src/HOL/BNF/BNF_GFP.thy
author traytel
Mon Mar 18 11:05:33 2013 +0100 (2013-03-18)
changeset 51446 a6ebb12cc003
parent 50058 bb1fadeba35e
child 51447 a19e973fa2cf
permissions -rw-r--r--
hide internal constants; tuned proofs
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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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header {* Greatest Fixed Point Operation on Bounded Natural Functors *}
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theory BNF_GFP
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imports BNF_FP Equiv_Relations_More "~~/src/HOL/Library/Sublist"
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keywords
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  "codata" :: thy_decl
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begin
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lemma sum_case_comp_Inl:
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"sum_case f g \<circ> Inl = f"
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unfolding comp_def by simp
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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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lemma converse_Times: "(A \<times> B) ^-1 = B \<times> A"
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by auto
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lemma equiv_triv1:
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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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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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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"
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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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(* 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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lemma diagI: "x \<in> A \<Longrightarrow> (x, x) \<in> diag A"
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unfolding diag_def by simp
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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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lemma diagE': "x \<in> diag A \<Longrightarrow> fst x = snd x"
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unfolding diag_def by auto
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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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lemma diag_UNIV: "diag UNIV = Id"
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unfolding diag_def by auto
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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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lemma diag_Comp: "diag A = diag A O diag A"
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unfolding diag_def by auto
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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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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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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})"
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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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definition prefCl where
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  "prefCl Kl = (\<forall> kl1 kl2. prefixeq kl1 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> prefixeq kl' 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. prefixeq kl1 kl2 \<and> kl2 \<in> Kl \<longrightarrow> kl1 \<in> Kl"
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    "prefixeq kl1 kl2" "k # kl2 \<in> Kl"
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  thus "k # kl1 \<in> Kl" using Cons_prefixeq_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 "prefixeq (k # []) (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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lemma fromCard_toCard:
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"\<lbrakk>|A| \<le>o r; Card_order r; b \<in> A\<rbrakk> \<Longrightarrow> fromCard A r (toCard A r b) = b"
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unfolding fromCard_def by (rule some_equality) (auto simp add: toCard_inj)
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(* pick according to the weak pullback *)
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definition pickWP where
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"pickWP A p1 p2 b1 b2 \<equiv> SOME a. a \<in> A \<and> p1 a = b1 \<and> p2 a = b2"
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lemma pickWP_pred:
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assumes "wpull A B1 B2 f1 f2 p1 p2" and
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"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
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shows "\<exists> a. a \<in> A \<and> p1 a = b1 \<and> p2 a = b2"
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using assms unfolding wpull_def by blast
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lemma pickWP:
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assumes "wpull A B1 B2 f1 f2 p1 p2" and
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"b1 \<in> B1" and "b2 \<in> B2" and "f1 b1 = f2 b2"
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shows "pickWP A p1 p2 b1 b2 \<in> A"
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      "p1 (pickWP A p1 p2 b1 b2) = b1"
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      "p2 (pickWP A p1 p2 b1 b2) = b2"
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unfolding pickWP_def using assms someI_ex[OF pickWP_pred] by fastforce+
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lemma Inl_Field_csum: "a \<in> Field r \<Longrightarrow> Inl a \<in> Field (r +c s)"
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unfolding Field_card_of csum_def by auto
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lemma Inr_Field_csum: "a \<in> Field s \<Longrightarrow> Inr a \<in> Field (r +c s)"
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unfolding Field_card_of csum_def by auto
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lemma nat_rec_0: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f 0 = f1"
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by auto
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lemma nat_rec_Suc: "f = nat_rec f1 (%n rec. f2 n rec) \<Longrightarrow> f (Suc n) = f2 n (f n)"
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by auto
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lemma list_rec_Nil: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f [] = f1"
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by auto
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lemma list_rec_Cons: "f = list_rec f1 (%x xs rec. f2 x xs rec) \<Longrightarrow> f (x # xs) = f2 x xs (f xs)"
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by auto
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lemma not_arg_cong_Inr: "x \<noteq> y \<Longrightarrow> Inr x \<noteq> Inr y"
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by simp
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ML_file "Tools/bnf_gfp_util.ML"
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ML_file "Tools/bnf_gfp_tactics.ML"
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ML_file "Tools/bnf_gfp.ML"
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end