author | paulson |
Tue, 26 Jun 2001 16:54:39 +0200 | |
changeset 11380 | e76366922751 |
parent 11317 | 7f9e4c389318 |
permissions | -rw-r--r-- |
1461 | 1 |
(* Title: ZF/AC/AC15_WO6.ML |
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ID: $Id$ |
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Author: Krzysztof Grabczewski |
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|
11380 | 5 |
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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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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Goalw [lepoll_def] "A\\<noteq>0 ==> B lepoll A*B"; |
|
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by (etac not_emptyE 1); |
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by (res_inst_tac [("x","\\<lambda>z \\<in> 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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Goal "0\\<notin>A ==> \\<forall>B \\<in> {cons(0,x*nat). x \\<in> 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 \\<in> A |] ==> \\<exists>B \\<in> C. a \\<in> B & B \\<subseteq> 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 \\<in> A; C \\<in> A; a \\<in> B; a \\<in> 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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"\\<forall>B \\<in> {cons(0, x*nat). x \\<in> 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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\ ==> \\<forall>B \\<in> A. \\<exists>! u. u \\<in> f`cons(0, B*nat) & u \\<subseteq> cons(0, B*nat) & \ |
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\ 0 \\<in> 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 \\<in> A} lepoll i"; |
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by (etac exE 1); |
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by (res_inst_tac |
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[("x", "\\<lambda>x \\<in> RepFun(A, P). LEAST j. \\<exists>a \\<in> 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 \\<in> nat; B \\<in> A; u(B) \\<subseteq> cons(0, B*nat); 0 \\<in> u(B); 2 lepoll u(B); \ |
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\ u(B) lepoll succ(n) |] \ |
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\ ==> (\\<lambda>x \\<in> A. {fst(x). x \\<in> u(x)-{0}})`B \\<noteq> 0 & \ |
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\ (\\<lambda>x \\<in> A. {fst(x). x \\<in> u(x)-{0}})`B \\<subseteq> B & \ |
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\ (\\<lambda>x \\<in> A. {fst(x). x \\<in> 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 "[| \\<exists>f. \\<forall>B \\<in> {cons(0, x*nat). x \\<in> 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 \\<in> nat |] \ |
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\ ==> \\<exists>f. \\<forall>B \\<in> A. f`B \\<noteq> 0 & f`B \\<subseteq> 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 \\<in> 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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Goal "Ord(x) ==> (\\<Union>a<x. F(a)) = (\\<Union>a \\<in> x. F(a))"; |
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by (fast_tac (claset() addSIs [ltI] addSDs [ltD]) 1); |
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qed "OUN_eq_UN"; |
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val [prem] = goal thy "\\<forall>x \\<in> Pow(A)-{0}. f`x\\<noteq>0 & f`x \\<subseteq> x & f`x lepoll m ==> \ |
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\ (\\<Union>i<LEAST x. HH(f,A,x)={A}. HH(f,A,i)) = A"; |
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by (simp_tac (simpset() addsimps [Ord_Least RS OUN_eq_UN]) 1); |
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by (rtac equalityI 1); |
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by (fast_tac (claset() addSDs [less_Least_subset_x]) 1); |
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by (fast_tac (claset() addSDs [prem RS bspec] |
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addSIs [f_subsets_imp_UN_HH_eq_x RS (Diff_eq_0_iff RS iffD1)]) 1); |
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val lemma1 = result(); |
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val [prem] = goal thy "\\<forall>x \\<in> Pow(A)-{0}. f`x\\<noteq>0 & f`x \\<subseteq> x & f`x lepoll m ==> \ |
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\ \\<forall>x<LEAST x. HH(f,A,x)={A}. HH(f,A,x) lepoll m"; |
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by (rtac oallI 1); |
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by (dresolve_tac [ltD RS less_Least_subset_x] 1); |
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by (ftac HH_subset_imp_eq 1); |
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by (etac ssubst 1); |
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by (fast_tac (claset() addIs [prem RS ballE] |
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addSDs [HH_subset_x_imp_subset_Diff_UN RS not_emptyI2]) 1); |
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val lemma2 = result(); |
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825877190618
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5137
diff
changeset
|
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Goalw [AC15_def, WO6_def] "AC15 ==> WO6"; |
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by (rtac allI 1); |
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by (eres_inst_tac [("x","Pow(A)-{0}")] allE 1); |
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by (etac impE 1); |
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by (Fast_tac 1); |
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by (REPEAT (eresolve_tac [bexE,conjE,exE] 1)); |
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by (rtac bexI 1 THEN (assume_tac 2)); |
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by (rtac conjI 1 THEN (assume_tac 1)); |
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by (res_inst_tac [("x","LEAST i. HH(f,A,i)={A}")] exI 1); |
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by (res_inst_tac [("x","\\<lambda>j \\<in> (LEAST i. HH(f,A,i)={A}). HH(f,A,j)")] exI 1); |
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by (Asm_full_simp_tac 1); |
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by (fast_tac (claset() addSIs [Ord_Least, lam_type RS domain_of_fun] |
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addSEs [less_Least_subset_x, lemma1, lemma2]) 1); |
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qed "AC15_WO6"; |
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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 \\<in> *) |
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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 \\<in> 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","\\<lambda>x \\<in> 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 \\<subseteq> 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 \\<subseteq> n *) |
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(* ********************************************************************** *) |
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Goalw AC_defs "[| n \\<in> 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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(* 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\\<noteq>0; A lepoll 1 |] ==> \\<exists>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 "\\<forall>B \\<in> A. f(B)\\<noteq>0 & f(B)<=B & f(B) lepoll 1 \ |
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\ ==> (\\<lambda>x \\<in> A. THE y. f(x)={y}) \\<in> (\\<Pi>X \\<in> 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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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"; |