author  lcp 
Fri, 23 Dec 1994 16:28:26 +0100  
changeset 826  190974c664a3 
parent 791  354a56e923ff 
child 862  ce99db6728ba 
permissions  rwrr 
735  1 
(* Title: ZF/Perm.ML 
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ID: $Id$ 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1991 University of Cambridge 

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735  6 
The theory underlying permutation groups 
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 Composition of relations, the identity relation 
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 Injections, surjections, bijections 

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 Lemmas for the SchroederBernstein Theorem 

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*) 

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open Perm; 

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(** Surjective function space **) 

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goalw Perm.thy [surj_def] "!!f A B. f: surj(A,B) ==> f: A>B"; 

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by (etac CollectD1 1); 

760  18 
qed "surj_is_fun"; 
0  19 

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goalw Perm.thy [surj_def] "!!f A B. f : Pi(A,B) ==> f: surj(A,range(f))"; 

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by (fast_tac (ZF_cs addIs [apply_equality] 

22 
addEs [range_of_fun,domain_type]) 1); 

760  23 
qed "fun_is_surj"; 
0  24 

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goalw Perm.thy [surj_def] "!!f A B. f: surj(A,B) ==> range(f)=B"; 

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by (best_tac (ZF_cs addIs [equalityI,apply_Pair] addEs [range_type]) 1); 

760  27 
qed "surj_range"; 
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(** A function with a right inverse is a surjection **) 
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val prems = goalw Perm.thy [surj_def] 
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"[ f: A>B; !!y. y:B ==> d(y): A; !!y. y:B ==> f`d(y) = y \ 
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\ ] ==> f: surj(A,B)"; 
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by (fast_tac (ZF_cs addIs prems) 1); 
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qed "f_imp_surjective"; 
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val prems = goal Perm.thy 
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"[ !!x. x:A ==> c(x): B; \ 
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\ !!y. y:B ==> d(y): A; \ 
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\ !!y. y:B ==> c(d(y)) = y \ 
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\ ] ==> (lam x:A.c(x)) : surj(A,B)"; 
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by (res_inst_tac [("d", "d")] f_imp_surjective 1); 
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by (ALLGOALS (asm_simp_tac (ZF_ss addsimps ([lam_type]@prems)) )); 
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qed "lam_surjective"; 
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735  46 
(*Cantor's theorem revisited*) 
47 
goalw Perm.thy [surj_def] "f ~: surj(A,Pow(A))"; 

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by (safe_tac ZF_cs); 

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by (cut_facts_tac [cantor] 1); 

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by (fast_tac subset_cs 1); 

760  51 
qed "cantor_surj"; 
735  52 

0  53 

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(** Injective function space **) 

55 

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goalw Perm.thy [inj_def] "!!f A B. f: inj(A,B) ==> f: A>B"; 

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by (etac CollectD1 1); 

760  58 
qed "inj_is_fun"; 
0  59 

60 
goalw Perm.thy [inj_def] 

61 
"!!f A B. [ <a,b>:f; <c,b>:f; f: inj(A,B) ] ==> a=c"; 

62 
by (REPEAT (eresolve_tac [asm_rl, Pair_mem_PiE, CollectE] 1)); 

63 
by (fast_tac ZF_cs 1); 

760  64 
qed "inj_equality"; 
0  65 

826  66 
goalw thy [inj_def] "!!A B f. [ f:inj(A,B); a:A; b:A; f`a=f`b ] ==> a=b"; 
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by (fast_tac ZF_cs 1); 

68 
val inj_apply_equality = result(); 

69 

484  70 
(** A function with a left inverse is an injection **) 
71 

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val prems = goal Perm.thy 

73 
"[ f: A>B; !!x. x:A ==> d(f`x)=x ] ==> f: inj(A,B)"; 

74 
by (asm_simp_tac (ZF_ss addsimps ([inj_def] @ prems)) 1); 

75 
by (safe_tac ZF_cs); 

76 
by (eresolve_tac [subst_context RS box_equals] 1); 

77 
by (REPEAT (ares_tac prems 1)); 

760  78 
qed "f_imp_injective"; 
484  79 

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val prems = goal Perm.thy 

81 
"[ !!x. x:A ==> c(x): B; \ 

82 
\ !!x. x:A ==> d(c(x)) = x \ 

83 
\ ] ==> (lam x:A.c(x)) : inj(A,B)"; 

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by (res_inst_tac [("d", "d")] f_imp_injective 1); 

85 
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps ([lam_type]@prems)) )); 

760  86 
qed "lam_injective"; 
484  87 

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(** Bijections **) 

0  89 

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goalw Perm.thy [bij_def] "!!f A B. f: bij(A,B) ==> f: inj(A,B)"; 

91 
by (etac IntD1 1); 

760  92 
qed "bij_is_inj"; 
0  93 

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goalw Perm.thy [bij_def] "!!f A B. f: bij(A,B) ==> f: surj(A,B)"; 

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by (etac IntD2 1); 

760  96 
qed "bij_is_surj"; 
0  97 

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(* f: bij(A,B) ==> f: A>B *) 

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bind_thm ("bij_is_fun", (bij_is_inj RS inj_is_fun)); 
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val prems = goalw Perm.thy [bij_def] 
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"[ !!x. x:A ==> c(x): B; \ 
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\ !!y. y:B ==> d(y): A; \ 
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\ !!x. x:A ==> d(c(x)) = x; \ 
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\ !!y. y:B ==> c(d(y)) = y \ 
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\ ] ==> (lam x:A.c(x)) : bij(A,B)"; 
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by (REPEAT (ares_tac (prems @ [IntI, lam_injective, lam_surjective]) 1)); 
760  108 
qed "lam_bijective"; 
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(** Identity function **) 
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val [prem] = goalw Perm.thy [id_def] "a:A ==> <a,a> : id(A)"; 

114 
by (rtac (prem RS lamI) 1); 

760  115 
qed "idI"; 
0  116 

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val major::prems = goalw Perm.thy [id_def] 

118 
"[ p: id(A); !!x.[ x:A; p=<x,x> ] ==> P \ 

119 
\ ] ==> P"; 

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by (rtac (major RS lamE) 1); 

121 
by (REPEAT (ares_tac prems 1)); 

760  122 
qed "idE"; 
0  123 

124 
goalw Perm.thy [id_def] "id(A) : A>A"; 

125 
by (rtac lam_type 1); 

126 
by (assume_tac 1); 

760  127 
qed "id_type"; 
0  128 

826  129 
goalw Perm.thy [id_def] "!!A x. x:A ==> id(A)`x = x"; 
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by (asm_simp_tac ZF_ss 1); 

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val id_conv = result(); 

132 

0  133 
val [prem] = goalw Perm.thy [id_def] "A<=B ==> id(A) <= id(B)"; 
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by (rtac (prem RS lam_mono) 1); 

760  135 
qed "id_mono"; 
0  136 

435  137 
goalw Perm.thy [inj_def,id_def] "!!A B. A<=B ==> id(A): inj(A,B)"; 
0  138 
by (REPEAT (ares_tac [CollectI,lam_type] 1)); 
435  139 
by (etac subsetD 1 THEN assume_tac 1); 
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by (simp_tac ZF_ss 1); 
760  141 
qed "id_subset_inj"; 
435  142 

143 
val id_inj = subset_refl RS id_subset_inj; 

0  144 

145 
goalw Perm.thy [id_def,surj_def] "id(A): surj(A,A)"; 

146 
by (fast_tac (ZF_cs addIs [lam_type,beta]) 1); 

760  147 
qed "id_surj"; 
0  148 

149 
goalw Perm.thy [bij_def] "id(A): bij(A,A)"; 

150 
by (fast_tac (ZF_cs addIs [id_inj,id_surj]) 1); 

760  151 
qed "id_bij"; 
0  152 

517  153 
goalw Perm.thy [id_def] "A <= B <> id(A) : A>B"; 
154 
by (safe_tac ZF_cs); 

155 
by (fast_tac (ZF_cs addSIs [lam_type]) 1); 

156 
by (dtac apply_type 1); 

157 
by (assume_tac 1); 

158 
by (asm_full_simp_tac ZF_ss 1); 

760  159 
qed "subset_iff_id"; 
517  160 

0  161 

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(*** Converse of a function ***) 
0  163 

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val [prem] = goal Perm.thy "f: inj(A,B) ==> converse(f) : range(f)>A"; 

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by (cut_facts_tac [prem] 1); 
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by (asm_full_simp_tac (ZF_ss addsimps [inj_def, Pi_iff, domain_converse]) 1); 
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by (resolve_tac [conjI] 1); 
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by (deepen_tac ZF_cs 0 2); 
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by (simp_tac (ZF_ss addsimps [function_def, converse_iff]) 1); 
0  170 
by (fast_tac (ZF_cs addEs [prem RSN (3,inj_equality)]) 1); 
760  171 
qed "inj_converse_fun"; 
0  172 

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(** Equations for converse(f) **) 
0  174 

175 
(*The premises are equivalent to saying that f is injective...*) 

176 
val prems = goal Perm.thy 

177 
"[ f: A>B; converse(f): C>A; a: A ] ==> converse(f)`(f`a) = a"; 

178 
by (fast_tac (ZF_cs addIs (prems@[apply_Pair,apply_equality,converseI])) 1); 

760  179 
qed "left_inverse_lemma"; 
0  180 

435  181 
goal Perm.thy 
182 
"!!f. [ f: inj(A,B); a: A ] ==> converse(f)`(f`a) = a"; 

183 
by (fast_tac (ZF_cs addIs [left_inverse_lemma,inj_converse_fun,inj_is_fun]) 1); 

760  184 
qed "left_inverse"; 
0  185 

435  186 
val left_inverse_bij = bij_is_inj RS left_inverse; 
187 

0  188 
val prems = goal Perm.thy 
189 
"[ f: A>B; converse(f): C>A; b: C ] ==> f`(converse(f)`b) = b"; 

190 
by (rtac (apply_Pair RS (converseD RS apply_equality)) 1); 

191 
by (REPEAT (resolve_tac prems 1)); 

760  192 
qed "right_inverse_lemma"; 
0  193 

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(*Should the premises be f:surj(A,B), b:B for symmetry with left_inverse? 
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No: they would not imply that converse(f) was a function! *) 
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goal Perm.thy "!!f. [ f: inj(A,B); b: range(f) ] ==> f`(converse(f)`b) = b"; 
0  197 
by (rtac right_inverse_lemma 1); 
435  198 
by (REPEAT (ares_tac [inj_converse_fun,inj_is_fun] 1)); 
760  199 
qed "right_inverse"; 
0  200 

435  201 
goalw Perm.thy [bij_def] 
202 
"!!f. [ f: bij(A,B); b: B ] ==> f`(converse(f)`b) = b"; 

203 
by (EVERY1 [etac IntE, etac right_inverse, 

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etac (surj_range RS ssubst), 

205 
assume_tac]); 

760  206 
qed "right_inverse_bij"; 
435  207 

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(** Converses of injections, surjections, bijections **) 
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goal Perm.thy "!!f A B. f: inj(A,B) ==> converse(f): inj(range(f), A)"; 
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by (resolve_tac [f_imp_injective] 1); 
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by (eresolve_tac [inj_converse_fun] 1); 
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by (resolve_tac [right_inverse] 1); 
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by (REPEAT (assume_tac 1)); 
760  215 
qed "inj_converse_inj"; 
0  216 

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goal Perm.thy "!!f A B. f: inj(A,B) ==> converse(f): surj(range(f), A)"; 
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by (REPEAT (ares_tac [f_imp_surjective, inj_converse_fun] 1)); 
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by (REPEAT (ares_tac [left_inverse] 2)); 
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by (REPEAT (ares_tac [inj_is_fun, range_of_fun RS apply_type] 1)); 
760  221 
qed "inj_converse_surj"; 
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0  223 
goalw Perm.thy [bij_def] "!!f A B. f: bij(A,B) ==> converse(f): bij(B,A)"; 
224 
by (etac IntE 1); 

225 
by (eresolve_tac [(surj_range RS subst)] 1); 

226 
by (rtac IntI 1); 

227 
by (etac inj_converse_inj 1); 

228 
by (etac inj_converse_surj 1); 

760  229 
qed "bij_converse_bij"; 
0  230 

231 

232 
(** Composition of two relations **) 

233 

791  234 
(*The inductive definition package could derive these theorems for (r O s)*) 
0  235 

236 
goalw Perm.thy [comp_def] "!!r s. [ <a,b>:s; <b,c>:r ] ==> <a,c> : r O s"; 

237 
by (fast_tac ZF_cs 1); 

760  238 
qed "compI"; 
0  239 

240 
val prems = goalw Perm.thy [comp_def] 

241 
"[ xz : r O s; \ 

242 
\ !!x y z. [ xz=<x,z>; <x,y>:s; <y,z>:r ] ==> P \ 

243 
\ ] ==> P"; 

244 
by (cut_facts_tac prems 1); 

245 
by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1 ORELSE ares_tac prems 1)); 

760  246 
qed "compE"; 
0  247 

248 
val compEpair = 

249 
rule_by_tactic (REPEAT_FIRST (etac Pair_inject ORELSE' bound_hyp_subst_tac) 

250 
THEN prune_params_tac) 

251 
(read_instantiate [("xz","<a,c>")] compE); 

252 

735  253 
val comp_cs = ZF_cs addSIs [idI] addIs [compI] addSEs [compE,idE]; 
0  254 

255 
(** Domain and Range  see Suppes, section 3.1 **) 

256 

257 
(*Boyer et al., Set Theory in FirstOrder Logic, JAR 2 (1986), 287327*) 

258 
goal Perm.thy "range(r O s) <= range(r)"; 

259 
by (fast_tac comp_cs 1); 

760  260 
qed "range_comp"; 
0  261 

262 
goal Perm.thy "!!r s. domain(r) <= range(s) ==> range(r O s) = range(r)"; 

263 
by (rtac (range_comp RS equalityI) 1); 

264 
by (fast_tac comp_cs 1); 

760  265 
qed "range_comp_eq"; 
0  266 

267 
goal Perm.thy "domain(r O s) <= domain(s)"; 

268 
by (fast_tac comp_cs 1); 

760  269 
qed "domain_comp"; 
0  270 

271 
goal Perm.thy "!!r s. range(s) <= domain(r) ==> domain(r O s) = domain(s)"; 

272 
by (rtac (domain_comp RS equalityI) 1); 

273 
by (fast_tac comp_cs 1); 

760  274 
qed "domain_comp_eq"; 
0  275 

218  276 
goal Perm.thy "(r O s)``A = r``(s``A)"; 
277 
by (fast_tac (comp_cs addIs [equalityI]) 1); 

760  278 
qed "image_comp"; 
218  279 

280 

0  281 
(** Other results **) 
282 

283 
goal Perm.thy "!!r s. [ r'<=r; s'<=s ] ==> (r' O s') <= (r O s)"; 

284 
by (fast_tac comp_cs 1); 

760  285 
qed "comp_mono"; 
0  286 

287 
(*composition preserves relations*) 

288 
goal Perm.thy "!!r s. [ s<=A*B; r<=B*C ] ==> (r O s) <= A*C"; 

289 
by (fast_tac comp_cs 1); 

760  290 
qed "comp_rel"; 
0  291 

292 
(*associative law for composition*) 

293 
goal Perm.thy "(r O s) O t = r O (s O t)"; 

294 
by (fast_tac (comp_cs addIs [equalityI]) 1); 

760  295 
qed "comp_assoc"; 
0  296 

297 
(*left identity of composition; provable inclusions are 

298 
id(A) O r <= r 

299 
and [ r<=A*B; B<=C ] ==> r <= id(C) O r *) 

300 
goal Perm.thy "!!r A B. r<=A*B ==> id(B) O r = r"; 

301 
by (fast_tac (comp_cs addIs [equalityI]) 1); 

760  302 
qed "left_comp_id"; 
0  303 

304 
(*right identity of composition; provable inclusions are 

305 
r O id(A) <= r 

306 
and [ r<=A*B; A<=C ] ==> r <= r O id(C) *) 

307 
goal Perm.thy "!!r A B. r<=A*B ==> r O id(A) = r"; 

308 
by (fast_tac (comp_cs addIs [equalityI]) 1); 

760  309 
qed "right_comp_id"; 
0  310 

311 

312 
(** Composition preserves functions, injections, and surjections **) 

313 

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goalw Perm.thy [function_def] 
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315 
"!!f g. [ function(g); function(f) ] ==> function(f O g)"; 
735  316 
by (fast_tac (ZF_cs addIs [compI] addSEs [compE, Pair_inject]) 1); 
760  317 
qed "comp_function"; 
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318 

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319 
goalw Perm.thy [Pi_def] 
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320 
"!!f g. [ g: A>B; f: B>C ] ==> (f O g) : A>C"; 
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321 
by (safe_tac subset_cs); 
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322 
by (asm_simp_tac (ZF_ss addsimps [comp_function]) 3); 
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323 
by (rtac (range_rel_subset RS domain_comp_eq RS ssubst) 2 THEN assume_tac 3); 
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324 
by (fast_tac ZF_cs 2); 
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325 
by (asm_simp_tac (ZF_ss addsimps [comp_rel]) 1); 
760  326 
qed "comp_fun"; 
0  327 

328 
goal Perm.thy "!!f g. [ g: A>B; f: B>C; a:A ] ==> (f O g)`a = f`(g`a)"; 

435  329 
by (REPEAT (ares_tac [comp_fun,apply_equality,compI, 
0  330 
apply_Pair,apply_type] 1)); 
760  331 
qed "comp_fun_apply"; 
0  332 

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goal Perm.thy "!!f g. [ g: inj(A,B); f: inj(B,C) ] ==> (f O g) : inj(A,C)"; 
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334 
by (res_inst_tac [("d", "%y. converse(g) ` (converse(f) ` y)")] 
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335 
f_imp_injective 1); 
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336 
by (REPEAT (ares_tac [comp_fun, inj_is_fun] 1)); 
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337 
by (asm_simp_tac (ZF_ss addsimps [comp_fun_apply, left_inverse] 
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338 
setsolver type_auto_tac [inj_is_fun, apply_type]) 1); 
760  339 
qed "comp_inj"; 
0  340 

341 
goalw Perm.thy [surj_def] 

342 
"!!f g. [ g: surj(A,B); f: surj(B,C) ] ==> (f O g) : surj(A,C)"; 

435  343 
by (best_tac (ZF_cs addSIs [comp_fun,comp_fun_apply]) 1); 
760  344 
qed "comp_surj"; 
0  345 

346 
goalw Perm.thy [bij_def] 

347 
"!!f g. [ g: bij(A,B); f: bij(B,C) ] ==> (f O g) : bij(A,C)"; 

348 
by (fast_tac (ZF_cs addIs [comp_inj,comp_surj]) 1); 

760  349 
qed "comp_bij"; 
0  350 

351 

352 
(** Dual properties of inj and surj  useful for proofs from 

353 
D Pastre. Automatic theorem proving in set theory. 

354 
Artificial Intelligence, 10:127, 1978. **) 

355 

356 
goalw Perm.thy [inj_def] 

357 
"!!f g. [ (f O g): inj(A,C); g: A>B; f: B>C ] ==> g: inj(A,B)"; 

358 
by (safe_tac comp_cs); 

359 
by (REPEAT (eresolve_tac [asm_rl, bspec RS bspec RS mp] 1)); 

435  360 
by (asm_simp_tac (FOL_ss addsimps [comp_fun_apply]) 1); 
760  361 
qed "comp_mem_injD1"; 
0  362 

363 
goalw Perm.thy [inj_def,surj_def] 

364 
"!!f g. [ (f O g): inj(A,C); g: surj(A,B); f: B>C ] ==> f: inj(B,C)"; 

365 
by (safe_tac comp_cs); 

366 
by (res_inst_tac [("x1", "x")] (bspec RS bexE) 1); 

367 
by (eres_inst_tac [("x1", "w")] (bspec RS bexE) 3); 

368 
by (REPEAT (assume_tac 1)); 

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369 
by (safe_tac comp_cs); 
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370 
by (res_inst_tac [("t", "op `(g)")] subst_context 1); 
0  371 
by (REPEAT (eresolve_tac [asm_rl, bspec RS bspec RS mp] 1)); 
435  372 
by (asm_simp_tac (FOL_ss addsimps [comp_fun_apply]) 1); 
760  373 
qed "comp_mem_injD2"; 
0  374 

375 
goalw Perm.thy [surj_def] 

376 
"!!f g. [ (f O g): surj(A,C); g: A>B; f: B>C ] ==> f: surj(B,C)"; 

435  377 
by (fast_tac (comp_cs addSIs [comp_fun_apply RS sym, apply_type]) 1); 
760  378 
qed "comp_mem_surjD1"; 
0  379 

380 
goal Perm.thy 

381 
"!!f g. [ (f O g)`a = c; g: A>B; f: B>C; a:A ] ==> f`(g`a) = c"; 

435  382 
by (REPEAT (ares_tac [comp_fun_apply RS sym RS trans] 1)); 
760  383 
qed "comp_fun_applyD"; 
0  384 

385 
goalw Perm.thy [inj_def,surj_def] 

386 
"!!f g. [ (f O g): surj(A,C); g: A>B; f: inj(B,C) ] ==> g: surj(A,B)"; 

387 
by (safe_tac comp_cs); 

388 
by (eres_inst_tac [("x1", "f`y")] (bspec RS bexE) 1); 

435  389 
by (REPEAT (ares_tac [apply_type] 1 ORELSE dtac comp_fun_applyD 1)); 
0  390 
by (best_tac (comp_cs addSIs [apply_type]) 1); 
760  391 
qed "comp_mem_surjD2"; 
0  392 

393 

394 
(** inverses of composition **) 

395 

396 
(*left inverse of composition; one inclusion is 

397 
f: A>B ==> id(A) <= converse(f) O f *) 

398 
val [prem] = goal Perm.thy 

399 
"f: inj(A,B) ==> converse(f) O f = id(A)"; 

400 
val injfD = prem RSN (3,inj_equality); 

401 
by (cut_facts_tac [prem RS inj_is_fun] 1); 

402 
by (fast_tac (comp_cs addIs [equalityI,apply_Pair] 

403 
addEs [domain_type, make_elim injfD]) 1); 

760  404 
qed "left_comp_inverse"; 
0  405 

406 
(*right inverse of composition; one inclusion is 

735  407 
f: A>B ==> f O converse(f) <= id(B) 
408 
*) 

0  409 
val [prem] = goalw Perm.thy [surj_def] 
410 
"f: surj(A,B) ==> f O converse(f) = id(B)"; 

411 
val appfD = (prem RS CollectD1) RSN (3,apply_equality2); 

412 
by (cut_facts_tac [prem] 1); 

413 
by (rtac equalityI 1); 

414 
by (best_tac (comp_cs addEs [domain_type, range_type, make_elim appfD]) 1); 

415 
by (best_tac (comp_cs addIs [apply_Pair]) 1); 

760  416 
qed "right_comp_inverse"; 
0  417 

435  418 
(** Proving that a function is a bijection **) 
419 

420 
goalw Perm.thy [id_def] 

421 
"!!f A B. [ f: A>B; g: B>A ] ==> \ 

422 
\ f O g = id(B) <> (ALL y:B. f`(g`y)=y)"; 

423 
by (safe_tac ZF_cs); 

424 
by (dres_inst_tac [("t", "%h.h`y ")] subst_context 1); 

425 
by (asm_full_simp_tac (ZF_ss addsimps [comp_fun_apply]) 1); 

437  426 
by (rtac fun_extension 1); 
435  427 
by (REPEAT (ares_tac [comp_fun, lam_type] 1)); 
428 
by (asm_simp_tac (ZF_ss addsimps [comp_fun_apply]) 1); 

760  429 
qed "comp_eq_id_iff"; 
435  430 

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431 
goalw Perm.thy [bij_def] 
435  432 
"!!f A B. [ f: A>B; g: B>A; f O g = id(B); g O f = id(A) \ 
433 
\ ] ==> f : bij(A,B)"; 

434 
by (asm_full_simp_tac (ZF_ss addsimps [comp_eq_id_iff]) 1); 

502
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435 
by (REPEAT (ares_tac [conjI, f_imp_injective, f_imp_surjective] 1 
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436 
ORELSE eresolve_tac [bspec, apply_type] 1)); 
760  437 
qed "fg_imp_bijective"; 
435  438 

439 
goal Perm.thy "!!f A. [ f: A>A; f O f = id(A) ] ==> f : bij(A,A)"; 

440 
by (REPEAT (ares_tac [fg_imp_bijective] 1)); 

760  441 
qed "nilpotent_imp_bijective"; 
435  442 

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443 
goal Perm.thy "!!f A B. [ converse(f): B>A; f: A>B ] ==> f : bij(A,B)"; 
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444 
by (asm_simp_tac (ZF_ss addsimps [fg_imp_bijective, comp_eq_id_iff, 
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445 
left_inverse_lemma, right_inverse_lemma]) 1); 
760  446 
qed "invertible_imp_bijective"; 
0  447 

448 
(** Unions of functions  cf similar theorems on func.ML **) 

449 

450 
goal Perm.thy "converse(r Un s) = converse(r) Un converse(s)"; 

451 
by (rtac equalityI 1); 

452 
by (DEPTH_SOLVE_1 

453 
(resolve_tac [Un_least,converse_mono, Un_upper1,Un_upper2] 2)); 

454 
by (fast_tac ZF_cs 1); 

791  455 
qed "converse_Un"; 
0  456 

457 
goalw Perm.thy [surj_def] 

458 
"!!f g. [ f: surj(A,B); g: surj(C,D); A Int C = 0 ] ==> \ 

459 
\ (f Un g) : surj(A Un C, B Un D)"; 

460 
by (DEPTH_SOLVE_1 (eresolve_tac [fun_disjoint_apply1, fun_disjoint_apply2] 1 

461 
ORELSE ball_tac 1 

462 
ORELSE (rtac trans 1 THEN atac 2) 

463 
ORELSE step_tac (ZF_cs addIs [fun_disjoint_Un]) 1)); 

760  464 
qed "surj_disjoint_Un"; 
0  465 

466 
(*A simple, highlevel proof; the version for injections follows from it, 

502
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467 
using f:inj(A,B) <> f:bij(A,range(f)) *) 
0  468 
goal Perm.thy 
469 
"!!f g. [ f: bij(A,B); g: bij(C,D); A Int C = 0; B Int D = 0 ] ==> \ 

470 
\ (f Un g) : bij(A Un C, B Un D)"; 

471 
by (rtac invertible_imp_bijective 1); 

791  472 
by (rtac (converse_Un RS ssubst) 1); 
0  473 
by (REPEAT (ares_tac [fun_disjoint_Un, bij_is_fun, bij_converse_bij] 1)); 
760  474 
qed "bij_disjoint_Un"; 
0  475 

476 

477 
(** Restrictions as surjections and bijections *) 

478 

479 
val prems = goalw Perm.thy [surj_def] 

480 
"f: Pi(A,B) ==> f: surj(A, f``A)"; 

481 
val rls = apply_equality :: (prems RL [apply_Pair,Pi_type]); 

482 
by (fast_tac (ZF_cs addIs rls) 1); 

760  483 
qed "surj_image"; 
0  484 

735  485 
goal Perm.thy "!!f. [ f: Pi(C,B); A<=C ] ==> restrict(f,A)``A = f``A"; 
0  486 
by (rtac equalityI 1); 
487 
by (SELECT_GOAL (rewtac restrict_def) 2); 

488 
by (REPEAT (eresolve_tac [imageE, apply_equality RS subst] 2 

489 
ORELSE ares_tac [subsetI,lamI,imageI] 2)); 

490 
by (REPEAT (ares_tac [image_mono,restrict_subset,subset_refl] 1)); 

760  491 
qed "restrict_image"; 
0  492 

493 
goalw Perm.thy [inj_def] 

494 
"!!f. [ f: inj(A,B); C<=A ] ==> restrict(f,C): inj(C,B)"; 

495 
by (safe_tac (ZF_cs addSEs [restrict_type2])); 

496 
by (REPEAT (eresolve_tac [asm_rl, bspec RS bspec RS mp, subsetD, 

497 
box_equals, restrict] 1)); 

760  498 
qed "restrict_inj"; 
0  499 

500 
val prems = goal Perm.thy 

501 
"[ f: Pi(A,B); C<=A ] ==> restrict(f,C): surj(C, f``C)"; 

502 
by (rtac (restrict_image RS subst) 1); 

503 
by (rtac (restrict_type2 RS surj_image) 3); 

504 
by (REPEAT (resolve_tac prems 1)); 

760  505 
qed "restrict_surj"; 
0  506 

507 
goalw Perm.thy [inj_def,bij_def] 

508 
"!!f. [ f: inj(A,B); C<=A ] ==> restrict(f,C): bij(C, f``C)"; 

509 
by (safe_tac ZF_cs); 

510 
by (REPEAT (eresolve_tac [bspec RS bspec RS mp, subsetD, 

511 
box_equals, restrict] 1 

512 
ORELSE ares_tac [surj_is_fun,restrict_surj] 1)); 

760  513 
qed "restrict_bij"; 
0  514 

515 

516 
(*** Lemmas for Ramsey's Theorem ***) 

517 

518 
goalw Perm.thy [inj_def] "!!f. [ f: inj(A,B); B<=D ] ==> f: inj(A,D)"; 

519 
by (fast_tac (ZF_cs addSEs [fun_weaken_type]) 1); 

760  520 
qed "inj_weaken_type"; 
0  521 

522 
val [major] = goal Perm.thy 

523 
"[ f: inj(succ(m), A) ] ==> restrict(f,m) : inj(m, A{f`m})"; 

524 
by (rtac (major RS restrict_bij RS bij_is_inj RS inj_weaken_type) 1); 

525 
by (fast_tac ZF_cs 1); 

526 
by (cut_facts_tac [major] 1); 

527 
by (rewtac inj_def); 

528 
by (safe_tac ZF_cs); 

529 
by (etac range_type 1); 

530 
by (assume_tac 1); 

531 
by (dtac apply_equality 1); 

532 
by (assume_tac 1); 

437  533 
by (res_inst_tac [("a","m")] mem_irrefl 1); 
0  534 
by (fast_tac ZF_cs 1); 
760  535 
qed "inj_succ_restrict"; 
0  536 

537 
goalw Perm.thy [inj_def] 

37  538 
"!!f. [ f: inj(A,B); a~:A; b~:B ] ==> \ 
0  539 
\ cons(<a,b>,f) : inj(cons(a,A), cons(b,B))"; 
540 
(*cannot use safe_tac: must preserve the implication*) 

541 
by (etac CollectE 1); 

542 
by (rtac CollectI 1); 

543 
by (etac fun_extend 1); 

544 
by (REPEAT (ares_tac [ballI] 1)); 

545 
by (REPEAT_FIRST (eresolve_tac [consE,ssubst])); 

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546 
(*Assumption ALL w:A. ALL x:A. f`w = f`x > w=x would make asm_simp_tac loop 
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547 
using ZF_ss! But FOL_ss ignores the assumption...*) 
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548 
by (ALLGOALS (asm_simp_tac 
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549 
(FOL_ss addsimps [fun_extend_apply2,fun_extend_apply1]))); 
0  550 
by (ALLGOALS (fast_tac (ZF_cs addIs [apply_type]))); 
760  551 
qed "inj_extend"; 