author | paulson |
Thu, 13 Jul 2000 13:05:58 +0200 | |
changeset 9305 | 3dfae8f90dcf |
parent 6070 | 032babd0120b |
child 11317 | 7f9e4c389318 |
permissions | -rw-r--r-- |
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(* Title: ZF/AC/AC10_AC15.ML |
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ID: $Id$ |
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Author: Krzysztof Grabczewski |
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The proofs needed to state that AC10, ..., AC15 are equivalent to the rest. |
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We need the following: |
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WO1 ==> AC10(n) ==> AC11 ==> AC12 ==> AC15 ==> WO6 |
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In order to add the formulations AC13 and AC14 we need: |
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AC10(succ(n)) ==> AC13(n) ==> AC14 ==> AC15 |
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or |
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AC1 ==> AC13(1); AC13(m) ==> AC13(n) ==> AC14 ==> AC15 (m le n) |
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|
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So we don't have to prove all implications of both cases. |
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Moreover we don't need to prove AC13(1) ==> AC1 and AC11 ==> AC14 as |
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Rubin & Rubin do. |
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*) |
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(* ********************************************************************** *) |
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(* Lemmas used in the proofs in which the conclusion is AC13, AC14 *) |
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(* or AC15 *) |
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(* - cons_times_nat_not_Finite *) |
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(* - ex_fun_AC13_AC15 *) |
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(* ********************************************************************** *) |
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||
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Goalw [lepoll_def] "A~=0 ==> B lepoll A*B"; |
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by (etac not_emptyE 1); |
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by (res_inst_tac [("x","lam z:B. <x,z>")] exI 1); |
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by (fast_tac (claset() addSIs [snd_conv, lam_injective]) 1); |
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qed "lepoll_Sigma"; |
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|
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Goal "0~:A ==> ALL B:{cons(0,x*nat). x:A}. ~Finite(B)"; |
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by (rtac ballI 1); |
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by (etac RepFunE 1); |
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by (hyp_subst_tac 1); |
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by (rtac notI 1); |
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by (dresolve_tac [subset_consI RS subset_imp_lepoll RS lepoll_Finite] 1); |
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by (resolve_tac [lepoll_Sigma RS lepoll_Finite RS (nat_not_Finite RS notE)] 1 |
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THEN (assume_tac 2)); |
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by (Fast_tac 1); |
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qed "cons_times_nat_not_Finite"; |
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Goal "[| Union(C)=A; a:A |] ==> EX B:C. a:B & B <= A"; |
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by (Fast_tac 1); |
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val lemma1 = result(); |
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Goalw [pairwise_disjoint_def] |
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"[| pairwise_disjoint(A); B:A; C:A; a:B; a:C |] ==> B=C"; |
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by (dtac IntI 1 THEN (assume_tac 1)); |
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by (dres_inst_tac [("A","B Int C")] not_emptyI 1); |
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by (Fast_tac 1); |
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val lemma2 = result(); |
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Goalw [sets_of_size_between_def] |
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"ALL B:{cons(0, x*nat). x:A}. pairwise_disjoint(f`B) & \ |
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\ sets_of_size_between(f`B, 2, n) & Union(f`B)=B \ |
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\ ==> ALL B:A. EX! u. u:f`cons(0, B*nat) & u <= cons(0, B*nat) & \ |
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\ 0:u & 2 lepoll u & u lepoll n"; |
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by (rtac ballI 1); |
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by (etac ballE 1); |
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by (Fast_tac 2); |
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by (REPEAT (etac conjE 1)); |
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by (dresolve_tac [consI1 RSN (2, lemma1)] 1); |
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by (etac bexE 1); |
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by (rtac ex1I 1); |
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by (Fast_tac 1); |
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by (REPEAT (etac conjE 1)); |
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by (rtac lemma2 1 THEN (REPEAT (assume_tac 1))); |
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val lemma3 = result(); |
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Goalw [lepoll_def] "[| A lepoll i; Ord(i) |] ==> {P(a). a:A} lepoll i"; |
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by (etac exE 1); |
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by (res_inst_tac |
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[("x", "lam x:RepFun(A, P). LEAST j. EX a:A. x=P(a) & f`a=j")] exI 1); |
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by (res_inst_tac [("d", "%y. P(converse(f)`y)")] lam_injective 1); |
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by (etac RepFunE 1); |
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by (forward_tac [inj_is_fun RS apply_type] 1 THEN (assume_tac 1)); |
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by (fast_tac (claset() addIs [LeastI2] |
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addSEs [Ord_in_Ord, inj_is_fun RS apply_type]) 1); |
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by (etac RepFunE 1); |
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by (rtac LeastI2 1); |
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by (Fast_tac 1); |
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by (fast_tac (claset() addSEs [Ord_in_Ord, inj_is_fun RS apply_type]) 1); |
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by (fast_tac (claset() addEs [sym, left_inverse RS ssubst]) 1); |
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val lemma4 = result(); |
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Goal "[| n:nat; B:A; u(B) <= cons(0, B*nat); 0:u(B); 2 lepoll u(B); \ |
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\ u(B) lepoll succ(n) |] \ |
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\ ==> (lam x:A. {fst(x). x:u(x)-{0}})`B ~= 0 & \ |
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\ (lam x:A. {fst(x). x:u(x)-{0}})`B <= B & \ |
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\ (lam x:A. {fst(x). x:u(x)-{0}})`B lepoll n"; |
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by (Asm_simp_tac 1); |
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by (rtac conjI 1); |
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by (fast_tac (empty_cs addSDs [RepFun_eq_0_iff RS iffD1] |
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addDs [lepoll_Diff_sing] |
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addEs [lepoll_trans RS succ_lepoll_natE, ssubst] |
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addSIs [notI, lepoll_refl, nat_0I]) 1); |
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by (rtac conjI 1); |
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by (fast_tac (claset() addSIs [fst_type] addSEs [consE]) 1); |
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by (fast_tac (claset() addSEs [equalityE, |
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Diff_lepoll RS (nat_into_Ord RSN (2, lemma4))]) 1); |
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val lemma5 = result(); |
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Goal "[| EX f. ALL B:{cons(0, x*nat). x:A}. \ |
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\ pairwise_disjoint(f`B) & \ |
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\ sets_of_size_between(f`B, 2, succ(n)) & \ |
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\ Union(f`B)=B; n:nat |] \ |
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\ ==> EX f. ALL B:A. f`B ~= 0 & f`B <= B & f`B lepoll n"; |
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by (fast_tac (empty_cs addSDs [lemma3, theI] addDs [bspec] |
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addSEs [exE, conjE] |
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addIs [exI, ballI, lemma5]) 1); |
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qed "ex_fun_AC13_AC15"; |
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(* ********************************************************************** *) |
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(* The target proofs *) |
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(* ********************************************************************** *) |
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(* ********************************************************************** *) |
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(* AC10(n) ==> AC11 *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "[| n:nat; 1 le n; AC10(n) |] ==> AC11"; |
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by (rtac bexI 1 THEN (assume_tac 2)); |
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by (Fast_tac 1); |
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qed "AC10_AC11"; |
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(* ********************************************************************** *) |
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(* AC11 ==> AC12 *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "AC11 ==> AC12"; |
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by (fast_tac (FOL_cs addSEs [bexE] addIs [bexI]) 1); |
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qed "AC11_AC12"; |
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(* ********************************************************************** *) |
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(* AC12 ==> AC15 *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "AC12 ==> AC15"; |
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by Safe_tac; |
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by (etac allE 1); |
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by (etac impE 1); |
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by (etac cons_times_nat_not_Finite 1); |
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by (fast_tac (claset() addSIs [ex_fun_AC13_AC15]) 1); |
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qed "AC12_AC15"; |
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(* ********************************************************************** *) |
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(* AC15 ==> WO6 *) |
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(* ********************************************************************** *) |
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(* in a separate file *) |
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(* ********************************************************************** *) |
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(* The proof needed in the first case, not in the second *) |
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(* ********************************************************************** *) |
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(* ********************************************************************** *) |
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(* AC10(n) ==> AC13(n-1) if 2 le n *) |
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(* *) |
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(* Because of the change to the formal definition of AC10(n) we prove *) |
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(* the following obviously equivalent theorem : *) |
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(* AC10(n) implies AC13(n) for (1 le n) *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "[| n:nat; 1 le n; AC10(n) |] ==> AC13(n)"; |
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by Safe_tac; |
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by (fast_tac (empty_cs addSEs [allE, cons_times_nat_not_Finite RSN (2, impE), |
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ex_fun_AC13_AC15]) 1); |
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qed "AC10_AC13"; |
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(* ********************************************************************** *) |
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(* The proofs needed in the second case, not in the first *) |
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(* ********************************************************************** *) |
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(* ********************************************************************** *) |
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(* AC1 ==> AC13(1) *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "AC1 ==> AC13(1)"; |
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by (rtac allI 1); |
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by (etac allE 1); |
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by (rtac impI 1); |
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by (mp_tac 1); |
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by (etac exE 1); |
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by (res_inst_tac [("x","lam x:A. {f`x}")] exI 1); |
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by (asm_simp_tac (simpset() addsimps |
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[singleton_eqpoll_1 RS eqpoll_imp_lepoll, |
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singletonI RS not_emptyI]) 1); |
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qed "AC1_AC13"; |
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(* ********************************************************************** *) |
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(* AC13(m) ==> AC13(n) for m <= n *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "[| m le n; AC13(m) |] ==> AC13(n)"; |
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by (dtac le_imp_lepoll 1); |
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by (fast_tac (claset() addSEs [lepoll_trans]) 1); |
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qed "AC13_mono"; |
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(* ********************************************************************** *) |
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(* The proofs necessary for both cases *) |
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(* ********************************************************************** *) |
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(* ********************************************************************** *) |
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(* AC13(n) ==> AC14 if 1 <= n *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "[| n:nat; 1 le n; AC13(n) |] ==> AC14"; |
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by (fast_tac (FOL_cs addIs [bexI]) 1); |
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qed "AC13_AC14"; |
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(* ********************************************************************** *) |
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(* AC14 ==> AC15 *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "AC14 ==> AC15"; |
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by (Fast_tac 1); |
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qed "AC14_AC15"; |
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|
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(* ********************************************************************** *) |
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(* The redundant proofs; however cited by Rubin & Rubin *) |
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(* ********************************************************************** *) |
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(* ********************************************************************** *) |
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(* AC13(1) ==> AC1 *) |
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(* ********************************************************************** *) |
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Goal "[| A~=0; A lepoll 1 |] ==> EX a. A={a}"; |
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by (fast_tac (claset() addSEs [not_emptyE, lepoll_1_is_sing]) 1); |
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qed "lemma_aux"; |
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Goal "ALL B:A. f(B)~=0 & f(B)<=B & f(B) lepoll 1 \ |
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\ ==> (lam x:A. THE y. f(x)={y}) : (PROD X:A. X)"; |
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by (rtac lam_type 1); |
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by (dtac bspec 1 THEN (assume_tac 1)); |
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by (REPEAT (etac conjE 1)); |
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by (eresolve_tac [lemma_aux RS exE] 1 THEN (assume_tac 1)); |
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by (asm_full_simp_tac (simpset() addsimps [the_element]) 1); |
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val lemma = result(); |
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Goalw AC_defs "AC13(1) ==> AC1"; |
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by (fast_tac (claset() addSEs [lemma]) 1); |
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qed "AC13_AC1"; |
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(* ********************************************************************** *) |
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(* AC11 ==> AC14 *) |
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(* ********************************************************************** *) |
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||
5147
825877190618
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5137
diff
changeset
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253 |
Goalw [AC11_def, AC14_def] "AC11 ==> AC14"; |
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by (fast_tac (claset() addSIs [AC10_AC13]) 1); |
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qed "AC11_AC14"; |