author | paulson |
Tue, 04 Aug 1998 16:05:19 +0200 | |
changeset 5241 | e5129172cb2b |
parent 5137 | 60205b0de9b9 |
child 5265 | 9d1d4c43c76d |
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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(* Case of finite ordinals *) |
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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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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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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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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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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)); |
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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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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 |
|
5068 | 306 |
Goal |
1196 | 307 |
"!!a. [| recfunAC16(f, fa, y, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)}; \ |
1461 | 308 |
\ Card(a); ~ Finite(a); A eqpoll a; \ |
309 |
\ k : nat; m : nat; y<a; \ |
|
310 |
\ fa : bij(a, {Y: Pow(A). Y eqpoll succ(k)}) |] \ |
|
311 |
\ ==> A - Union(recfunAC16(f, fa, y, a)) - fa`y eqpoll a"; |
|
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312 |
by (rtac dbl_Diff_eqpoll_Card 1 THEN (TRYALL assume_tac)); |
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313 |
by (rtac Union_recfunAC16_lesspoll 1 THEN (REPEAT (assume_tac 1))); |
1196 | 314 |
by (eresolve_tac [add_type RS nat_succI] 1 THEN (assume_tac 1)); |
315 |
by (resolve_tac [nat_succI RSN (2, bexI RS (Finite_def RS def_imp_iff RS |
|
1461 | 316 |
iffD2)) RS (Card_is_Ord RSN (3, Finite_lesspoll_infinite_Ord))] 1 |
317 |
THEN (TRYALL assume_tac)); |
|
1196 | 318 |
by (eresolve_tac [ltD RSN (2, bij_is_fun RS apply_type) RS CollectE] 1 |
1461 | 319 |
THEN (TRYALL assume_tac)); |
3731 | 320 |
qed "dbl_Diff_eqpoll"; |
1196 | 321 |
|
322 |
(* back to the proof *) |
|
323 |
||
324 |
val disj_Un_eqpoll_nat_sum = disj_Un_eqpoll_sum RS ( |
|
325 |
sum_eqpoll_cong RSN (2, |
|
326 |
nat_sum_eqpoll_sum RSN (3, |
|
327 |
eqpoll_trans RS eqpoll_trans))) |> standard; |
|
328 |
||
5137 | 329 |
|
330 |
Goal "[| x : Pow(A - B - fa`i); x eqpoll m; \ |
|
1461 | 331 |
\ fa : bij(a, {x: Pow(A) . x eqpoll k}); i<a; k:nat; m:nat |] \ |
332 |
\ ==> fa ` i Un x : {x: Pow(A) . x eqpoll k #+ m}"; |
|
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333 |
by (rtac CollectI 1); |
4091 | 334 |
by (fast_tac (claset() addSIs [PowD RS (PowD RSN (2, Un_least RS PowI))] |
1461 | 335 |
addSEs [ltD RSN (2, bij_is_fun RS apply_type RS CollectE)]) 1); |
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336 |
by (rtac disj_Un_eqpoll_nat_sum 1 |
1461 | 337 |
THEN (TRYALL assume_tac)); |
4091 | 338 |
by (fast_tac (claset() addSIs [equals0I]) 1); |
1196 | 339 |
by (eresolve_tac [ltD RSN (2, bij_is_fun RS apply_type RS CollectE)] 1 |
1461 | 340 |
THEN (REPEAT (assume_tac 1))); |
3731 | 341 |
qed "Un_in_Collect"; |
1196 | 342 |
|
343 |
(* ********************************************************************** *) |
|
1461 | 344 |
(* Lemmas simplifying assumptions *) |
1196 | 345 |
(* ********************************************************************** *) |
346 |
||
5137 | 347 |
|
348 |
Goal "[| ALL y<succ(j). F(y)<=X & (ALL x<a. x<y | P(x,y) \ |
|
1461 | 349 |
\ --> Q(x,y)); succ(j)<a |] \ |
350 |
\ ==> F(j)<=X & (ALL x<a. x<j | P(x,j) --> Q(x,j))"; |
|
4152 | 351 |
by (dtac ospec 1); |
2469 | 352 |
by (resolve_tac [lt_Ord RS (succI1 RS ltI RS lt_Ord RS le_refl)] 1 |
1461 | 353 |
THEN (REPEAT (assume_tac 1))); |
1196 | 354 |
val lemma6 = result(); |
355 |
||
5137 | 356 |
|
357 |
Goal "[| ALL x<a. x<j | P(x,j) --> Q(x,j); succ(j)<a |] \ |
|
358 |
\ ==> P(j,j) --> (ALL x<a. x le j | P(x,j) --> Q(x,j))"; |
|
4091 | 359 |
by (fast_tac (claset() addSEs [leE]) 1); |
1196 | 360 |
val lemma7 = result(); |
361 |
||
362 |
(* ********************************************************************** *) |
|
363 |
(* Lemmas needded to prove ex_next_set which means that for any successor *) |
|
1461 | 364 |
(* ordinal there is a set satisfying certain properties *) |
1196 | 365 |
(* ********************************************************************** *) |
366 |
||
5137 | 367 |
|
368 |
Goal "[| A eqpoll a; ~ Finite(a); Ord(a); m:nat |] \ |
|
1461 | 369 |
\ ==> EX X:Pow(A). X eqpoll m"; |
1196 | 370 |
by (eresolve_tac [Ord_nat RSN (2, ltI) RS |
1461 | 371 |
(nat_le_infinite_Ord RSN (2, lt_trans2)) RS |
372 |
leI RS le_imp_lepoll RS |
|
373 |
((eqpoll_sym RS eqpoll_imp_lepoll) RSN (2, lepoll_trans)) RS |
|
374 |
lepoll_imp_eqpoll_subset RS exE] 1 |
|
375 |
THEN REPEAT (assume_tac 1)); |
|
4091 | 376 |
by (fast_tac (claset() addSEs [eqpoll_sym]) 1); |
3731 | 377 |
qed "ex_subset_eqpoll"; |
1196 | 378 |
|
5137 | 379 |
|
380 |
Goal "[| A <= B Un C; A Int C = 0 |] ==> A <= B"; |
|
5241 | 381 |
by (fast_tac (claset() addEs [equals0E]) 1); |
3731 | 382 |
qed "subset_Un_disjoint"; |
1196 | 383 |
|
5137 | 384 |
|
385 |
Goal "[| X:Pow(A - Union(B) -C); T:B; F<=T |] ==> F Int X = 0"; |
|
4091 | 386 |
by (fast_tac (claset() addSIs [equals0I]) 1); |
3731 | 387 |
qed "Int_empty"; |
1196 | 388 |
|
389 |
(* ********************************************************************** *) |
|
1461 | 390 |
(* equipollent subset (and finite) is the whole set *) |
1196 | 391 |
(* ********************************************************************** *) |
392 |
||
5137 | 393 |
|
394 |
Goal "[| A <= B; a : A; A - {a} = B - {a} |] ==> A = B"; |
|
4091 | 395 |
by (fast_tac (claset() addSEs [equalityE]) 1); |
3731 | 396 |
qed "Diffs_eq_imp_eq"; |
1196 | 397 |
|
5137 | 398 |
|
399 |
Goal "m:nat ==> ALL A B. A <= B & m lepoll A & B lepoll m --> A=B"; |
|
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400 |
by (etac nat_induct 1); |
4091 | 401 |
by (fast_tac (claset() addSDs [lepoll_0_is_0]) 1); |
1196 | 402 |
by (REPEAT (resolve_tac [allI, impI] 1)); |
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|
403 |
by (REPEAT (etac conjE 1)); |
1196 | 404 |
by (resolve_tac [succ_lepoll_imp_not_empty RS not_emptyE] 1 |
1461 | 405 |
THEN (assume_tac 1)); |
1196 | 406 |
by (forward_tac [subsetD RS Diff_sing_lepoll] 1 |
1461 | 407 |
THEN REPEAT (assume_tac 1)); |
1196 | 408 |
by (forward_tac [lepoll_Diff_sing] 1); |
409 |
by (REPEAT (eresolve_tac [allE, impE] 1)); |
|
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|
410 |
by (rtac conjI 1); |
2469 | 411 |
by (Fast_tac 2); |
412 |
by (Fast_tac 1); |
|
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|
413 |
by (etac Diffs_eq_imp_eq 1 |
1461 | 414 |
THEN REPEAT (assume_tac 1)); |
3731 | 415 |
qed "subset_imp_eq_lemma"; |
1196 | 416 |
|
5137 | 417 |
|
418 |
Goal "[| A <= B; m lepoll A; B lepoll m; m:nat |] ==> A=B"; |
|
1196 | 419 |
by (fast_tac (FOL_cs addSDs [subset_imp_eq_lemma]) 1); |
3731 | 420 |
qed "subset_imp_eq"; |
1196 | 421 |
|
5137 | 422 |
|
423 |
Goal "[| f:bij(a, {Y:X. Y eqpoll succ(k)}); k:nat; f`b<=f`y; b<a; \ |
|
1461 | 424 |
\ y<a |] ==> b=y"; |
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|
425 |
by (dtac subset_imp_eq 1); |
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|
426 |
by (etac nat_succI 3); |
4091 | 427 |
by (fast_tac (claset() addSEs [bij_is_fun RS (ltD RSN (2, apply_type)) RS |
1461 | 428 |
CollectE, eqpoll_sym RS eqpoll_imp_lepoll]) 1); |
4091 | 429 |
by (fast_tac (claset() addSEs [bij_is_fun RS (ltD RSN (2, apply_type)) RS |
1461 | 430 |
CollectE, eqpoll_imp_lepoll]) 1); |
1196 | 431 |
by (rewrite_goals_tac [bij_def, inj_def]); |
4091 | 432 |
by (fast_tac (claset() addSDs [ltD]) 1); |
3731 | 433 |
qed "bij_imp_arg_eq"; |
1196 | 434 |
|
5137 | 435 |
|
5068 | 436 |
Goal |
1196 | 437 |
"!!a. [| recfunAC16(f, fa, y, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)}; \ |
1461 | 438 |
\ Card(a); ~ Finite(a); A eqpoll a; \ |
439 |
\ k : nat; m : nat; y<a; \ |
|
440 |
\ fa : bij(a, {Y: Pow(A). Y eqpoll succ(k)}); \ |
|
441 |
\ ~ (EX Y:recfunAC16(f, fa, y, a). fa`y <= Y) |] \ |
|
442 |
\ ==> EX X:{Y: Pow(A). Y eqpoll succ(k #+ m)}. fa`y <= X & \ |
|
443 |
\ (ALL b<a. fa`b <= X --> \ |
|
444 |
\ (ALL T:recfunAC16(f, fa, y, a). ~ fa`b <= T))"; |
|
1196 | 445 |
by (eresolve_tac [dbl_Diff_eqpoll RS ex_subset_eqpoll RS bexE] 1 |
1461 | 446 |
THEN REPEAT (assume_tac 1)); |
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|
447 |
by (etac Card_is_Ord 1); |
1196 | 448 |
by (forward_tac [Un_in_Collect] 2 THEN REPEAT (assume_tac 2)); |
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|
449 |
by (etac CollectE 4); |
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|
450 |
by (rtac bexI 4); |
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|
451 |
by (rtac CollectI 5); |
1196 | 452 |
by (assume_tac 5); |
453 |
by (eresolve_tac [add_succ RS subst] 5); |
|
454 |
by (assume_tac 1); |
|
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|
455 |
by (etac nat_succI 1); |
1196 | 456 |
by (assume_tac 1); |
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|
457 |
by (rtac conjI 1); |
2469 | 458 |
by (Fast_tac 1); |
1196 | 459 |
by (REPEAT (resolve_tac [ballI, impI, oallI, notI] 1)); |
460 |
by (dresolve_tac [Int_empty RSN (2, subset_Un_disjoint)] 1 |
|
1461 | 461 |
THEN REPEAT (assume_tac 1)); |
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|
462 |
by (dtac bij_imp_arg_eq 1 THEN REPEAT (assume_tac 1)); |
1196 | 463 |
by (hyp_subst_tac 1); |
464 |
by (eresolve_tac [bexI RSN (2, notE)] 1 THEN TRYALL assume_tac); |
|
3731 | 465 |
qed "ex_next_set"; |
1196 | 466 |
|
467 |
(* ********************************************************************** *) |
|
1461 | 468 |
(* Lemma ex_next_Ord states that for any successor *) |
469 |
(* ordinal there is a number of the set satisfying certain properties *) |
|
1196 | 470 |
(* ********************************************************************** *) |
471 |
||
5137 | 472 |
|
5068 | 473 |
Goal |
1196 | 474 |
"!!a. [| recfunAC16(f, fa, y, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)}; \ |
1461 | 475 |
\ Card(a); ~ Finite(a); A eqpoll a; \ |
476 |
\ k : nat; m : nat; y<a; \ |
|
477 |
\ fa : bij(a, {Y: Pow(A). Y eqpoll succ(k)}); \ |
|
478 |
\ f : bij(a, {Y: Pow(A). Y eqpoll succ(k #+ m)}); \ |
|
479 |
\ ~ (EX Y:recfunAC16(f, fa, y, a). fa`y <= Y) |] \ |
|
480 |
\ ==> EX c<a. fa`y <= f`c & \ |
|
481 |
\ (ALL b<a. fa`b <= f`c --> \ |
|
482 |
\ (ALL T:recfunAC16(f, fa, y, a). ~ fa`b <= T))"; |
|
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|
483 |
by (dtac ex_next_set 1 THEN REPEAT (assume_tac 1)); |
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|
484 |
by (etac bexE 1); |
1196 | 485 |
by (resolve_tac [bij_converse_bij RS bij_is_fun RS apply_type RS ltI RSN |
1461 | 486 |
(2, oexI)] 1); |
1196 | 487 |
by (resolve_tac [right_inverse_bij RS ssubst] 1 |
1461 | 488 |
THEN REPEAT (ares_tac [Card_is_Ord] 1)); |
3731 | 489 |
qed "ex_next_Ord"; |
1196 | 490 |
|
5137 | 491 |
|
492 |
Goal "[| EX! Y. Y:Z & P(Y); ~P(W) |] ==> EX! Y. Y: (Z Un {W}) & P(Y)"; |
|
2469 | 493 |
by (Fast_tac 1); |
3731 | 494 |
qed "ex1_in_Un_sing"; |
1196 | 495 |
|
496 |
(* ********************************************************************** *) |
|
1461 | 497 |
(* Lemma simplifying assumptions *) |
1196 | 498 |
(* ********************************************************************** *) |
499 |
||
5137 | 500 |
|
501 |
Goal "[| ALL x<a. x<j | (EX xa:F(j). P(x, xa)) \ |
|
1461 | 502 |
\ --> (EX! Y. Y:F(j) & P(x, Y)); F(j) <= X; \ |
503 |
\ L : X; P(j, L) & (ALL x<a. P(x, L) --> (ALL xa:F(j). ~P(x, xa))) |] \ |
|
504 |
\ ==> F(j) Un {L} <= X & \ |
|
2845 | 505 |
\ (ALL x<a. x le j | (EX xa: (F(j) Un {L}). P(x, xa)) --> \ |
506 |
\ (EX! Y. Y: (F(j) Un {L}) & P(x, Y)))"; |
|
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|
507 |
by (rtac conjI 1); |
4091 | 508 |
by (fast_tac (claset() addSIs [singleton_subsetI]) 1); |
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|
509 |
by (rtac oallI 1); |
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|
510 |
by (etac oallE 1 THEN (contr_tac 2)); |
5137 | 511 |
by (blast_tac (claset() addSEs [leE]) 1); |
1196 | 512 |
val lemma8 = result(); |
513 |
||
514 |
(* ********************************************************************** *) |
|
515 |
(* The main part of the proof: inductive proof of the property of T_gamma *) |
|
1461 | 516 |
(* lemma main_induct *) |
1196 | 517 |
(* ********************************************************************** *) |
518 |
||
5137 | 519 |
|
520 |
Goal "[| b < a; f : bij(a, {Y: Pow(A) . Y eqpoll succ(k #+ m)}); \ |
|
1461 | 521 |
\ fa : bij(a, {Y: Pow(A) . Y eqpoll succ(k)}); \ |
522 |
\ ~Finite(a); Card(a); A eqpoll a; k : nat; m : nat |] \ |
|
523 |
\ ==> recfunAC16(f, fa, b, a) <= {X: Pow(A) . X eqpoll succ(k #+ m)} & \ |
|
524 |
\ (ALL x<a. x < b | (EX Y:recfunAC16(f, fa, b, a). fa ` x <= Y) --> \ |
|
525 |
\ (EX! Y. Y:recfunAC16(f, fa, b, a) & fa ` x <= Y))"; |
|
2469 | 526 |
by (etac lt_induct 1); |
527 |
by (forward_tac [lt_Ord] 1); |
|
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|
528 |
by (etac Ord_cases 1); |
1196 | 529 |
(* case 0 *) |
4091 | 530 |
by (asm_simp_tac (simpset() addsimps [recfunAC16_0]) 1); |
1196 | 531 |
(* case Limit *) |
4091 | 532 |
by (asm_simp_tac (simpset() addsimps [recfunAC16_Limit]) 2); |
2469 | 533 |
by (rtac lemma5 2 THEN (REPEAT (assume_tac 2))); |
1196 | 534 |
by (fast_tac (FOL_cs addSEs [recfunAC16_mono]) 2); |
535 |
(* case succ *) |
|
536 |
by (hyp_subst_tac 1); |
|
537 |
by (eresolve_tac [lemma6 RS conjE] 1 THEN (assume_tac 1)); |
|
5137 | 538 |
by (asm_simp_tac (simpset() delsplits [split_if] |
539 |
addsimps [recfunAC16_succ]) 1); |
|
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|
540 |
by (resolve_tac [conjI RS (split_if RS iffD2)] 1); |
5137 | 541 |
by (Asm_simp_tac 1); |
542 |
by (etac lemma7 1 THEN assume_tac 1); |
|
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|
543 |
by (rtac impI 1); |
1196 | 544 |
by (resolve_tac [ex_next_Ord RS oexE] 1 |
5137 | 545 |
THEN REPEAT (ares_tac [le_refl RS lt_trans] 1)); |
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|
546 |
by (etac lemma8 1 THEN (assume_tac 1)); |
1196 | 547 |
by (resolve_tac [bij_is_fun RS apply_type] 1 THEN (assume_tac 1)); |
548 |
by (eresolve_tac [Least_le RS lt_trans2 RS ltD] 1 |
|
1461 | 549 |
THEN REPEAT (ares_tac [lt_Ord, succ_leI] 1)); |
5137 | 550 |
(*SLOW! 5 secs?*) |
1196 | 551 |
by (eresolve_tac [lt_Ord RSN (2, LeastI)] 1 THEN (assume_tac 1)); |
3731 | 552 |
qed "main_induct"; |
1196 | 553 |
|
554 |
(* ********************************************************************** *) |
|
1461 | 555 |
(* Lemma to simplify the inductive proof *) |
1200 | 556 |
(* - the desired property is a consequence of the inductive assumption *) |
1196 | 557 |
(* ********************************************************************** *) |
558 |
||
559 |
val [prem1, prem2, prem3, prem4] = goal thy |
|
1461 | 560 |
"[| (!!b. b<a ==> F(b) <= S & (ALL x<a. (x<b | (EX Y:F(b). f`x <= Y)) \ |
561 |
\ --> (EX! Y. Y : F(b) & f`x <= Y))); \ |
|
562 |
\ f:a->f``(a); Limit(a); (!!i j. i le j ==> F(i) <= F(j)) |] \ |
|
563 |
\ ==> (UN j<a. F(j)) <= S & \ |
|
564 |
\ (ALL x:f``a. EX! Y. Y : (UN j<a. F(j)) & x <= Y)"; |
|
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|
565 |
by (rtac conjI 1); |
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|
566 |
by (rtac subsetI 1); |
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|
567 |
by (etac OUN_E 1); |
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|
568 |
by (dtac prem1 1); |
2469 | 569 |
by (Fast_tac 1); |
2493 | 570 |
(** LEVEL 5 **) |
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|
571 |
by (rtac ballI 1); |
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|
572 |
by (etac imageE 1); |
1196 | 573 |
by (dresolve_tac [prem3 RS Limit_is_Ord RSN (2, ltI) RS |
1461 | 574 |
(prem3 RS Limit_has_succ)] 1); |
1196 | 575 |
by (forward_tac [prem1] 1); |
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|
576 |
by (etac conjE 1); |
2493 | 577 |
(** LEVEL 10 **) |
1196 | 578 |
by (dresolve_tac [leI RS succ_leE RSN (2, ospec)] 1 THEN (assume_tac 1)); |
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|
579 |
by (etac impE 1); |
4091 | 580 |
by (fast_tac (claset() addSEs [leI RS succ_leE RS lt_Ord RS le_refl]) 1); |
1196 | 581 |
by (dresolve_tac [prem2 RSN (2, apply_equality)] 1); |
582 |
by (REPEAT (eresolve_tac [conjE, ex1E] 1)); |
|
2493 | 583 |
(** LEVEL 15 **) |
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|
584 |
by (rtac ex1I 1); |
4091 | 585 |
by (fast_tac (claset() addSIs [OUN_I]) 1); |
1196 | 586 |
by (REPEAT (eresolve_tac [conjE, OUN_E] 1)); |
587 |
by (eresolve_tac [lt_Ord RSN (2, lt_Ord RS Ord_linear_le)] 1 THEN (assume_tac 1)); |
|
588 |
by (dresolve_tac [prem4 RS subsetD] 2 THEN (assume_tac 2)); |
|
2493 | 589 |
(** LEVEL 20 **) |
590 |
by (fast_tac FOL_cs 2); |
|
1196 | 591 |
by (forward_tac [prem1] 1); |
592 |
by (forward_tac [succ_leE] 1); |
|
593 |
by (dresolve_tac [prem4 RS subsetD] 1 THEN (assume_tac 1)); |
|
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|
594 |
by (etac conjE 1); |
2493 | 595 |
(** LEVEL 25 **) |
1196 | 596 |
by (dresolve_tac [lt_trans RSN (2, ospec)] 1 THEN (TRYALL assume_tac)); |
597 |
by (dresolve_tac [disjI1 RSN (2, mp)] 1 THEN (assume_tac 1)); |
|
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|
598 |
by (etac ex1_two_eq 1); |
2469 | 599 |
by (REPEAT (Fast_tac 1)); |
3731 | 600 |
qed "lemma_simp_induct"; |
1196 | 601 |
|
602 |
(* ********************************************************************** *) |
|
1461 | 603 |
(* The target theorem *) |
1196 | 604 |
(* ********************************************************************** *) |
605 |
||
5137 | 606 |
|
5068 | 607 |
Goalw [AC16_def] |
1461 | 608 |
"!!n k. [| WO2; 0<m; k:nat; m:nat |] ==> AC16(k #+ m,k)"; |
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|
609 |
by (rtac allI 1); |
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|
610 |
by (rtac impI 1); |
2493 | 611 |
by (forward_tac [WO2_infinite_subsets_eqpoll_X] 1 |
612 |
THEN (REPEAT (assume_tac 1))); |
|
1196 | 613 |
by (forw_inst_tac [("n","k #+ m")] (WO2_infinite_subsets_eqpoll_X) 1 |
1461 | 614 |
THEN (REPEAT (ares_tac [add_type] 1))); |
1196 | 615 |
by (forward_tac [WO2_imp_ex_Card] 1); |
616 |
by (REPEAT (eresolve_tac [exE,conjE] 1)); |
|
617 |
by (dresolve_tac [eqpoll_trans RS eqpoll_sym RS (eqpoll_def RS |
|
1461 | 618 |
def_imp_iff RS iffD1)] 1 THEN (assume_tac 1)); |
1196 | 619 |
by (dresolve_tac [eqpoll_trans RS eqpoll_sym RS (eqpoll_def RS |
1461 | 620 |
def_imp_iff RS iffD1)] 1 THEN (assume_tac 1)); |
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|
621 |
by (REPEAT (etac exE 1)); |
1196 | 622 |
by (res_inst_tac [("x","UN j<a. recfunAC16(fa,f,j,a)")] exI 1); |
623 |
by (res_inst_tac [("P","%z. ?Y & (ALL x:z. ?Z(x))")] |
|
1461 | 624 |
(bij_is_surj RS surj_image_eq RS subst) 1 |
625 |
THEN (assume_tac 1)); |
|
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|
626 |
by (rtac lemma_simp_induct 1); |
1196 | 627 |
by (eresolve_tac [bij_is_fun RS surj_image RS surj_is_fun] 2); |
628 |
by (eresolve_tac [eqpoll_imp_lepoll RS lepoll_infinite RS |
|
1461 | 629 |
infinite_Card_is_InfCard RS InfCard_is_Limit] 2 |
630 |
THEN REPEAT (assume_tac 2)); |
|
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|
631 |
by (etac recfunAC16_mono 2); |
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|
632 |
by (rtac main_induct 1 |
1461 | 633 |
THEN REPEAT (ares_tac [eqpoll_imp_lepoll RS lepoll_infinite] 1)); |
1196 | 634 |
qed "WO2_AC16"; |