author | blanchet |
Fri, 21 Sep 2012 16:34:40 +0200 | |
changeset 49509 | 163914705f8d |
parent 49328 | a1c10b46fecd |
permissions | -rw-r--r-- |
49509
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renamed top-level theory from "Codatatype" to "BNF"
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(* Title: HOL/BNF/BNF_GFP.thy |
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Author: Dmitriy Traytel, TU Muenchen |
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Copyright 2012 |
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Greatest fixed point operation on bounded natural functors. |
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*) |
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|
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header {* Greatest Fixed Point Operation on Bounded Natural Functors *} |
7f79f94a432c
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added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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theory BNF_GFP |
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imports BNF_FP Equiv_Relations_More "~~/src/HOL/Library/Prefix_Order" |
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keywords |
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reorganized dependencies so that the sugar does not depend on GFP -- this will be essential for bootstrapping
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"codata_raw" :: thy_decl and |
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reorganized dependencies so that the sugar does not depend on GFP -- this will be essential for bootstrapping
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"codata" :: thy_decl |
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begin |
7f79f94a432c
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|
49312 | 17 |
lemma sum_case_comp_Inl: |
18 |
"sum_case f g \<circ> Inl = f" |
|
19 |
unfolding comp_def by simp |
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||
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lemma sum_case_expand_Inr: "f o Inl = g \<Longrightarrow> f x = sum_case g (f o Inr) x" |
|
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by (auto split: sum.splits) |
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||
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lemma converse_Times: "(A \<times> B) ^-1 = B \<times> A" |
|
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by auto |
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||
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lemma equiv_triv1: |
|
28 |
assumes "equiv A R" and "(a, b) \<in> R" and "(a, c) \<in> R" |
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shows "(b, c) \<in> R" |
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using assms unfolding equiv_def sym_def trans_def by blast |
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||
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lemma equiv_triv2: |
|
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assumes "equiv A R" and "(a, b) \<in> R" and "(b, c) \<in> R" |
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shows "(a, c) \<in> R" |
|
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using assms unfolding equiv_def trans_def by blast |
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||
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lemma equiv_proj: |
|
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assumes e: "equiv A R" and "z \<in> R" |
|
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shows "(proj R o fst) z = (proj R o snd) z" |
|
40 |
proof - |
|
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from assms(2) have z: "(fst z, snd z) \<in> R" by auto |
|
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have P: "\<And>x. (fst z, x) \<in> R \<Longrightarrow> (snd z, x) \<in> R" by (erule equiv_triv1[OF e z]) |
|
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have "\<And>x. (snd z, x) \<in> R \<Longrightarrow> (fst z, x) \<in> R" by (erule equiv_triv2[OF e z]) |
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with P show ?thesis unfolding proj_def[abs_def] by auto |
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qed |
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||
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(* Operators: *) |
|
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definition diag where "diag A \<equiv> {(a,a) | a. a \<in> A}" |
|
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definition image2 where "image2 A f g = {(f a, g a) | a. a \<in> A}" |
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||
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lemma diagI: "x \<in> A \<Longrightarrow> (x, x) \<in> diag A" |
|
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unfolding diag_def by simp |
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||
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lemma diagE: "(a, b) \<in> diag A \<Longrightarrow> a = b" |
|
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unfolding diag_def by simp |
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||
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lemma diagE': "x \<in> diag A \<Longrightarrow> fst x = snd x" |
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unfolding diag_def by auto |
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||
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lemma diag_fst: "x \<in> diag A \<Longrightarrow> fst x \<in> A" |
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unfolding diag_def by auto |
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||
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lemma diag_UNIV: "diag UNIV = Id" |
|
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unfolding diag_def by auto |
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65 |
||
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lemma diag_converse: "diag A = (diag A) ^-1" |
|
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unfolding diag_def by auto |
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||
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lemma diag_Comp: "diag A = diag A O diag A" |
|
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unfolding diag_def by auto |
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||
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lemma diag_Gr: "diag A = Gr A id" |
|
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unfolding diag_def Gr_def by simp |
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||
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lemma diag_UNIV_I: "x = y \<Longrightarrow> (x, y) \<in> diag UNIV" |
|
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unfolding diag_def by auto |
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lemma image2_eqI: "\<lbrakk>b = f x; c = g x; x \<in> A\<rbrakk> \<Longrightarrow> (b, c) \<in> image2 A f g" |
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unfolding image2_def by auto |
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lemma Id_subset: "Id \<subseteq> {(a, b). P a b \<or> a = b}" |
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by auto |
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lemma IdD: "(a, b) \<in> Id \<Longrightarrow> a = b" |
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by auto |
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lemma image2_Gr: "image2 A f g = (Gr A f)^-1 O (Gr A g)" |
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unfolding image2_def Gr_def by auto |
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lemma GrI: "\<lbrakk>x \<in> A; f x = fx\<rbrakk> \<Longrightarrow> (x, fx) \<in> Gr A f" |
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unfolding Gr_def by simp |
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lemma GrE: "(x, fx) \<in> Gr A f \<Longrightarrow> (x \<in> A \<Longrightarrow> f x = fx \<Longrightarrow> P) \<Longrightarrow> P" |
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unfolding Gr_def by simp |
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lemma GrD1: "(x, fx) \<in> Gr A f \<Longrightarrow> x \<in> A" |
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unfolding Gr_def by simp |
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lemma GrD2: "(x, fx) \<in> Gr A f \<Longrightarrow> f x = fx" |
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unfolding Gr_def by simp |
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lemma Gr_incl: "Gr A f \<subseteq> A <*> B \<longleftrightarrow> f ` A \<subseteq> B" |
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unfolding Gr_def by auto |
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||
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definition relImage where |
|
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"relImage R f \<equiv> {(f a1, f a2) | a1 a2. (a1,a2) \<in> R}" |
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definition relInvImage where |
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"relInvImage A R f \<equiv> {(a1, a2) | a1 a2. a1 \<in> A \<and> a2 \<in> A \<and> (f a1, f a2) \<in> R}" |
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lemma relImage_Gr: |
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"\<lbrakk>R \<subseteq> A \<times> A\<rbrakk> \<Longrightarrow> relImage R f = (Gr A f)^-1 O R O Gr A f" |
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unfolding relImage_def Gr_def relcomp_def by auto |
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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" |
|
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unfolding Gr_def relcomp_def image_def relInvImage_def by auto |
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lemma relImage_mono: |
|
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"R1 \<subseteq> R2 \<Longrightarrow> relImage R1 f \<subseteq> relImage R2 f" |
|
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unfolding relImage_def by auto |
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lemma relInvImage_mono: |
|
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"R1 \<subseteq> R2 \<Longrightarrow> relInvImage A R1 f \<subseteq> relInvImage A R2 f" |
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unfolding relInvImage_def by auto |
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lemma relInvImage_diag: |
|
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"(\<And>a1 a2. f a1 = f a2 \<longleftrightarrow> a1 = a2) \<Longrightarrow> relInvImage A (diag B) f \<subseteq> Id" |
|
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unfolding relInvImage_def diag_def by auto |
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lemma relInvImage_UNIV_relImage: |
|
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"R \<subseteq> relInvImage UNIV (relImage R f) f" |
|
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unfolding relInvImage_def relImage_def by auto |
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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})" |
|
135 |
unfolding equiv_def refl_on_def Image_def by (auto intro: transD symD) |
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lemma relImage_proj: |
|
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assumes "equiv A R" |
|
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shows "relImage R (proj R) \<subseteq> diag (A//R)" |
|
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unfolding relImage_def diag_def apply safe |
|
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using proj_iff[OF assms] |
|
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by (metis assms equiv_Image proj_def proj_preserves) |
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lemma relImage_relInvImage: |
|
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assumes "R \<subseteq> f ` A <*> f ` A" |
|
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shows "relImage (relInvImage A R f) f = R" |
|
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using assms unfolding relImage_def relInvImage_def by fastforce |
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lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)" |
|
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by simp |
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lemma fst_diag_id: "(fst \<circ> (%x. (x, x))) z = id z" |
|
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by simp |
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lemma snd_diag_id: "(snd \<circ> (%x. (x, x))) z = id z" |
|
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by simp |
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lemma Collect_restrict': "{(x, y) | x y. phi x y \<and> P x y} \<subseteq> {(x, y) | x y. phi x y}" |
|
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by auto |
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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" |
|
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unfolding convol_def by auto |
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(*Extended Sublist*) |
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||
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definition prefCl where |
|
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"prefCl Kl = (\<forall> kl1 kl2. kl1 \<le> kl2 \<and> kl2 \<in> Kl \<longrightarrow> kl1 \<in> Kl)" |
|
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definition PrefCl where |
|
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"PrefCl A n = (\<forall>kl kl'. kl \<in> A n \<and> kl' \<le> kl \<longrightarrow> (\<exists>m\<le>n. kl' \<in> A m))" |
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lemma prefCl_UN: |
|
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"\<lbrakk>\<And>n. PrefCl A n\<rbrakk> \<Longrightarrow> prefCl (\<Union>n. A n)" |
|
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unfolding prefCl_def PrefCl_def by fastforce |
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definition Succ where "Succ Kl kl = {k . kl @ [k] \<in> Kl}" |
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definition Shift where "Shift Kl k = {kl. k # kl \<in> Kl}" |
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definition shift where "shift lab k = (\<lambda>kl. lab (k # kl))" |
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lemma empty_Shift: "\<lbrakk>[] \<in> Kl; k \<in> Succ Kl []\<rbrakk> \<Longrightarrow> [] \<in> Shift Kl k" |
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unfolding Shift_def Succ_def by simp |
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lemma Shift_clists: "Kl \<subseteq> Field (clists r) \<Longrightarrow> Shift Kl k \<subseteq> Field (clists r)" |
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unfolding Shift_def clists_def Field_card_of by auto |
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lemma Shift_prefCl: "prefCl Kl \<Longrightarrow> prefCl (Shift Kl k)" |
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unfolding prefCl_def Shift_def |
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proof safe |
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fix kl1 kl2 |
|
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assume "\<forall>kl1 kl2. kl1 \<le> kl2 \<and> kl2 \<in> Kl \<longrightarrow> kl1 \<in> Kl" |
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"kl1 \<le> kl2" "k # kl2 \<in> Kl" |
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thus "k # kl1 \<in> Kl" using Cons_prefix_Cons[of k kl1 k kl2] by blast |
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qed |
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lemma not_in_Shift: "kl \<notin> Shift Kl x \<Longrightarrow> x # kl \<notin> Kl" |
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unfolding Shift_def by simp |
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lemma prefCl_Succ: "\<lbrakk>prefCl Kl; k # kl \<in> Kl\<rbrakk> \<Longrightarrow> k \<in> Succ Kl []" |
|
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unfolding Succ_def proof |
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assume "prefCl Kl" "k # kl \<in> Kl" |
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moreover have "k # [] \<le> k # kl" by auto |
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ultimately have "k # [] \<in> Kl" unfolding prefCl_def by blast |
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thus "[] @ [k] \<in> Kl" by simp |
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qed |
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lemma SuccD: "k \<in> Succ Kl kl \<Longrightarrow> kl @ [k] \<in> Kl" |
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unfolding Succ_def by simp |
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lemmas SuccE = SuccD[elim_format] |
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lemma SuccI: "kl @ [k] \<in> Kl \<Longrightarrow> k \<in> Succ Kl kl" |
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unfolding Succ_def by simp |
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lemma ShiftD: "kl \<in> Shift Kl k \<Longrightarrow> k # kl \<in> Kl" |
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unfolding Shift_def by simp |
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lemma Succ_Shift: "Succ (Shift Kl k) kl = Succ Kl (k # kl)" |
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unfolding Succ_def Shift_def by auto |
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lemma ShiftI: "k # kl \<in> Kl \<Longrightarrow> kl \<in> Shift Kl k" |
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unfolding Shift_def by simp |
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lemma Func_cexp: "|Func A B| =o |B| ^c |A|" |
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unfolding cexp_def Field_card_of by (simp only: card_of_refl) |
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lemma clists_bound: "A \<in> Field (cpow (clists r)) - {{}} \<Longrightarrow> |A| \<le>o clists r" |
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unfolding cpow_def clists_def Field_card_of by (auto simp: card_of_mono1) |
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lemma cpow_clists_czero: "\<lbrakk>A \<in> Field (cpow (clists r)) - {{}}; |A| =o czero\<rbrakk> \<Longrightarrow> False" |
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unfolding cpow_def clists_def |
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by (auto simp add: card_of_ordIso_czero_iff_empty[symmetric]) |
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(erule notE, erule ordIso_transitive, rule czero_ordIso) |
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lemma incl_UNION_I: |
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assumes "i \<in> I" and "A \<subseteq> F i" |
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shows "A \<subseteq> UNION I F" |
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using assms by auto |
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lemma Nil_clists: "{[]} \<subseteq> Field (clists r)" |
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unfolding clists_def Field_card_of by auto |
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lemma Cons_clists: |
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"\<lbrakk>x \<in> Field r; xs \<in> Field (clists r)\<rbrakk> \<Longrightarrow> x # xs \<in> Field (clists r)" |
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unfolding clists_def Field_card_of by auto |
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lemma length_Cons: "length (x # xs) = Suc (length xs)" |
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by simp |
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lemma length_append_singleton: "length (xs @ [x]) = Suc (length xs)" |
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by simp |
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(*injection into the field of a cardinal*) |
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definition "toCard_pred A r f \<equiv> inj_on f A \<and> f ` A \<subseteq> Field r \<and> Card_order r" |
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definition "toCard A r \<equiv> SOME f. toCard_pred A r f" |
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lemma ex_toCard_pred: |
|
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"\<lbrakk>|A| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> \<exists> f. toCard_pred A r f" |
|
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unfolding toCard_pred_def |
|
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using card_of_ordLeq[of A "Field r"] |
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ordLeq_ordIso_trans[OF _ card_of_unique[of "Field r" r], of "|A|"] |
|
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by blast |
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lemma toCard_pred_toCard: |
|
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"\<lbrakk>|A| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> toCard_pred A r (toCard A r)" |
|
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unfolding toCard_def using someI_ex[OF ex_toCard_pred] . |
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lemma toCard_inj: "\<lbrakk>|A| \<le>o r; Card_order r; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> |
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toCard A r x = toCard A r y \<longleftrightarrow> x = y" |
|
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using toCard_pred_toCard unfolding inj_on_def toCard_pred_def by blast |
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lemma toCard: "\<lbrakk>|A| \<le>o r; Card_order r; b \<in> A\<rbrakk> \<Longrightarrow> toCard A r b \<in> Field r" |
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using toCard_pred_toCard unfolding toCard_pred_def by blast |
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definition "fromCard A r k \<equiv> SOME b. b \<in> A \<and> toCard A r b = k" |
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275 |
lemma fromCard_toCard: |
|
276 |
"\<lbrakk>|A| \<le>o r; Card_order r; b \<in> A\<rbrakk> \<Longrightarrow> fromCard A r (toCard A r b) = b" |
|
277 |
unfolding fromCard_def by (rule some_equality) (auto simp add: toCard_inj) |
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278 |
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279 |
(* pick according to the weak pullback *) |
|
280 |
definition pickWP_pred where |
|
49325 | 281 |
"pickWP_pred A p1 p2 b1 b2 a \<equiv> a \<in> A \<and> p1 a = b1 \<and> p2 a = b2" |
49312 | 282 |
|
283 |
definition pickWP where |
|
49325 | 284 |
"pickWP A p1 p2 b1 b2 \<equiv> SOME a. pickWP_pred A p1 p2 b1 b2 a" |
49312 | 285 |
|
286 |
lemma pickWP_pred: |
|
287 |
assumes "wpull A B1 B2 f1 f2 p1 p2" and |
|
288 |
"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2" |
|
289 |
shows "\<exists> a. pickWP_pred A p1 p2 b1 b2 a" |
|
290 |
using assms unfolding wpull_def pickWP_pred_def by blast |
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291 |
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292 |
lemma pickWP_pred_pickWP: |
|
293 |
assumes "wpull A B1 B2 f1 f2 p1 p2" and |
|
294 |
"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2" |
|
295 |
shows "pickWP_pred A p1 p2 b1 b2 (pickWP A p1 p2 b1 b2)" |
|
296 |
unfolding pickWP_def using assms by(rule someI_ex[OF pickWP_pred]) |
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297 |
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298 |
lemma pickWP: |
|
299 |
assumes "wpull A B1 B2 f1 f2 p1 p2" and |
|
300 |
"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2" |
|
301 |
shows "pickWP A p1 p2 b1 b2 \<in> A" |
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302 |
"p1 (pickWP A p1 p2 b1 b2) = b1" |
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303 |
"p2 (pickWP A p1 p2 b1 b2) = b2" |
|
304 |
using assms pickWP_pred_pickWP unfolding pickWP_pred_def by fastforce+ |
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305 |
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306 |
lemma Inl_Field_csum: "a \<in> Field r \<Longrightarrow> Inl a \<in> Field (r +c s)" |
|
307 |
unfolding Field_card_of csum_def by auto |
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308 |
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309 |
lemma Inr_Field_csum: "a \<in> Field s \<Longrightarrow> Inr a \<in> Field (r +c s)" |
|
310 |
unfolding Field_card_of csum_def by auto |
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311 |
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312 |
lemma nat_rec_0: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f 0 = f1" |
|
313 |
by auto |
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314 |
||
315 |
lemma nat_rec_Suc: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f (Suc n) = f2 n (f n)" |
|
316 |
by auto |
|
317 |
||
318 |
lemma list_rec_Nil: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f [] = f1" |
|
319 |
by auto |
|
320 |
||
321 |
lemma list_rec_Cons: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f (x # xs) = f2 x xs (f xs)" |
|
322 |
by auto |
|
323 |
||
324 |
lemma not_arg_cong_Inr: "x \<noteq> y \<Longrightarrow> Inr x \<noteq> Inr y" |
|
325 |
by simp |
|
326 |
||
49309
f20b24214ac2
split basic BNFs into really basic ones and others, and added Andreas Lochbihler's "option" BNF
blanchet
parents:
49308
diff
changeset
|
327 |
ML_file "Tools/bnf_gfp_util.ML" |
f20b24214ac2
split basic BNFs into really basic ones and others, and added Andreas Lochbihler's "option" BNF
blanchet
parents:
49308
diff
changeset
|
328 |
ML_file "Tools/bnf_gfp_tactics.ML" |
f20b24214ac2
split basic BNFs into really basic ones and others, and added Andreas Lochbihler's "option" BNF
blanchet
parents:
49308
diff
changeset
|
329 |
ML_file "Tools/bnf_gfp.ML" |
f20b24214ac2
split basic BNFs into really basic ones and others, and added Andreas Lochbihler's "option" BNF
blanchet
parents:
49308
diff
changeset
|
330 |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
331 |
end |