(*  Title:      HOL/Codatatype/BNF_FP.thy

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Composition of bounded natural functors.

 *)

 header {* Composition of Bounded Natural Functors *}

 theory BNF_FP

 imports BNF_Comp BNF_Wrap

 keywords

   "defaults"

 begin

 lemma case_unit: "(case u of () => f) = f"

 by (cases u) (hypsubst, rule unit.cases)

 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"

 by simp

 lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"

 by clarify

 lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"

 by auto

 lemma all_unit_eq: "(\<And>x. PROP P x) \<equiv> PROP P ()"

 by simp

 lemma all_prod_eq: "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"

 by clarsimp

 lemma rev_bspec: "a \<in> A \<Longrightarrow> \<forall>z \<in> A. P z \<Longrightarrow> P a"

 by simp

 lemma Un_cong: "\<lbrakk>A = B; C = D\<rbrakk> \<Longrightarrow> A \<union> C = B \<union> D"

 by simp

 definition convol ("<_ , _>") where

 "<f , g> \<equiv> %a. (f a, g a)"

 lemma fst_convol:

 "fst o <f , g> = f"

 apply(rule ext)

 unfolding convol_def by simp

 lemma snd_convol:

 "snd o <f , g> = g"

 apply(rule ext)

 unfolding convol_def by simp

 lemma pointfree_idE: "f o g = id \<Longrightarrow> f (g x) = x"

 unfolding o_def fun_eq_iff by simp

 lemma o_bij:

   assumes gf: "g o f = id" and fg: "f o g = id"

   shows "bij f"

 unfolding bij_def inj_on_def surj_def proof safe

   fix a1 a2 assume "f a1 = f a2"

   hence "g ( f a1) = g (f a2)" by simp

   thus "a1 = a2" using gf unfolding fun_eq_iff by simp

 next

   fix b

   have "b = f (g b)"

   using fg unfolding fun_eq_iff by simp

   thus "EX a. b = f a" by blast

 qed

 lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp

 lemma sum_case_step:

   "sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"

   "sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"

 by auto

 lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"

 by simp

 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"

 by blast

 lemma obj_sumE_f':

 "\<lbrakk>\<forall>x. s = f (Inl x) \<longrightarrow> P; \<forall>x. s = f (Inr x) \<longrightarrow> P\<rbrakk> \<Longrightarrow> s = f x \<longrightarrow> P"

 by (cases x) blast+

 lemma obj_sumE_f:

 "\<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"

 by (rule allI) (rule obj_sumE_f')

 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 obj_sum_step':

 "\<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"

 by (cases x) blast+

 lemma obj_sum_step:

 "\<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"

 by (rule allI) (rule obj_sum_step')

 lemma sum_case_if:

 "sum_case f g (if p then Inl x else Inr y) = (if p then f x else g y)"

 by simp

 lemma UN_compreh_bex_eq_empty:

 "\<Union>{y. \<exists>x\<in>A. y = {}} = {}"

 by blast

 lemma UN_compreh_bex_eq_singleton:

 "\<Union>{y. \<exists>x\<in>A. y = {f x}} = {y. \<exists>x\<in>A. y = f x}"

 by blast

 lemma mem_UN_comprehI:

 "z \<in> {y. \<exists>x\<in>A. y = f x} \<Longrightarrow> z \<in> \<Union>{y. \<exists>x\<in>A. y = {f x}}"

 "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"

 "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}"

 "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"

 by blast+

 lemma mem_UN_comprehI':

 "\<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'}}"

 by blast

 lemma mem_UN_compreh_eq: "(z : \<Union>{y. \<exists>x\<in>A. y = F x}) = (\<exists>x\<in>A. z : F x)"

 by blast

 lemma eq_UN_compreh_Un: "(xa = \<Union>{y. \<exists>x\<in>A. y = c_set1 x \<union> c_set2 x}) =

   (xa = (\<Union>{y. \<exists>x\<in>A. y = c_set1 x} \<union> \<Union>{y. \<exists>x\<in>A. y = c_set2 x}))"

 by blast

 lemma mem_compreh_eq_iff:

 "z \<in> {y. \<exists>x\<in>A. y = f x} = (\<exists> x. x \<in> A & z \<in> {f x})"

 by blast

 lemma ex_mem_singleton: "(\<exists>y. y \<in> A \<and> y \<in> {x}) = (x \<in> A)"

 by simp

 lemma induct_set_step:

 "\<lbrakk>b \<in> A; c \<in> F b\<rbrakk> \<Longrightarrow> \<exists>x. x \<in> A \<and> c \<in> F x"

 "\<lbrakk>B \<in> A; c \<in> f B\<rbrakk> \<Longrightarrow> \<exists>C. (\<exists>a \<in> A. C = f a) \<and>c \<in> C"

 by blast+

 ML_file "Tools/bnf_fp_util.ML"

 ML_file "Tools/bnf_fp_sugar_tactics.ML"

 ML_file "Tools/bnf_fp_sugar.ML"

 end