(* Title: HOL/BNF_Def.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012, 2013, 2014
Definition of bounded natural functors.
*)
section \<open>Definition of Bounded Natural Functors\<close>
theory BNF_Def
imports BNF_Cardinal_Arithmetic Fun_Def_Base
keywords
"print_bnfs" :: diag and
"bnf" :: thy_goal_defn
begin
lemma Collect_case_prodD: "x \ Collect (case_prod A) \ A (fst x) (snd x)"
by auto
inductive
rel_sum :: "('a \ 'c \ bool) \ ('b \ 'd \ bool) \ 'a + 'b \ 'c + 'd \ bool" for R1 R2
where
"R1 a c \ rel_sum R1 R2 (Inl a) (Inl c)"
| "R2 b d \ rel_sum R1 R2 (Inr b) (Inr d)"
definition
rel_fun :: "('a \ 'c \ bool) \ ('b \ 'd \ bool) \ ('a \ 'b) \ ('c \ 'd) \ bool"
where
"rel_fun A B = (\f g. \x y. A x y \ B (f x) (g y))"
lemma rel_funI [intro]:
assumes "\x y. A x y \ B (f x) (g y)"
shows "rel_fun A B f g"
using assms by (simp add: rel_fun_def)
lemma rel_funD:
assumes "rel_fun A B f g" and "A x y"
shows "B (f x) (g y)"
using assms by (simp add: rel_fun_def)
lemma rel_fun_mono:
"\ rel_fun X A f g; \x y. Y x y \ X x y; \x y. A x y \ B x y \ \ rel_fun Y B f g"
by(simp add: rel_fun_def)
lemma rel_fun_mono' [mono]:
"\ \x y. Y x y \ X x y; \x y. A x y \ B x y \ \ rel_fun X A f g \ rel_fun Y B f g"
by(simp add: rel_fun_def)
definition rel_set :: "('a \ 'b \ bool) \ 'a set \ 'b set \ bool"
where "rel_set R = (\A B. (\x\A. \y\B. R x y) \ (\y\B. \x\A. R x y))"
lemma rel_setI:
assumes "\x. x \ A \ \y\B. R x y"
assumes "\y. y \ B \ \x\A. R x y"
shows "rel_set R A B"
using assms unfolding rel_set_def by simp
lemma predicate2_transferD:
"\rel_fun R1 (rel_fun R2 (=)) P Q; a \ A; b \ B; A \ {(x, y). R1 x y}; B \ {(x, y). R2 x y}\ \
P (fst a) (fst b) \<longleftrightarrow> Q (snd a) (snd b)"
unfolding rel_fun_def by (blast dest!: Collect_case_prodD)
definition collect where
"collect F x = (\f \ F. f x)"
lemma fstI: "x = (y, z) \ fst x = y"
by simp
lemma sndI: "x = (y, z) \ snd x = z"
by simp
lemma bijI': "\\x y. (f x = f y) = (x = y); \y. \x. y = f x\ \ bij f"
unfolding bij_def inj_on_def by auto blast
(* Operator: *)
definition "Gr A f = {(a, f a) | a. a \ A}"
definition "Grp A f = (\a b. b = f a \ a \ A)"
definition vimage2p where
"vimage2p f g R = (\x y. R (f x) (g y))"
lemma collect_comp: "collect F \ g = collect ((\f. f \ g) ` F)"
by (rule ext) (simp add: collect_def)
definition convol ("\(_,/ _)\") where
"\f, g\ \ \a. (f a, g a)"
lemma fst_convol: "fst \ \f, g\ = f"
apply(rule ext)
unfolding convol_def by simp
lemma snd_convol: "snd \ \f, g\ = g"
apply(rule ext)
unfolding convol_def by simp
lemma convol_mem_GrpI:
"x \ A \ \id, g\ x \ (Collect (case_prod (Grp A g)))"
unfolding convol_def Grp_def by auto
definition csquare where
"csquare A f1 f2 p1 p2 \ (\ a \ A. f1 (p1 a) = f2 (p2 a))"
lemma eq_alt: "(=) = Grp UNIV id"
unfolding Grp_def by auto
lemma leq_conversepI: "R = (=) \ R \ R\\"
by auto
lemma leq_OOI: "R = (=) \ R \ R OO R"
by auto
lemma OO_Grp_alt: "(Grp A f)\\ OO Grp A g = (\x y. \z. z \ A \ f z = x \ g z = y)"
unfolding Grp_def by auto
lemma Grp_UNIV_id: "f = id \ (Grp UNIV f)\\ OO Grp UNIV f = Grp UNIV f"
unfolding Grp_def by auto
lemma Grp_UNIV_idI: "x = y \ Grp UNIV id x y"
unfolding Grp_def by auto
lemma Grp_mono: "A \ B \ Grp A f \ Grp B f"
unfolding Grp_def by auto
lemma GrpI: "\f x = y; x \ A\ \ Grp A f x y"
unfolding Grp_def by auto
lemma GrpE: "Grp A f x y \ (\f x = y; x \ A\ \ R) \ R"
unfolding Grp_def by auto
lemma Collect_case_prod_Grp_eqD: "z \ Collect (case_prod (Grp A f)) \ (f \ fst) z = snd z"
unfolding Grp_def comp_def by auto
lemma Collect_case_prod_Grp_in: "z \ Collect (case_prod (Grp A f)) \ fst z \ A"
unfolding Grp_def comp_def by auto
definition "pick_middlep P Q a c = (SOME b. P a b \ Q b c)"
lemma pick_middlep:
"(P OO Q) a c \ P a (pick_middlep P Q a c) \ Q (pick_middlep P Q a c) c"
unfolding pick_middlep_def apply(rule someI_ex) by auto
definition fstOp where
"fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))"
definition sndOp where
"sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))"
lemma fstOp_in: "ac \ Collect (case_prod (P OO Q)) \ fstOp P Q ac \ Collect (case_prod P)"
unfolding fstOp_def mem_Collect_eq
by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct1])
lemma fst_fstOp: "fst bc = (fst \ fstOp P Q) bc"
unfolding comp_def fstOp_def by simp
lemma snd_sndOp: "snd bc = (snd \ sndOp P Q) bc"
unfolding comp_def sndOp_def by simp
lemma sndOp_in: "ac \ Collect (case_prod (P OO Q)) \ sndOp P Q ac \ Collect (case_prod Q)"
unfolding sndOp_def mem_Collect_eq
by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct2])
lemma csquare_fstOp_sndOp:
"csquare (Collect (f (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)"
unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp
lemma snd_fst_flip: "snd xy = (fst \ (%(x, y). (y, x))) xy"
by (simp split: prod.split)
lemma fst_snd_flip: "fst xy = (snd \ (%(x, y). (y, x))) xy"
by (simp split: prod.split)
lemma flip_pred: "A \ Collect (case_prod (R \\)) \ (%(x, y). (y, x)) ` A \ Collect (case_prod R)"
by auto
lemma predicate2_eqD: "A = B \ A a b \ B a b"
by simp
lemma case_sum_o_inj: "case_sum f g \ Inl = f" "case_sum f g \ Inr = g"
by auto
lemma map_sum_o_inj: "map_sum f g \ Inl = Inl \ f" "map_sum f g \ Inr = Inr \ g"
by auto
lemma card_order_csum_cone_cexp_def:
"card_order r \ ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \ {Inr ()})|"
unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order)
lemma If_the_inv_into_in_Func:
"\inj_on g C; C \ B \ {x}\ \
(\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})"
unfolding Func_def by (auto dest: the_inv_into_into)
lemma If_the_inv_into_f_f:
"\i \ C; inj_on g C\ \ ((\i. if i \ g ` C then the_inv_into C g i else x) \ g) i = id i"
unfolding Func_def by (auto elim: the_inv_into_f_f)
lemma the_inv_f_o_f_id: "inj f \ (the_inv f \ f) z = id z"
by (simp add: the_inv_f_f)
lemma vimage2pI: "R (f x) (g y) \ vimage2p f g R x y"
unfolding vimage2p_def .
lemma rel_fun_iff_leq_vimage2p: "(rel_fun R S) f g = (R \ vimage2p f g S)"
unfolding rel_fun_def vimage2p_def by auto
lemma convol_image_vimage2p: "\f \ fst, g \ snd\ ` Collect (case_prod (vimage2p f g R)) \ Collect (case_prod R)"
unfolding vimage2p_def convol_def by auto
lemma vimage2p_Grp: "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)\\"
unfolding vimage2p_def Grp_def by auto
lemma subst_Pair: "P x y \ a = (x, y) \ P (fst a) (snd a)"
by simp
lemma comp_apply_eq: "f (g x) = h (k x) \ (f \ g) x = (h \ k) x"
unfolding comp_apply by assumption
lemma refl_ge_eq: "(\x. R x x) \ (=) \ R"
by auto
lemma ge_eq_refl: "(=) \ R \ R x x"
by auto
lemma reflp_eq: "reflp R = ((=) \ R)"
by (auto simp: reflp_def fun_eq_iff)
lemma transp_relcompp: "transp r \ r OO r \ r"
by (auto simp: transp_def)
lemma symp_conversep: "symp R = (R\\ \ R)"
by (auto simp: symp_def fun_eq_iff)
lemma diag_imp_eq_le: "(\x. x \ A \ R x x) \ \x y. x \ A \ y \ A \ x = y \ R x y"
by blast
definition eq_onp :: "('a \ bool) \ 'a \ 'a \ bool"
where "eq_onp R = (\x y. R x \ x = y)"
lemma eq_onp_Grp: "eq_onp P = BNF_Def.Grp (Collect P) id"
unfolding eq_onp_def Grp_def by auto
lemma eq_onp_to_eq: "eq_onp P x y \ x = y"
by (simp add: eq_onp_def)
lemma eq_onp_top_eq_eq: "eq_onp top = (=)"
by (simp add: eq_onp_def)
lemma eq_onp_same_args: "eq_onp P x x = P x"
by (auto simp add: eq_onp_def)
lemma eq_onp_eqD: "eq_onp P = Q \ P x = Q x x"
unfolding eq_onp_def by blast
lemma Ball_Collect: "Ball A P = (A \ (Collect P))"
by auto
lemma eq_onp_mono0: "\x\A. P x \ Q x \ \x\A. \y\A. eq_onp P x y \ eq_onp Q x y"
unfolding eq_onp_def by auto
lemma eq_onp_True: "eq_onp (\_. True) = (=)"
unfolding eq_onp_def by simp
lemma Ball_image_comp: "Ball (f ` A) g = Ball A (g \ f)"
by auto
lemma rel_fun_Collect_case_prodD:
"rel_fun A B f g \ X \ Collect (case_prod A) \ x \ X \ B ((f \ fst) x) ((g \ snd) x)"
unfolding rel_fun_def by auto
lemma eq_onp_mono_iff: "eq_onp P \ eq_onp Q \ P \ Q"
unfolding eq_onp_def by auto
ML_file \<open>Tools/BNF/bnf_util.ML\<close>
ML_file \<open>Tools/BNF/bnf_tactics.ML\<close>
ML_file \<open>Tools/BNF/bnf_def_tactics.ML\<close>
ML_file \<open>Tools/BNF/bnf_def.ML\<close>
end
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