--- a/src/HOL/BNF_GFP.thy Tue Sep 02 14:40:14 2014 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,300 +0,0 @@
-(* Title: HOL/BNF_GFP.thy
- Author: Dmitriy Traytel, TU Muenchen
- Author: Lorenz Panny, TU Muenchen
- Author: Jasmin Blanchette, TU Muenchen
- Copyright 2012, 2013, 2014
-
-Greatest fixed point operation on bounded natural functors.
-*)
-
-header {* Greatest Fixed Point Operation on Bounded Natural Functors *}
-
-theory BNF_GFP
-imports BNF_FP_Base String
-keywords
- "codatatype" :: thy_decl and
- "primcorecursive" :: thy_goal and
- "primcorec" :: thy_decl
-begin
-
-setup {*
-Sign.const_alias @{binding proj} @{const_name Equiv_Relations.proj}
-*}
-
-lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
- by simp
-
-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 not_TrueE: "\<not> True \<Longrightarrow> P"
- by (erule notE, rule TrueI)
-
-lemma neq_eq_eq_contradict: "\<lbrakk>t \<noteq> u; s = t; s = u\<rbrakk> \<Longrightarrow> P"
- by fast
-
-lemma case_sum_expand_Inr: "f o Inl = g \<Longrightarrow> f x = case_sum g (f o Inr) x"
- by (auto split: sum.splits)
-
-lemma case_sum_expand_Inr': "f o Inl = g \<Longrightarrow> h = f o Inr \<longleftrightarrow> case_sum g h = f"
- apply rule
- apply (rule ext, force split: sum.split)
- by (rule ext, metis case_sum_o_inj(2))
-
-lemma converse_Times: "(A \<times> B) ^-1 = B \<times> A"
- by fast
-
-lemma equiv_proj:
- assumes e: "equiv A R" and m: "z \<in> R"
- shows "(proj R o fst) z = (proj R o snd) z"
-proof -
- from m have z: "(fst z, snd z) \<in> R" by auto
- with e have "\<And>x. (fst z, x) \<in> R \<Longrightarrow> (snd z, x) \<in> R" "\<And>x. (snd z, x) \<in> R \<Longrightarrow> (fst z, x) \<in> R"
- unfolding equiv_def sym_def trans_def by blast+
- then show ?thesis unfolding proj_def[abs_def] by auto
-qed
-
-(* Operators: *)
-definition image2 where "image2 A f g = {(f a, g a) | a. a \<in> A}"
-
-lemma Id_on_Gr: "Id_on A = Gr A id"
- unfolding Id_on_def Gr_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 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 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_incl: "Gr A f \<subseteq> A <*> B \<longleftrightarrow> f ` A \<subseteq> B"
- unfolding Gr_def by auto
-
-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 in_rel_Collect_split_eq: "in_rel (Collect (split X)) = X"
- unfolding fun_eq_iff by auto
-
-lemma Collect_split_in_rel_leI: "X \<subseteq> Y \<Longrightarrow> X \<subseteq> Collect (split (in_rel Y))"
- by auto
-
-lemma Collect_split_in_rel_leE: "X \<subseteq> Collect (split (in_rel Y)) \<Longrightarrow> (X \<subseteq> Y \<Longrightarrow> R) \<Longrightarrow> R"
- by force
-
-lemma conversep_in_rel: "(in_rel R)\<inverse>\<inverse> = in_rel (R\<inverse>)"
- unfolding fun_eq_iff by auto
-
-lemma relcompp_in_rel: "in_rel R OO in_rel S = in_rel (R O S)"
- unfolding fun_eq_iff by auto
-
-lemma in_rel_Gr: "in_rel (Gr A f) = Grp A f"
- unfolding Gr_def Grp_def fun_eq_iff by auto
-
-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_Id_on:
- "(\<And>a1 a2. f a1 = f a2 \<longleftrightarrow> a1 = a2) \<Longrightarrow> relInvImage A (Id_on B) f \<subseteq> Id"
- unfolding relInvImage_def Id_on_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> Id_on (A//R)"
- unfolding relImage_def Id_on_def
- using proj_iff[OF assms] equiv_class_eq_iff[OF assms]
- by (auto simp: 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 fast
-
-lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)"
- by simp
-
-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_diag_fst: "fst o ((\<lambda>x. (x, x)) o fst) = fst" by auto
-lemma snd_diag_fst: "snd o ((\<lambda>x. (x, x)) o fst) = fst" by auto
-lemma fst_diag_snd: "fst o ((\<lambda>x. (x, x)) o snd) = snd" by auto
-lemma snd_diag_snd: "snd o ((\<lambda>x. (x, x)) o snd) = snd" by auto
-
-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))"
-
-lemma empty_Shift: "\<lbrakk>[] \<in> Kl; k \<in> Succ Kl []\<rbrakk> \<Longrightarrow> [] \<in> Shift Kl k"
- unfolding Shift_def Succ_def by simp
-
-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 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
-
-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)
-
-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 rec_nat_0_imp: "f = rec_nat f1 (%n rec. f2 n rec) \<Longrightarrow> f 0 = f1"
- by auto
-
-lemma rec_nat_Suc_imp: "f = rec_nat f1 (%n rec. f2 n rec) \<Longrightarrow> f (Suc n) = f2 n (f n)"
- by auto
-
-lemma rec_list_Nil_imp: "f = rec_list f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f [] = f1"
- by auto
-
-lemma rec_list_Cons_imp: "f = rec_list f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f (x # xs) = f2 x xs (f xs)"
- by auto
-
-lemma not_arg_cong_Inr: "x \<noteq> y \<Longrightarrow> Inr x \<noteq> Inr y"
- by simp
-
-definition image2p where
- "image2p f g R = (\<lambda>x y. \<exists>x' y'. R x' y' \<and> f x' = x \<and> g y' = y)"
-
-lemma image2pI: "R x y \<Longrightarrow> image2p f g R (f x) (g y)"
- unfolding image2p_def by blast
-
-lemma image2pE: "\<lbrakk>image2p f g R fx gy; (\<And>x y. fx = f x \<Longrightarrow> gy = g y \<Longrightarrow> R x y \<Longrightarrow> P)\<rbrakk> \<Longrightarrow> P"
- unfolding image2p_def by blast
-
-lemma rel_fun_iff_geq_image2p: "rel_fun R S f g = (image2p f g R \<le> S)"
- unfolding rel_fun_def image2p_def by auto
-
-lemma rel_fun_image2p: "rel_fun R (image2p f g R) f g"
- unfolding rel_fun_def image2p_def by auto
-
-
-subsection {* Equivalence relations, quotients, and Hilbert's choice *}
-
-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
-
-definition univ where "univ f X == f (Eps (%x. x \<in> X))"
-
-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"
-proof (unfold univ_def)
- 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_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 ECH RES univ_commute by fastforce
- thus "univ f X \<in> B" using x PRES by simp
-qed
-
-ML_file "Tools/BNF/bnf_gfp_util.ML"
-ML_file "Tools/BNF/bnf_gfp_tactics.ML"
-ML_file "Tools/BNF/bnf_gfp.ML"
-ML_file "Tools/BNF/bnf_gfp_rec_sugar_tactics.ML"
-ML_file "Tools/BNF/bnf_gfp_rec_sugar.ML"
-
-end