1.1 --- a/src/HOL/Library/Quotient_List.thy Tue Jun 22 19:46:16 2010 +0200
1.2 +++ b/src/HOL/Library/Quotient_List.thy Wed Jun 23 08:42:41 2010 +0200
1.3 @@ -8,15 +8,7 @@
1.4 imports Main Quotient_Syntax
1.5 begin
1.6
1.7 -fun
1.8 - list_rel
1.9 -where
1.10 - "list_rel R [] [] = True"
1.11 -| "list_rel R (x#xs) [] = False"
1.12 -| "list_rel R [] (x#xs) = False"
1.13 -| "list_rel R (x#xs) (y#ys) = (R x y \<and> list_rel R xs ys)"
1.14 -
1.15 -declare [[map list = (map, list_rel)]]
1.16 +declare [[map list = (map, list_all2)]]
1.17
1.18 lemma split_list_all:
1.19 shows "(\<forall>x. P x) \<longleftrightarrow> P [] \<and> (\<forall>x xs. P (x#xs))"
1.20 @@ -33,52 +25,47 @@
1.21 apply(simp_all)
1.22 done
1.23
1.24 +lemma list_all2_reflp:
1.25 + shows "equivp R \<Longrightarrow> list_all2 R xs xs"
1.26 + by (induct xs, simp_all add: equivp_reflp)
1.27
1.28 -lemma list_rel_reflp:
1.29 - shows "equivp R \<Longrightarrow> list_rel R xs xs"
1.30 - apply(induct xs)
1.31 - apply(simp_all add: equivp_reflp)
1.32 - done
1.33 -
1.34 -lemma list_rel_symp:
1.35 +lemma list_all2_symp:
1.36 assumes a: "equivp R"
1.37 - shows "list_rel R xs ys \<Longrightarrow> list_rel R ys xs"
1.38 - apply(induct xs ys rule: list_induct2')
1.39 + and b: "list_all2 R xs ys"
1.40 + shows "list_all2 R ys xs"
1.41 + using list_all2_lengthD[OF b] b
1.42 + apply(induct xs ys rule: list_induct2)
1.43 apply(simp_all)
1.44 apply(rule equivp_symp[OF a])
1.45 apply(simp)
1.46 done
1.47
1.48 -lemma list_rel_transp:
1.49 +thm list_induct3
1.50 +
1.51 +lemma list_all2_transp:
1.52 assumes a: "equivp R"
1.53 - shows "list_rel R xs1 xs2 \<Longrightarrow> list_rel R xs2 xs3 \<Longrightarrow> list_rel R xs1 xs3"
1.54 - using a
1.55 - apply(induct R xs1 xs2 arbitrary: xs3 rule: list_rel.induct)
1.56 - apply(simp)
1.57 - apply(simp)
1.58 - apply(simp)
1.59 - apply(case_tac xs3)
1.60 - apply(clarify)
1.61 - apply(simp (no_asm_use))
1.62 - apply(clarify)
1.63 - apply(simp (no_asm_use))
1.64 - apply(auto intro: equivp_transp)
1.65 + and b: "list_all2 R xs1 xs2"
1.66 + and c: "list_all2 R xs2 xs3"
1.67 + shows "list_all2 R xs1 xs3"
1.68 + using list_all2_lengthD[OF b] list_all2_lengthD[OF c] b c
1.69 + apply(induct rule: list_induct3)
1.70 + apply(simp_all)
1.71 + apply(auto intro: equivp_transp[OF a])
1.72 done
1.73
1.74 lemma list_equivp[quot_equiv]:
1.75 assumes a: "equivp R"
1.76 - shows "equivp (list_rel R)"
1.77 - apply(rule equivpI)
1.78 + shows "equivp (list_all2 R)"
1.79 + apply (intro equivpI)
1.80 unfolding reflp_def symp_def transp_def
1.81 - apply(subst split_list_all)
1.82 - apply(simp add: equivp_reflp[OF a] list_rel_reflp[OF a])
1.83 - apply(blast intro: list_rel_symp[OF a])
1.84 - apply(blast intro: list_rel_transp[OF a])
1.85 + apply(simp add: list_all2_reflp[OF a])
1.86 + apply(blast intro: list_all2_symp[OF a])
1.87 + apply(blast intro: list_all2_transp[OF a])
1.88 done
1.89
1.90 -lemma list_rel_rel:
1.91 +lemma list_all2_rel:
1.92 assumes q: "Quotient R Abs Rep"
1.93 - shows "list_rel R r s = (list_rel R r r \<and> list_rel R s s \<and> (map Abs r = map Abs s))"
1.94 + shows "list_all2 R r s = (list_all2 R r r \<and> list_all2 R s s \<and> (map Abs r = map Abs s))"
1.95 apply(induct r s rule: list_induct2')
1.96 apply(simp_all)
1.97 using Quotient_rel[OF q]
1.98 @@ -87,21 +74,16 @@
1.99
1.100 lemma list_quotient[quot_thm]:
1.101 assumes q: "Quotient R Abs Rep"
1.102 - shows "Quotient (list_rel R) (map Abs) (map Rep)"
1.103 + shows "Quotient (list_all2 R) (map Abs) (map Rep)"
1.104 unfolding Quotient_def
1.105 apply(subst split_list_all)
1.106 apply(simp add: Quotient_abs_rep[OF q] abs_o_rep[OF q] map_id)
1.107 - apply(rule conjI)
1.108 - apply(rule allI)
1.109 + apply(intro conjI allI)
1.110 apply(induct_tac a)
1.111 - apply(simp)
1.112 - apply(simp)
1.113 - apply(simp add: Quotient_rep_reflp[OF q])
1.114 - apply(rule allI)+
1.115 - apply(rule list_rel_rel[OF q])
1.116 + apply(simp_all add: Quotient_rep_reflp[OF q])
1.117 + apply(rule list_all2_rel[OF q])
1.118 done
1.119
1.120 -
1.121 lemma cons_prs_aux:
1.122 assumes q: "Quotient R Abs Rep"
1.123 shows "(map Abs) ((Rep h) # (map Rep t)) = h # t"
1.124 @@ -115,7 +97,7 @@
1.125
1.126 lemma cons_rsp[quot_respect]:
1.127 assumes q: "Quotient R Abs Rep"
1.128 - shows "(R ===> list_rel R ===> list_rel R) (op #) (op #)"
1.129 + shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)"
1.130 by (auto)
1.131
1.132 lemma nil_prs[quot_preserve]:
1.133 @@ -125,7 +107,7 @@
1.134
1.135 lemma nil_rsp[quot_respect]:
1.136 assumes q: "Quotient R Abs Rep"
1.137 - shows "list_rel R [] []"
1.138 + shows "list_all2 R [] []"
1.139 by simp
1.140
1.141 lemma map_prs_aux:
1.142 @@ -146,8 +128,8 @@
1.143 lemma map_rsp[quot_respect]:
1.144 assumes q1: "Quotient R1 Abs1 Rep1"
1.145 and q2: "Quotient R2 Abs2 Rep2"
1.146 - shows "((R1 ===> R2) ===> (list_rel R1) ===> list_rel R2) map map"
1.147 - and "((R1 ===> op =) ===> (list_rel R1) ===> op =) map map"
1.148 + shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map"
1.149 + and "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map"
1.150 apply simp_all
1.151 apply(rule_tac [!] allI)+
1.152 apply(rule_tac [!] impI)
1.153 @@ -183,53 +165,45 @@
1.154 by (simp only: expand_fun_eq fun_map_def foldl_prs_aux[OF a b])
1.155 (simp)
1.156
1.157 -lemma list_rel_empty:
1.158 - shows "list_rel R [] b \<Longrightarrow> length b = 0"
1.159 +lemma list_all2_empty:
1.160 + shows "list_all2 R [] b \<Longrightarrow> length b = 0"
1.161 by (induct b) (simp_all)
1.162
1.163 -lemma list_rel_len:
1.164 - shows "list_rel R a b \<Longrightarrow> length a = length b"
1.165 - apply (induct a arbitrary: b)
1.166 - apply (simp add: list_rel_empty)
1.167 - apply (case_tac b)
1.168 - apply simp_all
1.169 - done
1.170 -
1.171 (* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)
1.172 lemma foldl_rsp[quot_respect]:
1.173 assumes q1: "Quotient R1 Abs1 Rep1"
1.174 and q2: "Quotient R2 Abs2 Rep2"
1.175 - shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_rel R2 ===> R1) foldl foldl"
1.176 + shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 R2 ===> R1) foldl foldl"
1.177 apply(auto)
1.178 - apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")
1.179 + apply (subgoal_tac "R1 xa ya \<longrightarrow> list_all2 R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")
1.180 apply simp
1.181 apply (rule_tac x="xa" in spec)
1.182 apply (rule_tac x="ya" in spec)
1.183 apply (rule_tac xs="xb" and ys="yb" in list_induct2)
1.184 - apply (rule list_rel_len)
1.185 + apply (rule list_all2_lengthD)
1.186 apply (simp_all)
1.187 done
1.188
1.189 lemma foldr_rsp[quot_respect]:
1.190 assumes q1: "Quotient R1 Abs1 Rep1"
1.191 and q2: "Quotient R2 Abs2 Rep2"
1.192 - shows "((R1 ===> R2 ===> R2) ===> list_rel R1 ===> R2 ===> R2) foldr foldr"
1.193 + shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr"
1.194 apply auto
1.195 - apply(subgoal_tac "R2 xb yb \<longrightarrow> list_rel R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")
1.196 + apply(subgoal_tac "R2 xb yb \<longrightarrow> list_all2 R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")
1.197 apply simp
1.198 apply (rule_tac xs="xa" and ys="ya" in list_induct2)
1.199 - apply (rule list_rel_len)
1.200 + apply (rule list_all2_lengthD)
1.201 apply (simp_all)
1.202 done
1.203
1.204 -lemma list_rel_rsp:
1.205 +lemma list_all2_rsp:
1.206 assumes r: "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"
1.207 - and l1: "list_rel R x y"
1.208 - and l2: "list_rel R a b"
1.209 - shows "list_rel S x a = list_rel T y b"
1.210 + and l1: "list_all2 R x y"
1.211 + and l2: "list_all2 R a b"
1.212 + shows "list_all2 S x a = list_all2 T y b"
1.213 proof -
1.214 - have a: "length y = length x" by (rule list_rel_len[OF l1, symmetric])
1.215 - have c: "length a = length b" by (rule list_rel_len[OF l2])
1.216 + have a: "length y = length x" by (rule list_all2_lengthD[OF l1, symmetric])
1.217 + have c: "length a = length b" by (rule list_all2_lengthD[OF l2])
1.218 show ?thesis proof (cases "length x = length a")
1.219 case True
1.220 have b: "length x = length a" by fact
1.221 @@ -243,20 +217,20 @@
1.222 next
1.223 case False
1.224 have d: "length x \<noteq> length a" by fact
1.225 - then have e: "\<not>list_rel S x a" using list_rel_len by auto
1.226 + then have e: "\<not>list_all2 S x a" using list_all2_lengthD by auto
1.227 have "length y \<noteq> length b" using d a c by simp
1.228 - then have "\<not>list_rel T y b" using list_rel_len by auto
1.229 + then have "\<not>list_all2 T y b" using list_all2_lengthD by auto
1.230 then show ?thesis using e by simp
1.231 qed
1.232 qed
1.233
1.234 lemma[quot_respect]:
1.235 - "((R ===> R ===> op =) ===> list_rel R ===> list_rel R ===> op =) list_rel list_rel"
1.236 - by (simp add: list_rel_rsp)
1.237 + "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
1.238 + by (simp add: list_all2_rsp)
1.239
1.240 lemma[quot_preserve]:
1.241 assumes a: "Quotient R abs1 rep1"
1.242 - shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_rel = list_rel"
1.243 + shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
1.244 apply (simp add: expand_fun_eq)
1.245 apply clarify
1.246 apply (induct_tac xa xb rule: list_induct2')
1.247 @@ -265,29 +239,29 @@
1.248
1.249 lemma[quot_preserve]:
1.250 assumes a: "Quotient R abs1 rep1"
1.251 - shows "(list_rel ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
1.252 + shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
1.253 by (induct l m rule: list_induct2') (simp_all add: Quotient_rel_rep[OF a])
1.254
1.255 -lemma list_rel_eq[id_simps]:
1.256 - shows "(list_rel (op =)) = (op =)"
1.257 +lemma list_all2_eq[id_simps]:
1.258 + shows "(list_all2 (op =)) = (op =)"
1.259 unfolding expand_fun_eq
1.260 apply(rule allI)+
1.261 apply(induct_tac x xa rule: list_induct2')
1.262 apply(simp_all)
1.263 done
1.264
1.265 -lemma list_rel_find_element:
1.266 +lemma list_all2_find_element:
1.267 assumes a: "x \<in> set a"
1.268 - and b: "list_rel R a b"
1.269 + and b: "list_all2 R a b"
1.270 shows "\<exists>y. (y \<in> set b \<and> R x y)"
1.271 proof -
1.272 - have "length a = length b" using b by (rule list_rel_len)
1.273 + have "length a = length b" using b by (rule list_all2_lengthD)
1.274 then show ?thesis using a b by (induct a b rule: list_induct2) auto
1.275 qed
1.276
1.277 -lemma list_rel_refl:
1.278 +lemma list_all2_refl:
1.279 assumes a: "\<And>x y. R x y = (R x = R y)"
1.280 - shows "list_rel R x x"
1.281 + shows "list_all2 R x x"
1.282 by (induct x) (auto simp add: a)
1.283
1.284 end