src/HOL/BNF_FP_Base.thy
author blanchet
Tue Mar 04 18:57:17 2014 +0100 (2014-03-04)
changeset 55906 abf91ebd0820
parent 55854 ee270328a781
child 55930 25a90cebbbe5
permissions -rw-r--r--
simplify sets in BNF composition
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(*  Title:      HOL/BNF_FP_Base.thy
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    Author:     Lorenz Panny, TU Muenchen
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012, 2013
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Shared fixed point operations on bounded natural functors.
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*)
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header {* Shared Fixed Point Operations on Bounded Natural Functors *}
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theory BNF_FP_Base
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imports BNF_Comp
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begin
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lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
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by auto
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lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
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by blast
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lemma case_unit_Unity: "(case u of () \<Rightarrow> f) = f"
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by (cases u) (hypsubst, rule unit.case)
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lemma case_prod_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
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by simp
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lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
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by simp
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lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
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by clarify
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lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
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by auto
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lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
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unfolding comp_def fun_eq_iff by simp
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lemma o_bij:
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  assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
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  shows "bij f"
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unfolding bij_def inj_on_def surj_def proof safe
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  fix a1 a2 assume "f a1 = f a2"
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  hence "g ( f a1) = g (f a2)" by simp
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  thus "a1 = a2" using gf unfolding fun_eq_iff by simp
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next
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  fix b
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  have "b = f (g b)"
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  using fg unfolding fun_eq_iff by simp
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  thus "EX a. b = f a" by blast
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qed
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lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
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lemma case_sum_step:
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"case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
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"case_sum f (case_sum f' g') (Inr p) = case_sum f' g' p"
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by auto
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lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
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by simp
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lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
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by blast
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lemma type_copy_obj_one_point_absE:
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  assumes "type_definition Rep Abs UNIV" "\<forall>x. s = Abs x \<longrightarrow> P" shows P
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  using type_definition.Rep_inverse[OF assms(1)]
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  by (intro mp[OF spec[OF assms(2), of "Rep s"]]) simp
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lemma obj_sumE_f:
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  assumes "\<forall>x. s = f (Inl x) \<longrightarrow> P" "\<forall>x. s = f (Inr x) \<longrightarrow> P"
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  shows "\<forall>x. s = f x \<longrightarrow> P"
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proof
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  fix x from assms show "s = f x \<longrightarrow> P" by (cases x) auto
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qed
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lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
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by (cases s) auto
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lemma case_sum_if:
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"case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
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by simp
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lemma Inl_Inr_False: "(Inl x = Inr y) = False"
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by simp
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lemma prod_set_simps:
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"fsts (x, y) = {x}"
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"snds (x, y) = {y}"
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unfolding fsts_def snds_def by simp+
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lemma sum_set_simps:
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"setl (Inl x) = {x}"
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"setl (Inr x) = {}"
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"setr (Inl x) = {}"
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"setr (Inr x) = {x}"
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unfolding sum_set_defs by simp+
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lemma UN_compreh_eq_eq:
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"\<Union>{y. y = A} = A"
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by blast+
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lemma ex_in_single: "(\<exists>x \<in> {y}. P x) = P y"
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by blast
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lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
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by blast
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lemma rewriteR_comp_comp: "\<lbrakk>g o h = r\<rbrakk> \<Longrightarrow> f o g o h = f o r"
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  unfolding comp_def fun_eq_iff by auto
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lemma rewriteR_comp_comp2: "\<lbrakk>g o h = r1 o r2; f o r1 = l\<rbrakk> \<Longrightarrow> f o g o h = l o r2"
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  unfolding comp_def fun_eq_iff by auto
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lemma rewriteL_comp_comp: "\<lbrakk>f o g = l\<rbrakk> \<Longrightarrow> f o (g o h) = l o h"
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  unfolding comp_def fun_eq_iff by auto
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lemma rewriteL_comp_comp2: "\<lbrakk>f o g = l1 o l2; l2 o h = r\<rbrakk> \<Longrightarrow> f o (g o h) = l1 o r"
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  unfolding comp_def fun_eq_iff by auto
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lemma convol_o: "<f, g> o h = <f o h, g o h>"
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  unfolding convol_def by auto
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lemma map_pair_o_convol: "map_pair h1 h2 o <f, g> = <h1 o f, h2 o g>"
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  unfolding convol_def by auto
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lemma map_pair_o_convol_id: "(map_pair f id \<circ> <id , g>) x = <id \<circ> f , g> x"
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  unfolding map_pair_o_convol id_comp comp_id ..
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lemma o_case_sum: "h o case_sum f g = case_sum (h o f) (h o g)"
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  unfolding comp_def by (auto split: sum.splits)
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lemma case_sum_o_sum_map: "case_sum f g o sum_map h1 h2 = case_sum (f o h1) (g o h2)"
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  unfolding comp_def by (auto split: sum.splits)
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lemma case_sum_o_sum_map_id: "(case_sum id g o sum_map f id) x = case_sum (f o id) g x"
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  unfolding case_sum_o_sum_map id_comp comp_id ..
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lemma fun_rel_def_butlast:
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  "fun_rel R (fun_rel S T) f g = (\<forall>x y. R x y \<longrightarrow> (fun_rel S T) (f x) (g y))"
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  unfolding fun_rel_def ..
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lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
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  by auto
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lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
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  by auto
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lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
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  unfolding Grp_def id_apply by blast
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lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
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   (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
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  unfolding Grp_def by rule auto
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lemma vimage2p_mono: "vimage2p f g R x y \<Longrightarrow> R \<le> S \<Longrightarrow> vimage2p f g S x y"
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  unfolding vimage2p_def by blast
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lemma vimage2p_refl: "(\<And>x. R x x) \<Longrightarrow> vimage2p f f R x x"
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  unfolding vimage2p_def by auto
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lemma
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  assumes "type_definition Rep Abs UNIV"
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  shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs o Rep = id"
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  unfolding fun_eq_iff comp_apply id_apply
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    type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
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lemma type_copy_map_comp0_undo:
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  assumes "type_definition Rep Abs UNIV"
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          "type_definition Rep' Abs' UNIV"
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          "type_definition Rep'' Abs'' UNIV"
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  shows "Abs' o M o Rep'' = (Abs' o M1 o Rep) o (Abs o M2 o Rep'') \<Longrightarrow> M1 o M2 = M"
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  by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
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    type_definition.Abs_inverse[OF assms(1) UNIV_I]
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    type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
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lemma vimage2p_id: "vimage2p id id R = R"
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  unfolding vimage2p_def by auto
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lemma vimage2p_comp: "vimage2p (f1 o f2) (g1 o g2) = vimage2p f2 g2 o vimage2p f1 g1"
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  unfolding fun_eq_iff vimage2p_def o_apply by simp
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ML_file "Tools/BNF/bnf_fp_util.ML"
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ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
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ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
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ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
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ML_file "Tools/BNF/bnf_fp_n2m.ML"
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ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
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end