author | lcp |
Fri, 16 Dec 1994 18:07:12 +0100 | |
changeset 803 | 4c8333ab3eae |
parent 760 | f0200e91b272 |
child 804 | 02430d273ebf |
permissions | -rw-r--r-- |
485 | 1 |
(* Title: ZF/Zorn.ML |
2 |
ID: $Id$ |
|
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 |
Copyright 1994 University of Cambridge |
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5 |
||
516 | 6 |
Proofs from the paper |
485 | 7 |
Abrial & Laffitte, |
8 |
Towards the Mechanization of the Proofs of Some |
|
9 |
Classical Theorems of Set Theory. |
|
10 |
*) |
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11 |
||
516 | 12 |
open Zorn; |
485 | 13 |
|
516 | 14 |
(*** Section 1. Mathematical Preamble ***) |
15 |
||
16 |
goal ZF.thy "!!A B C. (ALL x:C. x<=A | B<=x) ==> Union(C)<=A | B<=Union(C)"; |
|
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by (fast_tac ZF_cs 1); |
|
760 | 18 |
qed "Union_lemma0"; |
516 | 19 |
|
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goal ZF.thy |
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"!!A B C. [| c:C; ALL x:C. A<=x | x<=B |] ==> A<=Inter(C) | Inter(C)<=B"; |
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by (fast_tac ZF_cs 1); |
|
760 | 23 |
qed "Inter_lemma0"; |
516 | 24 |
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25 |
||
26 |
(*** Section 2. The Transfinite Construction ***) |
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27 |
||
28 |
goalw Zorn.thy [increasing_def] |
|
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"!!f A. f: increasing(A) ==> f: Pow(A)->Pow(A)"; |
|
30 |
by (eresolve_tac [CollectD1] 1); |
|
760 | 31 |
qed "increasingD1"; |
516 | 32 |
|
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goalw Zorn.thy [increasing_def] |
|
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"!!f A. [| f: increasing(A); x<=A |] ==> x <= f`x"; |
|
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by (eresolve_tac [CollectD2 RS spec RS mp] 1); |
|
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by (assume_tac 1); |
|
760 | 37 |
qed "increasingD2"; |
516 | 38 |
|
485 | 39 |
(*Introduction rules*) |
516 | 40 |
val [TFin_nextI, Pow_TFin_UnionI] = TFin.intrs; |
485 | 41 |
val TFin_UnionI = PowI RS Pow_TFin_UnionI; |
42 |
||
516 | 43 |
val TFin_is_subset = TFin.dom_subset RS subsetD RS PowD; |
485 | 44 |
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45 |
||
46 |
(** Structural induction on TFin(S,next) **) |
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47 |
||
48 |
val major::prems = goal Zorn.thy |
|
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"[| n: TFin(S,next); \ |
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\ !!x. [| x : TFin(S,next); P(x); next: increasing(S) |] ==> P(next`x); \ |
|
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\ !!Y. [| Y <= TFin(S,next); ALL y:Y. P(y) |] ==> P(Union(Y)) \ |
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52 |
\ |] ==> P(n)"; |
|
516 | 53 |
by (rtac (major RS TFin.induct) 1); |
485 | 54 |
by (ALLGOALS (fast_tac (ZF_cs addIs prems))); |
760 | 55 |
qed "TFin_induct"; |
485 | 56 |
|
57 |
(*Perform induction on n, then prove the major premise using prems. *) |
|
58 |
fun TFin_ind_tac a prems i = |
|
59 |
EVERY [res_inst_tac [("n",a)] TFin_induct i, |
|
60 |
rename_last_tac a ["1"] (i+1), |
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rename_last_tac a ["2"] (i+2), |
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ares_tac prems i]; |
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63 |
||
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(*** Section 3. Some Properties of the Transfinite Construction ***) |
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65 |
||
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
760
diff
changeset
|
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bind_thm ("increasing_trans", |
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
760
diff
changeset
|
67 |
TFin_is_subset RSN (3, increasingD2 RSN (2,subset_trans))); |
485 | 68 |
|
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(*Lemma 1 of section 3.1*) |
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val major::prems = goal Zorn.thy |
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"[| n: TFin(S,next); m: TFin(S,next); \ |
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\ ALL x: TFin(S,next) . x<=m --> x=m | next`x<=m \ |
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\ |] ==> n<=m | next`m<=n"; |
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by (cut_facts_tac prems 1); |
|
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br (major RS TFin_induct) 1; |
|
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by (etac Union_lemma0 2); (*or just fast_tac ZF_cs*) |
|
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by (fast_tac (subset_cs addIs [increasing_trans]) 1); |
|
760 | 78 |
qed "TFin_linear_lemma1"; |
485 | 79 |
|
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(*Lemma 2 of section 3.2. Interesting in its own right! |
|
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Requires next: increasing(S) in the second induction step. *) |
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val [major,ninc] = goal Zorn.thy |
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"[| m: TFin(S,next); next: increasing(S) \ |
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\ |] ==> ALL n: TFin(S,next) . n<=m --> n=m | next`n<=m"; |
|
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br (major RS TFin_induct) 1; |
|
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br (impI RS ballI) 1; |
|
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(*case split using TFin_linear_lemma1*) |
|
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by (res_inst_tac [("n1","n"), ("m1","x")] |
|
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(TFin_linear_lemma1 RS disjE) 1 THEN REPEAT (assume_tac 1)); |
|
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by (dres_inst_tac [("x","n")] bspec 1 THEN assume_tac 1); |
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by (fast_tac (subset_cs addIs [increasing_trans]) 1); |
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by (REPEAT (ares_tac [disjI1,equalityI] 1)); |
|
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(*second induction step*) |
|
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br (impI RS ballI) 1; |
|
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br (Union_lemma0 RS disjE) 1; |
|
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be disjI2 3; |
|
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by (REPEAT (ares_tac [disjI1,equalityI] 2)); |
|
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br ballI 1; |
|
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by (ball_tac 1); |
|
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by (set_mp_tac 1); |
|
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by (res_inst_tac [("n1","n"), ("m1","x")] |
|
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(TFin_linear_lemma1 RS disjE) 1 THEN REPEAT (assume_tac 1)); |
|
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by (fast_tac subset_cs 1); |
|
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br (ninc RS increasingD2 RS subset_trans RS disjI1) 1; |
|
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by (REPEAT (ares_tac [TFin_is_subset] 1)); |
|
760 | 106 |
qed "TFin_linear_lemma2"; |
485 | 107 |
|
108 |
(*a more convenient form for Lemma 2*) |
|
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goal Zorn.thy |
|
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"!!m n. [| n<=m; m: TFin(S,next); n: TFin(S,next); next: increasing(S) \ |
|
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\ |] ==> n=m | next`n<=m"; |
|
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br (TFin_linear_lemma2 RS bspec RS mp) 1; |
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by (REPEAT (assume_tac 1)); |
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760 | 114 |
qed "TFin_subsetD"; |
485 | 115 |
|
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(*Consequences from section 3.3 -- Property 3.2, the ordering is total*) |
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goal Zorn.thy |
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"!!m n. [| m: TFin(S,next); n: TFin(S,next); next: increasing(S) \ |
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\ |] ==> n<=m | m<=n"; |
|
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br (TFin_linear_lemma2 RSN (3,TFin_linear_lemma1) RS disjE) 1; |
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by (REPEAT (assume_tac 1) THEN etac disjI2 1); |
|
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by (fast_tac (subset_cs addIs [increasingD2 RS subset_trans, |
|
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TFin_is_subset]) 1); |
|
760 | 124 |
qed "TFin_subset_linear"; |
485 | 125 |
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126 |
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127 |
(*Lemma 3 of section 3.3*) |
|
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val major::prems = goal Zorn.thy |
|
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"[| n: TFin(S,next); m: TFin(S,next); m = next`m |] ==> n<=m"; |
|
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by (cut_facts_tac prems 1); |
|
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br (major RS TFin_induct) 1; |
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bd TFin_subsetD 1; |
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by (REPEAT (assume_tac 1)); |
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by (fast_tac (ZF_cs addEs [ssubst]) 1); |
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by (fast_tac (subset_cs addIs [TFin_is_subset]) 1); |
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760 | 136 |
qed "equal_next_upper"; |
485 | 137 |
|
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(*Property 3.3 of section 3.3*) |
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goal Zorn.thy |
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"!!m. [| m: TFin(S,next); next: increasing(S) \ |
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\ |] ==> m = next`m <-> m = Union(TFin(S,next))"; |
|
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br iffI 1; |
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br (Union_upper RS equalityI) 1; |
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br (equal_next_upper RS Union_least) 2; |
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by (REPEAT (assume_tac 1)); |
|
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be ssubst 1; |
|
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by (rtac (increasingD2 RS equalityI) 1 THEN assume_tac 1); |
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by (ALLGOALS |
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(fast_tac (subset_cs addIs [TFin_UnionI, TFin_nextI, TFin_is_subset]))); |
|
760 | 150 |
qed "equal_next_Union"; |
485 | 151 |
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152 |
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(*** Section 4. Hausdorff's Theorem: every set contains a maximal chain ***) |
|
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(*** NB: We assume the partial ordering is <=, the subset relation! **) |
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155 |
||
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(** Defining the "next" operation for Hausdorff's Theorem **) |
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157 |
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158 |
goalw Zorn.thy [chain_def] "chain(A) <= Pow(A)"; |
|
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by (resolve_tac [Collect_subset] 1); |
|
760 | 160 |
qed "chain_subset_Pow"; |
485 | 161 |
|
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goalw Zorn.thy [super_def] "super(A,c) <= chain(A)"; |
|
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by (resolve_tac [Collect_subset] 1); |
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760 | 164 |
qed "super_subset_chain"; |
485 | 165 |
|
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goalw Zorn.thy [maxchain_def] "maxchain(A) <= chain(A)"; |
|
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by (resolve_tac [Collect_subset] 1); |
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760 | 168 |
qed "maxchain_subset_chain"; |
485 | 169 |
|
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goal Zorn.thy |
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"!!S. [| ch : (PROD X:Pow(chain(S)) - {0}. X); \ |
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\ X : chain(S); X ~: maxchain(S) \ |
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\ |] ==> ch ` super(S,X) : super(S,X)"; |
|
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by (eresolve_tac [apply_type] 1); |
|
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by (rewrite_goals_tac [super_def, maxchain_def]); |
|
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by (fast_tac ZF_cs 1); |
|
760 | 177 |
qed "choice_super"; |
485 | 178 |
|
179 |
goal Zorn.thy |
|
180 |
"!!S. [| ch : (PROD X:Pow(chain(S)) - {0}. X); \ |
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\ X : chain(S); X ~: maxchain(S) \ |
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\ |] ==> ch ` super(S,X) ~= X"; |
|
183 |
by (resolve_tac [notI] 1); |
|
184 |
by (dresolve_tac [choice_super] 1); |
|
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by (assume_tac 1); |
|
186 |
by (assume_tac 1); |
|
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by (asm_full_simp_tac (ZF_ss addsimps [super_def]) 1); |
|
760 | 188 |
qed "choice_not_equals"; |
485 | 189 |
|
190 |
(*This justifies Definition 4.4*) |
|
191 |
goal Zorn.thy |
|
192 |
"!!S. ch: (PROD X: Pow(chain(S))-{0}. X) ==> \ |
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\ EX next: increasing(S). ALL X: Pow(S). \ |
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\ next`X = if(X: chain(S)-maxchain(S), ch`super(S,X), X)"; |
|
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by (rtac bexI 1); |
|
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by (rtac ballI 1); |
|
197 |
by (resolve_tac [beta] 1); |
|
198 |
by (assume_tac 1); |
|
199 |
bw increasing_def; |
|
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by (rtac CollectI 1); |
|
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by (rtac lam_type 1); |
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by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1); |
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by (fast_tac (ZF_cs addSIs [super_subset_chain RS subsetD, |
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chain_subset_Pow RS subsetD, |
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choice_super]) 1); |
|
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(*Now, verify that it increases*) |
|
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by (resolve_tac [allI] 1); |
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208 |
by (resolve_tac [impI] 1); |
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by (asm_simp_tac (ZF_ss addsimps [Pow_iff, subset_refl] |
|
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setloop split_tac [expand_if]) 1); |
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by (safe_tac ZF_cs); |
|
212 |
by (dresolve_tac [choice_super] 1); |
|
213 |
by (REPEAT (assume_tac 1)); |
|
214 |
bw super_def; |
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215 |
by (fast_tac ZF_cs 1); |
|
760 | 216 |
qed "Hausdorff_next_exists"; |
485 | 217 |
|
218 |
(*Lemma 4*) |
|
219 |
goal Zorn.thy |
|
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"!!S. [| c: TFin(S,next); \ |
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\ ch: (PROD X: Pow(chain(S))-{0}. X); \ |
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\ next: increasing(S); \ |
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\ ALL X: Pow(S). next`X = \ |
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\ if(X: chain(S)-maxchain(S), ch`super(S,X), X) \ |
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\ |] ==> c: chain(S)"; |
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by (eresolve_tac [TFin_induct] 1); |
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by (asm_simp_tac |
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(ZF_ss addsimps [chain_subset_Pow RS subsetD, |
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choice_super RS (super_subset_chain RS subsetD)] |
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setloop split_tac [expand_if]) 1); |
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bw chain_def; |
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by (rtac CollectI 1 THEN fast_tac ZF_cs 1); |
|
233 |
(*Cannot use safe_tac: the disjunction must be left alone*) |
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by (REPEAT (rtac ballI 1 ORELSE etac UnionE 1)); |
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by (res_inst_tac [("m1","B"), ("n1","Ba")] (TFin_subset_linear RS disjE) 1); |
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(*fast_tac is just too slow here!*) |
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by (DEPTH_SOLVE (eresolve_tac [asm_rl, subsetD] 1 |
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ORELSE ball_tac 1 THEN etac (CollectD2 RS bspec RS bspec) 1)); |
|
760 | 239 |
qed "TFin_chain_lemma4"; |
485 | 240 |
|
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goal Zorn.thy "EX c. c : maxchain(S)"; |
|
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by (rtac (AC_Pi_Pow RS exE) 1); |
|
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by (rtac (Hausdorff_next_exists RS bexE) 1); |
|
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by (assume_tac 1); |
|
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by (rename_tac "ch next" 1); |
|
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by (subgoal_tac "Union(TFin(S,next)) : chain(S)" 1); |
|
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by (REPEAT (ares_tac [TFin_chain_lemma4, subset_refl RS TFin_UnionI] 2)); |
|
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by (res_inst_tac [("x", "Union(TFin(S,next))")] exI 1); |
|
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by (resolve_tac [classical] 1); |
|
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by (subgoal_tac "next ` Union(TFin(S,next)) = Union(TFin(S,next))" 1); |
|
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by (resolve_tac [equal_next_Union RS iffD2 RS sym] 2); |
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by (resolve_tac [subset_refl RS TFin_UnionI] 2); |
|
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by (assume_tac 2); |
|
254 |
by (resolve_tac [refl] 2); |
|
255 |
by (asm_full_simp_tac |
|
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(ZF_ss addsimps [subset_refl RS TFin_UnionI RS |
|
516 | 257 |
(TFin.dom_subset RS subsetD)] |
485 | 258 |
setloop split_tac [expand_if]) 1); |
259 |
by (eresolve_tac [choice_not_equals RS notE] 1); |
|
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by (REPEAT (assume_tac 1)); |
|
760 | 261 |
qed "Hausdorff"; |
485 | 262 |
|
263 |
||
264 |
(*** Section 5. Zorn's Lemma: if all chains in S have upper bounds in S |
|
265 |
then S contains a maximal element ***) |
|
266 |
||
267 |
(*Used in the proof of Zorn's Lemma*) |
|
268 |
goalw Zorn.thy [chain_def] |
|
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"!!c. [| c: chain(A); z: A; ALL x:c. x<=z |] ==> cons(z,c) : chain(A)"; |
|
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by (fast_tac ZF_cs 1); |
|
760 | 271 |
qed "chain_extend"; |
485 | 272 |
|
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goal Zorn.thy |
|
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"!!S. ALL c: chain(S). Union(c) : S ==> EX y:S. ALL z:S. y<=z --> y=z"; |
|
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by (resolve_tac [Hausdorff RS exE] 1); |
|
276 |
by (asm_full_simp_tac (ZF_ss addsimps [maxchain_def]) 1); |
|
277 |
by (rename_tac "c" 1); |
|
278 |
by (res_inst_tac [("x", "Union(c)")] bexI 1); |
|
279 |
by (fast_tac ZF_cs 2); |
|
280 |
by (safe_tac ZF_cs); |
|
281 |
by (rename_tac "z" 1); |
|
282 |
by (resolve_tac [classical] 1); |
|
283 |
by (subgoal_tac "cons(z,c): super(S,c)" 1); |
|
284 |
by (fast_tac (ZF_cs addEs [equalityE]) 1); |
|
285 |
bw super_def; |
|
286 |
by (safe_tac eq_cs); |
|
287 |
by (fast_tac (ZF_cs addEs [chain_extend]) 1); |
|
288 |
by (best_tac (ZF_cs addEs [equalityE]) 1); |
|
760 | 289 |
qed "Zorn"; |
485 | 290 |
|
291 |
||
292 |
(*** Section 6. Zermelo's Theorem: every set can be well-ordered ***) |
|
293 |
||
294 |
(*Lemma 5*) |
|
295 |
val major::prems = goal Zorn.thy |
|
296 |
"[| n: TFin(S,next); Z <= TFin(S,next); z:Z; ~ Inter(Z) : Z \ |
|
297 |
\ |] ==> ALL m:Z. n<=m"; |
|
298 |
by (cut_facts_tac prems 1); |
|
299 |
br (major RS TFin_induct) 1; |
|
300 |
by (fast_tac ZF_cs 2); (*second induction step is easy*) |
|
301 |
br ballI 1; |
|
302 |
br (bspec RS TFin_subsetD RS disjE) 1; |
|
303 |
by (REPEAT_SOME (eresolve_tac [asm_rl,subsetD])); |
|
304 |
by (subgoal_tac "x = Inter(Z)" 1); |
|
305 |
by (fast_tac ZF_cs 1); |
|
306 |
by (fast_tac eq_cs 1); |
|
760 | 307 |
qed "TFin_well_lemma5"; |
485 | 308 |
|
309 |
(*Well-ordering of TFin(S,next)*) |
|
310 |
goal Zorn.thy "!!Z. [| Z <= TFin(S,next); z:Z |] ==> Inter(Z) : Z"; |
|
311 |
br classical 1; |
|
312 |
by (subgoal_tac "Z = {Union(TFin(S,next))}" 1); |
|
313 |
by (asm_simp_tac (ZF_ss addsimps [Inter_singleton]) 1); |
|
314 |
be equal_singleton 1; |
|
315 |
br (Union_upper RS equalityI) 1; |
|
316 |
br (subset_refl RS TFin_UnionI RS TFin_well_lemma5 RS bspec) 2; |
|
317 |
by (REPEAT_SOME (eresolve_tac [asm_rl,subsetD])); |
|
760 | 318 |
qed "well_ord_TFin_lemma"; |
485 | 319 |
|
320 |
(*This theorem just packages the previous result*) |
|
321 |
goal Zorn.thy |
|
322 |
"!!S. next: increasing(S) ==> \ |
|
323 |
\ well_ord(TFin(S,next), Subset_rel(TFin(S,next)))"; |
|
324 |
by (resolve_tac [well_ordI] 1); |
|
325 |
by (rewrite_goals_tac [Subset_rel_def, linear_def]); |
|
326 |
(*Prove the linearity goal first*) |
|
327 |
by (REPEAT (rtac ballI 2)); |
|
328 |
by (excluded_middle_tac "x=y" 2); |
|
329 |
by (fast_tac ZF_cs 3); |
|
330 |
(*The x~=y case remains*) |
|
331 |
by (res_inst_tac [("n1","x"), ("m1","y")] |
|
332 |
(TFin_subset_linear RS disjE) 2 THEN REPEAT (assume_tac 2)); |
|
333 |
by (fast_tac ZF_cs 2); |
|
334 |
by (fast_tac ZF_cs 2); |
|
335 |
(*Now prove the well_foundedness goal*) |
|
336 |
by (resolve_tac [wf_onI] 1); |
|
337 |
by (forward_tac [well_ord_TFin_lemma] 1 THEN assume_tac 1); |
|
338 |
by (dres_inst_tac [("x","Inter(Z)")] bspec 1 THEN assume_tac 1); |
|
339 |
by (fast_tac eq_cs 1); |
|
760 | 340 |
qed "well_ord_TFin"; |
485 | 341 |
|
342 |
(** Defining the "next" operation for Zermelo's Theorem **) |
|
343 |
||
344 |
goal AC.thy |
|
345 |
"!!S. [| ch : (PROD X:Pow(S) - {0}. X); X<=S; X~=S \ |
|
346 |
\ |] ==> ch ` (S-X) : S-X"; |
|
347 |
by (eresolve_tac [apply_type] 1); |
|
348 |
by (fast_tac (eq_cs addEs [equalityE]) 1); |
|
760 | 349 |
qed "choice_Diff"; |
485 | 350 |
|
351 |
(*This justifies Definition 6.1*) |
|
352 |
goal Zorn.thy |
|
353 |
"!!S. ch: (PROD X: Pow(S)-{0}. X) ==> \ |
|
354 |
\ EX next: increasing(S). ALL X: Pow(S). \ |
|
355 |
\ next`X = if(X=S, S, cons(ch`(S-X), X))"; |
|
356 |
by (rtac bexI 1); |
|
357 |
by (rtac ballI 1); |
|
358 |
by (resolve_tac [beta] 1); |
|
359 |
by (assume_tac 1); |
|
360 |
bw increasing_def; |
|
361 |
by (rtac CollectI 1); |
|
362 |
by (rtac lam_type 1); |
|
363 |
(*Verify that it increases*) |
|
364 |
by (resolve_tac [allI] 2); |
|
365 |
by (resolve_tac [impI] 2); |
|
366 |
by (asm_simp_tac (ZF_ss addsimps [Pow_iff, subset_consI, subset_refl] |
|
367 |
setloop split_tac [expand_if]) 2); |
|
368 |
(*Type checking is surprisingly hard!*) |
|
369 |
by (asm_simp_tac (ZF_ss addsimps [Pow_iff, cons_subset_iff, subset_refl] |
|
370 |
setloop split_tac [expand_if]) 1); |
|
371 |
by (fast_tac (ZF_cs addSIs [choice_Diff RS DiffD1]) 1); |
|
760 | 372 |
qed "Zermelo_next_exists"; |
485 | 373 |
|
374 |
||
375 |
(*The construction of the injection*) |
|
376 |
goal Zorn.thy |
|
377 |
"!!S. [| ch: (PROD X: Pow(S)-{0}. X); \ |
|
378 |
\ next: increasing(S); \ |
|
379 |
\ ALL X: Pow(S). next`X = if(X=S, S, cons(ch`(S-X), X)) \ |
|
380 |
\ |] ==> (lam x:S. Union({y: TFin(S,next). x~: y})) \ |
|
381 |
\ : inj(S, TFin(S,next) - {S})"; |
|
382 |
by (res_inst_tac [("d", "%y. ch`(S-y)")] lam_injective 1); |
|
383 |
by (rtac DiffI 1); |
|
384 |
by (resolve_tac [Collect_subset RS TFin_UnionI] 1); |
|
385 |
by (fast_tac (ZF_cs addSIs [Collect_subset RS TFin_UnionI] |
|
386 |
addEs [equalityE]) 1); |
|
387 |
by (subgoal_tac "x ~: Union({y: TFin(S,next). x~: y})" 1); |
|
388 |
by (fast_tac (ZF_cs addEs [equalityE]) 2); |
|
389 |
by (subgoal_tac "Union({y: TFin(S,next). x~: y}) ~= S" 1); |
|
390 |
by (fast_tac (ZF_cs addEs [equalityE]) 2); |
|
391 |
(*For proving x : next`Union(...); |
|
392 |
Abrial & Laffitte's justification appears to be faulty.*) |
|
393 |
by (subgoal_tac "~ next ` Union({y: TFin(S,next). x~: y}) <= \ |
|
394 |
\ Union({y: TFin(S,next). x~: y})" 1); |
|
395 |
by (asm_simp_tac |
|
396 |
(ZF_ss addsimps [Collect_subset RS TFin_UnionI RS TFin_is_subset, |
|
397 |
Pow_iff, cons_subset_iff, subset_refl, |
|
398 |
choice_Diff RS DiffD2] |
|
399 |
setloop split_tac [expand_if]) 2); |
|
400 |
by (subgoal_tac "x : next ` Union({y: TFin(S,next). x~: y})" 1); |
|
401 |
by (fast_tac (subset_cs addSIs [Collect_subset RS TFin_UnionI, TFin_nextI]) 2); |
|
402 |
(*End of the lemmas!*) |
|
403 |
by (asm_full_simp_tac |
|
404 |
(ZF_ss addsimps [Collect_subset RS TFin_UnionI RS TFin_is_subset, |
|
405 |
Pow_iff, cons_subset_iff, subset_refl] |
|
406 |
setloop split_tac [expand_if]) 1); |
|
407 |
by (REPEAT (eresolve_tac [asm_rl, consE, sym, notE] 1)); |
|
760 | 408 |
qed "choice_imp_injection"; |
485 | 409 |
|
410 |
(*The wellordering theorem*) |
|
411 |
goal Zorn.thy "EX r. well_ord(S,r)"; |
|
412 |
by (rtac (AC_Pi_Pow RS exE) 1); |
|
413 |
by (rtac (Zermelo_next_exists RS bexE) 1); |
|
414 |
by (assume_tac 1); |
|
415 |
br exI 1; |
|
416 |
by (resolve_tac [well_ord_rvimage] 1); |
|
417 |
by (eresolve_tac [well_ord_TFin] 2); |
|
418 |
by (resolve_tac [choice_imp_injection RS inj_weaken_type] 1); |
|
419 |
by (REPEAT (ares_tac [Diff_subset] 1)); |
|
760 | 420 |
qed "AC_well_ord"; |