--- a/src/ZF/AC/WO6_WO1.ML Thu Apr 13 11:44:37 1995 +0200
+++ b/src/ZF/AC/WO6_WO1.ML Thu Apr 13 14:15:36 1995 +0200
@@ -2,12 +2,30 @@
ID: $Id$
Author: Krzysztof Gr`abczewski
-The proof of "WO6 ==> WO1".
+The proof of "WO6 ==> WO1". Simplified by L C Paulson.
From the book "Equivalents of the Axiom of Choice" by Rubin & Rubin,
pages 2-5
+
+Simplifier bug in proof of UN_gg2_eq????
*)
+open WO6_WO1;
+
+goal OrderType.thy
+ "!!i j k. [| k < i++j; Ord(i); Ord(j) |] ==> \
+\ k < i | (~ k<i & k = i ++ (k--i) & (k--i)<j)";
+by (res_inst_tac [("i","k"),("j","i")] Ord_linear2 1);
+by (dtac odiff_lt_mono2 4 THEN assume_tac 4);
+by (asm_full_simp_tac
+ (ZF_ss addsimps [oadd_odiff_inverse, odiff_oadd_inverse]) 4);
+by (safe_tac (ZF_cs addSEs [lt_Ord]));
+val lt_oadd_odiff_disj = result();
+
+(*The corresponding elimination rule*)
+val lt_oadd_odiff_cases = rule_by_tactic (safe_tac ZF_cs)
+ (lt_oadd_odiff_disj RS disjE);
+
(* ********************************************************************** *)
(* The most complicated part of the proof - lemma ii - p. 2-4 *)
(* ********************************************************************** *)
@@ -20,18 +38,12 @@
by (fast_tac ZF_cs 1);
val domain_uu_subset = result();
-goal thy "!!a. [| ALL b<a. f`b lepoll m; b<a |] \
-\ ==> domain(uu(f, b, g, d)) lepoll m";
+goal thy "!! a. ALL b<a. f`b lepoll m ==> \
+\ ALL b<a. ALL g<a. ALL d<a. domain(uu(f,b,g,d)) lepoll m";
by (fast_tac (AC_cs addSEs
[domain_uu_subset RS subset_imp_lepoll RS lepoll_trans]) 1);
-val domain_uu_lepoll_m = result();
-
-goal thy "!! a. ALL b<a. f`b lepoll m ==> \
-\ ALL b<a. ALL g<a. ALL d<a. domain(uu(f,b,g,d)) lepoll m";
-by (fast_tac (AC_cs addEs [domain_uu_lepoll_m]) 1);
val quant_domain_uu_lepoll_m = result();
-(* used in case 2 *)
goalw thy [uu_def] "uu(f,b,g,d) <= f`b * f`g";
by (fast_tac ZF_cs 1);
val uu_subset1 = result();
@@ -40,7 +52,7 @@
by (fast_tac ZF_cs 1);
val uu_subset2 = result();
-goal thy "!! a. [| ALL b<a. f`b lepoll m; d<a |] ==> uu(f,b,g,d) lepoll m";
+goal thy "!! a. [| ALL b<a. f`b lepoll m; d<a |] ==> uu(f,b,g,d) lepoll m";
by (fast_tac (AC_cs
addSEs [uu_subset2 RS subset_imp_lepoll RS lepoll_trans]) 1);
val uu_lepoll_m = result();
@@ -50,74 +62,55 @@
(* ********************************************************************** *)
goalw thy [lesspoll_def]
"!! a f u. ALL b<a. ALL g<a. ALL d<a. u(f,b,g,d) lepoll m ==> \
-\ (ALL b<a. f`b ~= 0 --> (EX g<a. EX d<a. u(f,b,g,d) ~= 0 & \
-\ u(f,b,g,d) lesspoll m)) | \
-\ (EX b<a. f`b ~= 0 & (ALL g<a. ALL d<a. u(f,b,g,d) ~= 0 --> \
-\ u(f,b,g,d) eqpoll m))";
+\ (ALL b<a. f`b ~= 0 --> (EX g<a. EX d<a. u(f,b,g,d) ~= 0 & \
+\ u(f,b,g,d) lesspoll m)) | \
+\ (EX b<a. f`b ~= 0 & (ALL g<a. ALL d<a. u(f,b,g,d) ~= 0 --> \
+\ u(f,b,g,d) eqpoll m))";
+by (asm_simp_tac OrdQuant_ss 1);
by (fast_tac AC_cs 1);
val cases = result();
(* ********************************************************************** *)
(* Lemmas used in both cases *)
(* ********************************************************************** *)
-goal thy "!!a f. Ord(a) ==> (UN b<a++a. f`b) = (UN b<a. f`b Un f`(a++b))";
-by (resolve_tac [equalityI] 1);
-by (fast_tac (AC_cs addIs [ltI, OUN_I] addSEs [OUN_E]
- addSDs [lt_oadd_disj]) 1);
-by (fast_tac (AC_cs addSEs [lt_oadd1, oadd_lt_mono2, OUN_E]
- addSIs [OUN_I]) 1);
+goal thy "!!a C. Ord(a) ==> (UN b<a++a. C(b)) = (UN b<a. C(b) Un C(a++b))";
+by (fast_tac (AC_cs addSIs [equalityI] addIs [ltI]
+ addSDs [lt_oadd_disj]
+ addSEs [lt_oadd1, oadd_lt_mono2]) 1);
val UN_oadd = result();
-val [prem] = goal thy
- "(!!b. b<a ==> P(b)=Q(b)) ==> (UN b<a. P(b)) = (UN b<a. Q(b))";
-by (fast_tac (ZF_cs addSIs [OUN_I, equalityI]
- addSEs [OUN_E, prem RS equalityD1 RS subsetD,
- prem RS sym RS equalityD1 RS subsetD]) 1);
-val UN_eq = result();
-
-goal thy "!!a. b<a ==> b = (THE l. l<a & a ++ b = a ++ l)";
-by (fast_tac (ZF_cs addSIs [the_equality RS sym]
- addIs [lt_Ord2, lt_Ord]
- addSEs [oadd_inject RS sym]) 1);
-val the_only_b = result();
-
-goal thy "!!A. B <= A ==> B Un (A-B) = A";
-by (fast_tac (ZF_cs addSIs [equalityI]) 1);
-val subset_imp_Un_Diff_eq = result();
(* ********************************************************************** *)
(* Case 1 : lemmas *)
(* ********************************************************************** *)
-goalw thy [vv1_def] "vv1(f,b,succ(m)) <= f`b";
-by (resolve_tac [expand_if RS iffD2] 1);
-by (fast_tac (ZF_cs addSIs [domain_uu_subset]) 1);
+goalw thy [vv1_def] "vv1(f,m,b) <= f`b";
+by (rtac (letI RS letI) 1);
+by (split_tac [expand_if] 1);
+by (simp_tac (ZF_ss addsimps [domain_uu_subset]) 1);
val vv1_subset = result();
(* ********************************************************************** *)
(* Case 1 : Union of images is the whole "y" *)
(* ********************************************************************** *)
-goal thy "!! a f y. [| (UN b<a. f`b) = y; Ord(a); m:nat |] ==> \
-\ (UN b<a++a. (lam b:a++a. if(b<a, vv1(f,b,succ(m)), \
-\ ww1(f, THE l. l<a & b=a++l, succ(m)))) ` b) = y";
-by (resolve_tac [UN_oadd RS ssubst] 1 THEN (atac 1));
-by (eresolve_tac [subst] 1);
-by (resolve_tac [UN_eq] 1);
-by (forw_inst_tac [("i","a")] lt_oadd1 1
- THEN (REPEAT (atac 1)));
-by (forw_inst_tac [("j","a")] oadd_lt_mono2 1
- THEN (REPEAT (atac 1)));
-by (asm_simp_tac (ZF_ss addsimps [ltD RS beta,
- oadd_le_self RS le_imp_not_lt RS if_not_P, lt_Ord]) 1);
-by (resolve_tac [the_only_b RS subst] 1 THEN (atac 1));
-by (asm_simp_tac (ZF_ss
- addsimps [vv1_subset RS subset_imp_Un_Diff_eq, ltD, ww1_def]) 1);
-val UN_eq_y = result();
+goalw thy [gg1_def]
+ "!! a f y. [| Ord(a); m:nat |] ==> \
+\ (UN b<a++a. gg1(f,a,m)`b) = (UN b<a. f`b)";
+by (asm_simp_tac
+ (OrdQuant_ss addsimps [UN_oadd, lt_oadd1,
+ oadd_le_self RS le_imp_not_lt, lt_Ord,
+ odiff_oadd_inverse, ltD,
+ vv1_subset RS Diff_partition, ww1_def]) 1);
+val UN_gg1_eq = result();
+
+goal thy "domain(gg1(f,a,m)) = a++a";
+by (simp_tac (ZF_ss addsimps [lam_funtype RS domain_of_fun, gg1_def]) 1);
+val domain_gg1 = result();
(* ********************************************************************** *)
(* every value of defined function is less than or equipollent to m *)
(* ********************************************************************** *)
-goal thy "!!a b. [| P(a, b); Ord(a); Ord(b); \
+goal thy "!!a b. [| P(a, b); Ord(a); Ord(b); \
\ Least_a = (LEAST a. EX x. Ord(x) & P(a, x)) |] \
\ ==> P(Least_a, LEAST b. P(Least_a, b))";
by (eresolve_tac [ssubst] 1);
@@ -125,69 +118,40 @@
by (REPEAT (fast_tac (ZF_cs addSEs [LeastI]) 1));
val nested_LeastI = result();
-val nested_Least_instance = read_instantiate
+val nested_Least_instance =
+ standard
+ (read_instantiate
[("P","%g d. domain(uu(f,b,g,d)) ~= 0 & \
-\ domain(uu(f,b,g,d)) lesspoll succ(m)")] nested_LeastI;
+\ domain(uu(f,b,g,d)) lepoll m")] nested_LeastI);
-goalw thy [vv1_def] "!!a. [| ALL b<a. f`b ~=0 --> \
-\ (EX g<a. EX d<a. domain(uu(f,b,g,d))~=0 & \
-\ domain(uu(f,b,g,d)) lesspoll succ(m)); m:nat; b<a |] \
-\ ==> vv1(f,b,succ(m)) lesspoll succ(m)";
-by (resolve_tac [expand_if RS iffD2] 1);
+goalw thy [gg1_def]
+ "!!a. [| Ord(a); m:nat; \
+\ ALL b<a. f`b ~=0 --> \
+\ (EX g<a. EX d<a. domain(uu(f,b,g,d)) ~= 0 & \
+\ domain(uu(f,b,g,d)) lepoll m); \
+\ ALL b<a. f`b lepoll succ(m); b<a++a \
+\ |] ==> gg1(f,a,m)`b lepoll m";
+by (asm_simp_tac OrdQuant_ss 1);
+by (safe_tac (OrdQuant_cs addSEs [lt_oadd_odiff_cases]));
+(*Case b<a : show vv1(f,m,b) lepoll m *)
+by (asm_simp_tac (ZF_ss addsimps [vv1_def, Let_def]
+ setloop split_tac [expand_if]) 1);
by (fast_tac (AC_cs addIs [nested_Least_instance RS conjunct2]
addSEs [lt_Ord]
- addSIs [empty_lepollI RS lepoll_imp_lesspoll_succ]) 1);
-val vv1_lesspoll_succ = result();
-
-goalw thy [vv1_def] "!!a. [| ALL b<a. f`b ~=0 --> \
-\ (EX g<a. EX d<a. domain(uu(f,b,g,d))~=0 & \
-\ domain(uu(f,b,g,d)) lesspoll succ(m)); m:nat; b<a; f`b ~= 0 |] \
-\ ==> vv1(f,b,succ(m)) ~= 0";
-by (resolve_tac [expand_if RS iffD2] 1);
-by (resolve_tac [conjI] 1);
-by (fast_tac ZF_cs 2);
-by (resolve_tac [impI] 1);
-by (eresolve_tac [oallE] 1);
-by (mp_tac 1);
-by (contr_tac 2);
-by (REPEAT (eresolve_tac [oexE] 1));
+ addSIs [empty_lepollI]) 1);
+(*Case a le b: show ww1(f,m,b--a) lepoll m *)
+by (asm_simp_tac (ZF_ss addsimps [ww1_def]) 1);
+by (excluded_middle_tac "f`(b--a) = 0" 1);
+by (asm_simp_tac (OrdQuant_ss addsimps [empty_lepollI]) 2);
+by (resolve_tac [Diff_lepoll] 1);
+by (fast_tac AC_cs 1);
+by (rtac vv1_subset 1);
+by (dtac (ospec RS mp) 1);
+by (REPEAT (eresolve_tac [asm_rl, oexE] 1));
by (asm_simp_tac (ZF_ss
- addsimps [lt_Ord, nested_Least_instance RS conjunct1]) 1);
-val vv1_not_0 = result();
-
-goalw thy [ww1_def] "!!a. [| ALL b<a. f`b ~=0 --> \
-\ (EX g<a. EX d<a. domain(uu(f,b,g,d))~=0 & \
-\ domain(uu(f,b,g,d)) lesspoll succ(m)); \
-\ ALL b<a. f`b lepoll succ(m); m:nat; b<a |] \
-\ ==> ww1(f,b,succ(m)) lesspoll succ(m)";
-by (excluded_middle_tac "f`b = 0" 1);
-by (asm_full_simp_tac (AC_ss
- addsimps [empty_lepollI RS lepoll_imp_lesspoll_succ]) 2);
-by (resolve_tac [Diff_lepoll RS lepoll_imp_lesspoll_succ] 1);
-by (fast_tac AC_cs 1);
-by (REPEAT (ares_tac [vv1_subset, vv1_not_0] 1));
-val ww1_lesspoll_succ = result();
-
-goal thy "!!a. [| Ord(a); m:nat; \
-\ ALL b<a. f`b ~=0 --> (EX g<a. EX d<a. domain(uu(f,b,g,d))~=0 & \
-\ domain(uu(f,b,g,d)) lesspoll succ(m)); \
-\ ALL b<a. f`b lepoll succ(m) |] \
-\ ==> ALL b<a++a. (lam b:a++a. if(b<a, vv1(f,b,succ(m)), \
-\ ww1(f,THE l. l<a & b = a ++ l,succ(m))))`b lepoll m";
-by (resolve_tac [oallI] 1);
-by (asm_full_simp_tac (ZF_ss addsimps [ltD RS beta]) 1);
-by (resolve_tac [lesspoll_succ_imp_lepoll] 1 THEN (atac 2));
-by (resolve_tac [expand_if RS iffD2] 1);
-by (resolve_tac [conjI] 1);
-by (resolve_tac [impI] 1);
-by (forward_tac [lt_oadd_disj1] 2 THEN (REPEAT (atac 2)));
-by (resolve_tac [impI] 2);
-by (eresolve_tac [disjE] 2 THEN (fast_tac (ZF_cs addSEs [ltE]) 2));
-by (asm_full_simp_tac (ZF_ss addsimps [vv1_lesspoll_succ]) 1);
-by (dresolve_tac [theI] 1);
-by (eresolve_tac [conjE] 1);
-by (resolve_tac [ww1_lesspoll_succ] 1 THEN (REPEAT (atac 1)));
-val all_sum_lepoll_m = result();
+ addsimps [vv1_def, Let_def, lt_Ord,
+ nested_Least_instance RS conjunct1]) 1);
+val gg1_lepoll_m = result();
(* ********************************************************************** *)
(* Case 2 : lemmas *)
@@ -197,9 +161,9 @@
(* Case 2 : vv2_subset *)
(* ********************************************************************** *)
-goalw thy [uu_def] "!!f. [| b<a; g<a; f`b~=0; f`g~=0; \
-\ y*y <= y; (UN b<a. f`b)=y |] \
-\ ==> EX d<a. uu(f,b,g,d)~=0";
+goalw thy [uu_def] "!!f. [| b<a; g<a; f`b~=0; f`g~=0; \
+\ y*y <= y; (UN b<a. f`b)=y \
+\ |] ==> EX d<a. uu(f,b,g,d) ~= 0";
by (fast_tac (AC_cs addSIs [not_emptyI]
addSDs [SigmaI RSN (2, subsetD)]
addSEs [not_emptyE]) 1);
@@ -208,12 +172,11 @@
goal thy "!!f. [| b<a; g<a; f`b~=0; f`g~=0; \
\ y*y<=y; (UN b<a. f`b)=y |] \
\ ==> uu(f,b,g,LEAST d. (uu(f,b,g,d) ~= 0)) ~= 0";
-by (dresolve_tac [ex_d_uu_not_empty] 1 THEN (REPEAT (atac 1)));
+by (dresolve_tac [ex_d_uu_not_empty] 1 THEN REPEAT (assume_tac 1));
by (fast_tac (AC_cs addSEs [LeastI, lt_Ord]) 1);
val uu_not_empty = result();
-(* moved from ZF_aux.ML *)
-goal thy "!!r. [| r<=A*B; r~=0 |] ==> domain(r)~=0";
+goal ZF.thy "!!r. [| r<=A*B; r~=0 |] ==> domain(r)~=0";
by (REPEAT (eresolve_tac [asm_rl, not_emptyE, subsetD RS SigmaE,
sym RSN (2, subst_elem) RS domainI RS not_emptyI] 1));
val not_empty_rel_imp_domain = result();
@@ -222,131 +185,125 @@
\ y*y <= y; (UN b<a. f`b)=y |] \
\ ==> (LEAST d. uu(f,b,g,d) ~= 0) < a";
by (eresolve_tac [ex_d_uu_not_empty RS oexE] 1
- THEN (REPEAT (atac 1)));
+ THEN REPEAT (assume_tac 1));
by (resolve_tac [Least_le RS lt_trans1] 1
THEN (REPEAT (ares_tac [lt_Ord] 1)));
val Least_uu_not_empty_lt_a = result();
-goal thy "!!B. [| B<=A; a~:B |] ==> B <= A-{a}";
+goal ZF.thy "!!B. [| B<=A; a~:B |] ==> B <= A-{a}";
by (fast_tac ZF_cs 1);
val subset_Diff_sing = result();
+(*Could this be proved more directly?*)
goal thy "!!A B. [| A lepoll m; m lepoll B; B <= A; m:nat |] ==> A=B";
by (eresolve_tac [natE] 1);
by (fast_tac (AC_cs addSDs [lepoll_0_is_0] addSIs [equalityI]) 1);
by (hyp_subst_tac 1);
by (resolve_tac [equalityI] 1);
-by (atac 2);
+by (assume_tac 2);
by (resolve_tac [subsetI] 1);
-by (excluded_middle_tac "?P" 1 THEN (atac 2));
+by (excluded_middle_tac "?P" 1 THEN (assume_tac 2));
by (eresolve_tac [subset_Diff_sing RS subset_imp_lepoll RSN (2,
diff_sing_lepoll RSN (3, lepoll_trans RS lepoll_trans)) RS
succ_lepoll_natE] 1
- THEN (REPEAT (atac 1)));
+ THEN REPEAT (assume_tac 1));
val supset_lepoll_imp_eq = result();
-goalw thy [vv2_def]
+goal thy
"!!a. [| ALL g<a. ALL d<a. domain(uu(f, b, g, d))~=0 --> \
\ domain(uu(f, b, g, d)) eqpoll succ(m); \
\ ALL b<a. f`b lepoll succ(m); y*y <= y; \
-\ (UN b<a. f`b)=y; b<a; g<a; d<a; f`b~=0; f`g~=0; m:nat; aa:f`b |] \
+\ (UN b<a. f`b)=y; b<a; g<a; d<a; f`b~=0; f`g~=0; m:nat; s:f`b |] \
\ ==> uu(f,b,g,LEAST d. uu(f,b,g,d)~=0) : f`b -> f`g";
by (eres_inst_tac [("x","g")] oallE 1 THEN (contr_tac 2));
by (eres_inst_tac [("P","%z. ?QQ(z) ~= 0 --> ?RR(z)")] oallE 1);
by (eresolve_tac [impE] 1);
-by (eresolve_tac [uu_not_empty RS (uu_subset1 RS
- not_empty_rel_imp_domain)] 1
- THEN (REPEAT (atac 1)));
+by (eresolve_tac [uu_not_empty RS (uu_subset1 RS not_empty_rel_imp_domain)] 1
+ THEN REPEAT (assume_tac 1));
by (eresolve_tac [Least_uu_not_empty_lt_a RSN (2, notE)] 2
- THEN (TRYALL atac));
+ THEN (TRYALL assume_tac));
by (resolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS
(Least_uu_not_empty_lt_a RSN (2, uu_lepoll_m) RSN (2,
uu_subset1 RSN (4, rel_is_fun)))] 1
- THEN (TRYALL atac));
+ THEN (TRYALL assume_tac));
by (resolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RSN (2,
supset_lepoll_imp_eq)] 1);
by (REPEAT (fast_tac (AC_cs addSIs [domain_uu_subset, nat_succI]) 1));
val uu_Least_is_fun = result();
goalw thy [vv2_def]
- "!!a. [| ALL g<a. ALL d<a. domain(uu(f, b, g, d))~=0 --> \
-\ domain(uu(f, b, g, d)) eqpoll succ(m); \
-\ ALL b<a. f`b lepoll succ(m); y*y <= y; \
-\ (UN b<a. f`b)=y; b<a; g<a; m:nat; aa:f`b |] \
-\ ==> vv2(f,b,g,aa) <= f`g";
-by (fast_tac (FOL_cs addIs [expand_if RS iffD2]
- addSEs [uu_Least_is_fun]
- addSIs [empty_subsetI, not_emptyI,
- singleton_subsetI, apply_type]) 1);
+ "!!a. [| ALL g<a. ALL d<a. domain(uu(f, b, g, d))~=0 --> \
+\ domain(uu(f, b, g, d)) eqpoll succ(m); \
+\ ALL b<a. f`b lepoll succ(m); y*y <= y; \
+\ (UN b<a. f`b)=y; b<a; g<a; m:nat; s:f`b \
+\ |] ==> vv2(f,b,g,s) <= f`g";
+by (split_tac [expand_if] 1);
+by (fast_tac (FOL_cs addSEs [uu_Least_is_fun]
+ addSIs [empty_subsetI, not_emptyI,
+ singleton_subsetI, apply_type]) 1);
val vv2_subset = result();
(* ********************************************************************** *)
(* Case 2 : Union of images is the whole "y" *)
(* ********************************************************************** *)
-goal thy "!!a. [| ALL g<a. ALL d<a. domain(uu(f,b,g,d)) ~= 0 --> \
-\ domain(uu(f,b,g,d)) eqpoll succ(m); \
-\ ALL b<a. f`b lepoll succ(m); y*y<=y; \
-\ (UN b<a.f`b)=y; Ord(a); m:nat; aa:f`b; b<a |] \
-\ ==> (UN g<a++a. (lam g:a++a. if(g<a, vv2(f,b,g,aa), \
-\ ww2(f,b,THE l. l<a & g=a++l,aa)))`g) = y";
-by (resolve_tac [UN_oadd RS ssubst] 1 THEN (atac 1));
-by (resolve_tac [subst] 1 THEN (atac 1));
-by (resolve_tac [UN_eq] 1);
-by (forw_inst_tac [("i","a"),("k","ba")] lt_oadd1 1
- THEN (REPEAT (atac 1)));
-by (forw_inst_tac [("j","a"),("k","ba")] oadd_lt_mono2 1
- THEN (REPEAT (atac 1)));
-by (asm_simp_tac (ZF_ss addsimps [ltD RS beta,
- oadd_le_self RS le_imp_not_lt RS if_not_P, lt_Ord]) 1);
-by (resolve_tac [the_only_b RS subst] 1 THEN (atac 1));
-by (asm_simp_tac (ZF_ss
- addsimps [vv2_subset RS subset_imp_Un_Diff_eq, ltI, ww2_def]) 1);
-val UN_eq_y_2 = result();
+goalw thy [gg2_def]
+ "!!a. [| ALL g<a. ALL d<a. domain(uu(f,b,g,d)) ~= 0 --> \
+\ domain(uu(f,b,g,d)) eqpoll succ(m); \
+\ ALL b<a. f`b lepoll succ(m); y*y<=y; \
+\ (UN b<a.f`b)=y; Ord(a); m:nat; s:f`b; b<a \
+\ |] ==> (UN g<a++a. gg2(f,a,b,s) ` g) = y";
+bd sym 1;
+by (asm_simp_tac
+ (OrdQuant_ss addsimps [UN_oadd, lt_oadd1,
+ oadd_le_self RS le_imp_not_lt, lt_Ord,
+ odiff_oadd_inverse, ww2_def,
+ standard (vv2_subset RS Diff_partition)]) 1);
+(*Omitting "standard" above causes "Failed congruence proof!" bug??*)
+val UN_gg2_eq = result();
+
+goal thy "domain(gg2(f,a,b,s)) = a++a";
+by (simp_tac (ZF_ss addsimps [lam_funtype RS domain_of_fun, gg2_def]) 1);
+val domain_gg2 = result();
(* ********************************************************************** *)
(* every value of defined function is less than or equipollent to m *)
(* ********************************************************************** *)
goalw thy [vv2_def]
- "!!m. [| m:nat; m~=0 |] ==> vv2(f,b,g,aa) lesspoll succ(m)";
-by (resolve_tac [conjI RS (expand_if RS iffD2)] 1);
-by (asm_simp_tac (AC_ss
- addsimps [empty_lepollI RS lepoll_imp_lesspoll_succ]) 2);
+ "!!m. [| m:nat; m~=0 |] ==> vv2(f,b,g,s) lepoll m";
+by (asm_simp_tac (OrdQuant_ss
+ addsimps [empty_lepollI]
+ setloop split_tac [expand_if]) 1);
by (fast_tac (AC_cs
addSDs [le_imp_subset RS subset_imp_lepoll RS lepoll_0_is_0]
- addSIs [singleton_eqpoll_1 RS eqpoll_imp_lepoll RS
- lepoll_trans RS lepoll_imp_lesspoll_succ,
+ addSIs [singleton_eqpoll_1 RS eqpoll_imp_lepoll RS lepoll_trans,
not_lt_imp_le RS le_imp_subset RS subset_imp_lepoll,
nat_into_Ord, nat_1I]) 1);
-val vv2_lesspoll_succ = result();
-
-goalw thy [ww2_def] "!!m. [| ALL b<a. f`b lepoll succ(m); g<a; m:nat; \
-\ vv2(f,b,g,d) <= f`g |] \
-\ ==> ww2(f,b,g,d) lesspoll succ(m)";
-by (excluded_middle_tac "f`g = 0" 1);
-by (asm_simp_tac (AC_ss
- addsimps [empty_lepollI RS lepoll_imp_lesspoll_succ]) 2);
-by (dresolve_tac [ospec] 1 THEN (atac 1));
-by (resolve_tac [Diff_lepoll RS lepoll_imp_lesspoll_succ] 1
- THEN (TRYALL atac));
-by (asm_simp_tac (AC_ss addsimps [vv2_def, expand_if, not_emptyI]) 1);
-val ww2_lesspoll_succ = result();
+val vv2_lepoll = result();
-goal thy "!!a. [| ALL g<a. ALL d<a. domain(uu(f,b,g,d)) ~= 0 --> \
-\ domain(uu(f,b,g,d)) eqpoll succ(m); \
-\ ALL b<a. f`b lepoll succ(m); y*y <= y; \
-\ (UN b<a. f`b)=y; b<a; aa:f`b; m:nat; m~= 0 |] \
-\ ==> ALL ba<a++a. (lam g:a++a. if(g<a, vv2(f,b,g,aa), \
-\ ww2(f,b,THE l. l<a & g=a++l,aa)))`ba lepoll m";
-by (resolve_tac [oallI] 1);
-by (asm_full_simp_tac AC_ss 1);
-by (resolve_tac [lesspoll_succ_imp_lepoll] 1 THEN (atac 2));
-by (resolve_tac [conjI RS (expand_if RS iffD2)] 1);
-by (asm_simp_tac (AC_ss addsimps [vv2_lesspoll_succ]) 1);
-by (forward_tac [lt_oadd_disj1] 1 THEN (REPEAT (ares_tac [lt_Ord2] 1)));
-by (fast_tac (FOL_cs addSIs [ww2_lesspoll_succ, vv2_subset]
- addSDs [theI]) 1);
-val all_sum_lepoll_m_2 = result();
+goalw thy [ww2_def]
+ "!!m. [| ALL b<a. f`b lepoll succ(m); g<a; m:nat; vv2(f,b,g,d) <= f`g \
+\ |] ==> ww2(f,b,g,d) lepoll m";
+by (excluded_middle_tac "f`g = 0" 1);
+by (asm_simp_tac (OrdQuant_ss
+ addsimps [empty_lepollI]) 2);
+by (dresolve_tac [ospec] 1 THEN (assume_tac 1));
+by (resolve_tac [Diff_lepoll] 1
+ THEN (TRYALL assume_tac));
+by (asm_simp_tac (OrdQuant_ss addsimps [vv2_def, expand_if, not_emptyI]) 1);
+val ww2_lepoll = result();
+
+goalw thy [gg2_def]
+ "!!a. [| ALL g<a. ALL d<a. domain(uu(f,b,g,d)) ~= 0 --> \
+\ domain(uu(f,b,g,d)) eqpoll succ(m); \
+\ ALL b<a. f`b lepoll succ(m); y*y <= y; \
+\ (UN b<a. f`b)=y; b<a; s:f`b; m:nat; m~= 0; g<a++a \
+\ |] ==> gg2(f,a,b,s) ` g lepoll m";
+by (asm_simp_tac OrdQuant_ss 1);
+by (safe_tac (OrdQuant_cs addSEs [lt_oadd_odiff_cases, lt_Ord2]));
+by (asm_simp_tac (OrdQuant_ss addsimps [vv2_lepoll]) 1);
+by (asm_simp_tac (ZF_ss addsimps [ww2_lepoll, vv2_subset]) 1);
+val gg2_lepoll_m = result();
(* ********************************************************************** *)
(* lemma ii *)
@@ -355,25 +312,23 @@
"!!y. [| succ(m) : NN(y); y*y <= y; m:nat; m~=0 |] ==> m : NN(y)";
by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1));
by (resolve_tac [quant_domain_uu_lepoll_m RS cases RS disjE] 1
- THEN (atac 1));
+ THEN (assume_tac 1));
(* case 1 *)
-by (resolve_tac [CollectI] 1);
-by (atac 1);
-by (res_inst_tac [("x","a ++ a")] exI 1);
-by (res_inst_tac [("x","lam b:a++a. if (b<a, vv1(f,b,succ(m)), \
-\ ww1(f,THE l. l<a & b=a++l,succ(m)))")] exI 1);
-by (fast_tac (FOL_cs addSIs [Ord_oadd, lam_funtype RS domain_of_fun,
- UN_eq_y, all_sum_lepoll_m]) 1);
+by (asm_full_simp_tac (ZF_ss addsimps [lesspoll_succ_iff]) 1);
+by (res_inst_tac [("x","a++a")] exI 1);
+by (fast_tac (OrdQuant_cs addSIs [Ord_oadd, domain_gg1, UN_gg1_eq,
+ gg1_lepoll_m]) 1);
(* case 2 *)
by (REPEAT (eresolve_tac [oexE, conjE] 1));
+by (res_inst_tac [("A","f`?B")] not_emptyE 1 THEN (assume_tac 1));
by (resolve_tac [CollectI] 1);
by (eresolve_tac [succ_natD] 1);
-by (res_inst_tac [("A","f`?B")] not_emptyE 1 THEN (atac 1));
by (res_inst_tac [("x","a++a")] exI 1);
-by (res_inst_tac [("x","lam g:a++a. if (g<a, vv2(f,b,g,x), \
-\ ww2(f,b,THE l. l<a & g=a++l,x))")] exI 1);
-by (fast_tac (FOL_cs addSIs [Ord_oadd, lam_funtype RS domain_of_fun,
- UN_eq_y_2, all_sum_lepoll_m_2]) 1);
+by (res_inst_tac [("x","gg2(f,a,b,x)")] exI 1);
+(*Calling fast_tac might get rid of the res_inst_tac calls, but it
+ is just too slow.*)
+by (asm_simp_tac (OrdQuant_ss addsimps
+ [Ord_oadd, domain_gg2, UN_gg2_eq, gg2_lepoll_m]) 1);
val lemma_ii = result();
@@ -392,7 +347,7 @@
goal thy "!!n. [| ALL n:nat. f(n)<=f(succ(n)); n le m; n : nat; m: nat |] \
\ ==> f(n)<=f(m)";
-by (res_inst_tac [("P","n le m")] impE 1 THEN (REPEAT (atac 2)));
+by (res_inst_tac [("P","n le m")] impE 1 THEN (REPEAT (assume_tac 2)));
by (res_inst_tac [("P","%z. n le z --> f(n) <= f(z)")] nat_induct 1);
by (REPEAT (fast_tac lt_cs 1));
val le_subsets = result();
@@ -400,26 +355,20 @@
goal thy "!!n m. [| n le m; m:nat |] ==> \
\ rec(n, x, %k r. r Un r*r) <= rec(m, x, %k r. r Un r*r)";
by (resolve_tac [z_n_subset_z_succ_n RS le_subsets] 1
- THEN (TRYALL atac));
+ THEN (TRYALL assume_tac));
by (eresolve_tac [Ord_nat RSN (2, ltI) RSN (2, lt_trans1) RS ltD] 1
- THEN (atac 1));
+ THEN (assume_tac 1));
val le_imp_rec_subset = result();
-goal thy "!!x. EX y. x Un y*y <= y";
+goal thy "EX y. x Un y*y <= y";
by (res_inst_tac [("x","UN n:nat. rec(n, x, %k r. r Un r*r)")] exI 1);
-by (resolve_tac [subsetI] 1);
-by (eresolve_tac [UnE] 1);
-by (resolve_tac [UN_I] 1);
-by (eresolve_tac [rec_0 RS ssubst] 2);
-by (resolve_tac [nat_0I] 1);
-by (eresolve_tac [SigmaE] 1);
-by (REPEAT (eresolve_tac [UN_E] 1));
+by (safe_tac ZF_cs);
+by (fast_tac (ZF_cs addSIs [nat_0I] addss nat_ss) 1);
by (res_inst_tac [("a","succ(n Un na)")] UN_I 1);
-by (eresolve_tac [Un_nat_type RS nat_succI] 1 THEN (atac 1));
-by (resolve_tac [rec_succ RS ssubst] 1);
+by (eresolve_tac [Un_nat_type RS nat_succI] 1 THEN (assume_tac 1));
by (fast_tac (ZF_cs addIs [le_imp_rec_subset RS subsetD]
addSIs [Un_upper1_le, Un_upper2_le, Un_nat_type]
- addSEs [nat_into_Ord]) 1);
+ addSEs [nat_into_Ord] addss nat_ss) 1);
val lemma_iv = result();
(* ********************************************************************** *)
@@ -453,7 +402,7 @@
goal thy "!!f. [| (UN b<a. f`b)=y; x:y; ALL b<a. f`b lepoll 1; Ord(a) |] \
\ ==> f` (LEAST i. f`i = {x}) = {x}";
-by (dresolve_tac [lemma1] 1 THEN (REPEAT (atac 1)));
+by (dresolve_tac [lemma1] 1 THEN REPEAT (assume_tac 1));
by (fast_tac (AC_cs addSEs [lt_Ord] addIs [LeastI]) 1);
val lemma2 = result();
@@ -462,11 +411,11 @@
by (REPEAT (eresolve_tac [exE, conjE] 1));
by (res_inst_tac [("x","a")] exI 1);
by (res_inst_tac [("x","lam x:y. LEAST i. f`i = {x}")] exI 1);
-by (resolve_tac [conjI] 1 THEN (atac 1));
+by (resolve_tac [conjI] 1 THEN (assume_tac 1));
by (res_inst_tac [("d","%i. THE x. x:f`i")] lam_injective 1);
-by (dresolve_tac [lemma1] 1 THEN (REPEAT (atac 1)));
+by (dresolve_tac [lemma1] 1 THEN REPEAT (assume_tac 1));
by (fast_tac (AC_cs addSEs [Least_le RS lt_trans1 RS ltD, lt_Ord]) 1);
-by (resolve_tac [lemma2 RS ssubst] 1 THEN (REPEAT (atac 1)));
+by (resolve_tac [lemma2 RS ssubst] 1 THEN REPEAT (assume_tac 1));
by (fast_tac (ZF_cs addSIs [the_equality]) 1);
val NN_imp_ex_inj = result();
@@ -495,7 +444,7 @@
\ !!m. [| m:nat; m~=0; P(succ(m)) |] ==> P(m) |] \
\ ==> P(1)";
by (resolve_tac [rev_induct_lemma RS impE] 1);
-by (eresolve_tac [impE] 4 THEN (atac 5));
+by (eresolve_tac [impE] 4 THEN (assume_tac 5));
by (REPEAT (ares_tac prems 1));
val rev_induct = result();
@@ -505,7 +454,7 @@
goal thy "!!n. [| n:NN(y); y*y <= y; n~=0 |] ==> 1:NN(y)";
by (resolve_tac [rev_induct] 1 THEN (REPEAT (ares_tac [NN_into_nat] 1)));
-by (resolve_tac [lemma_ii] 1 THEN (REPEAT (atac 1)));
+by (resolve_tac [lemma_ii] 1 THEN REPEAT (assume_tac 1));
val lemma3 = result();
(* ********************************************************************** *)