blanchet@50323: (*  Title:      HOL/Codatatype/BNF_FP.thy
blanchet@50323:     Author:     Dmitriy Traytel, TU Muenchen
blanchet@50323:     Author:     Jasmin Blanchette, TU Muenchen
blanchet@50323:     Copyright   2012
blanchet@50323: 
blanchet@50323: Composition of bounded natural functors.
blanchet@50323: *)
blanchet@50323: 
blanchet@50323: header {* Composition of Bounded Natural Functors *}
blanchet@50323: 
blanchet@50323: theory BNF_FP
blanchet@50323: imports BNF_Comp BNF_Wrap
blanchet@50323: keywords
blanchet@50323:   "defaults"
blanchet@50323: begin
blanchet@50323: 
blanchet@50327: lemma case_unit: "(case u of () => f) = f"
blanchet@50327: by (cases u) (hypsubst, rule unit.cases)
blanchet@50327: 
blanchet@50350: lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
blanchet@50350: by simp
blanchet@50350: 
blanchet@50350: lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
blanchet@50350: by clarify
blanchet@50350: 
blanchet@50350: lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
blanchet@50350: by auto
blanchet@50350: 
blanchet@50383: lemma all_unit_eq: "(\<And>x. PROP P x) \<equiv> PROP P ()"
blanchet@50383: by simp
blanchet@50327: 
blanchet@50383: lemma all_prod_eq: "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
blanchet@50383: by clarsimp
blanchet@50327: 
blanchet@50327: lemma rev_bspec: "a \<in> A \<Longrightarrow> \<forall>z \<in> A. P z \<Longrightarrow> P a"
blanchet@50327: by simp
blanchet@50327: 
blanchet@50327: lemma Un_cong: "\<lbrakk>A = B; C = D\<rbrakk> \<Longrightarrow> A \<union> C = B \<union> D"
blanchet@50327: by simp
blanchet@50327: 
blanchet@50327: definition convol ("<_ , _>") where
blanchet@50327: "<f , g> \<equiv> %a. (f a, g a)"
blanchet@50327: 
blanchet@50327: lemma fst_convol:
blanchet@50327: "fst o <f , g> = f"
blanchet@50327: apply(rule ext)
blanchet@50327: unfolding convol_def by simp
blanchet@50327: 
blanchet@50327: lemma snd_convol:
blanchet@50327: "snd o <f , g> = g"
blanchet@50327: apply(rule ext)
blanchet@50327: unfolding convol_def by simp
blanchet@50327: 
blanchet@50327: lemma pointfree_idE: "f o g = id \<Longrightarrow> f (g x) = x"
blanchet@50327: unfolding o_def fun_eq_iff by simp
blanchet@50327: 
blanchet@50327: lemma o_bij:
blanchet@50327:   assumes gf: "g o f = id" and fg: "f o g = id"
blanchet@50327:   shows "bij f"
blanchet@50327: unfolding bij_def inj_on_def surj_def proof safe
blanchet@50327:   fix a1 a2 assume "f a1 = f a2"
blanchet@50327:   hence "g ( f a1) = g (f a2)" by simp
blanchet@50327:   thus "a1 = a2" using gf unfolding fun_eq_iff by simp
blanchet@50327: next
blanchet@50327:   fix b
blanchet@50327:   have "b = f (g b)"
blanchet@50327:   using fg unfolding fun_eq_iff by simp
blanchet@50327:   thus "EX a. b = f a" by blast
blanchet@50327: qed
blanchet@50327: 
blanchet@50327: lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
blanchet@50327: 
blanchet@50327: lemma sum_case_step:
blanchet@50327:   "sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"
blanchet@50327:   "sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"
blanchet@50327: by auto
blanchet@50327: 
blanchet@50327: lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
blanchet@50327: by simp
blanchet@50327: 
blanchet@50327: lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
blanchet@50327: by blast
blanchet@50327: 
blanchet@50340: lemma obj_sumE_f':
blanchet@50340: "\<lbrakk>\<forall>x. s = f (Inl x) \<longrightarrow> P; \<forall>x. s = f (Inr x) \<longrightarrow> P\<rbrakk> \<Longrightarrow> s = f x \<longrightarrow> P"
blanchet@50340: by (cases x) blast+
blanchet@50340: 
blanchet@50327: lemma obj_sumE_f:
blanchet@50327: "\<lbrakk>\<forall>x. s = f (Inl x) \<longrightarrow> P; \<forall>x. s = f (Inr x) \<longrightarrow> P\<rbrakk> \<Longrightarrow> \<forall>x. s = f x \<longrightarrow> P"
blanchet@50340: by (rule allI) (rule obj_sumE_f')
blanchet@50327: 
blanchet@50327: lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
blanchet@50327: by (cases s) auto
blanchet@50327: 
blanchet@50340: lemma obj_sum_step':
blanchet@50340: "\<lbrakk>\<forall>x. s = f (Inr (Inl x)) \<longrightarrow> P; \<forall>x. s = f (Inr (Inr x)) \<longrightarrow> P\<rbrakk> \<Longrightarrow> s = f (Inr x) \<longrightarrow> P"
blanchet@50340: by (cases x) blast+
blanchet@50340: 
blanchet@50327: lemma obj_sum_step:
blanchet@50340: "\<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"
blanchet@50340: by (rule allI) (rule obj_sum_step')
blanchet@50327: 
blanchet@50327: lemma sum_case_if:
blanchet@50327: "sum_case f g (if p then Inl x else Inr y) = (if p then f x else g y)"
blanchet@50327: by simp
blanchet@50327: 
blanchet@50441: lemma UN_compreh_bex_eq_empty:
blanchet@50441: "\<Union>{y. \<exists>x\<in>A. y = {}} = {}"
blanchet@50441: by blast
blanchet@50441: 
blanchet@50441: lemma UN_compreh_bex_eq_singleton:
blanchet@50441: "\<Union>{y. \<exists>x\<in>A. y = {f x}} = {y. \<exists>x\<in>A. y = f x}"
blanchet@50441: by blast
blanchet@50441: 
blanchet@50441: lemma mem_UN_comprehI:
blanchet@50441: "z \<in> {y. \<exists>x\<in>A. y = f x} \<Longrightarrow> z \<in> \<Union>{y. \<exists>x\<in>A. y = {f x}}"
blanchet@50441: "z \<in> {y. \<exists>x\<in>A. y = f x} \<union> B \<Longrightarrow> z \<in> \<Union>{y. \<exists>x\<in>A. y = {f x}} \<union> B"
blanchet@50441: "z \<in> \<Union>{y. \<exists>x\<in>A. y = F x} \<union> \<Union>{y. \<exists>x\<in>A. y = G x} \<Longrightarrow> z \<in> \<Union>{y. \<exists>x\<in>A. y = F x \<union> G x}"
blanchet@50441: "z \<in> \<Union>{y. \<exists>x\<in>A. y = F x} \<union> (\<Union>{y. \<exists>x\<in>A. y = G x} \<union> B) \<Longrightarrow> z \<in> \<Union>{y. \<exists>x\<in>A. y = F x \<union> G x} \<union> B"
blanchet@50383: by blast+
blanchet@50383: 
blanchet@50441: lemma mem_UN_comprehI':
blanchet@50441: "\<exists>y. (\<exists>x\<in>A. y = F x) \<and> z \<in> y \<Longrightarrow> z \<in> \<Union>{y. \<exists>x\<in>A. y = {y. \<exists>y'\<in>F x. y = y'}}"
blanchet@50441: by blast
blanchet@50441: 
blanchet@50443: lemma mem_UN_compreh_eq: "(z : \<Union>{y. \<exists>x\<in>A. y = F x}) = (\<exists>x\<in>A. z : F x)"
blanchet@50443: by blast
blanchet@50443: 
blanchet@50443: lemma eq_UN_compreh_Un: "(xa = \<Union>{y. \<exists>x\<in>A. y = c_set1 x \<union> c_set2 x}) =
blanchet@50443:   (xa = (\<Union>{y. \<exists>x\<in>A. y = c_set1 x} \<union> \<Union>{y. \<exists>x\<in>A. y = c_set2 x}))"
blanchet@50443: by blast
blanchet@50443: 
blanchet@50442: lemma mem_compreh_eq_iff:
blanchet@50443: "z \<in> {y. \<exists>x\<in>A. y = f x} = (\<exists> x. x \<in> A & z \<in> {f x})"
blanchet@50442: by blast
blanchet@50442: 
blanchet@50442: lemma ex_mem_singleton: "(\<exists>y. y \<in> A \<and> y \<in> {x}) = (x \<in> A)"
blanchet@50442: by simp
blanchet@50442: 
blanchet@50444: lemma prod_set_simps:
blanchet@50444: "fsts (x, y) = {x}"
blanchet@50444: "snds (x, y) = {y}"
blanchet@50444: unfolding fsts_def snds_def by simp+
blanchet@50444: 
blanchet@50444: lemma sum_set_simps:
blanchet@50444: "sum_setl (Inl x) = {x}"
blanchet@50444: "sum_setl (Inr x) = {}"
blanchet@50444: "sum_setr (Inl x) = {}"
blanchet@50444: "sum_setr (Inr x) = {x}"
blanchet@50444: unfolding sum_setl_def sum_setr_def by simp+
blanchet@50444: 
blanchet@50442: lemma induct_set_step:
blanchet@50442: "\<lbrakk>b \<in> A; c \<in> F b\<rbrakk> \<Longrightarrow> \<exists>x. x \<in> A \<and> c \<in> F x"
blanchet@50442: "\<lbrakk>B \<in> A; c \<in> f B\<rbrakk> \<Longrightarrow> \<exists>C. (\<exists>a \<in> A. C = f a) \<and>c \<in> C"
blanchet@50442: by blast+
blanchet@50383: 
blanchet@50324: ML_file "Tools/bnf_fp_util.ML"
blanchet@50324: ML_file "Tools/bnf_fp_sugar_tactics.ML"
blanchet@50324: ML_file "Tools/bnf_fp_sugar.ML"
blanchet@50324: 
blanchet@50323: end