# Theory BNF_Fixpoint_Base

theory BNF_Fixpoint_Base
imports BNF_Composition Basic_BNFs
```(*  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

Shared fixpoint operations on bounded natural functors.
*)

section ‹Shared Fixpoint Operations on Bounded Natural Functors›

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 "EX 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: "op = ≤ (λa b. P a b ∨ a = b)"
by auto

lemma eq_le_Grp_id_iff: "(op = ≤ 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)) ≡
(⋀x. x ∈ P ⟹ f x ∈ 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 = (λx. g) ⟹ f (λx. 0) = g"
by (erule arg_cong)

lemma inj_on_convol_ident: "inj_on (λx. (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 (λl r. f l r y) x"
by (case_tac x) simp

lemma case_sum_map_sum: "case_sum l r (map_sum f g x) = case_sum (l ∘ f) (r ∘ g) x"
by (case_tac x) simp+

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 (λl r. h (f l) (g r)) x"
by (case_tac x) simp+

lemma case_prod_o_map_prod: "case_prod f ∘ map_prod g1 g2 = case_prod (λl 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)) (op ∘) (op ∘)"
unfolding rel_fun_def by simp

lemma If_transfer: "rel_fun (op =) (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 o f = g ⟹ ⟨g, snd o f⟩ = f"
unfolding convol_def by auto

lemma convol_expand_snd':
assumes "(fst o f = g)"
shows "h = snd o f ⟷ ⟨g, h⟩ = f"
proof -
from assms have *: "⟨g, snd o f⟩ = f" by (rule convol_expand_snd)
then have "h = snd o f ⟷ h = snd o ⟨g, snd o f⟩" by simp
moreover have "… ⟷ h = snd o 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 o Inl = g ⟹ case_sum g (f o Inr) = f"
by (auto split: sum.splits)

lemma case_sum_expand_Inr': "f o Inl = g ⟹ h = f o Inr ⟷ case_sum g h = f"
by (rule iffI) (auto simp add: fun_eq_iff split: sum.splits)

lemma case_sum_expand_Inr: "f o Inl = g ⟹ f x = case_sum g (f o 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 (λ_. c) = c"
unfolding Let_def ..

ML_file "Tools/BNF/bnf_fp_util_tactics.ML"
ML_file "Tools/BNF/bnf_fp_util.ML"
ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
ML_file "Tools/BNF/bnf_fp_n2m.ML"
ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"

end
```