--- a/src/ZF/Cardinal.ML Fri Jan 03 10:48:28 1997 +0100
+++ b/src/ZF/Cardinal.ML Fri Jan 03 15:01:55 1997 +0100
@@ -25,7 +25,7 @@
\ X - lfp(X, %W. X - g``(Y - f``W)) ";
by (res_inst_tac [("P", "%u. ?v = X-u")]
(decomp_bnd_mono RS lfp_Tarski RS ssubst) 1);
-by (simp_tac (ZF_ss addsimps [subset_refl, double_complement,
+by (simp_tac (!simpset addsimps [subset_refl, double_complement,
gfun RS fun_is_rel RS image_subset]) 1);
qed "Banach_last_equation";
@@ -45,7 +45,7 @@
"[| f: inj(X,Y); g: inj(Y,X) |] ==> EX h. h: bij(X,Y)";
by (cut_facts_tac prems 1);
by (cut_facts_tac [(prems RL [inj_is_fun]) MRS decomposition] 1);
-by (fast_tac (ZF_cs addSIs [restrict_bij,bij_disjoint_Un]
+by (fast_tac (!claset addSIs [restrict_bij,bij_disjoint_Un]
addIs [bij_converse_bij]) 1);
(* The instantiation of exI to "restrict(f,XA) Un converse(restrict(g,YB))"
is forced by the context!! *)
@@ -62,12 +62,12 @@
bind_thm ("eqpoll_refl", id_bij RS bij_imp_eqpoll);
goalw Cardinal.thy [eqpoll_def] "!!X Y. X eqpoll Y ==> Y eqpoll X";
-by (fast_tac (ZF_cs addIs [bij_converse_bij]) 1);
+by (fast_tac (!claset addIs [bij_converse_bij]) 1);
qed "eqpoll_sym";
goalw Cardinal.thy [eqpoll_def]
"!!X Y. [| X eqpoll Y; Y eqpoll Z |] ==> X eqpoll Z";
-by (fast_tac (ZF_cs addIs [comp_bij]) 1);
+by (fast_tac (!claset addIs [comp_bij]) 1);
qed "eqpoll_trans";
(** Le-pollence is a partial ordering **)
@@ -83,12 +83,12 @@
goalw Cardinal.thy [eqpoll_def, bij_def, lepoll_def]
"!!X Y. X eqpoll Y ==> X lepoll Y";
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "eqpoll_imp_lepoll";
goalw Cardinal.thy [lepoll_def]
"!!X Y. [| X lepoll Y; Y lepoll Z |] ==> X lepoll Z";
-by (fast_tac (ZF_cs addIs [comp_inj]) 1);
+by (fast_tac (!claset addIs [comp_inj]) 1);
qed "lepoll_trans";
(*Asymmetry law*)
@@ -106,7 +106,7 @@
qed "eqpollE";
goal Cardinal.thy "X eqpoll Y <-> X lepoll Y & Y lepoll X";
-by (fast_tac (ZF_cs addIs [eqpollI] addSEs [eqpollE]) 1);
+by (fast_tac (!claset addIs [eqpollI] addSEs [eqpollE]) 1);
qed "eqpoll_iff";
goalw Cardinal.thy [lepoll_def, inj_def] "!!A. A lepoll 0 ==> A = 0";
@@ -117,70 +117,70 @@
bind_thm ("empty_lepollI", empty_subsetI RS subset_imp_lepoll);
goal Cardinal.thy "A lepoll 0 <-> A=0";
-by (fast_tac (ZF_cs addIs [lepoll_0_is_0, lepoll_refl]) 1);
+by (fast_tac (!claset addIs [lepoll_0_is_0, lepoll_refl]) 1);
qed "lepoll_0_iff";
goalw Cardinal.thy [lepoll_def]
"!!A. [| A lepoll B; C lepoll D; B Int D = 0 |] ==> A Un C lepoll B Un D";
-by (fast_tac (ZF_cs addIs [inj_disjoint_Un]) 1);
+by (fast_tac (!claset addIs [inj_disjoint_Un]) 1);
qed "Un_lepoll_Un";
(*A eqpoll 0 ==> A=0*)
bind_thm ("eqpoll_0_is_0", eqpoll_imp_lepoll RS lepoll_0_is_0);
goal Cardinal.thy "A eqpoll 0 <-> A=0";
-by (fast_tac (ZF_cs addIs [eqpoll_0_is_0, eqpoll_refl]) 1);
+by (fast_tac (!claset addIs [eqpoll_0_is_0, eqpoll_refl]) 1);
qed "eqpoll_0_iff";
goalw Cardinal.thy [eqpoll_def]
"!!A. [| A eqpoll B; C eqpoll D; A Int C = 0; B Int D = 0 |] ==> \
\ A Un C eqpoll B Un D";
-by (fast_tac (ZF_cs addIs [bij_disjoint_Un]) 1);
+by (fast_tac (!claset addIs [bij_disjoint_Un]) 1);
qed "eqpoll_disjoint_Un";
(*** lesspoll: contributions by Krzysztof Grabczewski ***)
goalw Cardinal.thy [lesspoll_def] "!!A. A lesspoll B ==> A lepoll B";
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "lesspoll_imp_lepoll";
goalw Cardinal.thy [lepoll_def]
"!!A. [| A lepoll B; well_ord(B,r) |] ==> EX s. well_ord(A,s)";
-by (fast_tac (ZF_cs addSEs [well_ord_rvimage]) 1);
+by (fast_tac (!claset addSEs [well_ord_rvimage]) 1);
qed "lepoll_well_ord";
goalw Cardinal.thy [lesspoll_def] "A lepoll B <-> A lesspoll B | A eqpoll B";
-by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE]) 1);
+by (fast_tac (!claset addSIs [eqpollI] addSEs [eqpollE]) 1);
qed "lepoll_iff_leqpoll";
goalw Cardinal.thy [inj_def, surj_def]
"!!f. [| f : inj(A, succ(m)); f ~: surj(A, succ(m)) |] ==> EX f. f:inj(A,m)";
-by (safe_tac lemmas_cs);
+by (safe_tac (claset_of"ZF"));
by (swap_res_tac [exI] 1);
by (res_inst_tac [("a", "lam z:A. if(f`z=m, y, f`z)")] CollectI 1);
-by (fast_tac (ZF_cs addSIs [if_type RS lam_type]
- addEs [apply_funtype RS succE]) 1);
+by (fast_tac (!claset addSIs [if_type RS lam_type]
+ addEs [apply_funtype RS succE]) 1);
(*Proving it's injective*)
-by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1);
-by (fast_tac ZF_cs 1);
+by (asm_simp_tac (!simpset setloop split_tac [expand_if]) 1);
+by (Fast_tac 1);
qed "inj_not_surj_succ";
(** Variations on transitivity **)
goalw Cardinal.thy [lesspoll_def]
"!!X. [| X lesspoll Y; Y lesspoll Z |] ==> X lesspoll Z";
-by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1);
+by (fast_tac (!claset addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1);
qed "lesspoll_trans";
goalw Cardinal.thy [lesspoll_def]
"!!X. [| X lesspoll Y; Y lepoll Z |] ==> X lesspoll Z";
-by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1);
+by (fast_tac (!claset addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1);
qed "lesspoll_lepoll_lesspoll";
goalw Cardinal.thy [lesspoll_def]
"!!X. [| X lesspoll Y; Z lepoll X |] ==> Z lesspoll Y";
-by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1);
+by (fast_tac (!claset addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1);
qed "lepoll_lesspoll_lesspoll";
@@ -189,10 +189,10 @@
val [premP,premOrd,premNot] = goalw Cardinal.thy [Least_def]
"[| P(i); Ord(i); !!x. x<i ==> ~P(x) |] ==> (LEAST x.P(x)) = i";
by (rtac the_equality 1);
-by (fast_tac (ZF_cs addSIs [premP,premOrd,premNot]) 1);
+by (fast_tac (!claset addSIs [premP,premOrd,premNot]) 1);
by (REPEAT (etac conjE 1));
by (etac (premOrd RS Ord_linear_lt) 1);
-by (ALLGOALS (fast_tac (ZF_cs addSIs [premP] addSDs [premNot])));
+by (ALLGOALS (fast_tac (!claset addSIs [premP] addSDs [premNot])));
qed "Least_equality";
goal Cardinal.thy "!!i. [| P(i); Ord(i) |] ==> P(LEAST x.P(x))";
@@ -202,7 +202,7 @@
by (rtac classical 1);
by (EVERY1 [stac Least_equality, assume_tac, assume_tac]);
by (assume_tac 2);
-by (fast_tac (ZF_cs addSEs [ltE]) 1);
+by (fast_tac (!claset addSEs [ltE]) 1);
qed "LeastI";
(*Proof is almost identical to the one above!*)
@@ -213,7 +213,7 @@
by (rtac classical 1);
by (EVERY1 [stac Least_equality, assume_tac, assume_tac]);
by (etac le_refl 2);
-by (fast_tac (ZF_cs addEs [ltE, lt_trans1] addIs [leI, ltI]) 1);
+by (fast_tac (!claset addEs [ltE, lt_trans1] addIs [leI, ltI]) 1);
qed "Least_le";
(*LEAST really is the smallest*)
@@ -234,12 +234,12 @@
goalw Cardinal.thy [Least_def]
"!!P. [| ~ (EX i. Ord(i) & P(i)) |] ==> (LEAST x.P(x)) = 0";
by (rtac the_0 1);
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "Least_0";
goal Cardinal.thy "Ord(LEAST x.P(x))";
by (excluded_middle_tac "EX i. Ord(i) & P(i)" 1);
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (rtac (Least_le RS ltE) 2);
by (REPEAT_SOME assume_tac);
by (etac (Least_0 RS ssubst) 1);
@@ -252,14 +252,14 @@
(*Not needed for simplification, but helpful below*)
val prems = goal Cardinal.thy
"[| !!y. P(y) <-> Q(y) |] ==> (LEAST x.P(x)) = (LEAST x.Q(x))";
-by (simp_tac (FOL_ss addsimps prems) 1);
+by (simp_tac (!simpset addsimps prems) 1);
qed "Least_cong";
(*Need AC to get X lepoll Y ==> |X| le |Y|; see well_ord_lepoll_imp_Card_le
Converse also requires AC, but see well_ord_cardinal_eqE*)
goalw Cardinal.thy [eqpoll_def,cardinal_def] "!!X Y. X eqpoll Y ==> |X| = |Y|";
by (rtac Least_cong 1);
-by (fast_tac (ZF_cs addEs [comp_bij,bij_converse_bij]) 1);
+by (fast_tac (!claset addEs [comp_bij,bij_converse_bij]) 1);
qed "cardinal_cong";
(*Under AC, the premise becomes trivial; one consequence is ||A|| = |A|*)
@@ -277,12 +277,12 @@
"!!X Y. [| well_ord(X,r); well_ord(Y,s); |X| = |Y| |] ==> X eqpoll Y";
by (rtac (eqpoll_sym RS eqpoll_trans) 1);
by (etac well_ord_cardinal_eqpoll 1);
-by (asm_simp_tac (ZF_ss addsimps [well_ord_cardinal_eqpoll]) 1);
+by (asm_simp_tac (!simpset addsimps [well_ord_cardinal_eqpoll]) 1);
qed "well_ord_cardinal_eqE";
goal Cardinal.thy
"!!X Y. [| well_ord(X,r); well_ord(Y,s) |] ==> |X| = |Y| <-> X eqpoll Y";
-by (fast_tac (ZF_cs addSEs [cardinal_cong, well_ord_cardinal_eqE]) 1);
+by (fast_tac (!claset addSEs [cardinal_cong, well_ord_cardinal_eqE]) 1);
qed "well_ord_cardinal_eqpoll_iff";
@@ -309,7 +309,7 @@
qed "Card_is_Ord";
goal Cardinal.thy "!!K. Card(K) ==> K le |K|";
-by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
+by (asm_simp_tac (!simpset addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
qed "Card_cardinal_le";
goalw Cardinal.thy [cardinal_def] "Ord(|A|)";
@@ -318,8 +318,8 @@
(*The cardinals are the initial ordinals*)
goal Cardinal.thy "Card(K) <-> Ord(K) & (ALL j. j<K --> ~ j eqpoll K)";
-by (safe_tac (ZF_cs addSIs [CardI, Card_is_Ord]));
-by (fast_tac ZF_cs 2);
+by (safe_tac (!claset addSIs [CardI, Card_is_Ord]));
+by (Fast_tac 2);
by (rewrite_goals_tac [Card_def, cardinal_def]);
by (rtac less_LeastE 1);
by (etac subst 2);
@@ -328,21 +328,21 @@
goalw Cardinal.thy [lesspoll_def] "!!a. [| Card(a); i<a |] ==> i lesspoll a";
by (dresolve_tac [Card_iff_initial RS iffD1] 1);
-by (fast_tac (ZF_cs addSEs [leI RS le_imp_lepoll]) 1);
+by (fast_tac (!claset addSEs [leI RS le_imp_lepoll]) 1);
qed "lt_Card_imp_lesspoll";
goal Cardinal.thy "Card(0)";
by (rtac (Ord_0 RS CardI) 1);
-by (fast_tac (ZF_cs addSEs [ltE]) 1);
+by (fast_tac (!claset addSEs [ltE]) 1);
qed "Card_0";
val [premK,premL] = goal Cardinal.thy
"[| Card(K); Card(L) |] ==> Card(K Un L)";
by (rtac ([premK RS Card_is_Ord, premL RS Card_is_Ord] MRS Ord_linear_le) 1);
by (asm_simp_tac
- (ZF_ss addsimps [premL, le_imp_subset, subset_Un_iff RS iffD1]) 1);
+ (!simpset addsimps [premL, le_imp_subset, subset_Un_iff RS iffD1]) 1);
by (asm_simp_tac
- (ZF_ss addsimps [premK, le_imp_subset, subset_Un_iff2 RS iffD1]) 1);
+ (!simpset addsimps [premK, le_imp_subset, subset_Un_iff2 RS iffD1]) 1);
qed "Card_Un";
(*Infinite unions of cardinals? See Devlin, Lemma 6.7, page 98*)
@@ -351,7 +351,7 @@
by (excluded_middle_tac "EX i. Ord(i) & i eqpoll A" 1);
by (etac (Least_0 RS ssubst) 1 THEN rtac Card_0 1);
by (rtac (Ord_Least RS CardI) 1);
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (rtac less_LeastE 1);
by (assume_tac 2);
by (etac eqpoll_trans 1);
@@ -388,16 +388,16 @@
qed "cardinal_lt_imp_lt";
goal Cardinal.thy "!!i j. [| |i| < K; Ord(i); Card(K) |] ==> i < K";
-by (asm_simp_tac (ZF_ss addsimps
+by (asm_simp_tac (!simpset addsimps
[cardinal_lt_imp_lt, Card_is_Ord, Card_cardinal_eq]) 1);
qed "Card_lt_imp_lt";
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (|i| < K) <-> (i < K)";
-by (fast_tac (ZF_cs addEs [Card_lt_imp_lt, Ord_cardinal_le RS lt_trans1]) 1);
+by (fast_tac (!claset addEs [Card_lt_imp_lt, Ord_cardinal_le RS lt_trans1]) 1);
qed "Card_lt_iff";
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (K le |i|) <-> (K le i)";
-by (asm_simp_tac (ZF_ss addsimps
+by (asm_simp_tac (!simpset addsimps
[Card_lt_iff, Card_is_Ord, Ord_cardinal,
not_lt_iff_le RS iff_sym]) 1);
qed "Card_le_iff";
@@ -433,22 +433,22 @@
goalw Cardinal.thy [lepoll_def, inj_def]
"!!A B. [| cons(u,A) lepoll cons(v,B); u~:A; v~:B |] ==> A lepoll B";
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (res_inst_tac [("x", "lam x:A. if(f`x=v, f`u, f`x)")] exI 1);
by (rtac CollectI 1);
(*Proving it's in the function space A->B*)
by (rtac (if_type RS lam_type) 1);
-by (fast_tac (ZF_cs addEs [apply_funtype RS consE]) 1);
-by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [apply_funtype RS consE]) 1);
+by (fast_tac (!claset addEs [apply_funtype RS consE]) 1);
+by (fast_tac (!claset addSEs [mem_irrefl] addEs [apply_funtype RS consE]) 1);
(*Proving it's injective*)
-by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1);
-by (fast_tac ZF_cs 1);
+by (asm_simp_tac (!simpset setloop split_tac [expand_if]) 1);
+by (Fast_tac 1);
qed "cons_lepoll_consD";
goal Cardinal.thy
"!!A B. [| cons(u,A) eqpoll cons(v,B); u~:A; v~:B |] ==> A eqpoll B";
-by (asm_full_simp_tac (ZF_ss addsimps [eqpoll_iff]) 1);
-by (fast_tac (ZF_cs addIs [cons_lepoll_consD]) 1);
+by (asm_full_simp_tac (!simpset addsimps [eqpoll_iff]) 1);
+by (fast_tac (!claset addIs [cons_lepoll_consD]) 1);
qed "cons_eqpoll_consD";
(*Lemma suggested by Mike Fourman*)
@@ -460,12 +460,12 @@
val [prem] = goal Cardinal.thy
"m:nat ==> ALL n: nat. m lepoll n --> m le n";
by (nat_ind_tac "m" [prem] 1);
-by (fast_tac (ZF_cs addSIs [nat_0_le]) 1);
+by (fast_tac (!claset addSIs [nat_0_le]) 1);
by (rtac ballI 1);
by (eres_inst_tac [("n","n")] natE 1);
-by (asm_simp_tac (ZF_ss addsimps [lepoll_def, inj_def,
+by (asm_simp_tac (!simpset addsimps [lepoll_def, inj_def,
succI1 RS Pi_empty2]) 1);
-by (fast_tac (ZF_cs addSIs [succ_leI] addSDs [succ_lepoll_succD]) 1);
+by (fast_tac (!claset addSIs [succ_leI] addSDs [succ_lepoll_succD]) 1);
qed "nat_lepoll_imp_le_lemma";
bind_thm ("nat_lepoll_imp_le", nat_lepoll_imp_le_lemma RS bspec RS mp);
@@ -473,8 +473,8 @@
goal Cardinal.thy
"!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n";
by (rtac iffI 1);
-by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2);
-by (fast_tac (ZF_cs addIs [nat_lepoll_imp_le, le_anti_sym]
+by (asm_simp_tac (!simpset addsimps [eqpoll_refl]) 2);
+by (fast_tac (!claset addIs [nat_lepoll_imp_le, le_anti_sym]
addSEs [eqpollE]) 1);
qed "nat_eqpoll_iff";
@@ -483,8 +483,8 @@
"!!n. n: nat ==> Card(n)";
by (stac Least_equality 1);
by (REPEAT_FIRST (ares_tac [eqpoll_refl, nat_into_Ord, refl]));
-by (asm_simp_tac (ZF_ss addsimps [lt_nat_in_nat RS nat_eqpoll_iff]) 1);
-by (fast_tac (ZF_cs addSEs [lt_irrefl]) 1);
+by (asm_simp_tac (!simpset addsimps [lt_nat_in_nat RS nat_eqpoll_iff]) 1);
+by (fast_tac (!claset addSEs [lt_irrefl]) 1);
qed "nat_into_Card";
(*Part of Kunen's Lemma 10.6*)
@@ -499,7 +499,7 @@
goalw Cardinal.thy [lesspoll_def]
"!!m. [| A lepoll m; m:nat |] ==> A lesspoll succ(m)";
by (rtac conjI 1);
-by (fast_tac (ZF_cs addIs [subset_imp_lepoll RSN (2,lepoll_trans)]) 1);
+by (fast_tac (!claset addIs [subset_imp_lepoll RSN (2,lepoll_trans)]) 1);
by (rtac notI 1);
by (dresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll] 1);
by (dtac lepoll_trans 1 THEN assume_tac 1);
@@ -508,12 +508,12 @@
goalw Cardinal.thy [lesspoll_def, lepoll_def, eqpoll_def, bij_def]
"!!m. [| A lesspoll succ(m); m:nat |] ==> A lepoll m";
-by (step_tac ZF_cs 1);
-by (fast_tac (ZF_cs addSIs [inj_not_surj_succ]) 1);
+by (step_tac (!claset) 1);
+by (fast_tac (!claset addSIs [inj_not_surj_succ]) 1);
qed "lesspoll_succ_imp_lepoll";
goal Cardinal.thy "!!m. m:nat ==> A lesspoll succ(m) <-> A lepoll m";
-by (fast_tac (ZF_cs addSIs [lepoll_imp_lesspoll_succ,
+by (fast_tac (!claset addSIs [lepoll_imp_lesspoll_succ,
lesspoll_succ_imp_lepoll]) 1);
qed "lesspoll_succ_iff";
@@ -522,7 +522,7 @@
by (rtac disjCI 1);
by (rtac lesspoll_succ_imp_lepoll 1);
by (assume_tac 2);
-by (asm_simp_tac (ZF_ss addsimps [lesspoll_def]) 1);
+by (asm_simp_tac (!simpset addsimps [lesspoll_def]) 1);
qed "lepoll_succ_disj";
@@ -539,7 +539,7 @@
goal Cardinal.thy "!!i n. [| Ord(i); n:nat |] ==> i eqpoll n <-> i=n";
by (rtac iffI 1);
-by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2);
+by (asm_simp_tac (!simpset addsimps [eqpoll_refl]) 2);
by (rtac Ord_linear_lt 1);
by (REPEAT_SOME (eresolve_tac [asm_rl, nat_into_Ord]));
by (etac (lt_nat_in_nat RS nat_eqpoll_iff RS iffD1) 1 THEN
@@ -552,7 +552,7 @@
by (stac Least_equality 1);
by (REPEAT_FIRST (ares_tac [eqpoll_refl, Ord_nat, refl]));
by (etac ltE 1);
-by (asm_simp_tac (ZF_ss addsimps [eqpoll_iff, lt_not_lepoll, ltI]) 1);
+by (asm_simp_tac (!simpset addsimps [eqpoll_iff, lt_not_lepoll, ltI]) 1);
qed "Card_nat";
(*Allows showing that |i| is a limit cardinal*)
@@ -568,40 +568,40 @@
(*Congruence law for cons under equipollence*)
goalw Cardinal.thy [lepoll_def]
"!!A B. [| A lepoll B; b ~: B |] ==> cons(a,A) lepoll cons(b,B)";
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (res_inst_tac [("x", "lam y: cons(a,A).if(y=a, b, f`y)")] exI 1);
by (res_inst_tac [("d","%z.if(z:B, converse(f)`z, a)")]
lam_injective 1);
-by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_type, cons_iff]
+by (asm_simp_tac (!simpset addsimps [inj_is_fun RS apply_type, cons_iff]
setloop etac consE') 1);
-by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_type, left_inverse]
+by (asm_simp_tac (!simpset addsimps [inj_is_fun RS apply_type, left_inverse]
setloop etac consE') 1);
qed "cons_lepoll_cong";
goal Cardinal.thy
"!!A B. [| A eqpoll B; a ~: A; b ~: B |] ==> cons(a,A) eqpoll cons(b,B)";
-by (asm_full_simp_tac (ZF_ss addsimps [eqpoll_iff, cons_lepoll_cong]) 1);
+by (asm_full_simp_tac (!simpset addsimps [eqpoll_iff, cons_lepoll_cong]) 1);
qed "cons_eqpoll_cong";
goal Cardinal.thy
"!!A B. [| a ~: A; b ~: B |] ==> \
\ cons(a,A) lepoll cons(b,B) <-> A lepoll B";
-by (fast_tac (ZF_cs addIs [cons_lepoll_cong, cons_lepoll_consD]) 1);
+by (fast_tac (!claset addIs [cons_lepoll_cong, cons_lepoll_consD]) 1);
qed "cons_lepoll_cons_iff";
goal Cardinal.thy
"!!A B. [| a ~: A; b ~: B |] ==> \
\ cons(a,A) eqpoll cons(b,B) <-> A eqpoll B";
-by (fast_tac (ZF_cs addIs [cons_eqpoll_cong, cons_eqpoll_consD]) 1);
+by (fast_tac (!claset addIs [cons_eqpoll_cong, cons_eqpoll_consD]) 1);
qed "cons_eqpoll_cons_iff";
goalw Cardinal.thy [succ_def] "{a} eqpoll 1";
-by (fast_tac (ZF_cs addSIs [eqpoll_refl RS cons_eqpoll_cong]) 1);
+by (fast_tac (!claset addSIs [eqpoll_refl RS cons_eqpoll_cong]) 1);
qed "singleton_eqpoll_1";
goal Cardinal.thy "|{a}| = 1";
by (resolve_tac [singleton_eqpoll_1 RS cardinal_cong RS trans] 1);
-by (simp_tac (arith_ss addsimps [nat_into_Card RS Card_cardinal_eq]) 1);
+by (simp_tac (!simpset addsimps [nat_into_Card RS Card_cardinal_eq]) 1);
qed "cardinal_singleton";
(*Congruence law for succ under equipollence*)
@@ -613,13 +613,13 @@
(*Congruence law for + under equipollence*)
goalw Cardinal.thy [eqpoll_def]
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D";
-by (fast_tac (ZF_cs addSIs [sum_bij]) 1);
+by (fast_tac (!claset addSIs [sum_bij]) 1);
qed "sum_eqpoll_cong";
(*Congruence law for * under equipollence*)
goalw Cardinal.thy [eqpoll_def]
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D";
-by (fast_tac (ZF_cs addSIs [prod_bij]) 1);
+by (fast_tac (!claset addSIs [prod_bij]) 1);
qed "prod_eqpoll_cong";
goalw Cardinal.thy [eqpoll_def]
@@ -628,16 +628,16 @@
by (res_inst_tac [("c", "%x. if(x:A, f`x, x)"),
("d", "%y. if(y: range(f), converse(f)`y, y)")]
lam_bijective 1);
-by (fast_tac (ZF_cs addSIs [if_type, apply_type] addIs [inj_is_fun]) 1);
+by (fast_tac (!claset addSIs [if_type, apply_type] addIs [inj_is_fun]) 1);
by (asm_simp_tac
- (ZF_ss addsimps [inj_converse_fun RS apply_funtype]
+ (!simpset addsimps [inj_converse_fun RS apply_funtype]
setloop split_tac [expand_if]) 1);
-by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_rangeI, left_inverse]
+by (asm_simp_tac (!simpset addsimps [inj_is_fun RS apply_rangeI, left_inverse]
setloop etac UnE') 1);
by (asm_simp_tac
- (ZF_ss addsimps [inj_converse_fun RS apply_funtype, right_inverse]
+ (!simpset addsimps [inj_converse_fun RS apply_funtype, right_inverse]
setloop split_tac [expand_if]) 1);
-by (fast_tac (ZF_cs addEs [equals0D]) 1);
+by (fast_tac (!claset addEs [equals0D]) 1);
qed "inj_disjoint_eqpoll";
@@ -650,7 +650,7 @@
by (rtac cons_lepoll_consD 1);
by (rtac mem_not_refl 3);
by (eresolve_tac [cons_Diff RS ssubst] 1);
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
qed "Diff_sing_lepoll";
(*If A has at least n+1 elements then A-{a} has at least n.*)
@@ -658,12 +658,12 @@
"!!A a n. [| succ(n) lepoll A |] ==> n lepoll A - {a}";
by (rtac cons_lepoll_consD 1);
by (rtac mem_not_refl 2);
-by (fast_tac ZF_cs 2);
-by (fast_tac (ZF_cs addSEs [subset_imp_lepoll RSN (2, lepoll_trans)]) 1);
+by (Fast_tac 2);
+by (fast_tac (!claset addSEs [subset_imp_lepoll RSN (2, lepoll_trans)]) 1);
qed "lepoll_Diff_sing";
goal Cardinal.thy "!!A a n. [| a:A; A eqpoll succ(n) |] ==> A - {a} eqpoll n";
-by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE]
+by (fast_tac (!claset addSIs [eqpollI] addSEs [eqpollE]
addIs [Diff_sing_lepoll,lepoll_Diff_sing]) 1);
qed "Diff_sing_eqpoll";
@@ -678,8 +678,8 @@
by (res_inst_tac [("x","lam x: A Un B. if (x:A,Inl(x),Inr(x))")] exI 1);
by (res_inst_tac [("d","%z. snd(z)")] lam_injective 1);
by (split_tac [expand_if] 1);
-by (fast_tac (ZF_cs addSIs [InlI, InrI]) 1);
-by (asm_full_simp_tac (ZF_ss addsimps [Inl_def, Inr_def]
+by (fast_tac (!claset addSIs [InlI, InrI]) 1);
+by (asm_full_simp_tac (!simpset addsimps [Inl_def, Inr_def]
setloop split_tac [expand_if]) 1);
qed "Un_lepoll_sum";
@@ -687,20 +687,20 @@
(*** Finite and infinite sets ***)
goalw Cardinal.thy [Finite_def] "Finite(0)";
-by (fast_tac (ZF_cs addSIs [eqpoll_refl, nat_0I]) 1);
+by (fast_tac (!claset addSIs [eqpoll_refl, nat_0I]) 1);
qed "Finite_0";
goalw Cardinal.thy [Finite_def]
"!!A. [| A lepoll n; n:nat |] ==> Finite(A)";
by (etac rev_mp 1);
by (etac nat_induct 1);
-by (fast_tac (ZF_cs addSDs [lepoll_0_is_0] addSIs [eqpoll_refl,nat_0I]) 1);
-by (fast_tac (ZF_cs addSDs [lepoll_succ_disj] addSIs [nat_succI]) 1);
+by (fast_tac (!claset addSDs [lepoll_0_is_0] addSIs [eqpoll_refl,nat_0I]) 1);
+by (fast_tac (!claset addSDs [lepoll_succ_disj] addSIs [nat_succI]) 1);
qed "lepoll_nat_imp_Finite";
goalw Cardinal.thy [Finite_def]
"!!X. [| Y lepoll X; Finite(X) |] ==> Finite(Y)";
-by (fast_tac (ZF_cs addSEs [eqpollE]
+by (fast_tac (!claset addSEs [eqpollE]
addEs [lepoll_trans RS
rewrite_rule [Finite_def] lepoll_nat_imp_Finite]) 1);
qed "lepoll_Finite";
@@ -711,12 +711,12 @@
goalw Cardinal.thy [Finite_def] "!!x. Finite(x) ==> Finite(cons(y,x))";
by (excluded_middle_tac "y:x" 1);
-by (asm_simp_tac (ZF_ss addsimps [cons_absorb]) 2);
+by (asm_simp_tac (!simpset addsimps [cons_absorb]) 2);
by (etac bexE 1);
by (rtac bexI 1);
by (etac nat_succI 2);
by (asm_simp_tac
- (ZF_ss addsimps [succ_def, cons_eqpoll_cong, mem_not_refl]) 1);
+ (!simpset addsimps [succ_def, cons_eqpoll_cong, mem_not_refl]) 1);
qed "Finite_cons";
goalw Cardinal.thy [succ_def] "!!x. Finite(x) ==> Finite(succ(x))";
@@ -727,12 +727,12 @@
"!!i. [| Ord(i); ~ Finite(i) |] ==> nat le i";
by (eresolve_tac [Ord_nat RSN (2,Ord_linear2)] 1);
by (assume_tac 2);
-by (fast_tac (ZF_cs addSIs [eqpoll_refl] addSEs [ltE]) 1);
+by (fast_tac (!claset addSIs [eqpoll_refl] addSEs [ltE]) 1);
qed "nat_le_infinite_Ord";
goalw Cardinal.thy [Finite_def, eqpoll_def]
"!!A. Finite(A) ==> EX r. well_ord(A,r)";
-by (fast_tac (ZF_cs addIs [well_ord_rvimage, bij_is_inj, well_ord_Memrel,
+by (fast_tac (!claset addIs [well_ord_rvimage, bij_is_inj, well_ord_Memrel,
nat_into_Ord]) 1);
qed "Finite_imp_well_ord";
@@ -742,22 +742,22 @@
goal Nat.thy "!!n. n:nat ==> wf[n](converse(Memrel(n)))";
by (etac nat_induct 1);
-by (fast_tac (ZF_cs addIs [wf_onI]) 1);
+by (fast_tac (!claset addIs [wf_onI]) 1);
by (rtac wf_onI 1);
-by (asm_full_simp_tac (ZF_ss addsimps [wf_on_def, wf_def, Memrel_iff]) 1);
+by (asm_full_simp_tac (!simpset addsimps [wf_on_def, wf_def, Memrel_iff]) 1);
by (excluded_middle_tac "x:Z" 1);
by (dres_inst_tac [("x", "x")] bspec 2 THEN assume_tac 2);
-by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [mem_asym]) 2);
+by (fast_tac (!claset addSEs [mem_irrefl] addEs [mem_asym]) 2);
by (dres_inst_tac [("x", "Z")] spec 1);
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (dres_inst_tac [("x", "xa")] bspec 1 THEN assume_tac 1);
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "nat_wf_on_converse_Memrel";
goal Cardinal.thy "!!n. n:nat ==> well_ord(n,converse(Memrel(n)))";
by (forward_tac [Ord_nat RS Ord_in_Ord RS well_ord_Memrel] 1);
by (rewtac well_ord_def);
-by (fast_tac (ZF_cs addSIs [tot_ord_converse, nat_wf_on_converse_Memrel]) 1);
+by (fast_tac (!claset addSIs [tot_ord_converse, nat_wf_on_converse_Memrel]) 1);
qed "nat_well_ord_converse_Memrel";
goal Cardinal.thy
@@ -768,7 +768,7 @@
by (forward_tac [ordermap_bij RS bij_is_inj RS well_ord_rvimage] 1);
by (assume_tac 1);
by (asm_full_simp_tac
- (ZF_ss addsimps [rvimage_converse, converse_Int, converse_prod,
+ (!simpset addsimps [rvimage_converse, converse_Int, converse_prod,
ordertype_ord_iso RS ord_iso_rvimage_eq]) 1);
qed "well_ord_converse";
@@ -778,12 +778,12 @@
REPEAT (assume_tac 1));
by (rtac eqpoll_trans 1 THEN assume_tac 2);
by (rewtac eqpoll_def);
-by (fast_tac (ZF_cs addSIs [ordermap_bij RS bij_converse_bij]) 1);
+by (fast_tac (!claset addSIs [ordermap_bij RS bij_converse_bij]) 1);
qed "ordertype_eq_n";
goalw Cardinal.thy [Finite_def]
"!!A. [| Finite(A); well_ord(A,r) |] ==> well_ord(A,converse(r))";
by (rtac well_ord_converse 1 THEN assume_tac 1);
-by (fast_tac (ZF_cs addDs [ordertype_eq_n]
+by (fast_tac (!claset addDs [ordertype_eq_n]
addSIs [nat_well_ord_converse_Memrel]) 1);
qed "Finite_well_ord_converse";