author | lcp |
Tue, 16 Aug 1994 19:09:43 +0200 | |
changeset 536 | 5fbfa997f1b0 |
parent 14 | 1c0926788772 |
child 674 | 41b59e4c78c4 |
permissions | -rw-r--r-- |
0 | 1 |
(* Title: ZF/domrange |
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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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Converse, domain, range of a relation or function |
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*) |
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(*** converse ***) |
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val converseI = prove_goalw ZF.thy [converse_def] |
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"!!a b r. <a,b>:r ==> <b,a>:converse(r)" |
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(fn _ => [ (fast_tac pair_cs 1) ]); |
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val converseD = prove_goalw ZF.thy [converse_def] |
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"!!a b r. <a,b> : converse(r) ==> <b,a> : r" |
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(fn _ => [ (fast_tac pair_cs 1) ]); |
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val converseE = prove_goalw ZF.thy [converse_def] |
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"[| yx : converse(r); \ |
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\ !!x y. [| yx=<y,x>; <x,y>:r |] ==> P \ |
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\ |] ==> P" |
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(fn [major,minor]=> |
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[ (rtac (major RS ReplaceE) 1), |
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(REPEAT (eresolve_tac [exE, conjE, minor] 1)), |
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(hyp_subst_tac 1), |
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(assume_tac 1) ]); |
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val converse_cs = pair_cs addSIs [converseI] |
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addSEs [converseD,converseE]; |
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val converse_of_converse = prove_goal ZF.thy |
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"!!A B r. r<=Sigma(A,B) ==> converse(converse(r)) = r" |
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(fn _ => [ (fast_tac (converse_cs addSIs [equalityI]) 1) ]); |
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val converse_type = prove_goal ZF.thy "!!A B r. r<=A*B ==> converse(r)<=B*A" |
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(fn _ => [ (fast_tac converse_cs 1) ]); |
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val converse_of_prod = prove_goal ZF.thy "converse(A*B) = B*A" |
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(fn _ => [ (fast_tac (converse_cs addSIs [equalityI]) 1) ]); |
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val converse_empty = prove_goal ZF.thy "converse(0) = 0" |
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(fn _ => [ (fast_tac (converse_cs addSIs [equalityI]) 1) ]); |
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(*** domain ***) |
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val domain_iff = prove_goalw ZF.thy [domain_def] |
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"a: domain(r) <-> (EX y. <a,y>: r)" |
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(fn _=> [ (fast_tac pair_cs 1) ]); |
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val domainI = prove_goal ZF.thy "!!a b r. <a,b>: r ==> a: domain(r)" |
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(fn _ => [ (etac (exI RS (domain_iff RS iffD2)) 1) ]); |
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val domainE = prove_goal ZF.thy |
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"[| a : domain(r); !!y. <a,y>: r ==> P |] ==> P" |
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(fn prems=> |
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[ (rtac (domain_iff RS iffD1 RS exE) 1), |
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(REPEAT (ares_tac prems 1)) ]); |
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val domain_subset = prove_goal ZF.thy "domain(Sigma(A,B)) <= A" |
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536
5fbfa997f1b0
ZF/domrange/domain_of_prod, domain_empty, etc: moved to equalities.ML where
lcp
parents:
14
diff
changeset
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(fn _ => [ (rtac subsetI 1), (etac domainE 1), (etac SigmaD1 1) ]); |
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(*** range ***) |
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val rangeI = prove_goalw ZF.thy [range_def] "!!a b r.<a,b>: r ==> b : range(r)" |
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(fn _ => [ (etac (converseI RS domainI) 1) ]); |
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val rangeE = prove_goalw ZF.thy [range_def] |
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"[| b : range(r); !!x. <x,b>: r ==> P |] ==> P" |
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(fn major::prems=> |
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[ (rtac (major RS domainE) 1), |
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(resolve_tac prems 1), |
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(etac converseD 1) ]); |
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val range_subset = prove_goalw ZF.thy [range_def] "range(A*B) <= B" |
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(fn _ => |
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[ (rtac (converse_of_prod RS ssubst) 1), |
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(rtac domain_subset 1) ]); |
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(*** field ***) |
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val fieldI1 = prove_goalw ZF.thy [field_def] "<a,b>: r ==> a : field(r)" |
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(fn [prem]=> |
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[ (rtac (prem RS domainI RS UnI1) 1) ]); |
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val fieldI2 = prove_goalw ZF.thy [field_def] "<a,b>: r ==> b : field(r)" |
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(fn [prem]=> |
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[ (rtac (prem RS rangeI RS UnI2) 1) ]); |
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val fieldCI = prove_goalw ZF.thy [field_def] |
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"(~ <c,a>:r ==> <a,b>: r) ==> a : field(r)" |
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(fn [prem]=> |
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[ (rtac (prem RS domainI RS UnCI) 1), |
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(swap_res_tac [rangeI] 1), |
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(etac notnotD 1) ]); |
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val fieldE = prove_goalw ZF.thy [field_def] |
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"[| a : field(r); \ |
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\ !!x. <a,x>: r ==> P; \ |
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\ !!x. <x,a>: r ==> P |] ==> P" |
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(fn major::prems=> |
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[ (rtac (major RS UnE) 1), |
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(REPEAT (eresolve_tac (prems@[domainE,rangeE]) 1)) ]); |
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val field_subset = prove_goal ZF.thy "field(A*B) <= A Un B" |
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(fn _ => [ (fast_tac (pair_cs addIs [fieldCI] addSEs [fieldE]) 1) ]); |
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val domain_subset_field = prove_goalw ZF.thy [field_def] |
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"domain(r) <= field(r)" |
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(fn _ => [ (rtac Un_upper1 1) ]); |
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val range_subset_field = prove_goalw ZF.thy [field_def] |
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"range(r) <= field(r)" |
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(fn _ => [ (rtac Un_upper2 1) ]); |
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val domain_times_range = prove_goal ZF.thy |
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"!!A B r. r <= Sigma(A,B) ==> r <= domain(r)*range(r)" |
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(fn _ => [ (fast_tac (pair_cs addIs [domainI,rangeI]) 1) ]); |
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val field_times_field = prove_goal ZF.thy |
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"!!A B r. r <= Sigma(A,B) ==> r <= field(r)*field(r)" |
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(fn _ => [ (fast_tac (pair_cs addIs [fieldI1,fieldI2]) 1) ]); |
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(*** Image of a set under a function/relation ***) |
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val image_iff = prove_goalw ZF.thy [image_def] |
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"b : r``A <-> (EX x:A. <x,b>:r)" |
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(fn _ => [ fast_tac (pair_cs addIs [rangeI]) 1 ]); |
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val image_singleton_iff = prove_goal ZF.thy |
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"b : r``{a} <-> <a,b>:r" |
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(fn _ => [ rtac (image_iff RS iff_trans) 1, |
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fast_tac pair_cs 1 ]); |
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val imageI = prove_goalw ZF.thy [image_def] |
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"!!a b r. [| <a,b>: r; a:A |] ==> b : r``A" |
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(fn _ => [ (REPEAT (ares_tac [CollectI,rangeI,bexI] 1)) ]); |
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val imageE = prove_goalw ZF.thy [image_def] |
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"[| b: r``A; !!x.[| <x,b>: r; x:A |] ==> P |] ==> P" |
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(fn major::prems=> |
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[ (rtac (major RS CollectE) 1), |
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(REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ]); |
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val image_subset = prove_goal ZF.thy |
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14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
0
diff
changeset
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"!!A B r. r <= A*B ==> r``C <= B" |
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(fn _ => |
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[ (rtac subsetI 1), |
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(REPEAT (eresolve_tac [asm_rl, imageE, subsetD RS SigmaD2] 1)) ]); |
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(*** Inverse image of a set under a function/relation ***) |
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val vimage_iff = prove_goalw ZF.thy [vimage_def,image_def,converse_def] |
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"a : r-``B <-> (EX y:B. <a,y>:r)" |
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(fn _ => [ fast_tac (pair_cs addIs [rangeI]) 1 ]); |
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val vimage_singleton_iff = prove_goal ZF.thy |
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"a : r-``{b} <-> <a,b>:r" |
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(fn _ => [ rtac (vimage_iff RS iff_trans) 1, |
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fast_tac pair_cs 1 ]); |
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val vimageI = prove_goalw ZF.thy [vimage_def] |
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"!!A B r. [| <a,b>: r; b:B |] ==> a : r-``B" |
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(fn _ => [ (REPEAT (ares_tac [converseI RS imageI] 1)) ]); |
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val vimageE = prove_goalw ZF.thy [vimage_def] |
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"[| a: r-``B; !!x.[| <a,x>: r; x:B |] ==> P |] ==> P" |
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(fn major::prems=> |
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[ (rtac (major RS imageE) 1), |
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(REPEAT (etac converseD 1 ORELSE ares_tac prems 1)) ]); |
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val vimage_subset = prove_goalw ZF.thy [vimage_def] |
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14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
0
diff
changeset
|
177 |
"!!A B r. r <= A*B ==> r-``C <= A" |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
0
diff
changeset
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(fn _ => [ (etac (converse_type RS image_subset) 1) ]); |
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(** Theorem-proving for ZF set theory **) |
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val ZF_cs = pair_cs |
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addSIs [converseI] |
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addIs [imageI, vimageI, domainI, rangeI, fieldCI] |
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addSEs [imageE, vimageE, domainE, rangeE, fieldE, converseD, converseE]; |
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val eq_cs = ZF_cs addSIs [equalityI]; |
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(** The Union of a set of relations is a relation -- Lemma for fun_Union **) |
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goal ZF.thy "!!S. (ALL x:S. EX A B. x <= A*B) ==> \ |
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\ Union(S) <= domain(Union(S)) * range(Union(S))"; |
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by (fast_tac ZF_cs 1); |
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val rel_Union = result(); |
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(** The Union of 2 relations is a relation (Lemma for fun_Un) **) |
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val rel_Un = prove_goal ZF.thy |
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"!!r s. [| r <= A*B; s <= C*D |] ==> (r Un s) <= (A Un C) * (B Un D)" |
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(fn _ => [ (fast_tac ZF_cs 1) ]); |
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