author | paulson |
Thu, 13 Aug 1998 18:07:56 +0200 | |
changeset 5315 | c9ad6bbf3a34 |
parent 5268 | 59ef39008514 |
child 6068 | 2d8f3e1f1151 |
permissions | -rw-r--r-- |
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(* Title: ZF/AC/WO2_AC16.ML |
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ID: $Id$ |
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Author: Krzysztof Grabczewski |
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|
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The proof of WO2 ==> AC16(k #+ m, k) |
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The main part of the proof is the inductive reasoning concerning |
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properties of constructed family T_gamma. |
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The proof deals with three cases for ordinals: 0, succ and limit ordinal. |
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The first instance is trivial, the third not difficult, but the second |
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is very complicated requiring many lemmas. |
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We also need to prove that at any stage gamma the set |
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(s - Union(...) - k_gamma) (Rubin & Rubin page 15) |
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contains m distinct elements (in fact is equipollent to s) |
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*) |
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(* ********************************************************************** *) |
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(* case of limit ordinal *) |
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(* ********************************************************************** *) |
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Goal "[| ALL y<x. ALL z<a. z<y | (EX Y: F(y). f(z)<=Y) \ |
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\ --> (EX! Y. Y:F(y) & f(z)<=Y); \ |
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\ ALL i j. i le j --> F(i) <= F(j); j le i; i<x; z<a; \ |
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\ V: F(i); f(z)<=V; W:F(j); f(z)<=W |] \ |
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\ ==> V = W"; |
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by (REPEAT (eresolve_tac [asm_rl, allE, impE] 1)); |
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by (dtac subsetD 1 THEN (assume_tac 1)); |
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by (REPEAT (dtac ospec 1 THEN (assume_tac 1))); |
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by (eresolve_tac [disjI2 RSN (2, impE)] 1); |
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by (fast_tac (FOL_cs addSIs [bexI]) 1); |
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by (etac ex1_two_eq 1 THEN (REPEAT (ares_tac [conjI] 1))); |
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val lemma3_1 = result(); |
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Goal "[| ALL y<x. ALL z<a. z<y | (EX Y: F(y). f(z)<=Y) \ |
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\ --> (EX! Y. Y:F(y) & f(z)<=Y); \ |
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\ ALL i j. i le j --> F(i) <= F(j); i<x; j<x; z<a; \ |
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\ V: F(i); f(z)<=V; W:F(j); f(z)<=W |] \ |
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\ ==> V = W"; |
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by (res_inst_tac [("j","j")] (lt_Ord RS (lt_Ord RSN (2, Ord_linear_le))) 1 |
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THEN (REPEAT (assume_tac 1))); |
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by (eresolve_tac [lemma3_1 RS sym] 1 THEN (REPEAT (assume_tac 1))); |
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by (etac lemma3_1 1 THEN (REPEAT (assume_tac 1))); |
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val lemma3 = result(); |
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Goal "[| ALL y<x. F(y) <= X & \ |
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\ (ALL x<a. x < y | (EX Y:F(y). fa(x) <= Y) --> \ |
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\ (EX! Y. Y : F(y) & fa(x) <= Y)); x < a |] \ |
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\ ==> ALL y<x. ALL z<a. z < y | (EX Y:F(y). fa(z) <= Y) --> \ |
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\ (EX! Y. Y : F(y) & fa(z) <= Y)"; |
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by (REPEAT (resolve_tac [oallI, impI] 1)); |
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by (dtac ospec 1 THEN (assume_tac 1)); |
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by (fast_tac (FOL_cs addSEs [oallE]) 1); |
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val lemma4 = result(); |
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Goal "[| ALL y<x. F(y) <= X & \ |
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\ (ALL x<a. x < y | (EX Y:F(y). fa(x) <= Y) --> \ |
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\ (EX! Y. Y : F(y) & fa(x) <= Y)); \ |
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\ x < a; Limit(x); ALL i j. i le j --> F(i) <= F(j) |] \ |
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\ ==> (UN x<x. F(x)) <= X & \ |
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\ (ALL xa<a. xa < x | (EX x:UN x<x. F(x). fa(xa) <= x) \ |
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\ --> (EX! Y. Y : (UN x<x. F(x)) & fa(xa) <= Y))"; |
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by (rtac conjI 1); |
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by (rtac subsetI 1); |
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by (etac OUN_E 1); |
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by (dtac ospec 1 THEN (assume_tac 1)); |
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by (Fast_tac 1); |
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by (dtac lemma4 1 THEN (assume_tac 1)); |
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by (rtac oallI 1); |
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by (rtac impI 1); |
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by (etac disjE 1); |
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by (forward_tac [Limit_has_succ RSN (2, ospec)] 1 THEN (REPEAT (assume_tac 1))); |
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by (dres_inst_tac [("A","a"),("x","xa")] ospec 1 THEN (assume_tac 1)); |
|
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by (eresolve_tac [lt_Ord RS le_refl RSN (2, disjI1 RSN (2, impE))] 1 |
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THEN (assume_tac 1)); |
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by (REPEAT (eresolve_tac [ex1E, conjE] 1)); |
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by (rtac ex1I 1); |
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by (rtac conjI 1 THEN (assume_tac 2)); |
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by (eresolve_tac [Limit_has_succ RS OUN_I] 1 THEN (TRYALL assume_tac)); |
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by (REPEAT (eresolve_tac [conjE, OUN_E] 1)); |
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by (etac lemma3 1 THEN (TRYALL assume_tac)); |
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by (etac Limit_has_succ 1 THEN (assume_tac 1)); |
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by (etac bexE 1); |
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by (rtac ex1I 1); |
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by (etac conjI 1 THEN (assume_tac 1)); |
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by (REPEAT (eresolve_tac [conjE, OUN_E] 1)); |
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by (etac lemma3 1 THEN (TRYALL assume_tac)); |
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val lemma5 = result(); |
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(* ********************************************************************** *) |
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(* case of successor ordinal *) |
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(* ********************************************************************** *) |
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(* |
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First quite complicated proof of the fact used in the recursive construction |
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of the family T_gamma (WO2 ==> AC16(k #+ m, k)) - the fact that at any stage |
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gamma the set (s - Union(...) - k_gamma) is equipollent to s |
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(Rubin & Rubin page 15). |
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*) |
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(* ********************************************************************** *) |
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(* dbl_Diff_eqpoll_Card *) |
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(* ********************************************************************** *) |
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||
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Goal "[| A eqpoll a; Card(a); ~Finite(a); B lesspoll a; \ |
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\ C lesspoll a |] ==> A - B - C eqpoll a"; |
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by (rtac Diff_lesspoll_eqpoll_Card 1 THEN (REPEAT (assume_tac 1))); |
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by (rtac Diff_lesspoll_eqpoll_Card 1 THEN (REPEAT (assume_tac 1))); |
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qed "dbl_Diff_eqpoll_Card"; |
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|
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(* ********************************************************************** *) |
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(* Case of finite ordinals *) |
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(* ********************************************************************** *) |
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||
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Goalw [lesspoll_def] |
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"!!X. [| Finite(X); ~Finite(a); Ord(a) |] ==> X lesspoll a"; |
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by (rtac conjI 1); |
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by (dresolve_tac [nat_le_infinite_Ord RS le_imp_lepoll] 1 |
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THEN (assume_tac 1)); |
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by (rewtac Finite_def); |
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by (fast_tac (claset() addSEs [eqpoll_sym RS eqpoll_trans]) 2); |
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by (rtac lepoll_trans 1 THEN (assume_tac 2)); |
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by (fast_tac (claset() addSEs [Ord_nat RSN (2, ltI) RS leI RS le_imp_subset RS |
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subset_imp_lepoll RSN (2, eqpoll_imp_lepoll RS lepoll_trans)]) 1); |
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qed "Finite_lesspoll_infinite_Ord"; |
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|
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Goal "x:X ==> Union(X) = Union(X-{x}) Un x"; |
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by (Fast_tac 1); |
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qed "Union_eq_Un_Diff"; |
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Goal "n:nat ==> ALL X. X eqpoll n --> (ALL x:X. Finite(x)) \ |
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\ --> Finite(Union(X))"; |
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by (etac nat_induct 1); |
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by (fast_tac (claset() addSDs [eqpoll_imp_lepoll RS lepoll_0_is_0] |
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addSIs [nat_0I RS nat_into_Finite] addss (simpset())) 1); |
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by (REPEAT (resolve_tac [allI, impI] 1)); |
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by (resolve_tac [eqpoll_succ_imp_not_empty RS not_emptyE] 1 THEN (assume_tac 1)); |
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by (res_inst_tac [("P","%z. Finite(z)")] (Union_eq_Un_Diff RS ssubst) 1 |
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THEN (assume_tac 1)); |
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by (rtac Finite_Un 1); |
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by (Fast_tac 2); |
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by (fast_tac (claset() addSIs [Diff_sing_eqpoll]) 1); |
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qed "Finite_Union_lemma"; |
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Goal "[| ALL x:X. Finite(x); Finite(X) |] ==> Finite(Union(X))"; |
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by (eresolve_tac [Finite_def RS def_imp_iff RS iffD1 RS bexE] 1); |
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by (dtac Finite_Union_lemma 1); |
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by (Fast_tac 1); |
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qed "Finite_Union"; |
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|
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Goalw [Finite_def] "[| x lepoll n; n:nat |] ==> Finite(x)"; |
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by (fast_tac (claset() |
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addEs [nat_into_Ord RSN (2, lepoll_imp_ex_le_eqpoll) RS exE, |
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Ord_nat RSN (2, ltI) RSN (2, lt_trans1) RS ltD]) 1); |
|
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qed "lepoll_nat_num_imp_Finite"; |
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Goal "[| ALL x:X. x lepoll n & x <= T; well_ord(T, R); X lepoll b; \ |
|
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\ b<a; ~Finite(a); Card(a); n:nat |] \ |
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\ ==> Union(X) lesspoll a"; |
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by (excluded_middle_tac "Finite(X)" 1); |
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by (resolve_tac [Card_is_Ord RSN (3, Finite_lesspoll_infinite_Ord)] 2 |
|
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THEN (REPEAT (assume_tac 3))); |
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by (fast_tac (claset() addSEs [lepoll_nat_num_imp_Finite] |
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addSIs [Finite_Union]) 2); |
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by (dresolve_tac [lt_Ord RSN (2, lepoll_imp_ex_le_eqpoll)] 1 THEN (assume_tac 1)); |
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by (REPEAT (eresolve_tac [exE, conjE] 1)); |
|
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by (forward_tac [eqpoll_imp_lepoll RS lepoll_infinite] 1 THEN (assume_tac 1)); |
|
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by (eresolve_tac [eqpoll_sym RS (eqpoll_def RS def_imp_iff RS iffD1) RS |
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exE] 1); |
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by (forward_tac [bij_is_surj RS surj_image_eq] 1); |
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by (dresolve_tac [[bij_is_fun, subset_refl] MRS image_fun] 1); |
|
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by (dresolve_tac [sym RS trans] 1 THEN (assume_tac 1)); |
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by (hyp_subst_tac 1); |
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by (rtac lepoll_lesspoll_lesspoll 1); |
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by (eresolve_tac [lt_trans1 RSN (2, lt_Card_imp_lesspoll)] 1 |
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THEN REPEAT (assume_tac 1)); |
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by (rtac UN_lepoll 1 |
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THEN (TRYALL (fast_tac (claset() addSEs [lt_Ord])))); |
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qed "Union_lesspoll"; |
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|
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(* ********************************************************************** *) |
|
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(* recfunAC16_lepoll_index *) |
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(* ********************************************************************** *) |
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||
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Goal "A Un {a} = cons(a, A)"; |
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by (Fast_tac 1); |
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qed "Un_sing_eq_cons"; |
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Goal "A lepoll B ==> A Un {a} lepoll succ(B)"; |
|
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by (asm_simp_tac (simpset() addsimps [Un_sing_eq_cons, succ_def]) 1); |
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by (eresolve_tac [mem_not_refl RSN (2, cons_lepoll_cong)] 1); |
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qed "Un_lepoll_succ"; |
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Goal "Ord(a) ==> F(a) - (UN b<succ(a). F(b)) = 0"; |
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by (fast_tac (claset() addSIs [OUN_I, le_refl]) 1); |
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qed "Diff_UN_succ_empty"; |
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|
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Goal "Ord(a) ==> F(a) Un X - (UN b<succ(a). F(b)) <= X"; |
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by (fast_tac (claset() addSIs [OUN_I, le_refl]) 1); |
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qed "Diff_UN_succ_subset"; |
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|
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Goal "Ord(x) ==> \ |
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\ recfunAC16(f, g, x, a) - (UN i<x. recfunAC16(f, g, i, a)) lepoll 1"; |
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by (etac Ord_cases 1); |
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by (asm_simp_tac (simpset() addsimps [recfunAC16_0, |
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empty_subsetI RS subset_imp_lepoll]) 1); |
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by (asm_simp_tac (simpset() addsimps [recfunAC16_Limit, Diff_cancel, |
|
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empty_subsetI RS subset_imp_lepoll]) 2); |
|
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by (asm_simp_tac (simpset() addsimps [recfunAC16_succ]) 1); |
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by (rtac conjI 1); |
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by (fast_tac (claset() addSIs [empty_subsetI RS subset_imp_lepoll] |
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addSEs [Diff_UN_succ_empty RS ssubst]) 1); |
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by (fast_tac (claset() addSEs [Diff_UN_succ_subset RS subset_imp_lepoll RS |
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(singleton_eqpoll_1 RS eqpoll_imp_lepoll RSN (2, lepoll_trans))]) 1); |
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qed "recfunAC16_Diff_lepoll_1"; |
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Goal "[| z : F(x); Ord(x) |] \ |
|
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\ ==> z:F(LEAST i. z:F(i)) - (UN j<(LEAST i. z:F(i)). F(j))"; |
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by (fast_tac (claset() addEs [less_LeastE] addSEs [OUN_E, LeastI]) 1); |
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qed "in_Least_Diff"; |
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Goal "[| (LEAST i. w:F(i)) = (LEAST i. z:F(i)); \ |
|
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\ w:(UN i<a. F(i)); z:(UN i<a. F(i)) |] \ |
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\ ==> EX b<a. w:(F(b) - (UN c<b. F(c))) & z:(F(b) - (UN c<b. F(c)))"; |
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by (REPEAT (etac OUN_E 1)); |
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by (dresolve_tac [lt_Ord RSN (2, in_Least_Diff)] 1 THEN (assume_tac 1)); |
244 |
by (forward_tac [lt_Ord RSN (2, in_Least_Diff)] 1 THEN (assume_tac 1)); |
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by (rtac oexI 1); |
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by (rtac conjI 1 THEN (assume_tac 2)); |
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by (etac subst 1 THEN (assume_tac 1)); |
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by (eresolve_tac [lt_Ord RSN (2, Least_le) RS lt_trans1] 1 |
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THEN (REPEAT (assume_tac 1))); |
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qed "Least_eq_imp_ex"; |
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Goal "[| A lepoll 1; a:A; b:A |] ==> a=b"; |
|
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by (fast_tac (claset() addSDs [lepoll_1_is_sing]) 1); |
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qed "two_in_lepoll_1"; |
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|
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Goal "[| ALL i<a. F(i)-(UN j<i. F(j)) lepoll 1; Limit(a) |] \ |
|
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\ ==> (UN x<a. F(x)) lepoll a"; |
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by (resolve_tac [lepoll_def RS (def_imp_iff RS iffD2)] 1); |
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by (res_inst_tac [("x","lam z: (UN x<a. F(x)). LEAST i. z:F(i)")] exI 1); |
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by (rewtac inj_def); |
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263 |
by (rtac CollectI 1); |
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|
264 |
by (rtac lam_type 1); |
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|
265 |
by (etac OUN_E 1); |
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|
266 |
by (etac Least_in_Ord 1); |
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|
267 |
by (etac ltD 1); |
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|
268 |
by (etac lt_Ord2 1); |
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|
269 |
by (rtac ballI 1); |
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270 |
by (rtac ballI 1); |
2469 | 271 |
by (Asm_simp_tac 1); |
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|
272 |
by (rtac impI 1); |
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|
273 |
by (dtac Least_eq_imp_ex 1 THEN (REPEAT (assume_tac 1))); |
4091 | 274 |
by (fast_tac (claset() addSEs [two_in_lepoll_1]) 1); |
3731 | 275 |
qed "UN_lepoll_index"; |
1196 | 276 |
|
5137 | 277 |
|
278 |
Goal "Ord(y) ==> recfunAC16(f, fa, y, a) lepoll y"; |
|
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279 |
by (etac trans_induct 1); |
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280 |
by (etac Ord_cases 1); |
4091 | 281 |
by (asm_simp_tac (simpset() addsimps [recfunAC16_0, lepoll_refl]) 1); |
282 |
by (asm_simp_tac (simpset() addsimps [recfunAC16_succ]) 1); |
|
5137 | 283 |
by (fast_tac (claset() |
1461 | 284 |
addSDs [succI1 RSN (2, bspec)] |
285 |
addSEs [subset_succI RS subset_imp_lepoll RSN (2, lepoll_trans), |
|
286 |
Un_lepoll_succ]) 1); |
|
4091 | 287 |
by (asm_simp_tac (simpset() addsimps [recfunAC16_Limit]) 1); |
288 |
by (fast_tac (claset() addSEs [lt_Ord RS recfunAC16_Diff_lepoll_1] |
|
5137 | 289 |
addSIs [UN_lepoll_index]) 1); |
3731 | 290 |
qed "recfunAC16_lepoll_index"; |
1196 | 291 |
|
5137 | 292 |
|
293 |
Goal "[| recfunAC16(f,g,y,a) <= {X: Pow(A). X eqpoll n}; \ |
|
1461 | 294 |
\ A eqpoll a; y<a; ~Finite(a); Card(a); n:nat |] \ |
295 |
\ ==> Union(recfunAC16(f,g,y,a)) lesspoll a"; |
|
1196 | 296 |
by (eresolve_tac [eqpoll_def RS def_imp_iff RS iffD1 RS exE] 1); |
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297 |
by (rtac Union_lesspoll 1 THEN (TRYALL assume_tac)); |
1196 | 298 |
by (eresolve_tac [lt_Ord RS recfunAC16_lepoll_index] 3); |
299 |
by (eresolve_tac [[bij_is_inj, Card_is_Ord RS well_ord_Memrel] MRS |
|
5137 | 300 |
well_ord_rvimage] 2 |
301 |
THEN (assume_tac 2)); |
|
4091 | 302 |
by (fast_tac (claset() addSEs [eqpoll_imp_lepoll]) 1); |
3731 | 303 |
qed "Union_recfunAC16_lesspoll"; |
1196 | 304 |
|
5137 | 305 |
|
5268 | 306 |
Goal "[| recfunAC16(f, fa, y, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)}; \ |
1461 | 307 |
\ Card(a); ~ Finite(a); A eqpoll a; \ |
308 |
\ k : nat; m : nat; y<a; \ |
|
309 |
\ fa : bij(a, {Y: Pow(A). Y eqpoll succ(k)}) |] \ |
|
310 |
\ ==> A - Union(recfunAC16(f, fa, y, a)) - fa`y eqpoll a"; |
|
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311 |
by (rtac dbl_Diff_eqpoll_Card 1 THEN (TRYALL assume_tac)); |
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312 |
by (rtac Union_recfunAC16_lesspoll 1 THEN (REPEAT (assume_tac 1))); |
1196 | 313 |
by (eresolve_tac [add_type RS nat_succI] 1 THEN (assume_tac 1)); |
314 |
by (resolve_tac [nat_succI RSN (2, bexI RS (Finite_def RS def_imp_iff RS |
|
1461 | 315 |
iffD2)) RS (Card_is_Ord RSN (3, Finite_lesspoll_infinite_Ord))] 1 |
316 |
THEN (TRYALL assume_tac)); |
|
1196 | 317 |
by (eresolve_tac [ltD RSN (2, bij_is_fun RS apply_type) RS CollectE] 1 |
1461 | 318 |
THEN (TRYALL assume_tac)); |
3731 | 319 |
qed "dbl_Diff_eqpoll"; |
1196 | 320 |
|
321 |
(* back to the proof *) |
|
322 |
||
323 |
val disj_Un_eqpoll_nat_sum = disj_Un_eqpoll_sum RS ( |
|
324 |
sum_eqpoll_cong RSN (2, |
|
325 |
nat_sum_eqpoll_sum RSN (3, |
|
326 |
eqpoll_trans RS eqpoll_trans))) |> standard; |
|
327 |
||
5137 | 328 |
|
329 |
Goal "[| x : Pow(A - B - fa`i); x eqpoll m; \ |
|
1461 | 330 |
\ fa : bij(a, {x: Pow(A) . x eqpoll k}); i<a; k:nat; m:nat |] \ |
331 |
\ ==> fa ` i Un x : {x: Pow(A) . x eqpoll k #+ m}"; |
|
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|
332 |
by (rtac CollectI 1); |
4091 | 333 |
by (fast_tac (claset() addSIs [PowD RS (PowD RSN (2, Un_least RS PowI))] |
1461 | 334 |
addSEs [ltD RSN (2, bij_is_fun RS apply_type RS CollectE)]) 1); |
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|
335 |
by (rtac disj_Un_eqpoll_nat_sum 1 |
1461 | 336 |
THEN (TRYALL assume_tac)); |
4091 | 337 |
by (fast_tac (claset() addSIs [equals0I]) 1); |
1196 | 338 |
by (eresolve_tac [ltD RSN (2, bij_is_fun RS apply_type RS CollectE)] 1 |
1461 | 339 |
THEN (REPEAT (assume_tac 1))); |
3731 | 340 |
qed "Un_in_Collect"; |
1196 | 341 |
|
342 |
(* ********************************************************************** *) |
|
1461 | 343 |
(* Lemmas simplifying assumptions *) |
1196 | 344 |
(* ********************************************************************** *) |
345 |
||
5137 | 346 |
|
347 |
Goal "[| ALL y<succ(j). F(y)<=X & (ALL x<a. x<y | P(x,y) \ |
|
1461 | 348 |
\ --> Q(x,y)); succ(j)<a |] \ |
349 |
\ ==> F(j)<=X & (ALL x<a. x<j | P(x,j) --> Q(x,j))"; |
|
4152 | 350 |
by (dtac ospec 1); |
2469 | 351 |
by (resolve_tac [lt_Ord RS (succI1 RS ltI RS lt_Ord RS le_refl)] 1 |
1461 | 352 |
THEN (REPEAT (assume_tac 1))); |
1196 | 353 |
val lemma6 = result(); |
354 |
||
5137 | 355 |
|
356 |
Goal "[| ALL x<a. x<j | P(x,j) --> Q(x,j); succ(j)<a |] \ |
|
357 |
\ ==> P(j,j) --> (ALL x<a. x le j | P(x,j) --> Q(x,j))"; |
|
4091 | 358 |
by (fast_tac (claset() addSEs [leE]) 1); |
1196 | 359 |
val lemma7 = result(); |
360 |
||
361 |
(* ********************************************************************** *) |
|
362 |
(* Lemmas needded to prove ex_next_set which means that for any successor *) |
|
1461 | 363 |
(* ordinal there is a set satisfying certain properties *) |
1196 | 364 |
(* ********************************************************************** *) |
365 |
||
5137 | 366 |
|
367 |
Goal "[| A eqpoll a; ~ Finite(a); Ord(a); m:nat |] \ |
|
1461 | 368 |
\ ==> EX X:Pow(A). X eqpoll m"; |
1196 | 369 |
by (eresolve_tac [Ord_nat RSN (2, ltI) RS |
1461 | 370 |
(nat_le_infinite_Ord RSN (2, lt_trans2)) RS |
371 |
leI RS le_imp_lepoll RS |
|
372 |
((eqpoll_sym RS eqpoll_imp_lepoll) RSN (2, lepoll_trans)) RS |
|
373 |
lepoll_imp_eqpoll_subset RS exE] 1 |
|
374 |
THEN REPEAT (assume_tac 1)); |
|
4091 | 375 |
by (fast_tac (claset() addSEs [eqpoll_sym]) 1); |
3731 | 376 |
qed "ex_subset_eqpoll"; |
1196 | 377 |
|
5137 | 378 |
|
379 |
Goal "[| A <= B Un C; A Int C = 0 |] ==> A <= B"; |
|
5265
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380 |
by (Blast_tac 1); |
3731 | 381 |
qed "subset_Un_disjoint"; |
1196 | 382 |
|
5137 | 383 |
|
384 |
Goal "[| X:Pow(A - Union(B) -C); T:B; F<=T |] ==> F Int X = 0"; |
|
5265
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385 |
by (Blast_tac 1); |
3731 | 386 |
qed "Int_empty"; |
1196 | 387 |
|
388 |
(* ********************************************************************** *) |
|
1461 | 389 |
(* equipollent subset (and finite) is the whole set *) |
1196 | 390 |
(* ********************************************************************** *) |
391 |
||
5137 | 392 |
|
393 |
Goal "[| A <= B; a : A; A - {a} = B - {a} |] ==> A = B"; |
|
4091 | 394 |
by (fast_tac (claset() addSEs [equalityE]) 1); |
3731 | 395 |
qed "Diffs_eq_imp_eq"; |
1196 | 396 |
|
5137 | 397 |
|
398 |
Goal "m:nat ==> ALL A B. A <= B & m lepoll A & B lepoll m --> A=B"; |
|
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|
399 |
by (etac nat_induct 1); |
4091 | 400 |
by (fast_tac (claset() addSDs [lepoll_0_is_0]) 1); |
1196 | 401 |
by (REPEAT (resolve_tac [allI, impI] 1)); |
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|
402 |
by (REPEAT (etac conjE 1)); |
1196 | 403 |
by (resolve_tac [succ_lepoll_imp_not_empty RS not_emptyE] 1 |
1461 | 404 |
THEN (assume_tac 1)); |
1196 | 405 |
by (forward_tac [subsetD RS Diff_sing_lepoll] 1 |
1461 | 406 |
THEN REPEAT (assume_tac 1)); |
1196 | 407 |
by (forward_tac [lepoll_Diff_sing] 1); |
408 |
by (REPEAT (eresolve_tac [allE, impE] 1)); |
|
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|
409 |
by (rtac conjI 1); |
2469 | 410 |
by (Fast_tac 2); |
411 |
by (Fast_tac 1); |
|
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|
412 |
by (etac Diffs_eq_imp_eq 1 |
1461 | 413 |
THEN REPEAT (assume_tac 1)); |
3731 | 414 |
qed "subset_imp_eq_lemma"; |
1196 | 415 |
|
5137 | 416 |
|
417 |
Goal "[| A <= B; m lepoll A; B lepoll m; m:nat |] ==> A=B"; |
|
1196 | 418 |
by (fast_tac (FOL_cs addSDs [subset_imp_eq_lemma]) 1); |
3731 | 419 |
qed "subset_imp_eq"; |
1196 | 420 |
|
5137 | 421 |
|
422 |
Goal "[| f:bij(a, {Y:X. Y eqpoll succ(k)}); k:nat; f`b<=f`y; b<a; \ |
|
1461 | 423 |
\ y<a |] ==> b=y"; |
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|
424 |
by (dtac subset_imp_eq 1); |
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|
425 |
by (etac nat_succI 3); |
4091 | 426 |
by (fast_tac (claset() addSEs [bij_is_fun RS (ltD RSN (2, apply_type)) RS |
1461 | 427 |
CollectE, eqpoll_sym RS eqpoll_imp_lepoll]) 1); |
4091 | 428 |
by (fast_tac (claset() addSEs [bij_is_fun RS (ltD RSN (2, apply_type)) RS |
1461 | 429 |
CollectE, eqpoll_imp_lepoll]) 1); |
1196 | 430 |
by (rewrite_goals_tac [bij_def, inj_def]); |
4091 | 431 |
by (fast_tac (claset() addSDs [ltD]) 1); |
3731 | 432 |
qed "bij_imp_arg_eq"; |
1196 | 433 |
|
5137 | 434 |
|
5268 | 435 |
Goal "[| recfunAC16(f, fa, y, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)}; \ |
1461 | 436 |
\ Card(a); ~ Finite(a); A eqpoll a; \ |
437 |
\ k : nat; m : nat; y<a; \ |
|
438 |
\ fa : bij(a, {Y: Pow(A). Y eqpoll succ(k)}); \ |
|
439 |
\ ~ (EX Y:recfunAC16(f, fa, y, a). fa`y <= Y) |] \ |
|
440 |
\ ==> EX X:{Y: Pow(A). Y eqpoll succ(k #+ m)}. fa`y <= X & \ |
|
441 |
\ (ALL b<a. fa`b <= X --> \ |
|
442 |
\ (ALL T:recfunAC16(f, fa, y, a). ~ fa`b <= T))"; |
|
1196 | 443 |
by (eresolve_tac [dbl_Diff_eqpoll RS ex_subset_eqpoll RS bexE] 1 |
1461 | 444 |
THEN REPEAT (assume_tac 1)); |
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|
445 |
by (etac Card_is_Ord 1); |
1196 | 446 |
by (forward_tac [Un_in_Collect] 2 THEN REPEAT (assume_tac 2)); |
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|
447 |
by (etac CollectE 4); |
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|
448 |
by (rtac bexI 4); |
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|
449 |
by (rtac CollectI 5); |
1196 | 450 |
by (assume_tac 5); |
451 |
by (eresolve_tac [add_succ RS subst] 5); |
|
452 |
by (assume_tac 1); |
|
1208
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changeset
|
453 |
by (etac nat_succI 1); |
1196 | 454 |
by (assume_tac 1); |
1208
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|
455 |
by (rtac conjI 1); |
2469 | 456 |
by (Fast_tac 1); |
1196 | 457 |
by (REPEAT (resolve_tac [ballI, impI, oallI, notI] 1)); |
458 |
by (dresolve_tac [Int_empty RSN (2, subset_Un_disjoint)] 1 |
|
1461 | 459 |
THEN REPEAT (assume_tac 1)); |
1208
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|
460 |
by (dtac bij_imp_arg_eq 1 THEN REPEAT (assume_tac 1)); |
1196 | 461 |
by (hyp_subst_tac 1); |
462 |
by (eresolve_tac [bexI RSN (2, notE)] 1 THEN TRYALL assume_tac); |
|
3731 | 463 |
qed "ex_next_set"; |
1196 | 464 |
|
465 |
(* ********************************************************************** *) |
|
1461 | 466 |
(* Lemma ex_next_Ord states that for any successor *) |
467 |
(* ordinal there is a number of the set satisfying certain properties *) |
|
1196 | 468 |
(* ********************************************************************** *) |
469 |
||
5137 | 470 |
|
5268 | 471 |
Goal "[| recfunAC16(f, fa, y, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)}; \ |
1461 | 472 |
\ Card(a); ~ Finite(a); A eqpoll a; \ |
473 |
\ k : nat; m : nat; y<a; \ |
|
474 |
\ fa : bij(a, {Y: Pow(A). Y eqpoll succ(k)}); \ |
|
475 |
\ f : bij(a, {Y: Pow(A). Y eqpoll succ(k #+ m)}); \ |
|
476 |
\ ~ (EX Y:recfunAC16(f, fa, y, a). fa`y <= Y) |] \ |
|
477 |
\ ==> EX c<a. fa`y <= f`c & \ |
|
478 |
\ (ALL b<a. fa`b <= f`c --> \ |
|
479 |
\ (ALL T:recfunAC16(f, fa, y, a). ~ fa`b <= T))"; |
|
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changeset
|
480 |
by (dtac ex_next_set 1 THEN REPEAT (assume_tac 1)); |
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|
481 |
by (etac bexE 1); |
1196 | 482 |
by (resolve_tac [bij_converse_bij RS bij_is_fun RS apply_type RS ltI RSN |
1461 | 483 |
(2, oexI)] 1); |
1196 | 484 |
by (resolve_tac [right_inverse_bij RS ssubst] 1 |
1461 | 485 |
THEN REPEAT (ares_tac [Card_is_Ord] 1)); |
3731 | 486 |
qed "ex_next_Ord"; |
1196 | 487 |
|
5137 | 488 |
|
489 |
Goal "[| EX! Y. Y:Z & P(Y); ~P(W) |] ==> EX! Y. Y: (Z Un {W}) & P(Y)"; |
|
2469 | 490 |
by (Fast_tac 1); |
3731 | 491 |
qed "ex1_in_Un_sing"; |
1196 | 492 |
|
493 |
(* ********************************************************************** *) |
|
1461 | 494 |
(* Lemma simplifying assumptions *) |
1196 | 495 |
(* ********************************************************************** *) |
496 |
||
5137 | 497 |
|
498 |
Goal "[| ALL x<a. x<j | (EX xa:F(j). P(x, xa)) \ |
|
1461 | 499 |
\ --> (EX! Y. Y:F(j) & P(x, Y)); F(j) <= X; \ |
500 |
\ L : X; P(j, L) & (ALL x<a. P(x, L) --> (ALL xa:F(j). ~P(x, xa))) |] \ |
|
501 |
\ ==> F(j) Un {L} <= X & \ |
|
2845 | 502 |
\ (ALL x<a. x le j | (EX xa: (F(j) Un {L}). P(x, xa)) --> \ |
503 |
\ (EX! Y. Y: (F(j) Un {L}) & P(x, Y)))"; |
|
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changeset
|
504 |
by (rtac conjI 1); |
4091 | 505 |
by (fast_tac (claset() addSIs [singleton_subsetI]) 1); |
1208
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|
506 |
by (rtac oallI 1); |
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|
507 |
by (etac oallE 1 THEN (contr_tac 2)); |
5137 | 508 |
by (blast_tac (claset() addSEs [leE]) 1); |
1196 | 509 |
val lemma8 = result(); |
510 |
||
511 |
(* ********************************************************************** *) |
|
512 |
(* The main part of the proof: inductive proof of the property of T_gamma *) |
|
1461 | 513 |
(* lemma main_induct *) |
1196 | 514 |
(* ********************************************************************** *) |
515 |
||
5137 | 516 |
|
517 |
Goal "[| b < a; f : bij(a, {Y: Pow(A) . Y eqpoll succ(k #+ m)}); \ |
|
1461 | 518 |
\ fa : bij(a, {Y: Pow(A) . Y eqpoll succ(k)}); \ |
519 |
\ ~Finite(a); Card(a); A eqpoll a; k : nat; m : nat |] \ |
|
520 |
\ ==> recfunAC16(f, fa, b, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)} & \ |
|
521 |
\ (ALL x<a. x < b | (EX Y:recfunAC16(f, fa, b, a). fa ` x <= Y) --> \ |
|
522 |
\ (EX! Y. Y:recfunAC16(f, fa, b, a) & fa ` x <= Y))"; |
|
2469 | 523 |
by (etac lt_induct 1); |
524 |
by (forward_tac [lt_Ord] 1); |
|
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|
525 |
by (etac Ord_cases 1); |
1196 | 526 |
(* case 0 *) |
4091 | 527 |
by (asm_simp_tac (simpset() addsimps [recfunAC16_0]) 1); |
1196 | 528 |
(* case Limit *) |
4091 | 529 |
by (asm_simp_tac (simpset() addsimps [recfunAC16_Limit]) 2); |
2469 | 530 |
by (rtac lemma5 2 THEN (REPEAT (assume_tac 2))); |
1196 | 531 |
by (fast_tac (FOL_cs addSEs [recfunAC16_mono]) 2); |
532 |
(* case succ *) |
|
533 |
by (hyp_subst_tac 1); |
|
534 |
by (eresolve_tac [lemma6 RS conjE] 1 THEN (assume_tac 1)); |
|
5137 | 535 |
by (asm_simp_tac (simpset() delsplits [split_if] |
536 |
addsimps [recfunAC16_succ]) 1); |
|
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|
537 |
by (resolve_tac [conjI RS (split_if RS iffD2)] 1); |
5137 | 538 |
by (Asm_simp_tac 1); |
539 |
by (etac lemma7 1 THEN assume_tac 1); |
|
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|
540 |
by (rtac impI 1); |
1196 | 541 |
by (resolve_tac [ex_next_Ord RS oexE] 1 |
5137 | 542 |
THEN REPEAT (ares_tac [le_refl RS lt_trans] 1)); |
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|
543 |
by (etac lemma8 1 THEN (assume_tac 1)); |
1196 | 544 |
by (resolve_tac [bij_is_fun RS apply_type] 1 THEN (assume_tac 1)); |
545 |
by (eresolve_tac [Least_le RS lt_trans2 RS ltD] 1 |
|
1461 | 546 |
THEN REPEAT (ares_tac [lt_Ord, succ_leI] 1)); |
5137 | 547 |
(*SLOW! 5 secs?*) |
1196 | 548 |
by (eresolve_tac [lt_Ord RSN (2, LeastI)] 1 THEN (assume_tac 1)); |
3731 | 549 |
qed "main_induct"; |
1196 | 550 |
|
551 |
(* ********************************************************************** *) |
|
1461 | 552 |
(* Lemma to simplify the inductive proof *) |
1200 | 553 |
(* - the desired property is a consequence of the inductive assumption *) |
1196 | 554 |
(* ********************************************************************** *) |
555 |
||
556 |
val [prem1, prem2, prem3, prem4] = goal thy |
|
1461 | 557 |
"[| (!!b. b<a ==> F(b) <= S & (ALL x<a. (x<b | (EX Y:F(b). f`x <= Y)) \ |
558 |
\ --> (EX! Y. Y : F(b) & f`x <= Y))); \ |
|
559 |
\ f:a->f``(a); Limit(a); (!!i j. i le j ==> F(i) <= F(j)) |] \ |
|
560 |
\ ==> (UN j<a. F(j)) <= S & \ |
|
561 |
\ (ALL x:f``a. EX! Y. Y : (UN j<a. F(j)) & x <= Y)"; |
|
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562 |
by (rtac conjI 1); |
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|
563 |
by (rtac subsetI 1); |
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|
564 |
by (etac OUN_E 1); |
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|
565 |
by (dtac prem1 1); |
2469 | 566 |
by (Fast_tac 1); |
2493 | 567 |
(** LEVEL 5 **) |
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|
568 |
by (rtac ballI 1); |
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|
569 |
by (etac imageE 1); |
1196 | 570 |
by (dresolve_tac [prem3 RS Limit_is_Ord RSN (2, ltI) RS |
1461 | 571 |
(prem3 RS Limit_has_succ)] 1); |
1196 | 572 |
by (forward_tac [prem1] 1); |
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|
573 |
by (etac conjE 1); |
2493 | 574 |
(** LEVEL 10 **) |
1196 | 575 |
by (dresolve_tac [leI RS succ_leE RSN (2, ospec)] 1 THEN (assume_tac 1)); |
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|
576 |
by (etac impE 1); |
4091 | 577 |
by (fast_tac (claset() addSEs [leI RS succ_leE RS lt_Ord RS le_refl]) 1); |
1196 | 578 |
by (dresolve_tac [prem2 RSN (2, apply_equality)] 1); |
579 |
by (REPEAT (eresolve_tac [conjE, ex1E] 1)); |
|
2493 | 580 |
(** LEVEL 15 **) |
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|
581 |
by (rtac ex1I 1); |
4091 | 582 |
by (fast_tac (claset() addSIs [OUN_I]) 1); |
1196 | 583 |
by (REPEAT (eresolve_tac [conjE, OUN_E] 1)); |
584 |
by (eresolve_tac [lt_Ord RSN (2, lt_Ord RS Ord_linear_le)] 1 THEN (assume_tac 1)); |
|
585 |
by (dresolve_tac [prem4 RS subsetD] 2 THEN (assume_tac 2)); |
|
2493 | 586 |
(** LEVEL 20 **) |
587 |
by (fast_tac FOL_cs 2); |
|
1196 | 588 |
by (forward_tac [prem1] 1); |
589 |
by (forward_tac [succ_leE] 1); |
|
590 |
by (dresolve_tac [prem4 RS subsetD] 1 THEN (assume_tac 1)); |
|
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|
591 |
by (etac conjE 1); |
2493 | 592 |
(** LEVEL 25 **) |
1196 | 593 |
by (dresolve_tac [lt_trans RSN (2, ospec)] 1 THEN (TRYALL assume_tac)); |
594 |
by (dresolve_tac [disjI1 RSN (2, mp)] 1 THEN (assume_tac 1)); |
|
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|
595 |
by (etac ex1_two_eq 1); |
2469 | 596 |
by (REPEAT (Fast_tac 1)); |
3731 | 597 |
qed "lemma_simp_induct"; |
1196 | 598 |
|
599 |
(* ********************************************************************** *) |
|
1461 | 600 |
(* The target theorem *) |
1196 | 601 |
(* ********************************************************************** *) |
602 |
||
5137 | 603 |
|
5315 | 604 |
Goalw [AC16_def] "[| WO2; 0<m; k:nat; m:nat |] ==> AC16(k #+ m,k)"; |
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|
605 |
by (rtac allI 1); |
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|
606 |
by (rtac impI 1); |
2493 | 607 |
by (forward_tac [WO2_infinite_subsets_eqpoll_X] 1 |
608 |
THEN (REPEAT (assume_tac 1))); |
|
1196 | 609 |
by (forw_inst_tac [("n","k #+ m")] (WO2_infinite_subsets_eqpoll_X) 1 |
1461 | 610 |
THEN (REPEAT (ares_tac [add_type] 1))); |
1196 | 611 |
by (forward_tac [WO2_imp_ex_Card] 1); |
612 |
by (REPEAT (eresolve_tac [exE,conjE] 1)); |
|
613 |
by (dresolve_tac [eqpoll_trans RS eqpoll_sym RS (eqpoll_def RS |
|
1461 | 614 |
def_imp_iff RS iffD1)] 1 THEN (assume_tac 1)); |
1196 | 615 |
by (dresolve_tac [eqpoll_trans RS eqpoll_sym RS (eqpoll_def RS |
1461 | 616 |
def_imp_iff RS iffD1)] 1 THEN (assume_tac 1)); |
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|
617 |
by (REPEAT (etac exE 1)); |
1196 | 618 |
by (res_inst_tac [("x","UN j<a. recfunAC16(fa,f,j,a)")] exI 1); |
619 |
by (res_inst_tac [("P","%z. ?Y & (ALL x:z. ?Z(x))")] |
|
1461 | 620 |
(bij_is_surj RS surj_image_eq RS subst) 1 |
621 |
THEN (assume_tac 1)); |
|
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|
622 |
by (rtac lemma_simp_induct 1); |
1196 | 623 |
by (eresolve_tac [bij_is_fun RS surj_image RS surj_is_fun] 2); |
624 |
by (eresolve_tac [eqpoll_imp_lepoll RS lepoll_infinite RS |
|
1461 | 625 |
infinite_Card_is_InfCard RS InfCard_is_Limit] 2 |
626 |
THEN REPEAT (assume_tac 2)); |
|
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|
627 |
by (etac recfunAC16_mono 2); |
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|
628 |
by (rtac main_induct 1 |
1461 | 629 |
THEN REPEAT (ares_tac [eqpoll_imp_lepoll RS lepoll_infinite] 1)); |
1196 | 630 |
qed "WO2_AC16"; |