author | paulson |
Fri, 11 Aug 2000 13:26:40 +0200 | |
changeset 9577 | 9e66e8ed8237 |
parent 187 | 8729bfdcb638 |
permissions | -rw-r--r-- |
0 | 1 |
(* Title: ZF/univ |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1992 University of Cambridge |
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The cumulative hierarchy and a small universe for recursive types |
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*) |
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open Univ; |
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(*NOT SUITABLE FOR REWRITING -- RECURSIVE!*) |
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goal Univ.thy "Vfrom(A,i) = A Un (UN j:i. Pow(Vfrom(A,j)))"; |
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by (rtac (Vfrom_def RS def_transrec RS ssubst) 1); |
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6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
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by (simp_tac ZF_ss 1); |
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val Vfrom = result(); |
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(** Monotonicity **) |
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goal Univ.thy "!!A B. A<=B ==> ALL j. i<=j --> Vfrom(A,i) <= Vfrom(B,j)"; |
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by (eps_ind_tac "i" 1); |
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by (rtac (impI RS allI) 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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by (etac Un_mono 1); |
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by (rtac UN_mono 1); |
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by (assume_tac 1); |
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by (rtac Pow_mono 1); |
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by (etac (bspec RS spec RS mp) 1); |
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by (assume_tac 1); |
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by (rtac subset_refl 1); |
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val Vfrom_mono_lemma = result(); |
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(* [| A<=B; i<=x |] ==> Vfrom(A,i) <= Vfrom(B,x) *) |
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val Vfrom_mono = standard (Vfrom_mono_lemma RS spec RS mp); |
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(** A fundamental equality: Vfrom does not require ordinals! **) |
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goal Univ.thy "Vfrom(A,x) <= Vfrom(A,rank(x))"; |
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by (eps_ind_tac "x" 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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by (fast_tac (ZF_cs addSIs [rank_lt RS ltD]) 1); |
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val Vfrom_rank_subset1 = result(); |
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goal Univ.thy "Vfrom(A,rank(x)) <= Vfrom(A,x)"; |
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by (eps_ind_tac "x" 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (rtac (subset_refl RS Un_mono) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
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by (rtac UN_least 1); |
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(*expand rank(x1) = (UN y:x1. succ(rank(y))) in assumptions*) |
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by (etac (rank RS equalityD1 RS subsetD RS UN_E) 1); |
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1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (rtac subset_trans 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (etac UN_upper 2); |
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by (rtac (subset_refl RS Vfrom_mono RS subset_trans RS Pow_mono) 1); |
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by (etac (ltI RS le_imp_subset) 1); |
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by (rtac (Ord_rank RS Ord_succ) 1); |
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by (etac bspec 1); |
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by (assume_tac 1); |
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val Vfrom_rank_subset2 = result(); |
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goal Univ.thy "Vfrom(A,rank(x)) = Vfrom(A,x)"; |
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by (rtac equalityI 1); |
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by (rtac Vfrom_rank_subset2 1); |
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by (rtac Vfrom_rank_subset1 1); |
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val Vfrom_rank_eq = result(); |
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(*** Basic closure properties ***) |
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goal Univ.thy "!!x y. y:x ==> 0 : Vfrom(A,x)"; |
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by (rtac (Vfrom RS ssubst) 1); |
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by (fast_tac ZF_cs 1); |
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val zero_in_Vfrom = result(); |
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goal Univ.thy "i <= Vfrom(A,i)"; |
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by (eps_ind_tac "i" 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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by (fast_tac ZF_cs 1); |
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val i_subset_Vfrom = result(); |
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goal Univ.thy "A <= Vfrom(A,i)"; |
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by (rtac (Vfrom RS ssubst) 1); |
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by (rtac Un_upper1 1); |
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val A_subset_Vfrom = result(); |
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goal Univ.thy "!!A a i. a <= Vfrom(A,i) ==> a: Vfrom(A,succ(i))"; |
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by (rtac (Vfrom RS ssubst) 1); |
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by (fast_tac ZF_cs 1); |
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val subset_mem_Vfrom = result(); |
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(** Finite sets and ordered pairs **) |
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goal Univ.thy "!!a. a: Vfrom(A,i) ==> {a} : Vfrom(A,succ(i))"; |
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by (rtac subset_mem_Vfrom 1); |
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by (safe_tac ZF_cs); |
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val singleton_in_Vfrom = result(); |
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goal Univ.thy |
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"!!A. [| a: Vfrom(A,i); b: Vfrom(A,i) |] ==> {a,b} : Vfrom(A,succ(i))"; |
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by (rtac subset_mem_Vfrom 1); |
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by (safe_tac ZF_cs); |
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val doubleton_in_Vfrom = result(); |
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goalw Univ.thy [Pair_def] |
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"!!A. [| a: Vfrom(A,i); b: Vfrom(A,i) |] ==> \ |
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\ <a,b> : Vfrom(A,succ(succ(i)))"; |
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by (REPEAT (ares_tac [doubleton_in_Vfrom] 1)); |
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val Pair_in_Vfrom = result(); |
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val [prem] = goal Univ.thy |
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"a<=Vfrom(A,i) ==> succ(a) : Vfrom(A,succ(succ(i)))"; |
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by (REPEAT (resolve_tac [subset_mem_Vfrom, succ_subsetI] 1)); |
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by (rtac (Vfrom_mono RSN (2,subset_trans)) 2); |
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by (REPEAT (resolve_tac [prem, subset_refl, subset_succI] 1)); |
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val succ_in_Vfrom = result(); |
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(*** 0, successor and limit equations fof Vfrom ***) |
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goal Univ.thy "Vfrom(A,0) = A"; |
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by (rtac (Vfrom RS ssubst) 1); |
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by (fast_tac eq_cs 1); |
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val Vfrom_0 = result(); |
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goal Univ.thy "!!i. Ord(i) ==> Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"; |
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by (rtac (Vfrom RS trans) 1); |
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6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
|
128 |
by (rtac (succI1 RS RepFunI RS Union_upper RSN |
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
|
129 |
(2, equalityI RS subst_context)) 1); |
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by (rtac UN_least 1); |
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by (rtac (subset_refl RS Vfrom_mono RS Pow_mono) 1); |
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by (etac (ltI RS le_imp_subset) 1); |
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by (etac Ord_succ 1); |
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val Vfrom_succ_lemma = result(); |
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goal Univ.thy "Vfrom(A,succ(i)) = A Un Pow(Vfrom(A,i))"; |
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by (res_inst_tac [("x1", "succ(i)")] (Vfrom_rank_eq RS subst) 1); |
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by (res_inst_tac [("x1", "i")] (Vfrom_rank_eq RS subst) 1); |
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by (rtac (rank_succ RS ssubst) 1); |
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by (rtac (Ord_rank RS Vfrom_succ_lemma) 1); |
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val Vfrom_succ = result(); |
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(*The premise distinguishes this from Vfrom(A,0); allowing X=0 forces |
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the conclusion to be Vfrom(A,Union(X)) = A Un (UN y:X. Vfrom(A,y)) *) |
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val [prem] = goal Univ.thy "y:X ==> Vfrom(A,Union(X)) = (UN y:X. Vfrom(A,y))"; |
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by (rtac (Vfrom RS ssubst) 1); |
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by (rtac equalityI 1); |
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(*first inclusion*) |
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by (rtac Un_least 1); |
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1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (rtac (A_subset_Vfrom RS subset_trans) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (rtac (prem RS UN_upper) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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152 |
by (rtac UN_least 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (etac UnionE 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
154 |
by (rtac subset_trans 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (etac UN_upper 2); |
0 | 156 |
by (rtac (Vfrom RS ssubst) 1); |
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (etac ([UN_upper, Un_upper2] MRS subset_trans) 1); |
0 | 158 |
(*opposite inclusion*) |
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
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by (rtac UN_least 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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by (fast_tac ZF_cs 1); |
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val Vfrom_Union = result(); |
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(*** Limit ordinals -- general properties ***) |
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goalw Univ.thy [Limit_def] "!!i. Limit(i) ==> Union(i) = i"; |
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by (fast_tac (eq_cs addSIs [ltI] addSEs [ltE] addEs [Ord_trans]) 1); |
0 | 168 |
val Limit_Union_eq = result(); |
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goalw Univ.thy [Limit_def] "!!i. Limit(i) ==> Ord(i)"; |
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by (etac conjunct1 1); |
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val Limit_is_Ord = result(); |
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goalw Univ.thy [Limit_def] "!!i. Limit(i) ==> 0 < i"; |
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by (etac (conjunct2 RS conjunct1) 1); |
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val Limit_has_0 = result(); |
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goalw Univ.thy [Limit_def] "!!i. [| Limit(i); j<i |] ==> succ(j) < i"; |
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by (fast_tac ZF_cs 1); |
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val Limit_has_succ = result(); |
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goalw Univ.thy [Limit_def] "Limit(nat)"; |
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by (safe_tac (ZF_cs addSIs (ltI::nat_typechecks))); |
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by (etac ltD 1); |
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0 | 185 |
val Limit_nat = result(); |
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goalw Univ.thy [Limit_def] |
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"!!i. [| 0<i; ALL y. succ(y) ~= i |] ==> Limit(i)"; |
0 | 189 |
by (safe_tac subset_cs); |
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by (rtac (not_le_iff_lt RS iffD1) 2); |
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by (fast_tac (lt_cs addEs [lt_anti_sym]) 4); |
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by (REPEAT (eresolve_tac [asm_rl, ltE, Ord_succ] 1)); |
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0 | 193 |
val non_succ_LimitI = result(); |
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goal Univ.thy "!!i. Ord(i) ==> i=0 | (EX j. i=succ(j)) | Limit(i)"; |
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by (fast_tac (ZF_cs addSIs [non_succ_LimitI, Ord_0_lt]) 1); |
0 | 197 |
val Ord_cases_lemma = result(); |
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val major::prems = goal Univ.thy |
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"[| Ord(i); \ |
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\ i=0 ==> P; \ |
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\ !!j. i=succ(j) ==> P; \ |
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\ Limit(i) ==> P \ |
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\ |] ==> P"; |
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by (cut_facts_tac [major RS Ord_cases_lemma] 1); |
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by (REPEAT (eresolve_tac (prems@[disjE, exE]) 1)); |
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val Ord_cases = result(); |
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(*** Vfrom applied to Limit ordinals ***) |
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(*NB. limit ordinals are non-empty; |
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Vfrom(A,0) = A = A Un (UN y:0. Vfrom(A,y)) *) |
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val [limiti] = goal Univ.thy |
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"Limit(i) ==> Vfrom(A,i) = (UN y:i. Vfrom(A,y))"; |
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by (rtac (limiti RS (Limit_has_0 RS ltD) RS Vfrom_Union RS subst) 1); |
0 | 217 |
by (rtac (limiti RS Limit_Union_eq RS ssubst) 1); |
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by (rtac refl 1); |
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val Limit_Vfrom_eq = result(); |
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||
27 | 221 |
goal Univ.thy "!!a. [| a: Vfrom(A,j); Limit(i); j<i |] ==> a : Vfrom(A,i)"; |
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by (rtac (Limit_Vfrom_eq RS equalityD2 RS subsetD) 1); |
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by (REPEAT (ares_tac [ltD RS UN_I] 1)); |
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val Limit_VfromI = result(); |
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val prems = goal Univ.thy |
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"[| a: Vfrom(A,i); Limit(i); \ |
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\ !!x. [| x<i; a: Vfrom(A,x) |] ==> R \ |
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\ |] ==> R"; |
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by (rtac (Limit_Vfrom_eq RS equalityD1 RS subsetD RS UN_E) 1); |
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by (REPEAT (ares_tac (prems @ [ltI, Limit_is_Ord]) 1)); |
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val Limit_VfromE = result(); |
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0 | 233 |
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val [major,limiti] = goal Univ.thy |
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"[| a: Vfrom(A,i); Limit(i) |] ==> {a} : Vfrom(A,i)"; |
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by (rtac ([major,limiti] MRS Limit_VfromE) 1); |
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by (etac ([singleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); |
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0 | 238 |
by (etac (limiti RS Limit_has_succ) 1); |
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val singleton_in_Vfrom_limit = result(); |
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val Vfrom_UnI1 = Un_upper1 RS (subset_refl RS Vfrom_mono RS subsetD) |
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and Vfrom_UnI2 = Un_upper2 RS (subset_refl RS Vfrom_mono RS subsetD); |
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(*Hard work is finding a single j:i such that {a,b}<=Vfrom(A,j)*) |
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val [aprem,bprem,limiti] = goal Univ.thy |
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"[| a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) |] ==> \ |
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\ {a,b} : Vfrom(A,i)"; |
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27 | 248 |
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); |
249 |
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); |
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by (rtac ([doubleton_in_Vfrom, limiti] MRS Limit_VfromI) 1); |
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by (etac Vfrom_UnI1 1); |
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by (etac Vfrom_UnI2 1); |
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by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); |
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0 | 254 |
val doubleton_in_Vfrom_limit = result(); |
255 |
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256 |
val [aprem,bprem,limiti] = goal Univ.thy |
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"[| a: Vfrom(A,i); b: Vfrom(A,i); Limit(i) |] ==> \ |
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\ <a,b> : Vfrom(A,i)"; |
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(*Infer that a, b occur at ordinals x,xa < i.*) |
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27 | 260 |
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); |
261 |
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); |
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by (rtac ([Pair_in_Vfrom, limiti] MRS Limit_VfromI) 1); |
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0 | 263 |
(*Infer that succ(succ(x Un xa)) < i *) |
27 | 264 |
by (etac Vfrom_UnI1 1); |
265 |
by (etac Vfrom_UnI2 1); |
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266 |
by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt] 1)); |
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0 | 267 |
val Pair_in_Vfrom_limit = result(); |
268 |
||
269 |
||
270 |
(*** Properties assuming Transset(A) ***) |
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271 |
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272 |
goal Univ.thy "!!i A. Transset(A) ==> Transset(Vfrom(A,i))"; |
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by (eps_ind_tac "i" 1); |
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by (rtac (Vfrom RS ssubst) 1); |
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275 |
by (fast_tac (ZF_cs addSIs [Transset_Union_family, Transset_Un, |
|
276 |
Transset_Pow]) 1); |
|
277 |
val Transset_Vfrom = result(); |
|
278 |
||
279 |
goal Univ.thy "!!A i. Transset(A) ==> Vfrom(A, succ(i)) = Pow(Vfrom(A,i))"; |
|
280 |
by (rtac (Vfrom_succ RS trans) 1); |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
281 |
by (rtac (Un_upper2 RSN (2,equalityI)) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
282 |
by (rtac (subset_refl RSN (2,Un_least)) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
283 |
by (rtac (A_subset_Vfrom RS subset_trans) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
284 |
by (etac (Transset_Vfrom RS (Transset_iff_Pow RS iffD1)) 1); |
0 | 285 |
val Transset_Vfrom_succ = result(); |
286 |
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287 |
goalw Ord.thy [Pair_def,Transset_def] |
|
288 |
"!!C. [| <a,b> <= C; Transset(C) |] ==> a: C & b: C"; |
|
289 |
by (fast_tac ZF_cs 1); |
|
290 |
val Transset_Pair_subset = result(); |
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291 |
||
292 |
goal Univ.thy |
|
293 |
"!!a b.[| <a,b> <= Vfrom(A,i); Transset(A); Limit(i) |] ==> \ |
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294 |
\ <a,b> : Vfrom(A,i)"; |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
295 |
by (etac (Transset_Pair_subset RS conjE) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
296 |
by (etac Transset_Vfrom 1); |
0 | 297 |
by (REPEAT (ares_tac [Pair_in_Vfrom_limit] 1)); |
298 |
val Transset_Pair_subset_Vfrom_limit = result(); |
|
299 |
||
300 |
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301 |
(*** Closure under product/sum applied to elements -- thus Vfrom(A,i) |
|
302 |
is a model of simple type theory provided A is a transitive set |
|
303 |
and i is a limit ordinal |
|
304 |
***) |
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305 |
||
187 | 306 |
(*General theorem for membership in Vfrom(A,i) when i is a limit ordinal*) |
307 |
val [aprem,bprem,limiti,step] = goal Univ.thy |
|
308 |
"[| a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); \ |
|
309 |
\ !!x y j. [| j<i; 1:j; x: Vfrom(A,j); y: Vfrom(A,j) \ |
|
310 |
\ |] ==> EX k. h(x,y): Vfrom(A,k) & k<i |] ==> \ |
|
311 |
\ h(a,b) : Vfrom(A,i)"; |
|
312 |
(*Infer that a, b occur at ordinals x,xa < i.*) |
|
313 |
by (rtac ([aprem,limiti] MRS Limit_VfromE) 1); |
|
314 |
by (rtac ([bprem,limiti] MRS Limit_VfromE) 1); |
|
315 |
by (res_inst_tac [("j1", "x Un xa Un succ(1)")] (step RS exE) 1); |
|
316 |
by (DO_GOAL [etac conjE, etac Limit_VfromI, rtac limiti, atac] 5); |
|
317 |
by (etac (Vfrom_UnI2 RS Vfrom_UnI1) 4); |
|
318 |
by (etac (Vfrom_UnI1 RS Vfrom_UnI1) 3); |
|
319 |
by (rtac (succI1 RS UnI2) 2); |
|
320 |
by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt] 1)); |
|
321 |
val in_Vfrom_limit = result(); |
|
0 | 322 |
|
323 |
(** products **) |
|
324 |
||
325 |
goal Univ.thy |
|
187 | 326 |
"!!A. [| a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) |] ==> \ |
327 |
\ a*b : Vfrom(A, succ(succ(succ(j))))"; |
|
0 | 328 |
by (dtac Transset_Vfrom 1); |
329 |
by (rtac subset_mem_Vfrom 1); |
|
330 |
by (rewtac Transset_def); |
|
331 |
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); |
|
332 |
val prod_in_Vfrom = result(); |
|
333 |
||
334 |
val [aprem,bprem,limiti,transset] = goal Univ.thy |
|
335 |
"[| a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) |] ==> \ |
|
336 |
\ a*b : Vfrom(A,i)"; |
|
187 | 337 |
by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1); |
338 |
by (REPEAT (ares_tac [exI, conjI, prod_in_Vfrom, transset, |
|
339 |
limiti RS Limit_has_succ] 1)); |
|
0 | 340 |
val prod_in_Vfrom_limit = result(); |
341 |
||
342 |
(** Disjoint sums, aka Quine ordered pairs **) |
|
343 |
||
344 |
goalw Univ.thy [sum_def] |
|
187 | 345 |
"!!A. [| a: Vfrom(A,j); b: Vfrom(A,j); Transset(A); 1:j |] ==> \ |
346 |
\ a+b : Vfrom(A, succ(succ(succ(j))))"; |
|
0 | 347 |
by (dtac Transset_Vfrom 1); |
348 |
by (rtac subset_mem_Vfrom 1); |
|
349 |
by (rewtac Transset_def); |
|
350 |
by (fast_tac (ZF_cs addIs [zero_in_Vfrom, Pair_in_Vfrom, |
|
351 |
i_subset_Vfrom RS subsetD]) 1); |
|
352 |
val sum_in_Vfrom = result(); |
|
353 |
||
354 |
val [aprem,bprem,limiti,transset] = goal Univ.thy |
|
355 |
"[| a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) |] ==> \ |
|
356 |
\ a+b : Vfrom(A,i)"; |
|
187 | 357 |
by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1); |
358 |
by (REPEAT (ares_tac [exI, conjI, sum_in_Vfrom, transset, |
|
359 |
limiti RS Limit_has_succ] 1)); |
|
0 | 360 |
val sum_in_Vfrom_limit = result(); |
361 |
||
362 |
(** function space! **) |
|
363 |
||
364 |
goalw Univ.thy [Pi_def] |
|
187 | 365 |
"!!A. [| a: Vfrom(A,j); b: Vfrom(A,j); Transset(A) |] ==> \ |
366 |
\ a->b : Vfrom(A, succ(succ(succ(succ(j)))))"; |
|
0 | 367 |
by (dtac Transset_Vfrom 1); |
368 |
by (rtac subset_mem_Vfrom 1); |
|
369 |
by (rtac (Collect_subset RS subset_trans) 1); |
|
370 |
by (rtac (Vfrom RS ssubst) 1); |
|
371 |
by (rtac (subset_trans RS subset_trans) 1); |
|
372 |
by (rtac Un_upper2 3); |
|
373 |
by (rtac (succI1 RS UN_upper) 2); |
|
374 |
by (rtac Pow_mono 1); |
|
375 |
by (rewtac Transset_def); |
|
376 |
by (fast_tac (ZF_cs addIs [Pair_in_Vfrom]) 1); |
|
377 |
val fun_in_Vfrom = result(); |
|
378 |
||
379 |
val [aprem,bprem,limiti,transset] = goal Univ.thy |
|
380 |
"[| a: Vfrom(A,i); b: Vfrom(A,i); Limit(i); Transset(A) |] ==> \ |
|
381 |
\ a->b : Vfrom(A,i)"; |
|
187 | 382 |
by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1); |
383 |
by (REPEAT (ares_tac [exI, conjI, fun_in_Vfrom, transset, |
|
384 |
limiti RS Limit_has_succ] 1)); |
|
0 | 385 |
val fun_in_Vfrom_limit = result(); |
386 |
||
387 |
||
388 |
(*** The set Vset(i) ***) |
|
389 |
||
390 |
goal Univ.thy "Vset(i) = (UN j:i. Pow(Vset(j)))"; |
|
391 |
by (rtac (Vfrom RS ssubst) 1); |
|
392 |
by (fast_tac eq_cs 1); |
|
393 |
val Vset = result(); |
|
394 |
||
395 |
val Vset_succ = Transset_0 RS Transset_Vfrom_succ; |
|
396 |
||
397 |
val Transset_Vset = Transset_0 RS Transset_Vfrom; |
|
398 |
||
399 |
(** Characterisation of the elements of Vset(i) **) |
|
400 |
||
27 | 401 |
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. b : Vset(i) --> rank(b) < i"; |
0 | 402 |
by (rtac (ordi RS trans_induct) 1); |
403 |
by (rtac (Vset RS ssubst) 1); |
|
404 |
by (safe_tac ZF_cs); |
|
405 |
by (rtac (rank RS ssubst) 1); |
|
27 | 406 |
by (rtac UN_succ_least_lt 1); |
407 |
by (fast_tac ZF_cs 2); |
|
408 |
by (REPEAT (ares_tac [ltI] 1)); |
|
0 | 409 |
val Vset_rank_imp1 = result(); |
410 |
||
27 | 411 |
(* [| Ord(i); x : Vset(i) |] ==> rank(x) < i *) |
412 |
val VsetD = standard (Vset_rank_imp1 RS spec RS mp); |
|
0 | 413 |
|
414 |
val [ordi] = goal Univ.thy "Ord(i) ==> ALL b. rank(b) : i --> b : Vset(i)"; |
|
415 |
by (rtac (ordi RS trans_induct) 1); |
|
416 |
by (rtac allI 1); |
|
417 |
by (rtac (Vset RS ssubst) 1); |
|
27 | 418 |
by (fast_tac (ZF_cs addSIs [rank_lt RS ltD]) 1); |
0 | 419 |
val Vset_rank_imp2 = result(); |
420 |
||
27 | 421 |
goal Univ.thy "!!x i. rank(x)<i ==> x : Vset(i)"; |
422 |
by (etac ltE 1); |
|
423 |
by (etac (Vset_rank_imp2 RS spec RS mp) 1); |
|
424 |
by (assume_tac 1); |
|
425 |
val VsetI = result(); |
|
0 | 426 |
|
27 | 427 |
goal Univ.thy "!!i. Ord(i) ==> b : Vset(i) <-> rank(b) < i"; |
0 | 428 |
by (rtac iffI 1); |
27 | 429 |
by (REPEAT (eresolve_tac [asm_rl, VsetD, VsetI] 1)); |
0 | 430 |
val Vset_Ord_rank_iff = result(); |
431 |
||
27 | 432 |
goal Univ.thy "b : Vset(a) <-> rank(b) < rank(a)"; |
0 | 433 |
by (rtac (Vfrom_rank_eq RS subst) 1); |
434 |
by (rtac (Ord_rank RS Vset_Ord_rank_iff) 1); |
|
435 |
val Vset_rank_iff = result(); |
|
436 |
||
437 |
goal Univ.thy "!!i. Ord(i) ==> rank(Vset(i)) = i"; |
|
438 |
by (rtac (rank RS ssubst) 1); |
|
439 |
by (rtac equalityI 1); |
|
440 |
by (safe_tac ZF_cs); |
|
441 |
by (EVERY' [rtac UN_I, |
|
442 |
etac (i_subset_Vfrom RS subsetD), |
|
443 |
etac (Ord_in_Ord RS rank_of_Ord RS ssubst), |
|
444 |
assume_tac, |
|
445 |
rtac succI1] 3); |
|
27 | 446 |
by (REPEAT (eresolve_tac [asm_rl, VsetD RS ltD, Ord_trans] 1)); |
0 | 447 |
val rank_Vset = result(); |
448 |
||
449 |
(** Lemmas for reasoning about sets in terms of their elements' ranks **) |
|
450 |
||
451 |
goal Univ.thy "a <= Vset(rank(a))"; |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
452 |
by (rtac subsetI 1); |
27 | 453 |
by (etac (rank_lt RS VsetI) 1); |
0 | 454 |
val arg_subset_Vset_rank = result(); |
455 |
||
456 |
val [iprem] = goal Univ.thy |
|
457 |
"[| !!i. Ord(i) ==> a Int Vset(i) <= b |] ==> a <= b"; |
|
27 | 458 |
by (rtac ([subset_refl, arg_subset_Vset_rank] MRS |
459 |
Int_greatest RS subset_trans) 1); |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
460 |
by (rtac (Ord_rank RS iprem) 1); |
0 | 461 |
val Int_Vset_subset = result(); |
462 |
||
463 |
(** Set up an environment for simplification **) |
|
464 |
||
465 |
val rank_rls = [rank_Inl, rank_Inr, rank_pair1, rank_pair2]; |
|
27 | 466 |
val rank_trans_rls = rank_rls @ (rank_rls RLN (2, [lt_trans])); |
0 | 467 |
|
468 |
val rank_ss = ZF_ss |
|
27 | 469 |
addsimps [case_Inl, case_Inr, VsetI] |
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
|
470 |
addsimps rank_trans_rls; |
0 | 471 |
|
472 |
(** Recursion over Vset levels! **) |
|
473 |
||
474 |
(*NOT SUITABLE FOR REWRITING: recursive!*) |
|
475 |
goalw Univ.thy [Vrec_def] "Vrec(a,H) = H(a, lam x:Vset(rank(a)). Vrec(x,H))"; |
|
476 |
by (rtac (transrec RS ssubst) 1); |
|
27 | 477 |
by (simp_tac (ZF_ss addsimps [Ord_rank, Ord_succ, VsetD RS ltD RS beta, |
478 |
VsetI RS beta, le_refl]) 1); |
|
0 | 479 |
val Vrec = result(); |
480 |
||
481 |
(*This form avoids giant explosions in proofs. NOTE USE OF == *) |
|
482 |
val rew::prems = goal Univ.thy |
|
483 |
"[| !!x. h(x)==Vrec(x,H) |] ==> \ |
|
484 |
\ h(a) = H(a, lam x: Vset(rank(a)). h(x))"; |
|
485 |
by (rewtac rew); |
|
486 |
by (rtac Vrec 1); |
|
487 |
val def_Vrec = result(); |
|
488 |
||
489 |
||
490 |
(*** univ(A) ***) |
|
491 |
||
492 |
goalw Univ.thy [univ_def] "!!A B. A<=B ==> univ(A) <= univ(B)"; |
|
493 |
by (etac Vfrom_mono 1); |
|
494 |
by (rtac subset_refl 1); |
|
495 |
val univ_mono = result(); |
|
496 |
||
497 |
goalw Univ.thy [univ_def] "!!A. Transset(A) ==> Transset(univ(A))"; |
|
498 |
by (etac Transset_Vfrom 1); |
|
499 |
val Transset_univ = result(); |
|
500 |
||
501 |
(** univ(A) as a limit **) |
|
502 |
||
503 |
goalw Univ.thy [univ_def] "univ(A) = (UN i:nat. Vfrom(A,i))"; |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
504 |
by (rtac (Limit_nat RS Limit_Vfrom_eq) 1); |
0 | 505 |
val univ_eq_UN = result(); |
506 |
||
507 |
goal Univ.thy "!!c. c <= univ(A) ==> c = (UN i:nat. c Int Vfrom(A,i))"; |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
508 |
by (rtac (subset_UN_iff_eq RS iffD1) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
509 |
by (etac (univ_eq_UN RS subst) 1); |
0 | 510 |
val subset_univ_eq_Int = result(); |
511 |
||
512 |
val [aprem, iprem] = goal Univ.thy |
|
513 |
"[| a <= univ(X); \ |
|
514 |
\ !!i. i:nat ==> a Int Vfrom(X,i) <= b \ |
|
515 |
\ |] ==> a <= b"; |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
516 |
by (rtac (aprem RS subset_univ_eq_Int RS ssubst) 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
517 |
by (rtac UN_least 1); |
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
518 |
by (etac iprem 1); |
0 | 519 |
val univ_Int_Vfrom_subset = result(); |
520 |
||
521 |
val prems = goal Univ.thy |
|
522 |
"[| a <= univ(X); b <= univ(X); \ |
|
523 |
\ !!i. i:nat ==> a Int Vfrom(X,i) = b Int Vfrom(X,i) \ |
|
524 |
\ |] ==> a = b"; |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
525 |
by (rtac equalityI 1); |
0 | 526 |
by (ALLGOALS |
527 |
(resolve_tac (prems RL [univ_Int_Vfrom_subset]) THEN' |
|
528 |
eresolve_tac (prems RL [equalityD1,equalityD2] RL [subset_trans]) THEN' |
|
529 |
rtac Int_lower1)); |
|
530 |
val univ_Int_Vfrom_eq = result(); |
|
531 |
||
532 |
(** Closure properties **) |
|
533 |
||
534 |
goalw Univ.thy [univ_def] "0 : univ(A)"; |
|
535 |
by (rtac (nat_0I RS zero_in_Vfrom) 1); |
|
536 |
val zero_in_univ = result(); |
|
537 |
||
538 |
goalw Univ.thy [univ_def] "A <= univ(A)"; |
|
539 |
by (rtac A_subset_Vfrom 1); |
|
540 |
val A_subset_univ = result(); |
|
541 |
||
542 |
val A_into_univ = A_subset_univ RS subsetD; |
|
543 |
||
544 |
(** Closure under unordered and ordered pairs **) |
|
545 |
||
546 |
goalw Univ.thy [univ_def] "!!A a. a: univ(A) ==> {a} : univ(A)"; |
|
547 |
by (rtac singleton_in_Vfrom_limit 1); |
|
548 |
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); |
|
549 |
val singleton_in_univ = result(); |
|
550 |
||
551 |
goalw Univ.thy [univ_def] |
|
552 |
"!!A a. [| a: univ(A); b: univ(A) |] ==> {a,b} : univ(A)"; |
|
553 |
by (rtac doubleton_in_Vfrom_limit 1); |
|
554 |
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); |
|
555 |
val doubleton_in_univ = result(); |
|
556 |
||
557 |
goalw Univ.thy [univ_def] |
|
558 |
"!!A a. [| a: univ(A); b: univ(A) |] ==> <a,b> : univ(A)"; |
|
559 |
by (rtac Pair_in_Vfrom_limit 1); |
|
560 |
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); |
|
561 |
val Pair_in_univ = result(); |
|
562 |
||
563 |
goal Univ.thy "univ(A)*univ(A) <= univ(A)"; |
|
564 |
by (REPEAT (ares_tac [subsetI,Pair_in_univ] 1 |
|
565 |
ORELSE eresolve_tac [SigmaE, ssubst] 1)); |
|
566 |
val product_univ = result(); |
|
567 |
||
568 |
val Sigma_subset_univ = standard |
|
569 |
(Sigma_mono RS (product_univ RSN (2,subset_trans))); |
|
570 |
||
571 |
goalw Univ.thy [univ_def] |
|
572 |
"!!a b.[| <a,b> <= univ(A); Transset(A) |] ==> <a,b> : univ(A)"; |
|
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
573 |
by (etac Transset_Pair_subset_Vfrom_limit 1); |
0 | 574 |
by (REPEAT (ares_tac [Ord_nat,Limit_nat] 1)); |
575 |
val Transset_Pair_subset_univ = result(); |
|
576 |
||
577 |
||
578 |
(** The natural numbers **) |
|
579 |
||
580 |
goalw Univ.thy [univ_def] "nat <= univ(A)"; |
|
581 |
by (rtac i_subset_Vfrom 1); |
|
582 |
val nat_subset_univ = result(); |
|
583 |
||
584 |
(* n:nat ==> n:univ(A) *) |
|
585 |
val nat_into_univ = standard (nat_subset_univ RS subsetD); |
|
586 |
||
587 |
(** instances for 1 and 2 **) |
|
588 |
||
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset
|
589 |
goal Univ.thy "1 : univ(A)"; |
0 | 590 |
by (REPEAT (ares_tac [nat_into_univ, nat_0I, nat_succI] 1)); |
591 |
val one_in_univ = result(); |
|
592 |
||
593 |
(*unused!*) |
|
27 | 594 |
goal Univ.thy "succ(1) : univ(A)"; |
0 | 595 |
by (REPEAT (ares_tac [nat_into_univ, nat_0I, nat_succI] 1)); |
596 |
val two_in_univ = result(); |
|
597 |
||
598 |
goalw Univ.thy [bool_def] "bool <= univ(A)"; |
|
599 |
by (fast_tac (ZF_cs addSIs [zero_in_univ,one_in_univ]) 1); |
|
600 |
val bool_subset_univ = result(); |
|
601 |
||
602 |
val bool_into_univ = standard (bool_subset_univ RS subsetD); |
|
603 |
||
604 |
||
605 |
(** Closure under disjoint union **) |
|
606 |
||
607 |
goalw Univ.thy [Inl_def] "!!A a. a: univ(A) ==> Inl(a) : univ(A)"; |
|
608 |
by (REPEAT (ares_tac [zero_in_univ,Pair_in_univ] 1)); |
|
609 |
val Inl_in_univ = result(); |
|
610 |
||
611 |
goalw Univ.thy [Inr_def] "!!A b. b: univ(A) ==> Inr(b) : univ(A)"; |
|
612 |
by (REPEAT (ares_tac [one_in_univ, Pair_in_univ] 1)); |
|
613 |
val Inr_in_univ = result(); |
|
614 |
||
615 |
goal Univ.thy "univ(C)+univ(C) <= univ(C)"; |
|
616 |
by (REPEAT (ares_tac [subsetI,Inl_in_univ,Inr_in_univ] 1 |
|
617 |
ORELSE eresolve_tac [sumE, ssubst] 1)); |
|
618 |
val sum_univ = result(); |
|
619 |
||
620 |
val sum_subset_univ = standard |
|
621 |
(sum_mono RS (sum_univ RSN (2,subset_trans))); |
|
622 |
||
623 |
||
624 |
(** Closure under binary union -- use Un_least **) |
|
625 |
(** Closure under Collect -- use (Collect_subset RS subset_trans) **) |
|
626 |
(** Closure under RepFun -- use RepFun_subset **) |
|
627 |
||
628 |