(*  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 prod_set_simps:

"fsts (x, y) = {x}"

"snds (x, y) = {y}"

unfolding fsts_def snds_def by simp+

lemma sum_set_simps:

"sum_setl (Inl x) = {x}"

"sum_setl (Inr x) = {}"

"sum_setr (Inl x) = {}"

"sum_setr (Inr x) = {x}"

unfolding sum_setl_def sum_setr_def 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