blanchet@55059: (*  Title:      HOL/BNF_Def.thy
blanchet@48975:     Author:     Dmitriy Traytel, TU Muenchen
blanchet@57398:     Author:     Jasmin Blanchette, TU Muenchen
blanchet@57698:     Copyright   2012, 2013, 2014
blanchet@48975: 
blanchet@48975: Definition of bounded natural functors.
blanchet@48975: *)
blanchet@48975: 
wenzelm@58889: section {* Definition of Bounded Natural Functors *}
blanchet@48975: 
blanchet@48975: theory BNF_Def
blanchet@57398: imports BNF_Cardinal_Arithmetic Fun_Def_Base
blanchet@48975: keywords
blanchet@49286:   "print_bnfs" :: diag and
blanchet@51836:   "bnf" :: thy_goal
blanchet@48975: begin
blanchet@48975: 
desharna@58104: lemma Collect_splitD: "x \<in> Collect (split A) \<Longrightarrow> A (fst x) (snd x)"
desharna@58104:   by auto
desharna@58104: 
blanchet@57398: definition
desharna@58446:    rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
desharna@58446: where
desharna@58446:    "rel_sum R1 R2 x y =
desharna@58446:      (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
desharna@58446:      | (Inr x, Inr y) \<Rightarrow> R2 x y
desharna@58446:      | _ \<Rightarrow> False)"
desharna@58446: 
desharna@58446: definition
blanchet@57398:   rel_fun :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"
blanchet@57398: where
blanchet@57398:   "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
blanchet@57398: 
blanchet@57398: lemma rel_funI [intro]:
blanchet@57398:   assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)"
blanchet@57398:   shows "rel_fun A B f g"
blanchet@57398:   using assms by (simp add: rel_fun_def)
blanchet@57398: 
blanchet@57398: lemma rel_funD:
blanchet@57398:   assumes "rel_fun A B f g" and "A x y"
blanchet@57398:   shows "B (f x) (g y)"
blanchet@57398:   using assms by (simp add: rel_fun_def)
blanchet@57398: 
desharna@58104: definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
desharna@58104:   where "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
desharna@58104: 
desharna@58104: lemma rel_setI:
desharna@58104:   assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
desharna@58104:   assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
desharna@58104:   shows "rel_set R A B"
desharna@58104:   using assms unfolding rel_set_def by simp
desharna@58104: 
desharna@58104: lemma predicate2_transferD:
desharna@58104:    "\<lbrakk>rel_fun R1 (rel_fun R2 (op =)) P Q; a \<in> A; b \<in> B; A \<subseteq> {(x, y). R1 x y}; B \<subseteq> {(x, y). R2 x y}\<rbrakk> \<Longrightarrow>
desharna@58104:    P (fst a) (fst b) \<longleftrightarrow> Q (snd a) (snd b)"
desharna@58104:   unfolding rel_fun_def by (blast dest!: Collect_splitD)
desharna@58104: 
blanchet@57398: definition collect where
blanchet@58352:   "collect F x = (\<Union>f \<in> F. f x)"
blanchet@57398: 
blanchet@57398: lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y"
blanchet@58352:   by simp
blanchet@57398: 
blanchet@57398: lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z"
blanchet@58352:   by simp
blanchet@57398: 
blanchet@57398: lemma bijI': "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f"
blanchet@58352:   unfolding bij_def inj_on_def by auto blast
blanchet@57398: 
blanchet@57398: (* Operator: *)
blanchet@57398: definition "Gr A f = {(a, f a) | a. a \<in> A}"
blanchet@57398: 
blanchet@57398: definition "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
blanchet@57398: 
blanchet@57398: definition vimage2p where
blanchet@57398:   "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
blanchet@57398: 
blanchet@56635: lemma collect_comp: "collect F \<circ> g = collect ((\<lambda>f. f \<circ> g) ` F)"
blanchet@55066:   by (rule ext) (auto simp only: comp_apply collect_def)
traytel@51893: 
wenzelm@57641: definition convol ("\<langle>(_,/ _)\<rangle>") where
blanchet@58352:   "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)"
traytel@49495: 
blanchet@58352: lemma fst_convol: "fst \<circ> \<langle>f, g\<rangle> = f"
blanchet@58352:   apply(rule ext)
blanchet@58352:   unfolding convol_def by simp
traytel@49495: 
blanchet@58352: lemma snd_convol: "snd \<circ> \<langle>f, g\<rangle> = g"
blanchet@58352:   apply(rule ext)
blanchet@58352:   unfolding convol_def by simp
traytel@49495: 
traytel@51893: lemma convol_mem_GrpI:
blanchet@58352:   "x \<in> A \<Longrightarrow> \<langle>id, g\<rangle> x \<in> (Collect (split (Grp A g)))"
blanchet@58352:   unfolding convol_def Grp_def by auto
traytel@51893: 
blanchet@49312: definition csquare where
blanchet@58352:   "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
blanchet@49312: 
traytel@51893: lemma eq_alt: "op = = Grp UNIV id"
blanchet@58352:   unfolding Grp_def by auto
traytel@51893: 
traytel@51893: lemma leq_conversepI: "R = op = \<Longrightarrow> R \<le> R^--1"
traytel@51893:   by auto
traytel@51893: 
traytel@54841: lemma leq_OOI: "R = op = \<Longrightarrow> R \<le> R OO R"
traytel@51893:   by auto
traytel@51893: 
traytel@53561: lemma OO_Grp_alt: "(Grp A f)^--1 OO Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
traytel@53561:   unfolding Grp_def by auto
traytel@53561: 
traytel@51893: lemma Grp_UNIV_id: "f = id \<Longrightarrow> (Grp UNIV f)^--1 OO Grp UNIV f = Grp UNIV f"
blanchet@58352:   unfolding Grp_def by auto
traytel@51893: 
traytel@51893: lemma Grp_UNIV_idI: "x = y \<Longrightarrow> Grp UNIV id x y"
blanchet@58352:   unfolding Grp_def by auto
traytel@51893: 
traytel@51893: lemma Grp_mono: "A \<le> B \<Longrightarrow> Grp A f \<le> Grp B f"
blanchet@58352:   unfolding Grp_def by auto
traytel@51893: 
traytel@51893: lemma GrpI: "\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> Grp A f x y"
blanchet@58352:   unfolding Grp_def by auto
traytel@51893: 
traytel@51893: lemma GrpE: "Grp A f x y \<Longrightarrow> (\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
blanchet@58352:   unfolding Grp_def by auto
traytel@51893: 
traytel@51893: lemma Collect_split_Grp_eqD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> (f \<circ> fst) z = snd z"
blanchet@58352:   unfolding Grp_def comp_def by auto
traytel@51893: 
traytel@51893: lemma Collect_split_Grp_inD: "z \<in> Collect (split (Grp A f)) \<Longrightarrow> fst z \<in> A"
blanchet@58352:   unfolding Grp_def comp_def by auto
traytel@51893: 
traytel@51893: definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)"
traytel@51893: 
traytel@51893: lemma pick_middlep:
blanchet@58352:   "(P OO Q) a c \<Longrightarrow> P a (pick_middlep P Q a c) \<and> Q (pick_middlep P Q a c) c"
blanchet@58352:   unfolding pick_middlep_def apply(rule someI_ex) by auto
blanchet@49312: 
blanchet@58352: definition fstOp where
blanchet@58352:   "fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))"
blanchet@58352: 
blanchet@58352: definition sndOp where
blanchet@58352:   "sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))"
traytel@51893: 
traytel@51893: lemma fstOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> fstOp P Q ac \<in> Collect (split P)"
blanchet@58352:   unfolding fstOp_def mem_Collect_eq
blanchet@58352:   by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct1])
blanchet@49312: 
traytel@51893: lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc"
blanchet@58352:   unfolding comp_def fstOp_def by simp
traytel@51893: 
traytel@51893: lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc"
blanchet@58352:   unfolding comp_def sndOp_def by simp
traytel@51893: 
traytel@51893: lemma sndOp_in: "ac \<in> Collect (split (P OO Q)) \<Longrightarrow> sndOp P Q ac \<in> Collect (split Q)"
blanchet@58352:   unfolding sndOp_def mem_Collect_eq
blanchet@58352:   by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct2])
traytel@51893: 
traytel@51893: lemma csquare_fstOp_sndOp:
blanchet@58352:   "csquare (Collect (split (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)"
blanchet@58352:   unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp
traytel@51893: 
blanchet@56635: lemma snd_fst_flip: "snd xy = (fst \<circ> (%(x, y). (y, x))) xy"
blanchet@58352:   by (simp split: prod.split)
blanchet@49312: 
blanchet@56635: lemma fst_snd_flip: "fst xy = (snd \<circ> (%(x, y). (y, x))) xy"
blanchet@58352:   by (simp split: prod.split)
blanchet@49312: 
traytel@51893: lemma flip_pred: "A \<subseteq> Collect (split (R ^--1)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (split R)"
blanchet@58352:   by auto
traytel@51893: 
traytel@51893: lemma Collect_split_mono: "A \<le> B \<Longrightarrow> Collect (split A) \<subseteq> Collect (split B)"
traytel@51893:   by auto
traytel@51893: 
traytel@51916: lemma Collect_split_mono_strong: 
traytel@55163:   "\<lbrakk>X = fst ` A; Y = snd ` A; \<forall>a\<in>X. \<forall>b \<in> Y. P a b \<longrightarrow> Q a b; A \<subseteq> Collect (split P)\<rbrakk> \<Longrightarrow>
blanchet@58352:    A \<subseteq> Collect (split Q)"
traytel@51916:   by fastforce
traytel@51916: 
traytel@55163: 
traytel@51917: lemma predicate2_eqD: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b"
blanchet@58352:   by simp
blanchet@49537: 
blanchet@58352: lemma case_sum_o_inj: "case_sum f g \<circ> Inl = f" "case_sum f g \<circ> Inr = g"
blanchet@58352:   by auto
traytel@52635: 
blanchet@58352: lemma map_sum_o_inj: "map_sum f g o Inl = Inl o f" "map_sum f g o Inr = Inr o g"
blanchet@58352:   by auto
traytel@57802: 
traytel@52635: lemma card_order_csum_cone_cexp_def:
traytel@52635:   "card_order r \<Longrightarrow> ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \<union> {Inr ()})|"
traytel@52635:   unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order)
traytel@52635: 
traytel@52635: lemma If_the_inv_into_in_Func:
traytel@52635:   "\<lbrakk>inj_on g C; C \<subseteq> B \<union> {x}\<rbrakk> \<Longrightarrow>
blanchet@58352:    (\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})"
blanchet@58352:   unfolding Func_def by (auto dest: the_inv_into_into)
traytel@52635: 
traytel@52635: lemma If_the_inv_into_f_f:
blanchet@58352:   "\<lbrakk>i \<in> C; inj_on g C\<rbrakk> \<Longrightarrow> ((\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<circ> g) i = id i"
blanchet@58352:   unfolding Func_def by (auto elim: the_inv_into_f_f)
traytel@52635: 
blanchet@56635: lemma the_inv_f_o_f_id: "inj f \<Longrightarrow> (the_inv f \<circ> f) z = id z"
blanchet@56635:   by (simp add: the_inv_f_f)
blanchet@56635: 
traytel@52731: lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y"
traytel@52731:   unfolding vimage2p_def by -
traytel@52719: 
blanchet@55945: lemma rel_fun_iff_leq_vimage2p: "(rel_fun R S) f g = (R \<le> vimage2p f g S)"
blanchet@55945:   unfolding rel_fun_def vimage2p_def by auto
traytel@52719: 
wenzelm@57641: lemma convol_image_vimage2p: "\<langle>f \<circ> fst, g \<circ> snd\<rangle> ` Collect (split (vimage2p f g R)) \<subseteq> Collect (split R)"
traytel@52731:   unfolding vimage2p_def convol_def by auto
traytel@52719: 
traytel@54961: lemma vimage2p_Grp: "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)\<inverse>\<inverse>"
traytel@54961:   unfolding vimage2p_def Grp_def by auto
traytel@54961: 
desharna@58106: lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)"
desharna@58106:   by simp
desharna@58106: 
blanchet@58352: lemma comp_apply_eq: "f (g x) = h (k x) \<Longrightarrow> (f \<circ> g) x = (h \<circ> k) x"
blanchet@58352:   unfolding comp_apply by assumption
blanchet@58352: 
blanchet@57398: ML_file "Tools/BNF/bnf_util.ML"
blanchet@57398: ML_file "Tools/BNF/bnf_tactics.ML"
blanchet@55062: ML_file "Tools/BNF/bnf_def_tactics.ML"
blanchet@55062: ML_file "Tools/BNF/bnf_def.ML"
blanchet@49309: 
blanchet@48975: end