(* Title: HOL/BNF_Fixpoint_Base.thy
Author: Lorenz Panny, TU Muenchen
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Author: Martin Desharnais, TU Muenchen
Copyright 2012, 2013, 2014
Shared fixpoint operations on bounded natural functors.
*)
section \<open>Shared Fixpoint Operations on Bounded Natural Functors\<close>
theory BNF_Fixpoint_Base
imports BNF_Composition Basic_BNFs
begin
lemma conj_imp_eq_imp_imp: "(P \ Q \ PROP R) \ (P \ Q \ PROP R)"
by standard simp_all
lemma predicate2D_conj: "P \ Q \ R \ R \ (P x y \ Q x y)"
by blast
lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
by blast
lemma case_unit_Unity: "(case u of () \ f) = f"
by (cases u) (hypsubst, rule unit.case)
lemma case_prod_Pair_iden: "(case p of (x, y) \ (x, y)) = p"
by simp
lemma unit_all_impI: "(P () \ Q ()) \ \x. P x \ Q x"
by simp
lemma pointfree_idE: "f \ g = id \ f (g x) = x"
unfolding comp_def fun_eq_iff by simp
lemma o_bij:
assumes gf: "g \ f = id" and fg: "f \ 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 "\a. b = f a" by blast
qed
lemma case_sum_step:
"case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
"case_sum f (case_sum f' g') (Inr p) = case_sum f' g' p"
by auto
lemma obj_one_pointE: "\x. s = x \ P \ P"
by blast
lemma type_copy_obj_one_point_absE:
assumes "type_definition Rep Abs UNIV" "\x. s = Abs x \ P" shows P
using type_definition.Rep_inverse[OF assms(1)]
by (intro mp[OF spec[OF assms(2), of "Rep s"]]) simp
lemma obj_sumE_f:
assumes "\x. s = f (Inl x) \ P" "\x. s = f (Inr x) \ P"
shows "\x. s = f x \ P"
proof
fix x from assms show "s = f x \ P" by (cases x) auto
qed
lemma case_sum_if:
"case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
by simp
lemma prod_set_simps[simp]:
"fsts (x, y) = {x}"
"snds (x, y) = {y}"
unfolding prod_set_defs by simp+
lemma sum_set_simps[simp]:
"setl (Inl x) = {x}"
"setl (Inr x) = {}"
"setr (Inl x) = {}"
"setr (Inr x) = {x}"
unfolding sum_set_defs by simp+
lemma Inl_Inr_False: "(Inl x = Inr y) = False"
by simp
lemma Inr_Inl_False: "(Inr x = Inl y) = False"
by simp
lemma spec2: "\x y. P x y \ P x y"
by blast
lemma rewriteR_comp_comp: "\g \ h = r\ \ f \ g \ h = f \ r"
unfolding comp_def fun_eq_iff by auto
lemma rewriteR_comp_comp2: "\g \ h = r1 \ r2; f \ r1 = l\ \ f \ g \ h = l \ r2"
unfolding comp_def fun_eq_iff by auto
lemma rewriteL_comp_comp: "\f \ g = l\ \ f \ (g \ h) = l \ h"
unfolding comp_def fun_eq_iff by auto
lemma rewriteL_comp_comp2: "\f \ g = l1 \ l2; l2 \ h = r\ \ f \ (g \ h) = l1 \ r"
unfolding comp_def fun_eq_iff by auto
lemma convol_o: "\f, g\ \ h = \f \ h, g \ h\"
unfolding convol_def by auto
lemma map_prod_o_convol: "map_prod h1 h2 \ \f, g\ = \h1 \ f, h2 \ g\"
unfolding convol_def by auto
lemma map_prod_o_convol_id: "(map_prod f id \ \id, g\) x = \id \ f, g\ x"
unfolding map_prod_o_convol id_comp comp_id ..
lemma o_case_sum: "h \ case_sum f g = case_sum (h \ f) (h \ g)"
unfolding comp_def by (auto split: sum.splits)
lemma case_sum_o_map_sum: "case_sum f g \ map_sum h1 h2 = case_sum (f \ h1) (g \ h2)"
unfolding comp_def by (auto split: sum.splits)
lemma case_sum_o_map_sum_id: "(case_sum id g \ map_sum f id) x = case_sum (f \ id) g x"
unfolding case_sum_o_map_sum id_comp comp_id ..
lemma rel_fun_def_butlast:
"rel_fun R (rel_fun S T) f g = (\x y. R x y \ (rel_fun S T) (f x) (g y))"
unfolding rel_fun_def ..
lemma subst_eq_imp: "(\a b. a = b \ P a b) \ (\a. P a a)"
by auto
lemma eq_subset: "(=) \ (\a b. P a b \ a = b)"
by auto
lemma eq_le_Grp_id_iff: "((=) \ Grp (Collect R) id) = (All R)"
unfolding Grp_def id_apply by blast
lemma Grp_id_mono_subst: "(\x y. Grp P id x y \ Grp Q id (f x) (f y)) \
(\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
unfolding Grp_def by rule auto
lemma vimage2p_mono: "vimage2p f g R x y \ R \ S \ vimage2p f g S x y"
unfolding vimage2p_def by blast
lemma vimage2p_refl: "(\x. R x x) \ vimage2p f f R x x"
unfolding vimage2p_def by auto
lemma
assumes "type_definition Rep Abs UNIV"
shows type_copy_Rep_o_Abs: "Rep \ Abs = id" and type_copy_Abs_o_Rep: "Abs \ Rep = id"
unfolding fun_eq_iff comp_apply id_apply
type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
lemma type_copy_map_comp0_undo:
assumes "type_definition Rep Abs UNIV"
"type_definition Rep' Abs' UNIV"
"type_definition Rep'' Abs'' UNIV"
shows "Abs' \ M \ Rep'' = (Abs' \ M1 \ Rep) \ (Abs \ M2 \ Rep'') \ M1 \ M2 = M"
by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
type_definition.Abs_inverse[OF assms(1) UNIV_I]
type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
lemma vimage2p_id: "vimage2p id id R = R"
unfolding vimage2p_def by auto
lemma vimage2p_comp: "vimage2p (f1 \ f2) (g1 \ g2) = vimage2p f2 g2 \ vimage2p f1 g1"
unfolding fun_eq_iff vimage2p_def o_apply by simp
lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g"
unfolding rel_fun_def vimage2p_def by auto
lemma fun_cong_unused_0: "f = (\x. g) \ f (\x. 0) = g"
by (erule arg_cong)
lemma inj_on_convol_ident: "inj_on (\x. (x, f x)) X"
unfolding inj_on_def by simp
lemma map_sum_if_distrib_then:
"\f g e x y. map_sum f g (if e then Inl x else y) = (if e then Inl (f x) else map_sum f g y)"
"\f g e x y. map_sum f g (if e then Inr x else y) = (if e then Inr (g x) else map_sum f g y)"
by simp_all
lemma map_sum_if_distrib_else:
"\f g e x y. map_sum f g (if e then x else Inl y) = (if e then map_sum f g x else Inl (f y))"
"\f g e x y. map_sum f g (if e then x else Inr y) = (if e then map_sum f g x else Inr (g y))"
by simp_all
lemma case_prod_app: "case_prod f x y = case_prod (\l r. f l r y) x"
by (cases x) simp
lemma case_sum_map_sum: "case_sum l r (map_sum f g x) = case_sum (l \ f) (r \ g) x"
by (cases x) simp_all
lemma case_sum_transfer:
"rel_fun (rel_fun R T) (rel_fun (rel_fun S T) (rel_fun (rel_sum R S) T)) case_sum case_sum"
unfolding rel_fun_def by (auto split: sum.splits)
lemma case_prod_map_prod: "case_prod h (map_prod f g x) = case_prod (\l r. h (f l) (g r)) x"
by (cases x) simp_all
lemma case_prod_o_map_prod: "case_prod f \ map_prod g1 g2 = case_prod (\l r. f (g1 l) (g2 r))"
unfolding comp_def by auto
lemma case_prod_transfer:
"(rel_fun (rel_fun A (rel_fun B C)) (rel_fun (rel_prod A B) C)) case_prod case_prod"
unfolding rel_fun_def by simp
lemma eq_ifI: "(P \ t = u1) \ (\ P \ t = u2) \ t = (if P then u1 else u2)"
by simp
lemma comp_transfer:
"rel_fun (rel_fun B C) (rel_fun (rel_fun A B) (rel_fun A C)) (\) (\)"
unfolding rel_fun_def by simp
lemma If_transfer: "rel_fun (=) (rel_fun A (rel_fun A A)) If If"
unfolding rel_fun_def by simp
lemma Abs_transfer:
assumes type_copy1: "type_definition Rep1 Abs1 UNIV"
assumes type_copy2: "type_definition Rep2 Abs2 UNIV"
shows "rel_fun R (vimage2p Rep1 Rep2 R) Abs1 Abs2"
unfolding vimage2p_def rel_fun_def
type_definition.Abs_inverse[OF type_copy1 UNIV_I]
type_definition.Abs_inverse[OF type_copy2 UNIV_I] by simp
lemma Inl_transfer:
"rel_fun S (rel_sum S T) Inl Inl"
by auto
lemma Inr_transfer:
"rel_fun T (rel_sum S T) Inr Inr"
by auto
lemma Pair_transfer: "rel_fun A (rel_fun B (rel_prod A B)) Pair Pair"
unfolding rel_fun_def by simp
lemma eq_onp_live_step: "x = y \ eq_onp P a a \ x \ P a \ y"
by (simp only: eq_onp_same_args)
lemma top_conj: "top x \ P \ P" "P \ top x \ P"
by blast+
lemma fst_convol': "fst (\f, g\ x) = f x"
using fst_convol unfolding convol_def by simp
lemma snd_convol': "snd (\f, g\ x) = g x"
using snd_convol unfolding convol_def by simp
lemma convol_expand_snd: "fst \ f = g \ \g, snd \ f\ = f"
unfolding convol_def by auto
lemma convol_expand_snd':
assumes "(fst \ f = g)"
shows "h = snd \ f \ \g, h\ = f"
proof -
from assms have *: "\g, snd \ f\ = f" by (rule convol_expand_snd)
then have "h = snd \ f \ h = snd \ \g, snd \ f\" by simp
moreover have "\ \ h = snd \ f" by (simp add: snd_convol)
moreover have "\ \ \g, h\ = f" by (subst (2) *[symmetric]) (auto simp: convol_def fun_eq_iff)
ultimately show ?thesis by simp
qed
lemma case_sum_expand_Inr_pointfree: "f \ Inl = g \ case_sum g (f \ Inr) = f"
by (auto split: sum.splits)
lemma case_sum_expand_Inr': "f \ Inl = g \ h = f \ Inr \ case_sum g h = f"
by (rule iffI) (auto simp add: fun_eq_iff split: sum.splits)
lemma case_sum_expand_Inr: "f \ Inl = g \ f x = case_sum g (f \ Inr) x"
by (auto split: sum.splits)
lemma id_transfer: "rel_fun A A id id"
unfolding rel_fun_def by simp
lemma fst_transfer: "rel_fun (rel_prod A B) A fst fst"
unfolding rel_fun_def by simp
lemma snd_transfer: "rel_fun (rel_prod A B) B snd snd"
unfolding rel_fun_def by simp
lemma convol_transfer:
"rel_fun (rel_fun R S) (rel_fun (rel_fun R T) (rel_fun R (rel_prod S T))) BNF_Def.convol BNF_Def.convol"
unfolding rel_fun_def convol_def by auto
lemma Let_const: "Let x (\_. c) = c"
unfolding Let_def ..
ML_file \<open>Tools/BNF/bnf_fp_util_tactics.ML\<close>
ML_file \<open>Tools/BNF/bnf_fp_util.ML\<close>
ML_file \<open>Tools/BNF/bnf_fp_def_sugar_tactics.ML\<close>
ML_file \<open>Tools/BNF/bnf_fp_def_sugar.ML\<close>
ML_file \<open>Tools/BNF/bnf_fp_n2m_tactics.ML\<close>
ML_file \<open>Tools/BNF/bnf_fp_n2m.ML\<close>
ML_file \<open>Tools/BNF/bnf_fp_n2m_sugar.ML\<close>
end
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