1(* 2 * Copyright 2014, NICTA 3 * 4 * This software may be distributed and modified according to the terms of 5 * the BSD 2-Clause license. Note that NO WARRANTY is provided. 6 * See "LICENSE_BSD2.txt" for details. 7 * 8 * @TAG(NICTA_BSD) 9 *) 10 11theory LemmaBucket 12imports 13 HaskellLemmaBucket 14 SpecValid_R 15 SubMonadLib 16begin 17 18lemma corres_underlying_trivial: 19 "\<lbrakk> nf' \<Longrightarrow> no_fail P' f \<rbrakk> \<Longrightarrow> corres_underlying Id nf nf' (=) \<top> P' f f" 20 by (auto simp add: corres_underlying_def Id_def no_fail_def) 21 22lemma hoare_spec_gen_asm: 23 "\<lbrakk> F \<Longrightarrow> s \<turnstile> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace> \<rbrakk> \<Longrightarrow> s \<turnstile> \<lbrace>P and K F\<rbrace> f \<lbrace>Q\<rbrace>" 24 "\<lbrakk> F \<Longrightarrow> s \<turnstile> \<lbrace>P\<rbrace> f' \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace> \<rbrakk> \<Longrightarrow> s \<turnstile> \<lbrace>P and K F\<rbrace> f' \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>" 25 unfolding spec_valid_def spec_validE_def validE_def 26 apply (clarsimp simp only: pred_conj_def conj_assoc[symmetric] 27 intro!: hoare_gen_asm[unfolded pred_conj_def])+ 28 done 29 30lemma spec_validE_fail: 31 "s \<turnstile> \<lbrace>P\<rbrace> fail \<lbrace>Q\<rbrace>,\<lbrace>E\<rbrace>" 32 by wp+ 33 34lemma mresults_fail: "mresults fail = {}" 35 by (simp add: mresults_def fail_def) 36 37lemma gets_symb_exec_l: 38 "corres_underlying sr nf nf' dc P P' (gets f) (return x)" 39 by (simp add: corres_underlying_def return_def simpler_gets_def split_def) 40 41lemmas mapM_x_wp_inv = mapM_x_wp[where S=UNIV, simplified] 42 43lemma mapM_wp_inv: 44 "(\<And>x. \<lbrace>P\<rbrace> f x \<lbrace>\<lambda>rv. P\<rbrace>) \<Longrightarrow> \<lbrace>P\<rbrace> mapM f xs \<lbrace>\<lambda>rv. P\<rbrace>" 45 apply (rule valid_return_unit) 46 apply (fold mapM_x_mapM) 47 apply (erule mapM_x_wp_inv) 48 done 49 50lemmas mapM_x_wp' = mapM_x_wp [OF _ subset_refl] 51 52lemma corres_underlying_similar: 53 "\<lbrakk> a = a'; b = b'; nf' \<Longrightarrow> no_fail \<top> (f a b) \<rbrakk> 54 \<Longrightarrow> corres_underlying Id nf nf' dc \<top> \<top> (f a b) (f a' b')" 55 by (simp add: corres_underlying_def no_fail_def, blast) 56 57lemma corres_underlying_gets_pre_lhs: 58 "(\<And>x. corres_underlying S nf nf' r (P x) P' (g x) g') \<Longrightarrow> 59 corres_underlying S nf nf' r (\<lambda>s. P (f s) s) P' (gets f >>= (\<lambda>x. g x)) g'" 60 apply (simp add: simpler_gets_def bind_def split_def corres_underlying_def) 61 apply force 62 done 63 64lemma mapM_x_inv_wp: 65 assumes x: "\<And>s. I s \<Longrightarrow> Q s" 66 assumes y: "\<And>x. x \<in> set xs \<Longrightarrow> \<lbrace>P\<rbrace> m x \<lbrace>\<lambda>rv. I\<rbrace>" 67 assumes z: "\<And>s. I s \<Longrightarrow> P s" 68 shows "\<lbrace>I\<rbrace> mapM_x m xs \<lbrace>\<lambda>rv. Q\<rbrace>" 69 apply (rule hoare_post_imp) 70 apply (erule x) 71 apply (rule mapM_x_wp) 72 apply (rule hoare_pre_imp [OF _ y]) 73 apply (erule z) 74 apply assumption 75 apply simp 76 done 77 78 79lemma mapM_x_accumulate_checks': 80 assumes P: "\<And>x. x \<in> set xs' \<Longrightarrow> \<lbrace>\<top>\<rbrace> f x \<lbrace>\<lambda>rv. P x\<rbrace>" 81 assumes P': "\<And>x y. \<lbrakk> x \<in> set xs'; y \<in> set xs' \<rbrakk> 82 \<Longrightarrow> \<lbrace>P y\<rbrace> f x \<lbrace>\<lambda>rv. P y\<rbrace>" 83 shows "set xs \<subseteq> set xs' \<Longrightarrow> 84 \<lbrace>\<top>\<rbrace> mapM_x f xs \<lbrace>\<lambda>rv s. \<forall>x \<in> set xs. P x s\<rbrace>" 85 apply (induct xs) 86 apply (simp add: mapM_x_Nil) 87 apply (simp add: mapM_x_Cons) 88 apply (rule hoare_pre) 89 apply (wp mapM_x_wp'[OF P']) 90 apply blast 91 apply simp 92 apply assumption 93 apply simp 94 apply (rule P) 95 apply simp 96 apply simp 97 done 98 99lemmas mapM_x_accumulate_checks 100 = mapM_x_accumulate_checks'[OF _ _ subset_refl] 101 102(* Other *) 103 104lemma isRight_rel_sum_comb2: 105 "\<lbrakk> (f \<oplus> r) v v'; isRight v' \<rbrakk> 106 \<Longrightarrow> isRight v \<and> r (theRight v) (theRight v')" 107 by (clarsimp simp: isRight_def) 108 109lemma isRight_case_sum: "isRight x \<Longrightarrow> case_sum f g x = g (theRight x)" 110 by (clarsimp simp add: isRight_def) 111 112lemma enumerate_append:"enumerate i (xs @ ys) = enumerate i xs @ enumerate (i + length xs) ys" 113 apply (induct xs arbitrary:ys i) 114 apply clarsimp+ 115 done 116 117lemma enumerate_bound:"(a, b) \<in> set (enumerate n xs) \<Longrightarrow> a < n + length xs" 118 by (metis add.commute in_set_enumerate_eq prod.sel(1)) 119 120lemma enumerate_exceed:"(n + length xs, b) \<notin> set (enumerate n xs)" 121 by (metis enumerate_bound less_not_refl) 122 123lemma all_pair_unwrap:"(\<forall>a. P (fst a) (snd a)) = (\<forall>a b. P a b)" 124 by force 125 126lemma if_fold[simp]:"(if P then Q else if P then R else S) = (if P then Q else S)" 127 by presburger 128 129lemma disjoint_subset_both:"\<lbrakk>A' \<subseteq> A; B' \<subseteq> B; A \<inter> B = {}\<rbrakk> \<Longrightarrow> A' \<inter> B' = {}" 130 by blast 131 132lemma union_split: "\<lbrakk>A \<inter> C = {}; B \<inter> C = {}\<rbrakk> \<Longrightarrow> (A \<union> B) \<inter> C = {}" 133 by (simp add: inf_sup_distrib2) 134 135lemma dom_expand: "dom (\<lambda>x. if P x then Some y else None) = {x. P x}" 136 using if_option_Some by fastforce 137 138lemma range_translate: "(range f = range g) = ((\<forall>x. \<exists>y. f x = g y) \<and> (\<forall>x. \<exists>y. f y = g x))" 139 by (rule iffI, 140 rule conjI, 141 clarsimp, 142 blast, 143 clarsimp, 144 metis f_inv_into_f range_eqI, 145 clarsimp, 146 subst set_eq_subset, 147 rule conjI, 148 clarsimp, 149 rename_tac arg, 150 erule_tac x=arg and P="\<lambda>x. (\<exists>y. f x = g y)" in allE, 151 clarsimp, 152 clarsimp, 153 rename_tac arg, 154 erule_tac x=arg and P="\<lambda>x. (\<exists>y. f y = g x)" in allE, 155 clarsimp, 156 metis range_eqI) 157 158lemma ran_expand: "\<exists>x. P x \<Longrightarrow> ran (\<lambda>x. if P x then Some y else None) = {y}" 159 by (rule subset_antisym, 160 (clarsimp simp:ran_def)+) 161 162lemma map_upd_expand: "f(x \<mapsto> y) = f ++ (\<lambda>z. if z = x then Some y else None)" 163 by (rule ext, rename_tac w, 164 case_tac "w = x", 165 simp, 166 simp add:map_add_def) 167 168lemma map_upd_subI: "\<lbrakk>f \<subseteq>\<^sub>m g; f x = None\<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>m g(x \<mapsto> y)" 169 by (rule_tac f="\<lambda>i. if i = x then Some y else None" in map_add_le_mapE, 170 simp add:map_le_def, 171 rule ballI, rename_tac a, 172 rule conjI, 173 erule_tac x=x in ballE, 174 clarsimp, 175 erule disjE, 176 clarsimp, 177 clarsimp simp:map_add_def, 178 clarsimp, 179 erule disjE, 180 clarsimp, 181 clarsimp simp:map_add_def, 182 clarsimp simp:map_add_def, 183 erule_tac x=a in ballE, 184 erule disjE, 185 (case_tac "g a"; simp_all), 186 clarsimp+) 187 188lemma all_ext: "\<forall>x. f x = g x \<Longrightarrow> f = g" 189 by presburger 190 191lemma conjI2: "\<lbrakk>B; B \<longrightarrow> A\<rbrakk> \<Longrightarrow> A \<and> B" 192 by auto 193 194(* Trivial lemmas for dealing with messy CNode obligations. *) 195lemma Least2: "\<lbrakk>\<not>P 0; \<not>P 1; P (2::nat)\<rbrakk> \<Longrightarrow> Least P = 2" 196 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 197lemma Least3: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; P (3::nat)\<rbrakk> \<Longrightarrow> Least P = 3" 198 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 199lemma Least4: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; P (4::nat)\<rbrakk> \<Longrightarrow> Least P = 4" 200 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 201lemma Least5: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; P (5::nat)\<rbrakk> \<Longrightarrow> Least P = 5" 202 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 203lemma Least6: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; P (6::nat)\<rbrakk> \<Longrightarrow> Least P = 6" 204 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 205lemma Least7: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; P (7::nat)\<rbrakk> \<Longrightarrow> Least P = 7" 206 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 207lemma Least8: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; P (8::nat)\<rbrakk> \<Longrightarrow> Least P = 8" 208 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 209lemma Least9: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; P (9::nat)\<rbrakk> \<Longrightarrow> Least P = 9" 210 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 211lemma Least10: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; P (10::nat)\<rbrakk> \<Longrightarrow> Least P 212 = 10" 213 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 214lemma Least11: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; P (11::nat)\<rbrakk> \<Longrightarrow> 215 Least P = 11" 216 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 217lemma Least12: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; P 218 (12::nat)\<rbrakk> \<Longrightarrow> Least P = 12" 219 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 220lemma Least13: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; P 221 (13::nat)\<rbrakk> \<Longrightarrow> Least P = 13" 222 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 223lemma Least14: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 224 13; P (14::nat)\<rbrakk> \<Longrightarrow> Least P = 14" 225 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 226lemma Least15: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 227 13; \<not>P 14; P (15::nat)\<rbrakk> \<Longrightarrow> Least P = 15" 228 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 229lemma Least16: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 230 13; \<not>P 14; \<not>P 15; P (16::nat)\<rbrakk> \<Longrightarrow> Least P = 16" 231 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 232lemma Least17: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 233 13; \<not>P 14; \<not>P 15; \<not>P 16; P (17::nat)\<rbrakk> \<Longrightarrow> Least P = 17" 234 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 235lemma Least18: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 236 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; P (18::nat)\<rbrakk> \<Longrightarrow> Least P = 18" 237 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 238lemma Least19: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 239 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; P (19::nat)\<rbrakk> \<Longrightarrow> Least P = 19" 240 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 241lemma Least20: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 242 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; P (20::nat)\<rbrakk> \<Longrightarrow> Least P = 20" 243 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 244lemma Least21: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 245 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; P (21::nat)\<rbrakk> \<Longrightarrow> Least P = 21" 246 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 247lemma Least22: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 248 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; \<not>P 21; P (22::nat)\<rbrakk> \<Longrightarrow> Least P 249 = 22" 250 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 251lemma Least23: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 252 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; \<not>P 21; \<not>P 22; P (23::nat)\<rbrakk> \<Longrightarrow> 253 Least P = 23" 254 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 255lemma Least24: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 256 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; \<not>P 21; \<not>P 22; \<not>P 23; P 257 (24::nat)\<rbrakk> \<Longrightarrow> Least P = 24" 258 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 259lemma Least25: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 260 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; \<not>P 21; \<not>P 22; \<not>P 23; \<not>P 24; P 261 (25::nat)\<rbrakk> \<Longrightarrow> Least P = 25" 262 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 263lemma Least26: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 264 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; \<not>P 21; \<not>P 22; \<not>P 23; \<not>P 24; \<not>P 265 25; P (26::nat)\<rbrakk> \<Longrightarrow> Least P = 26" 266 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 267lemma Least27: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 268 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; \<not>P 21; \<not>P 22; \<not>P 23; \<not>P 24; \<not>P 269 25; \<not>P 26; P (27::nat)\<rbrakk> \<Longrightarrow> Least P = 27" 270 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 271lemma Least28: "\<lbrakk>\<not>P 0; \<not>P 1; \<not>P 2; \<not>P 3; \<not>P 4; \<not>P 5; \<not>P 6; \<not>P 7; \<not>P 8; \<not>P 9; \<not>P 10; \<not>P 11; \<not>P 12; \<not>P 272 13; \<not>P 14; \<not>P 15; \<not>P 16; \<not>P 17; \<not>P 18; \<not>P 19; \<not>P 20; \<not>P 21; \<not>P 22; \<not>P 23; \<not>P 24; \<not>P 273 25; \<not>P 26; \<not>P 27; P (28::nat)\<rbrakk> \<Longrightarrow> Least P = 28" 274 by (simp add: Least_Suc eval_nat_numeral(2) eval_nat_numeral(3)) 275 276lemma map_add_discard: "\<not> cond x \<Longrightarrow> (f ++ (\<lambda>x. if cond x then (g x) else None)) x = f x" 277 by (simp add: map_add_def) 278 279lemma dom_split:"\<lbrakk>\<forall>x \<in> S. \<exists>y. f x = Some y; \<forall>x. x \<notin> S \<longrightarrow> f x = None\<rbrakk> \<Longrightarrow> dom f = S" 280 by (auto simp:dom_def) 281 282lemma map_set_in: "x \<in> f ` S = (\<exists>y\<in>S. f y = x)" 283 by blast 284 285lemma map_length_split: 286 "map (length \<circ> (\<lambda>(a, b). P a b # map (f a b) (Q a b))) xs = map (\<lambda>(a, b). 1 + length (Q a b)) xs" 287 by clarsimp 288 289lemma sum_suc: "(\<Sum>x \<leftarrow> xs. Suc (f x)) = length xs + (\<Sum>x \<leftarrow> xs. f x)" 290 apply (induct xs) 291 by clarsimp+ 292 293lemma sum_suc_pair: "(\<Sum>(a, b) \<leftarrow> xs. Suc (f a b)) = length xs + (\<Sum>(a, b) \<leftarrow> xs. f a b)" 294 apply (induct xs) 295 by clarsimp+ 296 297lemma fold_add_sum: "fold (+) ((map (\<lambda>(a, b). f a b) xs)::nat list) 0 = (\<Sum>(a, b) \<leftarrow> xs. f a b)" 298 apply (subst fold_plus_sum_list_rev) 299 apply (subst sum_list_rev) 300 by clarsimp 301 302lemma set_of_enumerate:"card (set (enumerate n xs)) = length xs" 303 by (metis distinct_card distinct_enumerate length_enumerate) 304 305lemma collapse_fst: "fst ` (\<lambda>x. (f x, g x)) ` s = f ` s" 306 by force 307 308lemma collapse_fst2: "fst ` (\<lambda>(x, y). (f x, g y)) ` s = (\<lambda>x. f (fst x)) ` s" 309 by force 310 311lemma collapse_fst3: "(\<lambda>x. f (fst x)) ` set (enumerate n xs) = f ` set [n..<n + length xs]" 312 by (metis image_image list.set_map map_fst_enumerate) 313 314lemma card_of_dom_bounded: 315 fixes f :: "'a \<Rightarrow> 'b option" 316 assumes "finite (UNIV::'a set)" 317 shows "card (dom f) \<le> CARD('a)" 318 by (simp add: assms card_mono) 319 320lemma third_in: "(a, b, c) \<in> S \<Longrightarrow> c \<in> (snd \<circ> snd) ` S" 321 by (metis (erased, hide_lams) map_set_in image_comp snd_conv) 322 323lemma third_in2: "(a \<in> (snd \<circ> snd) ` (set (enumerate i xs))) = (a \<in> snd ` (set xs))" 324 by (metis map_map map_snd_enumerate set_map) 325 326lemma map_of_enum: "map_of (enumerate n xs) x = Some y \<Longrightarrow> y \<in> set xs" 327 apply (clarsimp) 328 by (metis enumerate_eq_zip in_set_zipE) 329 330lemma map_of_append: 331 "(map_of xs ++ map_of ys) x = (case map_of ys x of None \<Rightarrow> map_of xs x | Some x' \<Rightarrow> Some x')" 332 by (simp add: map_add_def) 333 334lemma map_of_append2: 335 "(map_of xs ++ map_of ys ++ map_of zs) x = 336 (case map_of zs x of None \<Rightarrow> (case map_of ys x of None \<Rightarrow> map_of xs x 337 | Some x' \<Rightarrow> Some x') 338 | Some x' \<Rightarrow> Some x')" 339 by (simp add: map_add_def) 340 341lemma map_of_in_set_map: "map_of (map (\<lambda>(n, y). (f n, y)) xs) x = Some z \<Longrightarrow> z \<in> snd ` set xs" 342 proof - 343 assume "map_of (map (\<lambda>(n, y). (f n, y)) xs) x = Some z" 344 hence "(x, z) \<in> (\<lambda>(uu, y). (f uu, y)) ` set xs" using map_of_SomeD by fastforce 345 thus "z \<in> snd ` set xs" using map_set_in by fastforce 346 qed 347 348lemma pair_in_enum: "(a, b) \<in> set (enumerate x ys) \<Longrightarrow> b \<in> set ys" 349 by (metis enumerate_eq_zip in_set_zip2) 350 351lemma distinct_inj: 352 "inj f \<Longrightarrow> distinct xs = distinct (map f xs)" 353 apply (induct xs) 354 apply simp 355 apply (simp add: inj_image_mem_iff) 356 done 357 358lemma distinct_map_via_ran: "distinct (map fst xs) \<Longrightarrow> ran (map_of xs) = set (map snd xs)" 359 apply (cut_tac xs="map fst xs" and ys="map snd xs" in ran_map_of_zip[symmetric]) 360 apply clarsimp+ 361 by (simp add: ran_distinct) 362 363lemma in_ran_in_set: "x \<in> ran (map_of xs) \<Longrightarrow> x \<in> set (map snd xs)" 364 by (metis (mono_tags, hide_lams) map_set_in map_of_SomeD ranE set_map snd_conv) 365 366lemma in_ran_map_app: "x \<in> ran (xs ++ ys ++ zs) \<Longrightarrow> x \<in> ran xs \<or> x \<in> ran ys \<or> x \<in> ran zs" 367 proof - 368 assume a1: "x \<in> ran (xs ++ ys ++ zs)" 369 obtain bb :: "'a \<Rightarrow> ('b \<Rightarrow> 'a option) \<Rightarrow> 'b" where 370 "\<forall>x0 x1. (\<exists>v2. x1 v2 = Some x0) = (x1 (bb x0 x1) = Some x0)" 371 by moura 372 hence f2: "\<forall>f a. (\<not> (\<exists>b. f b = Some a) \<or> f (bb a f) = Some a) \<and> ((\<exists>b. f b = Some a) \<or> (\<forall>b. f b \<noteq> Some a))" 373 by blast 374 have "\<exists>b. (xs ++ ys ++ zs) b = Some x" 375 using a1 by (simp add: ran_def) 376 hence f3: "(xs ++ ys ++ zs) (bb x (xs ++ ys ++ zs)) = Some x" 377 using f2 by meson 378 { assume "ys (bb x (xs ++ ys ++ zs)) \<noteq> None \<or> xs (bb x (xs ++ ys ++ zs)) \<noteq> Some x" 379 { assume "ys (bb x (xs ++ ys ++ zs)) \<noteq> Some x \<and> (ys (bb x (xs ++ ys ++ zs)) \<noteq> None \<or> xs (bb x (xs ++ ys ++ zs)) \<noteq> Some x)" 380 hence "\<exists>b. zs b = Some x" 381 using f3 by auto 382 hence ?thesis 383 by (simp add: ran_def) } 384 hence ?thesis 385 using ran_def by fastforce } 386 thus ?thesis 387 using ran_def by fastforce 388 qed 389 390lemma none_some_map: "None \<notin> S \<Longrightarrow> Some x \<in> S = (x \<in> the ` S)" 391 apply (rule iffI) 392 apply force 393 apply (subst in_these_eq[symmetric]) 394 apply (clarsimp simp:Option.these_def) 395 apply (case_tac "\<exists>y. xa = Some y") 396 by clarsimp+ 397 398lemma none_some_map2: "the ` Set.filter (\<lambda>s. \<not> Option.is_none s) (range f) = ran f" 399 apply (rule subset_antisym) 400 apply clarsimp 401 apply (case_tac "f x", simp_all) 402 apply (simp add: ranI) 403 apply clarsimp 404 apply (subst none_some_map[symmetric]) 405 apply clarsimp+ 406 apply (erule ranE) 407 by (metis range_eqI) 408 409lemma prop_map_of_prop:"\<lbrakk>\<forall>z \<in> set xs. P (g z); map_of (map (\<lambda>x. (f x, g x)) xs) y = Some a\<rbrakk> \<Longrightarrow> P a" 410 using map_of_SomeD by fastforce 411 412lemma range_subsetI2: "\<forall>y\<in>A. \<exists>x. f x = y \<Longrightarrow> A \<subseteq> range f" 413 by fast 414 415lemma insert_strip: "x \<noteq> y \<Longrightarrow> (x \<in> insert y S) = (x \<in> S)" 416 by simp 417 418lemma dom_map_add: "dom ys = A \<Longrightarrow> dom (xs ++ ys) = A \<union> dom xs" 419 by simp 420 421lemma set_compre_unwrap: "({x. P x} \<subseteq> S) = (\<forall>x. P x \<longrightarrow> x \<in> S)" 422 by blast 423 424lemma map_add_same: "\<lbrakk>xs = ys; zs = ws\<rbrakk> \<Longrightarrow> xs ++ zs = ys ++ ws" 425 by simp 426 427lemma map_add_find_left: "n k = None \<Longrightarrow> (m ++ n) k = m k" 428 by (simp add:map_add_def) 429 430lemma map_length_split_triple: 431 "map (length \<circ> (\<lambda>(a, b, c). P a b c # map (f a b c) (Q a b c))) xs = 432 map (\<lambda>(a, b, c). 1 + length (Q a b c)) xs" 433 by fastforce 434 435lemma sum_suc_triple: "(\<Sum>(a, b, c)\<leftarrow>xs. Suc (f a b c)) = length xs + (\<Sum>(a, b, c)\<leftarrow>xs. f a b c)" 436 by (induct xs; clarsimp) 437 438lemma sum_enumerate: "(\<Sum>(a, b)\<leftarrow>enumerate n xs. P b) = (\<Sum>b\<leftarrow>xs. P b)" 439 by (induct xs arbitrary:n; clarsimp) 440 441lemma dom_map_fold:"dom (fold (++) (map (\<lambda>x. [f x \<mapsto> g x]) xs) ms) = dom ms \<union> set (map f xs)" 442 by (induct xs arbitrary:f g ms; clarsimp) 443 444lemma list_ran_prop:"map_of (map (\<lambda>x. (f x, g x)) xs) i = Some t \<Longrightarrow> \<exists>x \<in> set xs. g x = t" 445 by (induct xs arbitrary:f g t i; clarsimp split:if_split_asm) 446 447lemma in_set_enumerate_eq2:"(a, b) \<in> set (enumerate n xs) \<Longrightarrow> (b = xs ! (a - n))" 448 by (simp add: in_set_enumerate_eq) 449 450lemma subset_eq_notI: "\<lbrakk>a\<in> B;a\<notin> C\<rbrakk> \<Longrightarrow> \<not> B \<subseteq> C" 451 by auto 452 453lemma nat_divide_less_eq: 454 fixes b :: nat 455 shows "0 < c \<Longrightarrow> (b div c < a) = (b < a * c)" 456 using td_gal_lt by blast 457 458lemma strengthen_imp_same_first_conj: 459 "(b \<and> (a \<longrightarrow> c) \<and> (a' \<longrightarrow> c')) \<Longrightarrow> ((a \<longrightarrow> b \<and> c) \<and> (a' \<longrightarrow> b \<and> c'))" 460 by blast 461 462lemma conj_impD: 463 "a \<and> b \<Longrightarrow> a \<longrightarrow> b" 464 by blast 465 466lemma set_list_mem_nonempty: 467 "x \<in> set xs \<Longrightarrow> xs \<noteq> []" 468 by auto 469 470lemma strenghten_False_imp: 471 "\<not>P \<Longrightarrow> P \<longrightarrow> Q" 472 by blast 473 474lemma foldl_fun_or_alt: 475 "foldl (\<lambda>x y. x \<or> f y) b ls = foldl (\<or>) b (map f ls)" 476 apply (induct ls) 477 apply clarsimp 478 apply clarsimp 479 by (simp add: foldl_map) 480 481end 482