1(* Title: HOL/BNF_Def.thy 2 Author: Dmitriy Traytel, TU Muenchen 3 Author: Jasmin Blanchette, TU Muenchen 4 Copyright 2012, 2013, 2014 5 6Definition of bounded natural functors. 7*) 8 9section \<open>Definition of Bounded Natural Functors\<close> 10 11theory BNF_Def 12imports BNF_Cardinal_Arithmetic Fun_Def_Base 13keywords 14 "print_bnfs" :: diag and 15 "bnf" :: thy_goal 16begin 17 18lemma Collect_case_prodD: "x \<in> Collect (case_prod A) \<Longrightarrow> A (fst x) (snd x)" 19 by auto 20 21inductive 22 rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool" for R1 R2 23where 24 "R1 a c \<Longrightarrow> rel_sum R1 R2 (Inl a) (Inl c)" 25| "R2 b d \<Longrightarrow> rel_sum R1 R2 (Inr b) (Inr d)" 26 27definition 28 rel_fun :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" 29where 30 "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))" 31 32lemma rel_funI [intro]: 33 assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" 34 shows "rel_fun A B f g" 35 using assms by (simp add: rel_fun_def) 36 37lemma rel_funD: 38 assumes "rel_fun A B f g" and "A x y" 39 shows "B (f x) (g y)" 40 using assms by (simp add: rel_fun_def) 41 42lemma rel_fun_mono: 43 "\<lbrakk> rel_fun X A f g; \<And>x y. Y x y \<longrightarrow> X x y; \<And>x y. A x y \<Longrightarrow> B x y \<rbrakk> \<Longrightarrow> rel_fun Y B f g" 44by(simp add: rel_fun_def) 45 46lemma rel_fun_mono' [mono]: 47 "\<lbrakk> \<And>x y. Y x y \<longrightarrow> X x y; \<And>x y. A x y \<longrightarrow> B x y \<rbrakk> \<Longrightarrow> rel_fun X A f g \<longrightarrow> rel_fun Y B f g" 48by(simp add: rel_fun_def) 49 50definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" 51 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))" 52 53lemma rel_setI: 54 assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y" 55 assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" 56 shows "rel_set R A B" 57 using assms unfolding rel_set_def by simp 58 59lemma predicate2_transferD: 60 "\<lbrakk>rel_fun R1 (rel_fun R2 (=)) P Q; a \<in> A; b \<in> B; A \<subseteq> {(x, y). R1 x y}; B \<subseteq> {(x, y). R2 x y}\<rbrakk> \<Longrightarrow> 61 P (fst a) (fst b) \<longleftrightarrow> Q (snd a) (snd b)" 62 unfolding rel_fun_def by (blast dest!: Collect_case_prodD) 63 64definition collect where 65 "collect F x = (\<Union>f \<in> F. f x)" 66 67lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y" 68 by simp 69 70lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z" 71 by simp 72 73lemma bijI': "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f" 74 unfolding bij_def inj_on_def by auto blast 75 76(* Operator: *) 77definition "Gr A f = {(a, f a) | a. a \<in> A}" 78 79definition "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)" 80 81definition vimage2p where 82 "vimage2p f g R = (\<lambda>x y. R (f x) (g y))" 83 84lemma collect_comp: "collect F \<circ> g = collect ((\<lambda>f. f \<circ> g) ` F)" 85 by (rule ext) (simp add: collect_def) 86 87definition convol ("\<langle>(_,/ _)\<rangle>") where 88 "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)" 89 90lemma fst_convol: "fst \<circ> \<langle>f, g\<rangle> = f" 91 apply(rule ext) 92 unfolding convol_def by simp 93 94lemma snd_convol: "snd \<circ> \<langle>f, g\<rangle> = g" 95 apply(rule ext) 96 unfolding convol_def by simp 97 98lemma convol_mem_GrpI: 99 "x \<in> A \<Longrightarrow> \<langle>id, g\<rangle> x \<in> (Collect (case_prod (Grp A g)))" 100 unfolding convol_def Grp_def by auto 101 102definition csquare where 103 "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))" 104 105lemma eq_alt: "(=) = Grp UNIV id" 106 unfolding Grp_def by auto 107 108lemma leq_conversepI: "R = (=) \<Longrightarrow> R \<le> R\<inverse>\<inverse>" 109 by auto 110 111lemma leq_OOI: "R = (=) \<Longrightarrow> R \<le> R OO R" 112 by auto 113 114lemma OO_Grp_alt: "(Grp A f)\<inverse>\<inverse> OO Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)" 115 unfolding Grp_def by auto 116 117lemma Grp_UNIV_id: "f = id \<Longrightarrow> (Grp UNIV f)\<inverse>\<inverse> OO Grp UNIV f = Grp UNIV f" 118 unfolding Grp_def by auto 119 120lemma Grp_UNIV_idI: "x = y \<Longrightarrow> Grp UNIV id x y" 121 unfolding Grp_def by auto 122 123lemma Grp_mono: "A \<le> B \<Longrightarrow> Grp A f \<le> Grp B f" 124 unfolding Grp_def by auto 125 126lemma GrpI: "\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> Grp A f x y" 127 unfolding Grp_def by auto 128 129lemma GrpE: "Grp A f x y \<Longrightarrow> (\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R" 130 unfolding Grp_def by auto 131 132lemma Collect_case_prod_Grp_eqD: "z \<in> Collect (case_prod (Grp A f)) \<Longrightarrow> (f \<circ> fst) z = snd z" 133 unfolding Grp_def comp_def by auto 134 135lemma Collect_case_prod_Grp_in: "z \<in> Collect (case_prod (Grp A f)) \<Longrightarrow> fst z \<in> A" 136 unfolding Grp_def comp_def by auto 137 138definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)" 139 140lemma pick_middlep: 141 "(P OO Q) a c \<Longrightarrow> P a (pick_middlep P Q a c) \<and> Q (pick_middlep P Q a c) c" 142 unfolding pick_middlep_def apply(rule someI_ex) by auto 143 144definition fstOp where 145 "fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))" 146 147definition sndOp where 148 "sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))" 149 150lemma fstOp_in: "ac \<in> Collect (case_prod (P OO Q)) \<Longrightarrow> fstOp P Q ac \<in> Collect (case_prod P)" 151 unfolding fstOp_def mem_Collect_eq 152 by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct1]) 153 154lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc" 155 unfolding comp_def fstOp_def by simp 156 157lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc" 158 unfolding comp_def sndOp_def by simp 159 160lemma sndOp_in: "ac \<in> Collect (case_prod (P OO Q)) \<Longrightarrow> sndOp P Q ac \<in> Collect (case_prod Q)" 161 unfolding sndOp_def mem_Collect_eq 162 by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct2]) 163 164lemma csquare_fstOp_sndOp: 165 "csquare (Collect (f (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)" 166 unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp 167 168lemma snd_fst_flip: "snd xy = (fst \<circ> (%(x, y). (y, x))) xy" 169 by (simp split: prod.split) 170 171lemma fst_snd_flip: "fst xy = (snd \<circ> (%(x, y). (y, x))) xy" 172 by (simp split: prod.split) 173 174lemma flip_pred: "A \<subseteq> Collect (case_prod (R \<inverse>\<inverse>)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (case_prod R)" 175 by auto 176 177lemma predicate2_eqD: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b" 178 by simp 179 180lemma case_sum_o_inj: "case_sum f g \<circ> Inl = f" "case_sum f g \<circ> Inr = g" 181 by auto 182 183lemma map_sum_o_inj: "map_sum f g \<circ> Inl = Inl \<circ> f" "map_sum f g \<circ> Inr = Inr \<circ> g" 184 by auto 185 186lemma card_order_csum_cone_cexp_def: 187 "card_order r \<Longrightarrow> ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \<union> {Inr ()})|" 188 unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order) 189 190lemma If_the_inv_into_in_Func: 191 "\<lbrakk>inj_on g C; C \<subseteq> B \<union> {x}\<rbrakk> \<Longrightarrow> 192 (\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})" 193 unfolding Func_def by (auto dest: the_inv_into_into) 194 195lemma If_the_inv_into_f_f: 196 "\<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" 197 unfolding Func_def by (auto elim: the_inv_into_f_f) 198 199lemma the_inv_f_o_f_id: "inj f \<Longrightarrow> (the_inv f \<circ> f) z = id z" 200 by (simp add: the_inv_f_f) 201 202lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y" 203 unfolding vimage2p_def . 204 205lemma rel_fun_iff_leq_vimage2p: "(rel_fun R S) f g = (R \<le> vimage2p f g S)" 206 unfolding rel_fun_def vimage2p_def by auto 207 208lemma convol_image_vimage2p: "\<langle>f \<circ> fst, g \<circ> snd\<rangle> ` Collect (case_prod (vimage2p f g R)) \<subseteq> Collect (case_prod R)" 209 unfolding vimage2p_def convol_def by auto 210 211lemma vimage2p_Grp: "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)\<inverse>\<inverse>" 212 unfolding vimage2p_def Grp_def by auto 213 214lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)" 215 by simp 216 217lemma comp_apply_eq: "f (g x) = h (k x) \<Longrightarrow> (f \<circ> g) x = (h \<circ> k) x" 218 unfolding comp_apply by assumption 219 220lemma refl_ge_eq: "(\<And>x. R x x) \<Longrightarrow> (=) \<le> R" 221 by auto 222 223lemma ge_eq_refl: "(=) \<le> R \<Longrightarrow> R x x" 224 by auto 225 226lemma reflp_eq: "reflp R = ((=) \<le> R)" 227 by (auto simp: reflp_def fun_eq_iff) 228 229lemma transp_relcompp: "transp r \<longleftrightarrow> r OO r \<le> r" 230 by (auto simp: transp_def) 231 232lemma symp_conversep: "symp R = (R\<inverse>\<inverse> \<le> R)" 233 by (auto simp: symp_def fun_eq_iff) 234 235lemma diag_imp_eq_le: "(\<And>x. x \<in> A \<Longrightarrow> R x x) \<Longrightarrow> \<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> x = y \<longrightarrow> R x y" 236 by blast 237 238definition eq_onp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 239 where "eq_onp R = (\<lambda>x y. R x \<and> x = y)" 240 241lemma eq_onp_Grp: "eq_onp P = BNF_Def.Grp (Collect P) id" 242 unfolding eq_onp_def Grp_def by auto 243 244lemma eq_onp_to_eq: "eq_onp P x y \<Longrightarrow> x = y" 245 by (simp add: eq_onp_def) 246 247lemma eq_onp_top_eq_eq: "eq_onp top = (=)" 248 by (simp add: eq_onp_def) 249 250lemma eq_onp_same_args: "eq_onp P x x = P x" 251 by (auto simp add: eq_onp_def) 252 253lemma eq_onp_eqD: "eq_onp P = Q \<Longrightarrow> P x = Q x x" 254 unfolding eq_onp_def by blast 255 256lemma Ball_Collect: "Ball A P = (A \<subseteq> (Collect P))" 257 by auto 258 259lemma eq_onp_mono0: "\<forall>x\<in>A. P x \<longrightarrow> Q x \<Longrightarrow> \<forall>x\<in>A. \<forall>y\<in>A. eq_onp P x y \<longrightarrow> eq_onp Q x y" 260 unfolding eq_onp_def by auto 261 262lemma eq_onp_True: "eq_onp (\<lambda>_. True) = (=)" 263 unfolding eq_onp_def by simp 264 265lemma Ball_image_comp: "Ball (f ` A) g = Ball A (g \<circ> f)" 266 by auto 267 268lemma rel_fun_Collect_case_prodD: 269 "rel_fun A B f g \<Longrightarrow> X \<subseteq> Collect (case_prod A) \<Longrightarrow> x \<in> X \<Longrightarrow> B ((f \<circ> fst) x) ((g \<circ> snd) x)" 270 unfolding rel_fun_def by auto 271 272lemma eq_onp_mono_iff: "eq_onp P \<le> eq_onp Q \<longleftrightarrow> P \<le> Q" 273 unfolding eq_onp_def by auto 274 275ML_file "Tools/BNF/bnf_util.ML" 276ML_file "Tools/BNF/bnf_tactics.ML" 277ML_file "Tools/BNF/bnf_def_tactics.ML" 278ML_file "Tools/BNF/bnf_def.ML" 279 280end 281