1 then "" else
+ string_of_int i)) b;
+ val str_init_name = Binding.name_of o str_init_bind;
+ val str_init_def_bind = rpair [] o Thm.def_binding o str_init_bind;
+
+ fun str_init_spec i =
+ let
+ val T = nth init_FTs (i - 1);
+ val init_xF = nth init_xFs (i - 1)
+ val select_s = nth select_ss (i - 1);
+ val map = mk_map_of_bnf (nth Dss (i - 1))
+ (passiveAs @ active_initTs) (passiveAs @ replicate n Asuc_bdT)
+ (nth bnfs (i - 1));
+ val map_args = passive_ids @ replicate n (mk_rapp iidx Asuc_bdT);
+ val str_initT = T --> IIT --> Asuc_bdT;
+
+ val lhs = Term.list_comb (Free (str_init_name i, str_initT), [init_xF, iidx]);
+ val rhs = select_s $ (Term.list_comb (map, map_args) $ init_xF);
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((str_init_frees, (_, str_init_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map (fn i => Specification.definition
+ (SOME (str_init_bind i, NONE, NoSyn), (str_init_def_bind i, str_init_spec i))) ks
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val str_inits =
+ map (Term.subst_atomic_types (map (`(Morphism.typ phi)) params') o Morphism.term phi)
+ str_init_frees;
+
+ val str_init_defs = map (Morphism.thm phi) str_init_def_frees;
+
+ val car_inits = map (mk_min_alg passive_UNIVs str_inits) ks;
+
+ (*TODO: replace with instantiate? (problem: figure out right type instantiation)*)
+ val alg_init_thm = Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_alg passive_UNIVs car_inits str_inits))
+ (K (rtac alg_min_alg_thm 1));
+
+ val alg_select_thm = Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_Ball II
+ (Term.absfree iidx' (mk_alg passive_UNIVs select_Bs select_ss))))
+ (mk_alg_select_tac Abs_IIT_inverse_thm)
+
+ val mor_select_thm =
+ let
+ val alg_prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val i_prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (iidx, II));
+ val mor_prem = HOLogic.mk_Trueprop (mk_mor select_Bs select_ss Bs ss Asuc_fs);
+ val prems = [alg_prem, i_prem, mor_prem];
+ val concl = HOLogic.mk_Trueprop
+ (mk_mor car_inits str_inits Bs ss
+ (map (fn f => HOLogic.mk_comp (f, mk_rapp iidx Asuc_bdT)) Asuc_fs));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (iidx :: Bs @ ss @ Asuc_fs) (Logic.list_implies (prems, concl)))
+ (K (mk_mor_select_tac mor_def mor_cong_thm mor_comp_thm mor_incl_min_alg_thm alg_def
+ alg_select_thm alg_set_thms set_natural'ss str_init_defs))
+ end;
+
+ val (init_ex_mor_thm, init_unique_mor_thms) =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val concl = HOLogic.mk_Trueprop
+ (list_exists_free init_fs (mk_mor car_inits str_inits Bs ss init_fs));
+ val ex_mor = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss) (Logic.mk_implies (prem, concl)))
+ (mk_init_ex_mor_tac Abs_IIT_inverse_thm ex_copy_alg_thm alg_min_alg_thm
+ card_of_min_alg_thms mor_comp_thm mor_select_thm mor_incl_min_alg_thm);
+
+ val prems = map2 (HOLogic.mk_Trueprop oo curry HOLogic.mk_mem) init_xs car_inits
+ val mor_prems = map HOLogic.mk_Trueprop
+ [mk_mor car_inits str_inits Bs ss init_fs,
+ mk_mor car_inits str_inits Bs ss init_fs_copy];
+ fun mk_fun_eq f g x = HOLogic.mk_eq (f $ x, g $ x);
+ val unique = HOLogic.mk_Trueprop
+ (Library.foldr1 HOLogic.mk_conj (map3 mk_fun_eq init_fs init_fs_copy init_xs));
+ val unique_mor = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (init_xs @ Bs @ ss @ init_fs @ init_fs_copy)
+ (Logic.list_implies (prems @ mor_prems, unique)))
+ (K (mk_init_unique_mor_tac m alg_def alg_init_thm least_min_alg_thms
+ in_mono'_thms alg_set_thms morE_thms map_congs));
+ in
+ (ex_mor, split_conj_thm unique_mor)
+ end;
+
+ val init_setss = mk_setss (passiveAs @ active_initTs);
+ val active_init_setss = map (drop m) init_setss;
+ val init_ins = map2 (fn sets => mk_in (passive_UNIVs @ car_inits) sets) init_setss init_FTs;
+
+ fun mk_closed phis =
+ let
+ fun mk_conjunct phi str_init init_sets init_in x x' =
+ let
+ val prem = Library.foldr1 HOLogic.mk_conj
+ (map2 (fn set => mk_Ball (set $ x)) init_sets phis);
+ val concl = phi $ (str_init $ x);
+ in
+ mk_Ball init_in (Term.absfree x' (HOLogic.mk_imp (prem, concl)))
+ end;
+ in
+ Library.foldr1 HOLogic.mk_conj
+ (map6 mk_conjunct phis str_inits active_init_setss init_ins init_xFs init_xFs')
+ end;
+
+ val init_induct_thm =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_closed init_phis);
+ val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map2 mk_Ball car_inits init_phis));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all init_phis (Logic.mk_implies (prem, concl)))
+ (K (mk_init_induct_tac m alg_def alg_init_thm least_min_alg_thms alg_set_thms))
+ end;
+
+ val timer = time (timer "Initiality definition & thms");
+
+ val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
+ lthy
+ |> fold_map2 (fn b => fn car_init => typedef true NONE (b, params, NoSyn) car_init NONE
+ (EVERY' [rtac ssubst, rtac @{thm ex_in_conv}, resolve_tac alg_not_empty_thms,
+ rtac alg_init_thm] 1)) bs car_inits
+ |>> apsnd split_list o split_list;
+
+ val Ts = map (fn name => Type (name, params')) T_names;
+ fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
+ val Ts' = mk_Ts passiveBs;
+ val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> initT)) T_glob_infos Ts;
+ val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, initT --> T)) T_glob_infos Ts;
+
+ val set_defs = map (the o #set_def) T_loc_infos;
+ val type_defs = map #type_definition T_loc_infos;
+ val Reps = map #Rep T_loc_infos;
+ val Rep_casess = map #Rep_cases T_loc_infos;
+ val Rep_injects = map #Rep_inject T_loc_infos;
+ val Rep_inverses = map #Rep_inverse T_loc_infos;
+ val Abs_inverses = map #Abs_inverse T_loc_infos;
+
+ val T_subset1s = map mk_T_subset1 set_defs;
+ val T_subset2s = map mk_T_subset2 set_defs;
+
+ fun mk_inver_thm mk_tac rep abs X thm =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_inver rep abs X))
+ (K (EVERY' [rtac ssubst, rtac @{thm inver_def}, rtac ballI, mk_tac thm] 1));
+
+ val inver_Reps = map4 (mk_inver_thm rtac) Abs_Ts Rep_Ts (map HOLogic.mk_UNIV Ts) Rep_inverses;
+ val inver_Abss = map4 (mk_inver_thm etac) Rep_Ts Abs_Ts car_inits
+ (map2 (curry op RS) T_subset1s Abs_inverses);
+
+ val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
+
+ val UNIVs = map HOLogic.mk_UNIV Ts;
+ val FTs = mk_FTs (passiveAs @ Ts);
+ val FTs' = mk_FTs (passiveBs @ Ts');
+ fun mk_set_Ts T = passiveAs @ replicate n (HOLogic.mk_setT T);
+ val setFTss = map (mk_FTs o mk_set_Ts) passiveAs;
+ val FTs_setss = mk_setss (passiveAs @ Ts);
+ val FTs'_setss = mk_setss (passiveBs @ Ts');
+ val map_FT_inits = map2 (fn Ds =>
+ mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ active_initTs)) Dss bnfs;
+ val fTs = map2 (curry op -->) Ts activeAs;
+ val iterT = Library.foldr1 HOLogic.mk_prodT (map2 (curry op -->) Ts activeAs);
+ val rec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) prod_sTs;
+ val rec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_fsts;
+ val rec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_fsts_rev;
+ val rec_fsts = map (Term.subst_atomic_types (activeBs ~~ Ts)) fsts;
+
+ val (((((((((((Izs, (Izs1, Izs1')), (Izs2, Izs2')), (xFs, xFs')), yFs), (AFss, AFss')),
+ (iter_f, iter_f')), fs), phis), phi2s), rec_ss), names_lthy) = names_lthy
+ |> mk_Frees "z" Ts
+ ||>> mk_Frees' "z1" Ts
+ ||>> mk_Frees' "z2" Ts'
+ ||>> mk_Frees' "x" FTs
+ ||>> mk_Frees "y" FTs'
+ ||>> mk_Freess' "z" setFTss
+ ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "f") iterT
+ ||>> mk_Frees "f" fTs
+ ||>> mk_Frees "phi" (map (fn T => T --> HOLogic.boolT) Ts)
+ ||>> mk_Frees "phi" (map2 (fn T => fn U => T --> U --> HOLogic.boolT) Ts Ts')
+ ||>> mk_Frees "s" rec_sTs;
+
+ fun fld_bind i = Binding.suffix_name ("_" ^ fldN) (nth bs (i - 1));
+ val fld_name = Binding.name_of o fld_bind;
+ val fld_def_bind = rpair [] o Thm.def_binding o fld_bind;
+
+ fun fld_spec i abs str map_FT_init x x' =
+ let
+ val fldT = nth FTs (i - 1) --> nth Ts (i - 1);
+
+ val lhs = Free (fld_name i, fldT);
+ val rhs = Term.absfree x' (abs $ (str $
+ (Term.list_comb (map_FT_init, map HOLogic.id_const passiveAs @ Rep_Ts) $ x)));
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((fld_frees, (_, fld_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map6 (fn i => fn abs => fn str => fn map => fn x => fn x' =>
+ Specification.definition
+ (SOME (fld_bind i, NONE, NoSyn), (fld_def_bind i, fld_spec i abs str map x x')))
+ ks Abs_Ts str_inits map_FT_inits xFs xFs'
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ fun mk_flds passive =
+ map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (deads @ passive)) o
+ Morphism.term phi) fld_frees;
+ val flds = mk_flds passiveAs;
+ val fld's = mk_flds passiveBs;
+ val fld_defs = map (Morphism.thm phi) fld_def_frees;
+
+ val (mor_Rep_thm, mor_Abs_thm) =
+ let
+ val copy = alg_init_thm RS copy_alg_thm;
+ fun mk_bij inj subset1 subset2 Rep cases = @{thm bij_betwI'} OF
+ [inj, Rep RS subset2, subset1 RS cases];
+ val bijs = map5 mk_bij Rep_injects T_subset1s T_subset2s Reps Rep_casess;
+ val mor_Rep =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor UNIVs flds car_inits str_inits Rep_Ts))
+ (mk_mor_Rep_tac fld_defs copy bijs inver_Abss inver_Reps);
+
+ val inv = mor_inv_thm OF [mor_Rep, talg_thm, alg_init_thm];
+ val mor_Abs =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor car_inits str_inits UNIVs flds Abs_Ts))
+ (K (mk_mor_Abs_tac inv inver_Abss inver_Reps));
+ in
+ (mor_Rep, mor_Abs)
+ end;
+
+ val timer = time (timer "fld definitions & thms");
+
+ val iter_fun = Term.absfree iter_f'
+ (mk_mor UNIVs flds active_UNIVs ss (map (mk_nthN n iter_f) ks));
+ val iter = HOLogic.choice_const iterT $ iter_fun;
+
+ fun iter_bind i = Binding.suffix_name ("_" ^ iterN) (nth bs (i - 1));
+ val iter_name = Binding.name_of o iter_bind;
+ val iter_def_bind = rpair [] o Thm.def_binding o iter_bind;
+
+ fun iter_spec i T AT =
+ let
+ val iterT = Library.foldr (op -->) (sTs, T --> AT);
+
+ val lhs = Term.list_comb (Free (iter_name i, iterT), ss);
+ val rhs = mk_nthN n iter i;
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((iter_frees, (_, iter_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn T => fn AT =>
+ Specification.definition
+ (SOME (iter_bind i, NONE, NoSyn), (iter_def_bind i, iter_spec i T AT)))
+ ks Ts activeAs
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val iters = map (fst o dest_Const o Morphism.term phi) iter_frees;
+ fun mk_iter Ts ss i = Term.list_comb (Const (nth iters (i - 1), Library.foldr (op -->)
+ (map fastype_of ss, nth Ts (i - 1) --> range_type (fastype_of (nth ss (i - 1))))), ss);
+ val iter_defs = map (Morphism.thm phi) iter_def_frees;
+
+ val mor_iter_thm =
+ let
+ val ex_mor = talg_thm RS init_ex_mor_thm;
+ val mor_cong = mor_cong_thm OF (map (mk_nth_conv n) ks);
+ val mor_comp = mor_Rep_thm RS mor_comp_thm;
+ val cT = certifyT lthy iterT;
+ val ct = certify lthy iter_fun
+ in
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor UNIVs flds active_UNIVs ss (map (mk_iter Ts ss) ks)))
+ (K (mk_mor_iter_tac cT ct iter_defs ex_mor (mor_comp RS mor_cong))))
+ end;
+
+ val iter_thms = map (fn morE => rule_by_tactic lthy
+ ((rtac CollectI THEN' CONJ_WRAP' (K (rtac @{thm subset_UNIV})) (1 upto m + n)) 1)
+ (mor_iter_thm RS morE)) morE_thms;
+
+ val (iter_unique_mor_thms, iter_unique_mor_thm) =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_mor UNIVs flds active_UNIVs ss fs);
+ fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_iter Ts ss i);
+ val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_fun_eq fs ks));
+ val unique_mor = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (ss @ fs) (Logic.mk_implies (prem, unique)))
+ (K (mk_iter_unique_mor_tac type_defs init_unique_mor_thms T_subset2s Reps
+ mor_comp_thm mor_Abs_thm mor_iter_thm));
+ in
+ `split_conj_thm unique_mor
+ end;
+
+ val (iter_unique_thms, iter_unique_thm) =
+ `split_conj_thm (mk_conjIN n RS
+ (mor_UNIV_thm RS @{thm ssubst[of _ _ "%x. x"]} RS iter_unique_mor_thm))
+
+ val iter_fld_thms =
+ map (fn thm => (mor_incl_thm OF replicate n @{thm subset_UNIV}) RS thm RS sym)
+ iter_unique_mor_thms;
+
+ val fld_o_iter_thms =
+ let
+ val mor = mor_comp_thm OF [mor_iter_thm, mor_str_thm];
+ in
+ map2 (fn unique => fn iter_fld =>
+ trans OF [mor RS unique, iter_fld]) iter_unique_mor_thms iter_fld_thms
+ end;
+
+ val timer = time (timer "iter definitions & thms");
+
+ val map_flds = map2 (fn Ds => fn bnf =>
+ Term.list_comb (mk_map_of_bnf Ds (passiveAs @ FTs) (passiveAs @ Ts) bnf,
+ map HOLogic.id_const passiveAs @ flds)) Dss bnfs;
+
+ fun unf_bind i = Binding.suffix_name ("_" ^ unfN) (nth bs (i - 1));
+ val unf_name = Binding.name_of o unf_bind;
+ val unf_def_bind = rpair [] o Thm.def_binding o unf_bind;
+
+ fun unf_spec i FT T =
+ let
+ val unfT = T --> FT;
+
+ val lhs = Free (unf_name i, unfT);
+ val rhs = mk_iter Ts map_flds i;
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((unf_frees, (_, unf_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn FT => fn T =>
+ Specification.definition
+ (SOME (unf_bind i, NONE, NoSyn), (unf_def_bind i, unf_spec i FT T))) ks FTs Ts
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ fun mk_unfs params =
+ map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
+ unf_frees;
+ val unfs = mk_unfs params';
+ val unf_defs = map (Morphism.thm phi) unf_def_frees;
+
+ val fld_o_unf_thms = map2 (Local_Defs.fold lthy o single) unf_defs fld_o_iter_thms;
+
+ val unf_o_fld_thms =
+ let
+ fun mk_goal unf fld FT =
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (unf, fld), HOLogic.id_const FT));
+ val goals = map3 mk_goal unfs flds FTs;
+ in
+ map5 (fn goal => fn unf_def => fn iter => fn map_comp_id => fn map_congL =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_unf_o_fld_tac unf_def iter map_comp_id map_congL fld_o_iter_thms)))
+ goals unf_defs iter_thms map_comp_id_thms map_congL_thms
+ end;
+
+ val unf_fld_thms = map (fn thm => thm RS @{thm pointfree_idE}) unf_o_fld_thms;
+ val fld_unf_thms = map (fn thm => thm RS @{thm pointfree_idE}) fld_o_unf_thms;
+
+ val bij_unf_thms =
+ map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) fld_o_unf_thms unf_o_fld_thms;
+ val inj_unf_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_unf_thms;
+ val surj_unf_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_unf_thms;
+ val unf_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_unf_thms;
+ val unf_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_unf_thms;
+ val unf_exhaust_thms = map (fn thm => thm RS @{thm exE}) unf_nchotomy_thms;
+
+ val bij_fld_thms =
+ map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) unf_o_fld_thms fld_o_unf_thms;
+ val inj_fld_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_fld_thms;
+ val surj_fld_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_fld_thms;
+ val fld_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_fld_thms;
+ val fld_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_fld_thms;
+ val fld_exhaust_thms = map (fn thm => thm RS @{thm exE}) fld_nchotomy_thms;
+
+ val timer = time (timer "unf definitions & thms");
+
+ val fst_rec_pair_thms =
+ let
+ val mor = mor_comp_thm OF [mor_iter_thm, mor_convol_thm];
+ in
+ map2 (fn unique => fn iter_fld =>
+ trans OF [mor RS unique, iter_fld]) iter_unique_mor_thms iter_fld_thms
+ end;
+
+ fun rec_bind i = Binding.suffix_name ("_" ^ recN) (nth bs (i - 1));
+ val rec_name = Binding.name_of o rec_bind;
+ val rec_def_bind = rpair [] o Thm.def_binding o rec_bind;
+
+ fun rec_spec i T AT =
+ let
+ val recT = Library.foldr (op -->) (rec_sTs, T --> AT);
+ val maps = map3 (fn fld => fn prod_s => fn map =>
+ mk_convol (HOLogic.mk_comp (fld, Term.list_comb (map, passive_ids @ rec_fsts)), prod_s))
+ flds rec_ss rec_maps;
+
+ val lhs = Term.list_comb (Free (rec_name i, recT), rec_ss);
+ val rhs = HOLogic.mk_comp (snd_const (HOLogic.mk_prodT (T, AT)), mk_iter Ts maps i);
+ in
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
+ end;
+
+ val ((rec_frees, (_, rec_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn T => fn AT =>
+ Specification.definition
+ (SOME (rec_bind i, NONE, NoSyn), (rec_def_bind i, rec_spec i T AT)))
+ ks Ts activeAs
+ |>> apsnd split_list o split_list
+ ||> `Local_Theory.restore;
+
+ (*transforms defined frees into consts*)
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val recs = map (fst o dest_Const o Morphism.term phi) rec_frees;
+ fun mk_rec ss i = Term.list_comb (Const (nth recs (i - 1), Library.foldr (op -->)
+ (map fastype_of ss, nth Ts (i - 1) --> range_type (fastype_of (nth ss (i - 1))))), ss);
+ val rec_defs = map (Morphism.thm phi) rec_def_frees;
+
+ val convols = map2 (fn T => fn i => mk_convol (HOLogic.id_const T, mk_rec rec_ss i)) Ts ks;
+ val rec_thms =
+ let
+ fun mk_goal i rec_s rec_map fld x =
+ let
+ val lhs = mk_rec rec_ss i $ (fld $ x);
+ val rhs = rec_s $ (Term.list_comb (rec_map, passive_ids @ convols) $ x);
+ in
+ fold_rev Logic.all (x :: rec_ss) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
+ end;
+ val goals = map5 mk_goal ks rec_ss rec_maps_rev flds xFs;
+ in
+ map2 (fn goal => fn iter =>
+ Skip_Proof.prove lthy [] [] goal (mk_rec_tac rec_defs iter fst_rec_pair_thms))
+ goals iter_thms
+ end;
+
+ val timer = time (timer "rec definitions & thms");
+
+ val (fld_induct_thm, induct_params) =
+ let
+ fun mk_prem phi fld sets x =
+ let
+ fun mk_IH phi set z =
+ let
+ val prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set $ x));
+ val concl = HOLogic.mk_Trueprop (phi $ z);
+ in
+ Logic.all z (Logic.mk_implies (prem, concl))
+ end;
+
+ val IHs = map3 mk_IH phis (drop m sets) Izs;
+ val concl = HOLogic.mk_Trueprop (phi $ (fld $ x));
+ in
+ Logic.all x (Logic.list_implies (IHs, concl))
+ end;
+
+ val prems = map4 mk_prem phis flds FTs_setss xFs;
+
+ fun mk_concl phi z = phi $ z;
+ val concl =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_concl phis Izs));
+
+ val goal = Logic.list_implies (prems, concl);
+ in
+ (Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (phis @ Izs) goal)
+ (K (mk_fld_induct_tac m set_natural'ss init_induct_thm morE_thms mor_Abs_thm
+ Rep_inverses Abs_inverses Reps T_subset1s T_subset2s)), rev (Term.add_tfrees goal []))
+ end;
+
+ val cTs = map (SOME o certifyT lthy o TFree) induct_params;
+
+ val weak_fld_induct_thms =
+ let fun insts i = (replicate (i - 1) TrueI) @ (@{thm asm_rl} :: replicate (n - i) TrueI);
+ in map (fn i => (fld_induct_thm OF insts i) RS mk_conjunctN n i) ks end;
+
+ val (fld_induct2_thm, induct2_params) =
+ let
+ fun mk_prem phi fld fld' sets sets' x y =
+ let
+ fun mk_IH phi set set' z1 z2 =
+ let
+ val prem1 = HOLogic.mk_Trueprop (HOLogic.mk_mem (z1, (set $ x)));
+ val prem2 = HOLogic.mk_Trueprop (HOLogic.mk_mem (z2, (set' $ y)));
+ val concl = HOLogic.mk_Trueprop (phi $ z1 $ z2);
+ in
+ fold_rev Logic.all [z1, z2] (Logic.list_implies ([prem1, prem2], concl))
+ end;
+
+ val IHs = map5 mk_IH phi2s (drop m sets) (drop m sets') Izs1 Izs2;
+ val concl = HOLogic.mk_Trueprop (phi $ (fld $ x) $ (fld' $ y));
+ in
+ fold_rev Logic.all [x, y] (Logic.list_implies (IHs, concl))
+ end;
+
+ val prems = map7 mk_prem phi2s flds fld's FTs_setss FTs'_setss xFs yFs;
+
+ fun mk_concl phi z1 z2 = phi $ z1 $ z2;
+ val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map3 mk_concl phi2s Izs1 Izs2));
+ fun mk_t phi (z1, z1') (z2, z2') =
+ Term.absfree z1' (HOLogic.mk_all (fst z2', snd z2', phi $ z1 $ z2));
+ val cts = map3 (SOME o certify lthy ooo mk_t) phi2s (Izs1 ~~ Izs1') (Izs2 ~~ Izs2');
+ val goal = Logic.list_implies (prems, concl);
+ in
+ (singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_fld_induct2_tac cTs cts fld_induct_thm weak_fld_induct_thms)),
+ rev (Term.add_tfrees goal []))
+ end;
+
+ val timer = time (timer "induction");
+
+ (*register new datatypes as BNFs*)
+ val lthy = if m = 0 then lthy else
+ let
+ val fTs = map2 (curry op -->) passiveAs passiveBs;
+ val f1Ts = map2 (curry op -->) passiveAs passiveYs;
+ val f2Ts = map2 (curry op -->) passiveBs passiveYs;
+ val p1Ts = map2 (curry op -->) passiveXs passiveAs;
+ val p2Ts = map2 (curry op -->) passiveXs passiveBs;
+ val uTs = map2 (curry op -->) Ts Ts';
+ val B1Ts = map HOLogic.mk_setT passiveAs;
+ val B2Ts = map HOLogic.mk_setT passiveBs;
+ val AXTs = map HOLogic.mk_setT passiveXs;
+ val XTs = mk_Ts passiveXs;
+ val YTs = mk_Ts passiveYs;
+ val IRTs = map2 (curry mk_relT) passiveAs passiveBs;
+ val IphiTs = map2 (fn T => fn U => T --> U --> HOLogic.boolT) passiveAs passiveBs;
+
+ val (((((((((((((((fs, fs'), fs_copy), us),
+ B1s), B2s), AXs), (xs, xs')), f1s), f2s), p1s), p2s), (ys, ys')), IRs), Iphis),
+ names_lthy) = names_lthy
+ |> mk_Frees' "f" fTs
+ ||>> mk_Frees "f" fTs
+ ||>> mk_Frees "u" uTs
+ ||>> mk_Frees "B1" B1Ts
+ ||>> mk_Frees "B2" B2Ts
+ ||>> mk_Frees "A" AXTs
+ ||>> mk_Frees' "x" XTs
+ ||>> mk_Frees "f1" f1Ts
+ ||>> mk_Frees "f2" f2Ts
+ ||>> mk_Frees "p1" p1Ts
+ ||>> mk_Frees "p2" p2Ts
+ ||>> mk_Frees' "y" passiveAs
+ ||>> mk_Frees "R" IRTs
+ ||>> mk_Frees "phi" IphiTs;
+
+ val map_FTFT's = map2 (fn Ds =>
+ mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+ fun mk_passive_maps ATs BTs Ts =
+ map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ Ts)) Dss bnfs;
+ fun mk_map_iter_arg fs Ts fld fmap =
+ HOLogic.mk_comp (fld, Term.list_comb (fmap, fs @ map HOLogic.id_const Ts));
+ fun mk_map Ts fs Ts' flds mk_maps =
+ mk_iter Ts (map2 (mk_map_iter_arg fs Ts') flds (mk_maps Ts'));
+ val pmapsABT' = mk_passive_maps passiveAs passiveBs;
+ val fs_maps = map (mk_map Ts fs Ts' fld's pmapsABT') ks;
+ val fs_copy_maps = map (mk_map Ts fs_copy Ts' fld's pmapsABT') ks;
+ val Yflds = mk_flds passiveYs;
+ val f1s_maps = map (mk_map Ts f1s YTs Yflds (mk_passive_maps passiveAs passiveYs)) ks;
+ val f2s_maps = map (mk_map Ts' f2s YTs Yflds (mk_passive_maps passiveBs passiveYs)) ks;
+ val p1s_maps = map (mk_map XTs p1s Ts flds (mk_passive_maps passiveXs passiveAs)) ks;
+ val p2s_maps = map (mk_map XTs p2s Ts' fld's (mk_passive_maps passiveXs passiveBs)) ks;
+
+ val (map_simp_thms, map_thms) =
+ let
+ fun mk_goal fs_map map fld fld' = fold_rev Logic.all fs
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (fs_map, fld),
+ HOLogic.mk_comp (fld', Term.list_comb (map, fs @ fs_maps)))));
+ val goals = map4 mk_goal fs_maps map_FTFT's flds fld's;
+ val maps = map4 (fn goal => fn iter => fn map_comp_id => fn map_cong =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_map_tac m n iter map_comp_id map_cong)))
+ goals iter_thms map_comp_id_thms map_congs;
+ in
+ map_split (fn thm => (thm RS @{thm pointfreeE}, thm)) maps
+ end;
+
+ val (map_unique_thms, map_unique_thm) =
+ let
+ fun mk_prem u map fld fld' =
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (u, fld),
+ HOLogic.mk_comp (fld', Term.list_comb (map, fs @ us))));
+ val prems = map4 mk_prem us map_FTFT's flds fld's;
+ val goal =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map2 (curry HOLogic.mk_eq) us fs_maps));
+ val unique = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (us @ fs) (Logic.list_implies (prems, goal)))
+ (K (mk_map_unique_tac m mor_def iter_unique_mor_thm map_comp_id_thms map_congs));
+ in
+ `split_conj_thm unique
+ end;
+
+ val timer = time (timer "map functions for the new datatypes");
+
+ val bd = mk_cpow sum_bd;
+ val bd_Cinfinite = sum_Cinfinite RS @{thm Cinfinite_cpow};
+ fun mk_cpow_bd thm = @{thm ordLeq_transitive} OF
+ [thm, sum_Card_order RS @{thm cpow_greater_eq}];
+ val set_bd_cpowss = map (map mk_cpow_bd) set_bd_sumss;
+
+ val timer = time (timer "bounds for the new datatypes");
+
+ val ls = 1 upto m;
+ val setsss = map (mk_setss o mk_set_Ts) passiveAs;
+ val map_setss = map (fn T => map2 (fn Ds =>
+ mk_map_of_bnf Ds (passiveAs @ Ts) (mk_set_Ts T)) Dss bnfs) passiveAs;
+
+ fun mk_col l T z z' sets =
+ let
+ fun mk_UN set = mk_Union T $ (set $ z);
+ in
+ Term.absfree z'
+ (mk_union (nth sets (l - 1) $ z,
+ Library.foldl1 mk_union (map mk_UN (drop m sets))))
+ end;
+
+ val colss = map5 (fn l => fn T => map3 (mk_col l T)) ls passiveAs AFss AFss' setsss;
+ val setss_by_range = map (fn cols => map (mk_iter Ts cols) ks) colss;
+ val setss_by_bnf = transpose setss_by_range;
+
+ val (set_simp_thmss, set_thmss) =
+ let
+ fun mk_goal sets fld set col map =
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_comp (set, fld),
+ HOLogic.mk_comp (col, Term.list_comb (map, passive_ids @ sets))));
+ val goalss =
+ map3 (fn sets => map4 (mk_goal sets) flds sets) setss_by_range colss map_setss;
+ val setss = map (map2 (fn iter => fn goal =>
+ Skip_Proof.prove lthy [] [] goal (K (mk_set_tac iter)))
+ iter_thms) goalss;
+
+ fun mk_simp_goal pas_set act_sets sets fld z set =
+ Logic.all z (HOLogic.mk_Trueprop (HOLogic.mk_eq (set $ (fld $ z),
+ mk_union (pas_set $ z,
+ Library.foldl1 mk_union (map2 (fn X => mk_UNION (X $ z)) act_sets sets)))));
+ val simp_goalss =
+ map2 (fn i => fn sets =>
+ map4 (fn Fsets => mk_simp_goal (nth Fsets (i - 1)) (drop m Fsets) sets)
+ FTs_setss flds xFs sets)
+ ls setss_by_range;
+
+ val set_simpss = map3 (fn i => map3 (fn set_nats => fn goal => fn set =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_set_simp_tac set (nth set_nats (i - 1)) (drop m set_nats))))
+ set_natural'ss) ls simp_goalss setss;
+ in
+ (set_simpss, setss)
+ end;
+
+ fun mk_set_thms set_simp = (@{thm xt1(3)} OF [set_simp, @{thm Un_upper1}]) ::
+ map (fn i => (@{thm xt1(3)} OF [set_simp, @{thm Un_upper2}]) RS
+ (mk_Un_upper n i RS subset_trans) RSN
+ (2, @{thm UN_upper} RS subset_trans))
+ (1 upto n);
+ val Fset_set_thmsss = transpose (map (map mk_set_thms) set_simp_thmss);
+
+ val timer = time (timer "set functions for the new datatypes");
+
+ val cxs = map (SOME o certify lthy) Izs;
+ val setss_by_bnf' =
+ map (map (Term.subst_atomic_types (passiveAs ~~ passiveBs))) setss_by_bnf;
+ val setss_by_range' = transpose setss_by_bnf';
+
+ val set_natural_thmss =
+ let
+ fun mk_set_natural f map z set set' =
+ HOLogic.mk_eq (mk_image f $ (set $ z), set' $ (map $ z));
+
+ fun mk_cphi f map z set set' = certify lthy
+ (Term.absfree (dest_Free z) (mk_set_natural f map z set set'));
+
+ val csetss = map (map (certify lthy)) setss_by_range';
+
+ val cphiss = map3 (fn f => fn sets => fn sets' =>
+ (map4 (mk_cphi f) fs_maps Izs sets sets')) fs setss_by_range setss_by_range';
+
+ val inducts = map (fn cphis =>
+ Drule.instantiate' cTs (map SOME cphis @ cxs) fld_induct_thm) cphiss;
+
+ val goals =
+ map3 (fn f => fn sets => fn sets' =>
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map4 (mk_set_natural f) fs_maps Izs sets sets')))
+ fs setss_by_range setss_by_range';
+
+ fun mk_tac induct = mk_set_nat_tac m (rtac induct) set_natural'ss map_simp_thms;
+ val thms = map5 (fn goal => fn csets => fn set_simps => fn induct => fn i =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_tac induct csets set_simps i)))
+ goals csetss set_simp_thmss inducts ls;
+ in
+ map split_conj_thm thms
+ end;
+
+ val set_bd_thmss =
+ let
+ fun mk_set_bd z set = mk_ordLeq (mk_card_of (set $ z)) bd;
+
+ fun mk_cphi z set = certify lthy (Term.absfree (dest_Free z) (mk_set_bd z set));
+
+ val cphiss = map (map2 mk_cphi Izs) setss_by_range;
+
+ val inducts = map (fn cphis =>
+ Drule.instantiate' cTs (map SOME cphis @ cxs) fld_induct_thm) cphiss;
+
+ val goals =
+ map (fn sets =>
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map2 mk_set_bd Izs sets))) setss_by_range;
+
+ fun mk_tac induct = mk_set_bd_tac m (rtac induct) bd_Cinfinite set_bd_cpowss;
+ val thms = map4 (fn goal => fn set_simps => fn induct => fn i =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal (mk_tac induct set_simps i)))
+ goals set_simp_thmss inducts ls;
+ in
+ map split_conj_thm thms
+ end;
+
+ val map_cong_thms =
+ let
+ fun mk_prem z set f g y y' =
+ mk_Ball (set $ z) (Term.absfree y' (HOLogic.mk_eq (f $ y, g $ y)));
+
+ fun mk_map_cong sets z fmap gmap =
+ HOLogic.mk_imp
+ (Library.foldr1 HOLogic.mk_conj (map5 (mk_prem z) sets fs fs_copy ys ys'),
+ HOLogic.mk_eq (fmap $ z, gmap $ z));
+
+ fun mk_cphi sets z fmap gmap =
+ certify lthy (Term.absfree (dest_Free z) (mk_map_cong sets z fmap gmap));
+
+ val cphis = map4 mk_cphi setss_by_bnf Izs fs_maps fs_copy_maps;
+
+ val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) fld_induct_thm;
+
+ val goal =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map4 mk_map_cong setss_by_bnf Izs fs_maps fs_copy_maps));
+
+ val thm = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_mcong_tac (rtac induct) Fset_set_thmsss map_congs map_simp_thms));
+ in
+ split_conj_thm thm
+ end;
+
+ val in_incl_min_alg_thms =
+ let
+ fun mk_prem z sets =
+ HOLogic.mk_mem (z, mk_in As sets (fastype_of z));
+
+ fun mk_incl z sets i =
+ HOLogic.mk_imp (mk_prem z sets, HOLogic.mk_mem (z, mk_min_alg As flds i));
+
+ fun mk_cphi z sets i =
+ certify lthy (Term.absfree (dest_Free z) (mk_incl z sets i));
+
+ val cphis = map3 mk_cphi Izs setss_by_bnf ks;
+
+ val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) fld_induct_thm;
+
+ val goal =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map3 mk_incl Izs setss_by_bnf ks));
+
+ val thm = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_incl_min_alg_tac (rtac induct) Fset_set_thmsss alg_set_thms alg_min_alg_thm));
+ in
+ split_conj_thm thm
+ end;
+
+ val Xsetss = map (map (Term.subst_atomic_types (passiveAs ~~ passiveXs))) setss_by_bnf;
+
+ val map_wpull_thms =
+ let
+ val cTs = map (SOME o certifyT lthy o TFree) induct2_params;
+ val cxs = map (SOME o certify lthy) (interleave Izs1 Izs2);
+
+ fun mk_prem z1 z2 sets1 sets2 map1 map2 =
+ HOLogic.mk_conj
+ (HOLogic.mk_mem (z1, mk_in B1s sets1 (fastype_of z1)),
+ HOLogic.mk_conj
+ (HOLogic.mk_mem (z2, mk_in B2s sets2 (fastype_of z2)),
+ HOLogic.mk_eq (map1 $ z1, map2 $ z2)));
+
+ val prems = map6 mk_prem Izs1 Izs2 setss_by_bnf setss_by_bnf' f1s_maps f2s_maps;
+
+ fun mk_concl z1 z2 sets map1 map2 T x x' =
+ mk_Bex (mk_in AXs sets T) (Term.absfree x'
+ (HOLogic.mk_conj (HOLogic.mk_eq (map1 $ x, z1), HOLogic.mk_eq (map2 $ x, z2))));
+
+ val concls = map8 mk_concl Izs1 Izs2 Xsetss p1s_maps p2s_maps XTs xs xs';
+
+ val goals = map2 (curry HOLogic.mk_imp) prems concls;
+
+ fun mk_cphi z1 z2 goal = certify lthy (Term.absfree z1 (Term.absfree z2 goal));
+
+ val cphis = map3 mk_cphi Izs1' Izs2' goals;
+
+ val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) fld_induct2_thm;
+
+ val goal = Logic.list_implies (map HOLogic.mk_Trueprop
+ (map8 mk_wpull AXs B1s B2s f1s f2s (replicate m NONE) p1s p2s),
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj goals));
+
+ val thm = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (K (mk_lfp_map_wpull_tac m (rtac induct) map_wpulls map_simp_thms
+ (transpose set_simp_thmss) Fset_set_thmsss fld_inject_thms)));
+ in
+ split_conj_thm thm
+ end;
+
+ val timer = time (timer "helpers for BNF properties");
+
+ val map_id_tacs = map (K o mk_map_id_tac map_ids) map_unique_thms;
+ val map_comp_tacs =
+ map2 (K oo mk_map_comp_tac map_comp's map_simp_thms) map_unique_thms ks;
+ val map_cong_tacs = map (mk_map_cong_tac m) map_cong_thms;
+ val set_nat_tacss = map (map (K o mk_set_natural_tac)) (transpose set_natural_thmss);
+ val bd_co_tacs = replicate n (K (mk_bd_card_order_tac bd_card_orders));
+ val bd_cinf_tacs = replicate n (K (rtac (bd_Cinfinite RS conjunct1) 1));
+ val set_bd_tacss = map (map (fn thm => K (rtac thm 1))) (transpose set_bd_thmss);
+ val in_bd_tacs = map2 (K oo mk_in_bd_tac sum_Card_order suc_bd_Cnotzero)
+ in_incl_min_alg_thms card_of_min_alg_thms;
+ val map_wpull_tacs = map (K o mk_wpull_tac) map_wpull_thms;
+
+ val tacss = map9 mk_tactics map_id_tacs map_comp_tacs map_cong_tacs set_nat_tacss bd_co_tacs
+ bd_cinf_tacs set_bd_tacss in_bd_tacs map_wpull_tacs;
+
+ val fld_witss =
+ let
+ val witss = map2 (fn Ds => fn bnf => mk_wits_of_bnf
+ (replicate (nwits_of_bnf bnf) Ds)
+ (replicate (nwits_of_bnf bnf) (passiveAs @ Ts)) bnf) Dss bnfs;
+ fun close_wit (I, wit) = fold_rev Term.absfree (map (nth ys') I) wit;
+ fun wit_apply (arg_I, arg_wit) (fun_I, fun_wit) =
+ (union (op =) arg_I fun_I, fun_wit $ arg_wit);
+
+ fun gen_arg support i =
+ if i < m then [([i], nth ys i)]
+ else maps (mk_wit support (nth flds (i - m)) (i - m)) (nth support (i - m))
+ and mk_wit support fld i (I, wit) =
+ let val args = map (gen_arg (nth_map i (remove (op =) (I, wit)) support)) I;
+ in
+ (args, [([], wit)])
+ |-> fold (map_product wit_apply)
+ |> map (apsnd (fn t => fld $ t))
+ |> minimize_wits
+ end;
+ in
+ map3 (fn fld => fn i => map close_wit o minimize_wits o maps (mk_wit witss fld i))
+ flds (0 upto n - 1) witss
+ end;
+
+ fun wit_tac _ = mk_wit_tac n (flat set_simp_thmss) (maps wit_thms_of_bnf bnfs);
+
+ val (Ibnfs, lthy) =
+ fold_map6 (fn tacs => fn b => fn map => fn sets => fn T => fn wits =>
+ add_bnf Dont_Inline user_policy I tacs wit_tac (SOME deads)
+ ((((b, fold_rev Term.absfree fs' map), sets), absdummy T bd), wits))
+ tacss bs fs_maps setss_by_bnf Ts fld_witss lthy;
+
+ val fold_maps = Local_Defs.fold lthy (map (fn bnf =>
+ mk_unabs_def m (map_def_of_bnf bnf RS @{thm meta_eq_to_obj_eq})) Ibnfs);
+
+ val fold_sets = Local_Defs.fold lthy (maps (fn bnf =>
+ map (fn thm => thm RS @{thm meta_eq_to_obj_eq}) (set_defs_of_bnf bnf)) Ibnfs);
+
+ val timer = time (timer "registered new datatypes as BNFs");
+
+ val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+ val Irels = map (mk_rel_of_bnf deads passiveAs passiveBs) Ibnfs;
+ val preds = map2 (fn Ds => mk_pred_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+ val Ipreds = map (mk_pred_of_bnf deads passiveAs passiveBs) Ibnfs;
+
+ val IrelRs = map (fn Irel => Term.list_comb (Irel, IRs)) Irels;
+ val relRs = map (fn rel => Term.list_comb (rel, IRs @ IrelRs)) rels;
+ val Ipredphis = map (fn Irel => Term.list_comb (Irel, Iphis)) Ipreds;
+ val predphis = map (fn rel => Term.list_comb (rel, Iphis @ Ipredphis)) preds;
+
+ val in_rels = map in_rel_of_bnf bnfs;
+ val in_Irels = map in_rel_of_bnf Ibnfs;
+ val pred_defs = map pred_def_of_bnf bnfs;
+ val Ipred_defs =
+ map (Drule.abs_def o (fn thm => thm RS @{thm eq_reflection}) o pred_def_of_bnf) Ibnfs;
+
+ val set_incl_thmss = map (map (fold_sets o hd)) Fset_set_thmsss;
+ val set_set_incl_thmsss = map (transpose o map (map fold_sets o tl)) Fset_set_thmsss;
+ val folded_map_simp_thms = map fold_maps map_simp_thms;
+ val folded_set_simp_thmss = map (map fold_sets) set_simp_thmss;
+ val folded_set_simp_thmss' = transpose folded_set_simp_thmss;
+
+ val Irel_unfold_thms =
+ let
+ fun mk_goal xF yF fld fld' IrelR relR = fold_rev Logic.all (xF :: yF :: IRs)
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (HOLogic.mk_mem (HOLogic.mk_prod (fld $ xF, fld' $ yF), IrelR),
+ HOLogic.mk_mem (HOLogic.mk_prod (xF, yF), relR))));
+ val goals = map6 mk_goal xFs yFs flds fld's IrelRs relRs;
+ in
+ map12 (fn i => fn goal => fn in_rel => fn map_comp => fn map_cong =>
+ fn map_simp => fn set_simps => fn fld_inject => fn fld_unf =>
+ fn set_naturals => fn set_incls => fn set_set_inclss =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_rel_unfold_tac in_Irels i in_rel map_comp map_cong map_simp set_simps
+ fld_inject fld_unf set_naturals set_incls set_set_inclss)))
+ ks goals in_rels map_comp's map_congs folded_map_simp_thms folded_set_simp_thmss'
+ fld_inject_thms fld_unf_thms set_natural'ss set_incl_thmss set_set_incl_thmsss
+ end;
+
+ val Ipred_unfold_thms =
+ let
+ fun mk_goal xF yF fld fld' Ipredphi predphi = fold_rev Logic.all (xF :: yF :: Iphis)
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (Ipredphi $ (fld $ xF) $ (fld' $ yF), predphi $ xF $ yF)));
+ val goals = map6 mk_goal xFs yFs flds fld's Ipredphis predphis;
+ in
+ map3 (fn goal => fn pred_def => fn Irel_unfold =>
+ Skip_Proof.prove lthy [] [] goal (mk_pred_unfold_tac pred_def Ipred_defs Irel_unfold))
+ goals pred_defs Irel_unfold_thms
+ end;
+
+ val timer = time (timer "additional properties");
+
+ val ls' = if m = 1 then [0] else ls
+ in
+ lthy
+ |> note map_uniqueN [fold_maps map_unique_thm]
+ |> notes map_simpsN (map single folded_map_simp_thms)
+ |> fold2 (fn i => notes (mk_set_simpsN i) o map single) ls' folded_set_simp_thmss
+ |> notes set_inclN set_incl_thmss
+ |> notes set_set_inclN (map flat set_set_incl_thmsss)
+ |> notes rel_unfoldN (map single Irel_unfold_thms)
+ |> notes pred_unfoldN (map single Ipred_unfold_thms)
+ end;
+
+ in
+ lthy
+ |> notes iterN (map single iter_thms)
+ |> notes iter_uniqueN (map single iter_unique_thms)
+ |> notes recN (map single rec_thms)
+ |> notes unf_fldN (map single unf_fld_thms)
+ |> notes fld_unfN (map single fld_unf_thms)
+ |> notes unf_injectN (map single unf_inject_thms)
+ |> notes unf_exhaustN (map single unf_exhaust_thms)
+ |> notes fld_injectN (map single fld_inject_thms)
+ |> notes fld_exhaustN (map single fld_exhaust_thms)
+ |> note fld_inductN [fld_induct_thm]
+ |> note fld_induct2N [fld_induct2_thm]
+ end;
+
+val _ =
+ Outer_Syntax.local_theory @{command_spec "bnf_data"} "least fixed points for BNF equations"
+ (Parse.and_list1
+ ((Parse.binding --| Parse.$$$ ":") -- (Parse.typ --| Parse.$$$ "=" -- Parse.typ)) >>
+ (fp_bnf_cmd bnf_lfp o apsnd split_list o split_list));
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_lfp_tactics.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,835 @@
+(* Title: HOL/Codatatype/Tools/bnf_lfp_tactics.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Tactics for the datatype construction.
+*)
+
+signature BNF_LFP_TACTICS =
+sig
+ val mk_alg_min_alg_tac: int -> thm -> thm list -> thm -> thm -> thm list list -> thm list ->
+ thm list -> tactic
+ val mk_alg_not_empty_tac: thm -> thm list -> thm list -> tactic
+ val mk_alg_select_tac: thm -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_alg_set_tac: thm -> tactic
+ val mk_bd_card_order_tac: thm list -> tactic
+ val mk_bd_limit_tac: int -> thm -> tactic
+ val mk_card_of_min_alg_tac: thm -> thm -> thm -> thm -> thm -> tactic
+ val mk_copy_alg_tac: thm list list -> thm list -> thm -> thm -> thm -> tactic
+ val mk_copy_str_tac: thm list list -> thm -> thm list -> tactic
+ val mk_ex_copy_alg_tac: int -> thm -> thm -> tactic
+ val mk_fld_induct2_tac: ctyp option list -> cterm option list -> thm -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_fld_induct_tac: int -> thm list list -> thm -> thm list -> thm -> thm list -> thm list ->
+ thm list -> thm list -> thm list -> tactic
+ val mk_in_bd_tac: thm -> thm -> thm -> thm -> tactic
+ val mk_incl_min_alg_tac: (int -> tactic) -> thm list list list -> thm list -> thm ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_init_ex_mor_tac: thm -> thm -> thm -> thm list -> thm -> thm -> thm ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_init_induct_tac: int -> thm -> thm -> thm list -> thm list -> tactic
+ val mk_init_unique_mor_tac: int -> thm -> thm -> thm list -> thm list -> thm list -> thm list ->
+ thm list -> tactic
+ val mk_iso_alt_tac: thm list -> thm -> tactic
+ val mk_iter_unique_mor_tac: thm list -> thm list -> thm list -> thm list -> thm -> thm -> thm ->
+ tactic
+ val mk_least_min_alg_tac: thm -> thm -> tactic
+ val mk_lfp_map_wpull_tac: int -> (int -> tactic) -> thm list -> thm list -> thm list list ->
+ thm list list list -> thm list -> tactic
+ val mk_map_comp_tac: thm list -> thm list -> thm -> int -> tactic
+ val mk_map_id_tac: thm list -> thm -> tactic
+ val mk_map_tac: int -> int -> thm -> thm -> thm -> tactic
+ val mk_map_unique_tac: int -> thm -> thm -> thm list -> thm list -> tactic
+ val mk_mcong_tac: (int -> tactic) -> thm list list list -> thm list -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_min_algs_card_of_tac: ctyp -> cterm -> int -> thm -> thm list -> thm list -> thm -> thm ->
+ thm -> thm -> thm -> thm -> thm -> thm -> tactic
+ val mk_min_algs_least_tac: ctyp -> cterm -> thm -> thm list -> thm list -> tactic
+ val mk_min_algs_mono_tac: thm -> tactic
+ val mk_min_algs_tac: thm -> thm list -> tactic
+ val mk_mor_Abs_tac: thm -> thm list -> thm list -> tactic
+ val mk_mor_Rep_tac: thm list -> thm -> thm list -> thm list -> thm list ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_mor_UNIV_tac: int -> thm list -> thm -> tactic
+ val mk_mor_comp_tac: thm -> thm list list -> thm list -> tactic
+ val mk_mor_convol_tac: 'a list -> thm -> tactic
+ val mk_mor_elim_tac: thm -> tactic
+ val mk_mor_incl_tac: thm -> thm list -> tactic
+ val mk_mor_inv_tac: thm -> thm -> thm list list -> thm list -> thm list -> thm list -> tactic
+ val mk_mor_iter_tac: ctyp -> cterm -> thm list -> thm -> thm -> tactic
+ val mk_mor_select_tac: thm -> thm -> thm -> thm -> thm -> thm -> thm list -> thm list list ->
+ thm list -> tactic
+ val mk_mor_str_tac: 'a list -> thm -> tactic
+ val mk_rec_tac: thm list -> thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_rel_unfold_tac: thm list -> int -> thm -> thm -> thm -> thm -> thm list -> thm ->
+ thm -> thm list -> thm list -> thm list list -> tactic
+ val mk_set_bd_tac: int -> (int -> tactic) -> thm -> thm list list -> thm list -> int ->
+ {prems: 'a, context: Proof.context} -> tactic
+ val mk_set_nat_tac: int -> (int -> tactic) -> thm list list -> thm list -> cterm list ->
+ thm list -> int -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_set_natural_tac: thm -> tactic
+ val mk_set_simp_tac: thm -> thm -> thm list -> tactic
+ val mk_set_tac: thm -> tactic
+ val mk_unf_o_fld_tac: thm -> thm -> thm -> thm -> thm list -> tactic
+ val mk_wit_tac: int -> thm list -> thm list -> tactic
+ val mk_wpull_tac: thm -> tactic
+end;
+
+structure BNF_LFP_Tactics : BNF_LFP_TACTICS =
+struct
+
+open BNF_Tactics
+open BNF_LFP_Util
+open BNF_Util
+
+fun mk_alg_set_tac alg_def =
+ dtac (alg_def RS @{thm subst[of _ _ "\<lambda>x. x"]}) 1 THEN
+ REPEAT_DETERM (etac conjE 1) THEN
+ EVERY' [etac bspec, rtac CollectI] 1 THEN
+ REPEAT_DETERM (etac conjI 1) THEN atac 1;
+
+fun mk_alg_not_empty_tac alg_set alg_sets wits =
+ (EVERY' [rtac notI, hyp_subst_tac, ftac alg_set] THEN'
+ REPEAT_DETERM o FIRST'
+ [rtac @{thm subset_UNIV},
+ EVERY' [rtac @{thm subset_emptyI}, eresolve_tac wits],
+ EVERY' [rtac subsetI, rtac FalseE, eresolve_tac wits],
+ EVERY' [rtac subsetI, dresolve_tac wits, hyp_subst_tac,
+ FIRST' (map (fn thm => rtac thm THEN' atac) alg_sets)]] THEN'
+ etac @{thm emptyE}) 1;
+
+fun mk_mor_elim_tac mor_def =
+ (dtac (subst OF [mor_def]) THEN'
+ REPEAT o etac conjE THEN'
+ TRY o rtac @{thm image_subsetI} THEN'
+ etac bspec THEN'
+ atac) 1;
+
+fun mk_mor_incl_tac mor_def map_id's =
+ (stac mor_def THEN'
+ rtac conjI THEN'
+ CONJ_WRAP' (K (EVERY' [rtac ballI, etac @{thm set_mp}, stac @{thm id_apply}, atac]))
+ map_id's THEN'
+ CONJ_WRAP' (fn thm =>
+ (EVERY' [rtac ballI, rtac trans, rtac @{thm id_apply}, stac thm, rtac refl])) map_id's) 1;
+
+fun mk_mor_comp_tac mor_def set_natural's map_comp_ids =
+ let
+ val fbetw_tac = EVERY' [rtac ballI, stac o_apply, etac bspec, etac bspec, atac];
+ fun mor_tac (set_natural', map_comp_id) =
+ EVERY' [rtac ballI, stac o_apply, rtac trans,
+ rtac trans, dtac @{thm rev_bspec}, atac, etac arg_cong,
+ REPEAT o eresolve_tac [CollectE, conjE], etac bspec, rtac CollectI] THEN'
+ CONJ_WRAP' (fn thm =>
+ FIRST' [rtac @{thm subset_UNIV},
+ (EVERY' [rtac @{thm ord_eq_le_trans}, rtac thm, rtac @{thm image_subsetI},
+ etac bspec, etac @{thm set_mp}, atac])]) set_natural' THEN'
+ rtac (map_comp_id RS arg_cong);
+ in
+ (dtac (mor_def RS subst) THEN' dtac (mor_def RS subst) THEN' stac mor_def THEN'
+ REPEAT o etac conjE THEN'
+ rtac conjI THEN'
+ CONJ_WRAP' (K fbetw_tac) set_natural's THEN'
+ CONJ_WRAP' mor_tac (set_natural's ~~ map_comp_ids)) 1
+ end;
+
+fun mk_mor_inv_tac alg_def mor_def set_natural's morEs map_comp_ids map_congLs =
+ let
+ val fbetw_tac = EVERY' [rtac ballI, etac @{thm set_mp}, etac imageI];
+ fun Collect_tac set_natural' =
+ CONJ_WRAP' (fn thm =>
+ FIRST' [rtac @{thm subset_UNIV},
+ (EVERY' [rtac @{thm ord_eq_le_trans}, rtac thm, rtac subset_trans,
+ etac @{thm image_mono}, atac])]) set_natural';
+ fun mor_tac (set_natural', ((morE, map_comp_id), map_congL)) =
+ EVERY' [rtac ballI, ftac @{thm rev_bspec}, atac,
+ REPEAT o eresolve_tac [CollectE, conjE], rtac sym, rtac trans, rtac sym,
+ etac @{thm inverE}, etac bspec, rtac CollectI, Collect_tac set_natural',
+ rtac trans, etac (morE RS arg_cong), rtac CollectI, Collect_tac set_natural',
+ rtac trans, rtac (map_comp_id RS arg_cong), rtac (map_congL RS arg_cong),
+ REPEAT_DETERM_N (length morEs) o
+ (EVERY' [rtac subst, rtac @{thm inver_pointfree}, etac @{thm inver_mono}, atac])];
+ in
+ (stac mor_def THEN'
+ dtac (alg_def RS @{thm subst[of _ _ "\<lambda>x. x"]}) THEN'
+ dtac (alg_def RS @{thm subst[of _ _ "\<lambda>x. x"]}) THEN'
+ REPEAT o etac conjE THEN'
+ rtac conjI THEN'
+ CONJ_WRAP' (K fbetw_tac) set_natural's THEN'
+ CONJ_WRAP' mor_tac (set_natural's ~~ (morEs ~~ map_comp_ids ~~ map_congLs))) 1
+ end;
+
+fun mk_mor_str_tac ks mor_def =
+ (stac mor_def THEN' rtac conjI THEN'
+ CONJ_WRAP' (K (EVERY' [rtac ballI, rtac UNIV_I])) ks THEN'
+ CONJ_WRAP' (K (EVERY' [rtac ballI, rtac refl])) ks) 1;
+
+fun mk_mor_convol_tac ks mor_def =
+ (stac mor_def THEN' rtac conjI THEN'
+ CONJ_WRAP' (K (EVERY' [rtac ballI, rtac UNIV_I])) ks THEN'
+ CONJ_WRAP' (K (EVERY' [rtac ballI, rtac trans, rtac @{thm fst_convol'}, rtac o_apply])) ks) 1;
+
+fun mk_mor_UNIV_tac m morEs mor_def =
+ let
+ val n = length morEs;
+ fun mor_tac morE = EVERY' [rtac ext, rtac trans, rtac o_apply, rtac trans, etac morE,
+ rtac CollectI, CONJ_WRAP' (K (rtac @{thm subset_UNIV})) (1 upto m + n),
+ rtac sym, rtac o_apply];
+ in
+ EVERY' [rtac iffI, CONJ_WRAP' mor_tac morEs,
+ stac mor_def, rtac conjI, CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) morEs,
+ REPEAT_DETERM o etac conjE, REPEAT_DETERM_N n o dtac (@{thm fun_eq_iff} RS subst),
+ CONJ_WRAP' (K (EVERY' [rtac ballI, REPEAT_DETERM o etac allE, rtac trans,
+ etac (o_apply RS subst), rtac o_apply])) morEs] 1
+ end;
+
+fun mk_iso_alt_tac mor_images mor_inv =
+ let
+ val n = length mor_images;
+ fun if_wrap_tac thm =
+ EVERY' [rtac ssubst, rtac @{thm bij_betw_iff_ex}, rtac exI, rtac conjI,
+ rtac @{thm inver_surj}, etac thm, etac thm, atac, etac conjI, atac]
+ val if_tac =
+ EVERY' [etac thin_rl, etac thin_rl, REPEAT o eresolve_tac [conjE, exE],
+ rtac conjI, atac, CONJ_WRAP' if_wrap_tac mor_images];
+ val only_if_tac =
+ EVERY' [rtac conjI, etac conjunct1, EVERY' (map (fn thm =>
+ EVERY' [rtac exE, rtac @{thm bij_betw_ex_weakE}, etac (conjunct2 RS thm)])
+ (map (mk_conjunctN n) (1 upto n))), REPEAT o rtac exI, rtac conjI, rtac mor_inv,
+ etac conjunct1, atac, atac, REPEAT_DETERM_N n o atac,
+ CONJ_WRAP' (K (etac conjunct2)) mor_images];
+ in
+ (rtac iffI THEN' if_tac THEN' only_if_tac) 1
+ end;
+
+fun mk_copy_str_tac set_natural's alg_def alg_sets =
+ let
+ val n = length alg_sets;
+ val bij_betw_inv_tac =
+ EVERY' [etac thin_rl, REPEAT_DETERM_N n o EVERY' [dtac @{thm bij_betwI}, atac, atac],
+ REPEAT_DETERM_N (2 * n) o etac thin_rl, REPEAT_DETERM_N (n - 1) o etac conjI, atac];
+ fun set_tac thms =
+ EVERY' [rtac @{thm ord_eq_le_trans}, resolve_tac thms, rtac subset_trans,
+ etac @{thm image_mono}, rtac equalityD1, etac @{thm bij_betw_imageE}];
+ val copy_str_tac =
+ CONJ_WRAP' (fn (thms, thm) =>
+ EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac @{thm set_mp},
+ rtac equalityD1, etac @{thm bij_betw_imageE}, rtac imageI, etac thm,
+ REPEAT_DETERM o rtac @{thm subset_UNIV}, REPEAT_DETERM_N n o (set_tac thms)])
+ (set_natural's ~~ alg_sets);
+ in
+ (rtac rev_mp THEN' DETERM o bij_betw_inv_tac THEN' rtac impI THEN'
+ stac alg_def THEN' copy_str_tac) 1
+ end;
+
+fun mk_copy_alg_tac set_natural's alg_sets mor_def iso_alt copy_str =
+ let
+ val n = length alg_sets;
+ val fbetw_tac = CONJ_WRAP' (K (etac @{thm bij_betwE})) alg_sets;
+ fun set_tac thms =
+ EVERY' [rtac @{thm ord_eq_le_trans}, resolve_tac thms, rtac subset_trans,
+ REPEAT_DETERM o etac conjE, etac @{thm image_mono},
+ rtac equalityD1, etac @{thm bij_betw_imageE}];
+ val mor_tac =
+ CONJ_WRAP' (fn (thms, thm) =>
+ EVERY' [rtac ballI, etac CollectE, etac @{thm inverE}, etac thm,
+ REPEAT_DETERM o rtac @{thm subset_UNIV}, REPEAT_DETERM_N n o (set_tac thms)])
+ (set_natural's ~~ alg_sets);
+ in
+ (rtac (iso_alt RS @{thm ssubst[of _ _ "%x. x"]}) THEN'
+ etac copy_str THEN' REPEAT_DETERM o atac THEN'
+ rtac conjI THEN' stac mor_def THEN' rtac conjI THEN' fbetw_tac THEN' mor_tac THEN'
+ CONJ_WRAP' (K atac) alg_sets) 1
+ end;
+
+fun mk_ex_copy_alg_tac n copy_str copy_alg =
+ EVERY' [REPEAT_DETERM_N n o rtac exI, rtac conjI, etac copy_str,
+ REPEAT_DETERM_N n o atac,
+ REPEAT_DETERM_N n o etac @{thm bij_betw_inver2},
+ REPEAT_DETERM_N n o etac @{thm bij_betw_inver1}, etac copy_alg,
+ REPEAT_DETERM_N n o atac,
+ REPEAT_DETERM_N n o etac @{thm bij_betw_inver2},
+ REPEAT_DETERM_N n o etac @{thm bij_betw_inver1}] 1;
+
+fun mk_bd_limit_tac n bd_Cinfinite =
+ EVERY' [REPEAT_DETERM o etac conjE, rtac rev_mp, rtac @{thm Cinfinite_limit_finite},
+ REPEAT_DETERM_N n o rtac @{thm finite.insertI}, rtac @{thm finite.emptyI},
+ REPEAT_DETERM_N n o etac @{thm insert_subsetI}, rtac @{thm empty_subsetI},
+ rtac bd_Cinfinite, rtac impI, etac bexE, rtac bexI,
+ CONJ_WRAP' (fn i =>
+ EVERY' [etac bspec, REPEAT_DETERM_N i o rtac @{thm insertI2}, rtac @{thm insertI1}])
+ (0 upto n - 1),
+ atac] 1;
+
+fun mk_min_algs_tac worel in_congs =
+ let
+ val minG_tac = EVERY' [rtac @{thm UN_cong}, rtac refl, dtac bspec, atac, etac arg_cong];
+ fun minH_tac thm =
+ EVERY' [rtac @{thm Un_cong}, minG_tac, rtac @{thm image_cong}, rtac thm,
+ REPEAT_DETERM_N (length in_congs) o minG_tac, rtac refl];
+ in
+ (rtac (worel RS (@{thm wo_rel.worec_fixpoint} RS fun_cong)) THEN' rtac ssubst THEN'
+ rtac meta_eq_to_obj_eq THEN' rtac (worel RS @{thm wo_rel.adm_wo_def}) THEN'
+ REPEAT_DETERM_N 3 o rtac allI THEN' rtac impI THEN'
+ CONJ_WRAP_GEN' (EVERY' [rtac ssubst, rtac @{thm Pair_eq}, rtac conjI]) minH_tac in_congs) 1
+ end;
+
+fun mk_min_algs_mono_tac min_algs = EVERY' [stac @{thm relChain_def}, rtac allI, rtac allI,
+ rtac impI, rtac @{thm case_split}, rtac @{thm xt1(3)}, rtac min_algs, etac @{thm FieldI2},
+ rtac subsetI, rtac UnI1, rtac @{thm UN_I}, etac @{thm underS_I}, atac, atac,
+ rtac equalityD1, dtac @{thm notnotD}, hyp_subst_tac, rtac refl] 1;
+
+fun mk_min_algs_card_of_tac cT ct m worel min_algs in_bds bd_Card_order bd_Cnotzero
+ suc_Card_order suc_Cinfinite suc_Cnotzero suc_Asuc Asuc_Cinfinite Asuc_Cnotzero =
+ let
+ val induct = worel RS
+ Drule.instantiate' [SOME cT] [NONE, SOME ct] @{thm well_order_induct_imp};
+ val src = 1 upto m + 1;
+ val dest = (m + 1) :: (1 upto m);
+ val absorbAs_tac = if m = 0 then K (all_tac)
+ else EVERY' [rtac @{thm ordIso_transitive}, rtac @{thm csum_cong1},
+ rtac @{thm ordIso_transitive},
+ BNF_Tactics.mk_rotate_eq_tac (rtac @{thm ordIso_refl} THEN'
+ FIRST' [rtac @{thm card_of_Card_order}, rtac @{thm Card_order_csum},
+ rtac @{thm Card_order_cexp}])
+ @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong}
+ src dest,
+ rtac @{thm csum_absorb1}, rtac Asuc_Cinfinite, rtac ctrans, rtac @{thm ordLeq_csum1},
+ FIRST' [rtac @{thm Card_order_csum}, rtac @{thm card_of_Card_order}],
+ rtac @{thm ordLeq_cexp1}, rtac suc_Cnotzero, rtac @{thm Card_order_csum}];
+
+ val minG_tac = EVERY' [rtac @{thm UNION_Cinfinite_bound}, rtac @{thm ordLess_imp_ordLeq},
+ rtac @{thm ordLess_transitive}, rtac @{thm card_of_underS}, rtac suc_Card_order,
+ atac, rtac suc_Asuc, rtac ballI, etac allE, dtac mp, etac @{thm underS_E},
+ dtac mp, etac @{thm underS_Field}, REPEAT o etac conjE, atac, rtac Asuc_Cinfinite]
+
+ fun mk_minH_tac (min_alg, in_bd) = EVERY' [rtac @{thm ordIso_ordLeq_trans},
+ rtac @{thm card_of_ordIso_subst}, etac min_alg, rtac @{thm Un_Cinfinite_bound},
+ minG_tac, rtac ctrans, rtac @{thm card_of_image}, rtac ctrans, rtac in_bd, rtac ctrans,
+ rtac @{thm cexp_mono1_Cnotzero}, rtac @{thm csum_mono1},
+ REPEAT_DETERM_N m o rtac @{thm csum_mono2},
+ CONJ_WRAP_GEN' (rtac @{thm csum_cinfinite_bound}) (K minG_tac) min_algs,
+ REPEAT_DETERM o FIRST'
+ [rtac @{thm card_of_Card_order}, rtac @{thm Card_order_csum}, rtac Asuc_Cinfinite],
+ rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac bd_Card_order,
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm cexp_cong1_Cnotzero}, absorbAs_tac,
+ rtac @{thm csum_absorb1}, rtac Asuc_Cinfinite, rtac @{thm ctwo_ordLeq_Cinfinite},
+ rtac Asuc_Cinfinite, rtac bd_Card_order,
+ rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac Asuc_Cnotzero,
+ rtac @{thm ordIso_imp_ordLeq}, rtac @{thm cexp_cprod_ordLeq},
+ TRY o rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac suc_Cinfinite,
+ rtac bd_Cnotzero, rtac @{thm cardSuc_ordLeq}, rtac bd_Card_order, rtac Asuc_Cinfinite];
+ in
+ (rtac induct THEN'
+ rtac impI THEN'
+ CONJ_WRAP' mk_minH_tac (min_algs ~~ in_bds)) 1
+ end;
+
+fun mk_min_algs_least_tac cT ct worel min_algs alg_sets =
+ let
+ val induct = worel RS
+ Drule.instantiate' [SOME cT] [NONE, SOME ct] @{thm well_order_induct_imp};
+
+ val minG_tac = EVERY' [rtac @{thm UN_least}, etac allE, dtac mp, etac @{thm underS_E},
+ dtac mp, etac @{thm underS_Field}, REPEAT_DETERM o etac conjE, atac];
+
+ fun mk_minH_tac (min_alg, alg_set) = EVERY' [rtac @{thm ord_eq_le_trans}, etac min_alg,
+ rtac @{thm Un_least}, minG_tac, rtac @{thm image_subsetI},
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE], etac alg_set,
+ REPEAT_DETERM o FIRST' [atac, etac subset_trans THEN' minG_tac]];
+ in
+ (rtac induct THEN'
+ rtac impI THEN'
+ CONJ_WRAP' mk_minH_tac (min_algs ~~ alg_sets)) 1
+ end;
+
+fun mk_alg_min_alg_tac m alg_def min_alg_defs bd_limit bd_Cinfinite
+ set_bdss min_algs min_alg_monos =
+ let
+ val n = length min_algs;
+ fun mk_cardSuc_UNION_tac set_bds (mono, def) = EVERY'
+ [rtac bexE, rtac @{thm cardSuc_UNION_Cinfinite}, rtac bd_Cinfinite, rtac mono,
+ etac (def RSN (2, @{thm subset_trans[OF _ equalityD1]})), resolve_tac set_bds];
+ fun mk_conjunct_tac (set_bds, (min_alg, min_alg_def)) =
+ EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ EVERY' (map (mk_cardSuc_UNION_tac set_bds) (min_alg_monos ~~ min_alg_defs)), rtac bexE,
+ rtac bd_limit, REPEAT_DETERM_N (n - 1) o etac conjI, atac,
+ rtac (min_alg_def RS @{thm set_mp[OF equalityD2]}),
+ rtac @{thm UN_I}, REPEAT_DETERM_N (m + 3 * n) o etac thin_rl, atac, rtac @{thm set_mp},
+ rtac equalityD2, rtac min_alg, atac, rtac UnI2, rtac @{thm image_eqI}, rtac refl,
+ rtac CollectI, REPEAT_DETERM_N m o dtac asm_rl, REPEAT_DETERM_N n o etac thin_rl,
+ REPEAT_DETERM o etac conjE,
+ CONJ_WRAP' (K (FIRST' [atac,
+ EVERY' [etac subset_trans, rtac subsetI, rtac @{thm UN_I},
+ etac @{thm underS_I}, atac, atac]]))
+ set_bds];
+ in
+ (rtac (alg_def RS @{thm ssubst[of _ _ "%x. x"]}) THEN'
+ CONJ_WRAP' mk_conjunct_tac (set_bdss ~~ (min_algs ~~ min_alg_defs))) 1
+ end;
+
+fun mk_card_of_min_alg_tac min_alg_def card_of suc_Card_order suc_Asuc Asuc_Cinfinite =
+ EVERY' [stac min_alg_def, rtac @{thm UNION_Cinfinite_bound},
+ rtac @{thm ordIso_ordLeq_trans}, rtac @{thm card_of_Field_ordIso}, rtac suc_Card_order,
+ rtac @{thm ordLess_imp_ordLeq}, rtac suc_Asuc, rtac ballI, dtac rev_mp, rtac card_of,
+ REPEAT_DETERM o etac conjE, atac, rtac Asuc_Cinfinite] 1;
+
+fun mk_least_min_alg_tac min_alg_def least =
+ EVERY' [stac min_alg_def, rtac @{thm UN_least}, dtac least, dtac mp, atac,
+ REPEAT_DETERM o etac conjE, atac] 1;
+
+fun mk_alg_select_tac Abs_inverse {context = ctxt, prems = _} =
+ EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac] 1 THEN
+ Local_Defs.unfold_tac ctxt (Abs_inverse :: @{thms fst_conv snd_conv}) THEN atac 1;
+
+fun mk_mor_select_tac mor_def mor_cong mor_comp mor_incl_min_alg alg_def alg_select
+ alg_sets set_natural's str_init_defs =
+ let
+ val n = length alg_sets;
+ val fbetw_tac =
+ CONJ_WRAP' (K (EVERY' [rtac ballI, etac @{thm rev_bspec}, etac CollectE, atac])) alg_sets;
+ val mor_tac =
+ CONJ_WRAP' (fn thm => EVERY' [rtac ballI, rtac thm]) str_init_defs;
+ fun alg_epi_tac ((alg_set, str_init_def), set_natural') =
+ EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
+ rtac ballI, ftac (alg_select RS bspec), stac str_init_def, etac alg_set,
+ REPEAT_DETERM o FIRST' [rtac @{thm subset_UNIV},
+ EVERY' [rtac @{thm ord_eq_le_trans}, resolve_tac set_natural', rtac @{thm subset_trans},
+ etac @{thm image_mono}, rtac @{thm image_Collect_subsetI}, etac bspec, atac]]];
+ in
+ (rtac mor_cong THEN' REPEAT_DETERM_N n o (rtac sym THEN' rtac @{thm o_id}) THEN'
+ rtac (Thm.permute_prems 0 1 mor_comp) THEN' etac (Thm.permute_prems 0 1 mor_comp) THEN'
+ stac mor_def THEN' rtac conjI THEN' fbetw_tac THEN' mor_tac THEN' rtac mor_incl_min_alg THEN'
+ stac alg_def THEN' CONJ_WRAP' alg_epi_tac ((alg_sets ~~ str_init_defs) ~~ set_natural's)) 1
+ end;
+
+fun mk_init_ex_mor_tac Abs_inverse copy_alg_ex alg_min_alg card_of_min_algs
+ mor_comp mor_select mor_incl_min_alg {context = ctxt, prems = _} =
+ let
+ val n = length card_of_min_algs;
+ val card_of_ordIso_tac = EVERY' [rtac ssubst, rtac @{thm card_of_ordIso},
+ rtac @{thm ordIso_symmetric}, rtac conjunct1, rtac conjunct2, atac];
+ fun internalize_tac card_of = EVERY' [rtac subst, rtac @{thm internalize_card_of_ordLeq2},
+ rtac @{thm ordLeq_ordIso_trans}, rtac card_of, rtac subst,
+ rtac @{thm Card_order_iff_ordIso_card_of}, rtac @{thm Card_order_cexp}];
+ in
+ (rtac rev_mp THEN'
+ REPEAT_DETERM_N (2 * n) o (rtac mp THEN' rtac @{thm ex_mono} THEN' rtac impI) THEN'
+ REPEAT_DETERM_N (n + 1) o etac thin_rl THEN' rtac (alg_min_alg RS copy_alg_ex) THEN'
+ REPEAT_DETERM_N n o atac THEN'
+ REPEAT_DETERM_N n o card_of_ordIso_tac THEN'
+ EVERY' (map internalize_tac card_of_min_algs) THEN'
+ rtac impI THEN'
+ REPEAT_DETERM o eresolve_tac [exE, conjE] THEN'
+ REPEAT_DETERM o rtac exI THEN'
+ rtac mor_select THEN' atac THEN' rtac CollectI THEN'
+ REPEAT_DETERM o rtac exI THEN'
+ rtac conjI THEN' rtac refl THEN' atac THEN'
+ K (Local_Defs.unfold_tac ctxt (Abs_inverse :: @{thms fst_conv snd_conv})) THEN'
+ etac mor_comp THEN' etac mor_incl_min_alg) 1
+ end;
+
+fun mk_init_unique_mor_tac m
+ alg_def alg_min_alg least_min_algs in_monos alg_sets morEs map_congs =
+ let
+ val n = length least_min_algs;
+ val ks = (1 upto n);
+
+ fun mor_tac morE in_mono = EVERY' [etac morE, rtac @{thm set_mp}, rtac in_mono,
+ REPEAT_DETERM_N n o rtac @{thm Collect_restrict}, rtac CollectI,
+ REPEAT_DETERM_N (m + n) o (TRY o rtac conjI THEN' atac)];
+ fun cong_tac map_cong = EVERY' [rtac (map_cong RS arg_cong),
+ REPEAT_DETERM_N m o rtac refl,
+ REPEAT_DETERM_N n o (etac @{thm prop_restrict} THEN' atac)];
+
+ fun mk_alg_tac (alg_set, (in_mono, (morE, map_cong))) = EVERY' [rtac ballI, rtac CollectI,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac (alg_min_alg RS alg_set),
+ REPEAT_DETERM_N m o rtac @{thm subset_UNIV},
+ REPEAT_DETERM_N n o (etac @{thm subset_trans} THEN' rtac @{thm Collect_restrict}),
+ rtac trans, mor_tac morE in_mono,
+ rtac trans, cong_tac map_cong,
+ rtac sym, mor_tac morE in_mono];
+
+ fun mk_unique_tac (k, least_min_alg) =
+ select_prem_tac n (etac @{thm prop_restrict}) k THEN' rtac least_min_alg THEN'
+ stac alg_def THEN'
+ CONJ_WRAP' mk_alg_tac (alg_sets ~~ (in_monos ~~ (morEs ~~ map_congs)));
+ in
+ CONJ_WRAP' mk_unique_tac (ks ~~ least_min_algs) 1
+ end;
+
+fun mk_init_induct_tac m alg_def alg_min_alg least_min_algs alg_sets =
+ let
+ val n = length least_min_algs;
+
+ fun mk_alg_tac alg_set = EVERY' [rtac ballI, rtac CollectI,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac (alg_min_alg RS alg_set),
+ REPEAT_DETERM_N m o rtac @{thm subset_UNIV},
+ REPEAT_DETERM_N n o (etac @{thm subset_trans} THEN' rtac @{thm Collect_restrict}),
+ rtac mp, etac bspec, rtac CollectI,
+ REPEAT_DETERM_N m o (rtac conjI THEN' atac),
+ CONJ_WRAP' (K (etac @{thm subset_trans} THEN' rtac @{thm Collect_restrict})) alg_sets,
+ CONJ_WRAP' (K (rtac ballI THEN' etac @{thm prop_restrict} THEN' atac)) alg_sets];
+
+ fun mk_induct_tac least_min_alg =
+ rtac ballI THEN' etac @{thm prop_restrict} THEN' rtac least_min_alg THEN'
+ stac alg_def THEN'
+ CONJ_WRAP' mk_alg_tac alg_sets;
+ in
+ CONJ_WRAP' mk_induct_tac least_min_algs 1
+ end;
+
+fun mk_mor_Rep_tac fld_defs copy bijs inver_Abss inver_Reps {context = ctxt, prems = _} =
+ (K (Local_Defs.unfold_tac ctxt fld_defs) THEN' rtac conjunct1 THEN' rtac copy THEN'
+ EVERY' (map (fn bij => EVERY' [rtac bij, atac, etac bexI, rtac UNIV_I]) bijs) THEN'
+ EVERY' (map rtac inver_Abss) THEN'
+ EVERY' (map rtac inver_Reps)) 1;
+
+fun mk_mor_Abs_tac inv inver_Abss inver_Reps =
+ (rtac inv THEN'
+ EVERY' (map2 (fn inver_Abs => fn inver_Rep =>
+ EVERY' [rtac conjI, rtac @{thm subset_UNIV}, rtac conjI, rtac inver_Rep, rtac inver_Abs])
+ inver_Abss inver_Reps)) 1;
+
+fun mk_mor_iter_tac cT ct iter_defs ex_mor mor =
+ (EVERY' (map stac iter_defs) THEN' EVERY' [rtac rev_mp, rtac ex_mor, rtac impI] THEN'
+ REPEAT_DETERM_N (length iter_defs) o etac exE THEN'
+ rtac (Drule.instantiate' [SOME cT] [SOME ct] @{thm someI}) THEN' etac mor) 1;
+
+fun mk_iter_unique_mor_tac type_defs init_unique_mors subsets Reps mor_comp mor_Abs mor_iter =
+ let
+ fun mk_subset subset Rep = etac subset ORELSE' rtac (Rep RS subset);
+ fun mk_unique type_def =
+ EVERY' [rtac @{thm surj_fun_eq}, rtac (type_def RS @{thm type_definition.Abs_image}),
+ rtac ballI, resolve_tac init_unique_mors, EVERY' (map2 mk_subset subsets Reps),
+ rtac mor_comp, rtac mor_Abs, atac,
+ rtac mor_comp, rtac mor_Abs, rtac mor_iter];
+ in
+ CONJ_WRAP' mk_unique type_defs 1
+ end;
+
+fun mk_unf_o_fld_tac unf_def iter map_comp_id map_congL fld_o_iters =
+ EVERY' [stac unf_def, rtac ext, rtac trans, rtac o_apply, rtac trans, rtac iter,
+ rtac trans, rtac map_comp_id, rtac trans, rtac map_congL,
+ EVERY' (map (fn thm =>
+ rtac ballI THEN' rtac (trans OF [thm RS fun_cong, @{thm id_apply}])) fld_o_iters),
+ rtac sym, rtac @{thm id_apply}] 1;
+
+fun mk_rec_tac rec_defs iter fst_recs {context = ctxt, prems = _}=
+ Local_Defs.unfold_tac ctxt
+ (rec_defs @ map (fn thm => thm RS @{thm convol_expand_snd}) fst_recs) THEN
+ EVERY' [rtac trans, rtac o_apply, rtac trans, rtac (iter RS @{thm arg_cong[of _ _ snd]}),
+ rtac @{thm snd_convol'}] 1;
+
+fun mk_fld_induct_tac m set_natural'ss init_induct morEs mor_Abs Rep_invs Abs_invs Reps
+ subset1s subset2s =
+ let
+ val n = length set_natural'ss;
+ val ks = 1 upto n;
+
+ fun mk_IH_tac Rep_inv Abs_inv set_natural' subset =
+ DETERM o EVERY' [dtac @{thm meta_mp}, rtac (Rep_inv RS arg_cong RS subst), etac bspec,
+ dtac @{thm set_rev_mp}, rtac equalityD1, rtac set_natural', etac imageE,
+ hyp_subst_tac, rtac (Abs_inv RS ssubst), rtac subset, etac @{thm set_mp},
+ atac, atac];
+
+ fun mk_closed_tac (k, (morE, set_natural's)) =
+ EVERY' [select_prem_tac n (dtac asm_rl) k, rtac ballI, rtac impI,
+ rtac (mor_Abs RS morE RS arg_cong RS ssubst), atac,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE], dtac @{thm meta_spec},
+ EVERY' (map4 mk_IH_tac Rep_invs Abs_invs (drop m set_natural's) subset1s), atac];
+
+ fun mk_induct_tac ((Rep, Rep_inv), subset) =
+ EVERY' [rtac (Rep_inv RS arg_cong RS subst), etac (Rep RS subset RSN (2, bspec))];
+ in
+ (rtac mp THEN' rtac impI THEN'
+ DETERM o CONJ_WRAP_GEN' (etac conjE THEN' rtac conjI) mk_induct_tac
+ ((Reps ~~ Rep_invs) ~~ subset2s) THEN'
+ rtac init_induct THEN'
+ DETERM o CONJ_WRAP' mk_closed_tac
+ (ks ~~ (morEs ~~ set_natural'ss))) 1
+ end;
+
+fun mk_fld_induct2_tac cTs cts fld_induct weak_fld_inducts {context = ctxt, prems = _} =
+ let
+ val n = length weak_fld_inducts;
+ val ks = 1 upto n;
+ fun mk_inner_induct_tac induct i =
+ EVERY' [rtac allI, fo_rtac induct ctxt,
+ select_prem_tac n (dtac @{thm meta_spec2}) i,
+ REPEAT_DETERM_N n o
+ EVERY' [dtac @{thm meta_mp} THEN_ALL_NEW Goal.norm_hhf_tac,
+ REPEAT_DETERM o dtac @{thm meta_spec}, etac (spec RS @{thm meta_mp}), atac],
+ atac];
+ in
+ EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts fld_induct),
+ EVERY' (map2 mk_inner_induct_tac weak_fld_inducts ks), rtac impI,
+ REPEAT_DETERM o eresolve_tac [conjE, allE],
+ CONJ_WRAP' (K atac) ks] 1
+ end;
+
+fun mk_map_tac m n iter map_comp_id map_cong =
+ EVERY' [rtac ext, rtac trans, rtac o_apply, rtac trans, rtac iter, rtac trans, rtac o_apply,
+ rtac trans, rtac (map_comp_id RS arg_cong), rtac trans, rtac (map_cong RS arg_cong),
+ REPEAT_DETERM_N m o rtac refl,
+ REPEAT_DETERM_N n o (EVERY' (map rtac [trans, o_apply, @{thm id_apply}])),
+ rtac sym, rtac o_apply] 1;
+
+fun mk_map_unique_tac m mor_def iter_unique_mor map_comp_ids map_congs =
+ let
+ val n = length map_congs;
+ fun mk_mor (comp_id, cong) = EVERY' [rtac ballI, rtac trans, etac @{thm pointfreeE},
+ rtac sym, rtac trans, rtac o_apply, rtac trans, rtac (comp_id RS arg_cong),
+ rtac (cong RS arg_cong),
+ REPEAT_DETERM_N m o rtac refl,
+ REPEAT_DETERM_N n o (EVERY' (map rtac [trans, o_apply, @{thm id_apply}]))];
+ in
+ EVERY' [rtac iter_unique_mor, rtac ssubst, rtac mor_def, rtac conjI,
+ CONJ_WRAP' (K (EVERY' [rtac ballI, rtac UNIV_I])) map_congs,
+ CONJ_WRAP' mk_mor (map_comp_ids ~~ map_congs)] 1
+ end;
+
+fun mk_set_tac iter = EVERY' [rtac ext, rtac trans, rtac o_apply,
+ rtac trans, rtac iter, rtac sym, rtac o_apply] 1;
+
+fun mk_set_simp_tac set set_natural' set_natural's =
+ let
+ val n = length set_natural's;
+ fun mk_UN thm = rtac (thm RS @{thm arg_cong[of _ _ Union]} RS trans) THEN'
+ rtac @{thm Union_image_eq};
+ in
+ EVERY' [rtac (set RS @{thm pointfreeE} RS trans), rtac @{thm Un_cong},
+ rtac (trans OF [set_natural', @{thm trans[OF fun_cong[OF image_id] id_apply]}]),
+ REPEAT_DETERM_N (n - 1) o rtac @{thm Un_cong},
+ EVERY' (map mk_UN set_natural's)] 1
+ end;
+
+fun mk_set_nat_tac m induct_tac set_natural'ss
+ map_simps csets set_simps i {context = ctxt, prems = _} =
+ let
+ val n = length map_simps;
+
+ fun useIH set_nat = EVERY' [rtac trans, rtac @{thm image_UN}, rtac trans, rtac @{thm UN_cong},
+ rtac refl, Goal.assume_rule_tac ctxt, rtac sym, rtac trans, rtac @{thm UN_cong},
+ rtac set_nat, rtac refl, rtac @{thm UN_simps(10)}];
+
+ fun mk_set_nat cset map_simp set_simp set_nats =
+ EVERY' [rtac trans, rtac @{thm image_cong}, rtac set_simp, rtac refl,
+ rtac sym, rtac (trans OF [map_simp RS HOL_arg_cong cset, set_simp RS trans]),
+ rtac sym, EVERY' (map rtac (trans :: @{thms image_Un Un_cong})),
+ rtac sym, rtac (nth set_nats (i - 1)),
+ REPEAT_DETERM_N (n - 1) o EVERY' (map rtac (trans :: @{thms image_Un Un_cong})),
+ EVERY' (map useIH (drop m set_nats))];
+ in
+ (induct_tac THEN' EVERY' (map4 mk_set_nat csets map_simps set_simps set_natural'ss)) 1
+ end;
+
+fun mk_set_bd_tac m induct_tac bd_Cinfinite set_bdss set_simps i {context = ctxt, prems = _} =
+ let
+ val n = length set_simps;
+
+ fun useIH set_bd = EVERY' [rtac @{thm UNION_Cinfinite_bound}, rtac set_bd, rtac ballI,
+ Goal.assume_rule_tac ctxt, rtac bd_Cinfinite];
+
+ fun mk_set_nat set_simp set_bds =
+ EVERY' [rtac @{thm ordIso_ordLeq_trans}, rtac @{thm card_of_ordIso_subst}, rtac set_simp,
+ rtac (bd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})), rtac (nth set_bds (i - 1)),
+ REPEAT_DETERM_N (n - 1) o rtac (bd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})),
+ EVERY' (map useIH (drop m set_bds))];
+ in
+ (induct_tac THEN' EVERY' (map2 mk_set_nat set_simps set_bdss)) 1
+ end;
+
+fun mk_mcong_tac induct_tac set_setsss map_congs map_simps {context = ctxt, prems = _} =
+ let
+ fun use_asm thm = EVERY' [etac bspec, etac @{thm set_rev_mp}, rtac thm];
+
+ fun useIH set_sets = EVERY' [rtac mp, Goal.assume_rule_tac ctxt,
+ CONJ_WRAP' (fn thm =>
+ EVERY' [rtac ballI, etac bspec, etac @{thm set_rev_mp}, etac thm]) set_sets];
+
+ fun mk_map_cong map_simp map_cong set_setss =
+ EVERY' [rtac impI, REPEAT_DETERM o etac conjE,
+ rtac trans, rtac map_simp, rtac trans, rtac (map_cong RS arg_cong),
+ EVERY' (map use_asm (map hd set_setss)),
+ EVERY' (map useIH (transpose (map tl set_setss))),
+ rtac sym, rtac map_simp];
+ in
+ (induct_tac THEN' EVERY' (map3 mk_map_cong map_simps map_congs set_setsss)) 1
+ end;
+
+fun mk_incl_min_alg_tac induct_tac set_setsss alg_sets alg_min_alg {context = ctxt, prems = _} =
+ let
+ fun use_asm thm = etac (thm RS subset_trans);
+
+ fun useIH set_sets = EVERY' [rtac subsetI, rtac mp, Goal.assume_rule_tac ctxt,
+ rtac CollectI, CONJ_WRAP' (fn thm => EVERY' [etac (thm RS subset_trans), atac]) set_sets];
+
+ fun mk_incl alg_set set_setss =
+ EVERY' [rtac impI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ rtac (alg_min_alg RS alg_set),
+ EVERY' (map use_asm (map hd set_setss)),
+ EVERY' (map useIH (transpose (map tl set_setss)))];
+ in
+ (induct_tac THEN' EVERY' (map2 mk_incl alg_sets set_setsss)) 1
+ end;
+
+fun mk_lfp_map_wpull_tac m induct_tac wpulls map_simps set_simpss set_setsss fld_injects =
+ let
+ val n = length wpulls;
+ val ks = 1 upto n;
+ val ls = 1 upto m;
+
+ fun use_pass_asm thm = rtac conjI THEN' etac (thm RS subset_trans);
+ fun use_act_asm thm = etac (thm RS subset_trans) THEN' atac;
+
+ fun useIH set_sets i = EVERY' [rtac ssubst, rtac @{thm wpull_def},
+ REPEAT_DETERM_N m o etac thin_rl, select_prem_tac n (dtac asm_rl) i,
+ rtac allI, rtac allI, rtac impI, REPEAT_DETERM o etac conjE,
+ REPEAT_DETERM o dtac @{thm meta_spec},
+ dtac @{thm meta_mp}, atac,
+ dtac @{thm meta_mp}, atac, etac mp,
+ rtac conjI, rtac CollectI, CONJ_WRAP' use_act_asm set_sets,
+ rtac conjI, rtac CollectI, CONJ_WRAP' use_act_asm set_sets,
+ atac];
+
+ fun mk_subset thm = EVERY' [rtac @{thm ord_eq_le_trans}, rtac thm, rtac @{thm Un_least}, atac,
+ REPEAT_DETERM_N (n - 1) o rtac @{thm Un_least},
+ REPEAT_DETERM_N n o
+ EVERY' [rtac @{thm UN_least}, rtac CollectE, etac @{thm set_rev_mp}, atac,
+ REPEAT_DETERM o etac conjE, atac]];
+
+ fun mk_wpull wpull map_simp set_simps set_setss fld_inject =
+ EVERY' [rtac impI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
+ rtac rev_mp, rtac wpull,
+ EVERY' (map (fn i => REPEAT_DETERM_N (i - 1) o etac thin_rl THEN' atac) ls),
+ EVERY' (map2 useIH (transpose (map tl set_setss)) ks),
+ rtac impI, REPEAT_DETERM_N (m + n) o etac thin_rl,
+ dtac @{thm subst[OF wpull_def, of "%x. x"]}, etac allE, etac allE, etac impE,
+ rtac conjI, rtac CollectI, EVERY' (map (use_pass_asm o hd) set_setss),
+ CONJ_WRAP' (K (rtac subset_refl)) ks,
+ rtac conjI, rtac CollectI, EVERY' (map (use_pass_asm o hd) set_setss),
+ CONJ_WRAP' (K (rtac subset_refl)) ks,
+ rtac subst, rtac fld_inject, rtac trans, rtac sym, rtac map_simp,
+ rtac trans, atac, rtac map_simp, REPEAT_DETERM o eresolve_tac [CollectE, conjE, bexE],
+ hyp_subst_tac, rtac bexI, rtac conjI, rtac map_simp, rtac map_simp, rtac CollectI,
+ CONJ_WRAP' mk_subset set_simps];
+ in
+ (induct_tac THEN' EVERY' (map5 mk_wpull wpulls map_simps set_simpss set_setsss fld_injects)) 1
+ end;
+
+(* BNF tactics *)
+
+fun mk_map_id_tac map_ids unique =
+ (rtac sym THEN' rtac unique THEN'
+ EVERY' (map (fn thm =>
+ EVERY' [rtac trans, rtac @{thm id_o}, rtac trans, rtac sym, rtac @{thm o_id},
+ rtac (thm RS sym RS arg_cong)]) map_ids)) 1;
+
+fun mk_map_comp_tac map_comps map_simps unique iplus1 =
+ let
+ val i = iplus1 - 1;
+ val unique' = Thm.permute_prems 0 i unique;
+ val map_comps' = drop i map_comps @ take i map_comps;
+ val map_simps' = drop i map_simps @ take i map_simps;
+ fun mk_comp comp simp =
+ EVERY' [rtac ext, rtac trans, rtac o_apply, rtac trans, rtac o_apply,
+ rtac trans, rtac (simp RS arg_cong), rtac trans, rtac simp,
+ rtac trans, rtac (comp RS arg_cong), rtac sym, rtac o_apply];
+ in
+ (rtac sym THEN' rtac unique' THEN' EVERY' (map2 mk_comp map_comps' map_simps')) 1
+ end;
+
+fun mk_set_natural_tac set_nat =
+ EVERY' (map rtac [ext, trans, o_apply, sym, trans, o_apply, set_nat]) 1;
+
+fun mk_in_bd_tac sum_Card_order sucbd_Cnotzero incl card_of_min_alg =
+ EVERY' [rtac ctrans, rtac @{thm card_of_mono1}, rtac subsetI, etac rev_mp,
+ rtac incl, rtac ctrans, rtac card_of_min_alg, rtac @{thm cexp_mono2_Cnotzero},
+ rtac @{thm cardSuc_ordLeq_cpow}, rtac sum_Card_order, rtac @{thm csum_Cnotzero2},
+ rtac @{thm ctwo_Cnotzero}, rtac sucbd_Cnotzero] 1;
+
+fun mk_bd_card_order_tac bd_card_orders =
+ (rtac @{thm card_order_cpow} THEN'
+ CONJ_WRAP_GEN' (rtac @{thm card_order_csum}) rtac bd_card_orders) 1;
+
+fun mk_wpull_tac wpull =
+ EVERY' [rtac ssubst, rtac @{thm wpull_def}, rtac allI, rtac allI,
+ rtac wpull, REPEAT_DETERM o atac] 1;
+
+fun mk_wit_tac n set_simp wit =
+ REPEAT_DETERM (atac 1 ORELSE
+ EVERY' [dtac @{thm set_rev_mp}, rtac equalityD1, resolve_tac set_simp,
+ REPEAT_DETERM o
+ (TRY o REPEAT_DETERM o etac UnE THEN' TRY o etac @{thm UN_E} THEN'
+ (eresolve_tac wit ORELSE'
+ (dresolve_tac wit THEN'
+ (etac FalseE ORELSE'
+ EVERY' [hyp_subst_tac, dtac @{thm set_rev_mp}, rtac equalityD1, resolve_tac set_simp,
+ REPEAT_DETERM_N n o etac UnE]))))] 1);
+
+fun mk_rel_unfold_tac in_Irels i in_rel map_comp map_cong map_simp set_simps fld_inject
+ fld_unf set_naturals set_incls set_set_inclss =
+ let
+ val m = length set_incls;
+ val n = length set_set_inclss;
+
+ val (passive_set_naturals, active_set_naturals) = chop m set_naturals;
+ val in_Irel = nth in_Irels (i - 1);
+ val le_arg_cong_fld_unf = fld_unf RS arg_cong RS @{thm ord_eq_le_trans};
+ val eq_arg_cong_fld_unf = fld_unf RS arg_cong RS trans;
+ val if_tac =
+ EVERY' [dtac (in_Irel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
+ rtac (in_rel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
+ EVERY' (map2 (fn set_natural => fn set_incl =>
+ EVERY' [rtac conjI, rtac @{thm ord_eq_le_trans}, rtac set_natural,
+ rtac @{thm ord_eq_le_trans}, rtac @{thm trans[OF fun_cong[OF image_id] id_apply]},
+ rtac (set_incl RS @{thm subset_trans}), etac le_arg_cong_fld_unf])
+ passive_set_naturals set_incls),
+ CONJ_WRAP' (fn (in_Irel, (set_natural, set_set_incls)) =>
+ EVERY' [rtac @{thm ord_eq_le_trans}, rtac set_natural, rtac @{thm image_subsetI},
+ rtac (in_Irel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
+ CONJ_WRAP' (fn thm =>
+ EVERY' (map etac [thm RS @{thm subset_trans}, le_arg_cong_fld_unf]))
+ set_set_incls,
+ rtac conjI, rtac refl, rtac refl])
+ (in_Irels ~~ (active_set_naturals ~~ set_set_inclss)),
+ CONJ_WRAP' (fn conv =>
+ EVERY' [rtac trans, rtac map_comp, rtac trans, rtac map_cong,
+ REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
+ REPEAT_DETERM_N n o EVERY' (map rtac [trans, o_apply, conv]),
+ rtac (fld_inject RS iffD1), rtac trans, rtac sym, rtac map_simp,
+ etac eq_arg_cong_fld_unf])
+ @{thms fst_conv snd_conv}];
+ val only_if_tac =
+ EVERY' [dtac (in_rel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
+ rtac (in_Irel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
+ CONJ_WRAP' (fn (set_simp, passive_set_natural) =>
+ EVERY' [rtac @{thm ord_eq_le_trans}, rtac set_simp, rtac @{thm Un_least},
+ rtac @{thm ord_eq_le_trans}, rtac @{thm box_equals[OF _ refl]},
+ rtac passive_set_natural, rtac @{thm trans[OF fun_cong[OF image_id] id_apply]},
+ atac,
+ CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
+ (fn (active_set_natural, in_Irel) => EVERY' [rtac @{thm ord_eq_le_trans},
+ rtac @{thm UN_cong[OF _ refl]}, rtac active_set_natural, rtac @{thm UN_least},
+ dtac @{thm set_rev_mp}, etac @{thm image_mono}, etac imageE,
+ dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Irel RS iffD1),
+ dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE,
+ dtac (Thm.permute_prems 0 1 @{thm ssubst_mem}), atac,
+ hyp_subst_tac, REPEAT_DETERM o eresolve_tac [CollectE, conjE], atac])
+ (rev (active_set_naturals ~~ in_Irels))])
+ (set_simps ~~ passive_set_naturals),
+ rtac conjI,
+ REPEAT_DETERM_N 2 o EVERY'[rtac trans, rtac map_simp, rtac (fld_inject RS iffD2),
+ rtac trans, rtac map_comp, rtac trans, rtac map_cong,
+ REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
+ EVERY' (map (fn in_Irel => EVERY' [rtac trans, rtac o_apply, dtac @{thm set_rev_mp}, atac,
+ dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Irel RS iffD1), dtac @{thm someI_ex},
+ REPEAT_DETERM o etac conjE, atac]) in_Irels),
+ atac]]
+ in
+ EVERY' [rtac iffI, if_tac, only_if_tac] 1
+ end;
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_lfp_util.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,76 @@
+(* Title: Codatatype_Tools/bnf_lfp_util.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+Library for the datatype construction.
+*)
+
+signature BNF_LFP_UTIL =
+sig
+ val HOL_arg_cong: cterm -> thm
+
+ val mk_bij_betw: term -> term -> term -> term
+ val mk_cardSuc: term -> term
+ val mk_convol: term * term -> term
+ val mk_cpow: term -> term
+ val mk_inver: term -> term -> term -> term
+ val mk_not_empty: term -> term
+ val mk_not_eq: term -> term -> term
+ val mk_rapp: term -> typ -> term
+ val mk_relChain: term -> term -> term
+ val mk_underS: term -> term
+ val mk_worec: term -> term -> term
+end;
+
+structure BNF_LFP_Util : BNF_LFP_UTIL =
+struct
+
+open BNF_Util
+
+fun HOL_arg_cong ct = Drule.instantiate'
+ (map SOME (Thm.dest_ctyp (Thm.ctyp_of_term ct))) [NONE, NONE, SOME ct] arg_cong;
+
+(*reverse application*)
+fun mk_rapp arg T = Term.absdummy (fastype_of arg --> T) (Bound 0 $ arg);
+
+fun mk_underS r =
+ let val T = fst (dest_relT (fastype_of r));
+ in Const (@{const_name rel.underS}, mk_relT (T, T) --> T --> HOLogic.mk_setT T) $ r end;
+
+fun mk_worec r f =
+ let val (A, AB) = apfst domain_type (dest_funT (fastype_of f));
+ in Const (@{const_name wo_rel.worec}, mk_relT (A, A) --> (AB --> AB) --> AB) $ r $ f end;
+
+fun mk_relChain r f =
+ let val (A, AB) = `domain_type (fastype_of f);
+ in Const (@{const_name relChain}, mk_relT (A, A) --> AB --> HOLogic.boolT) $ r $ f end;
+
+fun mk_cardSuc r =
+ let val T = fst (dest_relT (fastype_of r));
+ in Const (@{const_name cardSuc}, mk_relT (T, T) --> mk_relT (`I (HOLogic.mk_setT T))) $ r end;
+
+fun mk_cpow r =
+ let val T = fst (dest_relT (fastype_of r));
+ in Const (@{const_name cpow}, mk_relT (T, T) --> mk_relT (`I (HOLogic.mk_setT T))) $ r end;
+
+fun mk_bij_betw f A B =
+ Const (@{const_name bij_betw},
+ fastype_of f --> fastype_of A --> fastype_of B --> HOLogic.boolT) $ f $ A $ B;
+
+fun mk_inver f g A =
+ Const (@{const_name inver}, fastype_of f --> fastype_of g --> fastype_of A --> HOLogic.boolT) $
+ f $ g $ A;
+
+fun mk_not_eq x y = HOLogic.mk_not (HOLogic.mk_eq (x, y));
+
+fun mk_not_empty B = mk_not_eq B (HOLogic.mk_set (HOLogic.dest_setT (fastype_of B)) []);
+
+fun mk_convol (f, g) =
+ let
+ val (fU, fTU) = `range_type (fastype_of f);
+ val ((gT, gU), gTU) = `dest_funT (fastype_of g);
+ val convolT = fTU --> gTU --> gT --> HOLogic.mk_prodT (fU, gU);
+ in Const (@{const_name convol}, convolT) $ f $ g end;
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_tactics.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,317 @@
+(* Title: HOL/Codatatype/Tools/bnf_tactics.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Jasmin Blanchette, TU Muenchen
+ Copyright 2012
+
+General tactics for bounded natural functors.
+*)
+
+signature BNF_TACTICS =
+sig
+ val select_prem_tac: int -> (int -> tactic) -> int -> int -> tactic
+ val fo_rtac: thm -> Proof.context -> int -> tactic
+ val subst_tac: Proof.context -> thm list -> int -> tactic
+ val substs_tac: Proof.context -> thm list -> int -> tactic
+
+ val mk_flatten_assoc_tac: (int -> tactic) -> thm -> thm -> thm -> tactic
+ val mk_rotate_eq_tac: (int -> tactic) -> thm -> thm -> thm -> thm -> ''a list -> ''a list ->
+ int -> tactic
+
+ val mk_Abs_bij_thm: Proof.context -> thm -> thm -> thm -> thm
+ val mk_Abs_inj_thm: thm -> thm -> thm
+ val mk_Abs_inverse_thm: thm -> thm -> thm
+ val mk_T_I: thm -> thm
+ val mk_T_subset1: thm -> thm
+ val mk_T_subset2: thm -> thm
+
+ val mk_Card_order_tac: thm -> tactic
+ val mk_collect_set_natural_tac: Proof.context -> thm list -> tactic
+ val mk_id': thm -> thm
+ val mk_comp': thm -> thm
+ val mk_in_mono_tac: int -> tactic
+ val mk_map_comp_id_tac: thm -> tactic
+ val mk_map_cong_tac: int -> thm -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_map_congL_tac: int -> thm -> thm -> tactic
+ val mk_map_wppull_tac: thm -> thm -> thm -> thm -> thm list -> tactic
+ val mk_map_wpull_tac: thm -> thm list -> thm -> tactic
+ val mk_set_natural': thm -> thm
+ val mk_simple_wit_tac: thm list -> tactic
+
+ val mk_pred_unfold_tac: thm -> thm list -> thm -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_rel_Gr_tac: thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm list ->
+ {prems: thm list, context: Proof.context} -> tactic
+ val mk_rel_Id_tac: int -> thm -> thm -> {prems: 'a, context: Proof.context} -> tactic
+ val mk_rel_O_tac: thm -> thm -> thm -> thm -> thm -> thm list ->
+ {prems: thm list, context: Proof.context} -> tactic
+ val mk_in_rel_tac: thm -> int -> {prems: 'b, context: Proof.context} -> tactic
+ val mk_rel_converse_tac: thm -> tactic
+ val mk_rel_converse_le_tac: thm -> thm -> thm -> thm -> thm list ->
+ {prems: thm list, context: Proof.context} -> tactic
+ val mk_rel_mono_tac: thm -> thm -> {prems: 'a, context: Proof.context} -> tactic
+end;
+
+structure BNF_Tactics : BNF_TACTICS =
+struct
+
+open BNF_Util
+
+val set_mp = @{thm set_mp};
+
+fun select_prem_tac n tac k = DETERM o (EVERY' [REPEAT_DETERM_N (k - 1) o etac thin_rl,
+ tac, REPEAT_DETERM_N (n - k) o etac thin_rl]);
+
+(*stolen from Christian Urban's Cookbook*)
+fun fo_rtac thm = Subgoal.FOCUS (fn {concl, ...} =>
+ let
+ val concl_pat = Drule.strip_imp_concl (cprop_of thm)
+ val insts = Thm.first_order_match (concl_pat, concl)
+ in
+ rtac (Drule.instantiate_normalize insts thm) 1
+ end);
+
+(*unlike "unfold_tac", succeeds when the RHS contains schematic variables not in the LHS*)
+fun subst_tac ctxt = EqSubst.eqsubst_tac ctxt [0];
+fun substs_tac ctxt = REPEAT_DETERM oo subst_tac ctxt;
+
+
+
+(* Theorems for typedefs with UNIV as representing set *)
+
+(*equivalent to UNIV_I for the representing set of the particular type T*)
+fun mk_T_subset1 def = set_mp OF [def RS meta_eq_to_obj_eq RS equalityD2];
+fun mk_T_subset2 def = set_mp OF [def RS meta_eq_to_obj_eq RS equalityD1];
+fun mk_T_I def = UNIV_I RS mk_T_subset1 def;
+
+fun mk_Abs_inverse_thm def inv = mk_T_I def RS inv;
+fun mk_Abs_inj_thm def inj = inj OF (replicate 2 (mk_T_I def));
+fun mk_Abs_bij_thm ctxt def inj surj = rule_by_tactic ctxt ((rtac surj THEN' etac exI) 1)
+ (mk_Abs_inj_thm def inj RS @{thm bijI});
+
+
+
+(* General tactic generators *)
+
+(*applies assoc rule to the lhs of an equation as long as possible*)
+fun mk_flatten_assoc_tac refl_tac trans assoc cong = rtac trans 1 THEN
+ REPEAT_DETERM (CHANGED ((FIRST' [rtac trans THEN' rtac assoc, rtac cong THEN' refl_tac]) 1)) THEN
+ refl_tac 1;
+
+(*proves two sides of an equation to be equal assuming both are flattened and rhs can be obtained
+from lhs by the given permutation of monoms*)
+fun mk_rotate_eq_tac refl_tac trans assoc com cong =
+ let
+ fun gen_tac [] [] = K all_tac
+ | gen_tac [x] [y] = if x = y then refl_tac else error "mk_rotate_eq_tac: different lists"
+ | gen_tac (x :: xs) (y :: ys) = if x = y
+ then rtac cong THEN' refl_tac THEN' gen_tac xs ys
+ else rtac trans THEN' rtac com THEN'
+ K (mk_flatten_assoc_tac refl_tac trans assoc cong) THEN'
+ gen_tac (xs @ [x]) (y :: ys)
+ | gen_tac _ _ = error "mk_rotate_eq_tac: different lists";
+ in
+ gen_tac
+ end;
+
+fun mk_Card_order_tac card_order =
+ (rtac conjE THEN'
+ rtac @{thm card_order_on_Card_order} THEN'
+ rtac card_order THEN'
+ atac) 1;
+
+fun mk_id' id = mk_trans (fun_cong OF [id]) @{thm id_apply};
+fun mk_comp' comp = @{thm o_eq_dest_lhs} OF [mk_sym comp];
+fun mk_set_natural' set_natural = set_natural RS @{thm pointfreeE};
+fun mk_in_mono_tac n = if n = 0 then rtac @{thm subset_UNIV} 1
+ else (rtac subsetI THEN'
+ rtac CollectI) 1 THEN
+ REPEAT_DETERM (eresolve_tac [CollectE, conjE] 1) THEN
+ REPEAT_DETERM_N (n - 1)
+ ((rtac conjI THEN' etac subset_trans THEN' atac) 1) THEN
+ (etac subset_trans THEN' atac) 1;
+
+fun mk_map_wpull_tac comp_in_alt inner_map_wpulls outer_map_wpull =
+ (rtac (@{thm wpull_cong} OF (replicate 3 comp_in_alt)) THEN' rtac outer_map_wpull) 1 THEN
+ WRAP (fn thm => rtac thm 1 THEN REPEAT_DETERM (atac 1)) (K all_tac) inner_map_wpulls all_tac THEN
+ TRY (REPEAT_DETERM (atac 1 ORELSE rtac @{thm wpull_id} 1));
+
+fun mk_collect_set_natural_tac ctxt set_naturals =
+ substs_tac ctxt (@{thms collect_o image_insert image_empty} @ set_naturals) 1 THEN rtac refl 1;
+
+fun mk_simple_wit_tac wit_thms = ALLGOALS (atac ORELSE' eresolve_tac (@{thm emptyE} :: wit_thms));
+
+fun mk_map_comp_id_tac map_comp =
+ (rtac trans THEN' rtac map_comp) 1 THEN
+ REPEAT_DETERM (stac @{thm o_id} 1) THEN
+ rtac refl 1;
+
+fun mk_map_congL_tac passive map_cong map_id' =
+ (rtac trans THEN' rtac map_cong THEN' EVERY' (replicate passive (rtac refl))) 1 THEN
+ REPEAT_DETERM (EVERY' [rtac trans, etac bspec, atac, rtac sym, rtac @{thm id_apply}] 1) THEN
+ rtac map_id' 1;
+
+fun mk_map_wppull_tac map_id map_cong map_wpull map_comp set_naturals =
+ if null set_naturals then
+ EVERY' [rtac @{thm wppull_id}, rtac map_wpull, rtac map_id, rtac map_id] 1
+ else EVERY' [REPEAT_DETERM o etac conjE, REPEAT_DETERM o dtac @{thm wppull_thePull},
+ REPEAT_DETERM o etac exE, rtac @{thm wpull_wppull}, rtac map_wpull,
+ REPEAT_DETERM o rtac @{thm wpull_thePull}, rtac ballI,
+ REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac CollectI,
+ CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
+ rtac @{thm image_subsetI}, rtac conjunct1, etac bspec, etac set_mp, atac])
+ set_naturals,
+ CONJ_WRAP' (fn thm => EVERY' [rtac (map_comp RS trans), rtac (map_comp RS trans),
+ rtac (map_comp RS trans RS sym), rtac map_cong,
+ REPEAT_DETERM_N (length set_naturals) o EVERY' [rtac (o_apply RS trans),
+ rtac (o_apply RS trans RS sym), rtac (o_apply RS trans), rtac thm,
+ rtac conjunct2, etac bspec, etac set_mp, atac]]) [conjunct1, conjunct2]] 1;
+
+fun mk_rel_Gr_tac rel_def map_id map_cong map_wpull in_cong map_id' map_comp set_naturals
+ {context = ctxt, prems = _} =
+ let
+ val n = length set_naturals;
+ in
+ if null set_naturals then
+ Local_Defs.unfold_tac ctxt [rel_def] THEN EVERY' [rtac @{thm Gr_UNIV_id}, rtac map_id] 1
+ else Local_Defs.unfold_tac ctxt [rel_def, @{thm Gr_def}] THEN
+ EVERY' [rtac equalityI, rtac subsetI,
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm relcompE}, @{thm converseE}],
+ REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ REPEAT_DETERM o etac conjE, hyp_subst_tac,
+ rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
+ rtac sym, rtac trans, rtac map_comp, rtac map_cong,
+ REPEAT_DETERM_N n o EVERY' [dtac @{thm set_rev_mp}, atac,
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ rtac (o_apply RS trans), rtac (@{thm fst_conv} RS arg_cong RS trans),
+ rtac (@{thm snd_conv} RS sym)],
+ rtac CollectI,
+ CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
+ rtac @{thm image_subsetI}, dtac @{thm set_rev_mp}, atac,
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
+ stac @{thm fst_conv}, atac]) set_naturals,
+ rtac @{thm subrelI}, etac CollectE, REPEAT_DETERM o eresolve_tac [exE, conjE],
+ REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ REPEAT_DETERM o etac conjE, hyp_subst_tac,
+ rtac allE, rtac subst, rtac @{thm wpull_def}, rtac map_wpull,
+ REPEAT_DETERM_N n o rtac @{thm wpull_Gr}, etac allE, etac impE, rtac conjI, atac,
+ rtac conjI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
+ CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
+ rtac @{thm image_mono}, atac]) set_naturals,
+ rtac sym, rtac map_id', REPEAT_DETERM o eresolve_tac [bexE, conjE],
+ rtac @{thm relcompI}, rtac @{thm converseI},
+ REPEAT_DETERM_N 2 o EVERY' [rtac CollectI, rtac exI,
+ rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl, etac sym,
+ etac @{thm set_rev_mp}, rtac equalityD1, rtac in_cong,
+ REPEAT_DETERM_N n o rtac @{thm Gr_def}]] 1
+ end;
+
+fun mk_rel_Id_tac n rel_Gr map_id {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt [rel_Gr, @{thm Id_alt}] THEN
+ subst_tac ctxt [map_id] 1 THEN
+ (if n = 0 then rtac refl 1
+ else EVERY' [rtac @{thm arg_cong2[of _ _ _ _ Gr]},
+ rtac equalityI, rtac @{thm subset_UNIV}, rtac subsetI, rtac CollectI,
+ CONJ_WRAP' (K (rtac @{thm subset_UNIV})) (1 upto n), rtac refl] 1);
+
+fun mk_rel_mono_tac rel_def in_mono {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt [rel_def] THEN
+ EVERY' [rtac @{thm relcomp_mono}, rtac @{thm iffD2[OF converse_mono]},
+ rtac @{thm Gr_mono}, rtac in_mono, REPEAT_DETERM o atac,
+ rtac @{thm Gr_mono}, rtac in_mono, REPEAT_DETERM o atac] 1;
+
+fun mk_map_cong_tac m map_cong {context = ctxt, prems = _} =
+ EVERY' [rtac mp, rtac map_cong,
+ CONJ_WRAP' (K (rtac ballI THEN' Goal.assume_rule_tac ctxt)) (1 upto m)] 1;
+
+fun mk_rel_converse_le_tac rel_def rel_Id map_cong map_comp set_naturals
+ {context = ctxt, prems = _} =
+ let
+ val n = length set_naturals;
+ in
+ if null set_naturals then
+ Local_Defs.unfold_tac ctxt [rel_Id] THEN rtac equalityD2 1 THEN rtac @{thm converse_Id} 1
+ else Local_Defs.unfold_tac ctxt [rel_def, @{thm Gr_def}] THEN
+ EVERY' [rtac @{thm subrelI},
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm relcompE}, @{thm converseE}],
+ REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ REPEAT_DETERM o etac conjE, hyp_subst_tac, rtac @{thm converseI},
+ rtac @{thm relcompI}, rtac @{thm converseI},
+ EVERY' (map (fn thm => EVERY' [rtac CollectI, rtac exI,
+ rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl, rtac trans,
+ rtac map_cong, REPEAT_DETERM_N n o rtac thm,
+ rtac (map_comp RS sym), rtac CollectI,
+ CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
+ etac @{thm flip_rel}]) set_naturals]) [@{thm snd_fst_flip}, @{thm fst_snd_flip}])] 1
+ end;
+
+fun mk_rel_converse_tac le_converse =
+ EVERY' [rtac equalityI, rtac le_converse, rtac @{thm xt1(6)}, rtac @{thm converse_shift},
+ rtac le_converse, REPEAT_DETERM o stac @{thm converse_converse}, rtac subset_refl] 1;
+
+fun mk_rel_O_tac rel_def rel_Id map_cong map_wppull map_comp set_naturals
+ {context = ctxt, prems = _} =
+ let
+ val n = length set_naturals;
+ fun in_tac nthO_in = rtac CollectI THEN'
+ CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
+ rtac @{thm image_subsetI}, rtac nthO_in, etac set_mp, atac]) set_naturals;
+ in
+ if null set_naturals then Local_Defs.unfold_tac ctxt [rel_Id] THEN rtac (@{thm Id_O_R} RS sym) 1
+ else Local_Defs.unfold_tac ctxt [rel_def, @{thm Gr_def}] THEN
+ EVERY' [rtac equalityI, rtac @{thm subrelI},
+ REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm relcompE}, @{thm converseE}],
+ REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ REPEAT_DETERM o etac conjE, hyp_subst_tac,
+ rtac @{thm relcompI}, rtac @{thm relcompI}, rtac @{thm converseI},
+ rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
+ rtac sym, rtac trans, rtac map_comp, rtac sym, rtac map_cong,
+ REPEAT_DETERM_N n o rtac @{thm fst_fstO},
+ in_tac @{thm fstO_in},
+ rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
+ rtac sym, rtac trans, rtac map_comp, rtac map_cong,
+ REPEAT_DETERM_N n o EVERY' [rtac trans, rtac o_apply, rtac ballE, rtac subst,
+ rtac @{thm csquare_def}, rtac @{thm csquare_fstO_sndO}, atac, etac notE,
+ etac set_mp, atac],
+ in_tac @{thm fstO_in},
+ rtac @{thm relcompI}, rtac @{thm converseI},
+ rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
+ rtac sym, rtac trans, rtac map_comp, rtac map_cong,
+ REPEAT_DETERM_N n o rtac o_apply,
+ in_tac @{thm sndO_in},
+ rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
+ rtac sym, rtac trans, rtac map_comp, rtac sym, rtac map_cong,
+ REPEAT_DETERM_N n o rtac @{thm snd_sndO},
+ in_tac @{thm sndO_in},
+ rtac @{thm subrelI},
+ REPEAT_DETERM o eresolve_tac [CollectE, @{thm relcompE}, @{thm converseE}],
+ REPEAT_DETERM o eresolve_tac [exE, conjE],
+ REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
+ REPEAT_DETERM o etac conjE, hyp_subst_tac,
+ rtac allE, rtac subst, rtac @{thm wppull_def}, rtac map_wppull,
+ CONJ_WRAP' (K (rtac @{thm wppull_fstO_sndO})) set_naturals,
+ etac allE, etac impE, etac conjI, etac conjI, atac,
+ REPEAT_DETERM o eresolve_tac [bexE, conjE],
+ rtac @{thm relcompI}, rtac @{thm converseI},
+ EVERY' (map (fn thm => EVERY' [rtac CollectI, rtac exI,
+ rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl, rtac sym, rtac trans,
+ rtac trans, rtac map_cong, REPEAT_DETERM_N n o rtac thm,
+ rtac (map_comp RS sym), atac, atac]) [@{thm fst_fstO}, @{thm snd_sndO}])] 1
+ end;
+
+fun mk_in_rel_tac rel_def m {context = ctxt, prems = _} =
+ let
+ val ls' = replicate (Int.max (1, m)) ();
+ in
+ Local_Defs.unfold_tac ctxt (rel_def ::
+ @{thms Gr_def converse_unfold relcomp_unfold mem_Collect_eq prod.cases Pair_eq}) THEN
+ EVERY' [rtac iffI, REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac, rtac exI,
+ rtac conjI, CONJ_WRAP' (K atac) ls', rtac conjI, rtac refl, rtac refl,
+ REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI, rtac conjI,
+ REPEAT_DETERM_N 2 o EVERY' [rtac exI, rtac conjI, etac @{thm conjI[OF refl sym]},
+ CONJ_WRAP' (K atac) ls']] 1
+ end;
+
+fun mk_pred_unfold_tac pred_def pred_defs rel_unfold {context = ctxt, prems = _} =
+ Local_Defs.unfold_tac ctxt (@{thm mem_Collect_eq_split} :: pred_def :: pred_defs) THEN
+ rtac rel_unfold 1;
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Codatatype/Tools/bnf_util.ML Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,508 @@
+(* Title: HOL/Codatatype/Tools/bnf_util.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+General library functions.
+*)
+
+signature BNF_UTIL =
+sig
+ val map3: ('a -> 'b -> 'c -> 'd) -> 'a list -> 'b list -> 'c list -> 'd list
+ val map4: ('a -> 'b -> 'c -> 'd -> 'e) -> 'a list -> 'b list -> 'c list -> 'd list -> 'e list
+ val map5: ('a -> 'b -> 'c -> 'd -> 'e -> 'f) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list
+ val map6: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list
+ val map7: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list
+ val map8: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list -> 'i list
+ val map9: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->
+ 'i list -> 'j list
+ val map10: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->
+ 'i list -> 'j list -> 'k list
+ val map11: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k -> 'l) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->
+ 'i list -> 'j list -> 'k list -> 'l list
+ val map12: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k -> 'l -> 'm) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->
+ 'i list -> 'j list -> 'k list -> 'l list -> 'm list
+ val fold_map2: ('a -> 'b -> 'c -> 'd * 'c) -> 'a list -> 'b list -> 'c -> 'd list * 'c
+ val fold_map3: ('a -> 'b -> 'c -> 'd -> 'e * 'd) ->
+ 'a list -> 'b list -> 'c list -> 'd -> 'e list * 'd
+ val fold_map4: ('a -> 'b -> 'c -> 'd -> 'e -> 'f * 'e) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e -> 'f list * 'e
+ val fold_map5: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g * 'f) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f -> 'g list * 'f
+ val fold_map6: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h * 'g) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g -> 'h list * 'g
+ val fold_map7: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i * 'h) ->
+ 'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h -> 'i list * 'h
+ val transpose: 'a list list -> 'a list list
+
+ val mk_fresh_names: Proof.context -> int -> string -> string list * Proof.context
+ val mk_TFrees: int -> Proof.context -> typ list * Proof.context
+ val mk_TFreess: int list -> Proof.context -> typ list list * Proof.context
+ val mk_Frees: string -> typ list -> Proof.context -> term list * Proof.context
+ val mk_Freess: string -> typ list list -> Proof.context -> term list list * Proof.context
+ val mk_Frees': string -> typ list -> Proof.context ->
+ (term list * (string * typ) list) * Proof.context
+ val mk_Freess': string -> typ list list -> Proof.context ->
+ (term list list * (string * typ) list list) * Proof.context
+
+ val mk_optionT: typ -> typ
+ val mk_relT: typ * typ -> typ
+ val dest_relT: typ -> typ * typ
+ val mk_sumT: typ * typ -> typ
+
+ val ctwo: term
+ val fst_const: typ -> term
+ val snd_const: typ -> term
+ val Id_const: typ -> term
+
+ val mk_Ball: term -> term -> term
+ val mk_Bex: term -> term -> term
+ val mk_Card_order: term -> term
+ val mk_Field: term -> term
+ val mk_Gr: term -> term -> term
+ val mk_UNION: term -> term -> term
+ val mk_Union: typ -> term
+ val mk_card_binop: string -> (typ * typ -> typ) -> term -> term -> term
+ val mk_card_of: term -> term
+ val mk_card_order: term -> term
+ val mk_ccexp: term -> term -> term
+ val mk_cexp: term -> term -> term
+ val mk_cinfinite: term -> term
+ val mk_collect: term list -> typ -> term
+ val mk_converse: term -> term
+ val mk_cprod: term -> term -> term
+ val mk_csum: term -> term -> term
+ val mk_dir_image: term -> term -> term
+ val mk_image: term -> term
+ val mk_in: term list -> term list -> typ -> term
+ val mk_ordLeq: term -> term -> term
+ val mk_rel_comp: term * term -> term
+ val mk_subset: term -> term -> term
+ val mk_wpull: term -> term -> term -> term -> term -> (term * term) option -> term -> term -> term
+
+ (*parameterized terms*)
+ val mk_nthN: int -> term -> int -> term
+
+ (*parameterized thms*)
+ val mk_Un_upper: int -> int -> thm
+ val mk_conjIN: int -> thm
+ val mk_conjunctN: int -> int -> thm
+ val mk_nthI: int -> int -> thm
+ val mk_nth_conv: int -> int -> thm
+ val mk_ordLeq_csum: int -> int -> thm -> thm
+ val mk_UnN: int -> int -> thm
+
+ val ctrans: thm
+ val o_apply: thm
+ val mk_sym: thm -> thm
+ val mk_trans: thm -> thm -> thm
+ val mk_unabs_def: int -> thm -> thm
+
+ val mk_permute: ''a list -> ''a list -> 'b list -> 'b list
+ val find_indices: ''a list -> ''a list -> int list
+
+ val WRAP: ('a -> tactic) -> ('a -> tactic) -> 'a list -> tactic -> tactic
+ val WRAP': ('a -> int -> tactic) -> ('a -> int -> tactic) -> 'a list -> (int -> tactic) -> int ->
+ tactic
+ val CONJ_WRAP_GEN: tactic -> ('a -> tactic) -> 'a list -> tactic
+ val CONJ_WRAP_GEN': (int -> tactic) -> ('a -> int -> tactic) -> 'a list -> int -> tactic
+ val CONJ_WRAP: ('a -> tactic) -> 'a list -> tactic
+ val CONJ_WRAP': ('a -> int -> tactic) -> 'a list -> int -> tactic
+end;
+
+structure BNF_Util : BNF_UTIL =
+struct
+
+(* Library proper *)
+
+fun map3 _ [] [] [] = []
+ | map3 f (x1::x1s) (x2::x2s) (x3::x3s) = f x1 x2 x3 :: map3 f x1s x2s x3s
+ | map3 _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map4 _ [] [] [] [] = []
+ | map4 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) = f x1 x2 x3 x4 :: map4 f x1s x2s x3s x4s
+ | map4 _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map5 _ [] [] [] [] [] = []
+ | map5 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) =
+ f x1 x2 x3 x4 x5 :: map5 f x1s x2s x3s x4s x5s
+ | map5 _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map6 _ [] [] [] [] [] [] = []
+ | map6 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) =
+ f x1 x2 x3 x4 x5 x6 :: map6 f x1s x2s x3s x4s x5s x6s
+ | map6 _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map7 _ [] [] [] [] [] [] [] = []
+ | map7 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) =
+ f x1 x2 x3 x4 x5 x6 x7 :: map7 f x1s x2s x3s x4s x5s x6s x7s
+ | map7 _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map8 _ [] [] [] [] [] [] [] [] = []
+ | map8 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) (x8::x8s) =
+ f x1 x2 x3 x4 x5 x6 x7 x8 :: map8 f x1s x2s x3s x4s x5s x6s x7s x8s
+ | map8 _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map9 _ [] [] [] [] [] [] [] [] [] = []
+ | map9 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)
+ (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) =
+ f x1 x2 x3 x4 x5 x6 x7 x8 x9 :: map9 f x1s x2s x3s x4s x5s x6s x7s x8s x9s
+ | map9 _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map10 _ [] [] [] [] [] [] [] [] [] [] = []
+ | map10 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)
+ (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) (x10::x10s) =
+ f x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 :: map10 f x1s x2s x3s x4s x5s x6s x7s x8s x9s x10s
+ | map10 _ _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map11 _ [] [] [] [] [] [] [] [] [] [] [] = []
+ | map11 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)
+ (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) (x10::x10s) (x11::x11s) =
+ f x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 :: map11 f x1s x2s x3s x4s x5s x6s x7s x8s x9s x10s x11s
+ | map11 _ _ _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun map12 _ [] [] [] [] [] [] [] [] [] [] [] [] = []
+ | map12 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)
+ (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) (x10::x10s) (x11::x11s) (x12::x12s) =
+ f x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 ::
+ map12 f x1s x2s x3s x4s x5s x6s x7s x8s x9s x10s x11s x12s
+ | map12 _ _ _ _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun fold_map2 _ [] [] acc = ([], acc)
+ | fold_map2 f (x1::x1s) (x2::x2s) acc =
+ let
+ val (x, acc') = f x1 x2 acc;
+ val (xs, acc'') = fold_map2 f x1s x2s acc';
+ in (x :: xs, acc'') end
+ | fold_map2 _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun fold_map3 _ [] [] [] acc = ([], acc)
+ | fold_map3 f (x1::x1s) (x2::x2s) (x3::x3s) acc =
+ let
+ val (x, acc') = f x1 x2 x3 acc;
+ val (xs, acc'') = fold_map3 f x1s x2s x3s acc';
+ in (x :: xs, acc'') end
+ | fold_map3 _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun fold_map4 _ [] [] [] [] acc = ([], acc)
+ | fold_map4 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) acc =
+ let
+ val (x, acc') = f x1 x2 x3 x4 acc;
+ val (xs, acc'') = fold_map4 f x1s x2s x3s x4s acc';
+ in (x :: xs, acc'') end
+ | fold_map4 _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun fold_map5 _ [] [] [] [] [] acc = ([], acc)
+ | fold_map5 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) acc =
+ let
+ val (x, acc') = f x1 x2 x3 x4 x5 acc;
+ val (xs, acc'') = fold_map5 f x1s x2s x3s x4s x5s acc';
+ in (x :: xs, acc'') end
+ | fold_map5 _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun fold_map6 _ [] [] [] [] [] [] acc = ([], acc)
+ | fold_map6 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) acc =
+ let
+ val (x, acc') = f x1 x2 x3 x4 x5 x6 acc;
+ val (xs, acc'') = fold_map6 f x1s x2s x3s x4s x5s x6s acc';
+ in (x :: xs, acc'') end
+ | fold_map6 _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+fun fold_map7 _ [] [] [] [] [] [] [] acc = ([], acc)
+ | fold_map7 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) acc =
+ let
+ val (x, acc') = f x1 x2 x3 x4 x5 x6 x7 acc;
+ val (xs, acc'') = fold_map7 f x1s x2s x3s x4s x5s x6s x7s acc';
+ in (x :: xs, acc'') end
+ | fold_map7 _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;
+
+(*Tactical WRAP surrounds a static given tactic (core) with two deterministic chains of tactics*)
+fun WRAP gen_before gen_after xs core_tac =
+ fold_rev (fn x => fn tac => gen_before x THEN tac THEN gen_after x) xs core_tac;
+
+fun WRAP' gen_before gen_after xs core_tac =
+ fold_rev (fn x => fn tac => gen_before x THEN' tac THEN' gen_after x) xs core_tac;
+
+fun CONJ_WRAP_GEN conj_tac gen_tac xs =
+ let val (butlast, last) = split_last xs;
+ in WRAP (fn thm => conj_tac THEN gen_tac thm) (K all_tac) butlast (gen_tac last) end;
+
+fun CONJ_WRAP_GEN' conj_tac gen_tac xs =
+ let val (butlast, last) = split_last xs;
+ in WRAP' (fn thm => conj_tac THEN' gen_tac thm) (K (K all_tac)) butlast (gen_tac last) end;
+
+(*not eta-converted because of monotype restriction*)
+fun CONJ_WRAP gen_tac = CONJ_WRAP_GEN (rtac conjI 1) gen_tac;
+fun CONJ_WRAP' gen_tac = CONJ_WRAP_GEN' (rtac conjI) gen_tac;
+
+
+
+(* Term construction *)
+
+(** Fresh variables **)
+
+fun mk_TFrees n = apfst (map TFree) o Variable.invent_types (replicate n (HOLogic.typeS));
+fun mk_TFreess ns = apfst (map (map TFree)) o
+ fold_map Variable.invent_types (map (fn n => replicate n (HOLogic.typeS)) ns);
+
+fun mk_names n x = if n = 1 then [x]
+ else map (fn i => x ^ string_of_int i) (1 upto n);
+
+fun mk_fresh_names ctxt = (fn names => Variable.variant_fixes names ctxt) oo mk_names;
+fun mk_Frees x Ts ctxt = mk_fresh_names ctxt (length Ts) x
+ |>> (fn names => map2 (curry Free) names Ts);
+fun mk_Freess x Tss ctxt =
+ fold_map2 (fn name => fn Ts => fn ctxt =>
+ mk_fresh_names ctxt (length Ts) name) (mk_names (length Tss) x) Tss ctxt
+ |>> (fn namess => map2 (map2 (curry Free)) namess Tss);
+fun mk_Frees' x Ts ctxt = mk_fresh_names ctxt (length Ts) x
+ |>> (fn names => `(map Free) (names ~~ Ts));
+fun mk_Freess' x Tss ctxt =
+ fold_map2 (fn name => fn Ts => fn ctxt =>
+ mk_fresh_names ctxt (length Ts) name) (mk_names (length Tss) x) Tss ctxt
+ |>> (fn namess => map_split (`(map Free) o (op ~~)) (namess ~~ Tss));
+
+
+(** Types **)
+
+fun mk_optionT T = Type (@{type_name option}, [T]);
+val mk_relT = HOLogic.mk_setT o HOLogic.mk_prodT;
+val dest_relT = HOLogic.dest_prodT o HOLogic.dest_setT;
+fun mk_sumT (LT, RT) = Type (@{type_name Sum_Type.sum}, [LT, RT]);
+fun mk_partial_funT (ranT, domT) = domT --> mk_optionT ranT;
+
+
+(** Constants **)
+
+fun fst_const T = Const (@{const_name fst}, T --> fst (HOLogic.dest_prodT T));
+fun snd_const T = Const (@{const_name snd}, T --> snd (HOLogic.dest_prodT T));
+fun Id_const T = Const (@{const_name Id}, mk_relT (T, T));
+
+
+(** Operators **)
+
+fun mk_converse R =
+ let
+ val RT = dest_relT (fastype_of R);
+ val RST = mk_relT (snd RT, fst RT);
+ in Const (@{const_name converse}, fastype_of R --> RST) $ R end;
+
+fun mk_rel_comp (R, S) =
+ let
+ val RT = fastype_of R;
+ val ST = fastype_of S;
+ val RST = mk_relT (fst (dest_relT RT), snd (dest_relT ST));
+ in Const (@{const_name relcomp}, RT --> ST --> RST) $ R $ S end;
+
+fun mk_Gr A f =
+ let val ((AT, BT), FT) = `dest_funT (fastype_of f);
+ in Const (@{const_name Gr}, HOLogic.mk_setT AT --> FT --> mk_relT (AT, BT)) $ A $ f end;
+
+fun mk_image f =
+ let val (T, U) = dest_funT (fastype_of f);
+ in Const (@{const_name image},
+ (T --> U) --> (HOLogic.mk_setT T) --> (HOLogic.mk_setT U)) $ f end;
+
+fun mk_Ball X f =
+ Const (@{const_name Ball}, fastype_of X --> fastype_of f --> HOLogic.boolT) $ X $ f;
+
+fun mk_Bex X f =
+ Const (@{const_name Bex}, fastype_of X --> fastype_of f --> HOLogic.boolT) $ X $ f;
+
+fun mk_UNION X f =
+ let val (T, U) = dest_funT (fastype_of f);
+ in Const (@{const_name SUPR}, fastype_of X --> (T --> U) --> U) $ X $ f end;
+
+fun mk_Union T =
+ Const (@{const_name Sup}, HOLogic.mk_setT (HOLogic.mk_setT T) --> HOLogic.mk_setT T);
+
+fun mk_Field r =
+ let val T = fst (dest_relT (fastype_of r));
+ in Const (@{const_name Field}, mk_relT (T, T) --> HOLogic.mk_setT T) $ r end;
+
+fun mk_card_order bd =
+ let
+ val T = fastype_of bd;
+ val AT = fst (dest_relT T);
+ in
+ Const (@{const_name card_order_on}, HOLogic.mk_setT AT --> T --> HOLogic.boolT) $
+ (HOLogic.mk_UNIV AT) $ bd
+ end;
+
+fun mk_Card_order bd =
+ let
+ val T = fastype_of bd;
+ val AT = fst (dest_relT T);
+ in
+ Const (@{const_name card_order_on}, HOLogic.mk_setT AT --> T --> HOLogic.boolT) $
+ mk_Field bd $ bd
+ end;
+
+fun mk_cinfinite bd =
+ Const (@{const_name cinfinite}, fastype_of bd --> HOLogic.boolT) $ bd;
+
+fun mk_ordLeq t1 t2 =
+ HOLogic.mk_mem (HOLogic.mk_prod (t1, t2),
+ Const (@{const_name ordLeq}, mk_relT (fastype_of t1, fastype_of t2)));
+
+fun mk_card_of A =
+ let
+ val AT = fastype_of A;
+ val T = HOLogic.dest_setT AT;
+ in
+ Const (@{const_name card_of}, AT --> mk_relT (T, T)) $ A
+ end;
+
+fun mk_dir_image r f =
+ let val (T, U) = dest_funT (fastype_of f);
+ in Const (@{const_name dir_image}, mk_relT (T, T) --> (T --> U) --> mk_relT (U, U)) $ r $ f end;
+
+(*FIXME: "x"?*)
+(*(nth sets i) must be of type "T --> 'ai set"*)
+fun mk_in As sets T =
+ let
+ fun in_single set A =
+ let val AT = fastype_of A;
+ in Const (@{const_name less_eq},
+ AT --> AT --> HOLogic.boolT) $ (set $ Free ("x", T)) $ A end;
+ in
+ if length sets > 0
+ then HOLogic.mk_Collect ("x", T, foldr1 (HOLogic.mk_conj) (map2 in_single sets As))
+ else HOLogic.mk_UNIV T
+ end;
+
+fun mk_wpull A B1 B2 f1 f2 pseudo p1 p2 =
+ let
+ val AT = fastype_of A;
+ val BT1 = fastype_of B1;
+ val BT2 = fastype_of B2;
+ val FT1 = fastype_of f1;
+ val FT2 = fastype_of f2;
+ val PT1 = fastype_of p1;
+ val PT2 = fastype_of p2;
+ val T1 = HOLogic.dest_setT BT1;
+ val T2 = HOLogic.dest_setT BT2;
+ val domP = domain_type PT1;
+ val ranF = range_type FT1;
+ val _ = if is_some pseudo orelse
+ (HOLogic.dest_setT AT = domP andalso
+ domain_type FT1 = T1 andalso
+ domain_type FT2 = T2 andalso
+ domain_type PT2 = domP andalso
+ range_type PT1 = T1 andalso
+ range_type PT2 = T2 andalso
+ range_type FT2 = ranF)
+ then () else raise TYPE ("mk_wpull", [BT1, BT2, FT1, FT2, PT1, PT2], []);
+ in
+ (case pseudo of
+ NONE => Const (@{const_name wpull},
+ AT --> BT1 --> BT2 --> FT1 --> FT2 --> PT1 --> PT2 --> HOLogic.boolT) $
+ A $ B1 $ B2 $ f1 $ f2 $ p1 $ p2
+ | SOME (e1, e2) => Const (@{const_name wppull},
+ AT --> BT1 --> BT2 --> FT1 --> FT2 --> fastype_of e1 --> fastype_of e2 -->
+ PT1 --> PT2 --> HOLogic.boolT) $
+ A $ B1 $ B2 $ f1 $ f2 $ e1 $ e2 $ p1 $ p2)
+ end;
+
+fun mk_subset t1 t2 =
+ Const (@{const_name less_eq}, (fastype_of t1) --> (fastype_of t2) --> HOLogic.boolT) $ t1 $ t2;
+
+fun mk_card_binop binop typop t1 t2 =
+ let
+ val (T1, relT1) = `(fst o dest_relT) (fastype_of t1);
+ val (T2, relT2) = `(fst o dest_relT) (fastype_of t2);
+ in
+ Const (binop, relT1 --> relT2 --> mk_relT (typop (T1, T2), typop (T1, T2))) $ t1 $ t2
+ end;
+
+val mk_csum = mk_card_binop @{const_name csum} mk_sumT;
+val mk_cprod = mk_card_binop @{const_name cprod} HOLogic.mk_prodT;
+val mk_cexp = mk_card_binop @{const_name cexp} mk_partial_funT;
+val mk_ccexp = mk_card_binop @{const_name ccexp} mk_partial_funT;
+val ctwo = @{term ctwo};
+
+fun mk_collect xs defT =
+ let val T = (case xs of [] => defT | (x::_) => fastype_of x);
+ in Const (@{const_name collect}, HOLogic.mk_setT T --> T) $ (HOLogic.mk_set T xs) end;
+
+fun mk_permute src dest xs = map (nth xs o (fn x => find_index ((curry op =) x) src)) dest;
+
+fun find_indices xs ys = map_filter I
+ (map_index (fn (i, y) => if member (op =) xs y then SOME i else NONE) ys);
+
+fun mk_trans thm1 thm2 = trans OF [thm1, thm2];
+fun mk_sym thm = sym OF [thm];
+
+(*TODO: antiquote heavily used theorems once*)
+val ctrans = @{thm ordLeq_transitive};
+val o_apply = @{thm o_apply};
+
+fun mk_nthN 1 t 1 = t
+ | mk_nthN _ t 1 = HOLogic.mk_fst t
+ | mk_nthN 2 t 2 = HOLogic.mk_snd t
+ | mk_nthN n t m = mk_nthN (n - 1) (HOLogic.mk_snd t) (m - 1);
+
+fun mk_nth_conv n m =
+ let
+ fun thm b = if b then @{thm fst_snd} else @{thm snd_snd}
+ fun mk_nth_conv _ 1 1 = refl
+ | mk_nth_conv _ _ 1 = @{thm fst_conv}
+ | mk_nth_conv _ 2 2 = @{thm snd_conv}
+ | mk_nth_conv b _ 2 = @{thm snd_conv} RS thm b
+ | mk_nth_conv b n m = mk_nth_conv false (n - 1) (m - 1) RS thm b;
+ in mk_nth_conv (not (m = n)) n m end;
+
+fun mk_nthI 1 1 = @{thm TrueE[OF TrueI]}
+ | mk_nthI n m = fold (curry op RS) (replicate (m - 1) @{thm sndI})
+ (if m = n then @{thm TrueE[OF TrueI]} else @{thm fstI});
+
+fun mk_conjunctN 1 1 = @{thm TrueE[OF TrueI]}
+ | mk_conjunctN _ 1 = conjunct1
+ | mk_conjunctN 2 2 = conjunct2
+ | mk_conjunctN n m = conjunct2 RS (mk_conjunctN (n - 1) (m - 1));
+
+fun mk_conjIN 1 = @{thm TrueE[OF TrueI]}
+ | mk_conjIN n = mk_conjIN (n - 1) RSN (2, conjI);
+
+fun mk_ordLeq_csum 1 1 thm = thm
+ | mk_ordLeq_csum _ 1 thm = @{thm ordLeq_transitive} OF [thm, @{thm ordLeq_csum1}]
+ | mk_ordLeq_csum 2 2 thm = @{thm ordLeq_transitive} OF [thm, @{thm ordLeq_csum2}]
+ | mk_ordLeq_csum n m thm = @{thm ordLeq_transitive} OF
+ [mk_ordLeq_csum (n - 1) (m - 1) thm, @{thm ordLeq_csum2[OF Card_order_csum]}];
+
+local
+ fun mk_Un_upper' 0 = subset_refl
+ | mk_Un_upper' 1 = @{thm Un_upper1}
+ | mk_Un_upper' k = Library.foldr (op RS o swap)
+ (replicate (k - 1) @{thm subset_trans[OF Un_upper1]}, @{thm Un_upper1});
+in
+ fun mk_Un_upper 1 1 = subset_refl
+ | mk_Un_upper n 1 = mk_Un_upper' (n - 2) RS @{thm subset_trans[OF Un_upper1]}
+ | mk_Un_upper n m = mk_Un_upper' (n - m) RS @{thm subset_trans[OF Un_upper2]};
+end;
+
+local
+ fun mk_UnN' 0 = @{thm UnI2}
+ | mk_UnN' m = mk_UnN' (m - 1) RS @{thm UnI1};
+in
+ fun mk_UnN 1 1 = @{thm TrueE[OF TrueI]}
+ | mk_UnN n 1 = Library.foldr1 (op RS o swap) (replicate (n - 1) @{thm UnI1})
+ | mk_UnN n m = mk_UnN' (n - m)
+end;
+
+fun transpose [] = []
+ | transpose ([] :: xss) = transpose xss
+ | transpose xss = map hd xss :: transpose (map tl xss);
+
+fun mk_unabs_def 0 thm = thm
+ | mk_unabs_def n thm = mk_unabs_def (n - 1) thm RS @{thm spec[OF iffD1[OF fun_eq_iff]]};
+
+
+end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Cardinal_Arithmetic.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,878 @@
+(* Title: HOL/Ordinals_and_Cardinals/Cardinal_Arithmetic.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Cardinal arithmetic.
+*)
+
+header {* Cardinal Arithmetic *}
+
+theory Cardinal_Arithmetic
+imports Cardinal_Order_Relation_Base
+begin
+
+text {*
+ The following collection of lemmas should be seen as an user interface to the HOL Theory
+ of cardinals. It is not expected to be complete in any sense, since its
+ development was driven by demand arising from the development of the (co)datatype package.
+*}
+
+(*library candidate*)
+lemma dir_image: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); Card_order r\<rbrakk> \<Longrightarrow> r =o dir_image r f"
+by (rule dir_image_ordIso) (auto simp add: inj_on_def card_order_on_def)
+
+(*should supersede a weaker lemma from the library*)
+lemma dir_image_Field: "Field (dir_image r f) = f ` Field r"
+unfolding dir_image_def Field_def Range_def Domain_def by fastforce
+
+lemma card_order_dir_image:
+ assumes bij: "bij f" and co: "card_order r"
+ shows "card_order (dir_image r f)"
+proof -
+ from assms have "Field (dir_image r f) = UNIV"
+ using card_order_on_Card_order[of UNIV r] unfolding bij_def dir_image_Field by auto
+ moreover from bij have "\<And>x y. (f x = f y) = (x = y)" unfolding bij_def inj_on_def by auto
+ with co have "Card_order (dir_image r f)"
+ using card_order_on_Card_order[of UNIV r] Card_order_ordIso2[OF _ dir_image] by blast
+ ultimately show ?thesis by auto
+qed
+
+(*library candidate*)
+lemma ordIso_refl: "Card_order r \<Longrightarrow> r =o r"
+by (rule card_order_on_ordIso)
+
+(*library candidate*)
+lemma ordLeq_refl: "Card_order r \<Longrightarrow> r \<le>o r"
+by (rule ordIso_imp_ordLeq, rule card_order_on_ordIso)
+
+(*library candidate*)
+lemma card_of_ordIso_subst: "A = B \<Longrightarrow> |A| =o |B|"
+by (simp only: ordIso_refl card_of_Card_order)
+
+(*library candidate*)
+lemma card_of_Times_Plus_distrib:
+ "|A <*> (B <+> C)| =o |A <*> B <+> A <*> C|" (is "|?RHS| =o |?LHS|")
+proof -
+ let ?f = "\<lambda>(a, bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"
+ have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+(*library candidate*)
+lemma Field_card_order: "card_order r \<Longrightarrow> Field r = UNIV"
+using card_order_on_Card_order[of UNIV r] by simp
+
+subsection {* Zero *}
+
+definition czero where
+ "czero = card_of {}"
+
+lemma czero_ordIso:
+ "czero =o czero"
+using card_of_empty_ordIso by (simp add: czero_def)
+
+lemma card_of_ordIso_czero_iff_empty:
+ "|A| =o (czero :: 'a rel) \<longleftrightarrow> A = ({} :: 'a set)"
+unfolding czero_def by (rule iffI[OF card_of_empty2]) (auto simp: card_of_refl)
+
+(* A "not czero" Cardinal predicate *)
+abbreviation Cnotzero where
+ "Cnotzero (r :: 'a rel) \<equiv> \<not>(r =o (czero :: 'a rel)) \<and> Card_order r"
+
+(*helper*)
+lemma Cnotzero_imp_not_empty: "Cnotzero r \<Longrightarrow> Field r \<noteq> {}"
+by (metis Card_order_iff_ordIso_card_of czero_def)
+
+lemma czeroI:
+ "\<lbrakk>Card_order r; Field r = {}\<rbrakk> \<Longrightarrow> r =o czero"
+using Cnotzero_imp_not_empty ordIso_transitive[OF _ czero_ordIso] by blast
+
+lemma czeroE:
+ "r =o czero \<Longrightarrow> Field r = {}"
+unfolding czero_def
+by (drule card_of_cong) (simp only: Field_card_of card_of_empty2)
+
+lemma Cnotzero_mono:
+ "\<lbrakk>Cnotzero r; Card_order q; r \<le>o q\<rbrakk> \<Longrightarrow> Cnotzero q"
+apply (rule ccontr)
+apply auto
+apply (drule czeroE)
+apply (erule notE)
+apply (erule czeroI)
+apply (drule card_of_mono2)
+apply (simp only: card_of_empty3)
+done
+
+subsection {* Infinite cardinals *}
+
+definition cinfinite where
+ "cinfinite r = infinite (Field r)"
+
+abbreviation Cinfinite where
+ "Cinfinite r \<equiv> cinfinite r \<and> Card_order r"
+
+lemma natLeq_ordLeq_cinfinite:
+ assumes inf: "Cinfinite r"
+ shows "natLeq \<le>o r"
+proof -
+ from inf have "natLeq \<le>o |Field r|" by (simp add: cinfinite_def infinite_iff_natLeq_ordLeq)
+ also from inf have "|Field r| =o r" by (simp add: card_of_unique ordIso_symmetric)
+ finally show ?thesis .
+qed
+
+lemma cinfinite_not_czero: "cinfinite r \<Longrightarrow> \<not> (r =o (czero :: 'a rel))"
+unfolding cinfinite_def by (metis czeroE finite.emptyI)
+
+lemma Cinfinite_Cnotzero: "Cinfinite r \<Longrightarrow> Cnotzero r"
+by (metis cinfinite_not_czero)
+
+lemma Cinfinite_cong: "\<lbrakk>r1 =o r2; Cinfinite r1\<rbrakk> \<Longrightarrow> Cinfinite r2"
+by (metis Card_order_ordIso2 card_of_mono2 card_of_ordLeq_infinite cinfinite_def ordIso_iff_ordLeq)
+
+lemma cinfinite_mono: "\<lbrakk>r1 \<le>o r2; cinfinite r1\<rbrakk> \<Longrightarrow> cinfinite r2"
+by (metis card_of_mono2 card_of_ordLeq_infinite cinfinite_def)
+
+
+subsection {* Binary sum *}
+
+definition csum (infixr "+c" 65) where
+ "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
+
+lemma Card_order_csum:
+ "Card_order (r1 +c r2)"
+unfolding csum_def by (simp add: card_of_Card_order)
+
+lemma csum_Cnotzero1:
+ "Cnotzero r1 \<Longrightarrow> Cnotzero (r1 +c r2)"
+unfolding csum_def
+by (metis Cnotzero_imp_not_empty Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)
+
+lemma csum_Cnotzero2:
+ "Cnotzero r2 \<Longrightarrow> Cnotzero (r1 +c r2)"
+unfolding csum_def
+by (metis Cnotzero_imp_not_empty Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)
+
+lemma card_order_csum:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 +c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis unfolding csum_def by (auto simp: card_of_card_order_on)
+qed
+
+lemma cinfinite_csum:
+ "cinfinite r1 \<or> cinfinite r2 \<Longrightarrow> cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (auto simp: Field_card_of)
+
+lemma Cinfinite_csum:
+ "Cinfinite r1 \<or> Cinfinite r2 \<Longrightarrow> Cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
+
+lemma Cinfinite_csum_strong:
+ "\<lbrakk>Cinfinite r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 +c r2)"
+by (metis Cinfinite_csum)
+
+lemma Cinfinite_csum1:
+ "Cinfinite r1 \<Longrightarrow> Cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
+
+lemma csum_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 +c p2 =o r1 +c r2"
+by (simp only: csum_def ordIso_Plus_cong)
+
+lemma csum_cong1: "p1 =o r1 \<Longrightarrow> p1 +c q =o r1 +c q"
+by (simp only: csum_def ordIso_Plus_cong1)
+
+lemma csum_cong2: "p2 =o r2 \<Longrightarrow> q +c p2 =o q +c r2"
+by (simp only: csum_def ordIso_Plus_cong2)
+
+lemma csum_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 +c p2 \<le>o r1 +c r2"
+by (simp only: csum_def ordLeq_Plus_mono)
+
+lemma csum_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 +c q \<le>o r1 +c q"
+by (simp only: csum_def ordLeq_Plus_mono1)
+
+lemma csum_mono2: "p2 \<le>o r2 \<Longrightarrow> q +c p2 \<le>o q +c r2"
+by (simp only: csum_def ordLeq_Plus_mono2)
+
+lemma ordLeq_csum1: "Card_order p1 \<Longrightarrow> p1 \<le>o p1 +c p2"
+by (simp only: csum_def Card_order_Plus1)
+
+lemma ordLeq_csum2: "Card_order p2 \<Longrightarrow> p2 \<le>o p1 +c p2"
+by (simp only: csum_def Card_order_Plus2)
+
+lemma csum_com: "p1 +c p2 =o p2 +c p1"
+by (simp only: csum_def card_of_Plus_commute)
+
+lemma csum_assoc: "(p1 +c p2) +c p3 =o p1 +c p2 +c p3"
+by (simp only: csum_def Field_card_of card_of_Plus_assoc)
+
+lemma Plus_csum: "|A <+> B| =o |A| +c |B|"
+by (simp only: csum_def Field_card_of card_of_refl)
+
+lemma Un_csum: "|A \<union> B| \<le>o |A| +c |B|"
+using ordLeq_ordIso_trans[OF card_of_Un_Plus_ordLeq Plus_csum] by blast
+
+
+subsection {* One *}
+
+definition cone where
+ "cone = card_of {()}"
+
+lemma Card_order_cone: "Card_order cone"
+unfolding cone_def by (rule card_of_Card_order)
+
+lemma single_cone:
+ "|{x}| =o cone"
+proof -
+ let ?f = "\<lambda>x. ()"
+ have "bij_betw ?f {x} {()}" unfolding bij_betw_def by auto
+ thus ?thesis unfolding cone_def using card_of_ordIso by blast
+qed
+
+lemma cone_not_czero: "\<not> (cone =o czero)"
+unfolding czero_def cone_def by (metis empty_not_insert card_of_empty3[of "{()}"] ordIso_iff_ordLeq)
+
+lemma cone_Cnotzero: "Cnotzero cone"
+by (simp add: cone_not_czero Card_order_cone)
+
+lemma cone_ordLeq_Cnotzero: "Cnotzero r \<Longrightarrow> cone \<le>o r"
+unfolding cone_def by (metis Card_order_singl_ordLeq czeroI)
+
+
+subsection{* Two *}
+
+definition ctwo where
+ "ctwo = |UNIV :: bool set|"
+
+lemma Card_order_ctwo: "Card_order ctwo"
+unfolding ctwo_def by (rule card_of_Card_order)
+
+lemma cone_ordLeq_ctwo: "cone \<le>o ctwo"
+unfolding cone_def ctwo_def card_of_ordLeq[symmetric] by auto
+
+lemma ctwo_not_czero: "\<not> (ctwo =o czero)"
+using card_of_empty3[of "UNIV :: bool set"] ordIso_iff_ordLeq
+unfolding czero_def ctwo_def by (metis UNIV_not_empty)
+
+lemma ctwo_Cnotzero: "Cnotzero ctwo"
+by (simp add: ctwo_not_czero Card_order_ctwo)
+
+
+subsection {* Family sum *}
+
+definition Csum where
+ "Csum r rs \<equiv> |SIGMA i : Field r. Field (rs i)|"
+
+(* Similar setup to the one for SIGMA from theory Big_Operators: *)
+syntax "_Csum" ::
+ "pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
+ ("(3CSUM _:_. _)" [0, 51, 10] 10)
+
+translations
+ "CSUM i:r. rs" == "CONST Csum r (%i. rs)"
+
+lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
+by (auto simp: Csum_def Field_card_of)
+
+(* NB: Always, under the cardinal operator,
+operations on sets are reduced automatically to operations on cardinals.
+This should make cardinal reasoning more direct and natural. *)
+
+
+subsection {* Product *}
+
+definition cprod (infixr "*c" 80) where
+ "r1 *c r2 = |Field r1 <*> Field r2|"
+
+lemma Times_cprod: "|A \<times> B| =o |A| *c |B|"
+by (simp only: cprod_def Field_card_of card_of_refl)
+
+lemma card_order_cprod:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 *c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis by (auto simp: cprod_def card_of_card_order_on)
+qed
+
+lemma Card_order_cprod: "Card_order (r1 *c r2)"
+by (simp only: cprod_def Field_card_of card_of_card_order_on)
+
+lemma cprod_cong2: "p2 =o r2 \<Longrightarrow> q *c p2 =o q *c r2"
+by (simp only: cprod_def ordIso_Times_cong2)
+
+lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
+by (simp only: cprod_def ordLeq_Times_mono1)
+
+lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
+by (simp only: cprod_def ordLeq_Times_mono2)
+
+lemma ordLeq_cprod1: "\<lbrakk>Card_order p1; Cnotzero p2\<rbrakk> \<Longrightarrow> p1 \<le>o p1 *c p2"
+unfolding cprod_def by (metis Card_order_Times1 czeroI)
+
+lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
+unfolding cprod_def by (metis Card_order_Times2 czeroI)
+
+lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
+by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
+
+lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
+by (metis cinfinite_mono ordLeq_cprod2)
+
+lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
+by (blast intro: cinfinite_cprod2 Card_order_cprod)
+
+lemma cprod_com: "p1 *c p2 =o p2 *c p1"
+by (simp only: cprod_def card_of_Times_commute)
+
+lemma card_of_Csum_Times:
+ "\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
+by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_Times)
+
+lemma card_of_Csum_Times':
+ assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
+ shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
+proof -
+ from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
+ with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
+ hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
+ also from * have "|I| *c |Field r| \<le>o |I| *c r"
+ by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
+ finally show ?thesis .
+qed
+
+lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
+unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
+
+lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
+unfolding csum_def by (metis Card_order_Plus_infinite cinfinite_def cinfinite_mono)
+
+lemma csum_absorb1':
+ assumes card: "Card_order r2"
+ and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
+ shows "r2 +c r1 =o r2"
+by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
+
+lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
+by (rule csum_absorb1') auto
+
+lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
+unfolding cinfinite_def cprod_def
+by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
+
+lemma cprod_infinite: "Cinfinite r \<Longrightarrow> r *c r =o r"
+using cprod_infinite1' Cinfinite_Cnotzero ordLeq_refl by blast
+
+
+subsection {* Exponentiation *}
+
+definition cexp (infixr "^c" 80) where
+ "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
+
+definition ccexp (infixr "^^c" 80) where
+ "r1 ^^c r2 \<equiv> |Pfunc (Field r2) (Field r1)|"
+
+lemma cexp_ordLeq_ccexp: "r1 ^c r2 \<le>o r1 ^^c r2"
+unfolding cexp_def ccexp_def by (rule card_of_mono1) (rule Func_Pfunc)
+
+lemma card_order_ccexp:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 ^^c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis unfolding ccexp_def Pfunc_def
+ by (auto simp: card_of_card_order_on split: option.split)
+qed
+
+lemma Card_order_cexp: "Card_order (r1 ^c r2)"
+unfolding cexp_def by (rule card_of_Card_order)
+
+lemma cexp_mono':
+ assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
+ and n1: "Field p1 \<noteq> {} \<or> cone \<le>o r1 ^c r2"
+ and n2: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
+ shows "p1 ^c p2 \<le>o r1 ^c r2"
+proof(cases "Field p1 = {}")
+ case True
+ hence "|Field (p1 ^c p2)| \<le>o cone"
+ unfolding czero_def cone_def cexp_def Field_card_of
+ by (cases "Field p2 = {}", auto intro: card_of_ordLeqI2 simp: Func_empty)
+ (metis Func_is_emp card_of_empty ex_in_conv)
+ hence "p1 ^c p2 \<le>o cone" by (simp add: Field_card_of cexp_def)
+ thus ?thesis using True n1 ordLeq_transitive by auto
+next
+ case False
+ have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
+ using 1 2 by (auto simp: card_of_mono2)
+ obtain f1 where f1: "f1 ` Field r1 = Field p1"
+ using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
+ obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
+ using 2 unfolding card_of_ordLeq[symmetric] by blast
+ have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
+ unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n2, symmetric] .
+ have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
+ using False by simp
+ show ?thesis
+ using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
+qed
+
+lemma cexp_mono:
+ assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
+ and n1: "Cnotzero p1 \<or> cone \<le>o r1 ^c r2"
+ and n2: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
+ shows "p1 ^c p2 \<le>o r1 ^c r2"
+proof (rule cexp_mono'[OF 1 2])
+ show "Field p1 \<noteq> {} \<or> cone \<le>o r1 ^c r2"
+ proof (cases "Cnotzero p1")
+ case True show ?thesis using Cnotzero_imp_not_empty[OF True] by (rule disjI1)
+ next
+ case False with n1 show ?thesis by blast
+ qed
+qed (rule czeroI[OF card, THEN n2, THEN czeroE])
+
+lemma cexp_mono1:
+ assumes 1: "p1 \<le>o r1"
+ and n1: "Cnotzero p1 \<or> cone \<le>o r1 ^c q" and q: "Card_order q"
+ shows "p1 ^c q \<le>o r1 ^c q"
+using ordLeq_refl[OF q] by (rule cexp_mono[OF 1 _ n1]) (auto simp: q)
+
+lemma cexp_mono1_Cnotzero: "\<lbrakk>p1 \<le>o r1; Cnotzero p1; Card_order q\<rbrakk> \<Longrightarrow> p1 ^c q \<le>o r1 ^c q"
+by (simp add: cexp_mono1)
+
+lemma cexp_mono1_cone_ordLeq: "\<lbrakk>p1 \<le>o r1; cone \<le>o r1 ^c q; Card_order q\<rbrakk> \<Longrightarrow> p1 ^c q \<le>o r1 ^c q"
+using assms by (simp add: cexp_mono1)
+
+lemma cexp_mono2':
+ assumes 2: "p2 \<le>o r2" and q: "Card_order q"
+ and n1: "Field q \<noteq> {} \<or> cone \<le>o q ^c r2"
+ and n2: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
+ shows "q ^c p2 \<le>o q ^c r2"
+using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n1 n2]) auto
+
+lemma cexp_mono2:
+ assumes 2: "p2 \<le>o r2" and q: "Card_order q"
+ and n1: "Cnotzero q \<or> cone \<le>o q ^c r2"
+ and n2: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n1 n2 card]) auto
+
+lemma cexp_mono2_Cnotzero:
+ assumes "p2 \<le>o r2" "Cnotzero q" and n2: "Cnotzero p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+proof (rule cexp_mono)
+ assume *: "p2 =o czero"
+ have False using n2 czeroI czeroE[OF *] by blast
+ thus "r2 =o czero" by blast
+qed (auto simp add: assms ordLeq_refl)
+
+lemma cexp_cong:
+ assumes 1: "p1 =o r1" and 2: "p2 =o r2"
+ and p1: "Cnotzero p1 \<or> cone \<le>o r1 ^c r2" and Cr: "Card_order r2"
+ and r1: "Cnotzero r1 \<or> cone \<le>o p1 ^c p2" and Cp: "Card_order p2"
+ shows "p1 ^c p2 =o r1 ^c r2"
+proof -
+ obtain f where "bij_betw f (Field p2) (Field r2)"
+ using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
+ hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
+ have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
+ and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
+ using 0 Cr Cp czeroE czeroI by auto
+ show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
+ using r p cexp_mono[OF _ _ p1 _ Cp] cexp_mono[OF _ _ r1 _ Cr]
+ by blast
+qed
+
+lemma cexp_cong1:
+ assumes 1: "p1 =o r1" and q: "Card_order q"
+ and p1: "Cnotzero p1 \<or> cone \<le>o r1 ^c q"
+ and r1: "Cnotzero r1 \<or> cone \<le>o p1 ^c q"
+ shows "p1 ^c q =o r1 ^c q"
+by (rule cexp_cong[OF 1 _ p1 q r1 q]) (rule ordIso_refl[OF q])
+
+lemma cexp_cong1_Cnotzero:
+ assumes "p1 =o r1" "Card_order q" "Cnotzero p1" "Cnotzero r1"
+ shows "p1 ^c q =o r1 ^c q"
+by (rule cexp_cong1, auto simp add: assms)
+
+lemma cexp_cong2:
+ assumes 2: "p2 =o r2" and q: "Card_order q"
+ and p: "Card_order p2" and r: "Card_order r2"
+ shows "Cnotzero q \<or> (cone \<le>o q ^c p2 \<and> cone \<le>o q ^c r2) \<Longrightarrow>
+ q ^c p2 =o q ^c r2"
+by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl q p r)
+
+lemma cexp_cong2_Cnotzero:
+ assumes 2: "p2 =o r2" and q: "Cnotzero q"
+ and p: "Card_order p2"
+ shows "q ^c p2 =o q ^c r2"
+by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
+
+lemma cexp_czero: "r ^c czero =o cone"
+unfolding cexp_def czero_def Field_card_of Func_empty by (rule single_cone)
+
+lemma cexp_cone:
+ assumes "Card_order r"
+ shows "r ^c cone =o r"
+proof -
+ have "r ^c cone =o |Field r|"
+ unfolding cexp_def cone_def Field_card_of Func_empty
+ card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
+ by (rule exI[of _ "\<lambda>f. case f () of Some a \<Rightarrow> a"]) auto
+ also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
+ finally show ?thesis .
+qed
+
+lemma cexp_cprod:
+ assumes r1: "Cnotzero r1"
+ shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
+proof -
+ have "?L =o r1 ^c (r3 *c r2)"
+ unfolding cprod_def cexp_def Field_card_of
+ using card_of_Func_Times by(rule ordIso_symmetric)
+ also have "r1 ^c (r3 *c r2) =o ?R"
+ apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
+ finally show ?thesis .
+qed
+
+lemma cexp_cprod_ordLeq:
+ assumes r1: "Cnotzero r1" and r2: "Cinfinite r2"
+ and r3: "Cnotzero r3" "r3 \<le>o r2"
+ shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
+proof-
+ have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
+ also have "r1 ^c (r2 *c r3) =o ?R"
+ apply(rule cexp_cong2)
+ apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
+ finally show ?thesis .
+qed
+
+lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
+by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
+
+lemma Pow_cexp_ctwo:
+ "|Pow A| =o ctwo ^c |A|"
+unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+
+lemma ordLess_ctwo_cexp:
+ assumes "Card_order r"
+ shows "r <o ctwo ^c r"
+proof -
+ have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
+ also have "|Pow (Field r)| =o ctwo ^c r"
+ unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+ finally show ?thesis .
+qed
+
+lemma ordLeq_cexp1:
+ assumes "Cnotzero r" "Card_order q"
+ shows "q \<le>o q ^c r"
+proof (cases "q =o (czero :: 'a rel)")
+ case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
+next
+ case False
+ thus ?thesis
+ apply -
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule ordIso_symmetric)
+ apply (rule cexp_cone)
+ apply (rule assms(2))
+ apply (rule cexp_mono2)
+ apply (rule cone_ordLeq_Cnotzero)
+ apply (rule assms(1))
+ apply (rule assms(2))
+ apply (rule disjI1)
+ apply (rule conjI)
+ apply (rule notI)
+ apply (erule notE)
+ apply (rule ordIso_transitive)
+ apply assumption
+ apply (rule czero_ordIso)
+ apply (rule assms(2))
+ apply (rule notE)
+ apply (rule cone_not_czero)
+ apply assumption
+ apply (rule Card_order_cone)
+ done
+qed
+
+lemma ordLeq_cexp2:
+ assumes "ctwo \<le>o q" "Card_order r"
+ shows "r \<le>o q ^c r"
+proof (cases "r =o (czero :: 'a rel)")
+ case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
+next
+ case False thus ?thesis
+ apply -
+ apply (rule ordLess_imp_ordLeq)
+ apply (rule ordLess_ordLeq_trans)
+ apply (rule ordLess_ctwo_cexp)
+ apply (rule assms(2))
+ apply (rule cexp_mono1)
+ apply (rule assms(1))
+ apply (rule disjI1)
+ apply (rule ctwo_Cnotzero)
+ apply (rule assms(2))
+ done
+qed
+
+lemma Cnotzero_cexp:
+ assumes "Cnotzero q" "Card_order r"
+ shows "Cnotzero (q ^c r)"
+proof (cases "r =o czero")
+ case False thus ?thesis
+ apply -
+ apply (rule Cnotzero_mono)
+ apply (rule assms(1))
+ apply (rule Card_order_cexp)
+ apply (rule ordLeq_cexp1)
+ apply (rule conjI)
+ apply (rule notI)
+ apply (erule notE)
+ apply (rule ordIso_transitive)
+ apply assumption
+ apply (rule czero_ordIso)
+ apply (rule assms(2))
+ apply (rule conjunct2)
+ apply (rule assms(1))
+ done
+next
+ case True thus ?thesis
+ apply -
+ apply (rule Cnotzero_mono)
+ apply (rule cone_Cnotzero)
+ apply (rule Card_order_cexp)
+ apply (rule ordIso_imp_ordLeq)
+ apply (rule ordIso_symmetric)
+ apply (rule ordIso_transitive)
+ apply (rule cexp_cong2)
+ apply assumption
+ apply (rule conjunct2)
+ apply (rule assms(1))
+ apply (rule assms(2))
+ apply (simp only: card_of_Card_order czero_def)
+ apply (rule disjI1)
+ apply (rule assms(1))
+ apply (rule cexp_czero)
+ done
+qed
+
+lemma Cinfinite_ctwo_cexp:
+ "Cinfinite r \<Longrightarrow> Cinfinite (ctwo ^c r)"
+unfolding ctwo_def cexp_def cinfinite_def Field_card_of
+by (rule conjI, rule infinite_Func, auto, rule card_of_card_order_on)
+
+lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
+by (metis assms cinfinite_mono ordLeq_cexp2)
+
+lemma cinfinite_ccexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^^c r)"
+by (rule cinfinite_mono[OF cexp_ordLeq_ccexp cinfinite_cexp])
+
+lemma Cinfinite_cexp:
+ "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
+by (simp add: cinfinite_cexp Card_order_cexp)
+
+lemma ctwo_ordLess_natLeq:
+ "ctwo <o natLeq"
+unfolding ctwo_def using finite_iff_ordLess_natLeq finite_UNIV by fast
+
+lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
+by (metis ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite ordLess_ordLeq_trans)
+
+lemma ctwo_ordLeq_Cinfinite:
+ assumes "Cinfinite r"
+ shows "ctwo \<le>o r"
+by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
+
+lemma Cinfinite_ordLess_cexp:
+ assumes r: "Cinfinite r"
+ shows "r <o r ^c r"
+proof -
+ have "r <o ctwo ^c r" using r by (simp only: ordLess_ctwo_cexp)
+ also have "ctwo ^c r \<le>o r ^c r"
+ by (rule cexp_mono1[OF ctwo_ordLeq_Cinfinite]) (auto simp: r ctwo_not_czero Card_order_ctwo)
+ finally show ?thesis .
+qed
+
+lemma infinite_ordLeq_cexp:
+ assumes "Cinfinite r"
+ shows "r \<le>o r ^c r"
+by (rule ordLess_imp_ordLeq[OF Cinfinite_ordLess_cexp[OF assms]])
+
+lemma cone_ordLeq_iff_Field:
+ assumes "cone \<le>o r"
+ shows "Field r \<noteq> {}"
+proof (rule ccontr)
+ assume "\<not> Field r \<noteq> {}"
+ hence "Field r = {}" by simp
+ thus False using card_of_empty3
+ card_of_mono2[OF assms] Cnotzero_imp_not_empty[OF cone_Cnotzero] by auto
+qed
+
+lemma cone_ordLeq_cexp: "cone \<le>o r1 \<Longrightarrow> cone \<le>o r1 ^c r2"
+by (simp add: cexp_def cone_def Func_non_emp card_of_singl_ordLeq cone_ordLeq_iff_Field)
+
+lemma Card_order_czero: "Card_order czero"
+by (simp only: card_of_Card_order czero_def)
+
+lemma cexp_mono2'':
+ assumes 2: "p2 \<le>o r2"
+ and n1: "Cnotzero q"
+ and n2: "Card_order p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+proof (cases "p2 =o (czero :: 'a rel)")
+ case True
+ hence "q ^c p2 =o q ^c (czero :: 'a rel)" using n1 n2 cexp_cong2 Card_order_czero by blast
+ also have "q ^c (czero :: 'a rel) =o cone" using cexp_czero by blast
+ also have "cone \<le>o q ^c r2" using cone_ordLeq_cexp cone_ordLeq_Cnotzero n1 by blast
+ finally show ?thesis .
+next
+ case False thus ?thesis using assms cexp_mono2' czeroI by metis
+qed
+
+lemma Un_Cinfinite_bound: "\<lbrakk>|A| \<le>o r; |B| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |A \<union> B| \<le>o r"
+by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
+
+lemma UNION_Cinfinite_bound: "\<lbrakk>|I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |\<Union>i \<in> I. A i| \<le>o r"
+by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
+
+lemma csum_cinfinite_bound:
+ assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
+ shows "p +c q \<le>o r"
+proof -
+ from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
+ unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
+ with assms show ?thesis unfolding cinfinite_def csum_def
+ by (blast intro: card_of_Plus_ordLeq_infinite_Field)
+qed
+
+lemma csum_cexp: "\<lbrakk>Cinfinite r1; Cinfinite r2; Card_order q; ctwo \<le>o q\<rbrakk> \<Longrightarrow>
+ q ^c r1 +c q ^c r2 \<le>o q ^c (r1 +c r2)"
+apply (rule csum_cinfinite_bound)
+apply (rule cexp_mono2)
+apply (rule ordLeq_csum1)
+apply (erule conjunct2)
+apply assumption
+apply (rule disjI2)
+apply (rule ordLeq_transitive)
+apply (rule cone_ordLeq_ctwo)
+apply (rule ordLeq_transitive)
+apply assumption
+apply (rule ordLeq_cexp1)
+apply (rule Cinfinite_Cnotzero)
+apply (rule Cinfinite_csum)
+apply (rule disjI1)
+apply assumption
+apply assumption
+apply (rule notE)
+apply (rule cinfinite_not_czero[of r1])
+apply (erule conjunct1)
+apply assumption
+apply (erule conjunct2)
+apply (rule cexp_mono2)
+apply (rule ordLeq_csum2)
+apply (erule conjunct2)
+apply assumption
+apply (rule disjI2)
+apply (rule ordLeq_transitive)
+apply (rule cone_ordLeq_ctwo)
+apply (rule ordLeq_transitive)
+apply assumption
+apply (rule ordLeq_cexp1)
+apply (rule Cinfinite_Cnotzero)
+apply (rule Cinfinite_csum)
+apply (rule disjI1)
+apply assumption
+apply assumption
+apply (rule notE)
+apply (rule cinfinite_not_czero[of r2])
+apply (erule conjunct1)
+apply assumption
+apply (erule conjunct2)
+apply (rule Card_order_cexp)
+apply (rule Card_order_cexp)
+apply (rule Cinfinite_cexp)
+apply assumption
+apply (rule Cinfinite_csum)
+by (rule disjI1)
+
+lemma csum_cexp': "\<lbrakk>Cinfinite r; Card_order q; ctwo \<le>o q\<rbrakk> \<Longrightarrow> q +c r \<le>o q ^c r"
+apply (rule csum_cinfinite_bound)
+ apply (metis Cinfinite_Cnotzero ordLeq_cexp1)
+ apply (metis ordLeq_cexp2)
+ apply blast+
+by (metis Cinfinite_cexp)
+
+lemma cprod_cinfinite_bound:
+ assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
+ shows "p *c q \<le>o r"
+proof -
+ from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
+ unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
+ with assms show ?thesis unfolding cinfinite_def cprod_def
+ by (blast intro: card_of_Times_ordLeq_infinite_Field)
+qed
+
+lemma cprod_csum_cexp:
+ "r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
+unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
+proof -
+ let ?f = "\<lambda>(a, b). %x. if x then Some (Inl a) else Some (Inr b)"
+ have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
+ by (auto simp: inj_on_def fun_eq_iff split: bool.split)
+ moreover
+ have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
+ by (auto simp: Func_def)
+ ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
+qed
+
+lemma card_of_Sigma_ordLeq_Cinfinite:
+ "\<lbrakk>Cinfinite r; |I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r\<rbrakk> \<Longrightarrow> |SIGMA i : I. A i| \<le>o r"
+unfolding cinfinite_def by (blast intro: card_of_Sigma_ordLeq_infinite_Field)
+
+
+(* cardSuc *)
+
+lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
+by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
+
+lemma cardSuc_UNION_Cinfinite:
+ assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
+ shows "EX i : Field (cardSuc r). B \<le> As i"
+using cardSuc_UNION assms unfolding cinfinite_def by blast
+
+subsection {* Powerset *}
+
+definition cpow where "cpow r = |Pow (Field r)|"
+
+lemma card_order_cpow: "card_order r \<Longrightarrow> card_order (cpow r)"
+by (simp only: cpow_def Field_card_order Pow_UNIV card_of_card_order_on)
+
+lemma cpow_greater_eq: "Card_order r \<Longrightarrow> r \<le>o cpow r"
+by (rule ordLess_imp_ordLeq) (simp only: cpow_def Card_order_Pow)
+
+lemma Card_order_cpow: "Card_order (cpow r)"
+unfolding cpow_def by (rule card_of_Card_order)
+
+lemma Cinfinite_cpow: "Cinfinite r \<Longrightarrow> Cinfinite (cpow r)"
+unfolding cpow_def cinfinite_def by (metis Field_card_of card_of_Card_order infinite_Pow)
+
+lemma cardSuc_ordLeq_cpow: "Card_order r \<Longrightarrow> cardSuc r \<le>o cpow r"
+unfolding cpow_def by (metis Card_order_Pow cardSuc_ordLess_ordLeq card_of_Card_order)
+
+lemma cpow_cexp_ctwo: "cpow r =o ctwo ^c r"
+unfolding cpow_def ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+
+subsection {* Lists *}
+
+definition clists where "clists r = |lists (Field r)|"
+
+lemma clists_Cinfinite: "Cinfinite r \<Longrightarrow> clists r =o r"
+unfolding cinfinite_def clists_def by (blast intro: Card_order_lists_infinite)
+
+lemma Card_order_clists: "Card_order (clists r)"
+unfolding clists_def by (rule card_of_Card_order)
+
+lemma Cnotzero_clists: "Cnotzero (clists r)"
+by (simp add: clists_def card_of_ordIso_czero_iff_empty lists_not_empty) (rule card_of_Card_order)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Cardinal_Order_Relation.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1099 @@
+(* Title: HOL/Ordinals_and_Cardinals/Cardinal_Order_Relation.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Cardinal-order relations.
+*)
+
+header {* Cardinal-Order Relations *}
+
+theory Cardinal_Order_Relation
+imports
+ "../Ordinals_and_Cardinals_Base/Cardinal_Order_Relation_Base"
+ Constructions_on_Wellorders
+begin
+
+declare
+ card_order_on_well_order_on[simp]
+ card_of_card_order_on[simp]
+ card_of_well_order_on[simp]
+ Field_card_of[simp]
+ card_of_Card_order[simp]
+ card_of_Well_order[simp]
+ card_of_least[simp]
+ card_of_unique[simp]
+ card_of_mono1[simp]
+ card_of_mono2[simp]
+ card_of_cong[simp]
+ card_of_Field_ordLess[simp]
+ card_of_Field_ordIso[simp]
+ card_of_underS[simp]
+ ordLess_Field[simp]
+ card_of_empty[simp]
+ card_of_empty1[simp]
+ card_of_image[simp]
+ card_of_singl_ordLeq[simp]
+ Card_order_singl_ordLeq[simp]
+ card_of_Pow[simp]
+ Card_order_Pow[simp]
+ card_of_set_type[simp]
+ card_of_Plus1[simp]
+ Card_order_Plus1[simp]
+ card_of_Plus2[simp]
+ Card_order_Plus2[simp]
+ card_of_Plus_mono1[simp]
+ card_of_Plus_mono2[simp]
+ card_of_Plus_mono[simp]
+ card_of_Plus_cong2[simp]
+ card_of_Plus_cong[simp]
+ card_of_Un1[simp]
+ card_of_diff[simp]
+ card_of_Un_Plus_ordLeq[simp]
+ card_of_Times1[simp]
+ card_of_Times2[simp]
+ card_of_Times3[simp]
+ card_of_Times_mono1[simp]
+ card_of_Times_mono2[simp]
+ card_of_Times_cong1[simp]
+ card_of_Times_cong2[simp]
+ card_of_ordIso_finite[simp]
+ finite_ordLess_infinite2[simp]
+ card_of_Times_same_infinite[simp]
+ card_of_Times_infinite_simps[simp]
+ card_of_Plus_infinite1[simp]
+ card_of_Plus_infinite2[simp]
+ card_of_Plus_ordLess_infinite[simp]
+ card_of_Plus_ordLess_infinite_Field[simp]
+ card_of_lists_infinite[simp]
+ infinite_cartesian_product[simp]
+ cardSuc_Card_order[simp]
+ cardSuc_greater[simp]
+ cardSuc_ordLeq[simp]
+ cardSuc_ordLeq_ordLess[simp]
+ cardSuc_mono_ordLeq[simp]
+ cardSuc_invar_ordIso[simp]
+ card_of_cardSuc_finite[simp]
+ cardSuc_finite[simp]
+ card_of_Plus_ordLeq_infinite_Field[simp]
+ curr_in[intro, simp]
+ Func_empty[simp]
+ Func_map_empty[simp]
+ Func_is_emp[simp]
+
+
+subsection {* Cardinal of a set *}
+
+lemma card_of_inj_rel: assumes INJ: "!! x y y'. \<lbrakk>(x,y) : R; (x,y') : R\<rbrakk> \<Longrightarrow> y = y'"
+shows "|{y. EX x. (x,y) : R}| <=o |{x. EX y. (x,y) : R}|"
+proof-
+ let ?Y = "{y. EX x. (x,y) : R}" let ?X = "{x. EX y. (x,y) : R}"
+ let ?f = "% y. SOME x. (x,y) : R"
+ have "?f ` ?Y <= ?X" using someI by force (* FIXME: takes a bit long *)
+ moreover have "inj_on ?f ?Y"
+ unfolding inj_on_def proof(auto)
+ fix y1 x1 y2 x2
+ assume *: "(x1, y1) \<in> R" "(x2, y2) \<in> R" and **: "?f y1 = ?f y2"
+ hence "(?f y1,y1) : R" using someI[of "% x. (x,y1) : R"] by auto
+ moreover have "(?f y2,y2) : R" using * someI[of "% x. (x,y2) : R"] by auto
+ ultimately show "y1 = y2" using ** INJ by auto
+ qed
+ ultimately show "|?Y| <=o |?X|" using card_of_ordLeq by blast
+qed
+
+lemma card_of_unique2: "\<lbrakk>card_order_on B r; bij_betw f A B\<rbrakk> \<Longrightarrow> r =o |A|"
+using card_of_ordIso card_of_unique ordIso_equivalence by blast
+
+lemma internalize_card_of_ordLess:
+"( |A| <o r) = (\<exists>B < Field r. |A| =o |B| \<and> |B| <o r)"
+proof
+ assume "|A| <o r"
+ then obtain p where 1: "Field p < Field r \<and> |A| =o p \<and> p <o r"
+ using internalize_ordLess[of "|A|" r] by blast
+ hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
+ hence "|Field p| =o p" using card_of_Field_ordIso by blast
+ hence "|A| =o |Field p| \<and> |Field p| <o r"
+ using 1 ordIso_equivalence ordIso_ordLess_trans by blast
+ thus "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r" using 1 by blast
+next
+ assume "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r"
+ thus "|A| <o r" using ordIso_ordLess_trans by blast
+qed
+
+lemma internalize_card_of_ordLess2:
+"( |A| <o |C| ) = (\<exists>B < C. |A| =o |B| \<and> |B| <o |C| )"
+using internalize_card_of_ordLess[of "A" "|C|"] Field_card_of[of C] by auto
+
+lemma Card_order_omax:
+assumes "finite R" and "R \<noteq> {}" and "\<forall>r\<in>R. Card_order r"
+shows "Card_order (omax R)"
+proof-
+ have "\<forall>r\<in>R. Well_order r"
+ using assms unfolding card_order_on_def by simp
+ thus ?thesis using assms apply - apply(drule omax_in) by auto
+qed
+
+lemma Card_order_omax2:
+assumes "finite I" and "I \<noteq> {}"
+shows "Card_order (omax {|A i| | i. i \<in> I})"
+proof-
+ let ?R = "{|A i| | i. i \<in> I}"
+ have "finite ?R" and "?R \<noteq> {}" using assms by auto
+ moreover have "\<forall>r\<in>?R. Card_order r"
+ using card_of_Card_order by auto
+ ultimately show ?thesis by(rule Card_order_omax)
+qed
+
+
+subsection {* Cardinals versus set operations on arbitrary sets *}
+
+lemma subset_ordLeq_strict:
+assumes "A \<le> B" and "|A| <o |B|"
+shows "A < B"
+proof-
+ {assume "\<not>(A < B)"
+ hence "A = B" using assms(1) by blast
+ hence False using assms(2) not_ordLess_ordIso card_of_refl by blast
+ }
+ thus ?thesis by blast
+qed
+
+corollary subset_ordLeq_diff:
+assumes "A \<le> B" and "|A| <o |B|"
+shows "B - A \<noteq> {}"
+using assms subset_ordLeq_strict by blast
+
+lemma card_of_empty4:
+"|{}::'b set| <o |A::'a set| = (A \<noteq> {})"
+proof(intro iffI notI)
+ assume *: "|{}::'b set| <o |A|" and "A = {}"
+ hence "|A| =o |{}::'b set|"
+ using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
+ hence "|{}::'b set| =o |A|" using ordIso_symmetric by blast
+ with * show False using not_ordLess_ordIso[of "|{}::'b set|" "|A|"] by blast
+next
+ assume "A \<noteq> {}"
+ hence "(\<not> (\<exists>f. inj_on f A \<and> f ` A \<subseteq> {}))"
+ unfolding inj_on_def by blast
+ thus "| {} | <o | A |"
+ using card_of_ordLess by blast
+qed
+
+lemma card_of_empty5:
+"|A| <o |B| \<Longrightarrow> B \<noteq> {}"
+using card_of_empty not_ordLess_ordLeq by blast
+
+lemma Well_order_card_of_empty:
+"Well_order r \<Longrightarrow> |{}| \<le>o r" by simp
+
+lemma card_of_UNIV[simp]:
+"|A :: 'a set| \<le>o |UNIV :: 'a set|"
+using card_of_mono1[of A] by simp
+
+lemma card_of_UNIV2[simp]:
+"Card_order r \<Longrightarrow> (r :: 'a rel) \<le>o |UNIV :: 'a set|"
+using card_of_UNIV[of "Field r"] card_of_Field_ordIso
+ ordIso_symmetric ordIso_ordLeq_trans by blast
+
+lemma card_of_Pow_mono[simp]:
+assumes "|A| \<le>o |B|"
+shows "|Pow A| \<le>o |Pow B|"
+proof-
+ obtain f where "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A B] by auto
+ hence "inj_on (image f) (Pow A) \<and> (image f) ` (Pow A) \<le> (Pow B)"
+ by (auto simp add: inj_on_image_Pow image_Pow_mono)
+ thus ?thesis using card_of_ordLeq[of "Pow A"] by metis
+qed
+
+lemma ordIso_Pow_mono[simp]:
+assumes "r \<le>o r'"
+shows "|Pow(Field r)| \<le>o |Pow(Field r')|"
+using assms card_of_mono2 card_of_Pow_mono by blast
+
+lemma card_of_Pow_cong[simp]:
+assumes "|A| =o |B|"
+shows "|Pow A| =o |Pow B|"
+proof-
+ obtain f where "bij_betw f A B"
+ using assms card_of_ordIso[of A B] by auto
+ hence "bij_betw (image f) (Pow A) (Pow B)"
+ by (auto simp add: bij_betw_image_Pow)
+ thus ?thesis using card_of_ordIso[of "Pow A"] by auto
+qed
+
+lemma ordIso_Pow_cong[simp]:
+assumes "r =o r'"
+shows "|Pow(Field r)| =o |Pow(Field r')|"
+using assms card_of_cong card_of_Pow_cong by blast
+
+corollary Card_order_Plus_empty1:
+"Card_order r \<Longrightarrow> r =o |(Field r) <+> {}|"
+using card_of_Plus_empty1 card_of_Field_ordIso ordIso_equivalence by blast
+
+corollary Card_order_Plus_empty2:
+"Card_order r \<Longrightarrow> r =o |{} <+> (Field r)|"
+using card_of_Plus_empty2 card_of_Field_ordIso ordIso_equivalence by blast
+
+lemma Card_order_Un1:
+shows "Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<union> B| "
+using card_of_Un1 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto
+
+lemma card_of_Un2[simp]:
+shows "|A| \<le>o |B \<union> A|"
+using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
+
+lemma Card_order_Un2:
+shows "Card_order r \<Longrightarrow> |Field r| \<le>o |A \<union> (Field r)| "
+using card_of_Un2 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto
+
+lemma Un_Plus_bij_betw:
+assumes "A Int B = {}"
+shows "\<exists>f. bij_betw f (A \<union> B) (A <+> B)"
+proof-
+ let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
+ have "bij_betw ?f (A \<union> B) (A <+> B)"
+ using assms by(unfold bij_betw_def inj_on_def, auto)
+ thus ?thesis by blast
+qed
+
+lemma card_of_Un_Plus_ordIso:
+assumes "A Int B = {}"
+shows "|A \<union> B| =o |A <+> B|"
+using assms card_of_ordIso[of "A \<union> B"] Un_Plus_bij_betw[of A B] by auto
+
+lemma card_of_Un_Plus_ordIso1:
+"|A \<union> B| =o |A <+> (B - A)|"
+using card_of_Un_Plus_ordIso[of A "B - A"] by auto
+
+lemma card_of_Un_Plus_ordIso2:
+"|A \<union> B| =o |(A - B) <+> B|"
+using card_of_Un_Plus_ordIso[of "A - B" B] by auto
+
+lemma card_of_Times_singl1: "|A| =o |A \<times> {b}|"
+proof-
+ have "bij_betw fst (A \<times> {b}) A" unfolding bij_betw_def inj_on_def by force
+ thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+qed
+
+corollary Card_order_Times_singl1:
+"Card_order r \<Longrightarrow> r =o |(Field r) \<times> {b}|"
+using card_of_Times_singl1[of _ b] card_of_Field_ordIso ordIso_equivalence by blast
+
+lemma card_of_Times_singl2: "|A| =o |{b} \<times> A|"
+proof-
+ have "bij_betw snd ({b} \<times> A) A" unfolding bij_betw_def inj_on_def by force
+ thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+qed
+
+corollary Card_order_Times_singl2:
+"Card_order r \<Longrightarrow> r =o |{a} \<times> (Field r)|"
+using card_of_Times_singl2[of _ a] card_of_Field_ordIso ordIso_equivalence by blast
+
+lemma card_of_Times_assoc: "|(A \<times> B) \<times> C| =o |A \<times> B \<times> C|"
+proof -
+ let ?f = "\<lambda>((a,b),c). (a,(b,c))"
+ have "A \<times> B \<times> C \<subseteq> ?f ` ((A \<times> B) \<times> C)"
+ proof
+ fix x assume "x \<in> A \<times> B \<times> C"
+ then obtain a b c where *: "a \<in> A" "b \<in> B" "c \<in> C" "x = (a, b, c)" by blast
+ let ?x = "((a, b), c)"
+ from * have "?x \<in> (A \<times> B) \<times> C" "x = ?f ?x" by auto
+ thus "x \<in> ?f ` ((A \<times> B) \<times> C)" by blast
+ qed
+ hence "bij_betw ?f ((A \<times> B) \<times> C) (A \<times> B \<times> C)"
+ unfolding bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+corollary Card_order_Times3:
+"Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<times> (Field r)|"
+using card_of_Times3 card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
+
+lemma card_of_Times_mono[simp]:
+assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
+shows "|A \<times> C| \<le>o |B \<times> D|"
+using assms card_of_Times_mono1[of A B C] card_of_Times_mono2[of C D B]
+ ordLeq_transitive[of "|A \<times> C|"] by blast
+
+corollary ordLeq_Times_mono:
+assumes "r \<le>o r'" and "p \<le>o p'"
+shows "|(Field r) \<times> (Field p)| \<le>o |(Field r') \<times> (Field p')|"
+using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Times_mono by blast
+
+corollary ordIso_Times_cong1[simp]:
+assumes "r =o r'"
+shows "|(Field r) \<times> C| =o |(Field r') \<times> C|"
+using assms card_of_cong card_of_Times_cong1 by blast
+
+lemma card_of_Times_cong[simp]:
+assumes "|A| =o |B|" and "|C| =o |D|"
+shows "|A \<times> C| =o |B \<times> D|"
+using assms
+by (auto simp add: ordIso_iff_ordLeq)
+
+corollary ordIso_Times_cong:
+assumes "r =o r'" and "p =o p'"
+shows "|(Field r) \<times> (Field p)| =o |(Field r') \<times> (Field p')|"
+using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Times_cong by blast
+
+lemma card_of_Sigma_mono2:
+assumes "inj_on f (I::'i set)" and "f ` I \<le> (J::'j set)"
+shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| \<le>o |SIGMA j : J. A j|"
+proof-
+ let ?LEFT = "SIGMA i : I. A (f i)"
+ let ?RIGHT = "SIGMA j : J. A j"
+ obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast
+ have "inj_on u ?LEFT \<and> u `?LEFT \<le> ?RIGHT"
+ using assms unfolding u_def inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by blast
+qed
+
+lemma card_of_Sigma_mono:
+assumes INJ: "inj_on f I" and IM: "f ` I \<le> J" and
+ LEQ: "\<forall>j \<in> J. |A j| \<le>o |B j|"
+shows "|SIGMA i : I. A (f i)| \<le>o |SIGMA j : J. B j|"
+proof-
+ have "\<forall>i \<in> I. |A(f i)| \<le>o |B(f i)|"
+ using IM LEQ by blast
+ hence "|SIGMA i : I. A (f i)| \<le>o |SIGMA i : I. B (f i)|"
+ using card_of_Sigma_mono1[of I] by metis
+ moreover have "|SIGMA i : I. B (f i)| \<le>o |SIGMA j : J. B j|"
+ using INJ IM card_of_Sigma_mono2 by blast
+ ultimately show ?thesis using ordLeq_transitive by blast
+qed
+
+
+lemma ordLeq_Sigma_mono1:
+assumes "\<forall>i \<in> I. p i \<le>o r i"
+shows "|SIGMA i : I. Field(p i)| \<le>o |SIGMA i : I. Field(r i)|"
+using assms by (auto simp add: card_of_Sigma_mono1)
+
+
+lemma ordLeq_Sigma_mono:
+assumes "inj_on f I" and "f ` I \<le> J" and
+ "\<forall>j \<in> J. p j \<le>o r j"
+shows "|SIGMA i : I. Field(p(f i))| \<le>o |SIGMA j : J. Field(r j)|"
+using assms card_of_mono2 card_of_Sigma_mono
+ [of f I J "\<lambda> i. Field(p i)" "\<lambda> j. Field(r j)"] by metis
+
+
+lemma card_of_Sigma_cong1:
+assumes "\<forall>i \<in> I. |A i| =o |B i|"
+shows "|SIGMA i : I. A i| =o |SIGMA i : I. B i|"
+using assms by (auto simp add: card_of_Sigma_mono1 ordIso_iff_ordLeq)
+
+
+lemma card_of_Sigma_cong2:
+assumes "bij_betw f (I::'i set) (J::'j set)"
+shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| =o |SIGMA j : J. A j|"
+proof-
+ let ?LEFT = "SIGMA i : I. A (f i)"
+ let ?RIGHT = "SIGMA j : J. A j"
+ obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast
+ have "bij_betw u ?LEFT ?RIGHT"
+ using assms unfolding u_def bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+lemma card_of_Sigma_cong:
+assumes BIJ: "bij_betw f I J" and
+ ISO: "\<forall>j \<in> J. |A j| =o |B j|"
+shows "|SIGMA i : I. A (f i)| =o |SIGMA j : J. B j|"
+proof-
+ have "\<forall>i \<in> I. |A(f i)| =o |B(f i)|"
+ using ISO BIJ unfolding bij_betw_def by blast
+ hence "|SIGMA i : I. A (f i)| =o |SIGMA i : I. B (f i)|"
+ using card_of_Sigma_cong1 by metis
+ moreover have "|SIGMA i : I. B (f i)| =o |SIGMA j : J. B j|"
+ using BIJ card_of_Sigma_cong2 by blast
+ ultimately show ?thesis using ordIso_transitive by blast
+qed
+
+lemma ordIso_Sigma_cong1:
+assumes "\<forall>i \<in> I. p i =o r i"
+shows "|SIGMA i : I. Field(p i)| =o |SIGMA i : I. Field(r i)|"
+using assms by (auto simp add: card_of_Sigma_cong1)
+
+lemma ordLeq_Sigma_cong:
+assumes "bij_betw f I J" and
+ "\<forall>j \<in> J. p j =o r j"
+shows "|SIGMA i : I. Field(p(f i))| =o |SIGMA j : J. Field(r j)|"
+using assms card_of_cong card_of_Sigma_cong
+ [of f I J "\<lambda> j. Field(p j)" "\<lambda> j. Field(r j)"] by blast
+
+corollary ordLeq_Sigma_Times:
+"\<forall>i \<in> I. p i \<le>o r \<Longrightarrow> |SIGMA i : I. Field (p i)| \<le>o |I \<times> (Field r)|"
+by (auto simp add: card_of_Sigma_Times)
+
+lemma card_of_UNION_Sigma2:
+assumes
+"!! i j. \<lbrakk>{i,j} <= I; i ~= j\<rbrakk> \<Longrightarrow> A i Int A j = {}"
+shows
+"|\<Union>i\<in>I. A i| =o |Sigma I A|"
+proof-
+ let ?L = "\<Union>i\<in>I. A i" let ?R = "Sigma I A"
+ have "|?L| <=o |?R|" using card_of_UNION_Sigma .
+ moreover have "|?R| <=o |?L|"
+ proof-
+ have "inj_on snd ?R"
+ unfolding inj_on_def using assms by auto
+ moreover have "snd ` ?R <= ?L" by auto
+ ultimately show ?thesis using card_of_ordLeq by blast
+ qed
+ ultimately show ?thesis by(simp add: ordIso_iff_ordLeq)
+qed
+
+corollary Plus_into_Times:
+assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
+shows "\<exists>f. inj_on f (A <+> B) \<and> f ` (A <+> B) \<le> A \<times> B"
+using assms by (auto simp add: card_of_Plus_Times card_of_ordLeq)
+
+corollary Plus_into_Times_types:
+assumes A2: "(a1::'a) \<noteq> a2" and B2: "(b1::'b) \<noteq> b2"
+shows "\<exists>(f::'a + 'b \<Rightarrow> 'a * 'b). inj f"
+using assms Plus_into_Times[of a1 a2 UNIV b1 b2 UNIV]
+by auto
+
+corollary Times_same_infinite_bij_betw:
+assumes "infinite A"
+shows "\<exists>f. bij_betw f (A \<times> A) A"
+using assms by (auto simp add: card_of_ordIso)
+
+corollary Times_same_infinite_bij_betw_types:
+assumes INF: "infinite(UNIV::'a set)"
+shows "\<exists>(f::('a * 'a) => 'a). bij f"
+using assms Times_same_infinite_bij_betw[of "UNIV::'a set"]
+by auto
+
+corollary Times_infinite_bij_betw:
+assumes INF: "infinite A" and NE: "B \<noteq> {}" and INJ: "inj_on g B \<and> g ` B \<le> A"
+shows "(\<exists>f. bij_betw f (A \<times> B) A) \<and> (\<exists>h. bij_betw h (B \<times> A) A)"
+proof-
+ have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast
+ thus ?thesis using INF NE
+ by (auto simp add: card_of_ordIso card_of_Times_infinite)
+qed
+
+corollary Times_infinite_bij_betw_types:
+assumes INF: "infinite(UNIV::'a set)" and
+ BIJ: "inj(g::'b \<Rightarrow> 'a)"
+shows "(\<exists>(f::('b * 'a) => 'a). bij f) \<and> (\<exists>(h::('a * 'b) => 'a). bij h)"
+using assms Times_infinite_bij_betw[of "UNIV::'a set" "UNIV::'b set" g]
+by auto
+
+lemma card_of_Times_ordLeq_infinite:
+"\<lbrakk>infinite C; |A| \<le>o |C|; |B| \<le>o |C|\<rbrakk>
+ \<Longrightarrow> |A <*> B| \<le>o |C|"
+by(simp add: card_of_Sigma_ordLeq_infinite)
+
+corollary Plus_infinite_bij_betw:
+assumes INF: "infinite A" and INJ: "inj_on g B \<and> g ` B \<le> A"
+shows "(\<exists>f. bij_betw f (A <+> B) A) \<and> (\<exists>h. bij_betw h (B <+> A) A)"
+proof-
+ have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast
+ thus ?thesis using INF
+ by (auto simp add: card_of_ordIso)
+qed
+
+corollary Plus_infinite_bij_betw_types:
+assumes INF: "infinite(UNIV::'a set)" and
+ BIJ: "inj(g::'b \<Rightarrow> 'a)"
+shows "(\<exists>(f::('b + 'a) => 'a). bij f) \<and> (\<exists>(h::('a + 'b) => 'a). bij h)"
+using assms Plus_infinite_bij_betw[of "UNIV::'a set" g "UNIV::'b set"]
+by auto
+
+lemma card_of_Un_infinite_simps[simp]:
+"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |A \<union> B| =o |A|"
+"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |B \<union> A| =o |A|"
+using card_of_Un_infinite by auto
+
+corollary Card_order_Un_infinite:
+assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
+ LEQ: "p \<le>o r"
+shows "| (Field r) \<union> (Field p) | =o r \<and> | (Field p) \<union> (Field r) | =o r"
+proof-
+ have "| Field r \<union> Field p | =o | Field r | \<and>
+ | Field p \<union> Field r | =o | Field r |"
+ using assms by (auto simp add: card_of_Un_infinite)
+ thus ?thesis
+ using assms card_of_Field_ordIso[of r]
+ ordIso_transitive[of "|Field r \<union> Field p|"]
+ ordIso_transitive[of _ "|Field r|"] by blast
+qed
+
+corollary subset_ordLeq_diff_infinite:
+assumes INF: "infinite B" and SUB: "A \<le> B" and LESS: "|A| <o |B|"
+shows "infinite (B - A)"
+using assms card_of_Un_diff_infinite card_of_ordIso_finite by blast
+
+lemma card_of_Times_ordLess_infinite[simp]:
+assumes INF: "infinite C" and
+ LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+shows "|A \<times> B| <o |C|"
+proof(cases "A = {} \<or> B = {}")
+ assume Case1: "A = {} \<or> B = {}"
+ hence "A \<times> B = {}" by blast
+ moreover have "C \<noteq> {}" using
+ LESS1 card_of_empty5 by blast
+ ultimately show ?thesis by(auto simp add: card_of_empty4)
+next
+ assume Case2: "\<not>(A = {} \<or> B = {})"
+ {assume *: "|C| \<le>o |A \<times> B|"
+ hence "infinite (A \<times> B)" using INF card_of_ordLeq_finite by blast
+ hence 1: "infinite A \<or> infinite B" using finite_cartesian_product by blast
+ {assume Case21: "|A| \<le>o |B|"
+ hence "infinite B" using 1 card_of_ordLeq_finite by blast
+ hence "|A \<times> B| =o |B|" using Case2 Case21
+ by (auto simp add: card_of_Times_infinite)
+ hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+ }
+ moreover
+ {assume Case22: "|B| \<le>o |A|"
+ hence "infinite A" using 1 card_of_ordLeq_finite by blast
+ hence "|A \<times> B| =o |A|" using Case2 Case22
+ by (auto simp add: card_of_Times_infinite)
+ hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+ }
+ ultimately have False using ordLeq_total card_of_Well_order[of A]
+ card_of_Well_order[of B] by blast
+ }
+ thus ?thesis using ordLess_or_ordLeq[of "|A \<times> B|" "|C|"]
+ card_of_Well_order[of "A \<times> B"] card_of_Well_order[of "C"] by auto
+qed
+
+lemma card_of_Times_ordLess_infinite_Field[simp]:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LESS1: "|A| <o r" and LESS2: "|B| <o r"
+shows "|A \<times> B| <o r"
+proof-
+ let ?C = "Field r"
+ have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|A| <o |?C|" "|B| <o |?C|"
+ using LESS1 LESS2 ordLess_ordIso_trans by blast+
+ hence "|A <*> B| <o |?C|" using INF
+ card_of_Times_ordLess_infinite by blast
+ thus ?thesis using 1 ordLess_ordIso_trans by blast
+qed
+
+lemma card_of_Un_ordLess_infinite[simp]:
+assumes INF: "infinite C" and
+ LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+shows "|A \<union> B| <o |C|"
+using assms card_of_Plus_ordLess_infinite card_of_Un_Plus_ordLeq
+ ordLeq_ordLess_trans by blast
+
+lemma card_of_Un_ordLess_infinite_Field[simp]:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LESS1: "|A| <o r" and LESS2: "|B| <o r"
+shows "|A Un B| <o r"
+proof-
+ let ?C = "Field r"
+ have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|A| <o |?C|" "|B| <o |?C|"
+ using LESS1 LESS2 ordLess_ordIso_trans by blast+
+ hence "|A Un B| <o |?C|" using INF
+ card_of_Un_ordLess_infinite by blast
+ thus ?thesis using 1 ordLess_ordIso_trans by blast
+qed
+
+lemma card_of_Un_singl_ordLess_infinite1:
+assumes "infinite B" and "|A| <o |B|"
+shows "|{a} Un A| <o |B|"
+proof-
+ have "|{a}| <o |B|" using assms by auto
+ thus ?thesis using assms card_of_Un_ordLess_infinite[of B] by fastforce
+qed
+
+lemma card_of_Un_singl_ordLess_infinite:
+assumes "infinite B"
+shows "( |A| <o |B| ) = ( |{a} Un A| <o |B| )"
+using assms card_of_Un_singl_ordLess_infinite1[of B A]
+proof(auto)
+ assume "|insert a A| <o |B|"
+ moreover have "|A| <=o |insert a A|" using card_of_mono1[of A] by blast
+ ultimately show "|A| <o |B|" using ordLeq_ordLess_trans by blast
+qed
+
+
+subsection {* Cardinals versus lists *}
+
+lemma Card_order_lists: "Card_order r \<Longrightarrow> r \<le>o |lists(Field r) |"
+using card_of_lists card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
+lemma Union_set_lists:
+"Union(set ` (lists A)) = A"
+unfolding lists_def2 proof(auto)
+ fix a assume "a \<in> A"
+ hence "set [a] \<le> A \<and> a \<in> set [a]" by auto
+ thus "\<exists>l. set l \<le> A \<and> a \<in> set l" by blast
+qed
+
+lemma inj_on_map_lists:
+assumes "inj_on f A"
+shows "inj_on (map f) (lists A)"
+using assms Union_set_lists[of A] inj_on_mapI[of f "lists A"] by auto
+
+lemma map_lists_mono:
+assumes "f ` A \<le> B"
+shows "(map f) ` (lists A) \<le> lists B"
+using assms unfolding lists_def2 by (auto, blast) (* lethal combination of methods :) *)
+
+lemma map_lists_surjective:
+assumes "f ` A = B"
+shows "(map f) ` (lists A) = lists B"
+using assms unfolding lists_def2
+proof (auto, blast)
+ fix l' assume *: "set l' \<le> f ` A"
+ have "set l' \<le> f ` A \<longrightarrow> l' \<in> map f ` {l. set l \<le> A}"
+ proof(induct l', auto)
+ fix l a
+ assume "a \<in> A" and "set l \<le> A" and
+ IH: "f ` (set l) \<le> f ` A"
+ hence "set (a # l) \<le> A" by auto
+ hence "map f (a # l) \<in> map f ` {l. set l \<le> A}" by blast
+ thus "f a # map f l \<in> map f ` {l. set l \<le> A}" by auto
+ qed
+ thus "l' \<in> map f ` {l. set l \<le> A}" using * by auto
+qed
+
+lemma bij_betw_map_lists:
+assumes "bij_betw f A B"
+shows "bij_betw (map f) (lists A) (lists B)"
+using assms unfolding bij_betw_def
+by(auto simp add: inj_on_map_lists map_lists_surjective)
+
+lemma card_of_lists_mono[simp]:
+assumes "|A| \<le>o |B|"
+shows "|lists A| \<le>o |lists B|"
+proof-
+ obtain f where "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A B] by auto
+ hence "inj_on (map f) (lists A) \<and> (map f) ` (lists A) \<le> (lists B)"
+ by (auto simp add: inj_on_map_lists map_lists_mono)
+ thus ?thesis using card_of_ordLeq[of "lists A"] by metis
+qed
+
+lemma ordIso_lists_mono:
+assumes "r \<le>o r'"
+shows "|lists(Field r)| \<le>o |lists(Field r')|"
+using assms card_of_mono2 card_of_lists_mono by blast
+
+lemma card_of_lists_cong[simp]:
+assumes "|A| =o |B|"
+shows "|lists A| =o |lists B|"
+proof-
+ obtain f where "bij_betw f A B"
+ using assms card_of_ordIso[of A B] by auto
+ hence "bij_betw (map f) (lists A) (lists B)"
+ by (auto simp add: bij_betw_map_lists)
+ thus ?thesis using card_of_ordIso[of "lists A"] by auto
+qed
+
+lemma ordIso_lists_cong:
+assumes "r =o r'"
+shows "|lists(Field r)| =o |lists(Field r')|"
+using assms card_of_cong card_of_lists_cong by blast
+
+corollary lists_infinite_bij_betw:
+assumes "infinite A"
+shows "\<exists>f. bij_betw f (lists A) A"
+using assms card_of_lists_infinite card_of_ordIso by blast
+
+corollary lists_infinite_bij_betw_types:
+assumes "infinite(UNIV :: 'a set)"
+shows "\<exists>(f::'a list \<Rightarrow> 'a). bij f"
+using assms assms lists_infinite_bij_betw[of "UNIV::'a set"]
+using lists_UNIV by auto
+
+
+subsection {* Cardinals versus the set-of-finite-sets operator *}
+
+definition Fpow :: "'a set \<Rightarrow> 'a set set"
+where "Fpow A \<equiv> {X. X \<le> A \<and> finite X}"
+
+lemma Fpow_mono: "A \<le> B \<Longrightarrow> Fpow A \<le> Fpow B"
+unfolding Fpow_def by auto
+
+lemma empty_in_Fpow: "{} \<in> Fpow A"
+unfolding Fpow_def by auto
+
+lemma Fpow_not_empty: "Fpow A \<noteq> {}"
+using empty_in_Fpow by blast
+
+lemma Fpow_subset_Pow: "Fpow A \<le> Pow A"
+unfolding Fpow_def by auto
+
+lemma card_of_Fpow[simp]: "|A| \<le>o |Fpow A|"
+proof-
+ let ?h = "\<lambda> a. {a}"
+ have "inj_on ?h A \<and> ?h ` A \<le> Fpow A"
+ unfolding inj_on_def Fpow_def by auto
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+lemma Card_order_Fpow: "Card_order r \<Longrightarrow> r \<le>o |Fpow(Field r) |"
+using card_of_Fpow card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
+lemma Fpow_Pow_finite: "Fpow A = Pow A Int {A. finite A}"
+unfolding Fpow_def Pow_def by blast
+
+lemma inj_on_image_Fpow:
+assumes "inj_on f A"
+shows "inj_on (image f) (Fpow A)"
+using assms Fpow_subset_Pow[of A] subset_inj_on[of "image f" "Pow A"]
+ inj_on_image_Pow by blast
+
+lemma image_Fpow_mono:
+assumes "f ` A \<le> B"
+shows "(image f) ` (Fpow A) \<le> Fpow B"
+using assms by(unfold Fpow_def, auto)
+
+lemma image_Fpow_surjective:
+assumes "f ` A = B"
+shows "(image f) ` (Fpow A) = Fpow B"
+using assms proof(unfold Fpow_def, auto)
+ fix Y assume *: "Y \<le> f ` A" and **: "finite Y"
+ hence "\<forall>b \<in> Y. \<exists>a. a \<in> A \<and> f a = b" by auto
+ with bchoice[of Y "\<lambda>b a. a \<in> A \<and> f a = b"]
+ obtain g where 1: "\<forall>b \<in> Y. g b \<in> A \<and> f(g b) = b" by blast
+ obtain X where X_def: "X = g ` Y" by blast
+ have "f ` X = Y \<and> X \<le> A \<and> finite X"
+ by(unfold X_def, force simp add: ** 1)
+ thus "Y \<in> (image f) ` {X. X \<le> A \<and> finite X}" by auto
+qed
+
+lemma bij_betw_image_Fpow:
+assumes "bij_betw f A B"
+shows "bij_betw (image f) (Fpow A) (Fpow B)"
+using assms unfolding bij_betw_def
+by (auto simp add: inj_on_image_Fpow image_Fpow_surjective)
+
+lemma card_of_Fpow_mono[simp]:
+assumes "|A| \<le>o |B|"
+shows "|Fpow A| \<le>o |Fpow B|"
+proof-
+ obtain f where "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A B] by auto
+ hence "inj_on (image f) (Fpow A) \<and> (image f) ` (Fpow A) \<le> (Fpow B)"
+ by (auto simp add: inj_on_image_Fpow image_Fpow_mono)
+ thus ?thesis using card_of_ordLeq[of "Fpow A"] by auto
+qed
+
+lemma ordIso_Fpow_mono:
+assumes "r \<le>o r'"
+shows "|Fpow(Field r)| \<le>o |Fpow(Field r')|"
+using assms card_of_mono2 card_of_Fpow_mono by blast
+
+lemma card_of_Fpow_cong[simp]:
+assumes "|A| =o |B|"
+shows "|Fpow A| =o |Fpow B|"
+proof-
+ obtain f where "bij_betw f A B"
+ using assms card_of_ordIso[of A B] by auto
+ hence "bij_betw (image f) (Fpow A) (Fpow B)"
+ by (auto simp add: bij_betw_image_Fpow)
+ thus ?thesis using card_of_ordIso[of "Fpow A"] by auto
+qed
+
+lemma ordIso_Fpow_cong:
+assumes "r =o r'"
+shows "|Fpow(Field r)| =o |Fpow(Field r')|"
+using assms card_of_cong card_of_Fpow_cong by blast
+
+lemma card_of_Fpow_lists: "|Fpow A| \<le>o |lists A|"
+proof-
+ have "set ` (lists A) = Fpow A"
+ unfolding lists_def2 Fpow_def using finite_list finite_set by blast
+ thus ?thesis using card_of_ordLeq2[of "Fpow A"] Fpow_not_empty[of A] by blast
+qed
+
+lemma card_of_Fpow_infinite[simp]:
+assumes "infinite A"
+shows "|Fpow A| =o |A|"
+using assms card_of_Fpow_lists card_of_lists_infinite card_of_Fpow
+ ordLeq_ordIso_trans ordIso_iff_ordLeq by blast
+
+corollary Fpow_infinite_bij_betw:
+assumes "infinite A"
+shows "\<exists>f. bij_betw f (Fpow A) A"
+using assms card_of_Fpow_infinite card_of_ordIso by blast
+
+
+subsection {* The cardinal $\omega$ and the finite cardinals *}
+
+subsubsection {* First as well-orders *}
+
+lemma Field_natLess: "Field natLess = (UNIV::nat set)"
+by(unfold Field_def, auto)
+
+lemma natLeq_ofilter_less: "ofilter natLeq {0 ..< n}"
+by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
+ simp add: Field_natLeq, unfold rel.under_def, auto)
+
+lemma natLeq_ofilter_leq: "ofilter natLeq {0 .. n}"
+by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def,
+ simp add: Field_natLeq, unfold rel.under_def, auto)
+
+lemma natLeq_ofilter_iff:
+"ofilter natLeq A = (A = UNIV \<or> (\<exists>n. A = {0 ..< n}))"
+proof(rule iffI)
+ assume "ofilter natLeq A"
+ hence "\<forall>m n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
+ by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def rel.under_def)
+ thus "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
+next
+ assume "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
+ thus "ofilter natLeq A"
+ by(auto simp add: natLeq_ofilter_less natLeq_UNIV_ofilter)
+qed
+
+lemma natLeq_under_leq: "under natLeq n = {0 .. n}"
+unfolding rel.under_def by auto
+
+corollary natLeq_on_ofilter:
+"ofilter(natLeq_on n) {0 ..< n}"
+by (auto simp add: natLeq_on_ofilter_less_eq)
+
+lemma natLeq_on_ofilter_less:
+"n < m \<Longrightarrow> ofilter (natLeq_on m) {0 .. n}"
+by(auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def,
+ simp add: Field_natLeq_on, unfold rel.under_def, auto)
+
+lemma natLeq_on_ordLess_natLeq: "natLeq_on n <o natLeq"
+using Field_natLeq Field_natLeq_on[of n] nat_infinite
+ finite_ordLess_infinite[of "natLeq_on n" natLeq]
+ natLeq_Well_order natLeq_on_Well_order[of n] by auto
+
+lemma natLeq_on_injective:
+"natLeq_on m = natLeq_on n \<Longrightarrow> m = n"
+using Field_natLeq_on[of m] Field_natLeq_on[of n]
+ atLeastLessThan_injective[of m n] by auto
+
+lemma natLeq_on_injective_ordIso:
+"(natLeq_on m =o natLeq_on n) = (m = n)"
+proof(auto simp add: natLeq_on_Well_order ordIso_reflexive)
+ assume "natLeq_on m =o natLeq_on n"
+ then obtain f where "bij_betw f {0..<m} {0..<n}"
+ using Field_natLeq_on assms unfolding ordIso_def iso_def[abs_def] by auto
+ thus "m = n" using atLeastLessThan_injective2 by blast
+qed
+
+
+subsubsection {* Then as cardinals *}
+
+lemma ordIso_natLeq_infinite1:
+"|A| =o natLeq \<Longrightarrow> infinite A"
+using ordIso_symmetric ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast
+
+lemma ordIso_natLeq_infinite2:
+"natLeq =o |A| \<Longrightarrow> infinite A"
+using ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast
+
+lemma ordLeq_natLeq_on_imp_finite:
+assumes "|A| \<le>o natLeq_on n"
+shows "finite A"
+proof-
+ have "|A| \<le>o |{0 ..< n}|"
+ using assms card_of_less ordIso_symmetric ordLeq_ordIso_trans by blast
+ thus ?thesis by (auto simp add: card_of_ordLeq_finite)
+qed
+
+
+subsubsection {* "Backwards compatibility" with the numeric cardinal operator for finite sets *}
+
+lemma finite_card_of_iff_card:
+assumes FIN: "finite A" and FIN': "finite B"
+shows "( |A| =o |B| ) = (card A = card B)"
+using assms card_of_ordIso[of A B] bij_betw_iff_card[of A B] by blast
+
+lemma finite_card_of_iff_card3:
+assumes FIN: "finite A" and FIN': "finite B"
+shows "( |A| <o |B| ) = (card A < card B)"
+proof-
+ have "( |A| <o |B| ) = (~ ( |B| \<le>o |A| ))" by simp
+ also have "... = (~ (card B \<le> card A))"
+ using assms by(simp add: finite_card_of_iff_card2)
+ also have "... = (card A < card B)" by auto
+ finally show ?thesis .
+qed
+
+lemma card_Field_natLeq_on:
+"card(Field(natLeq_on n)) = n"
+using Field_natLeq_on card_atLeastLessThan by auto
+
+
+subsection {* The successor of a cardinal *}
+
+lemma embed_implies_ordIso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and EMB: "embed r' r f"
+shows "r' =o Restr r (f ` (Field r'))"
+using assms embed_implies_iso_Restr Well_order_Restr unfolding ordIso_def by blast
+
+lemma cardSuc_Well_order[simp]:
+"Card_order r \<Longrightarrow> Well_order(cardSuc r)"
+using cardSuc_Card_order unfolding card_order_on_def by blast
+
+lemma Field_cardSuc_not_empty:
+assumes "Card_order r"
+shows "Field (cardSuc r) \<noteq> {}"
+proof
+ assume "Field(cardSuc r) = {}"
+ hence "|Field(cardSuc r)| \<le>o r" using assms Card_order_empty[of r] by auto
+ hence "cardSuc r \<le>o r" using assms card_of_Field_ordIso
+ cardSuc_Card_order ordIso_symmetric ordIso_ordLeq_trans by blast
+ thus False using cardSuc_greater not_ordLess_ordLeq assms by blast
+qed
+
+lemma cardSuc_mono_ordLess[simp]:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(cardSuc r <o cardSuc r') = (r <o r')"
+proof-
+ have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
+ using assms by auto
+ thus ?thesis
+ using not_ordLeq_iff_ordLess not_ordLeq_iff_ordLess[of r r']
+ using cardSuc_mono_ordLeq[of r' r] assms by blast
+qed
+
+lemma card_of_Plus_ordLeq_infinite[simp]:
+assumes C: "infinite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|"
+shows "|A <+> B| \<le>o |C|"
+proof-
+ let ?r = "cardSuc |C|"
+ have "Card_order ?r \<and> infinite (Field ?r)" using assms by simp
+ moreover have "|A| <o ?r" and "|B| <o ?r" using A B by auto
+ ultimately have "|A <+> B| <o ?r"
+ using card_of_Plus_ordLess_infinite_Field by blast
+ thus ?thesis using C by simp
+qed
+
+lemma card_of_Un_ordLeq_infinite[simp]:
+assumes C: "infinite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|"
+shows "|A Un B| \<le>o |C|"
+using assms card_of_Plus_ordLeq_infinite card_of_Un_Plus_ordLeq
+ordLeq_transitive by metis
+
+
+subsection {* Others *}
+
+lemma under_mono[simp]:
+assumes "Well_order r" and "(i,j) \<in> r"
+shows "under r i \<subseteq> under r j"
+using assms unfolding rel.under_def order_on_defs
+trans_def by blast
+
+lemma underS_under:
+assumes "i \<in> Field r"
+shows "underS r i = under r i - {i}"
+using assms unfolding rel.underS_def rel.under_def by auto
+
+lemma relChain_under:
+assumes "Well_order r"
+shows "relChain r (\<lambda> i. under r i)"
+using assms unfolding relChain_def by auto
+
+lemma infinite_card_of_diff_singl:
+assumes "infinite A"
+shows "|A - {a}| =o |A|"
+by (metis assms card_of_infinite_diff_finitte finite.emptyI finite_insert)
+
+lemma card_of_vimage:
+assumes "B \<subseteq> range f"
+shows "|B| \<le>o |f -` B|"
+apply(rule surj_imp_ordLeq[of _ f])
+using assms by (metis Int_absorb2 image_vimage_eq order_refl)
+
+lemma surj_card_of_vimage:
+assumes "surj f"
+shows "|B| \<le>o |f -` B|"
+by (metis assms card_of_vimage subset_UNIV)
+
+(* bounded powerset *)
+definition Bpow where
+"Bpow r A \<equiv> {X . X \<subseteq> A \<and> |X| \<le>o r}"
+
+lemma Bpow_empty[simp]:
+assumes "Card_order r"
+shows "Bpow r {} = {{}}"
+using assms unfolding Bpow_def by auto
+
+lemma singl_in_Bpow:
+assumes rc: "Card_order r"
+and r: "Field r \<noteq> {}" and a: "a \<in> A"
+shows "{a} \<in> Bpow r A"
+proof-
+ have "|{a}| \<le>o r" using r rc by auto
+ thus ?thesis unfolding Bpow_def using a by auto
+qed
+
+lemma ordLeq_card_Bpow:
+assumes rc: "Card_order r" and r: "Field r \<noteq> {}"
+shows "|A| \<le>o |Bpow r A|"
+proof-
+ have "inj_on (\<lambda> a. {a}) A" unfolding inj_on_def by auto
+ moreover have "(\<lambda> a. {a}) ` A \<subseteq> Bpow r A"
+ using singl_in_Bpow[OF assms] by auto
+ ultimately show ?thesis unfolding card_of_ordLeq[symmetric] by blast
+qed
+
+lemma infinite_Bpow:
+assumes rc: "Card_order r" and r: "Field r \<noteq> {}"
+and A: "infinite A"
+shows "infinite (Bpow r A)"
+using ordLeq_card_Bpow[OF rc r]
+by (metis A card_of_ordLeq_infinite)
+
+lemma Bpow_ordLeq_Func_Field:
+assumes rc: "Card_order r" and r: "Field r \<noteq> {}" and A: "infinite A"
+shows "|Bpow r A| \<le>o |Func (Field r) A|"
+proof-
+ let ?F = "\<lambda> f. {x | x a. f a = Some x}"
+ {fix X assume "X \<in> Bpow r A - {{}}"
+ hence XA: "X \<subseteq> A" and "|X| \<le>o r"
+ and X: "X \<noteq> {}" unfolding Bpow_def by auto
+ hence "|X| \<le>o |Field r|" by (metis Field_card_of card_of_mono2)
+ then obtain F where 1: "X = F ` (Field r)"
+ using card_of_ordLeq2[OF X] by metis
+ def f \<equiv> "\<lambda> i. if i \<in> Field r then Some (F i) else None"
+ have "\<exists> f \<in> Func (Field r) A. X = ?F f"
+ apply (intro bexI[of _ f]) using 1 XA unfolding Func_def f_def by auto
+ }
+ hence "Bpow r A - {{}} \<subseteq> ?F ` (Func (Field r) A)" by auto
+ hence "|Bpow r A - {{}}| \<le>o |Func (Field r) A|"
+ by (rule surj_imp_ordLeq)
+ moreover
+ {have 2: "infinite (Bpow r A)" using infinite_Bpow[OF rc r A] .
+ have "|Bpow r A| =o |Bpow r A - {{}}|"
+ using card_of_infinite_diff_finitte
+ by (metis Pow_empty 2 finite_Pow_iff infinite_imp_nonempty ordIso_symmetric)
+ }
+ ultimately show ?thesis by (metis ordIso_ordLeq_trans)
+qed
+
+lemma Func_emp2[simp]: "A \<noteq> {} \<Longrightarrow> Func A {} = {}" by auto
+
+lemma empty_in_Func[simp]:
+"B \<noteq> {} \<Longrightarrow> empty \<in> Func {} B"
+unfolding Func_def by auto
+
+lemma Func_mono[simp]:
+assumes "B1 \<subseteq> B2"
+shows "Func A B1 \<subseteq> Func A B2"
+using assms unfolding Func_def by force
+
+lemma Pfunc_mono[simp]:
+assumes "A1 \<subseteq> A2" and "B1 \<subseteq> B2"
+shows "Pfunc A B1 \<subseteq> Pfunc A B2"
+using assms in_mono unfolding Pfunc_def apply safe
+apply(case_tac "x a", auto)
+by (metis in_mono option.simps(5))
+
+lemma card_of_Func_UNIV_UNIV:
+"|Func (UNIV::'a set) (UNIV::'b set)| =o |UNIV::('a \<Rightarrow> 'b) set|"
+using card_of_Func_UNIV[of "UNIV::'b set"] by auto
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Cardinal_Order_Relation_Base.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,2579 @@
+(* Title: HOL/Ordinals_and_Cardinals/Cardinal_Order_Relation_Base.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Cardinal-order relations (base).
+*)
+
+header {* Cardinal-Order Relations (Base) *}
+
+theory Cardinal_Order_Relation_Base
+imports Constructions_on_Wellorders_Base
+begin
+
+
+text{* In this section, we define cardinal-order relations to be minim well-orders
+on their field. Then we define the cardinal of a set to be {\em some} cardinal-order
+relation on that set, which will be unique up to order isomorphism. Then we study
+the connection between cardinals and:
+\begin{itemize}
+\item standard set-theoretic constructions: products,
+sums, unions, lists, powersets, set-of finite sets operator;
+\item finiteness and infiniteness (in particular, with the numeric cardinal operator
+for finite sets, @{text "card"}, from the theory @{text "Finite_Sets.thy"}).
+\end{itemize}
+%
+On the way, we define the canonical $\omega$ cardinal and finite cardinals. We also
+define (again, up to order isomorphism) the successor of a cardinal, and show that
+any cardinal admits a successor.
+
+Main results of this section are the existence of cardinal relations and the
+facts that, in the presence of infiniteness,
+most of the standard set-theoretic constructions (except for the powerset)
+{\em do not increase cardinality}. In particular, e.g., the set of words/lists over
+any infinite set has the same cardinality (hence, is in bijection) with that set.
+*}
+
+
+subsection {* Cardinal orders *}
+
+
+text{* A cardinal order in our setting shall be a well-order {\em minim} w.r.t. the
+order-embedding relation, @{text "\<le>o"} (which is the same as being {\em minimal} w.r.t. the
+strict order-embedding relation, @{text "<o"}), among all the well-orders on its field. *}
+
+definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
+where
+"card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
+
+
+abbreviation "Card_order r \<equiv> card_order_on (Field r) r"
+abbreviation "card_order r \<equiv> card_order_on UNIV r"
+
+
+lemma card_order_on_well_order_on:
+assumes "card_order_on A r"
+shows "well_order_on A r"
+using assms unfolding card_order_on_def by simp
+
+
+lemma card_order_on_Card_order:
+"card_order_on A r \<Longrightarrow> A = Field r \<and> Card_order r"
+unfolding card_order_on_def using rel.well_order_on_Field by blast
+
+
+text{* The existence of a cardinal relation on any given set (which will mean
+that any set has a cardinal) follows from two facts:
+\begin{itemize}
+\item Zermelo's theorem (proved in @{text "Zorn.thy"} as theorem @{text "well_order_on"}),
+which states that on any given set there exists a well-order;
+\item The well-founded-ness of @{text "<o"}, ensuring that then there exists a minimal
+such well-order, i.e., a cardinal order.
+\end{itemize}
+*}
+
+
+theorem card_order_on: "\<exists>r. card_order_on A r"
+proof-
+ obtain R where R_def: "R = {r. well_order_on A r}" by blast
+ have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
+ using well_order_on[of A] R_def rel.well_order_on_Well_order by blast
+ hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+ using exists_minim_Well_order[of R] by auto
+ thus ?thesis using R_def unfolding card_order_on_def by auto
+qed
+
+
+lemma card_order_on_ordIso:
+assumes CO: "card_order_on A r" and CO': "card_order_on A r'"
+shows "r =o r'"
+using assms unfolding card_order_on_def
+using ordIso_iff_ordLeq by blast
+
+
+lemma Card_order_ordIso:
+assumes CO: "Card_order r" and ISO: "r' =o r"
+shows "Card_order r'"
+using ISO unfolding ordIso_def
+proof(unfold card_order_on_def, auto)
+ fix p' assume "well_order_on (Field r') p'"
+ hence 0: "Well_order p' \<and> Field p' = Field r'"
+ using rel.well_order_on_Well_order by blast
+ obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"
+ using ISO unfolding ordIso_def by auto
+ hence 3: "inj_on f (Field r') \<and> f ` (Field r') = Field r"
+ by (auto simp add: iso_iff embed_inj_on)
+ let ?p = "dir_image p' f"
+ have 4: "p' =o ?p \<and> Well_order ?p"
+ using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)
+ moreover have "Field ?p = Field r"
+ using 0 3 by (auto simp add: dir_image_Field2 order_on_defs)
+ ultimately have "well_order_on (Field r) ?p" by auto
+ hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto
+ thus "r' \<le>o p'"
+ using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast
+qed
+
+
+lemma Card_order_ordIso2:
+assumes CO: "Card_order r" and ISO: "r =o r'"
+shows "Card_order r'"
+using assms Card_order_ordIso ordIso_symmetric by blast
+
+
+subsection {* Cardinal of a set *}
+
+
+text{* We define the cardinal of set to be {\em some} cardinal order on that set.
+We shall prove that this notion is unique up to order isomorphism, meaning
+that order isomorphism shall be the true identity of cardinals. *}
+
+
+definition card_of :: "'a set \<Rightarrow> 'a rel" ("|_|" )
+where "card_of A = (SOME r. card_order_on A r)"
+
+
+lemma card_of_card_order_on: "card_order_on A |A|"
+unfolding card_of_def by (auto simp add: card_order_on someI_ex)
+
+
+lemma card_of_well_order_on: "well_order_on A |A|"
+using card_of_card_order_on card_order_on_def by blast
+
+
+lemma Field_card_of: "Field |A| = A"
+using card_of_card_order_on[of A] unfolding card_order_on_def
+using rel.well_order_on_Field by blast
+
+
+lemma card_of_Card_order: "Card_order |A|"
+by (simp only: card_of_card_order_on Field_card_of)
+
+
+corollary ordIso_card_of_imp_Card_order:
+"r =o |A| \<Longrightarrow> Card_order r"
+using card_of_Card_order Card_order_ordIso by blast
+
+
+lemma card_of_Well_order: "Well_order |A|"
+using card_of_Card_order unfolding card_order_on_def by auto
+
+
+lemma card_of_refl: "|A| =o |A|"
+using card_of_Well_order ordIso_reflexive by blast
+
+
+lemma card_of_least: "well_order_on A r \<Longrightarrow> |A| \<le>o r"
+using card_of_card_order_on unfolding card_order_on_def by blast
+
+
+lemma card_of_ordIso:
+"(\<exists>f. bij_betw f A B) = ( |A| =o |B| )"
+proof(auto)
+ fix f assume *: "bij_betw f A B"
+ then obtain r where "well_order_on B r \<and> |A| =o r"
+ using Well_order_iso_copy card_of_well_order_on by blast
+ hence "|B| \<le>o |A|" using card_of_least
+ ordLeq_ordIso_trans ordIso_symmetric by blast
+ moreover
+ {let ?g = "inv_into A f"
+ have "bij_betw ?g B A" using * bij_betw_inv_into by blast
+ then obtain r where "well_order_on A r \<and> |B| =o r"
+ using Well_order_iso_copy card_of_well_order_on by blast
+ hence "|A| \<le>o |B|" using card_of_least
+ ordLeq_ordIso_trans ordIso_symmetric by blast
+ }
+ ultimately show "|A| =o |B|" using ordIso_iff_ordLeq by blast
+next
+ assume "|A| =o |B|"
+ then obtain f where "iso ( |A| ) ( |B| ) f"
+ unfolding ordIso_def by auto
+ hence "bij_betw f A B" unfolding iso_def Field_card_of by simp
+ thus "\<exists>f. bij_betw f A B" by auto
+qed
+
+
+lemma card_of_ordLeq:
+"(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
+proof(auto)
+ fix f assume *: "inj_on f A" and **: "f ` A \<le> B"
+ {assume "|B| <o |A|"
+ hence "|B| \<le>o |A|" using ordLeq_iff_ordLess_or_ordIso by blast
+ then obtain g where "embed ( |B| ) ( |A| ) g"
+ unfolding ordLeq_def by auto
+ hence 1: "inj_on g B \<and> g ` B \<le> A" using embed_inj_on[of "|B|" "|A|" "g"]
+ card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]
+ embed_Field[of "|B|" "|A|" g] by auto
+ obtain h where "bij_betw h A B"
+ using * ** 1 Cantor_Bernstein[of f] by fastforce
+ hence "|A| =o |B|" using card_of_ordIso by blast
+ hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto
+ }
+ thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
+ by (auto simp: card_of_Well_order)
+next
+ assume *: "|A| \<le>o |B|"
+ obtain f where "embed ( |A| ) ( |B| ) f"
+ using * unfolding ordLeq_def by auto
+ hence "inj_on f A \<and> f ` A \<le> B" using embed_inj_on[of "|A|" "|B|" f]
+ card_of_Well_order[of "A"] Field_card_of[of "A"] Field_card_of[of "B"]
+ embed_Field[of "|A|" "|B|" f] by auto
+ thus "\<exists>f. inj_on f A \<and> f ` A \<le> B" by auto
+qed
+
+
+lemma card_of_ordLeq2:
+"A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
+using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
+
+
+lemma card_of_ordLess:
+"(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
+proof-
+ have "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = (\<not> |A| \<le>o |B| )"
+ using card_of_ordLeq by blast
+ also have "\<dots> = ( |B| <o |A| )"
+ using card_of_Well_order[of A] card_of_Well_order[of B]
+ not_ordLeq_iff_ordLess by blast
+ finally show ?thesis .
+qed
+
+
+lemma card_of_ordLess2:
+"B \<noteq> {} \<Longrightarrow> (\<not>(\<exists>f. f ` A = B)) = ( |A| <o |B| )"
+using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto
+
+
+lemma card_of_ordIsoI:
+assumes "bij_betw f A B"
+shows "|A| =o |B|"
+using assms unfolding card_of_ordIso[symmetric] by auto
+
+
+lemma card_of_ordLeqI:
+assumes "inj_on f A" and "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
+shows "|A| \<le>o |B|"
+using assms unfolding card_of_ordLeq[symmetric] by auto
+
+
+lemma card_of_unique:
+"card_order_on A r \<Longrightarrow> r =o |A|"
+by (simp only: card_order_on_ordIso card_of_card_order_on)
+
+
+lemma card_of_mono1:
+"A \<le> B \<Longrightarrow> |A| \<le>o |B|"
+using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
+
+
+lemma card_of_mono2:
+assumes "r \<le>o r'"
+shows "|Field r| \<le>o |Field r'|"
+proof-
+ obtain f where
+ 1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
+ using assms unfolding ordLeq_def
+ by (auto simp add: rel.well_order_on_Well_order)
+ hence "inj_on f (Field r) \<and> f ` (Field r) \<le> Field r'"
+ by (auto simp add: embed_inj_on embed_Field)
+ thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast
+qed
+
+
+lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"
+by (simp add: ordIso_iff_ordLeq card_of_mono2)
+
+
+lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"
+using card_of_least card_of_well_order_on rel.well_order_on_Well_order by blast
+
+
+lemma card_of_Field_ordIso:
+assumes "Card_order r"
+shows "|Field r| =o r"
+proof-
+ have "card_order_on (Field r) r"
+ using assms card_order_on_Card_order by blast
+ moreover have "card_order_on (Field r) |Field r|"
+ using card_of_card_order_on by blast
+ ultimately show ?thesis using card_order_on_ordIso by blast
+qed
+
+
+lemma Card_order_iff_ordIso_card_of:
+"Card_order r = (r =o |Field r| )"
+using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast
+
+
+lemma Card_order_iff_ordLeq_card_of:
+"Card_order r = (r \<le>o |Field r| )"
+proof-
+ have "Card_order r = (r =o |Field r| )"
+ unfolding Card_order_iff_ordIso_card_of by simp
+ also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
+ unfolding ordIso_iff_ordLeq by simp
+ also have "... = (r \<le>o |Field r| )"
+ using card_of_Field_ordLess
+ by (auto simp: card_of_Field_ordLess ordLeq_Well_order_simp)
+ finally show ?thesis .
+qed
+
+
+lemma Card_order_iff_Restr_underS:
+assumes "Well_order r"
+shows "Card_order r = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o |Field r| )"
+using assms unfolding Card_order_iff_ordLeq_card_of
+using ordLeq_iff_ordLess_Restr card_of_Well_order by blast
+
+
+lemma card_of_underS:
+assumes r: "Card_order r" and a: "a : Field r"
+shows "|rel.underS r a| <o r"
+proof-
+ let ?A = "rel.underS r a" let ?r' = "Restr r ?A"
+ have 1: "Well_order r"
+ using r unfolding card_order_on_def by simp
+ have "Well_order ?r'" using 1 Well_order_Restr by auto
+ moreover have "card_order_on (Field ?r') |Field ?r'|"
+ using card_of_card_order_on .
+ ultimately have "|Field ?r'| \<le>o ?r'"
+ unfolding card_order_on_def by simp
+ moreover have "Field ?r' = ?A"
+ using 1 wo_rel.underS_ofilter Field_Restr_ofilter
+ unfolding wo_rel_def by fastforce
+ ultimately have "|?A| \<le>o ?r'" by simp
+ also have "?r' <o |Field r|"
+ using 1 a r Card_order_iff_Restr_underS by blast
+ also have "|Field r| =o r"
+ using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
+ finally show ?thesis .
+qed
+
+
+lemma ordLess_Field:
+assumes "r <o r'"
+shows "|Field r| <o r'"
+proof-
+ have "well_order_on (Field r) r" using assms unfolding ordLess_def
+ by (auto simp add: rel.well_order_on_Well_order)
+ hence "|Field r| \<le>o r" using card_of_least by blast
+ thus ?thesis using assms ordLeq_ordLess_trans by blast
+qed
+
+
+lemma internalize_card_of_ordLeq:
+"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
+proof
+ assume "|A| \<le>o r"
+ then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>o r"
+ using internalize_ordLeq[of "|A|" r] by blast
+ hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
+ hence "|Field p| =o p" using card_of_Field_ordIso by blast
+ hence "|A| =o |Field p| \<and> |Field p| \<le>o r"
+ using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
+ thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
+next
+ assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
+ thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast
+qed
+
+
+lemma internalize_card_of_ordLeq2:
+"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"
+using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto
+
+
+
+subsection {* Cardinals versus set operations on arbitrary sets *}
+
+
+text{* Here we embark in a long journey of simple results showing
+that the standard set-theoretic operations are well-behaved w.r.t. the notion of
+cardinal -- essentially, this means that they preserve the ``cardinal identity"
+@{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.
+*}
+
+
+lemma card_of_empty: "|{}| \<le>o |A|"
+using card_of_ordLeq inj_on_id by blast
+
+
+lemma card_of_empty1:
+assumes "Well_order r \<or> Card_order r"
+shows "|{}| \<le>o r"
+proof-
+ have "Well_order r" using assms unfolding card_order_on_def by auto
+ hence "|Field r| <=o r"
+ using assms card_of_Field_ordLess by blast
+ moreover have "|{}| \<le>o |Field r|" by (simp add: card_of_empty)
+ ultimately show ?thesis using ordLeq_transitive by blast
+qed
+
+
+corollary Card_order_empty:
+"Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)
+
+
+lemma card_of_empty2:
+assumes LEQ: "|A| =o |{}|"
+shows "A = {}"
+using assms card_of_ordIso[of A] bij_betw_empty2 by blast
+
+
+lemma card_of_empty3:
+assumes LEQ: "|A| \<le>o |{}|"
+shows "A = {}"
+using assms
+by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
+ ordLeq_Well_order_simp)
+
+
+lemma card_of_empty_ordIso:
+"|{}::'a set| =o |{}::'b set|"
+using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
+
+
+lemma card_of_image:
+"|f ` A| <=o |A|"
+proof(cases "A = {}", simp add: card_of_empty)
+ assume "A ~= {}"
+ hence "f ` A ~= {}" by auto
+ thus "|f ` A| \<le>o |A|"
+ using card_of_ordLeq2[of "f ` A" A] by auto
+qed
+
+
+lemma surj_imp_ordLeq:
+assumes "B <= f ` A"
+shows "|B| <=o |A|"
+proof-
+ have "|B| <=o |f ` A|" using assms card_of_mono1 by auto
+ thus ?thesis using card_of_image ordLeq_transitive by blast
+qed
+
+
+lemma card_of_ordLeqI2:
+assumes "B \<subseteq> f ` A"
+shows "|B| \<le>o |A|"
+using assms by (metis surj_imp_ordLeq)
+
+
+lemma card_of_singl_ordLeq:
+assumes "A \<noteq> {}"
+shows "|{b}| \<le>o |A|"
+proof-
+ obtain a where *: "a \<in> A" using assms by auto
+ let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
+ have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
+ using * unfolding inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_singl_ordLeq:
+"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"
+using card_of_singl_ordLeq[of "Field r" b]
+ card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast
+
+
+lemma card_of_Pow: "|A| <o |Pow A|"
+using card_of_ordLess2[of "Pow A" A] Cantors_paradox[of A]
+ Pow_not_empty[of A] by auto
+
+
+lemma infinite_Pow:
+assumes "infinite A"
+shows "infinite (Pow A)"
+proof-
+ have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq)
+ thus ?thesis by (metis assms finite_Pow_iff)
+qed
+
+
+corollary Card_order_Pow:
+"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
+using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
+
+
+corollary card_of_set_type: "|UNIV::'a set| <o |UNIV::'a set set|"
+using card_of_Pow[of "UNIV::'a set"] by simp
+
+
+lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
+proof-
+ have "Inl ` A \<le> A <+> B" by auto
+ thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_Plus1:
+"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
+using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
+proof-
+ have "Inr ` B \<le> A <+> B" by auto
+ thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_Plus2:
+"Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"
+using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"
+proof-
+ have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by auto
+qed
+
+
+lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"
+proof-
+ have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by auto
+qed
+
+
+lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"
+proof-
+ let ?f = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
+ | Inr b \<Rightarrow> Inl b"
+ have "bij_betw ?f (A <+> B) (B <+> A)"
+ unfolding bij_betw_def inj_on_def by force
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_assoc:
+fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
+shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
+proof -
+ def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).
+ case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a
+ |Inr b \<Rightarrow> Inr (Inl b))
+ |Inr c \<Rightarrow> Inr (Inr c)"
+ have "A <+> B <+> C \<subseteq> f ` ((A <+> B) <+> C)"
+ proof
+ fix x assume x: "x \<in> A <+> B <+> C"
+ show "x \<in> f ` ((A <+> B) <+> C)"
+ proof(cases x)
+ case (Inl a)
+ hence "a \<in> A" "x = f (Inl (Inl a))"
+ using x unfolding f_def by auto
+ thus ?thesis by auto
+ next
+ case (Inr bc) note 1 = Inr show ?thesis
+ proof(cases bc)
+ case (Inl b)
+ hence "b \<in> B" "x = f (Inl (Inr b))"
+ using x 1 unfolding f_def by auto
+ thus ?thesis by auto
+ next
+ case (Inr c)
+ hence "c \<in> C" "x = f (Inr c)"
+ using x 1 unfolding f_def by auto
+ thus ?thesis by auto
+ qed
+ qed
+ qed
+ hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"
+ unfolding bij_betw_def inj_on_def f_def by auto
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_mono1:
+assumes "|A| \<le>o |B|"
+shows "|A <+> C| \<le>o |B <+> C|"
+proof-
+ obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ obtain g where g_def:
+ "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
+ have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
+ proof-
+ {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
+ "g d1 = g d2"
+ hence "d1 = d2" using 1 unfolding inj_on_def
+ by(case_tac d1, case_tac d2, auto simp add: g_def)
+ }
+ moreover
+ {fix d assume "d \<in> A <+> C"
+ hence "g d \<in> B <+> C" using 1
+ by(case_tac d, auto simp add: g_def)
+ }
+ ultimately show ?thesis unfolding inj_on_def by auto
+ qed
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary ordLeq_Plus_mono1:
+assumes "r \<le>o r'"
+shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
+using assms card_of_mono2 card_of_Plus_mono1 by blast
+
+
+lemma card_of_Plus_mono2:
+assumes "|A| \<le>o |B|"
+shows "|C <+> A| \<le>o |C <+> B|"
+using assms card_of_Plus_mono1[of A B C]
+ card_of_Plus_commute[of C A] card_of_Plus_commute[of B C]
+ ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]
+by blast
+
+
+corollary ordLeq_Plus_mono2:
+assumes "r \<le>o r'"
+shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
+using assms card_of_mono2 card_of_Plus_mono2 by blast
+
+
+lemma card_of_Plus_mono:
+assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
+shows "|A <+> C| \<le>o |B <+> D|"
+using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
+ ordLeq_transitive[of "|A <+> C|"] by blast
+
+
+corollary ordLeq_Plus_mono:
+assumes "r \<le>o r'" and "p \<le>o p'"
+shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"
+using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast
+
+
+lemma card_of_Plus_cong1:
+assumes "|A| =o |B|"
+shows "|A <+> C| =o |B <+> C|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)
+
+
+corollary ordIso_Plus_cong1:
+assumes "r =o r'"
+shows "|(Field r) <+> C| =o |(Field r') <+> C|"
+using assms card_of_cong card_of_Plus_cong1 by blast
+
+
+lemma card_of_Plus_cong2:
+assumes "|A| =o |B|"
+shows "|C <+> A| =o |C <+> B|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)
+
+
+corollary ordIso_Plus_cong2:
+assumes "r =o r'"
+shows "|A <+> (Field r)| =o |A <+> (Field r')|"
+using assms card_of_cong card_of_Plus_cong2 by blast
+
+
+lemma card_of_Plus_cong:
+assumes "|A| =o |B|" and "|C| =o |D|"
+shows "|A <+> C| =o |B <+> D|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)
+
+
+corollary ordIso_Plus_cong:
+assumes "r =o r'" and "p =o p'"
+shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
+using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast
+
+
+lemma card_of_Un1:
+shows "|A| \<le>o |A \<union> B| "
+using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
+
+
+lemma card_of_diff:
+shows "|A - B| \<le>o |A|"
+using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce
+
+
+lemma card_of_Un_Plus_ordLeq:
+"|A \<union> B| \<le>o |A <+> B|"
+proof-
+ let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
+ have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
+ unfolding inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by blast
+qed
+
+
+lemma card_of_Times1:
+assumes "A \<noteq> {}"
+shows "|B| \<le>o |B \<times> A|"
+proof(cases "B = {}", simp add: card_of_empty)
+ assume *: "B \<noteq> {}"
+ have "fst `(B \<times> A) = B" unfolding image_def using assms by auto
+ thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
+ card_of_ordLeq[of B "B \<times> A"] * by blast
+qed
+
+
+corollary Card_order_Times1:
+"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
+using card_of_Times1[of B] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
+proof-
+ let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
+ have "bij_betw ?f (A \<times> B) (B \<times> A)"
+ unfolding bij_betw_def inj_on_def by auto
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Times2:
+assumes "A \<noteq> {}" shows "|B| \<le>o |A \<times> B|"
+using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
+ ordLeq_ordIso_trans by blast
+
+
+corollary Card_order_Times2:
+"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
+using card_of_Times2[of A] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Times3: "|A| \<le>o |A \<times> A|"
+using card_of_Times1[of A]
+by(cases "A = {}", simp add: card_of_empty, blast)
+
+
+lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \<times> (UNIV::bool set)|"
+proof-
+ let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
+ |Inr a \<Rightarrow> (a,False)"
+ have "bij_betw ?f (A <+> A) (A \<times> (UNIV::bool set))"
+ proof-
+ {fix c1 and c2 assume "?f c1 = ?f c2"
+ hence "c1 = c2"
+ by(case_tac "c1", case_tac "c2", auto, case_tac "c2", auto)
+ }
+ moreover
+ {fix c assume "c \<in> A <+> A"
+ hence "?f c \<in> A \<times> (UNIV::bool set)"
+ by(case_tac c, auto)
+ }
+ moreover
+ {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
+ have "(a,bl) \<in> ?f ` ( A <+> A)"
+ proof(cases bl)
+ assume bl hence "?f(Inl a) = (a,bl)" by auto
+ thus ?thesis using * by force
+ next
+ assume "\<not> bl" hence "?f(Inr a) = (a,bl)" by auto
+ thus ?thesis using * by force
+ qed
+ }
+ ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto
+ qed
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Times_mono1:
+assumes "|A| \<le>o |B|"
+shows "|A \<times> C| \<le>o |B \<times> C|"
+proof-
+ obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ obtain g where g_def:
+ "g = (\<lambda>(a,c::'c). (f a,c))" by blast
+ have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
+ using 1 unfolding inj_on_def using g_def by auto
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary ordLeq_Times_mono1:
+assumes "r \<le>o r'"
+shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
+using assms card_of_mono2 card_of_Times_mono1 by blast
+
+
+lemma card_of_Times_mono2:
+assumes "|A| \<le>o |B|"
+shows "|C \<times> A| \<le>o |C \<times> B|"
+using assms card_of_Times_mono1[of A B C]
+ card_of_Times_commute[of C A] card_of_Times_commute[of B C]
+ ordIso_ordLeq_trans[of "|C \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
+by blast
+
+
+corollary ordLeq_Times_mono2:
+assumes "r \<le>o r'"
+shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
+using assms card_of_mono2 card_of_Times_mono2 by blast
+
+
+lemma card_of_Times_cong1:
+assumes "|A| =o |B|"
+shows "|A \<times> C| =o |B \<times> C|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono1)
+
+
+lemma card_of_Times_cong2:
+assumes "|A| =o |B|"
+shows "|C \<times> A| =o |C \<times> B|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono2)
+
+
+corollary ordIso_Times_cong2:
+assumes "r =o r'"
+shows "|A \<times> (Field r)| =o |A \<times> (Field r')|"
+using assms card_of_cong card_of_Times_cong2 by blast
+
+
+lemma card_of_Sigma_mono1:
+assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
+shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
+proof-
+ have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
+ using assms by (auto simp add: card_of_ordLeq)
+ with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
+ obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i" by fastforce
+ obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
+ have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
+ using 1 unfolding inj_on_def using g_def by force
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary card_of_Sigma_Times:
+"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> |SIGMA i : I. A i| \<le>o |I \<times> B|"
+using card_of_Sigma_mono1[of I A "\<lambda>i. B"] .
+
+
+lemma card_of_UNION_Sigma:
+"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
+using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by metis
+
+
+lemma card_of_bool:
+assumes "a1 \<noteq> a2"
+shows "|UNIV::bool set| =o |{a1,a2}|"
+proof-
+ let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> a2"
+ have "bij_betw ?f UNIV {a1,a2}"
+ proof-
+ {fix bl1 and bl2 assume "?f bl1 = ?f bl2"
+ hence "bl1 = bl2" using assms by (case_tac bl1, case_tac bl2, auto)
+ }
+ moreover
+ {fix bl have "?f bl \<in> {a1,a2}" by (case_tac bl, auto)
+ }
+ moreover
+ {fix a assume *: "a \<in> {a1,a2}"
+ have "a \<in> ?f ` UNIV"
+ proof(cases "a = a1")
+ assume "a = a1"
+ hence "?f True = a" by auto thus ?thesis by blast
+ next
+ assume "a \<noteq> a1" hence "a = a2" using * by auto
+ hence "?f False = a" by auto thus ?thesis by blast
+ qed
+ }
+ ultimately show ?thesis unfolding bij_betw_def inj_on_def
+ by (metis image_subsetI order_eq_iff subsetI)
+ qed
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_Times_aux:
+assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ LEQ: "|A| \<le>o |B|"
+shows "|A <+> B| \<le>o |A \<times> B|"
+proof-
+ have 1: "|UNIV::bool set| \<le>o |A|"
+ using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
+ ordIso_ordLeq_trans[of "|UNIV::bool set|"] by metis
+ (* *)
+ have "|A <+> B| \<le>o |B <+> B|"
+ using LEQ card_of_Plus_mono1 by blast
+ moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
+ using card_of_Plus_Times_bool by blast
+ moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
+ using 1 by (simp add: card_of_Times_mono2)
+ moreover have " |B \<times> A| =o |A \<times> B|"
+ using card_of_Times_commute by blast
+ ultimately show "|A <+> B| \<le>o |A \<times> B|"
+ using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
+ ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
+ ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
+ by blast
+qed
+
+
+lemma card_of_Plus_Times:
+assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
+shows "|A <+> B| \<le>o |A \<times> B|"
+proof-
+ {assume "|A| \<le>o |B|"
+ hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
+ }
+ moreover
+ {assume "|B| \<le>o |A|"
+ hence "|B <+> A| \<le>o |B \<times> A|"
+ using assms by (auto simp add: card_of_Plus_Times_aux)
+ hence ?thesis
+ using card_of_Plus_commute card_of_Times_commute
+ ordIso_ordLeq_trans ordLeq_ordIso_trans by metis
+ }
+ ultimately show ?thesis
+ using card_of_Well_order[of A] card_of_Well_order[of B]
+ ordLeq_total[of "|A|"] by metis
+qed
+
+
+lemma card_of_ordLeq_finite:
+assumes "|A| \<le>o |B|" and "finite B"
+shows "finite A"
+using assms unfolding ordLeq_def
+using embed_inj_on[of "|A|" "|B|"] embed_Field[of "|A|" "|B|"]
+ Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce
+
+
+lemma card_of_ordLeq_infinite:
+assumes "|A| \<le>o |B|" and "infinite A"
+shows "infinite B"
+using assms card_of_ordLeq_finite by auto
+
+
+lemma card_of_ordIso_finite:
+assumes "|A| =o |B|"
+shows "finite A = finite B"
+using assms unfolding ordIso_def iso_def[abs_def]
+by (auto simp: bij_betw_finite Field_card_of)
+
+
+lemma card_of_ordIso_finite_Field:
+assumes "Card_order r" and "r =o |A|"
+shows "finite(Field r) = finite A"
+using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast
+
+
+subsection {* Cardinals versus set operations involving infinite sets *}
+
+
+text{* Here we show that, for infinite sets, most set-theoretic constructions
+do not increase the cardinality. The cornerstone for this is
+theorem @{text "Card_order_Times_same_infinite"}, which states that self-product
+does not increase cardinality -- the proof of this fact adapts a standard
+set-theoretic argument, as presented, e.g., in the proof of theorem 1.5.11
+at page 47 in \cite{card-book}. Then everything else follows fairly easily. *}
+
+
+lemma infinite_iff_card_of_nat:
+"infinite A = ( |UNIV::nat set| \<le>o |A| )"
+by (auto simp add: infinite_iff_countable_subset card_of_ordLeq)
+
+
+lemma finite_iff_cardOf_nat:
+"finite A = ( |A| <o |UNIV :: nat set| )"
+using infinite_iff_card_of_nat[of A]
+not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"]
+by (fastforce simp: card_of_Well_order)
+
+lemma finite_ordLess_infinite2:
+assumes "finite A" and "infinite B"
+shows "|A| <o |B|"
+using assms
+finite_ordLess_infinite[of "|A|" "|B|"]
+card_of_Well_order[of A] card_of_Well_order[of B]
+Field_card_of[of A] Field_card_of[of B] by auto
+
+
+text{* The next two results correspond to the ZF fact that all infinite cardinals are
+limit ordinals: *}
+
+lemma Card_order_infinite_not_under:
+assumes CARD: "Card_order r" and INF: "infinite (Field r)"
+shows "\<not> (\<exists>a. Field r = rel.under r a)"
+proof(auto)
+ have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
+ using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
+ fix a assume *: "Field r = rel.under r a"
+ show False
+ proof(cases "a \<in> Field r")
+ assume Case1: "a \<notin> Field r"
+ hence "rel.under r a = {}" unfolding Field_def rel.under_def by auto
+ thus False using INF * by auto
+ next
+ let ?r' = "Restr r (rel.underS r a)"
+ assume Case2: "a \<in> Field r"
+ hence 1: "rel.under r a = rel.underS r a \<union> {a} \<and> a \<notin> rel.underS r a"
+ using 0 rel.Refl_under_underS rel.underS_notIn by fastforce
+ have 2: "wo_rel.ofilter r (rel.underS r a) \<and> rel.underS r a < Field r"
+ using 0 wo_rel.underS_ofilter * 1 Case2 by auto
+ hence "?r' <o r" using 0 using ofilter_ordLess by blast
+ moreover
+ have "Field ?r' = rel.underS r a \<and> Well_order ?r'"
+ using 2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
+ ultimately have "|rel.underS r a| <o r" using ordLess_Field[of ?r'] by auto
+ moreover have "|rel.under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
+ ultimately have "|rel.underS r a| <o |rel.under r a|"
+ using ordIso_symmetric ordLess_ordIso_trans by blast
+ moreover
+ {have "\<exists>f. bij_betw f (rel.under r a) (rel.underS r a)"
+ using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
+ hence "|rel.under r a| =o |rel.underS r a|" using card_of_ordIso by blast
+ }
+ ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast
+ qed
+qed
+
+
+lemma infinite_Card_order_limit:
+assumes r: "Card_order r" and "infinite (Field r)"
+and a: "a : Field r"
+shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"
+proof-
+ have "Field r \<noteq> rel.under r a"
+ using assms Card_order_infinite_not_under by blast
+ moreover have "rel.under r a \<le> Field r"
+ using rel.under_Field .
+ ultimately have "rel.under r a < Field r" by blast
+ then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"
+ unfolding rel.under_def by blast
+ moreover have ba: "b \<noteq> a"
+ using 1 r unfolding card_order_on_def well_order_on_def
+ linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
+ ultimately have "(a,b) : r"
+ using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
+ total_on_def by blast
+ thus ?thesis using 1 ba by auto
+qed
+
+
+theorem Card_order_Times_same_infinite:
+assumes CO: "Card_order r" and INF: "infinite(Field r)"
+shows "|Field r \<times> Field r| \<le>o r"
+proof-
+ obtain phi where phi_def:
+ "phi = (\<lambda>r::'a rel. Card_order r \<and> infinite(Field r) \<and>
+ \<not> |Field r \<times> Field r| \<le>o r )" by blast
+ have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
+ unfolding phi_def card_order_on_def by auto
+ have Ft: "\<not>(\<exists>r. phi r)"
+ proof
+ assume "\<exists>r. phi r"
+ hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
+ using temp1 by auto
+ then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
+ 3: "Card_order r \<and> Well_order r"
+ using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
+ let ?A = "Field r" let ?r' = "bsqr r"
+ have 4: "Well_order ?r' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
+ using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
+ have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
+ using card_of_Card_order card_of_Well_order by blast
+ (* *)
+ have "r <o |?A \<times> ?A|"
+ using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
+ moreover have "|?A \<times> ?A| \<le>o ?r'"
+ using card_of_least[of "?A \<times> ?A"] 4 by auto
+ ultimately have "r <o ?r'" using ordLess_ordLeq_trans by auto
+ then obtain f where 6: "embed r ?r' f" and 7: "\<not> bij_betw f ?A (?A \<times> ?A)"
+ unfolding ordLess_def embedS_def[abs_def]
+ by (auto simp add: Field_bsqr)
+ let ?B = "f ` ?A"
+ have "|?A| =o |?B|"
+ using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
+ hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast
+ (* *)
+ have "wo_rel.ofilter ?r' ?B"
+ using 6 embed_Field_ofilter 3 4 by blast
+ hence "wo_rel.ofilter ?r' ?B \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
+ using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
+ hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
+ using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
+ have "\<not> (\<exists>a. Field r = rel.under r a)"
+ using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
+ then obtain A1 where temp3: "wo_rel.ofilter r A1 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
+ using temp2 3 bsqr_ofilter[of r ?B] by blast
+ hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
+ hence 10: "r \<le>o |A1 \<times> A1|" using 8 ordIso_ordLeq_trans by blast
+ let ?r1 = "Restr r A1"
+ have "?r1 <o r" using temp3 ofilter_ordLess 3 by blast
+ moreover
+ {have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast
+ hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast
+ }
+ ultimately have 11: "|A1| <o r" using ordLeq_ordLess_trans by blast
+ (* *)
+ have "infinite (Field r)" using 1 unfolding phi_def by simp
+ hence "infinite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
+ hence "infinite A1" using 9 infinite_super finite_cartesian_product by blast
+ moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
+ using card_of_Card_order[of A1] card_of_Well_order[of A1]
+ by (simp add: Field_card_of)
+ moreover have "\<not> r \<le>o | A1 |"
+ using temp4 11 3 using not_ordLeq_iff_ordLess by blast
+ ultimately have "infinite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
+ by (simp add: card_of_card_order_on)
+ hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
+ using 2 unfolding phi_def by blast
+ hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
+ hence "r \<le>o |A1|" using 10 ordLeq_transitive by blast
+ thus False using 11 not_ordLess_ordLeq by auto
+ qed
+ thus ?thesis using assms unfolding phi_def by blast
+qed
+
+
+corollary card_of_Times_same_infinite:
+assumes "infinite A"
+shows "|A \<times> A| =o |A|"
+proof-
+ let ?r = "|A|"
+ have "Field ?r = A \<and> Card_order ?r"
+ using Field_card_of card_of_Card_order[of A] by fastforce
+ hence "|A \<times> A| \<le>o |A|"
+ using Card_order_Times_same_infinite[of ?r] assms by auto
+ thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast
+qed
+
+
+lemma card_of_Times_infinite:
+assumes INF: "infinite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
+shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
+proof-
+ have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
+ using assms by (simp add: card_of_Times1 card_of_Times2)
+ moreover
+ {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
+ using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
+ moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
+ ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
+ using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
+ }
+ ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
+qed
+
+
+corollary card_of_Times_infinite_simps:
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
+by (auto simp add: card_of_Times_infinite ordIso_symmetric)
+
+
+corollary Card_order_Times_infinite:
+assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
+ NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
+shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
+proof-
+ have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"
+ using assms by (simp add: card_of_Times_infinite card_of_mono2)
+ thus ?thesis
+ using assms card_of_Field_ordIso[of r]
+ ordIso_transitive[of "|Field r \<times> Field p|"]
+ ordIso_transitive[of _ "|Field r|"] by blast
+qed
+
+
+lemma card_of_Sigma_ordLeq_infinite:
+assumes INF: "infinite B" and
+ LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+shows "|SIGMA i : I. A i| \<le>o |B|"
+proof(cases "I = {}", simp add: card_of_empty)
+ assume *: "I \<noteq> {}"
+ have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
+ using LEQ card_of_Sigma_Times by blast
+ moreover have "|I \<times> B| =o |B|"
+ using INF * LEQ_I by (auto simp add: card_of_Times_infinite)
+ ultimately show ?thesis using ordLeq_ordIso_trans by blast
+qed
+
+
+lemma card_of_Sigma_ordLeq_infinite_Field:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+shows "|SIGMA i : I. A i| \<le>o r"
+proof-
+ let ?B = "Field r"
+ have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ hence "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ
+ card_of_Sigma_ordLeq_infinite by blast
+ thus ?thesis using 1 ordLeq_ordIso_trans by blast
+qed
+
+
+lemma card_of_Times_ordLeq_infinite_Field:
+"\<lbrakk>infinite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
+ \<Longrightarrow> |A <*> B| \<le>o r"
+by(simp add: card_of_Sigma_ordLeq_infinite_Field)
+
+
+lemma card_of_UNION_ordLeq_infinite:
+assumes INF: "infinite B" and
+ LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+shows "|\<Union> i \<in> I. A i| \<le>o |B|"
+proof(cases "I = {}", simp add: card_of_empty)
+ assume *: "I \<noteq> {}"
+ have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
+ using card_of_UNION_Sigma by blast
+ moreover have "|SIGMA i : I. A i| \<le>o |B|"
+ using assms card_of_Sigma_ordLeq_infinite by blast
+ ultimately show ?thesis using ordLeq_transitive by blast
+qed
+
+
+corollary card_of_UNION_ordLeq_infinite_Field:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+shows "|\<Union> i \<in> I. A i| \<le>o r"
+proof-
+ let ?B = "Field r"
+ have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ hence "|\<Union> i \<in> I. A i| \<le>o |?B|" using INF LEQ
+ card_of_UNION_ordLeq_infinite by blast
+ thus ?thesis using 1 ordLeq_ordIso_trans by blast
+qed
+
+
+lemma card_of_Plus_infinite1:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A <+> B| =o |A|"
+proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
+ let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b" let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
+ assume *: "B \<noteq> {}"
+ then obtain b1 where 1: "b1 \<in> B" by blast
+ show ?thesis
+ proof(cases "B = {b1}")
+ assume Case1: "B = {b1}"
+ have 2: "bij_betw ?Inl A ((?Inl ` A))"
+ unfolding bij_betw_def inj_on_def by auto
+ hence 3: "infinite (?Inl ` A)"
+ using INF bij_betw_finite[of ?Inl A] by blast
+ let ?A' = "?Inl ` A \<union> {?Inr b1}"
+ obtain g where "bij_betw g (?Inl ` A) ?A'"
+ using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
+ moreover have "?A' = A <+> B" using Case1 by blast
+ ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
+ hence "bij_betw (g o ?Inl) A (A <+> B)"
+ using 2 by (auto simp add: bij_betw_trans)
+ thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+ next
+ assume Case2: "B \<noteq> {b1}"
+ with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
+ obtain f where "inj_on f B \<and> f ` B \<le> A"
+ using LEQ card_of_ordLeq[of B] by fastforce
+ with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
+ unfolding inj_on_def by auto
+ with 3 have "|A <+> B| \<le>o |A \<times> B|"
+ by (auto simp add: card_of_Plus_Times)
+ moreover have "|A \<times> B| =o |A|"
+ using assms * by (simp add: card_of_Times_infinite_simps)
+ ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by metis
+ thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
+ qed
+qed
+
+
+lemma card_of_Plus_infinite2:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|B <+> A| =o |A|"
+using assms card_of_Plus_commute card_of_Plus_infinite1
+ordIso_equivalence by blast
+
+
+lemma card_of_Plus_infinite:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
+using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
+
+
+corollary Card_order_Plus_infinite:
+assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
+ LEQ: "p \<le>o r"
+shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
+proof-
+ have "| Field r <+> Field p | =o | Field r | \<and>
+ | Field p <+> Field r | =o | Field r |"
+ using assms by (simp add: card_of_Plus_infinite card_of_mono2)
+ thus ?thesis
+ using assms card_of_Field_ordIso[of r]
+ ordIso_transitive[of "|Field r <+> Field p|"]
+ ordIso_transitive[of _ "|Field r|"] by blast
+qed
+
+
+lemma card_of_Un_infinite:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
+proof-
+ have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq)
+ moreover have "|A <+> B| =o |A|"
+ using assms by (metis card_of_Plus_infinite)
+ ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
+ hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast
+ thus ?thesis using Un_commute[of B A] by auto
+qed
+
+
+lemma card_of_Un_diff_infinite:
+assumes INF: "infinite A" and LESS: "|B| <o |A|"
+shows "|A - B| =o |A|"
+proof-
+ obtain C where C_def: "C = A - B" by blast
+ have "|A \<union> B| =o |A|"
+ using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast
+ moreover have "C \<union> B = A \<union> B" unfolding C_def by auto
+ ultimately have 1: "|C \<union> B| =o |A|" by auto
+ (* *)
+ {assume *: "|C| \<le>o |B|"
+ moreover
+ {assume **: "finite B"
+ hence "finite C"
+ using card_of_ordLeq_finite * by blast
+ hence False using ** INF card_of_ordIso_finite 1 by blast
+ }
+ hence "infinite B" by auto
+ ultimately have False
+ using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis
+ }
+ hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast
+ {assume *: "finite C"
+ hence "finite B" using card_of_ordLeq_finite 2 by blast
+ hence False using * INF card_of_ordIso_finite 1 by blast
+ }
+ hence "infinite C" by auto
+ hence "|C| =o |A|"
+ using card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis
+ thus ?thesis unfolding C_def .
+qed
+
+
+lemma card_of_Plus_ordLess_infinite:
+assumes INF: "infinite C" and
+ LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+shows "|A <+> B| <o |C|"
+proof(cases "A = {} \<or> B = {}")
+ assume Case1: "A = {} \<or> B = {}"
+ hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
+ using card_of_Plus_empty1 card_of_Plus_empty2 by blast
+ hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"
+ using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
+ thus ?thesis using LESS1 LESS2
+ ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
+ ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
+next
+ assume Case2: "\<not>(A = {} \<or> B = {})"
+ {assume *: "|C| \<le>o |A <+> B|"
+ hence "infinite (A <+> B)" using INF card_of_ordLeq_finite by blast
+ hence 1: "infinite A \<or> infinite B" using finite_Plus by blast
+ {assume Case21: "|A| \<le>o |B|"
+ hence "infinite B" using 1 card_of_ordLeq_finite by blast
+ hence "|A <+> B| =o |B|" using Case2 Case21
+ by (auto simp add: card_of_Plus_infinite)
+ hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+ }
+ moreover
+ {assume Case22: "|B| \<le>o |A|"
+ hence "infinite A" using 1 card_of_ordLeq_finite by blast
+ hence "|A <+> B| =o |A|" using Case2 Case22
+ by (auto simp add: card_of_Plus_infinite)
+ hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
+ }
+ ultimately have False using ordLeq_total card_of_Well_order[of A]
+ card_of_Well_order[of B] by blast
+ }
+ thus ?thesis using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]
+ card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
+qed
+
+
+lemma card_of_Plus_ordLess_infinite_Field:
+assumes INF: "infinite (Field r)" and r: "Card_order r" and
+ LESS1: "|A| <o r" and LESS2: "|B| <o r"
+shows "|A <+> B| <o r"
+proof-
+ let ?C = "Field r"
+ have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|A| <o |?C|" "|B| <o |?C|"
+ using LESS1 LESS2 ordLess_ordIso_trans by blast+
+ hence "|A <+> B| <o |?C|" using INF
+ card_of_Plus_ordLess_infinite by blast
+ thus ?thesis using 1 ordLess_ordIso_trans by blast
+qed
+
+
+lemma infinite_card_of_insert:
+assumes "infinite A"
+shows "|insert a A| =o |A|"
+proof-
+ have iA: "insert a A = A \<union> {a}" by simp
+ show ?thesis
+ using infinite_imp_bij_betw2[OF assms] unfolding iA
+ by (metis bij_betw_inv card_of_ordIso)
+qed
+
+
+subsection {* Cardinals versus lists *}
+
+
+text{* The next is an auxiliary operator, which shall be used for inductive
+proofs of facts concerning the cardinality of @{text "List"} : *}
+
+definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set"
+where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}"
+
+
+lemma lists_def2: "lists A = {l. set l \<le> A}"
+using in_listsI by blast
+
+
+lemma lists_UNION_nlists: "lists A = (\<Union> n. nlists A n)"
+unfolding lists_def2 nlists_def by blast
+
+
+lemma card_of_lists: "|A| \<le>o |lists A|"
+proof-
+ let ?h = "\<lambda> a. [a]"
+ have "inj_on ?h A \<and> ?h ` A \<le> lists A"
+ unfolding inj_on_def lists_def2 by auto
+ thus ?thesis by (metis card_of_ordLeq)
+qed
+
+
+lemma nlists_0: "nlists A 0 = {[]}"
+unfolding nlists_def by auto
+
+
+lemma nlists_not_empty:
+assumes "A \<noteq> {}"
+shows "nlists A n \<noteq> {}"
+proof(induct n, simp add: nlists_0)
+ fix n assume "nlists A n \<noteq> {}"
+ then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto
+ hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto
+ thus "nlists A (Suc n) \<noteq> {}" by auto
+qed
+
+
+lemma Nil_in_lists: "[] \<in> lists A"
+unfolding lists_def2 by auto
+
+
+lemma lists_not_empty: "lists A \<noteq> {}"
+using Nil_in_lists by blast
+
+
+lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+proof-
+ let ?B = "A \<times> (nlists A n)" let ?h = "\<lambda>(a,l). a # l"
+ have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)"
+ unfolding inj_on_def nlists_def by auto
+ moreover have "nlists A (Suc n) \<le> ?h ` ?B"
+ proof(auto)
+ fix l assume "l \<in> nlists A (Suc n)"
+ hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto
+ then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv)
+ hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto
+ thus "l \<in> ?h ` ?B" using 2 unfolding nlists_def by auto
+ qed
+ ultimately have "bij_betw ?h ?B (nlists A (Suc n))"
+ unfolding bij_betw_def by auto
+ thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+qed
+
+
+lemma card_of_nlists_infinite:
+assumes "infinite A"
+shows "|nlists A n| \<le>o |A|"
+proof(induct n)
+ have "A \<noteq> {}" using assms by auto
+ thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0 card_of_singl_ordLeq)
+next
+ fix n assume IH: "|nlists A n| \<le>o |A|"
+ have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+ using card_of_nlists_Succ by blast
+ moreover
+ {have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast
+ hence "|A \<times> (nlists A n)| =o |A|"
+ using assms IH by (auto simp add: card_of_Times_infinite)
+ }
+ ultimately show "|nlists A (Suc n)| \<le>o |A|"
+ using ordIso_transitive ordIso_iff_ordLeq by blast
+qed
+
+
+lemma card_of_lists_infinite:
+assumes "infinite A"
+shows "|lists A| =o |A|"
+proof-
+ have "|lists A| \<le>o |A|"
+ using assms
+ by (auto simp add: lists_UNION_nlists card_of_UNION_ordLeq_infinite
+ infinite_iff_card_of_nat card_of_nlists_infinite)
+ thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast
+qed
+
+
+lemma Card_order_lists_infinite:
+assumes "Card_order r" and "infinite(Field r)"
+shows "|lists(Field r)| =o r"
+using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast
+
+
+
+subsection {* The cardinal $\omega$ and the finite cardinals *}
+
+
+text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict
+order relation on
+@{text "nat"}, that we abbreviate by @{text "natLeq"}. The finite cardinals
+shall be the restrictions of these relations to the numbers smaller than
+fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}. *}
+
+abbreviation "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"
+abbreviation "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
+
+abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
+where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
+
+lemma infinite_cartesian_product:
+assumes "infinite A" "infinite B"
+shows "infinite (A \<times> B)"
+proof
+ assume "finite (A \<times> B)"
+ from assms(1) have "A \<noteq> {}" by auto
+ with `finite (A \<times> B)` have "finite B" using finite_cartesian_productD2 by auto
+ with assms(2) show False by simp
+qed
+
+
+
+subsubsection {* First as well-orders *}
+
+
+lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"
+by(unfold Field_def, auto)
+
+
+lemma natLeq_Refl: "Refl natLeq"
+unfolding refl_on_def Field_def by auto
+
+
+lemma natLeq_trans: "trans natLeq"
+unfolding trans_def by auto
+
+
+lemma natLeq_Preorder: "Preorder natLeq"
+unfolding preorder_on_def
+by (auto simp add: natLeq_Refl natLeq_trans)
+
+
+lemma natLeq_antisym: "antisym natLeq"
+unfolding antisym_def by auto
+
+
+lemma natLeq_Partial_order: "Partial_order natLeq"
+unfolding partial_order_on_def
+by (auto simp add: natLeq_Preorder natLeq_antisym)
+
+
+lemma natLeq_Total: "Total natLeq"
+unfolding total_on_def by auto
+
+
+lemma natLeq_Linear_order: "Linear_order natLeq"
+unfolding linear_order_on_def
+by (auto simp add: natLeq_Partial_order natLeq_Total)
+
+
+lemma natLeq_natLess_Id: "natLess = natLeq - Id"
+by auto
+
+
+lemma natLeq_Well_order: "Well_order natLeq"
+unfolding well_order_on_def
+using natLeq_Linear_order wf_less natLeq_natLess_Id by auto
+
+
+corollary natLeq_well_order_on: "well_order_on UNIV natLeq"
+using natLeq_Well_order Field_natLeq by auto
+
+
+lemma natLeq_wo_rel: "wo_rel natLeq"
+unfolding wo_rel_def using natLeq_Well_order .
+
+
+lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV"
+using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto
+
+
+lemma closed_nat_set_iff:
+assumes "\<forall>(m::nat) n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
+shows "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
+proof-
+ {assume "A \<noteq> UNIV" hence "\<exists>n. n \<notin> A" by blast
+ moreover obtain n where n_def: "n = (LEAST n. n \<notin> A)" by blast
+ ultimately have 1: "n \<notin> A \<and> (\<forall>m. m < n \<longrightarrow> m \<in> A)"
+ using LeastI_ex[of "\<lambda> n. n \<notin> A"] n_def Least_le[of "\<lambda> n. n \<notin> A"] by fastforce
+ have "A = {0 ..< n}"
+ proof(auto simp add: 1)
+ fix m assume *: "m \<in> A"
+ {assume "n \<le> m" with assms * have "n \<in> A" by blast
+ hence False using 1 by auto
+ }
+ thus "m < n" by fastforce
+ qed
+ hence "\<exists>n. A = {0 ..< n}" by blast
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma Field_natLeq_on: "Field (natLeq_on n) = {0 ..< n}"
+unfolding Field_def by auto
+
+
+lemma natLeq_underS_less: "rel.underS natLeq n = {0 ..< n}"
+unfolding rel.underS_def by auto
+
+
+lemma Restr_natLeq: "Restr natLeq {0 ..< n} = natLeq_on n"
+by auto
+
+
+lemma Restr_natLeq2:
+"Restr natLeq (rel.underS natLeq n) = natLeq_on n"
+by (auto simp add: Restr_natLeq natLeq_underS_less)
+
+
+lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"
+using Restr_natLeq[of n] natLeq_Well_order
+ Well_order_Restr[of natLeq "{0..<n}"] by auto
+
+
+corollary natLeq_on_well_order_on: "well_order_on {0 ..< n} (natLeq_on n)"
+using natLeq_on_Well_order Field_natLeq_on by auto
+
+
+lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"
+unfolding wo_rel_def using natLeq_on_Well_order .
+
+
+lemma natLeq_on_ofilter_less_eq:
+"n \<le> m \<Longrightarrow> wo_rel.ofilter (natLeq_on m) {0 ..< n}"
+by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def,
+ simp add: Field_natLeq_on, unfold rel.under_def, auto)
+
+
+lemma natLeq_on_ofilter_iff:
+"wo_rel.ofilter (natLeq_on m) A = (\<exists>n \<le> m. A = {0 ..< n})"
+proof(rule iffI)
+ assume *: "wo_rel.ofilter (natLeq_on m) A"
+ hence 1: "A \<le> {0..<m}"
+ by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def Field_natLeq_on)
+ hence "\<forall>n1 n2. n2 \<in> A \<and> n1 \<le> n2 \<longrightarrow> n1 \<in> A"
+ using * by(fastforce simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def)
+ hence "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
+ thus "\<exists>n \<le> m. A = {0 ..< n}" using 1 atLeastLessThan_less_eq by blast
+next
+ assume "(\<exists>n\<le>m. A = {0 ..< n})"
+ thus "wo_rel.ofilter (natLeq_on m) A" by (auto simp add: natLeq_on_ofilter_less_eq)
+qed
+
+
+
+subsubsection {* Then as cardinals *}
+
+
+lemma natLeq_Card_order: "Card_order natLeq"
+proof(auto simp add: natLeq_Well_order
+ Card_order_iff_Restr_underS Restr_natLeq2, simp add: Field_natLeq)
+ fix n have "finite(Field (natLeq_on n))"
+ unfolding Field_natLeq_on by auto
+ moreover have "infinite(UNIV::nat set)" by auto
+ ultimately show "natLeq_on n <o |UNIV::nat set|"
+ using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
+ Field_card_of[of "UNIV::nat set"]
+ card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto
+qed
+
+
+corollary card_of_Field_natLeq:
+"|Field natLeq| =o natLeq"
+using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
+ ordIso_symmetric[of natLeq] by blast
+
+
+corollary card_of_nat:
+"|UNIV::nat set| =o natLeq"
+using Field_natLeq card_of_Field_natLeq by auto
+
+
+corollary infinite_iff_natLeq_ordLeq:
+"infinite A = ( natLeq \<le>o |A| )"
+using infinite_iff_card_of_nat[of A] card_of_nat
+ ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
+
+
+corollary finite_iff_ordLess_natLeq:
+"finite A = ( |A| <o natLeq)"
+using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
+ card_of_Well_order natLeq_Well_order by blast
+
+
+lemma ordIso_natLeq_on_imp_finite:
+"|A| =o natLeq_on n \<Longrightarrow> finite A"
+unfolding ordIso_def iso_def[abs_def]
+by (auto simp: Field_natLeq_on bij_betw_finite Field_card_of)
+
+
+lemma natLeq_on_Card_order: "Card_order (natLeq_on n)"
+proof(unfold card_order_on_def,
+ auto simp add: natLeq_on_Well_order, simp add: Field_natLeq_on)
+ fix r assume "well_order_on {0..<n} r"
+ thus "natLeq_on n \<le>o r"
+ using finite_atLeastLessThan natLeq_on_well_order_on
+ finite_well_order_on_ordIso ordIso_iff_ordLeq by blast
+qed
+
+
+corollary card_of_Field_natLeq_on:
+"|Field (natLeq_on n)| =o natLeq_on n"
+using Field_natLeq_on natLeq_on_Card_order
+ Card_order_iff_ordIso_card_of[of "natLeq_on n"]
+ ordIso_symmetric[of "natLeq_on n"] by blast
+
+
+corollary card_of_less:
+"|{0 ..< n}| =o natLeq_on n"
+using Field_natLeq_on card_of_Field_natLeq_on by auto
+
+
+lemma natLeq_on_ordLeq_less_eq:
+"((natLeq_on m) \<le>o (natLeq_on n)) = (m \<le> n)"
+proof
+ assume "natLeq_on m \<le>o natLeq_on n"
+ then obtain f where "inj_on f {0..<m} \<and> f ` {0..<m} \<le> {0..<n}"
+ using Field_natLeq_on[of m] Field_natLeq_on[of n]
+ unfolding ordLeq_def using embed_inj_on[of "natLeq_on m" "natLeq_on n"]
+ embed_Field[of "natLeq_on m" "natLeq_on n"] using natLeq_on_Well_order[of m] by fastforce
+ thus "m \<le> n" using atLeastLessThan_less_eq2 by blast
+next
+ assume "m \<le> n"
+ hence "inj_on id {0..<m} \<and> id ` {0..<m} \<le> {0..<n}" unfolding inj_on_def by auto
+ hence "|{0..<m}| \<le>o |{0..<n}|" using card_of_ordLeq by blast
+ thus "natLeq_on m \<le>o natLeq_on n"
+ using card_of_less ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
+qed
+
+
+lemma natLeq_on_ordLeq_less:
+"((natLeq_on m) <o (natLeq_on n)) = (m < n)"
+using not_ordLeq_iff_ordLess[of "natLeq_on m" "natLeq_on n"]
+natLeq_on_Well_order natLeq_on_ordLeq_less_eq by auto
+
+
+
+subsubsection {* "Backwards compatibility" with the numeric cardinal operator for finite sets *}
+
+
+lemma finite_card_of_iff_card2:
+assumes FIN: "finite A" and FIN': "finite B"
+shows "( |A| \<le>o |B| ) = (card A \<le> card B)"
+using assms card_of_ordLeq[of A B] inj_on_iff_card_le[of A B] by blast
+
+
+lemma finite_imp_card_of_natLeq_on:
+assumes "finite A"
+shows "|A| =o natLeq_on (card A)"
+proof-
+ obtain h where "bij_betw h A {0 ..< card A}"
+ using assms ex_bij_betw_finite_nat by blast
+ thus ?thesis using card_of_ordIso card_of_less ordIso_equivalence by blast
+qed
+
+
+lemma finite_iff_card_of_natLeq_on:
+"finite A = (\<exists>n. |A| =o natLeq_on n)"
+using finite_imp_card_of_natLeq_on[of A]
+by(auto simp add: ordIso_natLeq_on_imp_finite)
+
+
+
+subsection {* The successor of a cardinal *}
+
+
+text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}
+being a successor cardinal of @{text "r"}. Although the definition does
+not require @{text "r"} to be a cardinal, only this case will be meaningful. *}
+
+
+definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
+where
+"isCardSuc r r' \<equiv>
+ Card_order r' \<and> r <o r' \<and>
+ (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
+
+
+text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},
+by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.
+Again, the picked item shall be proved unique up to order-isomorphism. *}
+
+
+definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
+where
+"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
+
+
+lemma exists_minim_Card_order:
+"\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+unfolding card_order_on_def using exists_minim_Well_order by blast
+
+
+lemma exists_isCardSuc:
+assumes "Card_order r"
+shows "\<exists>r'. isCardSuc r r'"
+proof-
+ let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
+ have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
+ by (simp add: card_of_Card_order Card_order_Pow)
+ then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"
+ using exists_minim_Card_order[of ?R] by blast
+ thus ?thesis unfolding isCardSuc_def by auto
+qed
+
+
+lemma cardSuc_isCardSuc:
+assumes "Card_order r"
+shows "isCardSuc r (cardSuc r)"
+unfolding cardSuc_def using assms
+by (simp add: exists_isCardSuc someI_ex)
+
+
+lemma cardSuc_Card_order:
+"Card_order r \<Longrightarrow> Card_order(cardSuc r)"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+lemma cardSuc_greater:
+"Card_order r \<Longrightarrow> r <o cardSuc r"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+lemma cardSuc_ordLeq:
+"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
+using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
+
+
+text{* The minimality property of @{text "cardSuc"} originally present in its definition
+is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}: *}
+
+lemma cardSuc_least_aux:
+"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+text{* But from this we can infer general minimality: *}
+
+lemma cardSuc_least:
+assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
+shows "cardSuc r \<le>o r'"
+proof-
+ let ?p = "cardSuc r"
+ have 0: "Well_order ?p \<and> Well_order r'"
+ using assms cardSuc_Card_order unfolding card_order_on_def by blast
+ {assume "r' <o ?p"
+ then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"
+ using internalize_ordLess[of r' ?p] by blast
+ (* *)
+ have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
+ moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast
+ ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast
+ hence False using 2 not_ordLess_ordLeq by blast
+ }
+ thus ?thesis using 0 ordLess_or_ordLeq by blast
+qed
+
+
+lemma cardSuc_ordLess_ordLeq:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(r <o r') = (cardSuc r \<le>o r')"
+proof(auto simp add: assms cardSuc_least)
+ assume "cardSuc r \<le>o r'"
+ thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast
+qed
+
+
+lemma cardSuc_ordLeq_ordLess:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(r' <o cardSuc r) = (r' \<le>o r)"
+proof-
+ have "Well_order r \<and> Well_order r'"
+ using assms unfolding card_order_on_def by auto
+ moreover have "Well_order(cardSuc r)"
+ using assms cardSuc_Card_order card_order_on_def by blast
+ ultimately show ?thesis
+ using assms cardSuc_ordLess_ordLeq[of r r']
+ not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast
+qed
+
+
+lemma cardSuc_mono_ordLeq:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
+using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast
+
+
+lemma cardSuc_invar_ordIso:
+assumes CARD: "Card_order r" and CARD': "Card_order r'"
+shows "(cardSuc r =o cardSuc r') = (r =o r')"
+proof-
+ have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
+ using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)
+ thus ?thesis
+ using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
+ using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
+qed
+
+
+lemma cardSuc_natLeq_on_Suc:
+"cardSuc(natLeq_on n) =o natLeq_on(Suc n)"
+proof-
+ obtain r r' p where r_def: "r = natLeq_on n" and
+ r'_def: "r' = cardSuc(natLeq_on n)" and
+ p_def: "p = natLeq_on(Suc n)" by blast
+ (* Preliminary facts: *)
+ have CARD: "Card_order r \<and> Card_order r' \<and> Card_order p" unfolding r_def r'_def p_def
+ using cardSuc_ordLess_ordLeq natLeq_on_Card_order cardSuc_Card_order by blast
+ hence WELL: "Well_order r \<and> Well_order r' \<and> Well_order p"
+ unfolding card_order_on_def by force
+ have FIELD: "Field r = {0..<n} \<and> Field p = {0..<(Suc n)}"
+ unfolding r_def p_def Field_natLeq_on by simp
+ hence FIN: "finite (Field r)" by force
+ have "r <o r'" using CARD unfolding r_def r'_def using cardSuc_greater by blast
+ hence "|Field r| <o r'" using CARD card_of_Field_ordIso ordIso_ordLess_trans by blast
+ hence LESS: "|Field r| <o |Field r'|"
+ using CARD card_of_Field_ordIso ordLess_ordIso_trans ordIso_symmetric by blast
+ (* Main proof: *)
+ have "r' \<le>o p" using CARD unfolding r_def r'_def p_def
+ using natLeq_on_ordLeq_less cardSuc_ordLess_ordLeq by blast
+ moreover have "p \<le>o r'"
+ proof-
+ {assume "r' <o p"
+ then obtain f where 0: "embedS r' p f" unfolding ordLess_def by force
+ let ?q = "Restr p (f ` Field r')"
+ have 1: "embed r' p f" using 0 unfolding embedS_def by force
+ hence 2: "f ` Field r' < {0..<(Suc n)}"
+ using WELL FIELD 0 by (auto simp add: embedS_iff)
+ have "wo_rel.ofilter p (f ` Field r')" using embed_Field_ofilter 1 WELL by blast
+ then obtain m where "m \<le> Suc n" and 3: "f ` (Field r') = {0..<m}"
+ unfolding p_def by (auto simp add: natLeq_on_ofilter_iff)
+ hence 4: "m \<le> n" using 2 by force
+ (* *)
+ have "bij_betw f (Field r') (f ` (Field r'))"
+ using 1 WELL embed_inj_on unfolding bij_betw_def by force
+ moreover have "finite(f ` (Field r'))" using 3 finite_atLeastLessThan[of 0 m] by force
+ ultimately have 5: "finite (Field r') \<and> card(Field r') = card (f ` (Field r'))"
+ using bij_betw_same_card bij_betw_finite by metis
+ hence "card(Field r') \<le> card(Field r)" using 3 4 FIELD by force
+ hence "|Field r'| \<le>o |Field r|" using FIN 5 finite_card_of_iff_card2 by blast
+ hence False using LESS not_ordLess_ordLeq by auto
+ }
+ thus ?thesis using WELL CARD by (fastforce simp: not_ordLess_iff_ordLeq)
+ qed
+ ultimately show ?thesis using ordIso_iff_ordLeq unfolding r'_def p_def by blast
+qed
+
+
+lemma card_of_cardSuc_finite:
+"finite(Field(cardSuc |A| )) = finite A"
+proof
+ assume *: "finite (Field (cardSuc |A| ))"
+ have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"
+ using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
+ hence "|A| \<le>o |Field(cardSuc |A| )|"
+ using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
+ ordLeq_ordIso_trans by blast
+ thus "finite A" using * card_of_ordLeq_finite by blast
+next
+ assume "finite A"
+ then obtain n where "|A| =o natLeq_on n" using finite_iff_card_of_natLeq_on by blast
+ hence "cardSuc |A| =o cardSuc(natLeq_on n)"
+ using card_of_Card_order cardSuc_invar_ordIso natLeq_on_Card_order by blast
+ hence "cardSuc |A| =o natLeq_on(Suc n)"
+ using cardSuc_natLeq_on_Suc ordIso_transitive by blast
+ hence "cardSuc |A| =o |{0..<(Suc n)}|" using card_of_less ordIso_equivalence by blast
+ moreover have "|Field (cardSuc |A| ) | =o cardSuc |A|"
+ using card_of_Field_ordIso cardSuc_Card_order card_of_Card_order by blast
+ ultimately have "|Field (cardSuc |A| ) | =o |{0..<(Suc n)}|"
+ using ordIso_equivalence by blast
+ thus "finite (Field (cardSuc |A| ))"
+ using card_of_ordIso_finite finite_atLeastLessThan by blast
+qed
+
+
+lemma cardSuc_finite:
+assumes "Card_order r"
+shows "finite (Field (cardSuc r)) = finite (Field r)"
+proof-
+ let ?A = "Field r"
+ have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)
+ hence "cardSuc |?A| =o cardSuc r" using assms
+ by (simp add: card_of_Card_order cardSuc_invar_ordIso)
+ moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"
+ by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
+ moreover
+ {have "|Field (cardSuc r) | =o cardSuc r"
+ using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
+ hence "cardSuc r =o |Field (cardSuc r) |"
+ using ordIso_symmetric by blast
+ }
+ ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"
+ using ordIso_transitive by blast
+ hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"
+ using card_of_ordIso_finite by blast
+ thus ?thesis by (simp only: card_of_cardSuc_finite)
+qed
+
+
+lemma card_of_Plus_ordLeq_infinite_Field:
+assumes r: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+and c: "Card_order r"
+shows "|A <+> B| \<le>o r"
+proof-
+ let ?r' = "cardSuc r"
+ have "Card_order ?r' \<and> infinite (Field ?r')" using assms
+ by (simp add: cardSuc_Card_order cardSuc_finite)
+ moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c
+ by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
+ ultimately have "|A <+> B| <o ?r'"
+ using card_of_Plus_ordLess_infinite_Field by blast
+ thus ?thesis using c r
+ by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
+qed
+
+
+lemma card_of_Un_ordLeq_infinite_Field:
+assumes C: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+and "Card_order r"
+shows "|A Un B| \<le>o r"
+using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
+ordLeq_transitive by blast
+
+
+
+subsection {* Regular cardinals *}
+
+
+definition cofinal where
+"cofinal A r \<equiv>
+ ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"
+
+
+definition regular where
+"regular r \<equiv>
+ ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
+
+
+definition relChain where
+"relChain r As \<equiv>
+ ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
+
+lemma regular_UNION:
+assumes r: "Card_order r" "regular r"
+and As: "relChain r As"
+and Bsub: "B \<le> (UN i : Field r. As i)"
+and cardB: "|B| <o r"
+shows "EX i : Field r. B \<le> As i"
+proof-
+ let ?phi = "%b j. j : Field r \<and> b : As j"
+ have "ALL b : B. EX j. ?phi b j" using Bsub by blast
+ then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"
+ using bchoice[of B ?phi] by blast
+ let ?K = "f ` B"
+ {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> As i"
+ have 2: "cofinal ?K r"
+ unfolding cofinal_def proof auto
+ fix i assume i: "i : Field r"
+ with 1 obtain b where b: "b : B \<and> b \<notin> As i" by blast
+ hence "i \<noteq> f b \<and> ~ (f b,i) : r"
+ using As f unfolding relChain_def by auto
+ hence "i \<noteq> f b \<and> (i, f b) : r" using r
+ unfolding card_order_on_def well_order_on_def linear_order_on_def
+ total_on_def using i f b by auto
+ with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
+ qed
+ moreover have "?K \<le> Field r" using f by blast
+ ultimately have "|?K| =o r" using 2 r unfolding regular_def by blast
+ moreover
+ {
+ have "|?K| <=o |B|" using card_of_image .
+ hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast
+ }
+ ultimately have False using not_ordLess_ordIso by blast
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma infinite_cardSuc_regular:
+assumes r_inf: "infinite (Field r)" and r_card: "Card_order r"
+shows "regular (cardSuc r)"
+proof-
+ let ?r' = "cardSuc r"
+ have r': "Card_order ?r'"
+ "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
+ using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
+ show ?thesis
+ unfolding regular_def proof auto
+ fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
+ hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)
+ also have 22: "|Field ?r'| =o ?r'"
+ using r' by (simp add: card_of_Field_ordIso[of ?r'])
+ finally have "|K| \<le>o ?r'" .
+ moreover
+ {let ?L = "UN j : K. rel.underS ?r' j"
+ let ?J = "Field r"
+ have rJ: "r =o |?J|"
+ using r_card card_of_Field_ordIso ordIso_symmetric by blast
+ assume "|K| <o ?r'"
+ hence "|K| <=o r" using r' card_of_Card_order[of K] by blast
+ hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
+ moreover
+ {have "ALL j : K. |rel.underS ?r' j| <o ?r'"
+ using r' 1 by (auto simp: card_of_underS)
+ hence "ALL j : K. |rel.underS ?r' j| \<le>o r"
+ using r' card_of_Card_order by blast
+ hence "ALL j : K. |rel.underS ?r' j| \<le>o |?J|"
+ using rJ ordLeq_ordIso_trans by blast
+ }
+ ultimately have "|?L| \<le>o |?J|"
+ using r_inf card_of_UNION_ordLeq_infinite by blast
+ hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
+ hence "|?L| <o ?r'" using r' card_of_Card_order by blast
+ moreover
+ {
+ have "Field ?r' \<le> ?L"
+ using 2 unfolding rel.underS_def cofinal_def by auto
+ hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
+ hence "?r' \<le>o |?L|"
+ using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
+ }
+ ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast
+ hence False using ordLess_irreflexive by blast
+ }
+ ultimately show "|K| =o ?r'"
+ unfolding ordLeq_iff_ordLess_or_ordIso by blast
+ qed
+qed
+
+lemma cardSuc_UNION:
+assumes r: "Card_order r" and "infinite (Field r)"
+and As: "relChain (cardSuc r) As"
+and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"
+and cardB: "|B| <=o r"
+shows "EX i : Field (cardSuc r). B \<le> As i"
+proof-
+ let ?r' = "cardSuc r"
+ have "Card_order ?r' \<and> |B| <o ?r'"
+ using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
+ card_of_Card_order by blast
+ moreover have "regular ?r'"
+ using assms by(simp add: infinite_cardSuc_regular)
+ ultimately show ?thesis
+ using As Bsub cardB regular_UNION by blast
+qed
+
+
+subsection {* Others *}
+
+(* FIXME: finitte ~> finite? *)
+lemma card_of_infinite_diff_finitte:
+assumes "infinite A" and "finite B"
+shows "|A - B| =o |A|"
+by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2)
+
+(* function space *)
+definition Func where
+"Func A B \<equiv>
+ {f. (\<forall> a. f a \<noteq> None \<longleftrightarrow> a \<in> A) \<and> (\<forall> a \<in> A. case f a of Some b \<Rightarrow> b \<in> B |None \<Rightarrow> True)}"
+
+lemma Func_empty:
+"Func {} B = {empty}"
+unfolding Func_def by auto
+
+lemma Func_elim:
+assumes "g \<in> Func A B" and "a \<in> A"
+shows "\<exists> b. b \<in> B \<and> g a = Some b"
+using assms unfolding Func_def by (cases "g a") force+
+
+definition curr where
+"curr A f \<equiv> \<lambda> a. if a \<in> A then Some (\<lambda> b. f (a,b)) else None"
+
+lemma curr_in:
+assumes f: "f \<in> Func (A <*> B) C"
+shows "curr A f \<in> Func A (Func B C)"
+using assms unfolding curr_def Func_def by auto
+
+lemma curr_inj:
+assumes "f1 \<in> Func (A <*> B) C" and "f2 \<in> Func (A <*> B) C"
+shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
+proof safe
+ assume c: "curr A f1 = curr A f2"
+ show "f1 = f2"
+ proof (rule ext, clarify)
+ fix a b show "f1 (a, b) = f2 (a, b)"
+ proof (cases "(a,b) \<in> A <*> B")
+ case False
+ thus ?thesis using assms unfolding Func_def
+ apply(cases "f1 (a,b)") apply(cases "f2 (a,b)", fastforce, fastforce)
+ apply(cases "f2 (a,b)") by auto
+ next
+ case True hence a: "a \<in> A" and b: "b \<in> B" by auto
+ thus ?thesis
+ using c unfolding curr_def fun_eq_iff
+ apply(elim allE[of _ a]) apply simp unfolding fun_eq_iff by auto
+ qed
+ qed
+qed
+
+lemma curr_surj:
+assumes "g \<in> Func A (Func B C)"
+shows "\<exists> f \<in> Func (A <*> B) C. curr A f = g"
+proof
+ let ?f = "\<lambda> ab. case g (fst ab) of None \<Rightarrow> None | Some g1 \<Rightarrow> g1 (snd ab)"
+ show "curr A ?f = g"
+ proof (rule ext)
+ fix a show "curr A ?f a = g a"
+ proof (cases "a \<in> A")
+ case False
+ hence "g a = None" using assms unfolding Func_def by auto
+ thus ?thesis unfolding curr_def using False by simp
+ next
+ case True
+ obtain g1 where "g1 \<in> Func B C" and "g a = Some g1"
+ using assms using Func_elim[OF assms True] by blast
+ thus ?thesis using True unfolding curr_def by auto
+ qed
+ qed
+ show "?f \<in> Func (A <*> B) C"
+ unfolding Func_def mem_Collect_eq proof(intro conjI allI ballI)
+ fix ab show "?f ab \<noteq> None \<longleftrightarrow> ab \<in> A \<times> B"
+ proof(cases "g (fst ab)")
+ case None
+ hence "fst ab \<notin> A" using assms unfolding Func_def by force
+ thus ?thesis using None by auto
+ next
+ case (Some g1)
+ hence fst: "fst ab \<in> A" and g1: "g1 \<in> Func B C"
+ using assms unfolding Func_def[of A] by force+
+ hence "?f ab \<noteq> None \<longleftrightarrow> g1 (snd ab) \<noteq> None" using Some by auto
+ also have "... \<longleftrightarrow> snd ab \<in> B" using g1 unfolding Func_def by auto
+ also have "... \<longleftrightarrow> ab \<in> A \<times> B" using fst by (cases ab, auto)
+ finally show ?thesis .
+ qed
+ next
+ fix ab assume ab: "ab \<in> A \<times> B"
+ hence "fst ab \<in> A" and "snd ab \<in> B" by(cases ab, auto)
+ then obtain g1 where "g1 \<in> Func B C" and "g (fst ab) = Some g1"
+ using assms using Func_elim[OF assms] by blast
+ thus "case ?f ab of Some c \<Rightarrow> c \<in> C |None \<Rightarrow> True"
+ unfolding Func_def by auto
+ qed
+qed
+
+(* FIXME: betwe ~> betw? *)
+lemma bij_betwe_curr:
+"bij_betw (curr A) (Func (A <*> B) C) (Func A (Func B C))"
+unfolding bij_betw_def inj_on_def image_def
+using curr_in curr_inj curr_surj by blast
+
+lemma card_of_Func_Times:
+"|Func (A <*> B) C| =o |Func A (Func B C)|"
+unfolding card_of_ordIso[symmetric]
+using bij_betwe_curr by blast
+
+definition Func_map where
+"Func_map B2 f1 f2 g b2 \<equiv>
+ if b2 \<in> B2 then case g (f2 b2) of None \<Rightarrow> None | Some a1 \<Rightarrow> Some (f1 a1)
+ else None"
+
+lemma Func_map:
+assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
+shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
+unfolding Func_def mem_Collect_eq proof(intro conjI allI ballI)
+ fix b2 show "Func_map B2 f1 f2 g b2 \<noteq> None \<longleftrightarrow> b2 \<in> B2"
+ proof(cases "b2 \<in> B2")
+ case True
+ hence "f2 b2 \<in> A2" using f2 by auto
+ then obtain a1 where "g (f2 b2) = Some a1" and "a1 \<in> A1"
+ using g unfolding Func_def by(cases "g (f2 b2)", fastforce+)
+ thus ?thesis unfolding Func_map_def using True by auto
+ qed(unfold Func_map_def, auto)
+next
+ fix b2 assume b2: "b2 \<in> B2"
+ hence "f2 b2 \<in> A2" using f2 by auto
+ then obtain a1 where "g (f2 b2) = Some a1" and "a1 \<in> A1"
+ using g unfolding Func_def by(cases "g (f2 b2)", fastforce+)
+ thus "case Func_map B2 f1 f2 g b2 of None \<Rightarrow> True | Some b1 \<Rightarrow> b1 \<in> B1"
+ unfolding Func_map_def using b2 f1 by auto
+qed
+
+lemma Func_map_empty:
+"Func_map B2 f1 f2 empty = empty"
+unfolding Func_map_def[abs_def] by (rule ext, auto)
+
+lemma Func_non_emp:
+assumes "B \<noteq> {}"
+shows "Func A B \<noteq> {}"
+proof-
+ obtain b where b: "b \<in> B" using assms by auto
+ hence "(\<lambda> a. if a \<in> A then Some b else None) \<in> Func A B"
+ unfolding Func_def by auto
+ thus ?thesis by blast
+qed
+
+lemma Func_is_emp:
+"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
+proof
+ assume L: ?L
+ moreover {assume "A = {}" hence False using L Func_empty by auto}
+ moreover {assume "B \<noteq> {}" hence False using L Func_non_emp by metis}
+ ultimately show ?R by blast
+next
+ assume R: ?R
+ moreover
+ {fix f assume "f \<in> Func A B"
+ moreover obtain a where "a \<in> A" using R by blast
+ ultimately obtain b where "b \<in> B" unfolding Func_def by(cases "f a", force+)
+ with R have False by auto
+ }
+ thus ?L by blast
+qed
+
+lemma Func_map_surj:
+assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
+and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
+shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
+proof(cases "B2 = {}")
+ case True
+ thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_empty)
+next
+ case False note B2 = False
+ show ?thesis
+proof safe
+ fix h assume h: "h \<in> Func B2 B1"
+ def j1 \<equiv> "inv_into A1 f1"
+ have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
+ then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2" by metis
+ {fix b2 assume b2: "b2 \<in> B2"
+ hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
+ moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
+ ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
+ } note kk = this
+ obtain b22 where b22: "b22 \<in> B2" using B2 by auto
+ def j2 \<equiv> "\<lambda> a2. if a2 \<in> f2 ` B2 then k a2 else b22"
+ have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
+ have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
+ using kk unfolding j2_def by auto
+ def g \<equiv> "Func_map A2 j1 j2 h"
+ have "Func_map B2 f1 f2 g = h"
+ proof (rule ext)
+ fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
+ proof(cases "b2 \<in> B2")
+ case True
+ show ?thesis
+ proof (cases "h b2")
+ case (Some b1)
+ hence b1: "b1 \<in> f1 ` A1" using True h unfolding B1 Func_def by auto
+ show ?thesis
+ using Some True A2 f_inv_into_f[OF b1]
+ unfolding g_def Func_map_def j1_def j2[OF True] by auto
+ qed(insert A2 True j2[OF True], unfold g_def Func_map_def, auto)
+ qed(insert h, unfold Func_def Func_map_def, auto)
+ qed
+ moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
+ using inv_into_into j2A2 B1 A2 inv_into_into
+ unfolding j1_def image_def by(force, force)
+ ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
+ unfolding Func_map_def[abs_def] unfolding image_def by auto
+qed(insert B1 Func_map[OF _ _ A2(2)], auto)
+qed
+
+(* partial-function space: *)
+definition Pfunc where
+"Pfunc A B \<equiv>
+ {f. (\<forall>a. f a \<noteq> None \<longrightarrow> a \<in> A) \<and>
+ (\<forall>a. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
+
+lemma Func_Pfunc:
+"Func A B \<subseteq> Pfunc A B"
+unfolding Func_def Pfunc_def by auto
+
+lemma Pfunc_Func:
+"Pfunc A B = (\<Union> A' \<in> Pow A. Func A' B)"
+proof safe
+ fix f assume f: "f \<in> Pfunc A B"
+ show "f \<in> (\<Union>A'\<in>Pow A. Func A' B)"
+ proof (intro UN_I)
+ let ?A' = "{a. f a \<noteq> None}"
+ show "?A' \<in> Pow A" using f unfolding Pow_def Pfunc_def by auto
+ show "f \<in> Func ?A' B" using f unfolding Func_def Pfunc_def by auto
+ qed
+next
+ fix f A' assume "f \<in> Func A' B" and "A' \<subseteq> A"
+ thus "f \<in> Pfunc A B" unfolding Func_def Pfunc_def by auto
+qed
+
+lemma card_of_Pow_Func:
+"|Pow A| =o |Func A (UNIV::bool set)|"
+proof-
+ def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then Some True else Some False)
+ else None"
+ have "bij_betw F (Pow A) (Func A (UNIV::bool set))"
+ unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
+ fix A1 A2 assume A1: "A1 \<in> Pow A" and A2: "A2 \<in> Pow A" and eq: "F A1 = F A2"
+ show "A1 = A2"
+ proof-
+ {fix a
+ have "a \<in> A1 \<longleftrightarrow> F A1 a = Some True" using A1 unfolding F_def Pow_def by auto
+ also have "... \<longleftrightarrow> F A2 a = Some True" unfolding eq ..
+ also have "... \<longleftrightarrow> a \<in> A2" using A2 unfolding F_def Pow_def by auto
+ finally have "a \<in> A1 \<longleftrightarrow> a \<in> A2" .
+ }
+ thus ?thesis by auto
+ qed
+ next
+ show "F ` Pow A = Func A UNIV"
+ proof safe
+ fix f assume f: "f \<in> Func A (UNIV::bool set)"
+ show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
+ let ?A1 = "{a \<in> A. f a = Some True}"
+ show "f = F ?A1" unfolding F_def apply(rule ext)
+ using f unfolding Func_def mem_Collect_eq by (auto,force)
+ qed auto
+ qed(unfold Func_def mem_Collect_eq F_def, auto)
+ qed
+ thus ?thesis unfolding card_of_ordIso[symmetric] by blast
+qed
+
+lemma card_of_Func_mono:
+fixes A1 A2 :: "'a set" and B :: "'b set"
+assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
+shows "|Func A1 B| \<le>o |Func A2 B|"
+proof-
+ obtain bb where bb: "bb \<in> B" using B by auto
+ def F \<equiv> "\<lambda> (f1::'a \<Rightarrow> 'b option) a. if a \<in> A2 then (if a \<in> A1 then f1 a else Some bb)
+ else None"
+ show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI)
+ show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe
+ fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g"
+ show "f = g"
+ proof(rule ext)
+ fix a show "f a = g a"
+ proof(cases "a \<in> A1")
+ case True
+ thus ?thesis using eq A12 unfolding F_def fun_eq_iff
+ by (elim allE[of _ a]) auto
+ qed(insert f g, unfold Func_def, fastforce)
+ qed
+ qed
+ qed(insert bb, unfold Func_def F_def, force)
+qed
+
+lemma card_of_Pfunc_Pow_Func:
+assumes "B \<noteq> {}"
+shows "|Pfunc A B| \<le>o |Pow A <*> Func A B|"
+proof-
+ have "|Pfunc A B| =o |\<Union> A' \<in> Pow A. Func A' B|" (is "_ =o ?K")
+ unfolding Pfunc_Func by(rule card_of_refl)
+ also have "?K \<le>o |Sigma (Pow A) (\<lambda> A'. Func A' B)|" using card_of_UNION_Sigma .
+ also have "|Sigma (Pow A) (\<lambda> A'. Func A' B)| \<le>o |Pow A <*> Func A B|"
+ apply(rule card_of_Sigma_mono1) using card_of_Func_mono[OF _ assms] by auto
+ finally show ?thesis .
+qed
+
+lemma ordLeq_Func:
+assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+shows "|A| \<le>o |Func A B|"
+unfolding card_of_ordLeq[symmetric] proof(intro exI conjI)
+ let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then Some b1 else Some b2)
+ else None"
+ show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto
+ show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def apply auto
+ by (metis option.simps(3))
+qed
+
+lemma infinite_Func:
+assumes A: "infinite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+shows "infinite (Func A B)"
+using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite)
+
+(* alternative function space avoiding the option type, with undefined instead of None *)
+definition Ffunc where
+"Ffunc A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
+
+lemma card_of_Func_Ffunc:
+"|Ffunc A B| =o |Func A B|"
+unfolding card_of_ordIso[symmetric] proof
+ let ?F = "\<lambda> f a. if a \<in> A then Some (f a) else None"
+ show "bij_betw ?F (Ffunc A B) (Func A B)"
+ unfolding bij_betw_def unfolding inj_on_def proof(intro conjI ballI impI)
+ fix f g assume f: "f \<in> Ffunc A B" and g: "g \<in> Ffunc A B" and eq: "?F f = ?F g"
+ show "f = g"
+ proof(rule ext)
+ fix a
+ show "f a = g a"
+ proof(cases "a \<in> A")
+ case True
+ have "Some (f a) = ?F f a" using True by auto
+ also have "... = ?F g a" using eq unfolding fun_eq_iff by(rule allE)
+ also have "... = Some (g a)" using True by auto
+ finally have "Some (f a) = Some (g a)" .
+ thus ?thesis by simp
+ qed(insert f g, unfold Ffunc_def, auto)
+ qed
+ next
+ show "?F ` Ffunc A B = Func A B"
+ proof safe
+ fix f assume f: "f \<in> Func A B"
+ def g \<equiv> "\<lambda> a. case f a of Some b \<Rightarrow> b | None \<Rightarrow> undefined"
+ have "g \<in> Ffunc A B"
+ using f unfolding g_def Func_def Ffunc_def by force+
+ moreover have "f = ?F g"
+ proof(rule ext)
+ fix a show "f a = ?F g a"
+ using f unfolding Func_def g_def by (cases "a \<in> A") force+
+ qed
+ ultimately show "f \<in> ?F ` (Ffunc A B)" by blast
+ qed(unfold Ffunc_def Func_def, auto)
+ qed
+qed
+
+lemma card_of_Func_UNIV:
+"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
+apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
+ let ?F = "\<lambda> f (a::'a). Some ((f a)::'b)"
+ show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
+ unfolding bij_betw_def inj_on_def proof safe
+ fix h :: "'a \<Rightarrow> 'b option" assume h: "h \<in> Func UNIV B"
+ hence "\<forall> a. \<exists> b. h a = Some b" unfolding Func_def by auto
+ then obtain f where f: "\<forall> a. h a = Some (f a)" by metis
+ hence "range f \<subseteq> B" using h unfolding Func_def by auto
+ thus "h \<in> (\<lambda>f a. Some (f a)) ` {f. range f \<subseteq> B}" using f unfolding image_def by auto
+ qed(unfold Func_def fun_eq_iff, auto)
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Constructions_on_Wellorders.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,797 @@
+(* Title: HOL/Ordinals_and_Cardinals/Constructions_on_Wellorders.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Constructions on wellorders.
+*)
+
+header {* Constructions on Wellorders *}
+
+theory Constructions_on_Wellorders
+imports
+ "../Ordinals_and_Cardinals_Base/Constructions_on_Wellorders_Base"
+ Wellorder_Embedding
+begin
+
+declare
+ ordLeq_Well_order_simp[simp]
+ ordLess_Well_order_simp[simp]
+ ordIso_Well_order_simp[simp]
+ not_ordLeq_iff_ordLess[simp]
+ not_ordLess_iff_ordLeq[simp]
+
+
+subsection {* Restriction to a set *}
+
+lemma Restr_incr2:
+"r <= r' \<Longrightarrow> Restr r A <= Restr r' A"
+by blast
+
+lemma Restr_incr:
+"\<lbrakk>r \<le> r'; A \<le> A'\<rbrakk> \<Longrightarrow> Restr r A \<le> Restr r' A'"
+by blast
+
+lemma Restr_Int:
+"Restr (Restr r A) B = Restr r (A Int B)"
+by blast
+
+lemma Restr_iff: "(a,b) : Restr r A = (a : A \<and> b : A \<and> (a,b) : r)"
+by (auto simp add: Field_def)
+
+lemma Restr_subset1: "Restr r A \<le> r"
+by auto
+
+lemma Restr_subset2: "Restr r A \<le> A \<times> A"
+by auto
+
+lemma wf_Restr:
+"wf r \<Longrightarrow> wf(Restr r A)"
+using wf_subset Restr_subset by blast
+
+lemma Restr_incr1:
+"A \<le> B \<Longrightarrow> Restr r A \<le> Restr r B"
+by blast
+
+
+subsection {* Order filters versus restrictions and embeddings *}
+
+lemma ofilter_Restr:
+assumes WELL: "Well_order r" and
+ OFA: "ofilter r A" and OFB: "ofilter r B" and SUB: "A \<le> B"
+shows "ofilter (Restr r B) A"
+proof-
+ let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (auto simp add: wo_rel.REFL)
+ hence Field: "Field ?rB = Field r Int B"
+ using Refl_Field_Restr by blast
+ have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+ by (auto simp add: Well_order_Restr wo_rel_def)
+ (* Main proof *)
+ show ?thesis
+ proof(auto simp add: WellB wo_rel.ofilter_def)
+ fix a assume "a \<in> A"
+ hence "a \<in> Field r \<and> a \<in> B" using assms Well
+ by (auto simp add: wo_rel.ofilter_def)
+ with Field show "a \<in> Field(Restr r B)" by auto
+ next
+ fix a b assume *: "a \<in> A" and "b \<in> under (Restr r B) a"
+ hence "b \<in> under r a"
+ using WELL OFB SUB ofilter_Restr_under[of r B a] by auto
+ thus "b \<in> A" using * Well OFA by(auto simp add: wo_rel.ofilter_def)
+ qed
+qed
+
+lemma ofilter_subset_iso:
+assumes WELL: "Well_order r" and
+ OFA: "ofilter r A" and OFB: "ofilter r B"
+shows "(A = B) = iso (Restr r A) (Restr r B) id"
+using assms
+by (auto simp add: ofilter_subset_embedS_iso)
+
+
+subsection {* Ordering the well-orders by existence of embeddings *}
+
+corollary ordLeq_refl_on: "refl_on {r. Well_order r} ordLeq"
+using ordLeq_reflexive unfolding ordLeq_def refl_on_def
+by blast
+
+corollary ordLeq_trans: "trans ordLeq"
+using trans_def[of ordLeq] ordLeq_transitive by blast
+
+corollary ordLeq_preorder_on: "preorder_on {r. Well_order r} ordLeq"
+by(auto simp add: preorder_on_def ordLeq_refl_on ordLeq_trans)
+
+corollary ordIso_refl_on: "refl_on {r. Well_order r} ordIso"
+using ordIso_reflexive unfolding refl_on_def ordIso_def
+by blast
+
+corollary ordIso_trans: "trans ordIso"
+using trans_def[of ordIso] ordIso_transitive by blast
+
+corollary ordIso_sym: "sym ordIso"
+by (auto simp add: sym_def ordIso_symmetric)
+
+corollary ordIso_equiv: "equiv {r. Well_order r} ordIso"
+by (auto simp add: equiv_def ordIso_sym ordIso_refl_on ordIso_trans)
+
+lemma ordLess_irrefl: "irrefl ordLess"
+by(unfold irrefl_def, auto simp add: ordLess_irreflexive)
+
+lemma ordLess_or_ordIso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "r <o r' \<or> r' <o r \<or> r =o r'"
+unfolding ordLess_def ordIso_def
+using assms embedS_or_iso[of r r'] by auto
+
+corollary ordLeq_ordLess_Un_ordIso:
+"ordLeq = ordLess \<union> ordIso"
+by (auto simp add: ordLeq_iff_ordLess_or_ordIso)
+
+lemma not_ordLeq_ordLess:
+"r \<le>o r' \<Longrightarrow> \<not> r' <o r"
+using not_ordLess_ordLeq by blast
+
+lemma ordIso_or_ordLess:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "r =o r' \<or> r <o r' \<or> r' <o r"
+using assms ordLess_or_ordLeq ordLeq_iff_ordLess_or_ordIso by blast
+
+lemmas ord_trans = ordIso_transitive ordLeq_transitive ordLess_transitive
+ ordIso_ordLeq_trans ordLeq_ordIso_trans
+ ordIso_ordLess_trans ordLess_ordIso_trans
+ ordLess_ordLeq_trans ordLeq_ordLess_trans
+
+lemma ofilter_ordLeq:
+assumes "Well_order r" and "ofilter r A"
+shows "Restr r A \<le>o r"
+proof-
+ have "A \<le> Field r" using assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+ thus ?thesis using assms
+ by (simp add: ofilter_subset_ordLeq wo_rel.Field_ofilter
+ wo_rel_def Restr_Field)
+qed
+
+corollary under_Restr_ordLeq:
+"Well_order r \<Longrightarrow> Restr r (under r a) \<le>o r"
+by (auto simp add: ofilter_ordLeq wo_rel.under_ofilter wo_rel_def)
+
+
+subsection {* Copy via direct images *}
+
+lemma Id_dir_image: "dir_image Id f \<le> Id"
+unfolding dir_image_def by auto
+
+lemma Un_dir_image:
+"dir_image (r1 \<union> r2) f = (dir_image r1 f) \<union> (dir_image r2 f)"
+unfolding dir_image_def by auto
+
+lemma Int_dir_image:
+assumes "inj_on f (Field r1 \<union> Field r2)"
+shows "dir_image (r1 Int r2) f = (dir_image r1 f) Int (dir_image r2 f)"
+proof
+ show "dir_image (r1 Int r2) f \<le> (dir_image r1 f) Int (dir_image r2 f)"
+ using assms unfolding dir_image_def inj_on_def by auto
+next
+ show "(dir_image r1 f) Int (dir_image r2 f) \<le> dir_image (r1 Int r2) f"
+ proof(clarify)
+ fix a' b'
+ assume "(a',b') \<in> dir_image r1 f" "(a',b') \<in> dir_image r2 f"
+ then obtain a1 b1 a2 b2
+ where 1: "a' = f a1 \<and> b' = f b1 \<and> a' = f a2 \<and> b' = f b2" and
+ 2: "(a1,b1) \<in> r1 \<and> (a2,b2) \<in> r2" and
+ 3: "{a1,b1} \<le> Field r1 \<and> {a2,b2} \<le> Field r2"
+ unfolding dir_image_def Field_def by blast
+ hence "a1 = a2 \<and> b1 = b2" using assms unfolding inj_on_def by auto
+ hence "a' = f a1 \<and> b' = f b1 \<and> (a1,b1) \<in> r1 Int r2 \<and> (a2,b2) \<in> r1 Int r2"
+ using 1 2 by auto
+ thus "(a',b') \<in> dir_image (r1 \<inter> r2) f"
+ unfolding dir_image_def by blast
+ qed
+qed
+
+
+subsection {* Ordinal-like sum of two (disjoint) well-orders *}
+
+text{* This is roughly obtained by ``concatenating" the two well-orders -- thus, all elements
+of the first will be smaller than all elements of the second. This construction
+only makes sense if the fields of the two well-order relations are disjoint. *}
+
+definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel" (infix "Osum" 60)
+where
+"r Osum r' = r \<union> r' \<union> {(a,a'). a \<in> Field r \<and> a' \<in> Field r'}"
+
+abbreviation Osum2 :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel" (infix "\<union>o" 60)
+where "r \<union>o r' \<equiv> r Osum r'"
+
+lemma Field_Osum: "Field(r Osum r') = Field r \<union> Field r'"
+unfolding Osum_def Field_def by blast
+
+lemma Osum_Refl:
+assumes FLD: "Field r Int Field r' = {}" and
+ REFL: "Refl r" and REFL': "Refl r'"
+shows "Refl (r Osum r')"
+using assms (* Need first unfold Field_Osum, only then Osum_def *)
+unfolding refl_on_def Field_Osum unfolding Osum_def by blast
+
+lemma Osum_trans:
+assumes FLD: "Field r Int Field r' = {}" and
+ TRANS: "trans r" and TRANS': "trans r'"
+shows "trans (r Osum r')"
+proof(unfold trans_def, auto)
+ fix x y z assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, z) \<in> r \<union>o r'"
+ show "(x, z) \<in> r \<union>o r'"
+ proof-
+ {assume Case1: "(x,y) \<in> r"
+ hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
+ have ?thesis
+ proof-
+ {assume Case11: "(y,z) \<in> r"
+ hence "(x,z) \<in> r" using Case1 TRANS trans_def[of r] by blast
+ hence ?thesis unfolding Osum_def by auto
+ }
+ moreover
+ {assume Case12: "(y,z) \<in> r'"
+ hence "y \<in> Field r'" unfolding Field_def by auto
+ hence False using FLD 1 by auto
+ }
+ moreover
+ {assume Case13: "z \<in> Field r'"
+ hence ?thesis using 1 unfolding Osum_def by auto
+ }
+ ultimately show ?thesis using ** unfolding Osum_def by blast
+ qed
+ }
+ moreover
+ {assume Case2: "(x,y) \<in> r'"
+ hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
+ have ?thesis
+ proof-
+ {assume Case21: "(y,z) \<in> r"
+ hence "y \<in> Field r" unfolding Field_def by auto
+ hence False using FLD 2 by auto
+ }
+ moreover
+ {assume Case22: "(y,z) \<in> r'"
+ hence "(x,z) \<in> r'" using Case2 TRANS' trans_def[of r'] by blast
+ hence ?thesis unfolding Osum_def by auto
+ }
+ moreover
+ {assume Case23: "y \<in> Field r"
+ hence False using FLD 2 by auto
+ }
+ ultimately show ?thesis using ** unfolding Osum_def by blast
+ qed
+ }
+ moreover
+ {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
+ have ?thesis
+ proof-
+ {assume Case31: "(y,z) \<in> r"
+ hence "y \<in> Field r" unfolding Field_def by auto
+ hence False using FLD Case3 by auto
+ }
+ moreover
+ {assume Case32: "(y,z) \<in> r'"
+ hence "z \<in> Field r'" unfolding Field_def by blast
+ hence ?thesis unfolding Osum_def using Case3 by auto
+ }
+ moreover
+ {assume Case33: "y \<in> Field r"
+ hence False using FLD Case3 by auto
+ }
+ ultimately show ?thesis using ** unfolding Osum_def by blast
+ qed
+ }
+ ultimately show ?thesis using * unfolding Osum_def by blast
+ qed
+qed
+
+lemma Osum_Preorder:
+"\<lbrakk>Field r Int Field r' = {}; Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r Osum r')"
+unfolding preorder_on_def using Osum_Refl Osum_trans by blast
+
+lemma Osum_antisym:
+assumes FLD: "Field r Int Field r' = {}" and
+ AN: "antisym r" and AN': "antisym r'"
+shows "antisym (r Osum r')"
+proof(unfold antisym_def, auto)
+ fix x y assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, x) \<in> r \<union>o r'"
+ show "x = y"
+ proof-
+ {assume Case1: "(x,y) \<in> r"
+ hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
+ have ?thesis
+ proof-
+ have "(y,x) \<in> r \<Longrightarrow> ?thesis"
+ using Case1 AN antisym_def[of r] by blast
+ moreover
+ {assume "(y,x) \<in> r'"
+ hence "y \<in> Field r'" unfolding Field_def by auto
+ hence False using FLD 1 by auto
+ }
+ moreover
+ have "x \<in> Field r' \<Longrightarrow> False" using FLD 1 by auto
+ ultimately show ?thesis using ** unfolding Osum_def by blast
+ qed
+ }
+ moreover
+ {assume Case2: "(x,y) \<in> r'"
+ hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
+ have ?thesis
+ proof-
+ {assume "(y,x) \<in> r"
+ hence "y \<in> Field r" unfolding Field_def by auto
+ hence False using FLD 2 by auto
+ }
+ moreover
+ have "(y,x) \<in> r' \<Longrightarrow> ?thesis"
+ using Case2 AN' antisym_def[of r'] by blast
+ moreover
+ {assume "y \<in> Field r"
+ hence False using FLD 2 by auto
+ }
+ ultimately show ?thesis using ** unfolding Osum_def by blast
+ qed
+ }
+ moreover
+ {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
+ have ?thesis
+ proof-
+ {assume "(y,x) \<in> r"
+ hence "y \<in> Field r" unfolding Field_def by auto
+ hence False using FLD Case3 by auto
+ }
+ moreover
+ {assume Case32: "(y,x) \<in> r'"
+ hence "x \<in> Field r'" unfolding Field_def by blast
+ hence False using FLD Case3 by auto
+ }
+ moreover
+ have "\<not> y \<in> Field r" using FLD Case3 by auto
+ ultimately show ?thesis using ** unfolding Osum_def by blast
+ qed
+ }
+ ultimately show ?thesis using * unfolding Osum_def by blast
+ qed
+qed
+
+lemma Osum_Partial_order:
+"\<lbrakk>Field r Int Field r' = {}; Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow>
+ Partial_order (r Osum r')"
+unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast
+
+lemma Osum_Total:
+assumes FLD: "Field r Int Field r' = {}" and
+ TOT: "Total r" and TOT': "Total r'"
+shows "Total (r Osum r')"
+using assms
+unfolding total_on_def Field_Osum unfolding Osum_def by blast
+
+lemma Osum_Linear_order:
+"\<lbrakk>Field r Int Field r' = {}; Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow>
+ Linear_order (r Osum r')"
+unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast
+
+lemma Osum_wf:
+assumes FLD: "Field r Int Field r' = {}" and
+ WF: "wf r" and WF': "wf r'"
+shows "wf (r Osum r')"
+unfolding wf_eq_minimal2 unfolding Field_Osum
+proof(intro allI impI, elim conjE)
+ fix A assume *: "A \<subseteq> Field r \<union> Field r'" and **: "A \<noteq> {}"
+ obtain B where B_def: "B = A Int Field r" by blast
+ show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r \<union>o r'"
+ proof(cases "B = {}")
+ assume Case1: "B \<noteq> {}"
+ hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto
+ then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"
+ using WF unfolding wf_eq_minimal2 by blast
+ hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto
+ (* *)
+ have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"
+ proof(intro ballI)
+ fix a1 assume **: "a1 \<in> A"
+ {assume Case11: "a1 \<in> Field r"
+ hence "(a1,a) \<notin> r" using B_def ** 2 by auto
+ moreover
+ have "(a1,a) \<notin> r'" using 3 by (auto simp add: Field_def)
+ ultimately have "(a1,a) \<notin> r Osum r'"
+ using 3 unfolding Osum_def by auto
+ }
+ moreover
+ {assume Case12: "a1 \<notin> Field r"
+ hence "(a1,a) \<notin> r" unfolding Field_def by auto
+ moreover
+ have "(a1,a) \<notin> r'" using 3 unfolding Field_def by auto
+ ultimately have "(a1,a) \<notin> r Osum r'"
+ using 3 unfolding Osum_def by auto
+ }
+ ultimately show "(a1,a) \<notin> r Osum r'" by blast
+ qed
+ thus ?thesis using 1 B_def by auto
+ next
+ assume Case2: "B = {}"
+ hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto
+ then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"
+ using WF' unfolding wf_eq_minimal2 by blast
+ hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast
+ (* *)
+ have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"
+ proof(unfold Osum_def, auto simp add: 3)
+ fix a1' assume "(a1', a') \<in> r"
+ thus False using 4 unfolding Field_def by blast
+ next
+ fix a1' assume "a1' \<in> A" and "a1' \<in> Field r"
+ thus False using Case2 B_def by auto
+ qed
+ thus ?thesis using 2 by blast
+ qed
+qed
+
+lemma Osum_minus_Id:
+assumes TOT: "Total r" and TOT': "Total r'" and
+ NID: "\<not> (r \<le> Id)" and NID': "\<not> (r' \<le> Id)"
+shows "(r Osum r') - Id \<le> (r - Id) Osum (r' - Id)"
+proof-
+ {fix a a' assume *: "(a,a') \<in> (r Osum r')" and **: "a \<noteq> a'"
+ have "(a,a') \<in> (r - Id) Osum (r' - Id)"
+ proof-
+ {assume "(a,a') \<in> r \<or> (a,a') \<in> r'"
+ with ** have ?thesis unfolding Osum_def by auto
+ }
+ moreover
+ {assume "a \<in> Field r \<and> a' \<in> Field r'"
+ hence "a \<in> Field(r - Id) \<and> a' \<in> Field (r' - Id)"
+ using assms rel.Total_Id_Field by blast
+ hence ?thesis unfolding Osum_def by auto
+ }
+ ultimately show ?thesis using * unfolding Osum_def by blast
+ qed
+ }
+ thus ?thesis by(auto simp add: Osum_def)
+qed
+
+
+lemma wf_Int_Times:
+assumes "A Int B = {}"
+shows "wf(A \<times> B)"
+proof(unfold wf_def, auto)
+ fix P x
+ assume *: "\<forall>x. (\<forall>y. y \<in> A \<and> x \<in> B \<longrightarrow> P y) \<longrightarrow> P x"
+ moreover have "\<forall>y \<in> A. P y" using assms * by blast
+ ultimately show "P x" using * by (case_tac "x \<in> B", auto)
+qed
+
+lemma Osum_minus_Id1:
+assumes "r \<le> Id"
+shows "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
+proof-
+ let ?Left = "(r Osum r') - Id"
+ let ?Right = "(r' - Id) \<union> (Field r \<times> Field r')"
+ {fix a::'a and b assume *: "(a,b) \<notin> Id"
+ {assume "(a,b) \<in> r"
+ with * have False using assms by auto
+ }
+ moreover
+ {assume "(a,b) \<in> r'"
+ with * have "(a,b) \<in> r' - Id" by auto
+ }
+ ultimately
+ have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
+ unfolding Osum_def by auto
+ }
+ thus ?thesis by auto
+qed
+
+lemma Osum_minus_Id2:
+assumes "r' \<le> Id"
+shows "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
+proof-
+ let ?Left = "(r Osum r') - Id"
+ let ?Right = "(r - Id) \<union> (Field r \<times> Field r')"
+ {fix a::'a and b assume *: "(a,b) \<notin> Id"
+ {assume "(a,b) \<in> r'"
+ with * have False using assms by auto
+ }
+ moreover
+ {assume "(a,b) \<in> r"
+ with * have "(a,b) \<in> r - Id" by auto
+ }
+ ultimately
+ have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
+ unfolding Osum_def by auto
+ }
+ thus ?thesis by auto
+qed
+
+lemma Osum_wf_Id:
+assumes TOT: "Total r" and TOT': "Total r'" and
+ FLD: "Field r Int Field r' = {}" and
+ WF: "wf(r - Id)" and WF': "wf(r' - Id)"
+shows "wf ((r Osum r') - Id)"
+proof(cases "r \<le> Id \<or> r' \<le> Id")
+ assume Case1: "\<not>(r \<le> Id \<or> r' \<le> Id)"
+ have "Field(r - Id) Int Field(r' - Id) = {}"
+ using FLD mono_Field[of "r - Id" r] mono_Field[of "r' - Id" r']
+ Diff_subset[of r Id] Diff_subset[of r' Id] by blast
+ thus ?thesis
+ using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"]
+ wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto
+next
+ have 1: "wf(Field r \<times> Field r')"
+ using FLD by (auto simp add: wf_Int_Times)
+ assume Case2: "r \<le> Id \<or> r' \<le> Id"
+ moreover
+ {assume Case21: "r \<le> Id"
+ hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
+ using Osum_minus_Id1[of r r'] by simp
+ moreover
+ {have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}"
+ using FLD unfolding Field_def by blast
+ hence "wf((r' - Id) \<union> (Field r \<times> Field r'))"
+ using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]
+ by (auto simp add: Un_commute)
+ }
+ ultimately have ?thesis by (auto simp add: wf_subset)
+ }
+ moreover
+ {assume Case22: "r' \<le> Id"
+ hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
+ using Osum_minus_Id2[of r' r] by simp
+ moreover
+ {have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}"
+ using FLD unfolding Field_def by blast
+ hence "wf((r - Id) \<union> (Field r \<times> Field r'))"
+ using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]
+ by (auto simp add: Un_commute)
+ }
+ ultimately have ?thesis by (auto simp add: wf_subset)
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma Osum_Well_order:
+assumes FLD: "Field r Int Field r' = {}" and
+ WELL: "Well_order r" and WELL': "Well_order r'"
+shows "Well_order (r Osum r')"
+proof-
+ have "Total r \<and> Total r'" using WELL WELL'
+ by (auto simp add: order_on_defs)
+ thus ?thesis using assms unfolding well_order_on_def
+ using Osum_Linear_order Osum_wf_Id by blast
+qed
+
+lemma Osum_embed:
+assumes FLD: "Field r Int Field r' = {}" and
+ WELL: "Well_order r" and WELL': "Well_order r'"
+shows "embed r (r Osum r') id"
+proof-
+ have 1: "Well_order (r Osum r')"
+ using assms by (auto simp add: Osum_Well_order)
+ moreover
+ have "compat r (r Osum r') id"
+ unfolding compat_def Osum_def by auto
+ moreover
+ have "inj_on id (Field r)" by simp
+ moreover
+ have "ofilter (r Osum r') (Field r)"
+ using 1 proof(auto simp add: wo_rel_def wo_rel.ofilter_def
+ Field_Osum rel.under_def)
+ fix a b assume 2: "a \<in> Field r" and 3: "(b,a) \<in> r Osum r'"
+ moreover
+ {assume "(b,a) \<in> r'"
+ hence "a \<in> Field r'" using Field_def[of r'] by blast
+ hence False using 2 FLD by blast
+ }
+ moreover
+ {assume "a \<in> Field r'"
+ hence False using 2 FLD by blast
+ }
+ ultimately
+ show "b \<in> Field r" by (auto simp add: Osum_def Field_def)
+ qed
+ ultimately show ?thesis
+ using assms by (auto simp add: embed_iff_compat_inj_on_ofilter)
+qed
+
+corollary Osum_ordLeq:
+assumes FLD: "Field r Int Field r' = {}" and
+ WELL: "Well_order r" and WELL': "Well_order r'"
+shows "r \<le>o r Osum r'"
+using assms Osum_embed Osum_Well_order
+unfolding ordLeq_def by blast
+
+lemma Well_order_embed_copy:
+assumes WELL: "well_order_on A r" and
+ INJ: "inj_on f A" and SUB: "f ` A \<le> B"
+shows "\<exists>r'. well_order_on B r' \<and> r \<le>o r'"
+proof-
+ have "bij_betw f A (f ` A)"
+ using INJ inj_on_imp_bij_betw by blast
+ then obtain r'' where "well_order_on (f ` A) r''" and 1: "r =o r''"
+ using WELL Well_order_iso_copy by blast
+ hence 2: "Well_order r'' \<and> Field r'' = (f ` A)"
+ using rel.well_order_on_Well_order by blast
+ (* *)
+ let ?C = "B - (f ` A)"
+ obtain r''' where "well_order_on ?C r'''"
+ using well_order_on by blast
+ hence 3: "Well_order r''' \<and> Field r''' = ?C"
+ using rel.well_order_on_Well_order by blast
+ (* *)
+ let ?r' = "r'' Osum r'''"
+ have "Field r'' Int Field r''' = {}"
+ using 2 3 by auto
+ hence "r'' \<le>o ?r'" using Osum_ordLeq[of r'' r'''] 2 3 by blast
+ hence 4: "r \<le>o ?r'" using 1 ordIso_ordLeq_trans by blast
+ (* *)
+ hence "Well_order ?r'" unfolding ordLeq_def by auto
+ moreover
+ have "Field ?r' = B" using 2 3 SUB by (auto simp add: Field_Osum)
+ ultimately show ?thesis using 4 by blast
+qed
+
+
+subsection {* The maxim among a finite set of ordinals *}
+
+text {* The correct phrasing would be ``a maxim of ...", as @{text "\<le>o"} is only a preorder. *}
+
+definition isOmax :: "'a rel set \<Rightarrow> 'a rel \<Rightarrow> bool"
+where
+"isOmax R r == r \<in> R \<and> (ALL r' : R. r' \<le>o r)"
+
+definition omax :: "'a rel set \<Rightarrow> 'a rel"
+where
+"omax R == SOME r. isOmax R r"
+
+lemma exists_isOmax:
+assumes "finite R" and "R \<noteq> {}" and "\<forall> r \<in> R. Well_order r"
+shows "\<exists> r. isOmax R r"
+proof-
+ have "finite R \<Longrightarrow> R \<noteq> {} \<longrightarrow> (\<forall> r \<in> R. Well_order r) \<longrightarrow> (\<exists> r. isOmax R r)"
+ apply(erule finite_induct) apply(simp add: isOmax_def)
+ proof(clarsimp)
+ fix r :: "('a \<times> 'a) set" and R assume *: "finite R" and **: "r \<notin> R"
+ and ***: "Well_order r" and ****: "\<forall>r\<in>R. Well_order r"
+ and IH: "R \<noteq> {} \<longrightarrow> (\<exists>p. isOmax R p)"
+ let ?R' = "insert r R"
+ show "\<exists>p'. (isOmax ?R' p')"
+ proof(cases "R = {}")
+ assume Case1: "R = {}"
+ thus ?thesis unfolding isOmax_def using ***
+ by (simp add: ordLeq_reflexive)
+ next
+ assume Case2: "R \<noteq> {}"
+ then obtain p where p: "isOmax R p" using IH by auto
+ hence 1: "Well_order p" using **** unfolding isOmax_def by simp
+ {assume Case21: "r \<le>o p"
+ hence "isOmax ?R' p" using p unfolding isOmax_def by simp
+ hence ?thesis by auto
+ }
+ moreover
+ {assume Case22: "p \<le>o r"
+ {fix r' assume "r' \<in> ?R'"
+ moreover
+ {assume "r' \<in> R"
+ hence "r' \<le>o p" using p unfolding isOmax_def by simp
+ hence "r' \<le>o r" using Case22 by(rule ordLeq_transitive)
+ }
+ moreover have "r \<le>o r" using *** by(rule ordLeq_reflexive)
+ ultimately have "r' \<le>o r" by auto
+ }
+ hence "isOmax ?R' r" unfolding isOmax_def by simp
+ hence ?thesis by auto
+ }
+ moreover have "r \<le>o p \<or> p \<le>o r"
+ using 1 *** ordLeq_total by auto
+ ultimately show ?thesis by blast
+ qed
+ qed
+ thus ?thesis using assms by auto
+qed
+
+lemma omax_isOmax:
+assumes "finite R" and "R \<noteq> {}" and "\<forall> r \<in> R. Well_order r"
+shows "isOmax R (omax R)"
+unfolding omax_def using assms
+by(simp add: exists_isOmax someI_ex)
+
+lemma omax_in:
+assumes "finite R" and "R \<noteq> {}" and "\<forall> r \<in> R. Well_order r"
+shows "omax R \<in> R"
+using assms omax_isOmax unfolding isOmax_def by blast
+
+lemma Well_order_omax:
+assumes "finite R" and "R \<noteq> {}" and "\<forall>r\<in>R. Well_order r"
+shows "Well_order (omax R)"
+using assms apply - apply(drule omax_in) by auto
+
+lemma omax_maxim:
+assumes "finite R" and "\<forall> r \<in> R. Well_order r" and "r \<in> R"
+shows "r \<le>o omax R"
+using assms omax_isOmax unfolding isOmax_def by blast
+
+lemma omax_ordLeq:
+assumes "finite R" and "R \<noteq> {}" and *: "\<forall> r \<in> R. r \<le>o p"
+shows "omax R \<le>o p"
+proof-
+ have "\<forall> r \<in> R. Well_order r" using * unfolding ordLeq_def by simp
+ thus ?thesis using assms omax_in by auto
+qed
+
+lemma omax_ordLess:
+assumes "finite R" and "R \<noteq> {}" and *: "\<forall> r \<in> R. r <o p"
+shows "omax R <o p"
+proof-
+ have "\<forall> r \<in> R. Well_order r" using * unfolding ordLess_def by simp
+ thus ?thesis using assms omax_in by auto
+qed
+
+lemma omax_ordLeq_elim:
+assumes "finite R" and "\<forall> r \<in> R. Well_order r"
+and "omax R \<le>o p" and "r \<in> R"
+shows "r \<le>o p"
+using assms omax_maxim[of R r] apply simp
+using ordLeq_transitive by blast
+
+lemma omax_ordLess_elim:
+assumes "finite R" and "\<forall> r \<in> R. Well_order r"
+and "omax R <o p" and "r \<in> R"
+shows "r <o p"
+using assms omax_maxim[of R r] apply simp
+using ordLeq_ordLess_trans by blast
+
+lemma ordLeq_omax:
+assumes "finite R" and "\<forall> r \<in> R. Well_order r"
+and "r \<in> R" and "p \<le>o r"
+shows "p \<le>o omax R"
+using assms omax_maxim[of R r] apply simp
+using ordLeq_transitive by blast
+
+lemma ordLess_omax:
+assumes "finite R" and "\<forall> r \<in> R. Well_order r"
+and "r \<in> R" and "p <o r"
+shows "p <o omax R"
+using assms omax_maxim[of R r] apply simp
+using ordLess_ordLeq_trans by blast
+
+lemma omax_ordLeq_mono:
+assumes P: "finite P" and R: "finite R"
+and NE_P: "P \<noteq> {}" and Well_R: "\<forall> r \<in> R. Well_order r"
+and LEQ: "\<forall> p \<in> P. \<exists> r \<in> R. p \<le>o r"
+shows "omax P \<le>o omax R"
+proof-
+ let ?mp = "omax P" let ?mr = "omax R"
+ {fix p assume "p : P"
+ then obtain r where r: "r : R" and "p \<le>o r"
+ using LEQ by blast
+ moreover have "r <=o ?mr"
+ using r R Well_R omax_maxim by blast
+ ultimately have "p <=o ?mr"
+ using ordLeq_transitive by blast
+ }
+ thus "?mp <=o ?mr"
+ using NE_P P using omax_ordLeq by blast
+qed
+
+lemma omax_ordLess_mono:
+assumes P: "finite P" and R: "finite R"
+and NE_P: "P \<noteq> {}" and Well_R: "\<forall> r \<in> R. Well_order r"
+and LEQ: "\<forall> p \<in> P. \<exists> r \<in> R. p <o r"
+shows "omax P <o omax R"
+proof-
+ let ?mp = "omax P" let ?mr = "omax R"
+ {fix p assume "p : P"
+ then obtain r where r: "r : R" and "p <o r"
+ using LEQ by blast
+ moreover have "r <=o ?mr"
+ using r R Well_R omax_maxim by blast
+ ultimately have "p <o ?mr"
+ using ordLess_ordLeq_trans by blast
+ }
+ thus "?mp <o ?mr"
+ using NE_P P omax_ordLess by blast
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Constructions_on_Wellorders_Base.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1633 @@
+(* Title: HOL/Ordinals_and_Cardinals/Constructions_on_Wellorders_Base.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Constructions on wellorders (base).
+*)
+
+header {* Constructions on Wellorders (Base) *}
+
+theory Constructions_on_Wellorders_Base
+imports Wellorder_Embedding_Base
+begin
+
+
+text {* In this section, we study basic constructions on well-orders, such as restriction to
+a set/order filter, copy via direct images, ordinal-like sum of disjoint well-orders,
+and bounded square. We also define between well-orders
+the relations @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"}),
+@{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"}), and
+@{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}). We study the
+connections between these relations, order filters, and the aforementioned constructions.
+A main result of this section is that @{text "<o"} is well-founded. *}
+
+
+subsection {* Restriction to a set *}
+
+
+abbreviation Restr :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a rel"
+where "Restr r A \<equiv> r Int (A \<times> A)"
+
+
+lemma Restr_subset:
+"A \<le> B \<Longrightarrow> Restr (Restr r B) A = Restr r A"
+by blast
+
+
+lemma Restr_Field: "Restr r (Field r) = r"
+unfolding Field_def by auto
+
+
+lemma Refl_Restr: "Refl r \<Longrightarrow> Refl(Restr r A)"
+unfolding refl_on_def Field_def by auto
+
+
+lemma antisym_Restr:
+"antisym r \<Longrightarrow> antisym(Restr r A)"
+unfolding antisym_def Field_def by auto
+
+
+lemma Total_Restr:
+"Total r \<Longrightarrow> Total(Restr r A)"
+unfolding total_on_def Field_def by auto
+
+
+lemma trans_Restr:
+"trans r \<Longrightarrow> trans(Restr r A)"
+unfolding trans_def Field_def by blast
+
+
+lemma Preorder_Restr:
+"Preorder r \<Longrightarrow> Preorder(Restr r A)"
+unfolding preorder_on_def by (simp add: Refl_Restr trans_Restr)
+
+
+lemma Partial_order_Restr:
+"Partial_order r \<Longrightarrow> Partial_order(Restr r A)"
+unfolding partial_order_on_def by (simp add: Preorder_Restr antisym_Restr)
+
+
+lemma Linear_order_Restr:
+"Linear_order r \<Longrightarrow> Linear_order(Restr r A)"
+unfolding linear_order_on_def by (simp add: Partial_order_Restr Total_Restr)
+
+
+lemma Well_order_Restr:
+assumes "Well_order r"
+shows "Well_order(Restr r A)"
+proof-
+ have "Restr r A - Id \<le> r - Id" using Restr_subset by blast
+ hence "wf(Restr r A - Id)" using assms
+ using well_order_on_def wf_subset by blast
+ thus ?thesis using assms unfolding well_order_on_def
+ by (simp add: Linear_order_Restr)
+qed
+
+
+lemma Field_Restr_subset: "Field(Restr r A) \<le> A"
+by (auto simp add: Field_def)
+
+
+lemma Refl_Field_Restr:
+"Refl r \<Longrightarrow> Field(Restr r A) = (Field r) Int A"
+by (auto simp add: refl_on_def Field_def)
+
+
+lemma Refl_Field_Restr2:
+"\<lbrakk>Refl r; A \<le> Field r\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: Refl_Field_Restr)
+
+
+lemma well_order_on_Restr:
+assumes WELL: "Well_order r" and SUB: "A \<le> Field r"
+shows "well_order_on A (Restr r A)"
+using assms
+using Well_order_Restr[of r A] Refl_Field_Restr2[of r A]
+ order_on_defs[of "Field r" r] by auto
+
+
+subsection {* Order filters versus restrictions and embeddings *}
+
+
+lemma Field_Restr_ofilter:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: wo_rel_def wo_rel.ofilter_def wo_rel.REFL Refl_Field_Restr2)
+
+
+lemma ofilter_Restr_under:
+assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and IN: "a \<in> A"
+shows "rel.under (Restr r A) a = rel.under r a"
+using assms wo_rel_def
+proof(auto simp add: wo_rel.ofilter_def rel.under_def)
+ fix b assume *: "a \<in> A" and "(b,a) \<in> r"
+ hence "b \<in> rel.under r a \<and> a \<in> Field r"
+ unfolding rel.under_def using Field_def by fastforce
+ thus "b \<in> A" using * assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_embed:
+assumes "Well_order r"
+shows "wo_rel.ofilter r A = (A \<le> Field r \<and> embed (Restr r A) r id)"
+proof
+ assume *: "wo_rel.ofilter r A"
+ show "A \<le> Field r \<and> embed (Restr r A) r id"
+ proof(unfold embed_def, auto)
+ fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
+ by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+ next
+ fix a assume "a \<in> Field (Restr r A)"
+ thus "bij_betw id (rel.under (Restr r A) a) (rel.under r a)" using assms *
+ by (simp add: ofilter_Restr_under Field_Restr_ofilter)
+ qed
+next
+ assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
+ hence "Field(Restr r A) \<le> Field r"
+ using assms embed_Field[of "Restr r A" r id] id_def
+ Well_order_Restr[of r] by auto
+ {fix a assume "a \<in> A"
+ hence "a \<in> Field(Restr r A)" using * assms
+ by (simp add: order_on_defs Refl_Field_Restr2)
+ hence "bij_betw id (rel.under (Restr r A) a) (rel.under r a)"
+ using * unfolding embed_def by auto
+ hence "rel.under r a \<le> rel.under (Restr r A) a"
+ unfolding bij_betw_def by auto
+ also have "\<dots> \<le> Field(Restr r A)" by (simp add: rel.under_Field)
+ also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
+ finally have "rel.under r a \<le> A" .
+ }
+ thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_Restr_Int:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
+shows "wo_rel.ofilter (Restr r B) (A Int B)"
+proof-
+ let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence Field: "Field ?rB = Field r Int B"
+ using Refl_Field_Restr by blast
+ have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ (* Main proof *)
+ show ?thesis using WellB assms
+ proof(auto simp add: wo_rel.ofilter_def rel.under_def)
+ fix a assume "a \<in> A" and *: "a \<in> B"
+ hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
+ with * show "a \<in> Field ?rB" using Field by auto
+ next
+ fix a b assume "a \<in> A" and "(b,a) \<in> r"
+ thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def rel.under_def)
+ qed
+qed
+
+
+lemma ofilter_Restr_subset:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
+shows "wo_rel.ofilter (Restr r B) A"
+proof-
+ have "A Int B = A" using SUB by blast
+ thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
+qed
+
+
+lemma ofilter_subset_embed:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence FieldA: "Field ?rA = Field r Int A"
+ using Refl_Field_Restr by blast
+ have FieldB: "Field ?rB = Field r Int B"
+ using Refl Refl_Field_Restr by blast
+ have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ (* Main proof *)
+ show ?thesis
+ proof
+ assume *: "A \<le> B"
+ hence "wo_rel.ofilter (Restr r B) A" using assms
+ by (simp add: ofilter_Restr_subset)
+ hence "embed (Restr ?rB A) (Restr r B) id"
+ using WellB ofilter_embed[of "?rB" A] by auto
+ thus "embed (Restr r A) (Restr r B) id"
+ using * by (simp add: Restr_subset)
+ next
+ assume *: "embed (Restr r A) (Restr r B) id"
+ {fix a assume **: "a \<in> A"
+ hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
+ with ** FieldA have "a \<in> Field ?rA" by auto
+ hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
+ hence "a \<in> B" using FieldB by auto
+ }
+ thus "A \<le> B" by blast
+ qed
+qed
+
+
+lemma ofilter_subset_embedS_iso:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
+ ((A = B) = (iso (Restr r A) (Restr r B) id))"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence "Field ?rA = Field r Int A"
+ using Refl_Field_Restr by blast
+ hence FieldA: "Field ?rA = A" using OFA Well
+ by (auto simp add: wo_rel.ofilter_def)
+ have "Field ?rB = Field r Int B"
+ using Refl Refl_Field_Restr by blast
+ hence FieldB: "Field ?rB = B" using OFB Well
+ by (auto simp add: wo_rel.ofilter_def)
+ (* Main proof *)
+ show ?thesis unfolding embedS_def iso_def
+ using assms ofilter_subset_embed[of r A B]
+ FieldA FieldB bij_betw_id_iff[of A B] by auto
+qed
+
+
+lemma ofilter_subset_embedS:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = embedS (Restr r A) (Restr r B) id"
+using assms by (simp add: ofilter_subset_embedS_iso)
+
+
+lemma embed_implies_iso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r' r f"
+shows "iso r' (Restr r (f ` (Field r'))) f"
+proof-
+ let ?A' = "Field r'"
+ let ?r'' = "Restr r (f ` ?A')"
+ have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
+ have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter by blast
+ hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
+ hence "bij_betw f ?A' (Field ?r'')"
+ using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
+ moreover
+ {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
+ unfolding Field_def by auto
+ hence "compat r' ?r'' f"
+ using assms embed_iff_compat_inj_on_ofilter
+ unfolding compat_def by blast
+ }
+ ultimately show ?thesis using WELL' 0 iso_iff3 by blast
+qed
+
+
+subsection {* The strict inclusion on proper ofilters is well-founded *}
+
+
+definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
+where
+"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
+ wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
+
+
+lemma wf_ofilterIncl:
+assumes WELL: "Well_order r"
+shows "wf(ofilterIncl r)"
+proof-
+ have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
+ hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
+ let ?h = "(\<lambda> A. wo_rel.suc r A)"
+ let ?rS = "r - Id"
+ have "wf ?rS" using WELL by (simp add: order_on_defs)
+ moreover
+ have "compat (ofilterIncl r) ?rS ?h"
+ proof(unfold compat_def ofilterIncl_def,
+ intro allI impI, simp, elim conjE)
+ fix A B
+ assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
+ **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
+ then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
+ 1: "A = rel.underS r a \<and> B = rel.underS r b"
+ using Well by (auto simp add: wo_rel.ofilter_underS_Field)
+ hence "a \<noteq> b" using *** by auto
+ moreover
+ have "(a,b) \<in> r" using 0 1 Lo ***
+ by (auto simp add: rel.underS_incl_iff)
+ moreover
+ have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
+ using Well 0 1 by (simp add: wo_rel.suc_underS)
+ ultimately
+ show "(wo_rel.suc r A, wo_rel.suc r B) \<in> r \<and> wo_rel.suc r A \<noteq> wo_rel.suc r B"
+ by simp
+ qed
+ ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
+qed
+
+
+
+subsection {* Ordering the well-orders by existence of embeddings *}
+
+
+text {* We define three relations between well-orders:
+\begin{itemize}
+\item @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"});
+\item @{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"});
+\item @{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).
+\end{itemize}
+%
+The prefix "ord" and the index "o" in these names stand for "ordinal-like".
+These relations shall be proved to be inter-connected in a similar fashion as the trio
+@{text "\<le>"}, @{text "<"}, @{text "="} associated to a total order on a set.
+*}
+
+
+definition ordLeq :: "('a rel * 'a' rel) set"
+where
+"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
+
+
+abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
+where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
+
+
+abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
+where "r \<le>o r' \<equiv> r <=o r'"
+
+
+definition ordLess :: "('a rel * 'a' rel) set"
+where
+"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
+
+abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
+where "r <o r' \<equiv> (r,r') \<in> ordLess"
+
+
+definition ordIso :: "('a rel * 'a' rel) set"
+where
+"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
+
+abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
+where "r =o r' \<equiv> (r,r') \<in> ordIso"
+
+
+lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
+
+lemma ordLeq_Well_order_simp:
+assumes "r \<le>o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordLeq_def by simp
+
+
+lemma ordLess_Well_order_simp:
+assumes "r <o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordLess_def by simp
+
+
+lemma ordIso_Well_order_simp:
+assumes "r =o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordIso_def by simp
+
+
+text{* Notice that the relations @{text "\<le>o"}, @{text "<o"}, @{text "=o"} connect well-orders
+on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
+restrict implicitly the type of these relations to @{text "(('a rel) * ('a rel)) set"} , i.e.,
+to @{text "'a rel rel"}. *}
+
+
+lemma ordLeq_reflexive:
+"Well_order r \<Longrightarrow> r \<le>o r"
+unfolding ordLeq_def using id_embed[of r] by blast
+
+
+lemma ordLeq_transitive[trans]:
+assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+ obtain f and f'
+ where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+ "embed r r' f" and "embed r' r'' f'"
+ using * ** unfolding ordLeq_def by blast
+ hence "embed r r'' (f' o f)"
+ using comp_embed[of r r' f r'' f'] by auto
+ thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
+qed
+
+
+lemma ordLeq_total:
+"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
+unfolding ordLeq_def using wellorders_totally_ordered by blast
+
+
+lemma ordIso_reflexive:
+"Well_order r \<Longrightarrow> r =o r"
+unfolding ordIso_def using id_iso[of r] by blast
+
+
+lemma ordIso_transitive[trans]:
+assumes *: "r =o r'" and **: "r' =o r''"
+shows "r =o r''"
+proof-
+ obtain f and f'
+ where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+ "iso r r' f" and 3: "iso r' r'' f'"
+ using * ** unfolding ordIso_def by auto
+ hence "iso r r'' (f' o f)"
+ using comp_iso[of r r' f r'' f'] by auto
+ thus "r =o r''" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma ordIso_symmetric:
+assumes *: "r =o r'"
+shows "r' =o r"
+proof-
+ obtain f where 1: "Well_order r \<and> Well_order r'" and
+ 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
+ using * by (auto simp add: ordIso_def iso_def)
+ let ?f' = "inv_into (Field r) f"
+ have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
+ using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
+ thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
+qed
+
+
+lemma ordLeq_ordLess_trans[trans]:
+assumes "r \<le>o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordLess_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordLess_def
+ using embed_comp_embedS by blast
+qed
+
+
+lemma ordLess_ordLeq_trans[trans]:
+assumes "r <o r'" and " r' \<le>o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordLess_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordLess_def
+ using embedS_comp_embed by blast
+qed
+
+
+lemma ordLeq_ordIso_trans[trans]:
+assumes "r \<le>o r'" and " r' =o r''"
+shows "r \<le>o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordIso_def
+ using embed_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLeq_trans[trans]:
+assumes "r =o r'" and " r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordIso_def
+ using iso_comp_embed by blast
+qed
+
+
+lemma ordLess_ordIso_trans[trans]:
+assumes "r <o r'" and " r' =o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLess_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLess_def ordIso_def
+ using embedS_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLess_trans[trans]:
+assumes "r =o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLess_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLess_def ordIso_def
+ using iso_comp_embedS by blast
+qed
+
+
+lemma ordLess_not_embed:
+assumes "r <o r'"
+shows "\<not>(\<exists>f'. embed r' r f')"
+proof-
+ obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
+ 3: " \<not> bij_betw f (Field r) (Field r')"
+ using assms unfolding ordLess_def by (auto simp add: embedS_def)
+ {fix f' assume *: "embed r' r f'"
+ hence "bij_betw f (Field r) (Field r')" using 1 2
+ by (simp add: embed_bothWays_Field_bij_betw)
+ with 3 have False by contradiction
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma ordLess_Field:
+assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
+shows "\<not> (f`(Field r1) = Field r2)"
+proof-
+ let ?A1 = "Field r1" let ?A2 = "Field r2"
+ obtain g where
+ 0: "Well_order r1 \<and> Well_order r2" and
+ 1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
+ using OL unfolding ordLess_def by (auto simp add: embedS_def)
+ hence "\<forall>a \<in> ?A1. f a = g a"
+ using 0 EMB embed_unique[of r1] by auto
+ hence "\<not>(bij_betw f ?A1 ?A2)"
+ using 1 bij_betw_cong[of ?A1] by blast
+ moreover
+ have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
+ ultimately show ?thesis by (simp add: bij_betw_def)
+qed
+
+
+lemma ordLess_iff:
+"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
+proof
+ assume *: "r <o r'"
+ hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
+ with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+ unfolding ordLess_def by auto
+next
+ assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+ then obtain f where 1: "embed r r' f"
+ using wellorders_totally_ordered[of r r'] by blast
+ moreover
+ {assume "bij_betw f (Field r) (Field r')"
+ with * 1 have "embed r' r (inv_into (Field r) f) "
+ using inv_into_Field_embed_bij_betw[of r r' f] by auto
+ with * have False by blast
+ }
+ ultimately show "(r,r') \<in> ordLess"
+ unfolding ordLess_def using * by (fastforce simp add: embedS_def)
+qed
+
+
+lemma ordLess_irreflexive: "\<not> r <o r"
+proof
+ assume "r <o r"
+ hence "Well_order r \<and> \<not>(\<exists>f. embed r r f)"
+ unfolding ordLess_iff ..
+ moreover have "embed r r id" using id_embed[of r] .
+ ultimately show False by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_or_ordIso:
+"r \<le>o r' = (r <o r' \<or> r =o r')"
+unfolding ordRels_def embedS_defs iso_defs by blast
+
+
+lemma ordIso_iff_ordLeq:
+"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
+proof
+ assume "r =o r'"
+ then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
+ embed r r' f \<and> bij_betw f (Field r) (Field r')"
+ unfolding ordIso_def iso_defs by auto
+ hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
+ by (simp add: inv_into_Field_embed_bij_betw)
+ thus "r \<le>o r' \<and> r' \<le>o r"
+ unfolding ordLeq_def using 1 by auto
+next
+ assume "r \<le>o r' \<and> r' \<le>o r"
+ then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
+ embed r r' f \<and> embed r' r g"
+ unfolding ordLeq_def by auto
+ hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
+ thus "r =o r'" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma not_ordLess_ordLeq:
+"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
+using ordLess_ordLeq_trans ordLess_irreflexive by blast
+
+
+lemma ordLess_or_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "r <o r' \<or> r' \<le>o r"
+proof-
+ have "r \<le>o r' \<or> r' \<le>o r"
+ using assms by (simp add: ordLeq_total)
+ moreover
+ {assume "\<not> r <o r' \<and> r \<le>o r'"
+ hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
+ hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma not_ordLess_ordIso:
+"r <o r' \<Longrightarrow> \<not> r =o r'"
+using assms ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
+
+
+lemma not_ordLeq_iff_ordLess:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' \<le>o r) = (r <o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma not_ordLess_iff_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' <o r) = (r \<le>o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma ordLess_transitive[trans]:
+"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
+using assms ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
+
+
+corollary ordLess_trans: "trans ordLess"
+unfolding trans_def using ordLess_transitive by blast
+
+
+lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
+
+
+lemma ordIso_imp_ordLeq:
+"r =o r' \<Longrightarrow> r \<le>o r'"
+using ordIso_iff_ordLeq by blast
+
+
+lemma ordLess_imp_ordLeq:
+"r <o r' \<Longrightarrow> r \<le>o r'"
+using ordLeq_iff_ordLess_or_ordIso by blast
+
+
+lemma ofilter_subset_ordLeq:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
+proof
+ assume "A \<le> B"
+ thus "Restr r A \<le>o Restr r B"
+ unfolding ordLeq_def using assms
+ Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
+next
+ assume *: "Restr r A \<le>o Restr r B"
+ then obtain f where "embed (Restr r A) (Restr r B) f"
+ unfolding ordLeq_def by blast
+ {assume "B < A"
+ hence "Restr r B <o Restr r A"
+ unfolding ordLess_def using assms
+ Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
+ hence False using * not_ordLess_ordLeq by blast
+ }
+ thus "A \<le> B" using OFA OFB WELL
+ wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
+qed
+
+
+lemma ofilter_subset_ordLess:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = (Restr r A <o Restr r B)"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have 1: "Well_order ?rA \<and> Well_order ?rB"
+ using WELL Well_order_Restr by blast
+ have "(A < B) = (\<not> B \<le> A)" using assms
+ wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
+ also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
+ using assms ofilter_subset_ordLeq by blast
+ also have "\<dots> = (Restr r A <o Restr r B)"
+ using 1 not_ordLeq_iff_ordLess by blast
+ finally show ?thesis .
+qed
+
+
+lemma ofilter_ordLess:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
+by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
+ wo_rel_def Restr_Field)
+
+
+corollary underS_Restr_ordLess:
+assumes "Well_order r" and "Field r \<noteq> {}"
+shows "Restr r (rel.underS r a) <o r"
+proof-
+ have "rel.underS r a < Field r" using assms
+ by (simp add: rel.underS_Field3)
+ thus ?thesis using assms
+ by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
+qed
+
+
+lemma embed_ordLess_ofilterIncl:
+assumes
+ OL12: "r1 <o r2" and OL23: "r2 <o r3" and
+ EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
+shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
+proof-
+ have OL13: "r1 <o r3"
+ using OL12 OL23 using ordLess_transitive by auto
+ let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A3 ="Field r3"
+ obtain f12 g23 where
+ 0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
+ 1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
+ 2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
+ using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
+ hence "\<forall>a \<in> ?A2. f23 a = g23 a"
+ using EMB23 embed_unique[of r2 r3] by blast
+ hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
+ using 2 bij_betw_cong[of ?A2 f23 g23] by blast
+ (* *)
+ have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
+ using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
+ have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
+ using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
+ have 6: "wo_rel.ofilter r3 (f13 ` ?A1) \<and> f13 ` ?A1 \<noteq> ?A3"
+ using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
+ (* *)
+ have "f12 ` ?A1 < ?A2"
+ using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+ moreover have "inj_on f23 ?A2"
+ using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
+ ultimately
+ have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
+ moreover
+ {have "embed r1 r3 (f23 o f12)"
+ using 1 EMB23 0 by (auto simp add: comp_embed)
+ hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
+ using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
+ hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
+ }
+ ultimately
+ have "f13 ` ?A1 < f23 ` ?A2" by simp
+ (* *)
+ with 5 6 show ?thesis
+ unfolding ofilterIncl_def by auto
+qed
+
+
+lemma ordLess_iff_ordIso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a))"
+proof(auto)
+ fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (rel.underS r a)"
+ hence "Restr r (rel.underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
+ thus "r' <o r" using ** ordIso_ordLess_trans by blast
+next
+ assume "r' <o r"
+ then obtain f where 1: "Well_order r \<and> Well_order r'" and
+ 2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
+ unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
+ hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
+ then obtain a where 3: "a \<in> Field r" and 4: "rel.underS r a = f ` (Field r')"
+ using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
+ have "iso r' (Restr r (f ` (Field r'))) f"
+ using embed_implies_iso_Restr 2 assms by blast
+ moreover have "Well_order (Restr r (f ` (Field r')))"
+ using WELL Well_order_Restr by blast
+ ultimately have "r' =o Restr r (f ` (Field r'))"
+ using WELL' unfolding ordIso_def by auto
+ hence "r' =o Restr r (rel.underS r a)" using 4 by auto
+ thus "\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a)" using 3 by auto
+qed
+
+
+lemma internalize_ordLess:
+"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
+proof
+ assume *: "r' <o r"
+ hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
+ with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (rel.underS r a)"
+ using ordLess_iff_ordIso_Restr by blast
+ let ?p = "Restr r (rel.underS r a)"
+ have "wo_rel.ofilter r (rel.underS r a)" using 0
+ by (simp add: wo_rel_def wo_rel.underS_ofilter)
+ hence "Field ?p = rel.underS r a" using 0 Field_Restr_ofilter by blast
+ hence "Field ?p < Field r" using rel.underS_Field2 1 by fastforce
+ moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
+ ultimately
+ show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
+next
+ assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
+ thus "r' <o r" using ordIso_ordLess_trans by blast
+qed
+
+
+lemma internalize_ordLeq:
+"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
+proof
+ assume *: "r' \<le>o r"
+ moreover
+ {assume "r' <o r"
+ then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
+ using internalize_ordLess[of r' r] by blast
+ hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ using ordLeq_iff_ordLess_or_ordIso by blast
+ }
+ moreover
+ have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
+ ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ using ordLeq_iff_ordLess_or_ordIso by blast
+next
+ assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r')"
+proof(auto)
+ assume *: "r \<le>o r'"
+ fix a assume "a \<in> Field r"
+ hence "Restr r (rel.underS r a) <o r"
+ using WELL underS_Restr_ordLess[of r] by blast
+ thus "Restr r (rel.underS r a) <o r'"
+ using * ordLess_ordLeq_trans by blast
+next
+ assume *: "\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r'"
+ {assume "r' <o r"
+ then obtain a where "a \<in> Field r \<and> r' =o Restr r (rel.underS r a)"
+ using assms ordLess_iff_ordIso_Restr by blast
+ hence False using * not_ordLess_ordIso ordIso_symmetric by blast
+ }
+ thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
+qed
+
+
+lemma finite_ordLess_infinite:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ FIN: "finite(Field r)" and INF: "infinite(Field r')"
+shows "r <o r'"
+proof-
+ {assume "r' \<le>o r"
+ then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
+ unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
+ hence False using finite_imageD finite_subset FIN INF by blast
+ }
+ thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
+qed
+
+
+lemma finite_well_order_on_ordIso:
+assumes FIN: "finite A" and
+ WELL: "well_order_on A r" and WELL': "well_order_on A r'"
+shows "r =o r'"
+proof-
+ have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+ using assms rel.well_order_on_Well_order by blast
+ moreover
+ have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
+ \<longrightarrow> r =o r'"
+ proof(clarify)
+ fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
+ have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+ using * ** rel.well_order_on_Well_order by blast
+ assume "r \<le>o r'"
+ then obtain f where 1: "embed r r' f" and
+ "inj_on f A \<and> f ` A \<le> A"
+ unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
+ hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
+ thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
+ qed
+ ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by blast
+qed
+
+
+subsection{* @{text "<o"} is well-founded *}
+
+
+text {* Of course, it only makes sense to state that the @{text "<o"} is well-founded
+on the restricted type @{text "'a rel rel"}. We prove this by first showing that, for any set
+of well-orders all embedded in a fixed well-order, the function mapping each well-order
+in the set to an order filter of the fixed well-order is compatible w.r.t. to @{text "<o"} versus
+{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded. *}
+
+
+definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
+where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
+
+
+lemma ord_to_filter_compat:
+"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
+ (ofilterIncl r0)
+ (ord_to_filter r0)"
+proof(unfold compat_def ord_to_filter_def, clarify)
+ fix r1::"'a rel" and r2::"'a rel"
+ let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A0 ="Field r0"
+ let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
+ let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
+ assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
+ hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
+ by (auto simp add: ordLess_def embedS_def)
+ hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
+ thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
+ using * ** by (simp add: embed_ordLess_ofilterIncl)
+qed
+
+
+theorem wf_ordLess: "wf ordLess"
+proof-
+ {fix r0 :: "('a \<times> 'a) set"
+ (* need to annotate here!*)
+ let ?ordLess = "ordLess::('d rel * 'd rel) set"
+ let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
+ {assume Case1: "Well_order r0"
+ hence "wf ?R"
+ using wf_ofilterIncl[of r0]
+ compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
+ ord_to_filter_compat[of r0] by auto
+ }
+ moreover
+ {assume Case2: "\<not> Well_order r0"
+ hence "?R = {}" unfolding ordLess_def by auto
+ hence "wf ?R" using wf_empty by simp
+ }
+ ultimately have "wf ?R" by blast
+ }
+ thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
+qed
+
+corollary exists_minim_Well_order:
+assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
+shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+proof-
+ obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
+ using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
+ equals0I[of R] by blast
+ with not_ordLeq_iff_ordLess WELL show ?thesis by blast
+qed
+
+
+
+subsection {* Copy via direct images *}
+
+
+text{* The direct image operator is the dual of the inverse image operator @{text "inv_image"}
+from @{text "Relation.thy"}. It is useful for transporting a well-order between
+different types. *}
+
+
+definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
+where
+"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
+
+
+lemma dir_image_Field:
+"Field(dir_image r f) \<le> f ` (Field r)"
+unfolding dir_image_def Field_def by auto
+
+
+lemma dir_image_minus_Id:
+"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
+unfolding inj_on_def Field_def dir_image_def by auto
+
+
+lemma Refl_dir_image:
+assumes "Refl r"
+shows "Refl(dir_image r f)"
+proof-
+ {fix a' b'
+ assume "(a',b') \<in> dir_image r f"
+ then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
+ unfolding dir_image_def by blast
+ hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
+ hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
+ with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
+ unfolding dir_image_def by auto
+ }
+ thus ?thesis
+ by(unfold refl_on_def Field_def Domain_def Range_def, auto)
+qed
+
+
+lemma trans_dir_image:
+assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
+shows "trans(dir_image r f)"
+proof(unfold trans_def, auto)
+ fix a' b' c'
+ assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
+ then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
+ 2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
+ unfolding dir_image_def by blast
+ hence "b1 \<in> Field r \<and> b2 \<in> Field r"
+ unfolding Field_def by auto
+ hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
+ hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
+ thus "(a',c') \<in> dir_image r f"
+ unfolding dir_image_def using 1 by auto
+qed
+
+
+lemma Preorder_dir_image:
+"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
+by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
+
+
+lemma antisym_dir_image:
+assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
+shows "antisym(dir_image r f)"
+proof(unfold antisym_def, auto)
+ fix a' b'
+ assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
+ then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
+ 2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
+ 3: "{a1,a2,b1,b2} \<le> Field r"
+ unfolding dir_image_def Field_def by blast
+ hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
+ hence "a1 = b2" using 2 AN unfolding antisym_def by auto
+ thus "a' = b'" using 1 by auto
+qed
+
+
+lemma Partial_order_dir_image:
+"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
+by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
+
+
+lemma Total_dir_image:
+assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
+shows "Total(dir_image r f)"
+proof(unfold total_on_def, intro ballI impI)
+ fix a' b'
+ assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
+ then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
+ using dir_image_Field[of r f] by blast
+ moreover assume "a' \<noteq> b'"
+ ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
+ hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
+ thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
+ using 1 unfolding dir_image_def by auto
+qed
+
+
+lemma Linear_order_dir_image:
+"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
+by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
+
+
+lemma wf_dir_image:
+assumes WF: "wf r" and INJ: "inj_on f (Field r)"
+shows "wf(dir_image r f)"
+proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
+ fix A'::"'b set"
+ assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
+ obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
+ have "A \<noteq> {} \<and> A \<le> Field r"
+ using A_def dir_image_Field[of r f] SUB NE by blast
+ then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
+ using WF unfolding wf_eq_minimal2 by blast
+ have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
+ proof(clarify)
+ fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
+ obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
+ 3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
+ using ** unfolding dir_image_def Field_def by blast
+ hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
+ hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
+ with 1 show False by auto
+ qed
+ thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
+ using A_def 1 by blast
+qed
+
+
+lemma Well_order_dir_image:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
+using assms unfolding well_order_on_def
+using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
+ dir_image_minus_Id[of f r]
+ subset_inj_on[of f "Field r" "Field(r - Id)"]
+ mono_Field[of "r - Id" r] by auto
+
+
+lemma dir_image_Field2:
+"Refl r \<Longrightarrow> Field(dir_image r f) = f ` (Field r)"
+unfolding Field_def dir_image_def refl_on_def Domain_def Range_def by blast
+
+
+lemma dir_image_bij_betw:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
+unfolding bij_betw_def
+by (simp add: dir_image_Field2 order_on_defs)
+
+
+lemma dir_image_compat:
+"compat r (dir_image r f) f"
+unfolding compat_def dir_image_def by auto
+
+
+lemma dir_image_iso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> iso r (dir_image r f) f"
+using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
+
+
+lemma dir_image_ordIso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> r =o dir_image r f"
+unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
+
+
+lemma Well_order_iso_copy:
+assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
+shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
+proof-
+ let ?r' = "dir_image r f"
+ have 1: "A = Field r \<and> Well_order r"
+ using WELL rel.well_order_on_Well_order by blast
+ hence 2: "iso r ?r' f"
+ using dir_image_iso using BIJ unfolding bij_betw_def by auto
+ hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
+ hence "Field ?r' = A'"
+ using 1 BIJ unfolding bij_betw_def by auto
+ moreover have "Well_order ?r'"
+ using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
+ ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
+qed
+
+
+
+subsection {* Bounded square *}
+
+
+text{* This construction essentially defines, for an order relation @{text "r"}, a lexicographic
+order @{text "bsqr r"} on @{text "(Field r) \<times> (Field r)"}, applying the
+following criteria (in this order):
+\begin{itemize}
+\item compare the maximums;
+\item compare the first components;
+\item compare the second components.
+\end{itemize}
+%
+The only application of this construction that we are aware of is
+at proving that the square of an infinite set has the same cardinal
+as that set. The essential property required there (and which is ensured by this
+construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
+in a product of proper filters on the original relation (assumed to be a well-order). *}
+
+
+definition bsqr :: "'a rel => ('a * 'a)rel"
+where
+"bsqr r = {((a1,a2),(b1,b2)).
+ {a1,a2,b1,b2} \<le> Field r \<and>
+ (a1 = b1 \<and> a2 = b2 \<or>
+ (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id
+ )}"
+
+
+lemma Field_bsqr:
+"Field (bsqr r) = Field r \<times> Field r"
+proof
+ show "Field (bsqr r) \<le> Field r \<times> Field r"
+ proof-
+ {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
+ moreover
+ have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
+ a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
+ ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
+ }
+ thus ?thesis unfolding Field_def by force
+ qed
+next
+ show "Field r \<times> Field r \<le> Field (bsqr r)"
+ proof(auto)
+ fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
+ hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
+ thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
+ qed
+qed
+
+
+lemma bsqr_Refl: "Refl(bsqr r)"
+by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
+
+
+lemma bsqr_Trans:
+assumes "Well_order r"
+shows "trans (bsqr r)"
+proof(unfold trans_def, auto)
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Trans: "trans r" using wo_rel.TRANS by auto
+ have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+ hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+ (* Main proof *)
+ fix a1 a2 b1 b2 c1 c2
+ assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
+ hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
+ have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ using * unfolding bsqr_def by auto
+ have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+ using ** unfolding bsqr_def by auto
+ show "((a1,a2),(c1,c2)) \<in> bsqr r"
+ proof-
+ {assume Case1: "a1 = b1 \<and> a2 = b2"
+ hence ?thesis using ** by simp
+ }
+ moreover
+ {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+ {assume Case21: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ using Case2 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
+ hence ?thesis using Case2 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+ {assume Case31: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence ?thesis using Case3 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+ hence "(a1,c1) \<in> r - Id"
+ using Case3 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
+ hence ?thesis using Case3 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ {assume Case41: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence ?thesis using Case4 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+ hence ?thesis using Case4 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+ hence "(a2,c2) \<in> r - Id"
+ using Case4 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ ultimately show ?thesis using 0 1 by auto
+ qed
+qed
+
+
+lemma bsqr_antisym:
+assumes "Well_order r"
+shows "antisym (bsqr r)"
+proof(unfold antisym_def, clarify)
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Trans: "trans r" using wo_rel.TRANS by auto
+ have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+ hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+ hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
+ using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
+ (* Main proof *)
+ fix a1 a2 b1 b2
+ assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
+ hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
+ have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ using * unfolding bsqr_def by auto
+ have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
+ using ** unfolding bsqr_def by auto
+ show "a1 = b1 \<and> a2 = b2"
+ proof-
+ {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+ {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case1 IrrS by blast
+ }
+ moreover
+ {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
+ hence False using Case1 by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+ {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case2 by auto
+ }
+ moreover
+ {assume Case22: "(b1,a1) \<in> r - Id"
+ hence False using Case2 IrrS by blast
+ }
+ moreover
+ {assume Case23: "b1 = a1"
+ hence False using Case2 by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ moreover
+ {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case3 by auto
+ }
+ moreover
+ {assume Case32: "(b1,a1) \<in> r - Id"
+ hence False using Case3 by auto
+ }
+ moreover
+ {assume Case33: "(b2,a2) \<in> r - Id"
+ hence False using Case3 IrrS by blast
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ ultimately show ?thesis using 0 1 by blast
+ qed
+qed
+
+
+lemma bsqr_Total:
+assumes "Well_order r"
+shows "Total(bsqr r)"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
+ using wo_rel.TOTALS by auto
+ (* Main proof *)
+ {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
+ hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
+ using Field_bsqr by blast
+ have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
+ proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
+ (* Why didn't clarsimp simp add: Well 0 do the same job? *)
+ assume Case1: "(a1,a2) \<in> r"
+ hence 1: "wo_rel.max2 r a1 a2 = a2"
+ using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case11: "(b1,b2) \<in> r"
+ hence 2: "wo_rel.max2 r b1 b2 = b2"
+ using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+ next
+ assume Case112: "a2 = b2"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
+ next
+ assume Case1122: "a1 = b1"
+ thus ?thesis using Case112 by auto
+ qed
+ qed
+ next
+ assume Case12: "(b2,b1) \<in> r"
+ hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+ next
+ assume Case122: "a2 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
+ next
+ assume Case1222: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
+ next
+ assume Case12222: "a2 = b2"
+ thus ?thesis using Case122 Case1222 by auto
+ qed
+ qed
+ qed
+ qed
+ next
+ assume Case2: "(a2,a1) \<in> r"
+ hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case21: "(b1,b2) \<in> r"
+ hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+ next
+ assume Case212: "a1 = b2"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
+ next
+ assume Case2122: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
+ next
+ assume Case21222: "a2 = b2"
+ thus ?thesis using Case2122 Case212 by auto
+ qed
+ qed
+ qed
+ next
+ assume Case22: "(b2,b1) \<in> r"
+ hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+ next
+ assume Case222: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
+ next
+ assume Case2222: "a2 = b2"
+ thus ?thesis using Case222 by auto
+ qed
+ qed
+ qed
+ qed
+ }
+ thus ?thesis unfolding total_on_def by fast
+qed
+
+
+lemma bsqr_Linear_order:
+assumes "Well_order r"
+shows "Linear_order(bsqr r)"
+unfolding order_on_defs
+using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
+
+
+lemma bsqr_Well_order:
+assumes "Well_order r"
+shows "Well_order(bsqr r)"
+using assms
+proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
+ have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+ using assms well_order_on_def Linear_order_Well_order_iff by blast
+ fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
+ hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
+ (* *)
+ obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
+ have "M \<noteq> {}" using 1 M_def ** by auto
+ moreover
+ have "M \<le> Field r" unfolding M_def
+ using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+ ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+ have "A1 \<le> Field r" unfolding A1_def using 1 by auto
+ moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
+ ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+ have "A2 \<le> Field r" unfolding A2_def using 1 by auto
+ moreover have "A2 \<noteq> {}" unfolding A2_def
+ using m_min a1_min unfolding A1_def M_def by blast
+ ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ have 2: "wo_rel.max2 r a1 a2 = m"
+ using a1_min a2_min unfolding A1_def A2_def by auto
+ have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
+ (* *)
+ moreover
+ {fix b1 b2 assume ***: "(b1,b2) \<in> D"
+ hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
+ have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+ using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
+ have "((a1,a2),(b1,b2)) \<in> bsqr r"
+ proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
+ assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
+ thus ?thesis unfolding bsqr_def using 4 5 by auto
+ next
+ assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
+ hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
+ hence 6: "(a1,b1) \<in> r" using a1_min by auto
+ show ?thesis
+ proof(cases "a1 = b1")
+ assume Case21: "a1 \<noteq> b1"
+ thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
+ next
+ assume Case22: "a1 = b1"
+ hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
+ hence 7: "(a2,b2) \<in> r" using a2_min by auto
+ thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
+ qed
+ qed
+ }
+ (* *)
+ ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
+qed
+
+
+lemma bsqr_max2:
+assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
+shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+proof-
+ have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
+ using LEQ unfolding Field_def by auto
+ hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+ hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
+ using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+ moreover have "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r \<or> wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
+ using LEQ unfolding bsqr_def by auto
+ ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
+qed
+
+
+lemma bsqr_ofilter:
+assumes WELL: "Well_order r" and
+ OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
+ NE: "\<not> (\<exists>a. Field r = rel.under r a)"
+shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
+proof-
+ let ?r' = "bsqr r"
+ have Well: "wo_rel r" using WELL wo_rel_def by blast
+ hence Trans: "trans r" using wo_rel.TRANS by blast
+ have Well': "Well_order ?r' \<and> wo_rel ?r'"
+ using WELL bsqr_Well_order wo_rel_def by blast
+ (* *)
+ have "D < Field ?r'" unfolding Field_bsqr using SUB .
+ with OF obtain a1 and a2 where
+ "(a1,a2) \<in> Field ?r'" and 1: "D = rel.underS ?r' (a1,a2)"
+ using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
+ hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
+ let ?m = "wo_rel.max2 r a1 a2"
+ have "D \<le> (rel.under r ?m) \<times> (rel.under r ?m)"
+ proof(unfold 1)
+ {fix b1 b2
+ let ?n = "wo_rel.max2 r b1 b2"
+ assume "(b1,b2) \<in> rel.underS ?r' (a1,a2)"
+ hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
+ unfolding rel.underS_def by blast
+ hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
+ moreover
+ {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
+ hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+ hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
+ using Well by (simp add: wo_rel.max2_greater)
+ }
+ ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
+ using Trans trans_def[of r] by blast
+ hence "(b1,b2) \<in> (rel.under r ?m) \<times> (rel.under r ?m)" unfolding rel.under_def by simp}
+ thus "rel.underS ?r' (a1,a2) \<le> (rel.under r ?m) \<times> (rel.under r ?m)" by auto
+ qed
+ moreover have "wo_rel.ofilter r (rel.under r ?m)"
+ using Well by (simp add: wo_rel.under_ofilter)
+ moreover have "rel.under r ?m < Field r"
+ using NE rel.under_Field[of r ?m] by blast
+ ultimately show ?thesis by blast
+qed
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Fun_More.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,183 @@
+(* Title: HOL/Ordinals_and_Cardinals/Fun_More.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+More on injections, bijections and inverses.
+*)
+
+header {* More on Injections, Bijections and Inverses *}
+
+theory Fun_More
+imports "../Ordinals_and_Cardinals_Base/Fun_More_Base"
+begin
+
+
+subsection {* Purely functional properties *}
+
+(* unused *)
+(*1*)lemma notIn_Un_bij_betw2:
+assumes NIN: "b \<notin> A" and NIN': "b' \<notin> A'" and
+ BIJ: "bij_betw f A A'"
+shows "bij_betw f (A \<union> {b}) (A' \<union> {b'}) = (f b = b')"
+proof
+ assume "f b = b'"
+ thus "bij_betw f (A \<union> {b}) (A' \<union> {b'})"
+ using assms notIn_Un_bij_betw[of b A f A'] by auto
+next
+ assume *: "bij_betw f (A \<union> {b}) (A' \<union> {b'})"
+ hence "f b \<in> A' \<union> {b'}"
+ unfolding bij_betw_def by auto
+ moreover
+ {assume "f b \<in> A'"
+ then obtain b1 where 1: "b1 \<in> A" and 2: "f b1 = f b" using BIJ
+ by (auto simp add: bij_betw_def)
+ hence "b = b1" using *
+ by (auto simp add: bij_betw_def inj_on_def)
+ with 1 NIN have False by auto
+ }
+ ultimately show "f b = b'" by blast
+qed
+
+(* unused *)
+(*1*)lemma bij_betw_ball:
+assumes BIJ: "bij_betw f A B"
+shows "(\<forall>b \<in> B. phi b) = (\<forall>a \<in> A. phi(f a))"
+using assms unfolding bij_betw_def inj_on_def by blast
+
+(* unused *)
+(*1*)lemma bij_betw_diff_singl:
+assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
+shows "bij_betw f (A - {a}) (A' - {f a})"
+proof-
+ let ?B = "A - {a}" let ?B' = "A' - {f a}"
+ have "f a \<in> A'" using IN BIJ unfolding bij_betw_def by blast
+ hence "a \<notin> ?B \<and> f a \<notin> ?B' \<and> A = ?B \<union> {a} \<and> A' = ?B' \<union> {f a}"
+ using IN by blast
+ thus ?thesis using notIn_Un_bij_betw3[of a ?B f ?B'] BIJ by simp
+qed
+
+
+subsection {* Properties involving finite and infinite sets *}
+
+(*3*)lemma inj_on_image_Pow:
+assumes "inj_on f A"
+shows "inj_on (image f) (Pow A)"
+unfolding Pow_def inj_on_def proof(clarsimp)
+ fix X Y assume *: "X \<le> A" and **: "Y \<le> A" and
+ ***: "f ` X = f ` Y"
+ show "X = Y"
+ proof(auto)
+ fix x assume ****: "x \<in> X"
+ with *** obtain y where "y \<in> Y \<and> f x = f y" by blast
+ with **** * ** assms show "x \<in> Y"
+ unfolding inj_on_def by auto
+ next
+ fix y assume ****: "y \<in> Y"
+ with *** obtain x where "x \<in> X \<and> f x = f y" by force
+ with **** * ** assms show "y \<in> X"
+ unfolding inj_on_def by auto
+ qed
+qed
+
+(*2*)lemma bij_betw_image_Pow:
+assumes "bij_betw f A B"
+shows "bij_betw (image f) (Pow A) (Pow B)"
+using assms unfolding bij_betw_def
+by (auto simp add: inj_on_image_Pow image_Pow_surj)
+
+(* unused *)
+(*1*)lemma bij_betw_inv_into_RIGHT:
+assumes BIJ: "bij_betw f A A'" and SUB: "B' \<le> A'"
+shows "f `((inv_into A f)`B') = B'"
+using assms
+proof(auto simp add: bij_betw_inv_into_right)
+ let ?f' = "(inv_into A f)"
+ fix a' assume *: "a' \<in> B'"
+ hence "a' \<in> A'" using SUB by auto
+ hence "a' = f (?f' a')"
+ using BIJ by (auto simp add: bij_betw_inv_into_right)
+ thus "a' \<in> f ` (?f' ` B')" using * by blast
+qed
+
+(* unused *)
+(*1*)lemma bij_betw_inv_into_RIGHT_LEFT:
+assumes BIJ: "bij_betw f A A'" and SUB: "B' \<le> A'" and
+ IM: "(inv_into A f) ` B' = B"
+shows "f ` B = B'"
+proof-
+ have "f`((inv_into A f)` B') = B'"
+ using assms bij_betw_inv_into_RIGHT[of f A A' B'] by auto
+ thus ?thesis using IM by auto
+qed
+
+(* unused *)
+(*2*)lemma bij_betw_inv_into_twice:
+assumes "bij_betw f A A'"
+shows "\<forall>a \<in> A. inv_into A' (inv_into A f) a = f a"
+proof
+ let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'"
+ have 1: "bij_betw ?f' A' A" using assms
+ by (auto simp add: bij_betw_inv_into)
+ fix a assume *: "a \<in> A"
+ then obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
+ using 1 unfolding bij_betw_def by force
+ hence "?f'' a = a'"
+ using * 1 3 by (auto simp add: bij_betw_inv_into_left)
+ moreover have "f a = a'" using assms 2 3
+ by (auto simp add: bij_betw_inv_into_right)
+ ultimately show "?f'' a = f a" by simp
+qed
+
+
+subsection {* Properties involving Hilbert choice *}
+
+
+subsection {* Other facts *}
+
+(*3*)lemma atLeastLessThan_injective:
+assumes "{0 ..< m::nat} = {0 ..< n}"
+shows "m = n"
+proof-
+ {assume "m < n"
+ hence "m \<in> {0 ..< n}" by auto
+ hence "{0 ..< m} < {0 ..< n}" by auto
+ hence False using assms by blast
+ }
+ moreover
+ {assume "n < m"
+ hence "n \<in> {0 ..< m}" by auto
+ hence "{0 ..< n} < {0 ..< m}" by auto
+ hence False using assms by blast
+ }
+ ultimately show ?thesis by force
+qed
+
+(*2*)lemma atLeastLessThan_injective2:
+"bij_betw f {0 ..< m::nat} {0 ..< n} \<Longrightarrow> m = n"
+using finite_atLeastLessThan[of m] finite_atLeastLessThan[of n]
+ card_atLeastLessThan[of m] card_atLeastLessThan[of n]
+ bij_betw_iff_card[of "{0 ..< m}" "{0 ..< n}"] by auto
+
+(* unused *)
+(*2*)lemma atLeastLessThan_less_eq3:
+"(\<exists>f. inj_on f {0..<(m::nat)} \<and> f ` {0..<m} \<le> {0..<n}) = (m \<le> n)"
+using atLeastLessThan_less_eq2
+proof(auto)
+ assume "m \<le> n"
+ hence "inj_on id {0..<m} \<and> id ` {0..<m} \<subseteq> {0..<n}" unfolding inj_on_def by force
+ thus "\<exists>f. inj_on f {0..<m} \<and> f ` {0..<m} \<subseteq> {0..<n}" by blast
+qed
+
+(* unused *)
+(*3*)lemma atLeastLessThan_less:
+"({0..<m} < {0..<n}) = ((m::nat) < n)"
+proof-
+ have "({0..<m} < {0..<n}) = ({0..<m} \<le> {0..<n} \<and> {0..<m} ~= {0..<n})"
+ using subset_iff_psubset_eq by blast
+ also have "\<dots> = (m \<le> n \<and> m ~= n)"
+ using atLeastLessThan_less_eq atLeastLessThan_injective by blast
+ also have "\<dots> = (m < n)" by auto
+ finally show ?thesis .
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Fun_More_Base.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,252 @@
+(* Title: HOL/Ordinals_and_Cardinals/Fun_More_Base.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+More on injections, bijections and inverses (base).
+*)
+
+header {* More on Injections, Bijections and Inverses (Base) *}
+
+theory Fun_More_Base
+imports "~~/src/HOL/Library/Infinite_Set"
+begin
+
+
+text {* This section proves more facts (additional to those in @{text "Fun.thy"},
+@{text "Hilbert_Choice.thy"}, @{text "Finite_Set.thy"} and @{text "Infinite_Set.thy"}),
+mainly concerning injections, bijections, inverses and (numeric) cardinals of
+finite sets. *}
+
+
+subsection {* Purely functional properties *}
+
+
+(*2*)lemma bij_betw_id_iff:
+"(A = B) = (bij_betw id A B)"
+by(simp add: bij_betw_def)
+
+
+(*2*)lemma bij_betw_byWitness:
+assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
+ RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
+ IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
+shows "bij_betw f A A'"
+using assms
+proof(unfold bij_betw_def inj_on_def, auto)
+ fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
+ have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
+ with ** show "a = b" by simp
+next
+ fix a' assume *: "a' \<in> A'"
+ hence "f' a' \<in> A" using IM2 by blast
+ moreover
+ have "a' = f(f' a')" using * RIGHT by simp
+ ultimately show "a' \<in> f ` A" by blast
+qed
+
+
+(*3*)corollary notIn_Un_bij_betw:
+assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
+ BIJ: "bij_betw f A A'"
+shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+proof-
+ have "bij_betw f {b} {f b}"
+ unfolding bij_betw_def inj_on_def by simp
+ with assms show ?thesis
+ using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
+qed
+
+
+(*1*)lemma notIn_Un_bij_betw3:
+assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
+shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+proof
+ assume "bij_betw f A A'"
+ thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+ using assms notIn_Un_bij_betw[of b A f A'] by blast
+next
+ assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+ have "f ` A = A'"
+ proof(auto)
+ fix a assume **: "a \<in> A"
+ hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
+ moreover
+ {assume "f a = f b"
+ hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
+ with NIN ** have False by blast
+ }
+ ultimately show "f a \<in> A'" by blast
+ next
+ fix a' assume **: "a' \<in> A'"
+ hence "a' \<in> f`(A \<union> {b})"
+ using * by (auto simp add: bij_betw_def)
+ then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
+ moreover
+ {assume "a = b" with 1 ** NIN' have False by blast
+ }
+ ultimately have "a \<in> A" by blast
+ with 1 show "a' \<in> f ` A" by blast
+ qed
+ thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
+qed
+
+
+subsection {* Properties involving finite and infinite sets *}
+
+
+(*3*)lemma inj_on_finite:
+assumes "inj_on f A" "f ` A \<le> B" "finite B"
+shows "finite A"
+using assms infinite_super by (fast dest: finite_imageD)
+
+
+(*3*)lemma infinite_imp_bij_betw:
+assumes INF: "infinite A"
+shows "\<exists>h. bij_betw h A (A - {a})"
+proof(cases "a \<in> A")
+ assume Case1: "a \<notin> A" hence "A - {a} = A" by blast
+ thus ?thesis using bij_betw_id[of A] by auto
+next
+ assume Case2: "a \<in> A"
+ have "infinite (A - {a})" using INF infinite_remove by auto
+ with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
+ where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
+ obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
+ obtain A' where A'_def: "A' = g ` UNIV" by blast
+ have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
+ have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
+ proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
+ case_tac "x = 0", auto simp add: 2)
+ fix y assume "a = (if y = 0 then a else f (Suc y))"
+ thus "y = 0" using temp by (case_tac "y = 0", auto)
+ next
+ fix x y
+ assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
+ thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
+ next
+ fix n show "f (Suc n) \<in> A" using 2 by blast
+ qed
+ hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
+ using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
+ hence 5: "bij_betw (inv g) A' UNIV"
+ by (auto simp add: bij_betw_inv_into)
+ (* *)
+ obtain n where "g n = a" using 3 by auto
+ hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
+ using 3 4 unfolding A'_def
+ by clarify (rule bij_betw_subset, auto simp: image_set_diff)
+ (* *)
+ obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
+ have 7: "bij_betw v UNIV (UNIV - {n})"
+ proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
+ fix m1 m2 assume "v m1 = v m2"
+ thus "m1 = m2"
+ by(case_tac "m1 < n", case_tac "m2 < n",
+ auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
+ next
+ show "v ` UNIV = UNIV - {n}"
+ proof(auto simp add: v_def)
+ fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
+ {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
+ then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
+ with 71 have "n \<le> m'" by auto
+ with 72 ** have False by auto
+ }
+ thus "m < n" by force
+ qed
+ qed
+ (* *)
+ obtain h' where h'_def: "h' = g o v o (inv g)" by blast
+ hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
+ by (auto simp add: bij_betw_trans)
+ (* *)
+ obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
+ have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
+ hence "bij_betw h A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
+ moreover
+ {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
+ hence "bij_betw h (A - A') (A - A')"
+ using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
+ }
+ moreover
+ have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
+ ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
+ using 4 by blast
+ ultimately have "bij_betw h A (A - {a})"
+ using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
+ thus ?thesis by blast
+qed
+
+
+(*3*)lemma infinite_imp_bij_betw2:
+assumes INF: "infinite A"
+shows "\<exists>h. bij_betw h A (A \<union> {a})"
+proof(cases "a \<in> A")
+ assume Case1: "a \<in> A" hence "A \<union> {a} = A" by blast
+ thus ?thesis using bij_betw_id[of A] by auto
+next
+ let ?A' = "A \<union> {a}"
+ assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
+ moreover have "infinite ?A'" using INF by auto
+ ultimately obtain f where "bij_betw f ?A' A"
+ using infinite_imp_bij_betw[of ?A' a] by auto
+ hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
+ thus ?thesis by auto
+qed
+
+
+subsection {* Properties involving Hilbert choice *}
+
+
+(*2*)lemma bij_betw_inv_into_left:
+assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
+shows "(inv_into A f) (f a) = a"
+using assms unfolding bij_betw_def
+by clarify (rule inv_into_f_f)
+
+(*2*)lemma bij_betw_inv_into_right:
+assumes "bij_betw f A A'" "a' \<in> A'"
+shows "f(inv_into A f a') = a'"
+using assms unfolding bij_betw_def using f_inv_into_f by force
+
+
+(*1*)lemma bij_betw_inv_into_LEFT:
+assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A"
+shows "(inv_into A f)`(f ` B) = B"
+using assms unfolding bij_betw_def using inv_into_image_cancel by force
+
+
+(*1*)lemma bij_betw_inv_into_LEFT_RIGHT:
+assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A" and
+ IM: "f ` B = B'"
+shows "(inv_into A f) ` B' = B"
+using assms bij_betw_inv_into_LEFT[of f A A' B] by fast
+
+
+(*1*)lemma bij_betw_inv_into_subset:
+assumes BIJ: "bij_betw f A A'" and
+ SUB: "B \<le> A" and IM: "f ` B = B'"
+shows "bij_betw (inv_into A f) B' B"
+using assms unfolding bij_betw_def
+by (auto intro: inj_on_inv_into)
+
+
+subsection {* Other facts *}
+
+
+(*2*)lemma atLeastLessThan_less_eq:
+"({0..<m} \<le> {0..<n}) = ((m::nat) \<le> n)"
+unfolding ivl_subset by arith
+
+
+(*2*)lemma atLeastLessThan_less_eq2:
+assumes "inj_on f {0..<(m::nat)} \<and> f ` {0..<m} \<le> {0..<n}"
+shows "m \<le> n"
+using assms
+using finite_atLeastLessThan[of m] finite_atLeastLessThan[of n]
+ card_atLeastLessThan[of m] card_atLeastLessThan[of n]
+ card_inj_on_le[of f "{0 ..< m}" "{0 ..< n}"] by auto
+
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Order_Relation_More.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,582 @@
+(* Title: HOL/Ordinals_and_Cardinals/Order_Relation_More.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Basics on order-like relations.
+*)
+
+header {* Basics on Order-Like Relations *}
+
+theory Order_Relation_More
+imports "../Ordinals_and_Cardinals_Base/Order_Relation_More_Base"
+begin
+
+
+subsection {* The upper and lower bounds operators *}
+
+lemma (in rel) aboveS_subset_above: "aboveS a \<le> above a"
+by(auto simp add: aboveS_def above_def)
+
+lemma (in rel) AboveS_subset_Above: "AboveS A \<le> Above A"
+by(auto simp add: AboveS_def Above_def)
+
+lemma (in rel) UnderS_disjoint: "A Int (UnderS A) = {}"
+by(auto simp add: UnderS_def)
+
+lemma (in rel) aboveS_notIn: "a \<notin> aboveS a"
+by(auto simp add: aboveS_def)
+
+lemma (in rel) Refl_above_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> above a"
+by(auto simp add: refl_on_def above_def)
+
+lemma (in rel) in_Above_under: "a \<in> Field r \<Longrightarrow> a \<in> Above (under a)"
+by(auto simp add: Above_def under_def)
+
+lemma (in rel) in_Under_above: "a \<in> Field r \<Longrightarrow> a \<in> Under (above a)"
+by(auto simp add: Under_def above_def)
+
+lemma (in rel) in_UnderS_aboveS: "a \<in> Field r \<Longrightarrow> a \<in> UnderS (aboveS a)"
+by(auto simp add: UnderS_def aboveS_def)
+
+lemma (in rel) subset_Above_Under: "B \<le> Field r \<Longrightarrow> B \<le> Above (Under B)"
+by(auto simp add: Above_def Under_def)
+
+lemma (in rel) subset_Under_Above: "B \<le> Field r \<Longrightarrow> B \<le> Under (Above B)"
+by(auto simp add: Under_def Above_def)
+
+lemma (in rel) subset_AboveS_UnderS: "B \<le> Field r \<Longrightarrow> B \<le> AboveS (UnderS B)"
+by(auto simp add: AboveS_def UnderS_def)
+
+lemma (in rel) subset_UnderS_AboveS: "B \<le> Field r \<Longrightarrow> B \<le> UnderS (AboveS B)"
+by(auto simp add: UnderS_def AboveS_def)
+
+lemma (in rel) Under_Above_Galois:
+"\<lbrakk>B \<le> Field r; C \<le> Field r\<rbrakk> \<Longrightarrow> (B \<le> Above C) = (C \<le> Under B)"
+by(unfold Above_def Under_def, blast)
+
+lemma (in rel) UnderS_AboveS_Galois:
+"\<lbrakk>B \<le> Field r; C \<le> Field r\<rbrakk> \<Longrightarrow> (B \<le> AboveS C) = (C \<le> UnderS B)"
+by(unfold AboveS_def UnderS_def, blast)
+
+lemma (in rel) Refl_above_aboveS:
+assumes REFL: "Refl r" and IN: "a \<in> Field r"
+shows "above a = aboveS a \<union> {a}"
+proof(unfold above_def aboveS_def, auto)
+ show "(a,a) \<in> r" using REFL IN refl_on_def[of _ r] by blast
+qed
+
+lemma (in rel) Linear_order_under_aboveS_Field:
+assumes LIN: "Linear_order r" and IN: "a \<in> Field r"
+shows "Field r = under a \<union> aboveS a"
+proof(unfold under_def aboveS_def, auto)
+ assume "a \<in> Field r" "(a, a) \<notin> r"
+ with LIN IN order_on_defs[of _ r] refl_on_def[of _ r]
+ show False by auto
+next
+ fix b assume "b \<in> Field r" "(b, a) \<notin> r"
+ with LIN IN order_on_defs[of "Field r" r] total_on_def[of "Field r" r]
+ have "(a,b) \<in> r \<or> a = b" by blast
+ thus "(a,b) \<in> r"
+ using LIN IN order_on_defs[of _ r] refl_on_def[of _ r] by auto
+next
+ fix b assume "(b, a) \<in> r"
+ thus "b \<in> Field r"
+ using LIN order_on_defs[of _ r] refl_on_def[of _ r] by blast
+next
+ fix b assume "b \<noteq> a" "(a, b) \<in> r"
+ thus "b \<in> Field r"
+ using LIN order_on_defs[of "Field r" r] refl_on_def[of "Field r" r] by blast
+qed
+
+lemma (in rel) Linear_order_underS_above_Field:
+assumes LIN: "Linear_order r" and IN: "a \<in> Field r"
+shows "Field r = underS a \<union> above a"
+proof(unfold underS_def above_def, auto)
+ assume "a \<in> Field r" "(a, a) \<notin> r"
+ with LIN IN order_on_defs[of _ r] refl_on_def[of _ r]
+ show False by auto
+next
+ fix b assume "b \<in> Field r" "(a, b) \<notin> r"
+ with LIN IN order_on_defs[of "Field r" r] total_on_def[of "Field r" r]
+ have "(b,a) \<in> r \<or> b = a" by blast
+ thus "(b,a) \<in> r"
+ using LIN IN order_on_defs[of _ r] refl_on_def[of _ r] by auto
+next
+ fix b assume "b \<noteq> a" "(b, a) \<in> r"
+ thus "b \<in> Field r"
+ using LIN order_on_defs[of _ r] refl_on_def[of _ r] by blast
+next
+ fix b assume "(a, b) \<in> r"
+ thus "b \<in> Field r"
+ using LIN order_on_defs[of "Field r" r] refl_on_def[of "Field r" r] by blast
+qed
+
+lemma (in rel) under_empty: "a \<notin> Field r \<Longrightarrow> under a = {}"
+unfolding Field_def under_def by auto
+
+lemma (in rel) above_Field: "above a \<le> Field r"
+by(unfold above_def Field_def, auto)
+
+lemma (in rel) aboveS_Field: "aboveS a \<le> Field r"
+by(unfold aboveS_def Field_def, auto)
+
+lemma (in rel) Above_Field: "Above A \<le> Field r"
+by(unfold Above_def Field_def, auto)
+
+lemma (in rel) Refl_under_Under:
+assumes REFL: "Refl r" and NE: "A \<noteq> {}"
+shows "Under A = (\<Inter> a \<in> A. under a)"
+proof
+ show "Under A \<subseteq> (\<Inter> a \<in> A. under a)"
+ by(unfold Under_def under_def, auto)
+next
+ show "(\<Inter> a \<in> A. under a) \<subseteq> Under A"
+ proof(auto)
+ fix x
+ assume *: "\<forall>xa \<in> A. x \<in> under xa"
+ hence "\<forall>xa \<in> A. (x,xa) \<in> r"
+ by (simp add: under_def)
+ moreover
+ {from NE obtain a where "a \<in> A" by blast
+ with * have "x \<in> under a" by simp
+ hence "x \<in> Field r"
+ using under_Field[of a] by auto
+ }
+ ultimately show "x \<in> Under A"
+ unfolding Under_def by auto
+ qed
+qed
+
+lemma (in rel) Refl_underS_UnderS:
+assumes REFL: "Refl r" and NE: "A \<noteq> {}"
+shows "UnderS A = (\<Inter> a \<in> A. underS a)"
+proof
+ show "UnderS A \<subseteq> (\<Inter> a \<in> A. underS a)"
+ by(unfold UnderS_def underS_def, auto)
+next
+ show "(\<Inter> a \<in> A. underS a) \<subseteq> UnderS A"
+ proof(auto)
+ fix x
+ assume *: "\<forall>xa \<in> A. x \<in> underS xa"
+ hence "\<forall>xa \<in> A. x \<noteq> xa \<and> (x,xa) \<in> r"
+ by (auto simp add: underS_def)
+ moreover
+ {from NE obtain a where "a \<in> A" by blast
+ with * have "x \<in> underS a" by simp
+ hence "x \<in> Field r"
+ using underS_Field[of a] by auto
+ }
+ ultimately show "x \<in> UnderS A"
+ unfolding UnderS_def by auto
+ qed
+qed
+
+lemma (in rel) Refl_above_Above:
+assumes REFL: "Refl r" and NE: "A \<noteq> {}"
+shows "Above A = (\<Inter> a \<in> A. above a)"
+proof
+ show "Above A \<subseteq> (\<Inter> a \<in> A. above a)"
+ by(unfold Above_def above_def, auto)
+next
+ show "(\<Inter> a \<in> A. above a) \<subseteq> Above A"
+ proof(auto)
+ fix x
+ assume *: "\<forall>xa \<in> A. x \<in> above xa"
+ hence "\<forall>xa \<in> A. (xa,x) \<in> r"
+ by (simp add: above_def)
+ moreover
+ {from NE obtain a where "a \<in> A" by blast
+ with * have "x \<in> above a" by simp
+ hence "x \<in> Field r"
+ using above_Field[of a] by auto
+ }
+ ultimately show "x \<in> Above A"
+ unfolding Above_def by auto
+ qed
+qed
+
+lemma (in rel) Refl_aboveS_AboveS:
+assumes REFL: "Refl r" and NE: "A \<noteq> {}"
+shows "AboveS A = (\<Inter> a \<in> A. aboveS a)"
+proof
+ show "AboveS A \<subseteq> (\<Inter> a \<in> A. aboveS a)"
+ by(unfold AboveS_def aboveS_def, auto)
+next
+ show "(\<Inter> a \<in> A. aboveS a) \<subseteq> AboveS A"
+ proof(auto)
+ fix x
+ assume *: "\<forall>xa \<in> A. x \<in> aboveS xa"
+ hence "\<forall>xa \<in> A. xa \<noteq> x \<and> (xa,x) \<in> r"
+ by (auto simp add: aboveS_def)
+ moreover
+ {from NE obtain a where "a \<in> A" by blast
+ with * have "x \<in> aboveS a" by simp
+ hence "x \<in> Field r"
+ using aboveS_Field[of a] by auto
+ }
+ ultimately show "x \<in> AboveS A"
+ unfolding AboveS_def by auto
+ qed
+qed
+
+lemma (in rel) under_Under_singl: "under a = Under {a}"
+by(unfold Under_def under_def, auto simp add: Field_def)
+
+lemma (in rel) underS_UnderS_singl: "underS a = UnderS {a}"
+by(unfold UnderS_def underS_def, auto simp add: Field_def)
+
+lemma (in rel) above_Above_singl: "above a = Above {a}"
+by(unfold Above_def above_def, auto simp add: Field_def)
+
+lemma (in rel) aboveS_AboveS_singl: "aboveS a = AboveS {a}"
+by(unfold AboveS_def aboveS_def, auto simp add: Field_def)
+
+lemma (in rel) Under_decr: "A \<le> B \<Longrightarrow> Under B \<le> Under A"
+by(unfold Under_def, auto)
+
+lemma (in rel) UnderS_decr: "A \<le> B \<Longrightarrow> UnderS B \<le> UnderS A"
+by(unfold UnderS_def, auto)
+
+lemma (in rel) Above_decr: "A \<le> B \<Longrightarrow> Above B \<le> Above A"
+by(unfold Above_def, auto)
+
+lemma (in rel) AboveS_decr: "A \<le> B \<Longrightarrow> AboveS B \<le> AboveS A"
+by(unfold AboveS_def, auto)
+
+lemma (in rel) under_incl_iff:
+assumes TRANS: "trans r" and REFL: "Refl r" and IN: "a \<in> Field r"
+shows "(under a \<le> under b) = ((a,b) \<in> r)"
+proof
+ assume "(a,b) \<in> r"
+ thus "under a \<le> under b" using TRANS
+ by (auto simp add: under_incr)
+next
+ assume "under a \<le> under b"
+ moreover
+ have "a \<in> under a" using REFL IN
+ by (auto simp add: Refl_under_in)
+ ultimately show "(a,b) \<in> r"
+ by (auto simp add: under_def)
+qed
+
+lemma (in rel) above_decr:
+assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
+shows "above b \<le> above a"
+proof(unfold above_def, auto)
+ fix x assume "(b,x) \<in> r"
+ with REL TRANS trans_def[of r]
+ show "(a,x) \<in> r" by blast
+qed
+
+lemma (in rel) aboveS_decr:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ REL: "(a,b) \<in> r"
+shows "aboveS b \<le> aboveS a"
+proof(unfold aboveS_def, auto)
+ assume *: "a \<noteq> b" and **: "(b,a) \<in> r"
+ with ANTISYM antisym_def[of r] REL
+ show False by auto
+next
+ fix x assume "x \<noteq> b" "(b,x) \<in> r"
+ with REL TRANS trans_def[of r]
+ show "(a,x) \<in> r" by blast
+qed
+
+lemma (in rel) under_trans:
+assumes TRANS: "trans r" and
+ IN1: "a \<in> under b" and IN2: "b \<in> under c"
+shows "a \<in> under c"
+proof-
+ have "(a,b) \<in> r \<and> (b,c) \<in> r"
+ using IN1 IN2 under_def by auto
+ hence "(a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ thus ?thesis unfolding under_def by simp
+qed
+
+lemma (in rel) under_underS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> under b" and IN2: "b \<in> underS c"
+shows "a \<in> underS c"
+proof-
+ have 0: "(a,b) \<in> r \<and> (b,c) \<in> r"
+ using IN1 IN2 under_def underS_def by auto
+ hence 1: "(a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ have 2: "b \<noteq> c" using IN2 underS_def by auto
+ have 3: "a \<noteq> c"
+ proof
+ assume "a = c" with 0 2 ANTISYM antisym_def[of r]
+ show False by auto
+ qed
+ from 1 3 show ?thesis unfolding underS_def by simp
+qed
+
+lemma (in rel) underS_under_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> underS b" and IN2: "b \<in> under c"
+shows "a \<in> underS c"
+proof-
+ have 0: "(a,b) \<in> r \<and> (b,c) \<in> r"
+ using IN1 IN2 under_def underS_def by auto
+ hence 1: "(a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ have 2: "a \<noteq> b" using IN1 underS_def by auto
+ have 3: "a \<noteq> c"
+ proof
+ assume "a = c" with 0 2 ANTISYM antisym_def[of r]
+ show False by auto
+ qed
+ from 1 3 show ?thesis unfolding underS_def by simp
+qed
+
+lemma (in rel) underS_underS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> underS b" and IN2: "b \<in> underS c"
+shows "a \<in> underS c"
+proof-
+ have "a \<in> under b"
+ using IN1 underS_subset_under by auto
+ with assms under_underS_trans show ?thesis by auto
+qed
+
+lemma (in rel) above_trans:
+assumes TRANS: "trans r" and
+ IN1: "b \<in> above a" and IN2: "c \<in> above b"
+shows "c \<in> above a"
+proof-
+ have "(a,b) \<in> r \<and> (b,c) \<in> r"
+ using IN1 IN2 above_def by auto
+ hence "(a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ thus ?thesis unfolding above_def by simp
+qed
+
+lemma (in rel) above_aboveS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "b \<in> above a" and IN2: "c \<in> aboveS b"
+shows "c \<in> aboveS a"
+proof-
+ have 0: "(a,b) \<in> r \<and> (b,c) \<in> r"
+ using IN1 IN2 above_def aboveS_def by auto
+ hence 1: "(a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ have 2: "b \<noteq> c" using IN2 aboveS_def by auto
+ have 3: "a \<noteq> c"
+ proof
+ assume "a = c" with 0 2 ANTISYM antisym_def[of r]
+ show False by auto
+ qed
+ from 1 3 show ?thesis unfolding aboveS_def by simp
+qed
+
+lemma (in rel) aboveS_above_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "b \<in> aboveS a" and IN2: "c \<in> above b"
+shows "c \<in> aboveS a"
+proof-
+ have 0: "(a,b) \<in> r \<and> (b,c) \<in> r"
+ using IN1 IN2 above_def aboveS_def by auto
+ hence 1: "(a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ have 2: "a \<noteq> b" using IN1 aboveS_def by auto
+ have 3: "a \<noteq> c"
+ proof
+ assume "a = c" with 0 2 ANTISYM antisym_def[of r]
+ show False by auto
+ qed
+ from 1 3 show ?thesis unfolding aboveS_def by simp
+qed
+
+lemma (in rel) aboveS_aboveS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "b \<in> aboveS a" and IN2: "c \<in> aboveS b"
+shows "c \<in> aboveS a"
+proof-
+ have "b \<in> above a"
+ using IN1 aboveS_subset_above by auto
+ with assms above_aboveS_trans show ?thesis by auto
+qed
+
+lemma (in rel) underS_Under_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> underS b" and IN2: "b \<in> Under C"
+shows "a \<in> UnderS C"
+proof-
+ from IN1 have "a \<in> under b"
+ using underS_subset_under[of b] by blast
+ with assms under_Under_trans
+ have "a \<in> Under C" by auto
+ (* *)
+ moreover
+ have "a \<notin> C"
+ proof
+ assume *: "a \<in> C"
+ have 1: "b \<noteq> a \<and> (a,b) \<in> r"
+ using IN1 underS_def[of b] by auto
+ have "\<forall>c \<in> C. (b,c) \<in> r"
+ using IN2 Under_def[of C] by auto
+ with * have "(b,a) \<in> r" by simp
+ with 1 ANTISYM antisym_def[of r]
+ show False by blast
+ qed
+ (* *)
+ ultimately
+ show ?thesis unfolding UnderS_def
+ using Under_def by auto
+qed
+
+lemma (in rel) underS_UnderS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> underS b" and IN2: "b \<in> UnderS C"
+shows "a \<in> UnderS C"
+proof-
+ from IN2 have "b \<in> Under C"
+ using UnderS_subset_Under[of C] by blast
+ with underS_Under_trans assms
+ show ?thesis by auto
+qed
+
+lemma (in rel) above_Above_trans:
+assumes TRANS: "trans r" and
+ IN1: "a \<in> above b" and IN2: "b \<in> Above C"
+shows "a \<in> Above C"
+proof-
+ have "(b,a) \<in> r \<and> (\<forall>c \<in> C. (c,b) \<in> r)"
+ using IN1 IN2 above_def Above_def by auto
+ hence "\<forall>c \<in> C. (c,a) \<in> r"
+ using TRANS trans_def[of r] by blast
+ moreover
+ have "a \<in> Field r" using IN1 Field_def above_def by force
+ ultimately
+ show ?thesis unfolding Above_def by auto
+qed
+
+lemma (in rel) aboveS_Above_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> aboveS b" and IN2: "b \<in> Above C"
+shows "a \<in> AboveS C"
+proof-
+ from IN1 have "a \<in> above b"
+ using aboveS_subset_above[of b] by blast
+ with assms above_Above_trans
+ have "a \<in> Above C" by auto
+ (* *)
+ moreover
+ have "a \<notin> C"
+ proof
+ assume *: "a \<in> C"
+ have 1: "b \<noteq> a \<and> (b,a) \<in> r"
+ using IN1 aboveS_def[of b] by auto
+ have "\<forall>c \<in> C. (c,b) \<in> r"
+ using IN2 Above_def[of C] by auto
+ with * have "(a,b) \<in> r" by simp
+ with 1 ANTISYM antisym_def[of r]
+ show False by blast
+ qed
+ (* *)
+ ultimately
+ show ?thesis unfolding AboveS_def
+ using Above_def by auto
+qed
+
+lemma (in rel) above_AboveS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> above b" and IN2: "b \<in> AboveS C"
+shows "a \<in> AboveS C"
+proof-
+ from IN2 have "b \<in> Above C"
+ using AboveS_subset_Above[of C] by blast
+ with assms above_Above_trans
+ have "a \<in> Above C" by auto
+ (* *)
+ moreover
+ have "a \<notin> C"
+ proof
+ assume *: "a \<in> C"
+ have 1: "(b,a) \<in> r"
+ using IN1 above_def[of b] by auto
+ have "\<forall>c \<in> C. b \<noteq> c \<and> (c,b) \<in> r"
+ using IN2 AboveS_def[of C] by auto
+ with * have "b \<noteq> a \<and> (a,b) \<in> r" by simp
+ with 1 ANTISYM antisym_def[of r]
+ show False by blast
+ qed
+ (* *)
+ ultimately
+ show ?thesis unfolding AboveS_def
+ using Above_def by auto
+qed
+
+lemma (in rel) aboveS_AboveS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> aboveS b" and IN2: "b \<in> AboveS C"
+shows "a \<in> AboveS C"
+proof-
+ from IN2 have "b \<in> Above C"
+ using AboveS_subset_Above[of C] by blast
+ with aboveS_Above_trans assms
+ show ?thesis by auto
+qed
+
+
+subsection {* Properties depending on more than one relation *}
+
+abbreviation "under \<equiv> rel.under"
+abbreviation "underS \<equiv> rel.underS"
+abbreviation "Under \<equiv> rel.Under"
+abbreviation "UnderS \<equiv> rel.UnderS"
+abbreviation "above \<equiv> rel.above"
+abbreviation "aboveS \<equiv> rel.aboveS"
+abbreviation "Above \<equiv> rel.Above"
+abbreviation "AboveS \<equiv> rel.AboveS"
+
+lemma under_incr2:
+"r \<le> r' \<Longrightarrow> under r a \<le> under r' a"
+unfolding rel.under_def by blast
+
+lemma underS_incr2:
+"r \<le> r' \<Longrightarrow> underS r a \<le> underS r' a"
+unfolding rel.underS_def by blast
+
+lemma Under_incr:
+"r \<le> r' \<Longrightarrow> Under r A \<le> Under r A"
+unfolding rel.Under_def by blast
+
+lemma UnderS_incr:
+"r \<le> r' \<Longrightarrow> UnderS r A \<le> UnderS r A"
+unfolding rel.UnderS_def by blast
+
+lemma Under_incr_decr:
+"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> Under r A \<le> Under r A'"
+unfolding rel.Under_def by blast
+
+lemma UnderS_incr_decr:
+"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> UnderS r A \<le> UnderS r A'"
+unfolding rel.UnderS_def by blast
+
+lemma above_incr2:
+"r \<le> r' \<Longrightarrow> above r a \<le> above r' a"
+unfolding rel.above_def by blast
+
+lemma aboveS_incr2:
+"r \<le> r' \<Longrightarrow> aboveS r a \<le> aboveS r' a"
+unfolding rel.aboveS_def by blast
+
+lemma Above_incr:
+"r \<le> r' \<Longrightarrow> Above r A \<le> Above r A"
+unfolding rel.Above_def by blast
+
+lemma AboveS_incr:
+"r \<le> r' \<Longrightarrow> AboveS r A \<le> AboveS r A"
+unfolding rel.AboveS_def by blast
+
+lemma Above_incr_decr:
+"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> Above r A \<le> Above r A'"
+unfolding rel.Above_def by blast
+
+lemma AboveS_incr_decr:
+"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> AboveS r A \<le> AboveS r A'"
+unfolding rel.AboveS_def by blast
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Order_Relation_More_Base.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,304 @@
+(* Title: HOL/Ordinals_and_Cardinals/Order_Relation_More_Base.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Basics on order-like relations (base).
+*)
+
+header {* Basics on Order-Like Relations (Base) *}
+
+theory Order_Relation_More_Base
+imports "~~/src/HOL/Library/Order_Relation"
+begin
+
+
+text{* In this section, we develop basic concepts and results pertaining
+to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
+total relations. The development is placed on top of the definitions
+from the theory @{text "Order_Relation"}. We also
+further define upper and lower bounds operators. *}
+
+
+locale rel = fixes r :: "'a rel"
+
+text{* The following context encompasses all this section, except
+for its last subsection. In other words, for the rest of this section except its last
+subsection, we consider a fixed relation @{text "r"}. *}
+
+context rel
+begin
+
+
+subsection {* Auxiliaries *}
+
+
+lemma refl_on_domain:
+"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
+by(auto simp add: refl_on_def)
+
+
+corollary well_order_on_domain:
+"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
+by(simp add: refl_on_domain order_on_defs)
+
+
+lemma well_order_on_Field:
+"well_order_on A r \<Longrightarrow> A = Field r"
+by(auto simp add: refl_on_def Field_def order_on_defs)
+
+
+lemma well_order_on_Well_order:
+"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
+using well_order_on_Field by simp
+
+
+lemma Total_Id_Field:
+assumes TOT: "Total r" and NID: "\<not> (r <= Id)"
+shows "Field r = Field(r - Id)"
+using mono_Field[of "r - Id" r] Diff_subset[of r Id]
+proof(auto)
+ have "r \<noteq> {}" using NID by fast
+ then obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" using NID by fast
+ hence 1: "b \<noteq> c \<and> {b,c} \<le> Field r" by (auto simp: Field_def)
+ (* *)
+ fix a assume *: "a \<in> Field r"
+ obtain d where 2: "d \<in> Field r" and 3: "d \<noteq> a"
+ using * 1 by blast
+ hence "(a,d) \<in> r \<or> (d,a) \<in> r" using * TOT
+ by (simp add: total_on_def)
+ thus "a \<in> Field(r - Id)" using 3 unfolding Field_def by blast
+qed
+
+
+lemma Total_subset_Id:
+assumes TOT: "Total r" and SUB: "r \<le> Id"
+shows "r = {} \<or> (\<exists>a. r = {(a,a)})"
+proof-
+ {assume "r \<noteq> {}"
+ then obtain a b where 1: "(a,b) \<in> r" by fast
+ hence "a = b" using SUB by blast
+ hence 2: "(a,a) \<in> r" using 1 by simp
+ {fix c d assume "(c,d) \<in> r"
+ hence "{a,c,d} \<le> Field r" using 1 unfolding Field_def by blast
+ hence "((a,c) \<in> r \<or> (c,a) \<in> r \<or> a = c) \<and>
+ ((a,d) \<in> r \<or> (d,a) \<in> r \<or> a = d)"
+ using TOT unfolding total_on_def by blast
+ hence "a = c \<and> a = d" using SUB by blast
+ }
+ hence "r \<le> {(a,a)}" by auto
+ with 2 have "\<exists>a. r = {(a,a)}" by blast
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma Linear_order_in_diff_Id:
+assumes LI: "Linear_order r" and
+ IN1: "a \<in> Field r" and IN2: "b \<in> Field r"
+shows "((a,b) \<in> r) = ((b,a) \<notin> r - Id)"
+using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force
+
+
+subsection {* The upper and lower bounds operators *}
+
+
+text{* Here we define upper (``above") and lower (``below") bounds operators.
+We think of @{text "r"} as a {\em non-strict} relation. The suffix ``S"
+at the names of some operators indicates that the bounds are strict -- e.g.,
+@{text "underS a"} is the set of all strict lower bounds of @{text "a"} (w.r.t. @{text "r"}).
+Capitalization of the first letter in the name reminds that the operator acts on sets, rather
+than on individual elements. *}
+
+definition under::"'a \<Rightarrow> 'a set"
+where "under a \<equiv> {b. (b,a) \<in> r}"
+
+definition underS::"'a \<Rightarrow> 'a set"
+where "underS a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}"
+
+definition Under::"'a set \<Rightarrow> 'a set"
+where "Under A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}"
+
+definition UnderS::"'a set \<Rightarrow> 'a set"
+where "UnderS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}"
+
+definition above::"'a \<Rightarrow> 'a set"
+where "above a \<equiv> {b. (a,b) \<in> r}"
+
+definition aboveS::"'a \<Rightarrow> 'a set"
+where "aboveS a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}"
+
+definition Above::"'a set \<Rightarrow> 'a set"
+where "Above A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}"
+
+definition AboveS::"'a set \<Rightarrow> 'a set"
+where "AboveS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a,b) \<in> r}"
+(* *)
+
+text{* Note: In the definitions of @{text "Above[S]"} and @{text "Under[S]"},
+ we bounded comprehension by @{text "Field r"} in order to properly cover
+ the case of @{text "A"} being empty. *}
+
+
+lemma UnderS_subset_Under: "UnderS A \<le> Under A"
+by(auto simp add: UnderS_def Under_def)
+
+
+lemma underS_subset_under: "underS a \<le> under a"
+by(auto simp add: underS_def under_def)
+
+
+lemma underS_notIn: "a \<notin> underS a"
+by(simp add: underS_def)
+
+
+lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under a"
+by(simp add: refl_on_def under_def)
+
+
+lemma AboveS_disjoint: "A Int (AboveS A) = {}"
+by(auto simp add: AboveS_def)
+
+
+lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS (underS a)"
+by(auto simp add: AboveS_def underS_def)
+
+
+lemma Refl_under_underS:
+assumes "Refl r" "a \<in> Field r"
+shows "under a = underS a \<union> {a}"
+unfolding under_def underS_def
+using assms refl_on_def[of _ r] by fastforce
+
+
+lemma underS_empty: "a \<notin> Field r \<Longrightarrow> underS a = {}"
+by (auto simp: Field_def underS_def)
+
+
+lemma under_Field: "under a \<le> Field r"
+by(unfold under_def Field_def, auto)
+
+
+lemma underS_Field: "underS a \<le> Field r"
+by(unfold underS_def Field_def, auto)
+
+
+lemma underS_Field2:
+"a \<in> Field r \<Longrightarrow> underS a < Field r"
+using assms underS_notIn underS_Field by blast
+
+
+lemma underS_Field3:
+"Field r \<noteq> {} \<Longrightarrow> underS a < Field r"
+by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty)
+
+
+lemma Under_Field: "Under A \<le> Field r"
+by(unfold Under_def Field_def, auto)
+
+
+lemma UnderS_Field: "UnderS A \<le> Field r"
+by(unfold UnderS_def Field_def, auto)
+
+
+lemma AboveS_Field: "AboveS A \<le> Field r"
+by(unfold AboveS_def Field_def, auto)
+
+
+lemma under_incr:
+assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
+shows "under a \<le> under b"
+proof(unfold under_def, auto)
+ fix x assume "(x,a) \<in> r"
+ with REL TRANS trans_def[of r]
+ show "(x,b) \<in> r" by blast
+qed
+
+
+lemma underS_incr:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ REL: "(a,b) \<in> r"
+shows "underS a \<le> underS b"
+proof(unfold underS_def, auto)
+ assume *: "b \<noteq> a" and **: "(b,a) \<in> r"
+ with ANTISYM antisym_def[of r] REL
+ show False by blast
+next
+ fix x assume "x \<noteq> a" "(x,a) \<in> r"
+ with REL TRANS trans_def[of r]
+ show "(x,b) \<in> r" by blast
+qed
+
+
+lemma underS_incl_iff:
+assumes LO: "Linear_order r" and
+ INa: "a \<in> Field r" and INb: "b \<in> Field r"
+shows "(underS a \<le> underS b) = ((a,b) \<in> r)"
+proof
+ assume "(a,b) \<in> r"
+ thus "underS a \<le> underS b" using LO
+ by (simp add: order_on_defs underS_incr)
+next
+ assume *: "underS a \<le> underS b"
+ {assume "a = b"
+ hence "(a,b) \<in> r" using assms
+ by (simp add: order_on_defs refl_on_def)
+ }
+ moreover
+ {assume "a \<noteq> b \<and> (b,a) \<in> r"
+ hence "b \<in> underS a" unfolding underS_def by blast
+ hence "b \<in> underS b" using * by blast
+ hence False by (simp add: underS_notIn)
+ }
+ ultimately
+ show "(a,b) \<in> r" using assms
+ order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast
+qed
+
+
+lemma under_Under_trans:
+assumes TRANS: "trans r" and
+ IN1: "a \<in> under b" and IN2: "b \<in> Under C"
+shows "a \<in> Under C"
+proof-
+ have "(a,b) \<in> r \<and> (\<forall>c \<in> C. (b,c) \<in> r)"
+ using IN1 IN2 under_def Under_def by blast
+ hence "\<forall>c \<in> C. (a,c) \<in> r"
+ using TRANS trans_def[of r] by blast
+ moreover
+ have "a \<in> Field r" using IN1 unfolding Field_def under_def by blast
+ ultimately
+ show ?thesis unfolding Under_def by blast
+qed
+
+
+lemma under_UnderS_trans:
+assumes TRANS: "trans r" and ANTISYM: "antisym r" and
+ IN1: "a \<in> under b" and IN2: "b \<in> UnderS C"
+shows "a \<in> UnderS C"
+proof-
+ from IN2 have "b \<in> Under C"
+ using UnderS_subset_Under[of C] by blast
+ with assms under_Under_trans
+ have "a \<in> Under C" by blast
+ (* *)
+ moreover
+ have "a \<notin> C"
+ proof
+ assume *: "a \<in> C"
+ have 1: "(a,b) \<in> r"
+ using IN1 under_def[of b] by auto
+ have "\<forall>c \<in> C. b \<noteq> c \<and> (b,c) \<in> r"
+ using IN2 UnderS_def[of C] by blast
+ with * have "b \<noteq> a \<and> (b,a) \<in> r" by blast
+ with 1 ANTISYM antisym_def[of r]
+ show False by blast
+ qed
+ (* *)
+ ultimately
+ show ?thesis unfolding UnderS_def Under_def by fast
+qed
+
+
+end (* context rel *)
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/README.txt Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,218 @@
+Authors: Andrei Popescu & Dmitriy Traytel
+
+
+PDF Documentation:
+-----------------
+
+See "document/root.pdf".
+
+
+
+Short description of the theories' main content:
+-----------------------------------------------
+
+The "minor" theories Fun_More, Wellfounded_More and Order_Relation_More are
+extensions of the existing theories Fun, Wellfounded and
+Order_Relation:
+-- Fun_More states more facts (mainly) concerning injections, bijections,
+inverses, and (numeric) cardinals of finite sets.
+-- Wellfounded_More states variations of well-founded
+recursion and well-founded recursion.
+-- Order_Relation_More fixes a relation, defines upper and lower bounds
+operators and proves many basic properties for these
+(depending on assumptions such as reflexivity or transitivity).
+
+The "major" theories are:
+-- Wellorder_Relation: Here one fixes a well-order relation, and then:
+----- 1) Defines the concepts of maximum (of two elements), minimum (of a set), supremum,
+ successor (of a set), and order filter (i.e., downwards closed set, a.k.a.
+ initial segment).
+-- Wellorder_Embedding:
+----- 2) For two well-order relations,
+ defines *well-order embeddings* as injective functions copying
+ the source into an order filter of the target and *compatible functions*
+ as those preserving the order. Also, *isomorphisms*
+ and *strict embeddings* are defined to be embeddings that are, and respectively
+ are not, bijections.
+
+-- Constructions_on_Wellorders:
+----- 1) Defines direct images, restrictions, disjoint unions and
+ bounded squares of well-orders.
+----- 2) Defines the relations "ordLeq", "ordLess" and "ordIso"
+ between well-order relations (concrete syntax: r <=o r', r <o r' and
+ r =o r', respectively), defined by the existence of an embedding,
+ strict embedding and isomorphism, respectively between the two members.
+ Among the properties proved for these relations:
+--------- ordLeq is total;
+--------- ordLess (on a fixed type) is well-founded.
+
+-- Cardinal_Order_Relation:
+---- 1) Defines a *cardinal order* to be a well-order minim w.r.t. "ordLeq"
+ (or, equivalently, minimal w.r.t. "ordLess").
+ ordLess being well-founded together with the well-ordering theorem (from theory Zorn.thy)
+ ensures the existence of a cardinal relation on any given set. In addition,
+ a cardinal relation on a set is unique up to order isomorphism.
+---- 2) Defines the cardinal of a set A, |A|, to be SOME cardinal
+ order on it (unique up to =o, according to the above).
+---- 3) Proves properties of cardinals, including their
+ interactions with sums, products, unions, lists,
+ powersets, sets of finite sets. Among them, nontrivial
+ facts concerning the invariance of infinite cardinals
+ under some of these constructs -- e.g., if A is infinite,
+ than the cardinal of lists/words over A is the same (up to
+ the "cardinal equality" =o) as that of A.
+---- 5) Studies the connection between the introduced (order-based) notion
+ of cardinal and the numeric one previously defined for
+ finite sets (operator "card"). On the way, one introduces the cardinal omega
+ (natLeq) and the finite cardinals (natLeq_on n).
+---- 6) Defines and proves the existence of successor cardinals.
+
+-- Cardinal_Arithmetic
+
+
+Here is a list of names of proved facts concerning cardinalities that are
+expressed independently of notions of order, and of potential interest
+for "working mathematicians":
+--- one_set_greater, one_type_greater (their proofs use the
+ fact that ordinals are totally ordered)
+--- Plus_into_Times, Plus_into_Times_types,
+ Plus_infinite_bij_betw, Plus_infinite_bij_betw_types,
+ Times_same_infinite_bij_betw, Times_same_infinite_bij_betw_types,
+ Times_infinite_bij_betw, Times_infinite_bij_betw_types
+ inj_on_UNION_infinite, List_infinite_bij_betw, List_infinite_bij_betw_types
+ Fpow_infinite_bij_betw
+ (their proofs employ cardinals)
+
+
+
+
+Minor technicalities and naming issues:
+---------------------------------------
+
+1. Most of the definitions and theorems are proved in files suffixed with
+"_Base". Bootstrapping considerations (for the (co)datatype package) made this
+division desirable.
+
+
+2. Even though we would have preferred to use "initial segment" instead of
+"order filter", we chose the latter to avoid terminological clash with
+the operator "init_seg_of" from Zorn.thy. The latter expresses a related, but different
+concept -- it considers a relation, rather than a set, as initial segment of a relation.
+
+
+3. We prefer to define the upper-bound operations under, underS,
+etc., as opposed to working with combinations of relation image,
+converse and diagonal, because the former seem more intuitive
+notations when we think of orderings (but of course we cannot
+define them as abbreviations, as this would have a global
+effect, also affecting cases where one does not think of relations
+as orders). Moreover, in my locales the relation parameter r for
+under, underS etc. is fixed, hence these operations can keep r
+implicit. To get a concrete glimpse at my aesthetic reason for
+introducing these operations: otherwise, instead of "underS a",
+we would have to write "(r - Id)^-1 `` {a}" or "r^-1 `` {a} - Id".
+
+
+4. Even though the main focus of this development are
+the well-order relations, we prove the basic results on order relations
+and bounds as generally as possible.
+To the contrary, the results concerning minima, suprema and successors
+are stated for well-order relations, not maximally generally.
+
+
+5. "Refl_on A r" requires in particular that "r <= A <*> A",
+and therefore whenever "Refl_on A r", we have that necessarily
+"A = Field r". This means that in a theory of orders the domain
+A would be redundant -- we decided not to include it explicitly
+for most of the tehory.
+
+
+6. An infinite ordinal/cardinal is one for which the field is infinite.
+We always prefer the slightly more verbose "finite (Field r)" to the more
+compact but less standard equivalent condition "finite r".
+
+
+7. After we proved lots of facts about injections and
+bijections, we discovered that a couple of
+fancier (set-restricted) version of some of them are proved in
+the theory FuncSet. However, we did not need here restricted
+abstraction and such, and felt we should not import the whole
+theory for just a couple of minor facts.
+
+
+
+
+
+
+
+Notes for anyone who would like to enrich these theories in the future
+--------------------------------------------------------------------------------------
+
+Theory Fun_More (and Fun_More_Base):
+- Careful: "inj" is an abbreviation for "inj_on UNIV", while
+ "bij" is not an abreviation for "bij_betw UNIV UNIV", but
+ a defined constant; there is no "surj_betw", but only "surj".
+ "inv" is an abbreviation for "inv_into UNIV"
+- In subsection "Purely functional properties":
+-- Recall lemma "comp_inj_on".
+-- A lemma for inj_on corresponding to "bij_betw_same_card" already exists, and is called "card_inj_on_le".
+- In subsection "Properties involving Hilbert choice":
+-- One should refrain from trying to prove "intuitive" properties of f conditioned
+ by properties of (inv_into f A), such as "bij_betw A' A (inv_into f A) ==> bij_betw A A' f".
+ They usually do not hold, since one cannot usually infer the well-definedness of "inv_into f A".
+- A lemma "bij_betw_inv_into_LEFT" -- why didn't
+"proof(auto simp add: bij_betw_inv_into_left)" finish the proof?
+-- Recall lemma "bij_betw_inv_into".
+- In subsection "Other facts":
+-- Does the lemma "atLeastLessThan_injective" already exist anywhere?
+
+Theory Order_Relation_More (and Order_Relation_More_Base):
+- In subsection "Auxiliaries":
+-- Recall the lemmas "order_on_defs", "Field_def", "Domain_def", "Range_def", "converse_def".
+-- Recall that "refl_on r A" forces r to not be defined outside A.
+ This is why "partial_order_def"
+ can afford to use non-parameterized versions of antisym and trans.
+-- Recall the ASCII notation for "converse r": "r ^-1".
+-- Recall the abbreviations:
+ abbreviation "Refl r ≡ refl_on (Field r) r"
+ abbreviation "Preorder r ≡ preorder_on (Field r) r"
+ abbreviation "Partial_order r ≡ partial_order_on (Field r) r"
+ abbreviation "Total r ≡ total_on (Field r) r"
+ abbreviation "Linear_order r ≡ linear_order_on (Field r) r"
+ abbreviation "Well_order r ≡ well_order_on (Field r) r"
+
+Theory Wellorder_Relation (and Wellorder_Relation_Base):
+- In subsection "Auxiliaries": recall lemmas "order_on_defs"
+- In subsection "The notions of maximum, minimum, supremum, successor and order filter":
+ Should we define all constants from "wo_rel" in "rel" instead,
+ so that their outside definition not be conditional in "wo_rel r"?
+
+Theory Wellfounded_More (and Wellfounded_More_Base):
+ Recall the lemmas "wfrec" and "wf_induct".
+
+Theory Wellorder_Embedding (and Wellorder_Embedding_Base):
+- Recall "inj_on_def" and "bij_betw_def".
+- Line 5 in the proof of lemma embed_in_Field: we have to figure out for this and many other
+ situations: Why did it work without annotations to Refl_under_in?
+- At the proof of theorem "wellorders_totally_ordered" (and, similarly, elsewhere):
+ Had we used metavariables instead of local definitions for H, f, g and test, the
+ goals (in the goal window) would have become unreadable,
+ making impossible to debug theorem instantiations.
+- At lemma "embed_unique": If we add the attribute "rule format" at lemma, we get an error at qed.
+
+Theory Constructions_on_Wellorders (and Constructions_on_Wellorders_Base):
+- Some of the lemmas in this section are about more general kinds of relations than
+ well-orders, but it is not clear whether they are useful in such more general contexts.
+- Recall that "equiv" does not have the "equiv_on" and "Equiv" versions,
+ like the order relation. "equiv" corresponds, for instance, to "well_order_on".
+- The lemmas "ord_trans" are not clearly useful together, as their employment within blast or auto
+tends to diverge.
+
+Theory Cardinal_Order_Relation (and Cardinal_Order_Relation_Base):
+- Careful: if "|..|" meets an outer parehthesis, an extra space needs to be inserted, as in
+ "( |A| )".
+- At lemma like ordLeq_Sigma_mono1: Not worth stating something like ordLeq_Sigma_mono2 --
+ would be a mere instance of card_of_Sigma_mono2.
+- At lemma ordLeq_Sigma_cong2: Again, no reason for stating something like ordLeq_Sigma_cong2.
+- At lemma Fpow_Pow_finite: why wouldn't a version of this lemma with "... Int finite"
+also be proved by blast?
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/TODO.txt Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,59 @@
+-1. At the new Isabelle release (after 2009-2), replace the operator List
+from Cardinal_Order_Relation with "lists" from List.thy in standard library.
+A lemma for "lists" is the actual definition of "List".
+
+0. Add:
+
+lemma finite_iff_cardOf_nat:
+"finite A = ( |A| <o |UNIV :: nat set| )"
+using infinite_iff_card_of_nat[of A]
+by (auto simp add: card_of_Well_order ordLess_iff_not_ordLeq)
+
+
+
+
+1. Remember that the version from my computer has a few more theorems than the one
+from the Isabelle site.
+
+2. Many basic cardinal arithmetic facts can be listed as simps!
+The the proofs can be simplified.
+
+3. Add:
+
+lemma card_of_Plus_ordLeq_infinite:
+assumes "infinite C" and "|A| );£o |C|" and "|B| £o |C|"-A
+shows "|A <+> B| );£o |C|"-A
+proof-
+ let ?phi = "(EX a1 a2. a1 ~= a2 );Ù {a1,a2} <= A) Ù -A
+ (EX b1 b2. b1 ~= b2 );Ù {b1,b2} <= B)"-A
+ show ?thesis
+ proof(cases ?phi, auto)
+ fix a1 b1 a2 b2
+ assume "a1 );¹ a2" "b1 ¹ b2" "a1 Î A" "a2 Î A" "b1 Î B" "b2 Î B"-A
+ hence "|A <+> B| <=o |A <*> B|"
+ apply - apply(rule card_of_Plus_Times) by auto
+ moreover have "|A <*> B| );£o |C|"-A
+ using assms by (auto simp add: card_of_Times_ordLeq_infinite)
+ ultimately show ?thesis using ordLeq_transitive by blast
+ next
+ assume "-=Õa1. a1 );Î A (<_(B (-=Õa2. a1 = a2 );Ú a2 Ï A)"-A
+ then obtain a where "A <= {a}" by blast
+ {assume "A = {}"
+ hence "|A <+> B| =o |B|"
+ using ordIso_symmetric card_of_Plus_empty2 by blast
+ hence ?thesis using assms ordIso_ordLeq_trans by blast
+ }
+ moreover
+ {assume "A = {a}"
+
+ }
+ qed
+qed
+
+
+lemma card_of_Plus_ordLess_infinite:
+assumes "infinite C" and "|A| <o |C|" and "|B| <o |C|"
+shows "|A <+> B| <o |C|"
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Wellfounded_More.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,52 @@
+(* Title: HOL/Ordinals_and_Cardinals/Wellfounded_More.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+More on well-founded relations.
+*)
+
+header {* More on Well-Founded Relations *}
+
+theory Wellfounded_More
+imports
+ "../Ordinals_and_Cardinals/Wellfounded_More_Base"
+ Order_Relation_More
+begin
+
+
+subsection {* Well-founded recursion via genuine fixpoints *}
+
+(*2*)lemma adm_wf_unique_fixpoint:
+fixes r :: "('a * 'a) set" and
+ H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" and
+ f :: "'a \<Rightarrow> 'b" and g :: "'a \<Rightarrow> 'b"
+assumes WF: "wf r" and ADM: "adm_wf r H" and fFP: "f = H f" and gFP: "g = H g"
+shows "f = g"
+proof-
+ {fix x
+ have "f x = g x"
+ proof(rule wf_induct[of r "(\<lambda>x. f x = g x)"],
+ auto simp add: WF)
+ fix x assume "\<forall>y. (y, x) \<in> r \<longrightarrow> f y = g y"
+ hence "H f x = H g x" using ADM adm_wf_def[of r H] by auto
+ thus "f x = g x" using fFP and gFP by simp
+ qed
+ }
+ thus ?thesis by (simp add: ext)
+qed
+
+(*2*)lemma wfrec_unique_fixpoint:
+fixes r :: "('a * 'a) set" and
+ H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" and
+ f :: "'a \<Rightarrow> 'b"
+assumes WF: "wf r" and ADM: "adm_wf r H" and
+ fp: "f = H f"
+shows "f = wfrec r H"
+proof-
+ have "H (wfrec r H) = wfrec r H"
+ using assms wfrec_fixpoint[of r H] by simp
+ thus ?thesis
+ using assms adm_wf_unique_fixpoint[of r H "wfrec r H"] by simp
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Wellfounded_More_Base.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,194 @@
+(* Title: HOL/Ordinals_and_Cardinals/Wellfounded_More_Base.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+More on well-founded relations (base).
+*)
+
+header {* More on Well-Founded Relations (Base) *}
+
+theory Wellfounded_More_Base
+imports Wellfounded Order_Relation_More_Base "~~/src/HOL/Library/Wfrec"
+begin
+
+
+text {* This section contains some variations of results in the theory
+@{text "Wellfounded.thy"}:
+\begin{itemize}
+\item means for slightly more direct definitions by well-founded recursion;
+\item variations of well-founded induction;
+\item means for proving a linear order to be a well-order.
+\end{itemize} *}
+
+
+subsection {* Well-founded recursion via genuine fixpoints *}
+
+
+(*2*)lemma wfrec_fixpoint:
+fixes r :: "('a * 'a) set" and
+ H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+assumes WF: "wf r" and ADM: "adm_wf r H"
+shows "wfrec r H = H (wfrec r H)"
+proof(rule ext)
+ fix x
+ have "wfrec r H x = H (cut (wfrec r H) r x) x"
+ using wfrec[of r H] WF by simp
+ also
+ {have "\<And> y. (y,x) : r \<Longrightarrow> (cut (wfrec r H) r x) y = (wfrec r H) y"
+ by (auto simp add: cut_apply)
+ hence "H (cut (wfrec r H) r x) x = H (wfrec r H) x"
+ using ADM adm_wf_def[of r H] by auto
+ }
+ finally show "wfrec r H x = H (wfrec r H) x" .
+qed
+
+
+
+subsection {* Characterizations of well-founded-ness *}
+
+
+text {* A transitive relation is well-founded iff it is ``locally" well-founded,
+i.e., iff its restriction to the lower bounds of of any element is well-founded. *}
+
+(*3*)lemma trans_wf_iff:
+assumes "trans r"
+shows "wf r = (\<forall>a. wf(r Int (r^-1``{a} \<times> r^-1``{a})))"
+proof-
+ obtain R where R_def: "R = (\<lambda> a. r Int (r^-1``{a} \<times> r^-1``{a}))" by blast
+ {assume *: "wf r"
+ {fix a
+ have "wf(R a)"
+ using * R_def wf_subset[of r "R a"] by auto
+ }
+ }
+ (* *)
+ moreover
+ {assume *: "\<forall>a. wf(R a)"
+ have "wf r"
+ proof(unfold wf_def, clarify)
+ fix phi a
+ assume **: "\<forall>a. (\<forall>b. (b,a) \<in> r \<longrightarrow> phi b) \<longrightarrow> phi a"
+ obtain chi where chi_def: "chi = (\<lambda>b. (b,a) \<in> r \<longrightarrow> phi b)" by blast
+ with * have "wf (R a)" by auto
+ hence "(\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b) \<longrightarrow> (\<forall>b. chi b)"
+ unfolding wf_def by blast
+ moreover
+ have "\<forall>b. (\<forall>c. (c,b) \<in> R a \<longrightarrow> chi c) \<longrightarrow> chi b"
+ proof(auto simp add: chi_def R_def)
+ fix b
+ assume 1: "(b,a) \<in> r" and 2: "\<forall>c. (c, b) \<in> r \<and> (c, a) \<in> r \<longrightarrow> phi c"
+ hence "\<forall>c. (c, b) \<in> r \<longrightarrow> phi c"
+ using assms trans_def[of r] by blast
+ thus "phi b" using ** by blast
+ qed
+ ultimately have "\<forall>b. chi b" by (rule mp)
+ with ** chi_def show "phi a" by blast
+ qed
+ }
+ ultimately show ?thesis using R_def by blast
+qed
+
+
+text {* The next lemma is a variation of @{text "wf_eq_minimal"} from Wellfounded,
+allowing one to assume the set included in the field. *}
+
+(*2*)lemma wf_eq_minimal2:
+"wf r = (\<forall>A. A <= Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r))"
+proof-
+ let ?phi = "\<lambda> A. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. \<not> (a',a) \<in> r)"
+ have "wf r = (\<forall>A. ?phi A)"
+ by (auto simp: ex_in_conv [THEN sym], erule wfE_min, assumption, blast)
+ (rule wfI_min, metis)
+ (* *)
+ also have "(\<forall>A. ?phi A) = (\<forall>B \<le> Field r. ?phi B)"
+ proof
+ assume "\<forall>A. ?phi A"
+ thus "\<forall>B \<le> Field r. ?phi B" by simp
+ next
+ assume *: "\<forall>B \<le> Field r. ?phi B"
+ show "\<forall>A. ?phi A"
+ proof(clarify)
+ fix A::"'a set" assume **: "A \<noteq> {}"
+ obtain B where B_def: "B = A Int (Field r)" by blast
+ show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r"
+ proof(cases "B = {}")
+ assume Case1: "B = {}"
+ obtain a where 1: "a \<in> A \<and> a \<notin> Field r"
+ using ** Case1 unfolding B_def by blast
+ hence "\<forall>a' \<in> A. (a',a) \<notin> r" using 1 unfolding Field_def by blast
+ thus ?thesis using 1 by blast
+ next
+ assume Case2: "B \<noteq> {}" have 1: "B \<le> Field r" unfolding B_def by blast
+ obtain a where 2: "a \<in> B \<and> (\<forall>a' \<in> B. (a',a) \<notin> r)"
+ using Case2 1 * by blast
+ have "\<forall>a' \<in> A. (a',a) \<notin> r"
+ proof(clarify)
+ fix a' assume "a' \<in> A" and **: "(a',a) \<in> r"
+ hence "a' \<in> B" unfolding B_def Field_def by blast
+ thus False using 2 ** by blast
+ qed
+ thus ?thesis using 2 unfolding B_def by blast
+ qed
+ qed
+ qed
+ finally show ?thesis by blast
+qed
+
+subsection {* Characterizations of well-founded-ness *}
+
+text {* The next lemma and its corollary enable one to prove that
+a linear order is a well-order in a way which is more standard than
+via well-founded-ness of the strict version of the relation. *}
+
+(*3*)
+lemma Linear_order_wf_diff_Id:
+assumes LI: "Linear_order r"
+shows "wf(r - Id) = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
+proof(cases "r \<le> Id")
+ assume Case1: "r \<le> Id"
+ hence temp: "r - Id = {}" by blast
+ hence "wf(r - Id)" by (simp add: temp)
+ moreover
+ {fix A assume *: "A \<le> Field r" and **: "A \<noteq> {}"
+ obtain a where 1: "r = {} \<or> r = {(a,a)}" using LI
+ unfolding order_on_defs using Case1 rel.Total_subset_Id by metis
+ hence "A = {a} \<and> r = {(a,a)}" using * ** unfolding Field_def by blast
+ hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" using 1 by blast
+ }
+ ultimately show ?thesis by blast
+next
+ assume Case2: "\<not> r \<le> Id"
+ hence 1: "Field r = Field(r - Id)" using rel.Total_Id_Field LI
+ unfolding order_on_defs by blast
+ show ?thesis
+ proof
+ assume *: "wf(r - Id)"
+ show "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+ proof(clarify)
+ fix A assume **: "A \<le> Field r" and ***: "A \<noteq> {}"
+ hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id"
+ using 1 * unfolding wf_eq_minimal2 by simp
+ moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
+ using rel.Linear_order_in_diff_Id[of r] ** LI by blast
+ ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r" by blast
+ qed
+ next
+ assume *: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+ show "wf(r - Id)"
+ proof(unfold wf_eq_minimal2, clarify)
+ fix A assume **: "A \<le> Field(r - Id)" and ***: "A \<noteq> {}"
+ hence "\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r"
+ using 1 * by simp
+ moreover have "\<forall>a \<in> A. \<forall>a' \<in> A. ((a,a') \<in> r) = ((a',a) \<notin> r - Id)"
+ using rel.Linear_order_in_diff_Id[of r] ** LI mono_Field[of "r - Id" r] by blast
+ ultimately show "\<exists>a \<in> A. \<forall>a' \<in> A. (a',a) \<notin> r - Id" by blast
+ qed
+ qed
+qed
+
+(*3*)corollary Linear_order_Well_order_iff:
+assumes "Linear_order r"
+shows "Well_order r = (\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r))"
+using assms unfolding well_order_on_def using Linear_order_wf_diff_Id[of r] by blast
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Wellorder_Embedding.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,178 @@
+(* Title: HOL/Ordinals_and_Cardinals/Wellorder_Embedding.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Well-order embeddings (full).
+*)
+
+header {* Well-Order Embeddings (Full) *}
+
+theory Wellorder_Embedding
+imports
+ "../Ordinals_and_Cardinals_Base/Wellorder_Embedding_Base"
+ Fun_More
+ Wellorder_Relation
+begin
+
+
+subsection {* Auxiliaries *}
+
+lemma UNION_bij_betw_ofilter:
+assumes WELL: "Well_order r" and
+ OF: "\<And> i. i \<in> I \<Longrightarrow> ofilter r (A i)" and
+ BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
+shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
+proof-
+ have "wo_rel r" using WELL by (simp add: wo_rel_def)
+ hence "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i"
+ using wo_rel.ofilter_linord[of r] OF by blast
+ with WELL BIJ show ?thesis
+ by (auto simp add: bij_betw_UNION_chain)
+qed
+
+
+subsection {* (Well-order) embeddings, strict embeddings, isomorphisms and order-compatible
+functions *}
+
+lemma embed_halfcong:
+assumes EQ: "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a" and
+ EMB: "embed r r' f"
+shows "embed r r' g"
+proof(unfold embed_def, auto)
+ fix a assume *: "a \<in> Field r"
+ hence "bij_betw f (under r a) (under r' (f a))"
+ using EMB unfolding embed_def by simp
+ moreover
+ {have "under r a \<le> Field r"
+ by (auto simp add: rel.under_Field)
+ hence "\<And> b. b \<in> under r a \<Longrightarrow> f b = g b"
+ using EQ by blast
+ }
+ moreover have "f a = g a" using * EQ by auto
+ ultimately show "bij_betw g (under r a) (under r' (g a))"
+ using bij_betw_cong[of "under r a" f g "under r' (f a)"] by auto
+qed
+
+lemma embed_cong[fundef_cong]:
+assumes "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a"
+shows "embed r r' f = embed r r' g"
+using assms embed_halfcong[of r f g r']
+ embed_halfcong[of r g f r'] by auto
+
+lemma embedS_cong[fundef_cong]:
+assumes "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a"
+shows "embedS r r' f = embedS r r' g"
+unfolding embedS_def using assms
+embed_cong[of r f g r'] bij_betw_cong[of "Field r" f g "Field r'"] by blast
+
+lemma iso_cong[fundef_cong]:
+assumes "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a"
+shows "iso r r' f = iso r r' g"
+unfolding iso_def using assms
+embed_cong[of r f g r'] bij_betw_cong[of "Field r" f g "Field r'"] by blast
+
+lemma id_compat: "compat r r id"
+by(auto simp add: id_def compat_def)
+
+lemma comp_compat:
+"\<lbrakk>compat r r' f; compat r' r'' f'\<rbrakk> \<Longrightarrow> compat r r'' (f' o f)"
+by(auto simp add: comp_def compat_def)
+
+corollary one_set_greater:
+"(\<exists>f::'a \<Rightarrow> 'a'. f ` A \<le> A' \<and> inj_on f A) \<or> (\<exists>g::'a' \<Rightarrow> 'a. g ` A' \<le> A \<and> inj_on g A')"
+proof-
+ obtain r where "well_order_on A r" by (fastforce simp add: well_order_on)
+ hence 1: "A = Field r \<and> Well_order r"
+ using rel.well_order_on_Well_order by auto
+ obtain r' where 2: "well_order_on A' r'" by (fastforce simp add: well_order_on)
+ hence 2: "A' = Field r' \<and> Well_order r'"
+ using rel.well_order_on_Well_order by auto
+ hence "(\<exists>f. embed r r' f) \<or> (\<exists>g. embed r' r g)"
+ using 1 2 by (auto simp add: wellorders_totally_ordered)
+ moreover
+ {fix f assume "embed r r' f"
+ hence "f`A \<le> A' \<and> inj_on f A"
+ using 1 2 by (auto simp add: embed_Field embed_inj_on)
+ }
+ moreover
+ {fix g assume "embed r' r g"
+ hence "g`A' \<le> A \<and> inj_on g A'"
+ using 1 2 by (auto simp add: embed_Field embed_inj_on)
+ }
+ ultimately show ?thesis by blast
+qed
+
+corollary one_type_greater:
+"(\<exists>f::'a \<Rightarrow> 'a'. inj f) \<or> (\<exists>g::'a' \<Rightarrow> 'a. inj g)"
+using one_set_greater[of UNIV UNIV] by auto
+
+
+subsection {* Uniqueness of embeddings *}
+
+lemma comp_embedS:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embedS r r' f" and EMB': "embedS r' r'' f'"
+shows "embedS r r'' (f' o f)"
+proof-
+ have "embed r' r'' f'" using EMB' unfolding embedS_def by simp
+ thus ?thesis using assms by (auto simp add: embedS_comp_embed)
+qed
+
+lemma iso_iff4:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "iso r r' f = (embed r r' f \<and> embed r' r (inv_into (Field r) f))"
+using assms embed_bothWays_iso
+by(unfold iso_def, auto simp add: inv_into_Field_embed_bij_betw)
+
+lemma embed_embedS_iso:
+"embed r r' f = (embedS r r' f \<or> iso r r' f)"
+unfolding embedS_def iso_def by blast
+
+lemma not_embedS_iso:
+"\<not> (embedS r r' f \<and> iso r r' f)"
+unfolding embedS_def iso_def by blast
+
+lemma embed_embedS_iff_not_iso:
+assumes "embed r r' f"
+shows "embedS r r' f = (\<not> iso r r' f)"
+using assms unfolding embedS_def iso_def by blast
+
+lemma iso_inv_into:
+assumes WELL: "Well_order r" and ISO: "iso r r' f"
+shows "iso r' r (inv_into (Field r) f)"
+using assms unfolding iso_def
+using bij_betw_inv_into inv_into_Field_embed_bij_betw by blast
+
+lemma embedS_or_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<exists>g. embedS r r' g) \<or> (\<exists>h. embedS r' r h) \<or> (\<exists>f. iso r r' f)"
+proof-
+ {fix f assume *: "embed r r' f"
+ {assume "bij_betw f (Field r) (Field r')"
+ hence ?thesis using * by (auto simp add: iso_def)
+ }
+ moreover
+ {assume "\<not> bij_betw f (Field r) (Field r')"
+ hence ?thesis using * by (auto simp add: embedS_def)
+ }
+ ultimately have ?thesis by auto
+ }
+ moreover
+ {fix f assume *: "embed r' r f"
+ {assume "bij_betw f (Field r') (Field r)"
+ hence "iso r' r f" using * by (auto simp add: iso_def)
+ hence "iso r r' (inv_into (Field r') f)"
+ using WELL' by (auto simp add: iso_inv_into)
+ hence ?thesis by blast
+ }
+ moreover
+ {assume "\<not> bij_betw f (Field r') (Field r)"
+ hence ?thesis using * by (auto simp add: embedS_def)
+ }
+ ultimately have ?thesis by auto
+ }
+ ultimately show ?thesis using WELL WELL'
+ wellorders_totally_ordered[of r r'] by blast
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Wellorder_Embedding_Base.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1149 @@
+(* Title: HOL/Ordinals_and_Cardinals/Wellorder_Embedding_Base.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Well-order embeddings (base).
+*)
+
+header {* Well-Order Embeddings (Base) *}
+
+theory Wellorder_Embedding_Base
+imports
+ "~~/src/HOL/Library/Zorn"
+ Fun_More_Base
+ Wellorder_Relation_Base
+begin
+
+
+text{* In this section, we introduce well-order {\em embeddings} and {\em isomorphisms} and
+prove their basic properties. The notion of embedding is considered from the point
+of view of the theory of ordinals, and therefore requires the source to be injected
+as an {\em initial segment} (i.e., {\em order filter}) of the target. A main result
+of this section is the existence of embeddings (in one direction or another) between
+any two well-orders, having as a consequence the fact that, given any two sets on
+any two types, one is smaller than (i.e., can be injected into) the other. *}
+
+
+subsection {* Auxiliaries *}
+
+lemma UNION_inj_on_ofilter:
+assumes WELL: "Well_order r" and
+ OF: "\<And> i. i \<in> I \<Longrightarrow> wo_rel.ofilter r (A i)" and
+ INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
+shows "inj_on f (\<Union> i \<in> I. A i)"
+proof-
+ have "wo_rel r" using WELL by (simp add: wo_rel_def)
+ hence "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i <= A j \<or> A j <= A i"
+ using wo_rel.ofilter_linord[of r] OF by blast
+ with WELL INJ show ?thesis
+ by (auto simp add: inj_on_UNION_chain)
+qed
+
+
+lemma under_underS_bij_betw:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ IN: "a \<in> Field r" and IN': "f a \<in> Field r'" and
+ BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
+proof-
+ have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
+ unfolding rel.underS_def by auto
+ moreover
+ {have "Refl r \<and> Refl r'" using WELL WELL'
+ by (auto simp add: order_on_defs)
+ hence "rel.under r a = rel.underS r a \<union> {a} \<and>
+ rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
+ using IN IN' by(auto simp add: rel.Refl_under_underS)
+ }
+ ultimately show ?thesis
+ using BIJ notIn_Un_bij_betw[of a "rel.underS r a" f "rel.underS r' (f a)"] by auto
+qed
+
+
+
+subsection {* (Well-order) embeddings, strict embeddings, isomorphisms and order-compatible
+functions *}
+
+
+text{* Standardly, a function is an embedding of a well-order in another if it injectively and
+order-compatibly maps the former into an order filter of the latter.
+Here we opt for a more succinct definition (operator @{text "embed"}),
+asking that, for any element in the source, the function should be a bijection
+between the set of strict lower bounds of that element
+and the set of strict lower bounds of its image. (Later we prove equivalence with
+the standard definition -- lemma @{text "embed_iff_compat_inj_on_ofilter"}.)
+A {\em strict embedding} (operator @{text "embedS"}) is a non-bijective embedding
+and an isomorphism (operator @{text "iso"}) is a bijective embedding. *}
+
+
+definition embed :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"embed r r' f \<equiv> \<forall>a \<in> Field r. bij_betw f (rel.under r a) (rel.under r' (f a))"
+
+
+lemmas embed_defs = embed_def embed_def[abs_def]
+
+
+text {* Strict embeddings: *}
+
+definition embedS :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
+
+
+lemmas embedS_defs = embedS_def embedS_def[abs_def]
+
+
+definition iso :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
+
+
+lemmas iso_defs = iso_def iso_def[abs_def]
+
+
+definition compat :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> bool"
+where
+"compat r r' f \<equiv> \<forall>a b. (a,b) \<in> r \<longrightarrow> (f a, f b) \<in> r'"
+
+
+lemma compat_wf:
+assumes CMP: "compat r r' f" and WF: "wf r'"
+shows "wf r"
+proof-
+ have "r \<le> inv_image r' f"
+ unfolding inv_image_def using CMP
+ by (auto simp add: compat_def)
+ with WF show ?thesis
+ using wf_inv_image[of r' f] wf_subset[of "inv_image r' f"] by auto
+qed
+
+
+lemma id_embed: "embed r r id"
+by(auto simp add: id_def embed_def bij_betw_def)
+
+
+lemma id_iso: "iso r r id"
+by(auto simp add: id_def embed_def iso_def bij_betw_def)
+
+
+lemma embed_in_Field:
+assumes WELL: "Well_order r" and
+ EMB: "embed r r' f" and IN: "a \<in> Field r"
+shows "f a \<in> Field r'"
+proof-
+ have Well: "wo_rel r"
+ using WELL by (auto simp add: wo_rel_def)
+ hence 1: "Refl r"
+ by (auto simp add: wo_rel.REFL)
+ hence "a \<in> rel.under r a" using IN rel.Refl_under_in by fastforce
+ hence "f a \<in> rel.under r' (f a)"
+ using EMB IN by (auto simp add: embed_def bij_betw_def)
+ thus ?thesis unfolding Field_def
+ by (auto simp: rel.under_def)
+qed
+
+
+lemma comp_embed:
+assumes WELL: "Well_order r" and
+ EMB: "embed r r' f" and EMB': "embed r' r'' f'"
+shows "embed r r'' (f' o f)"
+proof(unfold embed_def, auto)
+ fix a assume *: "a \<in> Field r"
+ hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using embed_def[of r] EMB by auto
+ moreover
+ {have "f a \<in> Field r'"
+ using EMB WELL * by (auto simp add: embed_in_Field)
+ hence "bij_betw f' (rel.under r' (f a)) (rel.under r'' (f' (f a)))"
+ using embed_def[of r'] EMB' by auto
+ }
+ ultimately
+ show "bij_betw (f' \<circ> f) (rel.under r a) (rel.under r'' (f'(f a)))"
+ by(auto simp add: bij_betw_trans)
+qed
+
+
+lemma comp_iso:
+assumes WELL: "Well_order r" and
+ EMB: "iso r r' f" and EMB': "iso r' r'' f'"
+shows "iso r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: comp_embed bij_betw_trans)
+
+
+text{* That @{text "embedS"} is also preserved by function composition shall be proved only later. *}
+
+
+lemma embed_Field:
+"\<lbrakk>Well_order r; embed r r' f\<rbrakk> \<Longrightarrow> f`(Field r) \<le> Field r'"
+by (auto simp add: embed_in_Field)
+
+
+lemma embed_preserves_ofilter:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and OF: "wo_rel.ofilter r A"
+shows "wo_rel.ofilter r' (f`A)"
+proof-
+ (* Preliminary facts *)
+ from WELL have Well: "wo_rel r" unfolding wo_rel_def .
+ from WELL' have Well': "wo_rel r'" unfolding wo_rel_def .
+ from OF have 0: "A \<le> Field r" by(auto simp add: Well wo_rel.ofilter_def)
+ (* Main proof *)
+ show ?thesis using Well' WELL EMB 0 embed_Field[of r r' f]
+ proof(unfold wo_rel.ofilter_def, auto simp add: image_def)
+ fix a b'
+ assume *: "a \<in> A" and **: "b' \<in> rel.under r' (f a)"
+ hence "a \<in> Field r" using 0 by auto
+ hence "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using * EMB by (auto simp add: embed_def)
+ hence "f`(rel.under r a) = rel.under r' (f a)"
+ by (simp add: bij_betw_def)
+ with ** image_def[of f "rel.under r a"] obtain b where
+ 1: "b \<in> rel.under r a \<and> b' = f b" by blast
+ hence "b \<in> A" using Well * OF
+ by (auto simp add: wo_rel.ofilter_def)
+ with 1 show "\<exists>b \<in> A. b' = f b" by blast
+ qed
+qed
+
+
+lemma embed_Field_ofilter:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f"
+shows "wo_rel.ofilter r' (f`(Field r))"
+proof-
+ have "wo_rel.ofilter r (Field r)"
+ using WELL by (auto simp add: wo_rel_def wo_rel.Field_ofilter)
+ with WELL WELL' EMB
+ show ?thesis by (auto simp add: embed_preserves_ofilter)
+qed
+
+
+lemma embed_compat:
+assumes EMB: "embed r r' f"
+shows "compat r r' f"
+proof(unfold compat_def, clarify)
+ fix a b
+ assume *: "(a,b) \<in> r"
+ hence 1: "b \<in> Field r" using Field_def[of r] by blast
+ have "a \<in> rel.under r b"
+ using * rel.under_def[of r] by simp
+ hence "f a \<in> rel.under r' (f b)"
+ using EMB embed_def[of r r' f]
+ bij_betw_def[of f "rel.under r b" "rel.under r' (f b)"]
+ image_def[of f "rel.under r b"] 1 by auto
+ thus "(f a, f b) \<in> r'"
+ by (auto simp add: rel.under_def)
+qed
+
+
+lemma embed_inj_on:
+assumes WELL: "Well_order r" and EMB: "embed r r' f"
+shows "inj_on f (Field r)"
+proof(unfold inj_on_def, clarify)
+ (* Preliminary facts *)
+ from WELL have Well: "wo_rel r" unfolding wo_rel_def .
+ with wo_rel.TOTAL[of r]
+ have Total: "Total r" by simp
+ from Well wo_rel.REFL[of r]
+ have Refl: "Refl r" by simp
+ (* Main proof *)
+ fix a b
+ assume *: "a \<in> Field r" and **: "b \<in> Field r" and
+ ***: "f a = f b"
+ hence 1: "a \<in> Field r \<and> b \<in> Field r"
+ unfolding Field_def by auto
+ {assume "(a,b) \<in> r"
+ hence "a \<in> rel.under r b \<and> b \<in> rel.under r b"
+ using Refl by(auto simp add: rel.under_def refl_on_def)
+ hence "a = b"
+ using EMB 1 ***
+ by (auto simp add: embed_def bij_betw_def inj_on_def)
+ }
+ moreover
+ {assume "(b,a) \<in> r"
+ hence "a \<in> rel.under r a \<and> b \<in> rel.under r a"
+ using Refl by(auto simp add: rel.under_def refl_on_def)
+ hence "a = b"
+ using EMB 1 ***
+ by (auto simp add: embed_def bij_betw_def inj_on_def)
+ }
+ ultimately
+ show "a = b" using Total 1
+ by (auto simp add: total_on_def)
+qed
+
+
+lemma embed_underS:
+assumes WELL: "Well_order r" and WELL: "Well_order r'" and
+ EMB: "embed r r' f" and IN: "a \<in> Field r"
+shows "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+proof-
+ have "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using assms by (auto simp add: embed_def)
+ moreover
+ {have "f a \<in> Field r'" using assms embed_Field[of r r' f] by auto
+ hence "rel.under r a = rel.underS r a \<union> {a} \<and>
+ rel.under r' (f a) = rel.underS r' (f a) \<union> {f a}"
+ using assms by (auto simp add: order_on_defs rel.Refl_under_underS)
+ }
+ moreover
+ {have "a \<notin> rel.underS r a \<and> f a \<notin> rel.underS r' (f a)"
+ unfolding rel.underS_def by blast
+ }
+ ultimately show ?thesis
+ by (auto simp add: notIn_Un_bij_betw3)
+qed
+
+
+lemma embed_iff_compat_inj_on_ofilter:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "embed r r' f = (compat r r' f \<and> inj_on f (Field r) \<and> wo_rel.ofilter r' (f`(Field r)))"
+using assms
+proof(auto simp add: embed_compat embed_inj_on embed_Field_ofilter,
+ unfold embed_def, auto) (* get rid of one implication *)
+ fix a
+ assume *: "inj_on f (Field r)" and
+ **: "compat r r' f" and
+ ***: "wo_rel.ofilter r' (f`(Field r))" and
+ ****: "a \<in> Field r"
+ (* Preliminary facts *)
+ have Well: "wo_rel r"
+ using WELL wo_rel_def[of r] by simp
+ hence Refl: "Refl r"
+ using wo_rel.REFL[of r] by simp
+ have Total: "Total r"
+ using Well wo_rel.TOTAL[of r] by simp
+ have Well': "wo_rel r'"
+ using WELL' wo_rel_def[of r'] by simp
+ hence Antisym': "antisym r'"
+ using wo_rel.ANTISYM[of r'] by simp
+ have "(a,a) \<in> r"
+ using **** Well wo_rel.REFL[of r]
+ refl_on_def[of _ r] by auto
+ hence "(f a, f a) \<in> r'"
+ using ** by(auto simp add: compat_def)
+ hence 0: "f a \<in> Field r'"
+ unfolding Field_def by auto
+ have "f a \<in> f`(Field r)"
+ using **** by auto
+ hence 2: "rel.under r' (f a) \<le> f`(Field r)"
+ using Well' *** wo_rel.ofilter_def[of r' "f`(Field r)"] by fastforce
+ (* Main proof *)
+ show "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ proof(unfold bij_betw_def, auto)
+ show "inj_on f (rel.under r a)"
+ using *
+ by (auto simp add: rel.under_Field subset_inj_on)
+ next
+ fix b assume "b \<in> rel.under r a"
+ thus "f b \<in> rel.under r' (f a)"
+ unfolding rel.under_def using **
+ by (auto simp add: compat_def)
+ next
+ fix b' assume *****: "b' \<in> rel.under r' (f a)"
+ hence "b' \<in> f`(Field r)"
+ using 2 by auto
+ with Field_def[of r] obtain b where
+ 3: "b \<in> Field r" and 4: "b' = f b" by auto
+ have "(b,a): r"
+ proof-
+ {assume "(a,b) \<in> r"
+ with ** 4 have "(f a, b'): r'"
+ by (auto simp add: compat_def)
+ with ***** Antisym' have "f a = b'"
+ by(auto simp add: rel.under_def antisym_def)
+ with 3 **** 4 * have "a = b"
+ by(auto simp add: inj_on_def)
+ }
+ moreover
+ {assume "a = b"
+ hence "(b,a) \<in> r" using Refl **** 3
+ by (auto simp add: refl_on_def)
+ }
+ ultimately
+ show ?thesis using Total **** 3 by (fastforce simp add: total_on_def)
+ qed
+ with 4 show "b' \<in> f`(rel.under r a)"
+ unfolding rel.under_def by auto
+ qed
+qed
+
+
+lemma inv_into_ofilter_embed:
+assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and
+ BIJ: "\<forall>b \<in> A. bij_betw f (rel.under r b) (rel.under r' (f b))" and
+ IMAGE: "f ` A = Field r'"
+shows "embed r' r (inv_into A f)"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r"
+ using WELL wo_rel_def[of r] by simp
+ have Refl: "Refl r"
+ using Well wo_rel.REFL[of r] by simp
+ have Total: "Total r"
+ using Well wo_rel.TOTAL[of r] by simp
+ (* Main proof *)
+ have 1: "bij_betw f A (Field r')"
+ proof(unfold bij_betw_def inj_on_def, auto simp add: IMAGE)
+ fix b1 b2
+ assume *: "b1 \<in> A" and **: "b2 \<in> A" and
+ ***: "f b1 = f b2"
+ have 11: "b1 \<in> Field r \<and> b2 \<in> Field r"
+ using * ** Well OF by (auto simp add: wo_rel.ofilter_def)
+ moreover
+ {assume "(b1,b2) \<in> r"
+ hence "b1 \<in> rel.under r b2 \<and> b2 \<in> rel.under r b2"
+ unfolding rel.under_def using 11 Refl
+ by (auto simp add: refl_on_def)
+ hence "b1 = b2" using BIJ * ** ***
+ by (auto simp add: bij_betw_def inj_on_def)
+ }
+ moreover
+ {assume "(b2,b1) \<in> r"
+ hence "b1 \<in> rel.under r b1 \<and> b2 \<in> rel.under r b1"
+ unfolding rel.under_def using 11 Refl
+ by (auto simp add: refl_on_def)
+ hence "b1 = b2" using BIJ * ** ***
+ by (auto simp add: bij_betw_def inj_on_def)
+ }
+ ultimately
+ show "b1 = b2"
+ using Total by (auto simp add: total_on_def)
+ qed
+ (* *)
+ let ?f' = "(inv_into A f)"
+ (* *)
+ have 2: "\<forall>b \<in> A. bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
+ proof(clarify)
+ fix b assume *: "b \<in> A"
+ hence "rel.under r b \<le> A"
+ using Well OF by(auto simp add: wo_rel.ofilter_def)
+ moreover
+ have "f ` (rel.under r b) = rel.under r' (f b)"
+ using * BIJ by (auto simp add: bij_betw_def)
+ ultimately
+ show "bij_betw ?f' (rel.under r' (f b)) (rel.under r b)"
+ using 1 by (auto simp add: bij_betw_inv_into_subset)
+ qed
+ (* *)
+ have 3: "\<forall>b' \<in> Field r'. bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
+ proof(clarify)
+ fix b' assume *: "b' \<in> Field r'"
+ have "b' = f (?f' b')" using * 1
+ by (auto simp add: bij_betw_inv_into_right)
+ moreover
+ {obtain b where 31: "b \<in> A" and "f b = b'" using IMAGE * by force
+ hence "?f' b' = b" using 1 by (auto simp add: bij_betw_inv_into_left)
+ with 31 have "?f' b' \<in> A" by auto
+ }
+ ultimately
+ show "bij_betw ?f' (rel.under r' b') (rel.under r (?f' b'))"
+ using 2 by auto
+ qed
+ (* *)
+ thus ?thesis unfolding embed_def .
+qed
+
+
+lemma inv_into_underS_embed:
+assumes WELL: "Well_order r" and
+ BIJ: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
+ IN: "a \<in> Field r" and
+ IMAGE: "f ` (rel.underS r a) = Field r'"
+shows "embed r' r (inv_into (rel.underS r a) f)"
+using assms
+by(auto simp add: wo_rel_def wo_rel.underS_ofilter inv_into_ofilter_embed)
+
+
+lemma inv_into_Field_embed:
+assumes WELL: "Well_order r" and EMB: "embed r r' f" and
+ IMAGE: "Field r' \<le> f ` (Field r)"
+shows "embed r' r (inv_into (Field r) f)"
+proof-
+ have "(\<forall>b \<in> Field r. bij_betw f (rel.under r b) (rel.under r' (f b)))"
+ using EMB by (auto simp add: embed_def)
+ moreover
+ have "f ` (Field r) \<le> Field r'"
+ using EMB WELL by (auto simp add: embed_Field)
+ ultimately
+ show ?thesis using assms
+ by(auto simp add: wo_rel_def wo_rel.Field_ofilter inv_into_ofilter_embed)
+qed
+
+
+lemma inv_into_Field_embed_bij_betw:
+assumes WELL: "Well_order r" and
+ EMB: "embed r r' f" and BIJ: "bij_betw f (Field r) (Field r')"
+shows "embed r' r (inv_into (Field r) f)"
+proof-
+ have "Field r' \<le> f ` (Field r)"
+ using BIJ by (auto simp add: bij_betw_def)
+ thus ?thesis using assms
+ by(auto simp add: inv_into_Field_embed)
+qed
+
+
+
+
+
+subsection {* Given any two well-orders, one can be embedded in the other *}
+
+
+text{* Here is an overview of the proof of of this fact, stated in theorem
+@{text "wellorders_totally_ordered"}:
+
+ Fix the well-orders @{text "r::'a rel"} and @{text "r'::'a' rel"}.
+ Attempt to define an embedding @{text "f::'a \<Rightarrow> 'a'"} from @{text "r"} to @{text "r'"} in the
+ natural way by well-order recursion ("hoping" that @{text "Field r"} turns out to be smaller
+ than @{text "Field r'"}), but also record, at the recursive step, in a function
+ @{text "g::'a \<Rightarrow> bool"}, the extra information of whether @{text "Field r'"}
+ gets exhausted or not.
+
+ If @{text "Field r'"} does not get exhausted, then @{text "Field r"} is indeed smaller
+ and @{text "f"} is the desired embedding from @{text "r"} to @{text "r'"}
+ (lemma @{text "wellorders_totally_ordered_aux"}).
+
+ Otherwise, it means that @{text "Field r'"} is the smaller one, and the inverse of
+ (the "good" segment of) @{text "f"} is the desired embedding from @{text "r'"} to @{text "r"}
+ (lemma @{text "wellorders_totally_ordered_aux2"}).
+*}
+
+
+lemma wellorders_totally_ordered_aux:
+fixes r ::"'a rel" and r'::"'a' rel" and
+ f :: "'a \<Rightarrow> 'a'" and a::'a
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and IN: "a \<in> Field r" and
+ IH: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))" and
+ NOT: "f ` (rel.underS r a) \<noteq> Field r'" and SUC: "f a = wo_rel.suc r' (f`(rel.underS r a))"
+shows "bij_betw f (rel.under r a) (rel.under r' (f a))"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using WELL unfolding wo_rel_def .
+ hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
+ have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
+ have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
+ have OF: "wo_rel.ofilter r (rel.underS r a)"
+ by (auto simp add: Well wo_rel.underS_ofilter)
+ hence UN: "rel.underS r a = (\<Union> b \<in> rel.underS r a. rel.under r b)"
+ using Well wo_rel.ofilter_under_UNION[of r "rel.underS r a"] by blast
+ (* Gather facts about elements of rel.underS r a *)
+ {fix b assume *: "b \<in> rel.underS r a"
+ hence t0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
+ have t1: "b \<in> Field r"
+ using * rel.underS_Field[of r a] by auto
+ have t2: "f`(rel.under r b) = rel.under r' (f b)"
+ using IH * by (auto simp add: bij_betw_def)
+ hence t3: "wo_rel.ofilter r' (f`(rel.under r b))"
+ using Well' by (auto simp add: wo_rel.under_ofilter)
+ have "f`(rel.under r b) \<le> Field r'"
+ using t2 by (auto simp add: rel.under_Field)
+ moreover
+ have "b \<in> rel.under r b"
+ using t1 by(auto simp add: Refl rel.Refl_under_in)
+ ultimately
+ have t4: "f b \<in> Field r'" by auto
+ have "f`(rel.under r b) = rel.under r' (f b) \<and>
+ wo_rel.ofilter r' (f`(rel.under r b)) \<and>
+ f b \<in> Field r'"
+ using t2 t3 t4 by auto
+ }
+ hence bFact:
+ "\<forall>b \<in> rel.underS r a. f`(rel.under r b) = rel.under r' (f b) \<and>
+ wo_rel.ofilter r' (f`(rel.under r b)) \<and>
+ f b \<in> Field r'" by blast
+ (* *)
+ have subField: "f`(rel.underS r a) \<le> Field r'"
+ using bFact by blast
+ (* *)
+ have OF': "wo_rel.ofilter r' (f`(rel.underS r a))"
+ proof-
+ have "f`(rel.underS r a) = f`(\<Union> b \<in> rel.underS r a. rel.under r b)"
+ using UN by auto
+ also have "\<dots> = (\<Union> b \<in> rel.underS r a. f`(rel.under r b))" by blast
+ also have "\<dots> = (\<Union> b \<in> rel.underS r a. (rel.under r' (f b)))"
+ using bFact by auto
+ finally
+ have "f`(rel.underS r a) = (\<Union> b \<in> rel.underS r a. (rel.under r' (f b)))" .
+ thus ?thesis
+ using Well' bFact
+ wo_rel.ofilter_UNION[of r' "rel.underS r a" "\<lambda> b. rel.under r' (f b)"] by fastforce
+ qed
+ (* *)
+ have "f`(rel.underS r a) \<union> rel.AboveS r' (f`(rel.underS r a)) = Field r'"
+ using Well' OF' by (auto simp add: wo_rel.ofilter_AboveS_Field)
+ hence NE: "rel.AboveS r' (f`(rel.underS r a)) \<noteq> {}"
+ using subField NOT by blast
+ (* Main proof *)
+ have INCL1: "f`(rel.underS r a) \<le> rel.underS r' (f a) "
+ proof(auto)
+ fix b assume *: "b \<in> rel.underS r a"
+ have "f b \<noteq> f a \<and> (f b, f a) \<in> r'"
+ using subField Well' SUC NE *
+ wo_rel.suc_greater[of r' "f`(rel.underS r a)" "f b"] by auto
+ thus "f b \<in> rel.underS r' (f a)"
+ unfolding rel.underS_def by simp
+ qed
+ (* *)
+ have INCL2: "rel.underS r' (f a) \<le> f`(rel.underS r a)"
+ proof
+ fix b' assume "b' \<in> rel.underS r' (f a)"
+ hence "b' \<noteq> f a \<and> (b', f a) \<in> r'"
+ unfolding rel.underS_def by simp
+ thus "b' \<in> f`(rel.underS r a)"
+ using Well' SUC NE OF'
+ wo_rel.suc_ofilter_in[of r' "f ` rel.underS r a" b'] by auto
+ qed
+ (* *)
+ have INJ: "inj_on f (rel.underS r a)"
+ proof-
+ have "\<forall>b \<in> rel.underS r a. inj_on f (rel.under r b)"
+ using IH by (auto simp add: bij_betw_def)
+ moreover
+ have "\<forall>b. wo_rel.ofilter r (rel.under r b)"
+ using Well by (auto simp add: wo_rel.under_ofilter)
+ ultimately show ?thesis
+ using WELL bFact UN
+ UNION_inj_on_ofilter[of r "rel.underS r a" "\<lambda>b. rel.under r b" f]
+ by auto
+ qed
+ (* *)
+ have BIJ: "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+ unfolding bij_betw_def
+ using INJ INCL1 INCL2 by auto
+ (* *)
+ have "f a \<in> Field r'"
+ using Well' subField NE SUC
+ by (auto simp add: wo_rel.suc_inField)
+ thus ?thesis
+ using WELL WELL' IN BIJ under_underS_bij_betw[of r r' a f] by auto
+qed
+
+
+lemma wellorders_totally_ordered_aux2:
+fixes r ::"'a rel" and r'::"'a' rel" and
+ f :: "'a \<Rightarrow> 'a'" and g :: "'a \<Rightarrow> bool" and a::'a
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+MAIN1:
+ "\<And> a. (False \<notin> g`(rel.underS r a) \<and> f`(rel.underS r a) \<noteq> Field r'
+ \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True)
+ \<and>
+ (\<not>(False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')
+ \<longrightarrow> g a = False)" and
+MAIN2: "\<And> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
+ bij_betw f (rel.under r a) (rel.under r' (f a))" and
+Case: "a \<in> Field r \<and> False \<in> g`(rel.under r a)"
+shows "\<exists>f'. embed r' r f'"
+proof-
+ have Well: "wo_rel r" using WELL unfolding wo_rel_def .
+ hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
+ have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
+ have Antisym: "antisym r" using Well wo_rel.ANTISYM[of r] by auto
+ have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
+ (* *)
+ have 0: "rel.under r a = rel.underS r a \<union> {a}"
+ using Refl Case by(auto simp add: rel.Refl_under_underS)
+ (* *)
+ have 1: "g a = False"
+ proof-
+ {assume "g a \<noteq> False"
+ with 0 Case have "False \<in> g`(rel.underS r a)" by blast
+ with MAIN1 have "g a = False" by blast}
+ thus ?thesis by blast
+ qed
+ let ?A = "{a \<in> Field r. g a = False}"
+ let ?a = "(wo_rel.minim r ?A)"
+ (* *)
+ have 2: "?A \<noteq> {} \<and> ?A \<le> Field r" using Case 1 by blast
+ (* *)
+ have 3: "False \<notin> g`(rel.underS r ?a)"
+ proof
+ assume "False \<in> g`(rel.underS r ?a)"
+ then obtain b where "b \<in> rel.underS r ?a" and 31: "g b = False" by auto
+ hence 32: "(b,?a) \<in> r \<and> b \<noteq> ?a"
+ by (auto simp add: rel.underS_def)
+ hence "b \<in> Field r" unfolding Field_def by auto
+ with 31 have "b \<in> ?A" by auto
+ hence "(?a,b) \<in> r" using wo_rel.minim_least 2 Well by fastforce
+ (* again: why worked without type annotations? *)
+ with 32 Antisym show False
+ by (auto simp add: antisym_def)
+ qed
+ have temp: "?a \<in> ?A"
+ using Well 2 wo_rel.minim_in[of r ?A] by auto
+ hence 4: "?a \<in> Field r" by auto
+ (* *)
+ have 5: "g ?a = False" using temp by blast
+ (* *)
+ have 6: "f`(rel.underS r ?a) = Field r'"
+ using MAIN1[of ?a] 3 5 by blast
+ (* *)
+ have 7: "\<forall>b \<in> rel.underS r ?a. bij_betw f (rel.under r b) (rel.under r' (f b))"
+ proof
+ fix b assume as: "b \<in> rel.underS r ?a"
+ moreover
+ have "wo_rel.ofilter r (rel.underS r ?a)"
+ using Well by (auto simp add: wo_rel.underS_ofilter)
+ ultimately
+ have "False \<notin> g`(rel.under r b)" using 3 Well by (auto simp add: wo_rel.ofilter_def)
+ moreover have "b \<in> Field r"
+ unfolding Field_def using as by (auto simp add: rel.underS_def)
+ ultimately
+ show "bij_betw f (rel.under r b) (rel.under r' (f b))"
+ using MAIN2 by auto
+ qed
+ (* *)
+ have "embed r' r (inv_into (rel.underS r ?a) f)"
+ using WELL WELL' 7 4 6 inv_into_underS_embed[of r ?a f r'] by auto
+ thus ?thesis
+ unfolding embed_def by blast
+qed
+
+
+theorem wellorders_totally_ordered:
+fixes r ::"'a rel" and r'::"'a' rel"
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<exists>f. embed r r' f) \<or> (\<exists>f'. embed r' r f')"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using WELL unfolding wo_rel_def .
+ hence Refl: "Refl r" using wo_rel.REFL[of r] by auto
+ have Trans: "trans r" using Well wo_rel.TRANS[of r] by auto
+ have Well': "wo_rel r'" using WELL' unfolding wo_rel_def .
+ (* Main proof *)
+ obtain H where H_def: "H =
+ (\<lambda>h a. if False \<notin> (snd o h)`(rel.underS r a) \<and> (fst o h)`(rel.underS r a) \<noteq> Field r'
+ then (wo_rel.suc r' ((fst o h)`(rel.underS r a)), True)
+ else (undefined, False))" by blast
+ have Adm: "wo_rel.adm_wo r H"
+ using Well
+ proof(unfold wo_rel.adm_wo_def, clarify)
+ fix h1::"'a \<Rightarrow> 'a' * bool" and h2::"'a \<Rightarrow> 'a' * bool" and x
+ assume "\<forall>y\<in>rel.underS r x. h1 y = h2 y"
+ hence "\<forall>y\<in>rel.underS r x. (fst o h1) y = (fst o h2) y \<and>
+ (snd o h1) y = (snd o h2) y" by auto
+ hence "(fst o h1)`(rel.underS r x) = (fst o h2)`(rel.underS r x) \<and>
+ (snd o h1)`(rel.underS r x) = (snd o h2)`(rel.underS r x)"
+ by (auto simp add: image_def)
+ thus "H h1 x = H h2 x" by (simp add: H_def)
+ qed
+ (* More constant definitions: *)
+ obtain h::"'a \<Rightarrow> 'a' * bool" and f::"'a \<Rightarrow> 'a'" and g::"'a \<Rightarrow> bool"
+ where h_def: "h = wo_rel.worec r H" and
+ f_def: "f = fst o h" and g_def: "g = snd o h" by blast
+ obtain test where test_def:
+ "test = (\<lambda> a. False \<notin> (g`(rel.underS r a)) \<and> f`(rel.underS r a) \<noteq> Field r')" by blast
+ (* *)
+ have *: "\<And> a. h a = H h a"
+ using Adm Well wo_rel.worec_fixpoint[of r H] by (simp add: h_def)
+ have Main1:
+ "\<And> a. (test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
+ (\<not>(test a) \<longrightarrow> g a = False)"
+ proof- (* How can I prove this withou fixing a? *)
+ fix a show "(test a \<longrightarrow> f a = wo_rel.suc r' (f`(rel.underS r a)) \<and> g a = True) \<and>
+ (\<not>(test a) \<longrightarrow> g a = False)"
+ using *[of a] test_def f_def g_def H_def by auto
+ qed
+ (* *)
+ let ?phi = "\<lambda> a. a \<in> Field r \<and> False \<notin> g`(rel.under r a) \<longrightarrow>
+ bij_betw f (rel.under r a) (rel.under r' (f a))"
+ have Main2: "\<And> a. ?phi a"
+ proof-
+ fix a show "?phi a"
+ proof(rule wo_rel.well_order_induct[of r ?phi],
+ simp only: Well, clarify)
+ fix a
+ assume IH: "\<forall>b. b \<noteq> a \<and> (b,a) \<in> r \<longrightarrow> ?phi b" and
+ *: "a \<in> Field r" and
+ **: "False \<notin> g`(rel.under r a)"
+ have 1: "\<forall>b \<in> rel.underS r a. bij_betw f (rel.under r b) (rel.under r' (f b))"
+ proof(clarify)
+ fix b assume ***: "b \<in> rel.underS r a"
+ hence 0: "(b,a) \<in> r \<and> b \<noteq> a" unfolding rel.underS_def by auto
+ moreover have "b \<in> Field r"
+ using *** rel.underS_Field[of r a] by auto
+ moreover have "False \<notin> g`(rel.under r b)"
+ using 0 ** Trans rel.under_incr[of r b a] by auto
+ ultimately show "bij_betw f (rel.under r b) (rel.under r' (f b))"
+ using IH by auto
+ qed
+ (* *)
+ have 21: "False \<notin> g`(rel.underS r a)"
+ using ** rel.underS_subset_under[of r a] by auto
+ have 22: "g`(rel.under r a) \<le> {True}" using ** by auto
+ moreover have 23: "a \<in> rel.under r a"
+ using Refl * by (auto simp add: rel.Refl_under_in)
+ ultimately have 24: "g a = True" by blast
+ have 2: "f`(rel.underS r a) \<noteq> Field r'"
+ proof
+ assume "f`(rel.underS r a) = Field r'"
+ hence "g a = False" using Main1 test_def by blast
+ with 24 show False using ** by blast
+ qed
+ (* *)
+ have 3: "f a = wo_rel.suc r' (f`(rel.underS r a))"
+ using 21 2 Main1 test_def by blast
+ (* *)
+ show "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using WELL WELL' 1 2 3 *
+ wellorders_totally_ordered_aux[of r r' a f] by auto
+ qed
+ qed
+ (* *)
+ let ?chi = "(\<lambda> a. a \<in> Field r \<and> False \<in> g`(rel.under r a))"
+ show ?thesis
+ proof(cases "\<exists>a. ?chi a")
+ assume "\<not> (\<exists>a. ?chi a)"
+ hence "\<forall>a \<in> Field r. bij_betw f (rel.under r a) (rel.under r' (f a))"
+ using Main2 by blast
+ thus ?thesis unfolding embed_def by blast
+ next
+ assume "\<exists>a. ?chi a"
+ then obtain a where "?chi a" by blast
+ hence "\<exists>f'. embed r' r f'"
+ using wellorders_totally_ordered_aux2[of r r' g f a]
+ WELL WELL' Main1 Main2 test_def by blast
+ thus ?thesis by blast
+ qed
+qed
+
+
+subsection {* Uniqueness of embeddings *}
+
+
+text{* Here we show a fact complementary to the one from the previous subsection -- namely,
+that between any two well-orders there is {\em at most} one embedding, and is the one
+definable by the expected well-order recursive equation. As a consequence, any two
+embeddings of opposite directions are mutually inverse. *}
+
+
+lemma embed_determined:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and IN: "a \<in> Field r"
+shows "f a = wo_rel.suc r' (f`(rel.underS r a))"
+proof-
+ have "bij_betw f (rel.underS r a) (rel.underS r' (f a))"
+ using assms by (auto simp add: embed_underS)
+ hence "f`(rel.underS r a) = rel.underS r' (f a)"
+ by (auto simp add: bij_betw_def)
+ moreover
+ {have "f a \<in> Field r'" using IN
+ using EMB WELL embed_Field[of r r' f] by auto
+ hence "f a = wo_rel.suc r' (rel.underS r' (f a))"
+ using WELL' by (auto simp add: wo_rel_def wo_rel.suc_underS)
+ }
+ ultimately show ?thesis by simp
+qed
+
+
+lemma embed_unique:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMBf: "embed r r' f" and EMBg: "embed r r' g"
+shows "a \<in> Field r \<longrightarrow> f a = g a"
+proof(rule wo_rel.well_order_induct[of r], auto simp add: WELL wo_rel_def)
+ fix a
+ assume IH: "\<forall>b. b \<noteq> a \<and> (b,a): r \<longrightarrow> b \<in> Field r \<longrightarrow> f b = g b" and
+ *: "a \<in> Field r"
+ hence "\<forall>b \<in> rel.underS r a. f b = g b"
+ unfolding rel.underS_def by (auto simp add: Field_def)
+ hence "f`(rel.underS r a) = g`(rel.underS r a)" by force
+ thus "f a = g a"
+ using assms * embed_determined[of r r' f a] embed_determined[of r r' g a] by auto
+qed
+
+
+lemma embed_bothWays_inverse:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r f'"
+shows "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
+proof-
+ have "embed r r (f' o f)" using assms
+ by(auto simp add: comp_embed)
+ moreover have "embed r r id" using assms
+ by (auto simp add: id_embed)
+ ultimately have "\<forall>a \<in> Field r. f'(f a) = a"
+ using assms embed_unique[of r r "f' o f" id] id_def by auto
+ moreover
+ {have "embed r' r' (f o f')" using assms
+ by(auto simp add: comp_embed)
+ moreover have "embed r' r' id" using assms
+ by (auto simp add: id_embed)
+ ultimately have "\<forall>a' \<in> Field r'. f(f' a') = a'"
+ using assms embed_unique[of r' r' "f o f'" id] id_def by auto
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma embed_bothWays_bij_betw:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r g"
+shows "bij_betw f (Field r) (Field r')"
+proof-
+ let ?A = "Field r" let ?A' = "Field r'"
+ have "embed r r (g o f) \<and> embed r' r' (f o g)"
+ using assms by (auto simp add: comp_embed)
+ hence 1: "(\<forall>a \<in> ?A. g(f a) = a) \<and> (\<forall>a' \<in> ?A'. f(g a') = a')"
+ using WELL id_embed[of r] embed_unique[of r r "g o f" id]
+ WELL' id_embed[of r'] embed_unique[of r' r' "f o g" id]
+ id_def by auto
+ have 2: "(\<forall>a \<in> ?A. f a \<in> ?A') \<and> (\<forall>a' \<in> ?A'. g a' \<in> ?A)"
+ using assms embed_Field[of r r' f] embed_Field[of r' r g] by blast
+ (* *)
+ show ?thesis
+ proof(unfold bij_betw_def inj_on_def, auto simp add: 2)
+ fix a b assume *: "a \<in> ?A" "b \<in> ?A" and **: "f a = f b"
+ have "a = g(f a) \<and> b = g(f b)" using * 1 by auto
+ with ** show "a = b" by auto
+ next
+ fix a' assume *: "a' \<in> ?A'"
+ hence "g a' \<in> ?A \<and> f(g a') = a'" using 1 2 by auto
+ thus "a' \<in> f ` ?A" by force
+ qed
+qed
+
+
+lemma embed_bothWays_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r g"
+shows "iso r r' f"
+unfolding iso_def using assms by (auto simp add: embed_bothWays_bij_betw)
+
+
+subsection {* More properties of embeddings, strict embeddings and isomorphisms *}
+
+
+lemma embed_bothWays_Field_bij_betw:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r r' f" and EMB': "embed r' r f'"
+shows "bij_betw f (Field r) (Field r')"
+proof-
+ have "(\<forall>a \<in> Field r. f'(f a) = a) \<and> (\<forall>a' \<in> Field r'. f(f' a') = a')"
+ using assms by (auto simp add: embed_bothWays_inverse)
+ moreover
+ have "f`(Field r) \<le> Field r' \<and> f' ` (Field r') \<le> Field r"
+ using assms by (auto simp add: embed_Field)
+ ultimately
+ show ?thesis using bij_betw_byWitness[of "Field r" f' f "Field r'"] by auto
+qed
+
+
+lemma embedS_comp_embed:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embedS r r' f" and EMB': "embed r' r'' f'"
+shows "embedS r r'' (f' o f)"
+proof-
+ let ?g = "(f' o f)" let ?h = "inv_into (Field r) ?g"
+ have 1: "embed r r' f \<and> \<not> (bij_betw f (Field r) (Field r'))"
+ using EMB by (auto simp add: embedS_def)
+ hence 2: "embed r r'' ?g"
+ using WELL EMB' comp_embed[of r r' f r'' f'] by auto
+ moreover
+ {assume "bij_betw ?g (Field r) (Field r'')"
+ hence "embed r'' r ?h" using 2 WELL
+ by (auto simp add: inv_into_Field_embed_bij_betw)
+ hence "embed r' r (?h o f')" using WELL' EMB'
+ by (auto simp add: comp_embed)
+ hence "bij_betw f (Field r) (Field r')" using WELL WELL' 1
+ by (auto simp add: embed_bothWays_Field_bij_betw)
+ with 1 have False by blast
+ }
+ ultimately show ?thesis unfolding embedS_def by auto
+qed
+
+
+lemma embed_comp_embedS:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embed r r' f" and EMB': "embedS r' r'' f'"
+shows "embedS r r'' (f' o f)"
+proof-
+ let ?g = "(f' o f)" let ?h = "inv_into (Field r) ?g"
+ have 1: "embed r' r'' f' \<and> \<not> (bij_betw f' (Field r') (Field r''))"
+ using EMB' by (auto simp add: embedS_def)
+ hence 2: "embed r r'' ?g"
+ using WELL EMB comp_embed[of r r' f r'' f'] by auto
+ moreover
+ {assume "bij_betw ?g (Field r) (Field r'')"
+ hence "embed r'' r ?h" using 2 WELL
+ by (auto simp add: inv_into_Field_embed_bij_betw)
+ hence "embed r'' r' (f o ?h)" using WELL'' EMB
+ by (auto simp add: comp_embed)
+ hence "bij_betw f' (Field r') (Field r'')" using WELL' WELL'' 1
+ by (auto simp add: embed_bothWays_Field_bij_betw)
+ with 1 have False by blast
+ }
+ ultimately show ?thesis unfolding embedS_def by auto
+qed
+
+
+lemma embed_comp_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embed r r' f" and EMB': "iso r' r'' f'"
+shows "embed r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: comp_embed)
+
+
+lemma iso_comp_embed:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "iso r r' f" and EMB': "embed r' r'' f'"
+shows "embed r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: comp_embed)
+
+
+lemma embedS_comp_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "embedS r r' f" and EMB': "iso r' r'' f'"
+shows "embedS r r'' (f' o f)"
+using assms unfolding iso_def
+by (auto simp add: embedS_comp_embed)
+
+
+lemma iso_comp_embedS:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+ and EMB: "iso r r' f" and EMB': "embedS r' r'' f'"
+shows "embedS r r'' (f' o f)"
+using assms unfolding iso_def using embed_comp_embedS
+by (auto simp add: embed_comp_embedS)
+
+
+lemma embedS_Field:
+assumes WELL: "Well_order r" and EMB: "embedS r r' f"
+shows "f ` (Field r) < Field r'"
+proof-
+ have "f`(Field r) \<le> Field r'" using assms
+ by (auto simp add: embed_Field embedS_def)
+ moreover
+ {have "inj_on f (Field r)" using assms
+ by (auto simp add: embedS_def embed_inj_on)
+ hence "f`(Field r) \<noteq> Field r'" using EMB
+ by (auto simp add: embedS_def bij_betw_def)
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma embedS_iff:
+assumes WELL: "Well_order r" and ISO: "embed r r' f"
+shows "embedS r r' f = (f ` (Field r) < Field r')"
+proof
+ assume "embedS r r' f"
+ thus "f ` Field r \<subset> Field r'"
+ using WELL by (auto simp add: embedS_Field)
+next
+ assume "f ` Field r \<subset> Field r'"
+ hence "\<not> bij_betw f (Field r) (Field r')"
+ unfolding bij_betw_def by blast
+ thus "embedS r r' f" unfolding embedS_def
+ using ISO by auto
+qed
+
+
+lemma iso_Field:
+"iso r r' f \<Longrightarrow> f ` (Field r) = Field r'"
+using assms by (auto simp add: iso_def bij_betw_def)
+
+
+lemma iso_iff:
+assumes "Well_order r"
+shows "iso r r' f = (embed r r' f \<and> f ` (Field r) = Field r')"
+proof
+ assume "iso r r' f"
+ thus "embed r r' f \<and> f ` (Field r) = Field r'"
+ by (auto simp add: iso_Field iso_def)
+next
+ assume *: "embed r r' f \<and> f ` Field r = Field r'"
+ hence "inj_on f (Field r)" using assms by (auto simp add: embed_inj_on)
+ with * have "bij_betw f (Field r) (Field r')"
+ unfolding bij_betw_def by simp
+ with * show "iso r r' f" unfolding iso_def by auto
+qed
+
+
+lemma iso_iff2:
+assumes "Well_order r"
+shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and>
+ (\<forall>a \<in> Field r. \<forall>b \<in> Field r.
+ (((a,b) \<in> r) = ((f a, f b) \<in> r'))))"
+using assms
+proof(auto simp add: iso_def)
+ fix a b
+ assume "embed r r' f"
+ hence "compat r r' f" using embed_compat[of r] by auto
+ moreover assume "(a,b) \<in> r"
+ ultimately show "(f a, f b) \<in> r'" using compat_def[of r] by auto
+next
+ let ?f' = "inv_into (Field r) f"
+ assume "embed r r' f" and 1: "bij_betw f (Field r) (Field r')"
+ hence "embed r' r ?f'" using assms
+ by (auto simp add: inv_into_Field_embed_bij_betw)
+ hence 2: "compat r' r ?f'" using embed_compat[of r'] by auto
+ fix a b assume *: "a \<in> Field r" "b \<in> Field r" and **: "(f a,f b) \<in> r'"
+ hence "?f'(f a) = a \<and> ?f'(f b) = b" using 1
+ by (auto simp add: bij_betw_inv_into_left)
+ thus "(a,b) \<in> r" using ** 2 compat_def[of r' r ?f'] by fastforce
+next
+ assume *: "bij_betw f (Field r) (Field r')" and
+ **: "\<forall>a\<in>Field r. \<forall>b\<in>Field r. ((a, b) \<in> r) = ((f a, f b) \<in> r')"
+ have 1: "\<And> a. rel.under r a \<le> Field r \<and> rel.under r' (f a) \<le> Field r'"
+ by (auto simp add: rel.under_Field)
+ have 2: "inj_on f (Field r)" using * by (auto simp add: bij_betw_def)
+ {fix a assume ***: "a \<in> Field r"
+ have "bij_betw f (rel.under r a) (rel.under r' (f a))"
+ proof(unfold bij_betw_def, auto)
+ show "inj_on f (rel.under r a)"
+ using 1 2 by (auto simp add: subset_inj_on)
+ next
+ fix b assume "b \<in> rel.under r a"
+ hence "a \<in> Field r \<and> b \<in> Field r \<and> (b,a) \<in> r"
+ unfolding rel.under_def by (auto simp add: Field_def Range_def Domain_def)
+ with 1 ** show "f b \<in> rel.under r' (f a)"
+ unfolding rel.under_def by auto
+ next
+ fix b' assume "b' \<in> rel.under r' (f a)"
+ hence 3: "(b',f a) \<in> r'" unfolding rel.under_def by simp
+ hence "b' \<in> Field r'" unfolding Field_def by auto
+ with * obtain b where "b \<in> Field r \<and> f b = b'"
+ unfolding bij_betw_def by force
+ with 3 ** ***
+ show "b' \<in> f ` (rel.under r a)" unfolding rel.under_def by blast
+ qed
+ }
+ thus "embed r r' f" unfolding embed_def using * by auto
+qed
+
+
+lemma iso_iff3:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> compat r r' f)"
+proof
+ assume "iso r r' f"
+ thus "bij_betw f (Field r) (Field r') \<and> compat r r' f"
+ unfolding compat_def using WELL by (auto simp add: iso_iff2 Field_def)
+next
+ have Well: "wo_rel r \<and> wo_rel r'" using WELL WELL'
+ by (auto simp add: wo_rel_def)
+ assume *: "bij_betw f (Field r) (Field r') \<and> compat r r' f"
+ thus "iso r r' f"
+ unfolding "compat_def" using assms
+ proof(auto simp add: iso_iff2)
+ fix a b assume **: "a \<in> Field r" "b \<in> Field r" and
+ ***: "(f a, f b) \<in> r'"
+ {assume "(b,a) \<in> r \<or> b = a"
+ hence "(b,a): r"using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
+ hence "(f b, f a) \<in> r'" using * unfolding compat_def by auto
+ hence "f a = f b"
+ using Well *** wo_rel.ANTISYM[of r'] antisym_def[of r'] by blast
+ hence "a = b" using * ** unfolding bij_betw_def inj_on_def by auto
+ hence "(a,b) \<in> r" using Well ** wo_rel.REFL[of r] refl_on_def[of _ r] by blast
+ }
+ thus "(a,b) \<in> r"
+ using Well ** wo_rel.TOTAL[of r] total_on_def[of _ r] by blast
+ qed
+qed
+
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Wellorder_Relation.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,513 @@
+(* Title: HOL/Ordinals_and_Cardinals/Wellorder_Relation.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Well-order relations.
+*)
+
+header {* Well-Order Relations *}
+
+theory Wellorder_Relation
+imports
+ "../Ordinals_and_Cardinals_Base/Wellorder_Relation_Base"
+ Wellfounded_More
+begin
+
+
+subsection {* Auxiliaries *}
+
+lemma (in wo_rel) PREORD: "Preorder r"
+using WELL order_on_defs[of _ r] by auto
+
+lemma (in wo_rel) PARORD: "Partial_order r"
+using WELL order_on_defs[of _ r] by auto
+
+lemma (in wo_rel) cases_Total2:
+"\<And> phi a b. \<lbrakk>{a,b} \<le> Field r; ((a,b) \<in> r - Id \<Longrightarrow> phi a b);
+ ((b,a) \<in> r - Id \<Longrightarrow> phi a b); (a = b \<Longrightarrow> phi a b)\<rbrakk>
+ \<Longrightarrow> phi a b"
+using TOTALS by auto
+
+
+subsection {* Well-founded induction and recursion adapted to non-strict well-order relations *}
+
+lemma (in wo_rel) worec_unique_fixpoint:
+assumes ADM: "adm_wo H" and fp: "f = H f"
+shows "f = worec H"
+proof-
+ have "adm_wf (r - Id) H"
+ unfolding adm_wf_def
+ using ADM adm_wo_def[of H] underS_def by auto
+ hence "f = wfrec (r - Id) H"
+ using fp WF wfrec_unique_fixpoint[of "r - Id" H] by simp
+ thus ?thesis unfolding worec_def .
+qed
+
+
+subsubsection {* Properties of max2 *}
+
+lemma (in wo_rel) max2_iff:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "((max2 a b, c) \<in> r) = ((a,c) \<in> r \<and> (b,c) \<in> r)"
+proof
+ assume "(max2 a b, c) \<in> r"
+ thus "(a,c) \<in> r \<and> (b,c) \<in> r"
+ using assms max2_greater[of a b] TRANS trans_def[of r] by blast
+next
+ assume "(a,c) \<in> r \<and> (b,c) \<in> r"
+ thus "(max2 a b, c) \<in> r"
+ using assms max2_among[of a b] by auto
+qed
+
+
+subsubsection{* Properties of minim *}
+
+lemma (in wo_rel) minim_Under:
+"\<lbrakk>B \<le> Field r; B \<noteq> {}\<rbrakk> \<Longrightarrow> minim B \<in> Under B"
+by(auto simp add: Under_def
+minim_in
+minim_inField
+minim_least
+under_ofilter
+underS_ofilter
+Field_ofilter
+ofilter_Under
+ofilter_UnderS
+ofilter_Un
+)
+
+lemma (in wo_rel) equals_minim_Under:
+"\<lbrakk>B \<le> Field r; a \<in> B; a \<in> Under B\<rbrakk>
+ \<Longrightarrow> a = minim B"
+by(auto simp add: Under_def equals_minim)
+
+lemma (in wo_rel) minim_iff_In_Under:
+assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
+shows "(a = minim B) = (a \<in> B \<and> a \<in> Under B)"
+proof
+ assume "a = minim B"
+ thus "a \<in> B \<and> a \<in> Under B"
+ using assms minim_in minim_Under by simp
+next
+ assume "a \<in> B \<and> a \<in> Under B"
+ thus "a = minim B"
+ using assms equals_minim_Under by simp
+qed
+
+lemma (in wo_rel) minim_Under_under:
+assumes NE: "A \<noteq> {}" and SUB: "A \<le> Field r"
+shows "Under A = under (minim A)"
+proof-
+ (* Preliminary facts *)
+ have 1: "minim A \<in> A"
+ using assms minim_in by auto
+ have 2: "\<forall>x \<in> A. (minim A, x) \<in> r"
+ using assms minim_least by auto
+ (* Main proof *)
+ have "Under A \<le> under (minim A)"
+ proof
+ fix x assume "x \<in> Under A"
+ with 1 Under_def have "(x,minim A) \<in> r" by auto
+ thus "x \<in> under(minim A)" unfolding under_def by simp
+ qed
+ (* *)
+ moreover
+ (* *)
+ have "under (minim A) \<le> Under A"
+ proof
+ fix x assume "x \<in> under(minim A)"
+ hence 11: "(x,minim A) \<in> r" unfolding under_def by simp
+ hence "x \<in> Field r" unfolding Field_def by auto
+ moreover
+ {fix a assume "a \<in> A"
+ with 2 have "(minim A, a) \<in> r" by simp
+ with 11 have "(x,a) \<in> r"
+ using TRANS trans_def[of r] by blast
+ }
+ ultimately show "x \<in> Under A" by (unfold Under_def, auto)
+ qed
+ (* *)
+ ultimately show ?thesis by blast
+qed
+
+lemma (in wo_rel) minim_UnderS_underS:
+assumes NE: "A \<noteq> {}" and SUB: "A \<le> Field r"
+shows "UnderS A = underS (minim A)"
+proof-
+ (* Preliminary facts *)
+ have 1: "minim A \<in> A"
+ using assms minim_in by auto
+ have 2: "\<forall>x \<in> A. (minim A, x) \<in> r"
+ using assms minim_least by auto
+ (* Main proof *)
+ have "UnderS A \<le> underS (minim A)"
+ proof
+ fix x assume "x \<in> UnderS A"
+ with 1 UnderS_def have "x \<noteq> minim A \<and> (x,minim A) \<in> r" by auto
+ thus "x \<in> underS(minim A)" unfolding underS_def by simp
+ qed
+ (* *)
+ moreover
+ (* *)
+ have "underS (minim A) \<le> UnderS A"
+ proof
+ fix x assume "x \<in> underS(minim A)"
+ hence 11: "x \<noteq> minim A \<and> (x,minim A) \<in> r" unfolding underS_def by simp
+ hence "x \<in> Field r" unfolding Field_def by auto
+ moreover
+ {fix a assume "a \<in> A"
+ with 2 have 3: "(minim A, a) \<in> r" by simp
+ with 11 have "(x,a) \<in> r"
+ using TRANS trans_def[of r] by blast
+ moreover
+ have "x \<noteq> a"
+ proof
+ assume "x = a"
+ with 11 3 ANTISYM antisym_def[of r]
+ show False by auto
+ qed
+ ultimately
+ have "x \<noteq> a \<and> (x,a) \<in> r" by simp
+ }
+ ultimately show "x \<in> UnderS A" by (unfold UnderS_def, auto)
+ qed
+ (* *)
+ ultimately show ?thesis by blast
+qed
+
+
+subsubsection{* Properties of supr *}
+
+lemma (in wo_rel) supr_Above:
+assumes SUB: "B \<le> Field r" and ABOVE: "Above B \<noteq> {}"
+shows "supr B \<in> Above B"
+proof(unfold supr_def)
+ have "Above B \<le> Field r"
+ using Above_Field by auto
+ thus "minim (Above B) \<in> Above B"
+ using assms by (simp add: minim_in)
+qed
+
+lemma (in wo_rel) supr_greater:
+assumes SUB: "B \<le> Field r" and ABOVE: "Above B \<noteq> {}" and
+ IN: "b \<in> B"
+shows "(b, supr B) \<in> r"
+proof-
+ from assms supr_Above
+ have "supr B \<in> Above B" by simp
+ with IN Above_def show ?thesis by simp
+qed
+
+lemma (in wo_rel) supr_least_Above:
+assumes SUB: "B \<le> Field r" and
+ ABOVE: "a \<in> Above B"
+shows "(supr B, a) \<in> r"
+proof(unfold supr_def)
+ have "Above B \<le> Field r"
+ using Above_Field by auto
+ thus "(minim (Above B), a) \<in> r"
+ using assms minim_least
+ by simp
+qed
+
+lemma (in wo_rel) supr_least:
+"\<lbrakk>B \<le> Field r; a \<in> Field r; (\<And> b. b \<in> B \<Longrightarrow> (b,a) \<in> r)\<rbrakk>
+ \<Longrightarrow> (supr B, a) \<in> r"
+by(auto simp add: supr_least_Above Above_def)
+
+lemma (in wo_rel) equals_supr_Above:
+assumes SUB: "B \<le> Field r" and ABV: "a \<in> Above B" and
+ MINIM: "\<And> a'. a' \<in> Above B \<Longrightarrow> (a,a') \<in> r"
+shows "a = supr B"
+proof(unfold supr_def)
+ have "Above B \<le> Field r"
+ using Above_Field by auto
+ thus "a = minim (Above B)"
+ using assms equals_minim by simp
+qed
+
+lemma (in wo_rel) equals_supr:
+assumes SUB: "B \<le> Field r" and IN: "a \<in> Field r" and
+ ABV: "\<And> b. b \<in> B \<Longrightarrow> (b,a) \<in> r" and
+ MINIM: "\<And> a'. \<lbrakk> a' \<in> Field r; \<And> b. b \<in> B \<Longrightarrow> (b,a') \<in> r\<rbrakk> \<Longrightarrow> (a,a') \<in> r"
+shows "a = supr B"
+proof-
+ have "a \<in> Above B"
+ unfolding Above_def using ABV IN by simp
+ moreover
+ have "\<And> a'. a' \<in> Above B \<Longrightarrow> (a,a') \<in> r"
+ unfolding Above_def using MINIM by simp
+ ultimately show ?thesis
+ using equals_supr_Above SUB by auto
+qed
+
+lemma (in wo_rel) supr_inField:
+assumes "B \<le> Field r" and "Above B \<noteq> {}"
+shows "supr B \<in> Field r"
+proof-
+ have "supr B \<in> Above B" using supr_Above assms by simp
+ thus ?thesis using assms Above_Field by auto
+qed
+
+lemma (in wo_rel) supr_above_Above:
+assumes SUB: "B \<le> Field r" and ABOVE: "Above B \<noteq> {}"
+shows "Above B = above (supr B)"
+proof(unfold Above_def above_def, auto)
+ fix a assume "a \<in> Field r" "\<forall>b \<in> B. (b,a) \<in> r"
+ with supr_least assms
+ show "(supr B, a) \<in> r" by auto
+next
+ fix b assume "(supr B, b) \<in> r"
+ thus "b \<in> Field r"
+ using REFL refl_on_def[of _ r] by auto
+next
+ fix a b
+ assume 1: "(supr B, b) \<in> r" and 2: "a \<in> B"
+ with assms supr_greater
+ have "(a,supr B) \<in> r" by auto
+ thus "(a,b) \<in> r"
+ using 1 TRANS trans_def[of r] by blast
+qed
+
+lemma (in wo_rel) supr_under:
+assumes IN: "a \<in> Field r"
+shows "a = supr (under a)"
+proof-
+ have "under a \<le> Field r"
+ using under_Field by auto
+ moreover
+ have "under a \<noteq> {}"
+ using IN Refl_under_in REFL by auto
+ moreover
+ have "a \<in> Above (under a)"
+ using in_Above_under IN by auto
+ moreover
+ have "\<forall>a' \<in> Above (under a). (a,a') \<in> r"
+ proof(unfold Above_def under_def, auto)
+ fix a'
+ assume "\<forall>aa. (aa, a) \<in> r \<longrightarrow> (aa, a') \<in> r"
+ hence "(a,a) \<in> r \<longrightarrow> (a,a') \<in> r" by blast
+ moreover have "(a,a) \<in> r"
+ using REFL IN by (auto simp add: refl_on_def)
+ ultimately
+ show "(a, a') \<in> r" by (rule mp)
+ qed
+ ultimately show ?thesis
+ using equals_supr_Above by auto
+qed
+
+
+subsubsection{* Properties of successor *}
+
+lemma (in wo_rel) suc_least:
+"\<lbrakk>B \<le> Field r; a \<in> Field r; (\<And> b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b,a) \<in> r)\<rbrakk>
+ \<Longrightarrow> (suc B, a) \<in> r"
+by(auto simp add: suc_least_AboveS AboveS_def)
+
+lemma (in wo_rel) equals_suc:
+assumes SUB: "B \<le> Field r" and IN: "a \<in> Field r" and
+ ABVS: "\<And> b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b,a) \<in> r" and
+ MINIM: "\<And> a'. \<lbrakk>a' \<in> Field r; \<And> b. b \<in> B \<Longrightarrow> a' \<noteq> b \<and> (b,a') \<in> r\<rbrakk> \<Longrightarrow> (a,a') \<in> r"
+shows "a = suc B"
+proof-
+ have "a \<in> AboveS B"
+ unfolding AboveS_def using ABVS IN by simp
+ moreover
+ have "\<And> a'. a' \<in> AboveS B \<Longrightarrow> (a,a') \<in> r"
+ unfolding AboveS_def using MINIM by simp
+ ultimately show ?thesis
+ using equals_suc_AboveS SUB by auto
+qed
+
+lemma (in wo_rel) suc_above_AboveS:
+assumes SUB: "B \<le> Field r" and
+ ABOVE: "AboveS B \<noteq> {}"
+shows "AboveS B = above (suc B)"
+proof(unfold AboveS_def above_def, auto)
+ fix a assume "a \<in> Field r" "\<forall>b \<in> B. a \<noteq> b \<and> (b,a) \<in> r"
+ with suc_least assms
+ show "(suc B,a) \<in> r" by auto
+next
+ fix b assume "(suc B, b) \<in> r"
+ thus "b \<in> Field r"
+ using REFL refl_on_def[of _ r] by auto
+next
+ fix a b
+ assume 1: "(suc B, b) \<in> r" and 2: "a \<in> B"
+ with assms suc_greater[of B a]
+ have "(a,suc B) \<in> r" by auto
+ thus "(a,b) \<in> r"
+ using 1 TRANS trans_def[of r] by blast
+next
+ fix a
+ assume 1: "(suc B, a) \<in> r" and 2: "a \<in> B"
+ with assms suc_greater[of B a]
+ have "(a,suc B) \<in> r" by auto
+ moreover have "suc B \<in> Field r"
+ using assms suc_inField by simp
+ ultimately have "a = suc B"
+ using 1 2 SUB ANTISYM antisym_def[of r] by auto
+ thus False
+ using assms suc_greater[of B a] 2 by auto
+qed
+
+lemma (in wo_rel) suc_singl_pred:
+assumes IN: "a \<in> Field r" and ABOVE_NE: "aboveS a \<noteq> {}" and
+ REL: "(a',suc {a}) \<in> r" and DIFF: "a' \<noteq> suc {a}"
+shows "a' = a \<or> (a',a) \<in> r"
+proof-
+ have *: "suc {a} \<in> Field r \<and> a' \<in> Field r"
+ using WELL REL well_order_on_domain by auto
+ {assume **: "a' \<noteq> a"
+ hence "(a,a') \<in> r \<or> (a',a) \<in> r"
+ using TOTAL IN * by (auto simp add: total_on_def)
+ moreover
+ {assume "(a,a') \<in> r"
+ with ** * assms WELL suc_least[of "{a}" a']
+ have "(suc {a},a') \<in> r" by auto
+ with REL DIFF * ANTISYM antisym_def[of r]
+ have False by simp
+ }
+ ultimately have "(a',a) \<in> r"
+ by blast
+ }
+ thus ?thesis by blast
+qed
+
+lemma (in wo_rel) under_underS_suc:
+assumes IN: "a \<in> Field r" and ABV: "aboveS a \<noteq> {}"
+shows "underS (suc {a}) = under a"
+proof-
+ have 1: "AboveS {a} \<noteq> {}"
+ using ABV aboveS_AboveS_singl by auto
+ have 2: "a \<noteq> suc {a} \<and> (a,suc {a}) \<in> r"
+ using suc_greater[of "{a}" a] IN 1 by auto
+ (* *)
+ have "underS (suc {a}) \<le> under a"
+ proof(unfold underS_def under_def, auto)
+ fix x assume *: "x \<noteq> suc {a}" and **: "(x,suc {a}) \<in> r"
+ with suc_singl_pred[of a x] IN ABV
+ have "x = a \<or> (x,a) \<in> r" by auto
+ with REFL refl_on_def[of _ r] IN
+ show "(x,a) \<in> r" by auto
+ qed
+ (* *)
+ moreover
+ (* *)
+ have "under a \<le> underS (suc {a})"
+ proof(unfold underS_def under_def, auto)
+ assume "(suc {a}, a) \<in> r"
+ with 2 ANTISYM antisym_def[of r]
+ show False by auto
+ next
+ fix x assume *: "(x,a) \<in> r"
+ with 2 TRANS trans_def[of r]
+ show "(x,suc {a}) \<in> r" by blast
+ (* blast is often better than auto/auto for transitivity-like properties *)
+ qed
+ (* *)
+ ultimately show ?thesis by blast
+qed
+
+
+subsubsection {* Properties of order filters *}
+
+lemma (in wo_rel) ofilter_INTER:
+"\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> ofilter(A i)\<rbrakk> \<Longrightarrow> ofilter (\<Inter> i \<in> I. A i)"
+unfolding ofilter_def by blast
+
+lemma (in wo_rel) ofilter_Inter:
+"\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> ofilter A\<rbrakk> \<Longrightarrow> ofilter (Inter S)"
+unfolding ofilter_def by blast
+
+lemma (in wo_rel) ofilter_Union:
+"(\<And> A. A \<in> S \<Longrightarrow> ofilter A) \<Longrightarrow> ofilter (Union S)"
+unfolding ofilter_def by blast
+
+lemma (in wo_rel) ofilter_under_Union:
+"ofilter A \<Longrightarrow> A = Union {under a| a. a \<in> A}"
+using ofilter_under_UNION[of A]
+by(unfold Union_eq, auto)
+
+
+subsubsection{* Other properties *}
+
+lemma (in wo_rel) Trans_Under_regressive:
+assumes NE: "A \<noteq> {}" and SUB: "A \<le> Field r"
+shows "Under(Under A) \<le> Under A"
+proof
+ let ?a = "minim A"
+ (* Preliminary facts *)
+ have 1: "minim A \<in> Under A"
+ using assms minim_Under by auto
+ have 2: "\<forall>y \<in> A. (minim A, y) \<in> r"
+ using assms minim_least by auto
+ (* Main proof *)
+ fix x assume "x \<in> Under(Under A)"
+ with 1 have 1: "(x,minim A) \<in> r"
+ using Under_def by auto
+ with Field_def have "x \<in> Field r" by fastforce
+ moreover
+ {fix y assume *: "y \<in> A"
+ hence "(x,y) \<in> r"
+ using 1 2 TRANS trans_def[of r] by blast
+ with Field_def have "(x,y) \<in> r" by auto
+ }
+ ultimately
+ show "x \<in> Under A" unfolding Under_def by auto
+qed
+
+lemma (in wo_rel) ofilter_suc_Field:
+assumes OF: "ofilter A" and NE: "A \<noteq> Field r"
+shows "ofilter (A \<union> {suc A})"
+proof-
+ (* Preliminary facts *)
+ have 1: "A \<le> Field r" using OF ofilter_def by auto
+ hence 2: "AboveS A \<noteq> {}"
+ using ofilter_AboveS_Field NE OF by blast
+ from 1 2 suc_inField
+ have 3: "suc A \<in> Field r" by auto
+ (* Main proof *)
+ show ?thesis
+ proof(unfold ofilter_def, auto simp add: 1 3)
+ fix a x
+ assume "a \<in> A" "x \<in> under a" "x \<notin> A"
+ with OF ofilter_def have False by auto
+ thus "x = suc A" by simp
+ next
+ fix x assume *: "x \<in> under (suc A)" and **: "x \<notin> A"
+ hence "x \<in> Field r" using under_def Field_def by fastforce
+ with ** have "x \<in> AboveS A"
+ using ofilter_AboveS_Field[of A] OF by auto
+ hence "(suc A,x) \<in> r"
+ using suc_least_AboveS by auto
+ moreover
+ have "(x,suc A) \<in> r" using * under_def by auto
+ ultimately show "x = suc A"
+ using ANTISYM antisym_def[of r] by auto
+ qed
+qed
+
+(* FIXME: needed? *)
+declare (in wo_rel)
+ minim_in[simp]
+ minim_inField[simp]
+ minim_least[simp]
+ under_ofilter[simp]
+ underS_ofilter[simp]
+ Field_ofilter[simp]
+ ofilter_Under[simp]
+ ofilter_UnderS[simp]
+ ofilter_Int[simp]
+ ofilter_Un[simp]
+
+abbreviation "worec \<equiv> wo_rel.worec"
+abbreviation "adm_wo \<equiv> wo_rel.adm_wo"
+abbreviation "isMinim \<equiv> wo_rel.isMinim"
+abbreviation "minim \<equiv> wo_rel.minim"
+abbreviation "max2 \<equiv> wo_rel.max2"
+abbreviation "supr \<equiv> wo_rel.supr"
+abbreviation "suc \<equiv> wo_rel.suc"
+abbreviation "ofilter \<equiv> wo_rel.ofilter"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Wellorder_Relation_Base.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,669 @@
+(* Title: HOL/Ordinals_and_Cardinals/Wellorder_Relation_Base.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Well-order relations (base).
+*)
+
+header {* Well-Order Relations (Base) *}
+
+theory Wellorder_Relation_Base
+imports Wellfounded_More_Base
+begin
+
+
+text{* In this section, we develop basic concepts and results pertaining
+to well-order relations. Note that we consider well-order relations
+as {\em non-strict relations},
+i.e., as containing the diagonals of their fields. *}
+
+
+locale wo_rel = rel + assumes WELL: "Well_order r"
+begin
+
+text{* The following context encompasses all this section. In other words,
+for the whole section, we consider a fixed well-order relation @{term "r"}. *}
+
+(* context wo_rel *)
+
+
+subsection {* Auxiliaries *}
+
+
+lemma REFL: "Refl r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma TRANS: "trans r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma ANTISYM: "antisym r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma TOTAL: "Total r"
+using WELL order_on_defs[of _ r] by auto
+
+
+lemma TOTALS: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
+using REFL TOTAL refl_on_def[of _ r] total_on_def[of _ r] by force
+
+
+lemma LIN: "Linear_order r"
+using WELL well_order_on_def[of _ r] by auto
+
+
+lemma WF: "wf (r - Id)"
+using WELL well_order_on_def[of _ r] by auto
+
+
+lemma cases_Total:
+"\<And> phi a b. \<lbrakk>{a,b} <= Field r; ((a,b) \<in> r \<Longrightarrow> phi a b); ((b,a) \<in> r \<Longrightarrow> phi a b)\<rbrakk>
+ \<Longrightarrow> phi a b"
+using TOTALS by auto
+
+
+lemma cases_Total3:
+"\<And> phi a b. \<lbrakk>{a,b} \<le> Field r; ((a,b) \<in> r - Id \<or> (b,a) \<in> r - Id \<Longrightarrow> phi a b);
+ (a = b \<Longrightarrow> phi a b)\<rbrakk> \<Longrightarrow> phi a b"
+using TOTALS by auto
+
+
+subsection {* Well-founded induction and recursion adapted to non-strict well-order relations *}
+
+
+text{* Here we provide induction and recursion principles specific to {\em non-strict}
+well-order relations.
+Although minor variations of those for well-founded relations, they will be useful
+for doing away with the tediousness of
+having to take out the diagonal each time in order to switch to a well-founded relation. *}
+
+
+lemma well_order_induct:
+assumes IND: "\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x"
+shows "P a"
+proof-
+ have "\<And>x. \<forall>y. (y, x) \<in> r - Id \<longrightarrow> P y \<Longrightarrow> P x"
+ using IND by blast
+ thus "P a" using WF wf_induct[of "r - Id" P a] by blast
+qed
+
+
+definition
+worec :: "(('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+where
+"worec F \<equiv> wfrec (r - Id) F"
+
+
+definition
+adm_wo :: "(('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool"
+where
+"adm_wo H \<equiv> \<forall>f g x. (\<forall>y \<in> underS x. f y = g y) \<longrightarrow> H f x = H g x"
+
+
+lemma worec_fixpoint:
+assumes ADM: "adm_wo H"
+shows "worec H = H (worec H)"
+proof-
+ let ?rS = "r - Id"
+ have "adm_wf (r - Id) H"
+ unfolding adm_wf_def
+ using ADM adm_wo_def[of H] underS_def by auto
+ hence "wfrec ?rS H = H (wfrec ?rS H)"
+ using WF wfrec_fixpoint[of ?rS H] by simp
+ thus ?thesis unfolding worec_def .
+qed
+
+
+subsection {* The notions of maximum, minimum, supremum, successor and order filter *}
+
+
+text{*
+We define the successor {\em of a set}, and not of an element (the latter is of course
+a particular case). Also, we define the maximum {\em of two elements}, @{text "max2"},
+and the minimum {\em of a set}, @{text "minim"} -- we chose these variants since we
+consider them the most useful for well-orders. The minimum is defined in terms of the
+auxiliary relational operator @{text "isMinim"}. Then, supremum and successor are
+defined in terms of minimum as expected.
+The minimum is only meaningful for non-empty sets, and the successor is only
+meaningful for sets for which strict upper bounds exist.
+Order filters for well-orders are also known as ``initial segments". *}
+
+
+definition max2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+where "max2 a b \<equiv> if (a,b) \<in> r then b else a"
+
+
+definition isMinim :: "'a set \<Rightarrow> 'a \<Rightarrow> bool"
+where "isMinim A b \<equiv> b \<in> A \<and> (\<forall>a \<in> A. (b,a) \<in> r)"
+
+definition minim :: "'a set \<Rightarrow> 'a"
+where "minim A \<equiv> THE b. isMinim A b"
+
+
+definition supr :: "'a set \<Rightarrow> 'a"
+where "supr A \<equiv> minim (Above A)"
+
+definition suc :: "'a set \<Rightarrow> 'a"
+where "suc A \<equiv> minim (AboveS A)"
+
+definition ofilter :: "'a set \<Rightarrow> bool"
+where
+"ofilter A \<equiv> (A \<le> Field r) \<and> (\<forall>a \<in> A. under a \<le> A)"
+
+
+subsubsection {* Properties of max2 *}
+
+
+lemma max2_greater_among:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(a, max2 a b) \<in> r \<and> (b, max2 a b) \<in> r \<and> max2 a b \<in> {a,b}"
+proof-
+ {assume "(a,b) \<in> r"
+ hence ?thesis using max2_def assms REFL refl_on_def
+ by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume "a = b"
+ hence "(a,b) \<in> r" using REFL assms
+ by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume *: "a \<noteq> b \<and> (b,a) \<in> r"
+ hence "(a,b) \<notin> r" using ANTISYM
+ by (auto simp add: antisym_def)
+ hence ?thesis using * max2_def assms REFL refl_on_def
+ by (auto simp add: refl_on_def)
+ }
+ ultimately show ?thesis using assms TOTAL
+ total_on_def[of "Field r" r] by blast
+qed
+
+
+lemma max2_greater:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(a, max2 a b) \<in> r \<and> (b, max2 a b) \<in> r"
+using assms by (auto simp add: max2_greater_among)
+
+
+lemma max2_among:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "max2 a b \<in> {a, b}"
+using assms max2_greater_among[of a b] by simp
+
+
+lemma max2_equals1:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(max2 a b = a) = ((b,a) \<in> r)"
+using assms ANTISYM unfolding antisym_def using TOTALS
+by(auto simp add: max2_def max2_among)
+
+
+lemma max2_equals2:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "(max2 a b = b) = ((a,b) \<in> r)"
+using assms ANTISYM unfolding antisym_def using TOTALS
+unfolding max2_def by auto
+
+
+subsubsection {* Existence and uniqueness for isMinim and well-definedness of minim *}
+
+
+lemma isMinim_unique:
+assumes MINIM: "isMinim B a" and MINIM': "isMinim B a'"
+shows "a = a'"
+proof-
+ {have "a \<in> B"
+ using MINIM isMinim_def by simp
+ hence "(a',a) \<in> r"
+ using MINIM' isMinim_def by simp
+ }
+ moreover
+ {have "a' \<in> B"
+ using MINIM' isMinim_def by simp
+ hence "(a,a') \<in> r"
+ using MINIM isMinim_def by simp
+ }
+ ultimately
+ show ?thesis using ANTISYM antisym_def[of r] by blast
+qed
+
+
+lemma Well_order_isMinim_exists:
+assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
+shows "\<exists>b. isMinim B b"
+proof-
+ from WF wf_eq_minimal[of "r - Id"] NE Id_def obtain b where
+ *: "b \<in> B \<and> (\<forall>b'. b' \<noteq> b \<and> (b',b) \<in> r \<longrightarrow> b' \<notin> B)" by force
+ show ?thesis
+ proof(simp add: isMinim_def, rule exI[of _ b], auto)
+ show "b \<in> B" using * by simp
+ next
+ fix b' assume As: "b' \<in> B"
+ hence **: "b \<in> Field r \<and> b' \<in> Field r" using As SUB * by auto
+ (* *)
+ from As * have "b' = b \<or> (b',b) \<notin> r" by auto
+ moreover
+ {assume "b' = b"
+ hence "(b,b') \<in> r"
+ using ** REFL by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume "b' \<noteq> b \<and> (b',b) \<notin> r"
+ hence "(b,b') \<in> r"
+ using ** TOTAL by (auto simp add: total_on_def)
+ }
+ ultimately show "(b,b') \<in> r" by blast
+ qed
+qed
+
+
+lemma minim_isMinim:
+assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
+shows "isMinim B (minim B)"
+proof-
+ let ?phi = "(\<lambda> b. isMinim B b)"
+ from assms Well_order_isMinim_exists
+ obtain b where *: "?phi b" by blast
+ moreover
+ have "\<And> b'. ?phi b' \<Longrightarrow> b' = b"
+ using isMinim_unique * by auto
+ ultimately show ?thesis
+ unfolding minim_def using theI[of ?phi b] by blast
+qed
+
+
+subsubsection{* Properties of minim *}
+
+
+lemma minim_in:
+assumes "B \<le> Field r" and "B \<noteq> {}"
+shows "minim B \<in> B"
+proof-
+ from minim_isMinim[of B] assms
+ have "isMinim B (minim B)" by simp
+ thus ?thesis by (simp add: isMinim_def)
+qed
+
+
+lemma minim_inField:
+assumes "B \<le> Field r" and "B \<noteq> {}"
+shows "minim B \<in> Field r"
+proof-
+ have "minim B \<in> B" using assms by (simp add: minim_in)
+ thus ?thesis using assms by blast
+qed
+
+
+lemma minim_least:
+assumes SUB: "B \<le> Field r" and IN: "b \<in> B"
+shows "(minim B, b) \<in> r"
+proof-
+ from minim_isMinim[of B] assms
+ have "isMinim B (minim B)" by auto
+ thus ?thesis by (auto simp add: isMinim_def IN)
+qed
+
+
+lemma equals_minim:
+assumes SUB: "B \<le> Field r" and IN: "a \<in> B" and
+ LEAST: "\<And> b. b \<in> B \<Longrightarrow> (a,b) \<in> r"
+shows "a = minim B"
+proof-
+ from minim_isMinim[of B] assms
+ have "isMinim B (minim B)" by auto
+ moreover have "isMinim B a" using IN LEAST isMinim_def by auto
+ ultimately show ?thesis
+ using isMinim_unique by auto
+qed
+
+
+subsubsection{* Properties of successor *}
+
+
+lemma suc_AboveS:
+assumes SUB: "B \<le> Field r" and ABOVES: "AboveS B \<noteq> {}"
+shows "suc B \<in> AboveS B"
+proof(unfold suc_def)
+ have "AboveS B \<le> Field r"
+ using AboveS_Field by auto
+ thus "minim (AboveS B) \<in> AboveS B"
+ using assms by (simp add: minim_in)
+qed
+
+
+lemma suc_greater:
+assumes SUB: "B \<le> Field r" and ABOVES: "AboveS B \<noteq> {}" and
+ IN: "b \<in> B"
+shows "suc B \<noteq> b \<and> (b,suc B) \<in> r"
+proof-
+ from assms suc_AboveS
+ have "suc B \<in> AboveS B" by simp
+ with IN AboveS_def show ?thesis by simp
+qed
+
+
+lemma suc_least_AboveS:
+assumes ABOVES: "a \<in> AboveS B"
+shows "(suc B,a) \<in> r"
+proof(unfold suc_def)
+ have "AboveS B \<le> Field r"
+ using AboveS_Field by auto
+ thus "(minim (AboveS B),a) \<in> r"
+ using assms minim_least by simp
+qed
+
+
+lemma suc_inField:
+assumes "B \<le> Field r" and "AboveS B \<noteq> {}"
+shows "suc B \<in> Field r"
+proof-
+ have "suc B \<in> AboveS B" using suc_AboveS assms by simp
+ thus ?thesis
+ using assms AboveS_Field by auto
+qed
+
+
+lemma equals_suc_AboveS:
+assumes SUB: "B \<le> Field r" and ABV: "a \<in> AboveS B" and
+ MINIM: "\<And> a'. a' \<in> AboveS B \<Longrightarrow> (a,a') \<in> r"
+shows "a = suc B"
+proof(unfold suc_def)
+ have "AboveS B \<le> Field r"
+ using AboveS_Field[of B] by auto
+ thus "a = minim (AboveS B)"
+ using assms equals_minim
+ by simp
+qed
+
+
+lemma suc_underS:
+assumes IN: "a \<in> Field r"
+shows "a = suc (underS a)"
+proof-
+ have "underS a \<le> Field r"
+ using underS_Field by auto
+ moreover
+ have "a \<in> AboveS (underS a)"
+ using in_AboveS_underS IN by auto
+ moreover
+ have "\<forall>a' \<in> AboveS (underS a). (a,a') \<in> r"
+ proof(clarify)
+ fix a'
+ assume *: "a' \<in> AboveS (underS a)"
+ hence **: "a' \<in> Field r"
+ using AboveS_Field by auto
+ {assume "(a,a') \<notin> r"
+ hence "a' = a \<or> (a',a) \<in> r"
+ using TOTAL IN ** by (auto simp add: total_on_def)
+ moreover
+ {assume "a' = a"
+ hence "(a,a') \<in> r"
+ using REFL IN ** by (auto simp add: refl_on_def)
+ }
+ moreover
+ {assume "a' \<noteq> a \<and> (a',a) \<in> r"
+ hence "a' \<in> underS a"
+ unfolding underS_def by simp
+ hence "a' \<notin> AboveS (underS a)"
+ using AboveS_disjoint by blast
+ with * have False by simp
+ }
+ ultimately have "(a,a') \<in> r" by blast
+ }
+ thus "(a, a') \<in> r" by blast
+ qed
+ ultimately show ?thesis
+ using equals_suc_AboveS by auto
+qed
+
+
+subsubsection {* Properties of order filters *}
+
+
+lemma under_ofilter:
+"ofilter (under a)"
+proof(unfold ofilter_def under_def, auto simp add: Field_def)
+ fix aa x
+ assume "(aa,a) \<in> r" "(x,aa) \<in> r"
+ thus "(x,a) \<in> r"
+ using TRANS trans_def[of r] by blast
+qed
+
+
+lemma underS_ofilter:
+"ofilter (underS a)"
+proof(unfold ofilter_def underS_def under_def, auto simp add: Field_def)
+ fix aa assume "(a, aa) \<in> r" "(aa, a) \<in> r" and DIFF: "aa \<noteq> a"
+ thus False
+ using ANTISYM antisym_def[of r] by blast
+next
+ fix aa x
+ assume "(aa,a) \<in> r" "aa \<noteq> a" "(x,aa) \<in> r"
+ thus "(x,a) \<in> r"
+ using TRANS trans_def[of r] by blast
+qed
+
+
+lemma Field_ofilter:
+"ofilter (Field r)"
+by(unfold ofilter_def under_def, auto simp add: Field_def)
+
+
+lemma ofilter_underS_Field:
+"ofilter A = ((\<exists>a \<in> Field r. A = underS a) \<or> (A = Field r))"
+proof
+ assume "(\<exists>a\<in>Field r. A = underS a) \<or> A = Field r"
+ thus "ofilter A"
+ by (auto simp: underS_ofilter Field_ofilter)
+next
+ assume *: "ofilter A"
+ let ?One = "(\<exists>a\<in>Field r. A = underS a)"
+ let ?Two = "(A = Field r)"
+ show "?One \<or> ?Two"
+ proof(cases ?Two, simp)
+ let ?B = "(Field r) - A"
+ let ?a = "minim ?B"
+ assume "A \<noteq> Field r"
+ moreover have "A \<le> Field r" using * ofilter_def by simp
+ ultimately have 1: "?B \<noteq> {}" by blast
+ hence 2: "?a \<in> Field r" using minim_inField[of ?B] by blast
+ have 3: "?a \<in> ?B" using minim_in[of ?B] 1 by blast
+ hence 4: "?a \<notin> A" by blast
+ have 5: "A \<le> Field r" using * ofilter_def[of A] by auto
+ (* *)
+ moreover
+ have "A = underS ?a"
+ proof
+ show "A \<le> underS ?a"
+ proof(unfold underS_def, auto simp add: 4)
+ fix x assume **: "x \<in> A"
+ hence 11: "x \<in> Field r" using 5 by auto
+ have 12: "x \<noteq> ?a" using 4 ** by auto
+ have 13: "under x \<le> A" using * ofilter_def ** by auto
+ {assume "(x,?a) \<notin> r"
+ hence "(?a,x) \<in> r"
+ using TOTAL total_on_def[of "Field r" r]
+ 2 4 11 12 by auto
+ hence "?a \<in> under x" using under_def by auto
+ hence "?a \<in> A" using ** 13 by blast
+ with 4 have False by simp
+ }
+ thus "(x,?a) \<in> r" by blast
+ qed
+ next
+ show "underS ?a \<le> A"
+ proof(unfold underS_def, auto)
+ fix x
+ assume **: "x \<noteq> ?a" and ***: "(x,?a) \<in> r"
+ hence 11: "x \<in> Field r" using Field_def by fastforce
+ {assume "x \<notin> A"
+ hence "x \<in> ?B" using 11 by auto
+ hence "(?a,x) \<in> r" using 3 minim_least[of ?B x] by blast
+ hence False
+ using ANTISYM antisym_def[of r] ** *** by auto
+ }
+ thus "x \<in> A" by blast
+ qed
+ qed
+ ultimately have ?One using 2 by blast
+ thus ?thesis by simp
+ qed
+qed
+
+
+lemma ofilter_Under:
+assumes "A \<le> Field r"
+shows "ofilter(Under A)"
+proof(unfold ofilter_def, auto)
+ fix x assume "x \<in> Under A"
+ thus "x \<in> Field r"
+ using Under_Field assms by auto
+next
+ fix a x
+ assume "a \<in> Under A" and "x \<in> under a"
+ thus "x \<in> Under A"
+ using TRANS under_Under_trans by auto
+qed
+
+
+lemma ofilter_UnderS:
+assumes "A \<le> Field r"
+shows "ofilter(UnderS A)"
+proof(unfold ofilter_def, auto)
+ fix x assume "x \<in> UnderS A"
+ thus "x \<in> Field r"
+ using UnderS_Field assms by auto
+next
+ fix a x
+ assume "a \<in> UnderS A" and "x \<in> under a"
+ thus "x \<in> UnderS A"
+ using TRANS ANTISYM under_UnderS_trans by auto
+qed
+
+
+lemma ofilter_Int: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A Int B)"
+unfolding ofilter_def by blast
+
+
+lemma ofilter_Un: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A \<union> B)"
+unfolding ofilter_def by blast
+
+
+lemma ofilter_UNION:
+"(\<And> i. i \<in> I \<Longrightarrow> ofilter(A i)) \<Longrightarrow> ofilter (\<Union> i \<in> I. A i)"
+unfolding ofilter_def by blast
+
+
+lemma ofilter_under_UNION:
+assumes "ofilter A"
+shows "A = (\<Union> a \<in> A. under a)"
+proof
+ have "\<forall>a \<in> A. under a \<le> A"
+ using assms ofilter_def by auto
+ thus "(\<Union> a \<in> A. under a) \<le> A" by blast
+next
+ have "\<forall>a \<in> A. a \<in> under a"
+ using REFL Refl_under_in assms ofilter_def by blast
+ thus "A \<le> (\<Union> a \<in> A. under a)" by blast
+qed
+
+
+subsubsection{* Other properties *}
+
+
+lemma ofilter_linord:
+assumes OF1: "ofilter A" and OF2: "ofilter B"
+shows "A \<le> B \<or> B \<le> A"
+proof(cases "A = Field r")
+ assume Case1: "A = Field r"
+ hence "B \<le> A" using OF2 ofilter_def by auto
+ thus ?thesis by simp
+next
+ assume Case2: "A \<noteq> Field r"
+ with ofilter_underS_Field OF1 obtain a where
+ 1: "a \<in> Field r \<and> A = underS a" by auto
+ show ?thesis
+ proof(cases "B = Field r")
+ assume Case21: "B = Field r"
+ hence "A \<le> B" using OF1 ofilter_def by auto
+ thus ?thesis by simp
+ next
+ assume Case22: "B \<noteq> Field r"
+ with ofilter_underS_Field OF2 obtain b where
+ 2: "b \<in> Field r \<and> B = underS b" by auto
+ have "a = b \<or> (a,b) \<in> r \<or> (b,a) \<in> r"
+ using 1 2 TOTAL total_on_def[of _ r] by auto
+ moreover
+ {assume "a = b" with 1 2 have ?thesis by auto
+ }
+ moreover
+ {assume "(a,b) \<in> r"
+ with underS_incr TRANS ANTISYM 1 2
+ have "A \<le> B" by auto
+ hence ?thesis by auto
+ }
+ moreover
+ {assume "(b,a) \<in> r"
+ with underS_incr TRANS ANTISYM 1 2
+ have "B \<le> A" by auto
+ hence ?thesis by auto
+ }
+ ultimately show ?thesis by blast
+ qed
+qed
+
+
+lemma ofilter_AboveS_Field:
+assumes "ofilter A"
+shows "A \<union> (AboveS A) = Field r"
+proof
+ show "A \<union> (AboveS A) \<le> Field r"
+ using assms ofilter_def AboveS_Field by auto
+next
+ {fix x assume *: "x \<in> Field r" and **: "x \<notin> A"
+ {fix y assume ***: "y \<in> A"
+ with ** have 1: "y \<noteq> x" by auto
+ {assume "(y,x) \<notin> r"
+ moreover
+ have "y \<in> Field r" using assms ofilter_def *** by auto
+ ultimately have "(x,y) \<in> r"
+ using 1 * TOTAL total_on_def[of _ r] by auto
+ with *** assms ofilter_def under_def have "x \<in> A" by auto
+ with ** have False by contradiction
+ }
+ hence "(y,x) \<in> r" by blast
+ with 1 have "y \<noteq> x \<and> (y,x) \<in> r" by auto
+ }
+ with * have "x \<in> AboveS A" unfolding AboveS_def by auto
+ }
+ thus "Field r \<le> A \<union> (AboveS A)" by blast
+qed
+
+
+lemma suc_ofilter_in:
+assumes OF: "ofilter A" and ABOVE_NE: "AboveS A \<noteq> {}" and
+ REL: "(b,suc A) \<in> r" and DIFF: "b \<noteq> suc A"
+shows "b \<in> A"
+proof-
+ have *: "suc A \<in> Field r \<and> b \<in> Field r"
+ using WELL REL well_order_on_domain by auto
+ {assume **: "b \<notin> A"
+ hence "b \<in> AboveS A"
+ using OF * ofilter_AboveS_Field by auto
+ hence "(suc A, b) \<in> r"
+ using suc_least_AboveS by auto
+ hence False using REL DIFF ANTISYM *
+ by (auto simp add: antisym_def)
+ }
+ thus ?thesis by blast
+qed
+
+
+
+end (* context wo_rel *)
+
+
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/document/intro.tex Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,123 @@
+\newcommand{\eqo}{\mbox{$=\!\!o$}}
+\newcommand{\leqo}{\mbox{$\leq\!\!o$}}
+\newcommand{\lesso}{\mbox{$<\!\!o$}}
+
+
+\begin{abstract}
+We develop a basic theory of ordinals and cardinals in Isabelle/HOL, up to the
+point where some cardinality facts relevant for the ``working mathematician" become available.
+Unlike in set theory, here we do not have at hand canonical notions of ordinal and cardinal.
+Therefore, here an ordinal is merely a well-order relation and a cardinal is an
+ordinal minim w.r.t. order embedding on its field.
+\end{abstract}
+
+
+
+\section{Introduction}
+
+In set theory (under formalizations such as Zermelo-Fraenkel or Von Neumann-Bernays-G\"{o}del), an
+{\em ordinal} is a special kind of well-order, namely one
+whose strict version is the restriction of the membership relation to a set. In particular,
+the field of a set-theoretic ordinal is a transitive set, and the non-strict version
+of an ordinal relation is set inclusion. Set-theoretic ordinals enjoy the nice properties
+of membership on transitive sets, while at the same time forming a complete class of
+representatives for well-orders (since any well-order turns out isomorphic to an ordinal).
+Moreover, the class of ordinals is itself transitive and well-ordered by membership as the strict relation
+and inclusion as the non-strict relation.
+Also knowing that any set can be well-ordered (in the presence of the axiom of choice), one then defines
+the {\em cardinal} of a set to be the smallest ordinal isomorphic to a well-order on that set.
+This makes the class of cardinals a complete set of representatives for the intuitive notion
+of set cardinality.\footnote{The ``intuitive" cardinality of a set $A$ is the class of all
+sets equipollent to $A$, i.e., being in bijection with $A$.}
+The ability to produce {\em canonical well-orders} from the membership relation (having the aforementioned
+convenient properties)
+allows for a harmonious development of the theory of cardinals in set-theoretic settings.
+Non-trivial cardinality results, such as $A$ being equipollent to $A \times A$ for any infinite $A$,
+follow rather quickly within this theory.
+
+However, a canonical notion of well-order is {\em not} available in HOL.
+Here, one has to do with well-order ``as is", but otherwise has all the necessary infrastructure
+(including Hilbert choice) to ``climb" well-orders recursively and to well-oder arbitrary sets.
+
+The current work, formalized in Isabelle/HOL, develops the basic theory of ordinals and cardinals
+up to the point where there are inferred a collection of non-trivial cardinality facts useful
+for the ``working mathematician", among which:
+\begin{itemize}
+\item Given any two sets (on any two given types)\footnote{Recall that, in HOL, a set
+on a type $\alpha$ is modeled, just like a predicate, as a function from $\alpha$ to \textsf{bool}.},
+one is injectable in the other.
+\item If at least one of two sets is infinite, then their sum and their Cartesian product are equipollent
+to the larger of the two.
+\item The set of lists (and also the set of finite sets) with element from an
+infinite set is equipollent with that set.
+\end{itemize}
+
+Our development emulates the standard one from set-theory, but keeps everything
+{\em up to order isomorphism}.
+An (HOL) ordinal is merely a well-order. An {\em ordinal embedding} is an
+injective and order-compatible function which maps its source into an initial segment
+(i.e., order filter) of its target. Now, a {\em cardinal} (called in this work a {\em cardinal order})
+is an ordinal minim w.r.t. the existence of embeddings among all
+well-orders on its field. After showing the existence of cardinals on any given set,
+we define the cardinal of a set $A$, denoted $|A|$, to be {\em some} cardinal order
+on $A$. This concept is unique only up to order isomorphism (denoted $\eqo$), but meets
+its purpose: any two sets $A$ and $B$ (laying at potentially distinct types)
+are in bijection if and only if $|A|\;\eqo\;|B|$. Moreover, we also show that numeric cardinals
+assigned to finite sets\footnote{Numeric cardinals of finite sets are already formalized in
+Isabelle/HOL.}
+are {\em conservatively extended} by our general (order-theoretic) notion of
+cardinal. We study the interaction of cardinals with standard set-theoretic
+constructions such as powersets, products, sums and lists. These constructions are shown
+to preserve the ``cardinal identity" $\eqo$ and also to be monotonic w.r.t. $\leqo$, the ordinal
+embedding relation. By studying the interaction between these constructions, infinite sets and
+cardinals, we obtain the
+aforementioned results for ``working mathematicians".
+
+For this development, we did not follow closely any particular textbook, and in fact are not
+aware of such basic theory of cardinals previously
+developed in HOL.\footnote{After writing this formalization, we became aware of
+Paul Taylor's membership-free development of the theory of ordinals \cite{taylor-ordinals}.} On
+the other hand,
+the set-theoretic versions of the facts proved here are folklore in set theory, and can be found,
+e.g., in the textbook \cite{card-book}. Beyond taking care of some locality aspects
+concerning the spreading of our concepts throughout types, we have not departed
+much from the techniques used in set theory for establishing these facts -- for instance,
+in the proof of one of our major theorems,
+\textit{Card-order-Times-same-infinite} from Section 8.4,\footnote{This theorem states that, for any
+infinite cardinal $r$ on a set $A$, $|A\times A|$ is not larger than $r$.}
+we have essentially applied the technique described, e.g., in the proof of
+theorem 1.5.11 from \cite{card-book}, page 47.
+
+Here is the structure of the rest of this document.
+
+The next three sections, 2-4, develop some
+mathematical prerequisites.
+In Section 2, a large collection of simple facts about
+injections, bijections, inverses, (in)finite sets and numeric cardinals are proved,
+making life easier
+for later, when proving less trivial facts.
+Section 3 introduces upper and lower
+bounds operators for order-like relations and studies their basic properties.
+Section 4 states some useful variations of well-founded recursion and induction principles.
+
+Then come the major sections, 5-8.
+Section 5 defines and studies, in the context of a well-order relation,
+the notions of minimum (of a set), maximum (of two elements), supremum, successor (of a set),
+and order filter (i.e., initial segment, i.e., downward-closed set).
+Section 6 defines and studies (well-order) embeddings, strict embeddings, isomorphisms, and
+compatible functions.
+Section 7 deals with various constructions on well-orders, and with the relations
+$\leqo$, $\lesso$ and $\eqo$ of well-order embedding, strict embedding, and isomorphism, respectively.
+Section 8 defines and studies cardinal order relations, the cardinal of a set, the connection
+of cardinals with set-theoretic constructs,
+the canonical cardinal of natural numbers and finite cardinals, the successor
+of a cardinal, as well as regular cardinals. (The latter play a crucial role in the development of
+a new (co)datatype package in HOL.)
+
+Finally, section 9 provides an abstraction of the previous developments on
+cardinals, to provide a simpler user interface to cardinals, which in most of
+the cases allows to forget that cardinals are represented by orders and use them
+through defined arithmetic operators.
+
+More informal details are provided at the beginning of each section, and also at the
+beginning of some of the subsections.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/document/isabelle.sty Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,233 @@
+%%
+%% macros for Isabelle generated LaTeX output
+%%
+
+%%% Simple document preparation (based on theory token language and symbols)
+
+% isabelle environments
+
+\newcommand{\isabellecontext}{UNKNOWN}
+
+\newcommand{\isastyle}{\UNDEF}
+\newcommand{\isastyleminor}{\UNDEF}
+\newcommand{\isastylescript}{\UNDEF}
+\newcommand{\isastyletext}{\normalsize\rm}
+\newcommand{\isastyletxt}{\rm}
+\newcommand{\isastylecmt}{\rm}
+
+\newcommand{\isaspacing}{%
+ \sfcode 42 1000 % .
+ \sfcode 63 1000 % ?
+ \sfcode 33 1000 % !
+ \sfcode 58 1000 % :
+ \sfcode 59 1000 % ;
+ \sfcode 44 1000 % ,
+}
+
+%symbol markup -- \emph achieves decent spacing via italic corrections
+\newcommand{\isamath}[1]{\emph{$#1$}}
+\newcommand{\isatext}[1]{\emph{#1}}
+\DeclareRobustCommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isaspacing\isastylescript##1}}}
+\newcommand{\isactrlsub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
+\newcommand{\isactrlsup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
+\newcommand{\isactrlisub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
+\newcommand{\isactrlisup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
+\DeclareRobustCommand{\isactrlbsub}{\emph\bgroup\math{}\sb\bgroup\mbox\bgroup\isaspacing\isastylescript}
+\DeclareRobustCommand{\isactrlesub}{\egroup\egroup\endmath\egroup}
+\DeclareRobustCommand{\isactrlbsup}{\emph\bgroup\math{}\sp\bgroup\mbox\bgroup\isaspacing\isastylescript}
+\DeclareRobustCommand{\isactrlesup}{\egroup\egroup\endmath\egroup}
+\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
+
+\newenvironment{isaantiq}{{\isacharat\isacharbraceleft}}{{\isacharbraceright}}
+\newcommand{\isaantiqopen}{\isakeyword{\isacharbraceleft}}
+\newcommand{\isaantiqclose}{\isakeyword{\isacharbraceright}}
+
+\newdimen\isa@parindent\newdimen\isa@parskip
+
+\newenvironment{isabellebody}{%
+\isamarkuptrue\par%
+\isa@parindent\parindent\parindent0pt%
+\isa@parskip\parskip\parskip0pt%
+\isaspacing\isastyle}{\par}
+
+\newenvironment{isabelle}
+{\begin{trivlist}\begin{isabellebody}\item\relax}
+{\end{isabellebody}\end{trivlist}}
+
+\newcommand{\isa}[1]{\emph{\isaspacing\isastyleminor #1}}
+
+\newcommand{\isaindent}[1]{\hphantom{#1}}
+\newcommand{\isanewline}{\mbox{}\par\mbox{}}
+\newcommand{\isasep}{}
+\newcommand{\isadigit}[1]{#1}
+
+\newcommand{\isachardefaults}{%
+\chardef\isacharbang=`\!%
+\chardef\isachardoublequote=`\"%
+\chardef\isachardoublequoteopen=`\"%
+\chardef\isachardoublequoteclose=`\"%
+\chardef\isacharhash=`\#%
+\chardef\isachardollar=`\$%
+\chardef\isacharpercent=`\%%
+\chardef\isacharampersand=`\&%
+\chardef\isacharprime=`\'%
+\chardef\isacharparenleft=`\(%
+\chardef\isacharparenright=`\)%
+\chardef\isacharasterisk=`\*%
+\chardef\isacharplus=`\+%
+\chardef\isacharcomma=`\,%
+\chardef\isacharminus=`\-%
+\chardef\isachardot=`\.%
+\chardef\isacharslash=`\/%
+\chardef\isacharcolon=`\:%
+\chardef\isacharsemicolon=`\;%
+\chardef\isacharless=`\<%
+\chardef\isacharequal=`\=%
+\chardef\isachargreater=`\>%
+\chardef\isacharquery=`\?%
+\chardef\isacharat=`\@%
+\chardef\isacharbrackleft=`\[%
+\chardef\isacharbackslash=`\\%
+\chardef\isacharbrackright=`\]%
+\chardef\isacharcircum=`\^%
+\chardef\isacharunderscore=`\_%
+\def\isacharunderscorekeyword{\_}%
+\chardef\isacharbackquote=`\`%
+\chardef\isacharbackquoteopen=`\`%
+\chardef\isacharbackquoteclose=`\`%
+\chardef\isacharbraceleft=`\{%
+\chardef\isacharbar=`\|%
+\chardef\isacharbraceright=`\}%
+\chardef\isachartilde=`\~%
+\def\isacharverbatimopen{\isacharbraceleft\isacharasterisk}%
+\def\isacharverbatimclose{\isacharasterisk\isacharbraceright}%
+}
+
+\newcommand{\isaliteral}[2]{#2}
+\newcommand{\isanil}{}
+
+
+% keyword and section markup
+
+\newcommand{\isakeyword}[1]
+{\emph{\bf\def\isachardot{.}\def\isacharunderscore{\isacharunderscorekeyword}%
+\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
+\newcommand{\isacommand}[1]{\isakeyword{#1}}
+
+\newcommand{\isamarkupheader}[1]{\section{#1}}
+\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
+\newcommand{\isamarkupsection}[1]{\section{#1}}
+\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
+\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
+\newcommand{\isamarkupsect}[1]{\section{#1}}
+\newcommand{\isamarkupsubsect}[1]{\subsection{#1}}
+\newcommand{\isamarkupsubsubsect}[1]{\subsubsection{#1}}
+
+\newif\ifisamarkup
+\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
+\newcommand{\isaendpar}{\par\medskip}
+\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
+\newenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}}{\end{isapar}}
+\newenvironment{isamarkuptxt}{\par\isastyletxt\begin{isapar}}{\end{isapar}}
+\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
+
+
+% styles
+
+\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
+
+\newcommand{\isabellestyledefault}{%
+\def\isastyle{\small\tt\slshape}%
+\def\isastyleminor{\small\tt\slshape}%
+\def\isastylescript{\footnotesize\tt\slshape}%
+\isachardefaults%
+}
+\isabellestyledefault
+
+\newcommand{\isabellestylett}{%
+\def\isastyle{\small\tt}%
+\def\isastyleminor{\small\tt}%
+\def\isastylescript{\footnotesize\tt}%
+\isachardefaults%
+}
+
+\newcommand{\isabellestyleit}{%
+\def\isastyle{\small\it}%
+\def\isastyleminor{\it}%
+\def\isastylescript{\footnotesize\it}%
+\isachardefaults%
+\def\isacharunderscorekeyword{\mbox{-}}%
+\def\isacharbang{\isamath{!}}%
+\def\isachardoublequote{\isanil}%
+\def\isachardoublequoteopen{\isanil}%
+\def\isachardoublequoteclose{\isanil}%
+\def\isacharhash{\isamath{\#}}%
+\def\isachardollar{\isamath{\$}}%
+\def\isacharpercent{\isamath{\%}}%
+\def\isacharampersand{\isamath{\&}}%
+\def\isacharprime{\isamath{\mskip2mu{'}\mskip-2mu}}%
+\def\isacharparenleft{\isamath{(}}%
+\def\isacharparenright{\isamath{)}}%
+\def\isacharasterisk{\isamath{*}}%
+\def\isacharplus{\isamath{+}}%
+\def\isacharcomma{\isamath{\mathord,}}%
+\def\isacharminus{\isamath{-}}%
+\def\isachardot{\isamath{\mathord.}}%
+\def\isacharslash{\isamath{/}}%
+\def\isacharcolon{\isamath{\mathord:}}%
+\def\isacharsemicolon{\isamath{\mathord;}}%
+\def\isacharless{\isamath{<}}%
+\def\isacharequal{\isamath{=}}%
+\def\isachargreater{\isamath{>}}%
+\def\isacharat{\isamath{@}}%
+\def\isacharbrackleft{\isamath{[}}%
+\def\isacharbackslash{\isamath{\backslash}}%
+\def\isacharbrackright{\isamath{]}}%
+\def\isacharunderscore{\mbox{-}}%
+\def\isacharbraceleft{\isamath{\{}}%
+\def\isacharbar{\isamath{\mid}}%
+\def\isacharbraceright{\isamath{\}}}%
+\def\isachartilde{\isamath{{}\sp{\sim}}}%
+\def\isacharbackquoteopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
+\def\isacharbackquoteclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
+\def\isacharverbatimopen{\isamath{\langle\!\langle}}%
+\def\isacharverbatimclose{\isamath{\rangle\!\rangle}}%
+}
+
+\newcommand{\isabellestyleitunderscore}{%
+\isabellestyleit%
+\def\isacharunderscore{\_}%
+\def\isacharunderscorekeyword{\_}%
+}
+
+\newcommand{\isabellestylesl}{%
+\isabellestyleit%
+\def\isastyle{\small\sl}%
+\def\isastyleminor{\sl}%
+\def\isastylescript{\footnotesize\sl}%
+}
+
+
+% tagged regions
+
+%plain TeX version of comment package -- much faster!
+\let\isafmtname\fmtname\def\fmtname{plain}
+\usepackage{comment}
+\let\fmtname\isafmtname
+
+\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
+
+\newcommand{\isakeeptag}[1]%
+{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
+\newcommand{\isadroptag}[1]%
+{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
+\newcommand{\isafoldtag}[1]%
+{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
+
+\isakeeptag{theory}
+\isakeeptag{proof}
+\isakeeptag{ML}
+\isakeeptag{visible}
+\isadroptag{invisible}
+
+\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/document/isabellesym.sty Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,360 @@
+%%
+%% definitions of standard Isabelle symbols
+%%
+
+\newcommand{\isasymzero}{\isamath{\mathbf{0}}} %requires amssymb
+\newcommand{\isasymone}{\isamath{\mathbf{1}}} %requires amssymb
+\newcommand{\isasymtwo}{\isamath{\mathbf{2}}} %requires amssymb
+\newcommand{\isasymthree}{\isamath{\mathbf{3}}} %requires amssymb
+\newcommand{\isasymfour}{\isamath{\mathbf{4}}} %requires amssymb
+\newcommand{\isasymfive}{\isamath{\mathbf{5}}} %requires amssymb
+\newcommand{\isasymsix}{\isamath{\mathbf{6}}} %requires amssymb
+\newcommand{\isasymseven}{\isamath{\mathbf{7}}} %requires amssymb
+\newcommand{\isasymeight}{\isamath{\mathbf{8}}} %requires amssymb
+\newcommand{\isasymnine}{\isamath{\mathbf{9}}} %requires amssymb
+\newcommand{\isasymA}{\isamath{\mathcal{A}}}
+\newcommand{\isasymB}{\isamath{\mathcal{B}}}
+\newcommand{\isasymC}{\isamath{\mathcal{C}}}
+\newcommand{\isasymD}{\isamath{\mathcal{D}}}
+\newcommand{\isasymE}{\isamath{\mathcal{E}}}
+\newcommand{\isasymF}{\isamath{\mathcal{F}}}
+\newcommand{\isasymG}{\isamath{\mathcal{G}}}
+\newcommand{\isasymH}{\isamath{\mathcal{H}}}
+\newcommand{\isasymI}{\isamath{\mathcal{I}}}
+\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
+\newcommand{\isasymK}{\isamath{\mathcal{K}}}
+\newcommand{\isasymL}{\isamath{\mathcal{L}}}
+\newcommand{\isasymM}{\isamath{\mathcal{M}}}
+\newcommand{\isasymN}{\isamath{\mathcal{N}}}
+\newcommand{\isasymO}{\isamath{\mathcal{O}}}
+\newcommand{\isasymP}{\isamath{\mathcal{P}}}
+\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
+\newcommand{\isasymR}{\isamath{\mathcal{R}}}
+\newcommand{\isasymS}{\isamath{\mathcal{S}}}
+\newcommand{\isasymT}{\isamath{\mathcal{T}}}
+\newcommand{\isasymU}{\isamath{\mathcal{U}}}
+\newcommand{\isasymV}{\isamath{\mathcal{V}}}
+\newcommand{\isasymW}{\isamath{\mathcal{W}}}
+\newcommand{\isasymX}{\isamath{\mathcal{X}}}
+\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
+\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
+\newcommand{\isasyma}{\isamath{\mathrm{a}}}
+\newcommand{\isasymb}{\isamath{\mathrm{b}}}
+\newcommand{\isasymc}{\isamath{\mathrm{c}}}
+\newcommand{\isasymd}{\isamath{\mathrm{d}}}
+\newcommand{\isasyme}{\isamath{\mathrm{e}}}
+\newcommand{\isasymf}{\isamath{\mathrm{f}}}
+\newcommand{\isasymg}{\isamath{\mathrm{g}}}
+\newcommand{\isasymh}{\isamath{\mathrm{h}}}
+\newcommand{\isasymi}{\isamath{\mathrm{i}}}
+\newcommand{\isasymj}{\isamath{\mathrm{j}}}
+\newcommand{\isasymk}{\isamath{\mathrm{k}}}
+\newcommand{\isasyml}{\isamath{\mathrm{l}}}
+\newcommand{\isasymm}{\isamath{\mathrm{m}}}
+\newcommand{\isasymn}{\isamath{\mathrm{n}}}
+\newcommand{\isasymo}{\isamath{\mathrm{o}}}
+\newcommand{\isasymp}{\isamath{\mathrm{p}}}
+\newcommand{\isasymq}{\isamath{\mathrm{q}}}
+\newcommand{\isasymr}{\isamath{\mathrm{r}}}
+\newcommand{\isasyms}{\isamath{\mathrm{s}}}
+\newcommand{\isasymt}{\isamath{\mathrm{t}}}
+\newcommand{\isasymu}{\isamath{\mathrm{u}}}
+\newcommand{\isasymv}{\isamath{\mathrm{v}}}
+\newcommand{\isasymw}{\isamath{\mathrm{w}}}
+\newcommand{\isasymx}{\isamath{\mathrm{x}}}
+\newcommand{\isasymy}{\isamath{\mathrm{y}}}
+\newcommand{\isasymz}{\isamath{\mathrm{z}}}
+\newcommand{\isasymAA}{\isamath{\mathfrak{A}}} %requires eufrak
+\newcommand{\isasymBB}{\isamath{\mathfrak{B}}} %requires eufrak
+\newcommand{\isasymCC}{\isamath{\mathfrak{C}}} %requires eufrak
+\newcommand{\isasymDD}{\isamath{\mathfrak{D}}} %requires eufrak
+\newcommand{\isasymEE}{\isamath{\mathfrak{E}}} %requires eufrak
+\newcommand{\isasymFF}{\isamath{\mathfrak{F}}} %requires eufrak
+\newcommand{\isasymGG}{\isamath{\mathfrak{G}}} %requires eufrak
+\newcommand{\isasymHH}{\isamath{\mathfrak{H}}} %requires eufrak
+\newcommand{\isasymII}{\isamath{\mathfrak{I}}} %requires eufrak
+\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}} %requires eufrak
+\newcommand{\isasymKK}{\isamath{\mathfrak{K}}} %requires eufrak
+\newcommand{\isasymLL}{\isamath{\mathfrak{L}}} %requires eufrak
+\newcommand{\isasymMM}{\isamath{\mathfrak{M}}} %requires eufrak
+\newcommand{\isasymNN}{\isamath{\mathfrak{N}}} %requires eufrak
+\newcommand{\isasymOO}{\isamath{\mathfrak{O}}} %requires eufrak
+\newcommand{\isasymPP}{\isamath{\mathfrak{P}}} %requires eufrak
+\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}} %requires eufrak
+\newcommand{\isasymRR}{\isamath{\mathfrak{R}}} %requires eufrak
+\newcommand{\isasymSS}{\isamath{\mathfrak{S}}} %requires eufrak
+\newcommand{\isasymTT}{\isamath{\mathfrak{T}}} %requires eufrak
+\newcommand{\isasymUU}{\isamath{\mathfrak{U}}} %requires eufrak
+\newcommand{\isasymVV}{\isamath{\mathfrak{V}}} %requires eufrak
+\newcommand{\isasymWW}{\isamath{\mathfrak{W}}} %requires eufrak
+\newcommand{\isasymXX}{\isamath{\mathfrak{X}}} %requires eufrak
+\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}} %requires eufrak
+\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}} %requires eufrak
+\newcommand{\isasymaa}{\isamath{\mathfrak{a}}} %requires eufrak
+\newcommand{\isasymbb}{\isamath{\mathfrak{b}}} %requires eufrak
+\newcommand{\isasymcc}{\isamath{\mathfrak{c}}} %requires eufrak
+\newcommand{\isasymdd}{\isamath{\mathfrak{d}}} %requires eufrak
+\newcommand{\isasymee}{\isamath{\mathfrak{e}}} %requires eufrak
+\newcommand{\isasymff}{\isamath{\mathfrak{f}}} %requires eufrak
+\newcommand{\isasymgg}{\isamath{\mathfrak{g}}} %requires eufrak
+\newcommand{\isasymhh}{\isamath{\mathfrak{h}}} %requires eufrak
+\newcommand{\isasymii}{\isamath{\mathfrak{i}}} %requires eufrak
+\newcommand{\isasymjj}{\isamath{\mathfrak{j}}} %requires eufrak
+\newcommand{\isasymkk}{\isamath{\mathfrak{k}}} %requires eufrak
+\newcommand{\isasymll}{\isamath{\mathfrak{l}}} %requires eufrak
+\newcommand{\isasymmm}{\isamath{\mathfrak{m}}} %requires eufrak
+\newcommand{\isasymnn}{\isamath{\mathfrak{n}}} %requires eufrak
+\newcommand{\isasymoo}{\isamath{\mathfrak{o}}} %requires eufrak
+\newcommand{\isasympp}{\isamath{\mathfrak{p}}} %requires eufrak
+\newcommand{\isasymqq}{\isamath{\mathfrak{q}}} %requires eufrak
+\newcommand{\isasymrr}{\isamath{\mathfrak{r}}} %requires eufrak
+\newcommand{\isasymss}{\isamath{\mathfrak{s}}} %requires eufrak
+\newcommand{\isasymtt}{\isamath{\mathfrak{t}}} %requires eufrak
+\newcommand{\isasymuu}{\isamath{\mathfrak{u}}} %requires eufrak
+\newcommand{\isasymvv}{\isamath{\mathfrak{v}}} %requires eufrak
+\newcommand{\isasymww}{\isamath{\mathfrak{w}}} %requires eufrak
+\newcommand{\isasymxx}{\isamath{\mathfrak{x}}} %requires eufrak
+\newcommand{\isasymyy}{\isamath{\mathfrak{y}}} %requires eufrak
+\newcommand{\isasymzz}{\isamath{\mathfrak{z}}} %requires eufrak
+\newcommand{\isasymalpha}{\isamath{\alpha}}
+\newcommand{\isasymbeta}{\isamath{\beta}}
+\newcommand{\isasymgamma}{\isamath{\gamma}}
+\newcommand{\isasymdelta}{\isamath{\delta}}
+\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
+\newcommand{\isasymzeta}{\isamath{\zeta}}
+\newcommand{\isasymeta}{\isamath{\eta}}
+\newcommand{\isasymtheta}{\isamath{\vartheta}}
+\newcommand{\isasymiota}{\isamath{\iota}}
+\newcommand{\isasymkappa}{\isamath{\kappa}}
+\newcommand{\isasymlambda}{\isamath{\lambda}}
+\newcommand{\isasymmu}{\isamath{\mu}}
+\newcommand{\isasymnu}{\isamath{\nu}}
+\newcommand{\isasymxi}{\isamath{\xi}}
+\newcommand{\isasympi}{\isamath{\pi}}
+\newcommand{\isasymrho}{\isamath{\varrho}}
+\newcommand{\isasymsigma}{\isamath{\sigma}}
+\newcommand{\isasymtau}{\isamath{\tau}}
+\newcommand{\isasymupsilon}{\isamath{\upsilon}}
+\newcommand{\isasymphi}{\isamath{\varphi}}
+\newcommand{\isasymchi}{\isamath{\chi}}
+\newcommand{\isasympsi}{\isamath{\psi}}
+\newcommand{\isasymomega}{\isamath{\omega}}
+\newcommand{\isasymGamma}{\isamath{\Gamma}}
+\newcommand{\isasymDelta}{\isamath{\Delta}}
+\newcommand{\isasymTheta}{\isamath{\Theta}}
+\newcommand{\isasymLambda}{\isamath{\Lambda}}
+\newcommand{\isasymXi}{\isamath{\Xi}}
+\newcommand{\isasymPi}{\isamath{\Pi}}
+\newcommand{\isasymSigma}{\isamath{\Sigma}}
+\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
+\newcommand{\isasymPhi}{\isamath{\Phi}}
+\newcommand{\isasymPsi}{\isamath{\Psi}}
+\newcommand{\isasymOmega}{\isamath{\Omega}}
+\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
+\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
+\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
+\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
+\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
+\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
+\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
+\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
+\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
+\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
+\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
+\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
+\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
+\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
+\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
+\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
+\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
+\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
+\newcommand{\isasymmapsto}{\isamath{\mapsto}}
+\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
+\newcommand{\isasymmidarrow}{\isamath{\relbar}}
+\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
+\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
+\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
+\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
+\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
+\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
+\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
+\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
+\newcommand{\isasymleadsto}{\isamath{\leadsto}} %requires amssymb
+\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}} %requires amssymb
+\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}} %requires amssymb
+\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}} %requires amssymb
+\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}} %requires amssymb
+\newcommand{\isasymrestriction}{\isamath{\restriction}} %requires amssymb
+\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
+\newcommand{\isasymup}{\isamath{\uparrow}}
+\newcommand{\isasymUp}{\isamath{\Uparrow}}
+\newcommand{\isasymdown}{\isamath{\downarrow}}
+\newcommand{\isasymDown}{\isamath{\Downarrow}}
+\newcommand{\isasymupdown}{\isamath{\updownarrow}}
+\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
+\newcommand{\isasymlangle}{\isamath{\langle}}
+\newcommand{\isasymrangle}{\isamath{\rangle}}
+\newcommand{\isasymlceil}{\isamath{\lceil}}
+\newcommand{\isasymrceil}{\isamath{\rceil}}
+\newcommand{\isasymlfloor}{\isamath{\lfloor}}
+\newcommand{\isasymrfloor}{\isamath{\rfloor}}
+\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
+\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
+\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
+\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
+\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
+\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
+\newcommand{\isasymguillemotleft}{\isatext{\flqq}} %requires babel
+\newcommand{\isasymguillemotright}{\isatext{\frqq}} %requires babel
+\newcommand{\isasymbottom}{\isamath{\bot}}
+\newcommand{\isasymtop}{\isamath{\top}}
+\newcommand{\isasymand}{\isamath{\wedge}}
+\newcommand{\isasymAnd}{\isamath{\bigwedge}}
+\newcommand{\isasymor}{\isamath{\vee}}
+\newcommand{\isasymOr}{\isamath{\bigvee}}
+\newcommand{\isasymforall}{\isamath{\forall\,}}
+\newcommand{\isasymexists}{\isamath{\exists\,}}
+\newcommand{\isasymnexists}{\isamath{\nexists\,}} %requires amssymb
+\newcommand{\isasymnot}{\isamath{\neg}}
+\newcommand{\isasymbox}{\isamath{\Box}} %requires amssymb
+\newcommand{\isasymdiamond}{\isamath{\Diamond}} %requires amssymb
+\newcommand{\isasymturnstile}{\isamath{\vdash}}
+\newcommand{\isasymTurnstile}{\isamath{\models}}
+\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
+\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
+\newcommand{\isasymstileturn}{\isamath{\dashv}}
+\newcommand{\isasymsurd}{\isamath{\surd}}
+\newcommand{\isasymle}{\isamath{\le}}
+\newcommand{\isasymge}{\isamath{\ge}}
+\newcommand{\isasymlless}{\isamath{\ll}}
+\newcommand{\isasymggreater}{\isamath{\gg}}
+\newcommand{\isasymlesssim}{\isamath{\lesssim}} %requires amssymb
+\newcommand{\isasymgreatersim}{\isamath{\gtrsim}} %requires amssymb
+\newcommand{\isasymlessapprox}{\isamath{\lessapprox}} %requires amssymb
+\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}} %requires amssymb
+\newcommand{\isasymin}{\isamath{\in}}
+\newcommand{\isasymnotin}{\isamath{\notin}}
+\newcommand{\isasymsubset}{\isamath{\subset}}
+\newcommand{\isasymsupset}{\isamath{\supset}}
+\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
+\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
+\newcommand{\isasymsqsubset}{\isamath{\sqsubset}} %requires amssymb
+\newcommand{\isasymsqsupset}{\isamath{\sqsupset}} %requires amssymb
+\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
+\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
+\newcommand{\isasyminter}{\isamath{\cap}}
+\newcommand{\isasymInter}{\isamath{\bigcap\,}}
+\newcommand{\isasymunion}{\isamath{\cup}}
+\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
+\newcommand{\isasymsqunion}{\isamath{\sqcup}}
+\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
+\newcommand{\isasymsqinter}{\isamath{\sqcap}}
+\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}} %requires stmaryrd
+\newcommand{\isasymsetminus}{\isamath{\setminus}}
+\newcommand{\isasympropto}{\isamath{\propto}}
+\newcommand{\isasymuplus}{\isamath{\uplus}}
+\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
+\newcommand{\isasymnoteq}{\isamath{\not=}}
+\newcommand{\isasymsim}{\isamath{\sim}}
+\newcommand{\isasymdoteq}{\isamath{\doteq}}
+\newcommand{\isasymsimeq}{\isamath{\simeq}}
+\newcommand{\isasymapprox}{\isamath{\approx}}
+\newcommand{\isasymasymp}{\isamath{\asymp}}
+\newcommand{\isasymcong}{\isamath{\cong}}
+\newcommand{\isasymsmile}{\isamath{\smile}}
+\newcommand{\isasymequiv}{\isamath{\equiv}}
+\newcommand{\isasymfrown}{\isamath{\frown}}
+\newcommand{\isasymJoin}{\isamath{\Join}} %requires amssymb
+\newcommand{\isasymbowtie}{\isamath{\bowtie}}
+\newcommand{\isasymprec}{\isamath{\prec}}
+\newcommand{\isasymsucc}{\isamath{\succ}}
+\newcommand{\isasympreceq}{\isamath{\preceq}}
+\newcommand{\isasymsucceq}{\isamath{\succeq}}
+\newcommand{\isasymparallel}{\isamath{\parallel}}
+\newcommand{\isasymbar}{\isamath{\mid}}
+\newcommand{\isasymplusminus}{\isamath{\pm}}
+\newcommand{\isasymminusplus}{\isamath{\mp}}
+\newcommand{\isasymtimes}{\isamath{\times}}
+\newcommand{\isasymdiv}{\isamath{\div}}
+\newcommand{\isasymcdot}{\isamath{\cdot}}
+\newcommand{\isasymstar}{\isamath{\star}}
+\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
+\newcommand{\isasymcirc}{\isamath{\circ}}
+\newcommand{\isasymdagger}{\isamath{\dagger}}
+\newcommand{\isasymddagger}{\isamath{\ddagger}}
+\newcommand{\isasymlhd}{\isamath{\lhd}} %requires amssymb
+\newcommand{\isasymrhd}{\isamath{\rhd}} %requires amssymb
+\newcommand{\isasymunlhd}{\isamath{\unlhd}} %requires amssymb
+\newcommand{\isasymunrhd}{\isamath{\unrhd}} %requires amssymb
+\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
+\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
+\newcommand{\isasymtriangle}{\isamath{\triangle}}
+\newcommand{\isasymtriangleq}{\isamath{\triangleq}} %requires amssymb
+\newcommand{\isasymoplus}{\isamath{\oplus}}
+\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
+\newcommand{\isasymotimes}{\isamath{\otimes}}
+\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
+\newcommand{\isasymodot}{\isamath{\odot}}
+\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
+\newcommand{\isasymominus}{\isamath{\ominus}}
+\newcommand{\isasymoslash}{\isamath{\oslash}}
+\newcommand{\isasymdots}{\isamath{\dots}}
+\newcommand{\isasymcdots}{\isamath{\cdots}}
+\newcommand{\isasymSum}{\isamath{\sum\,}}
+\newcommand{\isasymProd}{\isamath{\prod\,}}
+\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
+\newcommand{\isasyminfinity}{\isamath{\infty}}
+\newcommand{\isasymintegral}{\isamath{\int\,}}
+\newcommand{\isasymointegral}{\isamath{\oint\,}}
+\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
+\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
+\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
+\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
+\newcommand{\isasymaleph}{\isamath{\aleph}}
+\newcommand{\isasymemptyset}{\isamath{\emptyset}}
+\newcommand{\isasymnabla}{\isamath{\nabla}}
+\newcommand{\isasympartial}{\isamath{\partial}}
+\newcommand{\isasymRe}{\isamath{\Re}}
+\newcommand{\isasymIm}{\isamath{\Im}}
+\newcommand{\isasymflat}{\isamath{\flat}}
+\newcommand{\isasymnatural}{\isamath{\natural}}
+\newcommand{\isasymsharp}{\isamath{\sharp}}
+\newcommand{\isasymangle}{\isamath{\angle}}
+\newcommand{\isasymcopyright}{\isatext{\rm\copyright}}
+\newcommand{\isasymregistered}{\isatext{\rm\textregistered}}
+\newcommand{\isasymhyphen}{\isatext{\rm-}}
+\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
+\newcommand{\isasymonesuperior}{\isamath{{}^1}}
+\newcommand{\isasymonequarter}{\isatext{\rm\textonequarter}} %requires textcomp
+\newcommand{\isasymtwosuperior}{\isamath{{}^2}}
+\newcommand{\isasymonehalf}{\isatext{\rm\textonehalf}} %requires textcomp
+\newcommand{\isasymthreesuperior}{\isamath{{}^3}}
+\newcommand{\isasymthreequarters}{\isatext{\rm\textthreequarters}} %requires textcomp
+\newcommand{\isasymordfeminine}{\isatext{\rm\textordfeminine}}
+\newcommand{\isasymordmasculine}{\isatext{\rm\textordmasculine}}
+\newcommand{\isasymsection}{\isatext{\rm\S}}
+\newcommand{\isasymparagraph}{\isatext{\rm\P}}
+\newcommand{\isasymexclamdown}{\isatext{\rm\textexclamdown}}
+\newcommand{\isasymquestiondown}{\isatext{\rm\textquestiondown}}
+\newcommand{\isasymeuro}{\isatext{\textgreek{\euro}}} %requires greek babel
+\newcommand{\isasympounds}{\isamath{\pounds}}
+\newcommand{\isasymyen}{\isatext{\yen}} %requires amssymb
+\newcommand{\isasymcent}{\isatext{\textcent}} %requires textcomp
+\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
+\newcommand{\isasymdegree}{\isatext{\rm\textdegree}} %requires textcomp
+\newcommand{\isasymamalg}{\isamath{\amalg}}
+\newcommand{\isasymmho}{\isamath{\mho}} %requires amssymb
+\newcommand{\isasymlozenge}{\isamath{\lozenge}} %requires amssymb
+\newcommand{\isasymwp}{\isamath{\wp}}
+\newcommand{\isasymwrong}{\isamath{\wr}}
+\newcommand{\isasymstruct}{\isamath{\diamond}}
+\newcommand{\isasymacute}{\isatext{\'\relax}}
+\newcommand{\isasymindex}{\isatext{\i}}
+\newcommand{\isasymdieresis}{\isatext{\"\relax}}
+\newcommand{\isasymcedilla}{\isatext{\c\relax}}
+\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
+\newcommand{\isasymspacespace}{\isamath{~~}}
+\newcommand{\isasymmodule}{\isamath{\langle}\isakeyword{module}\isamath{\rangle}}
+\newcommand{\isasymsome}{\isamath{\epsilon\,}}
+\newcommand{\isasymbind}{\isamath{\mathbin{>\!\!\!>\mkern-6.7mu=}}}
+\newcommand{\isasymthen}{\isamath{\mathbin{>\!\!\!>}}}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/document/pdfsetup.sty Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,7 @@
+%%
+%% default hyperref setup (both for pdf and dvi output)
+%%
+
+\usepackage{color}
+\definecolor{linkcolor}{rgb}{0,0,0.5}
+\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,filecolor=linkcolor,pagecolor=linkcolor,urlcolor=linkcolor]{hyperref}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/document/railsetup.sty Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,1202 @@
+% rail.sty - style file to support railroad diagrams
+%
+% 09-Jul-90 L. Rooijakkers
+% 08-Oct-90 L. Rooijakkers fixed centering bug when \rail@tmpc<0.
+% 07-Feb-91 L. Rooijakkers added \railoptions command, indexing
+% 08-Feb-91 L. Rooijakkers minor fixes
+% 28-Jun-94 K. Barthelmann turned into LaTeX2e package
+% 08-Dec-96 K. Barthelmann replaced \@writefile
+% 13-Dec-96 K. Barthelmann cleanup
+% 22-Feb-98 K. Barthelmann fixed catcodes of special characters
+% 18-Apr-98 K. Barthelmann fixed \par handling
+% 19-May-98 J. Olsson Added new macros to support arrow heads.
+% 26-Jul-98 K. Barthelmann changed \par to output newlines
+% 02-May-11 M. Wenzel default setup for Isabelle
+%
+% This style file needs to be used in conjunction with the 'rail'
+% program. Running LaTeX as 'latex file' produces file.rai, which should be
+% processed by Rail with 'rail file'. This produces file.rao, which will
+% be picked up by LaTeX on the next 'latex file' run.
+%
+% LaTeX will warn if there is no file.rao or it's out of date.
+%
+% The macros in this file thus consist of two parts: those that read and
+% write the .rai and .rao files, and those that do the actual formatting
+% of the railroad diagrams.
+
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{rail}[1998/05/19]
+
+% railroad diagram formatting parameters (user level)
+% all of these are copied into their internal versions by \railinit
+%
+% \railunit : \unitlength within railroad diagrams
+% \railextra : extra length at outside of diagram
+% \railboxheight : height of ovals and frames
+% \railboxskip : vertical space between lines
+% \railboxleft : space to the left of a box
+% \railboxright : space to the right of a box
+% \railovalspace : extra space around contents of oval
+% \railframespace : extra space around contents of frame
+% \railtextleft : space to the left of text
+% \railtextright : space to the right of text
+% \railtextup : space to lift text up
+% \railjoinsize : circle size of join/split arcs
+% \railjoinadjust : space to adjust join
+%
+% \railnamesep : separator between name and rule body
+
+\newlength\railunit
+\newlength\railextra
+\newlength\railboxheight
+\newlength\railboxskip
+\newlength\railboxleft
+\newlength\railboxright
+\newlength\railovalspace
+\newlength\railframespace
+\newlength\railtextleft
+\newlength\railtextright
+\newlength\railtextup
+\newlength\railjoinsize
+\newlength\railjoinadjust
+\newlength\railnamesep
+
+% initialize the parameters
+
+\setlength\railunit{1sp}
+\setlength\railextra{4ex}
+\setlength\railboxleft{1ex}
+\setlength\railboxright{1ex}
+\setlength\railovalspace{2ex}
+\setlength\railframespace{2ex}
+\setlength\railtextleft{1ex}
+\setlength\railtextright{1ex}
+\setlength\railjoinadjust{0pt}
+\setlength\railnamesep{1ex}
+
+\DeclareOption{10pt}{
+ \setlength\railboxheight{16pt}
+ \setlength\railboxskip{24pt}
+ \setlength\railtextup{5pt}
+ \setlength\railjoinsize{16pt}
+}
+\DeclareOption{11pt}{
+ \setlength\railboxheight{16pt}
+ \setlength\railboxskip{24pt}
+ \setlength\railtextup{5pt}
+ \setlength\railjoinsize{16pt}
+}
+\DeclareOption{12pt}{
+ \setlength\railboxheight{20pt}
+ \setlength\railboxskip{28pt}
+ \setlength\railtextup{6pt}
+ \setlength\railjoinsize{20pt}
+}
+
+\ExecuteOptions{10pt}
+\ProcessOptions
+
+% internal versions of the formatting parameters
+%
+% \rail@extra : \railextra
+% \rail@boxht : \railboxheight
+% \rail@boxsp : \railboxskip
+% \rail@boxlf : \railboxleft
+% \rail@boxrt : \railboxright
+% \rail@boxhht : \railboxheight / 2
+% \rail@ovalsp : \railovalspace
+% \rail@framesp : \railframespace
+% \rail@textlf : \railtextleft
+% \rail@textrt : \railtextright
+% \rail@textup : \railtextup
+% \rail@joinsz : \railjoinsize
+% \rail@joinhsz : \railjoinsize / 2
+% \rail@joinadj : \railjoinadjust
+%
+% \railinit : internalize all of the parameters.
+
+\newcount\rail@extra
+\newcount\rail@boxht
+\newcount\rail@boxsp
+\newcount\rail@boxlf
+\newcount\rail@boxrt
+\newcount\rail@boxhht
+\newcount\rail@ovalsp
+\newcount\rail@framesp
+\newcount\rail@textlf
+\newcount\rail@textrt
+\newcount\rail@textup
+\newcount\rail@joinsz
+\newcount\rail@joinhsz
+\newcount\rail@joinadj
+
+\newcommand\railinit{
+\rail@extra=\railextra
+\divide\rail@extra by \railunit
+\rail@boxht=\railboxheight
+\divide\rail@boxht by \railunit
+\rail@boxsp=\railboxskip
+\divide\rail@boxsp by \railunit
+\rail@boxlf=\railboxleft
+\divide\rail@boxlf by \railunit
+\rail@boxrt=\railboxright
+\divide\rail@boxrt by \railunit
+\rail@boxhht=\railboxheight
+\divide\rail@boxhht by \railunit
+\divide\rail@boxhht by 2
+\rail@ovalsp=\railovalspace
+\divide\rail@ovalsp by \railunit
+\rail@framesp=\railframespace
+\divide\rail@framesp by \railunit
+\rail@textlf=\railtextleft
+\divide\rail@textlf by \railunit
+\rail@textrt=\railtextright
+\divide\rail@textrt by \railunit
+\rail@textup=\railtextup
+\divide\rail@textup by \railunit
+\rail@joinsz=\railjoinsize
+\divide\rail@joinsz by \railunit
+\rail@joinhsz=\railjoinsize
+\divide\rail@joinhsz by \railunit
+\divide\rail@joinhsz by 2
+\rail@joinadj=\railjoinadjust
+\divide\rail@joinadj by \railunit
+}
+
+\AtBeginDocument{\railinit}
+
+% \rail@param : declarations for list environment
+%
+% \railparam{TEXT} : sets \rail@param to TEXT
+%
+% \rail@reserved : characters reserved for grammar
+
+\newcommand\railparam[1]{
+\def\rail@param{
+ \setlength\leftmargin{0pt}\setlength\rightmargin{0pt}
+ \setlength\labelwidth{0pt}\setlength\labelsep{0pt}
+ \setlength\itemindent{0pt}\setlength\listparindent{0pt}
+ #1
+}
+}
+\railparam{}
+
+\newtoks\rail@reserved
+\rail@reserved={:;|*+?[]()'"}
+
+% \rail@termfont : format setup for terminals
+%
+% \rail@nontfont : format setup for nonterminals
+%
+% \rail@annofont : format setup for annotations
+%
+% \rail@rulefont : format setup for rule names
+%
+% \rail@indexfont : format setup for index entry
+%
+% \railtermfont{TEXT} : set terminal format setup to TEXT
+%
+% \railnontermfont{TEXT} : set nonterminal format setup to TEXT
+%
+% \railannotatefont{TEXT} : set annotation format setup to TEXT
+%
+% \railnamefont{TEXT} : set rule name format setup to TEXT
+%
+% \railindexfont{TEXT} : set index entry format setup to TEXT
+
+\def\rail@termfont{\ttfamily\upshape}
+\def\rail@nontfont{\rmfamily\upshape}
+\def\rail@annofont{\rmfamily\itshape}
+\def\rail@namefont{\rmfamily\itshape}
+\def\rail@indexfont{\rmfamily\itshape}
+
+\newcommand\railtermfont[1]{
+\def\rail@termfont{#1}
+}
+
+\newcommand\railnontermfont[1]{
+\def\rail@nontfont{#1}
+}
+
+\newcommand\railannotatefont[1]{
+\def\rail@annofont{#1}
+}
+
+\newcommand\railnamefont[1]{
+\def\rail@namefont{#1}
+}
+
+\newcommand\railindexfont[1]{
+\def\rail@indexfont{#1}
+}
+
+% railroad read/write macros
+%
+% \begin{rail} TEXT \end{rail} : TEXT is written out to the .rai file,
+% as \rail@i{NR}{TEXT}. Then the matching
+% \rail@o{NR}{FMT} from the .rao file is
+% executed (if defined).
+%
+% \railoptions{OPTIONS} : OPTIONS are written out to the .rai file,
+% as \rail@p{OPTIONS}.
+%
+% \railterm{IDENT,IDENT,...} : format IDENT as terminals. writes out
+% \rail@t{IDENT} to the .rai file
+%
+% \railalias{IDENT}{TEXT} : format IDENT as TEXT. defines \rail@t@IDENT as
+% TEXT.
+%
+% \railtoken{IDENT}{TEXT} : abbreviates \railalias{IDENT}{TEXT}\railterm{IDENT}
+% (for backward compatibility)
+%
+% \rail@setcodes : guards special characters
+%
+% \rail@makeother{CHARACTER} : sets \catcode of CHARACTER to "other"
+% used inside a loop for \rail@setcodes
+%
+% \rail@nr : railroad diagram counter
+%
+% \ifrail@match : current \rail@i{NR}{TEXT} matches
+%
+% \rail@first : actions to be done first. read in .rao file,
+% open .rai file if \@filesw true, undefine \rail@first.
+% executed from \begin{rail}, \railoptions and \railterm.
+%
+% \rail@i{NR}{TEXT} : defines \rail@i@NR as TEXT. written to the .rai
+% file by \rail, read from the .rao file by
+% \rail@first
+%
+% \rail@t{IDENT} : tells Rail that IDENT is to be custom formatted,
+% written to the .rai file by \railterm.
+%
+% \rail@o{NR}{TEXT} : defines \rail@o@NR as TEXT, read from the .rao
+% file by \rail@first.
+%
+% \rail@p{OPTIONS} : pass options to rail, written to the .rai file by
+% \railoptions
+%
+% \rail@write{TEXT} : write TEXT to the .rai file
+%
+% \rail@warn : warn user for mismatching diagrams
+%
+% \rail@endwarn : either \relax or \rail@warn
+%
+% \ifrail@all : checked at the end of the document
+
+\def\rail@makeother#1{
+ \expandafter\catcode\expandafter`\csname\string #1\endcsname=12
+}
+
+\def\rail@setcodes{
+\let\par=\relax
+\let\\=\relax
+\expandafter\@tfor\expandafter\rail@symbol\expandafter:\expandafter=%
+ \the\rail@reserved
+\do{\expandafter\rail@makeother\rail@symbol}
+}
+
+\newcount\rail@nr
+
+\newif\ifrail@all
+\rail@alltrue
+
+\newif\ifrail@match
+
+\def\rail@first{
+\begingroup
+\makeatletter
+\rail@setcodes
+\InputIfFileExists{\jobname.rao}{}{\PackageInfo{rail}{No file \jobname.rao}}
+\makeatother
+\endgroup
+\if@filesw
+\newwrite\tf@rai
+\immediate\openout\tf@rai=\jobname.rai
+\fi
+\global\let\rail@first=\relax
+}
+
+\long\def\rail@body#1\end{
+{
+\newlinechar=`^^J
+\def\par{\string\par^^J}
+\rail@write{\string\rail@i{\number\rail@nr}{#1}}
+}
+\xdef\rail@i@{#1}
+\end
+}
+
+\newenvironment{rail}{
+\global\advance\rail@nr by 1
+\rail@first
+\begingroup
+\rail@setcodes
+\rail@body
+}{
+\endgroup
+\rail@matchtrue
+\@ifundefined{rail@o@\number\rail@nr}{\rail@matchfalse}{}
+\expandafter\ifx\csname rail@i@\number\rail@nr\endcsname\rail@i@
+\else
+\rail@matchfalse
+\fi
+\ifrail@match
+\csname rail@o@\number\rail@nr\endcsname
+\else
+\PackageWarning{rail}{Railroad diagram {\number\rail@nr} doesn't match}
+\global\let\rail@endwarn=\rail@warn
+\begin{list}{}{\rail@param}
+\rail@begin{1}{}
+\rail@setbox{\bfseries ???}
+\rail@oval
+\rail@end
+\end{list}
+\fi
+}
+
+\newcommand\railoptions[1]{
+\rail@first
+\rail@write{\string\rail@p{#1}}
+}
+
+\newcommand\railterm[1]{
+\rail@first
+\@for\rail@@:=#1\do{
+\rail@write{\string\rail@t{\rail@@}}
+}
+}
+
+\newcommand\railalias[2]{
+\expandafter\def\csname rail@t@#1\endcsname{#2}
+}
+
+\newcommand\railtoken[2]{\railalias{#1}{#2}\railterm{#1}}
+
+\long\def\rail@i#1#2{
+\expandafter\gdef\csname rail@i@#1\endcsname{#2}
+}
+
+\def\rail@o#1#2{
+\expandafter\gdef\csname rail@o@#1\endcsname{
+\begin{list}{}{\rail@param}
+#2
+\end{list}
+}
+}
+
+\def\rail@t#1{}
+
+\def\rail@p#1{}
+
+\long\def\rail@write#1{\@ifundefined{tf@rai}{}{\immediate\write\tf@rai{#1}}}
+
+\def\rail@warn{
+\PackageWarningNoLine{rail}{Railroad diagram(s) may have changed.
+ Use 'rail' and rerun}
+}
+
+\let\rail@endwarn=\relax
+
+\AtEndDocument{\rail@endwarn}
+
+% index entry macro
+%
+% \rail@index{IDENT} : add index entry for IDENT
+
+\def\rail@index#1{
+\index{\rail@indexfont#1}
+}
+
+% railroad formatting primitives
+%
+% \rail@x : current x
+% \rail@y : current y
+% \rail@ex : current end x
+% \rail@sx : starting x for \rail@cr
+% \rail@rx : rightmost previous x for \rail@cr
+%
+% \rail@tmpa : temporary count
+% \rail@tmpb : temporary count
+% \rail@tmpc : temporary count
+%
+% \rail@put : put at (\rail@x,\rail@y)
+% \rail@vput : put vector at (\rail@x,\rail@y)
+%
+% \rail@eline : end line by drawing from \rail@ex to \rail@x
+%
+% \rail@vreline : end line by drawing a vector from \rail@x to \rail@ex
+%
+% \rail@vleline : end line by drawing a vector from \rail@ex to \rail@x
+%
+% \rail@sety{LEVEL} : set \rail@y to level LEVEL
+
+\newcount\rail@x
+\newcount\rail@y
+\newcount\rail@ex
+\newcount\rail@sx
+\newcount\rail@rx
+
+\newcount\rail@tmpa
+\newcount\rail@tmpb
+\newcount\rail@tmpc
+
+\def\rail@put{\put(\number\rail@x,\number\rail@y)}
+
+\def\rail@vput{\put(\number\rail@ex,\number\rail@y)}
+
+\def\rail@eline{
+\rail@tmpb=\rail@x
+\advance\rail@tmpb by -\rail@ex
+\rail@put{\line(-1,0){\number\rail@tmpb}}
+}
+
+\def\rail@vreline{
+\rail@tmpb=\rail@x
+\advance\rail@tmpb by -\rail@ex
+\rail@vput{\vector(1,0){\number\rail@tmpb}}
+}
+
+\def\rail@vleline{
+\rail@tmpb=\rail@x
+\advance\rail@tmpb by -\rail@ex
+\rail@put{\vector(-1,0){\number\rail@tmpb}}
+}
+
+\def\rail@sety#1{
+\rail@y=#1
+\multiply\rail@y by -\rail@boxsp
+\advance\rail@y by -\rail@boxht
+}
+
+% \rail@begin{HEIGHT}{NAME} : begin a railroad diagram of height HEIGHT
+%
+% \rail@end : end a railroad diagram
+%
+% \rail@expand{IDENT} : expand IDENT
+
+\def\rail@begin#1#2{
+\item
+\begin{minipage}[t]{\linewidth}
+\ifx\@empty#2\else
+{\rail@namefont \rail@expand{#2}}\\*[\railnamesep]
+\fi
+\unitlength=\railunit
+\rail@tmpa=#1
+\multiply\rail@tmpa by \rail@boxsp
+\begin{picture}(0,\number\rail@tmpa)(0,-\number\rail@tmpa)
+\rail@ex=0
+\rail@rx=0
+\rail@x=\rail@extra
+\rail@sx=\rail@x
+\rail@sety{0}
+}
+
+\def\rail@end{
+\advance\rail@x by \rail@extra
+\rail@eline
+\end{picture}
+\end{minipage}
+}
+
+\def\rail@vend{
+\advance\rail@x by \rail@extra
+\rail@vreline
+\end{picture}
+\end{minipage}
+}
+
+\def\rail@expand#1{\@ifundefined{rail@t@#1}{#1}{\csname rail@t@#1\endcsname}}
+
+% \rail@token{TEXT}[ANNOT] : format token TEXT with annotation
+% \rail@ltoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow left
+% \rail@rtoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow right
+%
+% \rail@ctoken{TEXT}[ANNOT] : format token TEXT centered with annotation
+% \rail@lctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow left
+% \rail@rctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow right
+%
+% \rail@nont{TEXT}[ANNOT] : format nonterminal TEXT with annotation
+% \rail@lnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation, arrow left
+% \rail@rnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation. arrow right
+%
+% \rail@cnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation
+% \rail@lcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
+% arrow left
+% \rail@rcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
+% arrow right
+%
+% \rail@term{TEXT}[ANNOT] : format terminal TEXT with annotation
+% \rail@lterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow left
+% \rail@rterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow right
+%
+% \rail@cterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation
+% \rail@lcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation, arrow left
+% \rail@rcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation,
+% arrow right
+%
+% \rail@annote[TEXT] : format TEXT as annotation
+
+\def\rail@token#1[#2]{
+\rail@setbox{%
+{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@oval
+}
+
+\def\rail@ltoken#1[#2]{
+\rail@setbox{%
+{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vloval
+}
+
+\def\rail@rtoken#1[#2]{
+\rail@setbox{%
+{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vroval
+}
+
+\def\rail@ctoken#1[#2]{
+\rail@setbox{%
+{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@coval
+}
+
+\def\rail@lctoken#1[#2]{
+\rail@setbox{%
+{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vlcoval
+}
+
+\def\rail@rctoken#1[#2]{
+\rail@setbox{%
+{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vrcoval
+}
+
+\def\rail@nont#1[#2]{
+\rail@setbox{%
+{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@frame
+}
+
+\def\rail@lnont#1[#2]{
+\rail@setbox{%
+{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vlframe
+}
+
+\def\rail@rnont#1[#2]{
+\rail@setbox{%
+{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vrframe
+}
+
+\def\rail@cnont#1[#2]{
+\rail@setbox{%
+{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@cframe
+}
+
+\def\rail@lcnont#1[#2]{
+\rail@setbox{%
+{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vlcframe
+}
+
+\def\rail@rcnont#1[#2]{
+\rail@setbox{%
+{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vrcframe
+}
+
+\def\rail@term#1[#2]{
+\rail@setbox{%
+{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@oval
+}
+
+\def\rail@lterm#1[#2]{
+\rail@setbox{%
+{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vloval
+}
+
+\def\rail@rterm#1[#2]{
+\rail@setbox{%
+{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vroval
+}
+
+\def\rail@cterm#1[#2]{
+\rail@setbox{%
+{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@coval
+}
+
+\def\rail@lcterm#1[#2]{
+\rail@setbox{%
+{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vlcoval
+}
+
+\def\rail@rcterm#1[#2]{
+\rail@setbox{%
+{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
+}
+\rail@vrcoval
+}
+
+\def\rail@annote[#1]{
+\rail@setbox{\rail@annofont #1}
+\rail@text
+}
+
+% \rail@box : temporary box for \rail@oval and \rail@frame
+%
+% \rail@setbox{TEXT} : set \rail@box to TEXT, set \rail@tmpa to width
+%
+% \rail@oval : format \rail@box of width \rail@tmpa inside an oval
+% \rail@vloval : format \rail@box of width \rail@tmpa inside an oval, vector left
+% \rail@vroval : format \rail@box of width \rail@tmpa inside an oval, vector right
+%
+% \rail@coval : same as \rail@oval, but centered between \rail@x and
+% \rail@mx
+% \rail@vlcoval : same as \rail@oval, but centered between \rail@x and
+% \rail@mx, vector left
+% \rail@vrcoval : same as \rail@oval, but centered between \rail@x and
+% \rail@mx, vector right
+%
+% \rail@frame : format \rail@box of width \rail@tmpa inside a frame
+% \rail@vlframe : format \rail@box of width \rail@tmpa inside a frame, vector left
+% \rail@vrframe : format \rail@box of width \rail@tmpa inside a frame, vector right
+%
+% \rail@cframe : same as \rail@frame, but centered between \rail@x and
+% \rail@mx
+% \rail@vlcframe : same as \rail@frame, but centered between \rail@x and
+% \rail@mx, vector left
+% \rail@vrcframe : same as \rail@frame, but centered between \rail@x and
+% \rail@mx, vector right
+%
+% \rail@text : format \rail@box of width \rail@tmpa above the line
+
+\newbox\rail@box
+
+\def\rail@setbox#1{
+\setbox\rail@box\hbox{\strut#1}
+\rail@tmpa=\wd\rail@box
+\divide\rail@tmpa by \railunit
+}
+
+\def\rail@oval{
+\advance\rail@x by \rail@boxlf
+\rail@eline
+\advance\rail@tmpa by \rail@ovalsp
+\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
+\rail@tmpb=\rail@tmpa
+\divide\rail@tmpb by 2
+\advance\rail@y by -\rail@boxhht
+\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
+\advance\rail@y by \rail@boxhht
+\advance\rail@x by \rail@tmpb
+\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
+\advance\rail@x by \rail@tmpb
+\rail@ex=\rail@x
+\advance\rail@x by \rail@boxrt
+}
+
+\def\rail@vloval{
+\advance\rail@x by \rail@boxlf
+\rail@eline
+\advance\rail@tmpa by \rail@ovalsp
+\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
+\rail@tmpb=\rail@tmpa
+\divide\rail@tmpb by 2
+\advance\rail@y by -\rail@boxhht
+\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
+\advance\rail@y by \rail@boxhht
+\advance\rail@x by \rail@tmpb
+\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
+\advance\rail@x by \rail@tmpb
+\rail@ex=\rail@x
+\advance\rail@x by \rail@boxrt
+\rail@vleline
+}
+
+\def\rail@vroval{
+\advance\rail@x by \rail@boxlf
+\rail@vreline
+\advance\rail@tmpa by \rail@ovalsp
+\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
+\rail@tmpb=\rail@tmpa
+\divide\rail@tmpb by 2
+\advance\rail@y by -\rail@boxhht
+\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
+\advance\rail@y by \rail@boxhht
+\advance\rail@x by \rail@tmpb
+\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
+\advance\rail@x by \rail@tmpb
+\rail@ex=\rail@x
+\advance\rail@x by \rail@boxrt
+}
+
+\def\rail@coval{
+\rail@tmpb=\rail@tmpa
+\advance\rail@tmpb by \rail@ovalsp
+\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
+\advance\rail@tmpb by \rail@boxlf
+\advance\rail@tmpb by \rail@boxrt
+\rail@tmpc=\rail@mx
+\advance\rail@tmpc by -\rail@x
+\advance\rail@tmpc by -\rail@tmpb
+\divide\rail@tmpc by 2
+\ifnum\rail@tmpc>0
+\advance\rail@x by \rail@tmpc
+\fi
+\rail@oval
+}
+
+\def\rail@vlcoval{
+\rail@tmpb=\rail@tmpa
+\advance\rail@tmpb by \rail@ovalsp
+\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
+\advance\rail@tmpb by \rail@boxlf
+\advance\rail@tmpb by \rail@boxrt
+\rail@tmpc=\rail@mx
+\advance\rail@tmpc by -\rail@x
+\advance\rail@tmpc by -\rail@tmpb
+\divide\rail@tmpc by 2
+\ifnum\rail@tmpc>0
+\advance\rail@x by \rail@tmpc
+\fi
+\rail@vloval
+}
+
+\def\rail@vrcoval{
+\rail@tmpb=\rail@tmpa
+\advance\rail@tmpb by \rail@ovalsp
+\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
+\advance\rail@tmpb by \rail@boxlf
+\advance\rail@tmpb by \rail@boxrt
+\rail@tmpc=\rail@mx
+\advance\rail@tmpc by -\rail@x
+\advance\rail@tmpc by -\rail@tmpb
+\divide\rail@tmpc by 2
+\ifnum\rail@tmpc>0
+\advance\rail@x by \rail@tmpc
+\fi
+\rail@vroval
+}
+
+\def\rail@frame{
+\advance\rail@x by \rail@boxlf
+\rail@eline
+\advance\rail@tmpa by \rail@framesp
+\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
+\advance\rail@y by -\rail@boxhht
+\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
+\advance\rail@y by \rail@boxhht
+\advance\rail@x by \rail@tmpa
+\rail@ex=\rail@x
+\advance\rail@x by \rail@boxrt
+}
+
+\def\rail@vlframe{
+\advance\rail@x by \rail@boxlf
+\rail@eline
+\advance\rail@tmpa by \rail@framesp
+\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
+\advance\rail@y by -\rail@boxhht
+\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
+\advance\rail@y by \rail@boxhht
+\advance\rail@x by \rail@tmpa
+\rail@ex=\rail@x
+\advance\rail@x by \rail@boxrt
+\rail@vleline
+}
+
+\def\rail@vrframe{
+\advance\rail@x by \rail@boxlf
+\rail@vreline
+\advance\rail@tmpa by \rail@framesp
+\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
+\advance\rail@y by -\rail@boxhht
+\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
+\advance\rail@y by \rail@boxhht
+\advance\rail@x by \rail@tmpa
+\rail@ex=\rail@x
+\advance\rail@x by \rail@boxrt
+}
+
+\def\rail@cframe{
+\rail@tmpb=\rail@tmpa
+\advance\rail@tmpb by \rail@framesp
+\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
+\advance\rail@tmpb by \rail@boxlf
+\advance\rail@tmpb by \rail@boxrt
+\rail@tmpc=\rail@mx
+\advance\rail@tmpc by -\rail@x
+\advance\rail@tmpc by -\rail@tmpb
+\divide\rail@tmpc by 2
+\ifnum\rail@tmpc>0
+\advance\rail@x by \rail@tmpc
+\fi
+\rail@frame
+}
+
+\def\rail@vlcframe{
+\rail@tmpb=\rail@tmpa
+\advance\rail@tmpb by \rail@framesp
+\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
+\advance\rail@tmpb by \rail@boxlf
+\advance\rail@tmpb by \rail@boxrt
+\rail@tmpc=\rail@mx
+\advance\rail@tmpc by -\rail@x
+\advance\rail@tmpc by -\rail@tmpb
+\divide\rail@tmpc by 2
+\ifnum\rail@tmpc>0
+\advance\rail@x by \rail@tmpc
+\fi
+\rail@vlframe
+}
+
+\def\rail@vrcframe{
+\rail@tmpb=\rail@tmpa
+\advance\rail@tmpb by \rail@framesp
+\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
+\advance\rail@tmpb by \rail@boxlf
+\advance\rail@tmpb by \rail@boxrt
+\rail@tmpc=\rail@mx
+\advance\rail@tmpc by -\rail@x
+\advance\rail@tmpc by -\rail@tmpb
+\divide\rail@tmpc by 2
+\ifnum\rail@tmpc>0
+\advance\rail@x by \rail@tmpc
+\fi
+\rail@vrframe
+}
+
+\def\rail@text{
+\advance\rail@x by \rail@textlf
+\advance\rail@y by \rail@textup
+\rail@put{\box\rail@box}
+\advance\rail@y by -\rail@textup
+\advance\rail@x by \rail@tmpa
+\advance\rail@x by \rail@textrt
+}
+
+% alternatives
+%
+% \rail@jx \rail@jy : current join point
+%
+% \rail@gx \rail@gy \rail@gex \rail@grx : global versions of \rail@x etc,
+% to pass values over group closings
+%
+% \rail@mx : maximum x so far
+%
+% \rail@sy : starting \rail@y for alternatives
+%
+% \rail@jput : put at (\rail@jx,\rail@jy)
+%
+% \rail@joval[PART] : put \oval[PART] with adjust
+
+\newcount\rail@jx
+\newcount\rail@jy
+
+\newcount\rail@gx
+\newcount\rail@gy
+\newcount\rail@gex
+\newcount\rail@grx
+
+\newcount\rail@sy
+\newcount\rail@mx
+
+\def\rail@jput{
+\put(\number\rail@jx,\number\rail@jy)
+}
+
+\def\rail@joval[#1]{
+\advance\rail@jx by \rail@joinadj
+\rail@jput{\oval(\number\rail@joinsz,\number\rail@joinsz)[#1]}
+\advance\rail@jx by -\rail@joinadj
+}
+
+% \rail@barsplit : incoming split for '|'
+%
+% \rail@plussplit : incoming split for '+'
+%
+
+\def\rail@barsplit{
+\advance\rail@jy by -\rail@joinhsz
+\rail@joval[tr]
+\advance\rail@jx by \rail@joinhsz
+}
+
+\def\rail@plussplit{
+\advance\rail@jy by -\rail@joinhsz
+\advance\rail@jx by \rail@joinsz
+\rail@joval[tl]
+\advance\rail@jx by -\rail@joinhsz
+}
+
+% \rail@alt{SPLIT} : start alternatives with incoming split SPLIT
+%
+
+\def\rail@alt#1{
+\rail@sy=\rail@y
+\rail@jx=\rail@x
+\rail@jy=\rail@y
+\advance\rail@x by \rail@joinsz
+\rail@mx=0
+\let\rail@list=\@empty
+\let\rail@comma=\@empty
+\let\rail@split=#1
+\begingroup
+\rail@sx=\rail@x
+\rail@rx=0
+}
+
+% \rail@nextalt{FIX}{Y} : start next alternative at vertical position Y
+% and fix-up FIX
+%
+
+\def\rail@nextalt#1#2{
+\global\rail@gx=\rail@x
+\global\rail@gy=\rail@y
+\global\rail@gex=\rail@ex
+\global\rail@grx=\rail@rx
+\endgroup
+#1
+\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
+\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
+\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
+\def\rail@comma{,}
+\rail@split
+\let\rail@split=\@empty
+\rail@sety{#2}
+\rail@tmpa=\rail@jy
+\advance\rail@tmpa by -\rail@y
+\advance\rail@tmpa by -\rail@joinhsz
+\rail@jput{\line(0,-1){\number\rail@tmpa}}
+\rail@jy=\rail@y
+\advance\rail@jy by \rail@joinhsz
+\advance\rail@jx by \rail@joinhsz
+\rail@joval[bl]
+\advance\rail@jx by -\rail@joinhsz
+\rail@ex=\rail@x
+\begingroup
+\rail@sx=\rail@x
+\rail@rx=0
+}
+
+% \rail@barjoin : outgoing join for first '|' alternative
+%
+% \rail@plusjoin : outgoing join for first '+' alternative
+%
+% \rail@altjoin : join for subsequent alternative
+%
+
+\def\rail@barjoin{
+\ifnum\rail@y<\rail@sy
+\global\rail@gex=\rail@jx
+\else
+\global\rail@gex=\rail@ex
+\fi
+\advance\rail@jy by -\rail@joinhsz
+\rail@joval[tl]
+\advance\rail@jx by -\rail@joinhsz
+\ifnum\rail@y<\rail@sy
+\rail@altjoin
+\fi
+}
+
+\def\rail@plusjoin{
+\global\rail@gex=\rail@ex
+\advance\rail@jy by -\rail@joinhsz
+\advance\rail@jx by -\rail@joinsz
+\rail@joval[tr]
+\advance\rail@jx by \rail@joinhsz
+}
+
+\def\rail@altjoin{
+\rail@eline
+\rail@tmpa=\rail@jy
+\advance\rail@tmpa by -\rail@y
+\advance\rail@tmpa by -\rail@joinhsz
+\rail@jput{\line(0,-1){\number\rail@tmpa}}
+\rail@jy=\rail@y
+\advance\rail@jy by \rail@joinhsz
+\advance\rail@jx by -\rail@joinhsz
+\rail@joval[br]
+\advance\rail@jx by \rail@joinhsz
+}
+
+% \rail@eltsplit EX:Y; : split EX:Y into \rail@ex \rail@y
+%
+% \rail@endalt{JOIN} : end alternatives with outgoing join JOIN
+
+\def\rail@eltsplit#1:#2;{\rail@ex=#1\rail@y=#2}
+
+\def\rail@endalt#1{
+\global\rail@gx=\rail@x
+\global\rail@gy=\rail@y
+\global\rail@gex=\rail@ex
+\global\rail@grx=\rail@rx
+\endgroup
+\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
+\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
+\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
+\rail@x=\rail@mx
+\rail@jx=\rail@x
+\rail@jy=\rail@sy
+\advance\rail@jx by \rail@joinsz
+\let\rail@join=#1
+\@for\rail@elt:=\rail@list\do{
+\expandafter\rail@eltsplit\rail@elt;
+\rail@join
+\let\rail@join=\rail@altjoin
+}
+\rail@x=\rail@mx
+\rail@y=\rail@sy
+\rail@ex=\rail@gex
+\advance\rail@x by \rail@joinsz
+}
+
+% \rail@bar : start '|' alternatives
+%
+% \rail@nextbar : next '|' alternative
+%
+% \rail@endbar : end '|' alternatives
+%
+
+\def\rail@bar{
+\rail@alt\rail@barsplit
+}
+
+\def\rail@nextbar{
+\rail@nextalt\relax
+}
+
+\def\rail@endbar{
+\rail@endalt\rail@barjoin
+}
+
+% \rail@plus : start '+' alternatives
+%
+% \rail@nextplus: next '+' alternative
+%
+% \rail@endplus : end '+' alternatives
+%
+
+\def\rail@plus{
+\rail@alt\rail@plussplit
+}
+
+\def\rail@nextplus{
+\rail@nextalt\rail@fixplus
+}
+
+\def\rail@fixplus{
+\ifnum\rail@gy<\rail@sy
+\begingroup
+\rail@x=\rail@gx
+\rail@y=\rail@gy
+\rail@ex=\rail@gex
+\rail@rx=\rail@grx
+\ifnum\rail@x<\rail@rx
+\rail@x=\rail@rx
+\fi
+\rail@eline
+\rail@jx=\rail@x
+\rail@jy=\rail@y
+\advance\rail@jy by \rail@joinhsz
+\rail@joval[br]
+\advance\rail@jx by \rail@joinhsz
+\rail@tmpa=\rail@sy
+\advance\rail@tmpa by -\rail@joinhsz
+\advance\rail@tmpa by -\rail@jy
+\rail@jput{\line(0,1){\number\rail@tmpa}}
+\rail@jy=\rail@sy
+\advance\rail@jy by -\rail@joinhsz
+\advance\rail@jx by \rail@joinhsz
+\rail@joval[tl]
+\advance\rail@jy by \rail@joinhsz
+\global\rail@gx=\rail@jx
+\global\rail@gy=\rail@jy
+\global\rail@gex=\rail@gx
+\global\rail@grx=\rail@rx
+\endgroup
+\fi
+}
+
+\def\rail@endplus{
+\rail@endalt\rail@plusjoin
+}
+
+% \rail@cr{Y} : carriage return to vertical position Y
+
+\def\rail@cr#1{
+\rail@tmpa=\rail@sx
+\advance\rail@tmpa by \rail@joinsz
+\ifnum\rail@x<\rail@tmpa\rail@x=\rail@tmpa\fi
+\rail@eline
+\rail@jx=\rail@x
+\rail@jy=\rail@y
+\advance\rail@x by \rail@joinsz
+\ifnum\rail@x>\rail@rx\rail@rx=\rail@x\fi
+\advance\rail@jy by -\rail@joinhsz
+\rail@joval[tr]
+\advance\rail@jx by \rail@joinhsz
+\rail@sety{#1}
+\rail@tmpa=\rail@jy
+\advance\rail@tmpa by -\rail@y
+\advance\rail@tmpa by -\rail@boxsp
+\advance\rail@tmpa by -\rail@joinhsz
+\rail@jput{\line(0,-1){\number\rail@tmpa}}
+\rail@jy=\rail@y
+\advance\rail@jy by \rail@boxsp
+\advance\rail@jy by \rail@joinhsz
+\advance\rail@jx by -\rail@joinhsz
+\rail@joval[br]
+\advance\rail@jy by -\rail@joinhsz
+\rail@tmpa=\rail@jx
+\advance\rail@tmpa by -\rail@sx
+\advance\rail@tmpa by -\rail@joinhsz
+\rail@jput{\line(-1,0){\number\rail@tmpa}}
+\rail@jx=\rail@sx
+\advance\rail@jx by \rail@joinhsz
+\advance\rail@jy by -\rail@joinhsz
+\rail@joval[tl]
+\advance\rail@jx by -\rail@joinhsz
+\rail@tmpa=\rail@boxsp
+\advance\rail@tmpa by -\rail@joinsz
+\rail@jput{\line(0,-1){\number\rail@tmpa}}
+\advance\rail@jy by -\rail@tmpa
+\advance\rail@jx by \rail@joinhsz
+\rail@joval[bl]
+\rail@x=\rail@jx
+\rail@ex=\rail@x
+}
+
+% default setup for Isabelle
+\newenvironment{railoutput}%
+{\begin{list}{}{\rail@param}\def\rail@expand{\relax}\makeatletter}{\makeatother\end{list}}
+
+\def\rail@termfont{\isabellestyle{tt}}
+\def\rail@nontfont{\isabellestyle{it}}
+\def\rail@namefont{\isabellestyle{it}}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/document/root.bib Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,17 @@
+ @book{card-book,
+ title = {Introduction to {C}ardinal {A}rithmetic},
+ author = {M. Holz and K. Steffens and E. Weitz},
+ publisher = "Birkh{\"{a}}user",
+ year = 1999,
+ }
+
+ @article{taylor-ordinals,
+ author = {Paul Taylor},
+ title = {Intuitionistic Sets and Ordinals},
+ journal = {J. Symb. Log.},
+ volume = {61},
+ number = {3},
+ year = {1996},
+ pages = {705-744},
+ bibsource = {DBLP, http://dblp.uni-trier.de}
+}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/document/root.tex Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,68 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage{isabelle,isabellesym}
+
+% further packages required for unusual symbols (see also
+% isabellesym.sty), use only when needed
+
+%\usepackage{amssymb}
+ %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
+ %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
+ %\<triangleq>, \<yen>, \<lozenge>
+
+%\usepackage[greek,english]{babel}
+ %option greek for \<euro>
+ %option english (default language) for \<guillemotleft>, \<guillemotright>
+
+%\usepackage[latin1]{inputenc}
+ %for \<onesuperior>, \<onequarter>, \<twosuperior>, \<onehalf>,
+ %\<threesuperior>, \<threequarters>, \<degree>
+
+%\usepackage[only,bigsqcap]{stmaryrd}
+ %for \<Sqinter>
+
+%\usepackage{eufrak}
+ %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
+
+%\usepackage{textcomp}
+ %for \<cent>, \<currency>
+
+% this should be the last package used
+\usepackage{pdfsetup}
+
+% urls in roman style, theory text in math-similar italics
+\urlstyle{rm}
+\isabellestyle{it}
+
+% for uniform font size
+%\renewcommand{\isastyle}{\isastyleminor}
+
+\bibliographystyle{plain}
+\begin{document}
+
+\title{Ordinals and cardinals in HOL}
+\author{Andrei Popescu \& Dmitriy Traytel}
+\date{}
+\maketitle
+
+\tableofcontents
+
+% sane default for proof documents
+\parindent 0pt\parskip 0.5ex
+
+\input{intro.tex}
+
+\bibliography{root}
+
+% generated text of all theories
+\input{session}
+
+% optional bibliography
+%\bibliographystyle{abbrv}
+%\bibliography{root}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
--- a/src/HOL/ROOT Tue Aug 28 16:33:06 2012 +0200
+++ b/src/HOL/ROOT Tue Aug 28 17:16:00 2012 +0200
@@ -597,6 +597,37 @@
theories Nominal_Examples
theories [quick_and_dirty] VC_Condition
+session "HOL-Ordinals_and_Cardinals-Base" in Ordinals_and_Cardinals = HOL +
+ options [document = false]
+ theories Cardinal_Arithmetic
+
+session "HOL-Ordinals_and_Cardinals" in Ordinals_and_Cardinals =
+ "HOL-Ordinals_and_Cardinals-Base" +
+ options [document = false]
+ theories Cardinal_Order_Relation
+
+session "HOL-Codatatype" in Codatatype = "HOL-Ordinals_and_Cardinals-Base" +
+ options [document = false]
+ theories Codatatype
+
+session "HOL-Codatatype-Examples" in "Codatatype/Examples" = "HOL-Codatatype" +
+ options [document = false]
+ theories
+ HFset
+ Lambda_Term
+ Process
+ TreeFsetI
+ (* FIXME: shouldn't require "parallel_proofs = 0";
+ with parallel proofs enabled, the build of this session
+ takes 10 times longer *)
+ theories [parallel_proofs = 0]
+ "Infinite_Derivation_Trees/Gram_Lang"
+ "Infinite_Derivation_Trees/Parallel"
+ Stream
+ theories [parallel_proofs = 0, condition = ISABELLE_FULL_TEST]
+ Misc_Codata
+ Misc_Data
+
session "HOL-Word" in Word = HOL +
options [document_graph]
theories Word