(*  Title:      HOL/BNF/BNF_FP_Base.thy

     Author:     Lorenz Panny, TU Muenchen

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012, 2013

 Shared fixed point operations on bounded natural functors, including

 *)

 header {* Shared Fixed Point Operations on Bounded Natural Functors *}

 theory BNF_FP_Base

 imports BNF_Comp BNF_Ctr_Sugar

 begin

 lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"

 by auto

 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"

 by blast

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

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

 lemma prod_case_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"

 by simp

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

 unfolding o_def fun_eq_iff by simp

 lemma o_bij:

   assumes gf: "g \<circ> f = id" and fg: "f \<circ> 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> \<forall>x. s = f x \<longrightarrow> P"

 by (rule allI) (metis sumE)

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

 by blast

 lemma UN_compreh_eq_eq:

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

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

 by blast+

 lemma Inl_Inr_False: "(Inl x = Inr y) = False"

 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:

 "setl (Inl x) = {x}"

 "setl (Inr x) = {}"

 "setr (Inl x) = {}"

 "setr (Inr x) = {x}"

 unfolding sum_set_defs by simp+

 lemma prod_rel_simp:

 "prod_rel P Q (x, y) (x', y') \<longleftrightarrow> P x x' \<and> Q y y'"

 unfolding prod_rel_def by simp

 lemma sum_rel_simps:

 "sum_rel P Q (Inl x) (Inl x') \<longleftrightarrow> P x x'"

 "sum_rel P Q (Inr y) (Inr y') \<longleftrightarrow> Q y y'"

 "sum_rel P Q (Inl x) (Inr y') \<longleftrightarrow> False"

 "sum_rel P Q (Inr y) (Inl x') \<longleftrightarrow> False"

 unfolding sum_rel_def by simp+

 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"

 by blast

 lemma rewriteR_comp_comp: "\<lbrakk>g o h = r\<rbrakk> \<Longrightarrow> f o g o h = f o r"

   unfolding o_def fun_eq_iff by auto

 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"

   unfolding o_def fun_eq_iff by auto

 lemma rewriteL_comp_comp: "\<lbrakk>f o g = l\<rbrakk> \<Longrightarrow> f o (g o h) = l o h"

   unfolding o_def fun_eq_iff by auto

 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"

   unfolding o_def fun_eq_iff by auto

 lemma convol_o: "<f, g> o h = <f o h, g o h>"

   unfolding convol_def by auto

 lemma map_pair_o_convol: "map_pair h1 h2 o <f, g> = <h1 o f, h2 o g>"

   unfolding convol_def by auto

 lemma map_pair_o_convol_id: "(map_pair f id \<circ> <id , g>) x = <id \<circ> f , g> x"

   unfolding map_pair_o_convol id_o o_id ..

 lemma o_sum_case: "h o sum_case f g = sum_case (h o f) (h o g)"

   unfolding o_def by (auto split: sum.splits)

 lemma sum_case_o_sum_map: "sum_case f g o sum_map h1 h2 = sum_case (f o h1) (g o h2)"

   unfolding o_def by (auto split: sum.splits)

 lemma sum_case_o_sum_map_id: "(sum_case id g o sum_map f id) x = sum_case (f o id) g x"

   unfolding sum_case_o_sum_map id_o o_id ..

 lemma fun_rel_def_butlast:

   "(fun_rel R (fun_rel S T)) f g = (\<forall>x y. R x y \<longrightarrow> (fun_rel S T) (f x) (g y))"

   unfolding fun_rel_def ..

 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"

   by auto

 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"

   by auto

 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"

   unfolding Grp_def id_apply by blast

 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>

    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"

   unfolding Grp_def by rule auto

 lemma eq_ifI: "\<lbrakk>b \<Longrightarrow> t = x; \<not> b \<Longrightarrow> t = y\<rbrakk> \<Longrightarrow> t = (if b then x else y)"

   by fastforce

 lemma if_if_True:

   "(if (if b then True else b') then (if b then x else x') else f (if b then y else y')) =

    (if b then x else if b' then x' else f y')"

   by simp

 lemma if_if_False:

   "(if (if b then False else b') then (if b then x else x') else f (if b then y else y')) =

    (if b then f y else if b' then x' else f y')"

   by simp

 lemma if_then_else_True_False:

   "(if p then False else r) \<longleftrightarrow> \<not> p \<and> r"

   "(if p then True else r) \<longleftrightarrow> p \<or> r"

   "(if p then q else False) \<longleftrightarrow> p \<and> q"

   "(if p then q else True) \<longleftrightarrow> \<not> p \<or> q"

 by simp_all

 lemmas bool_if_simps = bool_simps(7,8,15-17,19,21-24,29-32) if_then_else_True_False

 ML_file "Tools/bnf_fp_util.ML"

 ML_file "Tools/bnf_fp_def_sugar_tactics.ML"

 ML_file "Tools/bnf_fp_def_sugar.ML"

 ML_file "Tools/bnf_fp_n2m_tactics.ML"

 ML_file "Tools/bnf_fp_n2m.ML"

 ML_file "Tools/bnf_fp_n2m_sugar.ML"

 ML_file "Tools/bnf_fp_rec_sugar_util.ML"

 ML_file "Tools/bnf_fp_rec_sugar_tactics.ML"

 ML_file "Tools/bnf_fp_rec_sugar.ML"

 end