--- a/src/ZF/Cardinal.ML Mon Jul 13 16:42:27 1998 +0200
+++ b/src/ZF/Cardinal.ML Mon Jul 13 16:43:57 1998 +0200
@@ -54,25 +54,25 @@
(** Equipollence is an equivalence relation **)
-Goalw [eqpoll_def] "!!f A B. f: bij(A,B) ==> A eqpoll B";
+Goalw [eqpoll_def] "f: bij(A,B) ==> A eqpoll B";
by (etac exI 1);
qed "bij_imp_eqpoll";
(*A eqpoll A*)
bind_thm ("eqpoll_refl", id_bij RS bij_imp_eqpoll);
-Goalw [eqpoll_def] "!!X Y. X eqpoll Y ==> Y eqpoll X";
+Goalw [eqpoll_def] "X eqpoll Y ==> Y eqpoll X";
by (blast_tac (claset() addIs [bij_converse_bij]) 1);
qed "eqpoll_sym";
Goalw [eqpoll_def]
- "!!X Y. [| X eqpoll Y; Y eqpoll Z |] ==> X eqpoll Z";
+ "[| X eqpoll Y; Y eqpoll Z |] ==> X eqpoll Z";
by (blast_tac (claset() addIs [comp_bij]) 1);
qed "eqpoll_trans";
(** Le-pollence is a partial ordering **)
-Goalw [lepoll_def] "!!X Y. X<=Y ==> X lepoll Y";
+Goalw [lepoll_def] "X<=Y ==> X lepoll Y";
by (rtac exI 1);
by (etac id_subset_inj 1);
qed "subset_imp_lepoll";
@@ -82,18 +82,18 @@
bind_thm ("le_imp_lepoll", le_imp_subset RS subset_imp_lepoll);
Goalw [eqpoll_def, bij_def, lepoll_def]
- "!!X Y. X eqpoll Y ==> X lepoll Y";
+ "X eqpoll Y ==> X lepoll Y";
by (Blast_tac 1);
qed "eqpoll_imp_lepoll";
Goalw [lepoll_def]
- "!!X Y. [| X lepoll Y; Y lepoll Z |] ==> X lepoll Z";
+ "[| X lepoll Y; Y lepoll Z |] ==> X lepoll Z";
by (blast_tac (claset() addIs [comp_inj]) 1);
qed "lepoll_trans";
(*Asymmetry law*)
Goalw [lepoll_def,eqpoll_def]
- "!!X Y. [| X lepoll Y; Y lepoll X |] ==> X eqpoll Y";
+ "[| X lepoll Y; Y lepoll X |] ==> X eqpoll Y";
by (REPEAT (etac exE 1));
by (rtac schroeder_bernstein 1);
by (REPEAT (assume_tac 1));
@@ -121,7 +121,7 @@
qed "lepoll_0_iff";
Goalw [lepoll_def]
- "!!A. [| A lepoll B; C lepoll D; B Int D = 0 |] ==> A Un C lepoll B Un D";
+ "[| A lepoll B; C lepoll D; B Int D = 0 |] ==> A Un C lepoll B Un D";
by (blast_tac (claset() addIs [inj_disjoint_Un]) 1);
qed "Un_lepoll_Un";
@@ -133,7 +133,7 @@
qed "eqpoll_0_iff";
Goalw [eqpoll_def]
- "!!A. [| A eqpoll B; C eqpoll D; A Int C = 0; B Int D = 0 |] ==> \
+ "[| A eqpoll B; C eqpoll D; A Int C = 0; B Int D = 0 |] ==> \
\ A Un C eqpoll B Un D";
by (blast_tac (claset() addIs [bij_disjoint_Un]) 1);
qed "eqpoll_disjoint_Un";
@@ -141,12 +141,12 @@
(*** lesspoll: contributions by Krzysztof Grabczewski ***)
-Goalw [lesspoll_def] "!!A. A lesspoll B ==> A lepoll B";
+Goalw [lesspoll_def] "A lesspoll B ==> A lepoll B";
by (Blast_tac 1);
qed "lesspoll_imp_lepoll";
Goalw [lepoll_def]
- "!!A. [| A lepoll B; well_ord(B,r) |] ==> EX s. well_ord(A,s)";
+ "[| A lepoll B; well_ord(B,r) |] ==> EX s. well_ord(A,s)";
by (blast_tac (claset() addIs [well_ord_rvimage]) 1);
qed "lepoll_well_ord";
@@ -155,31 +155,31 @@
qed "lepoll_iff_leqpoll";
Goalw [inj_def, surj_def]
- "!!f. [| f : inj(A, succ(m)); f ~: surj(A, succ(m)) |] ==> EX f. f:inj(A,m)";
+ "[| f : inj(A, succ(m)); f ~: surj(A, succ(m)) |] ==> EX f. f:inj(A,m)";
by (safe_tac (claset_of ZF.thy));
by (swap_res_tac [exI] 1);
by (res_inst_tac [("a", "lam z:A. if(f`z=m, y, f`z)")] CollectI 1);
by (best_tac (claset() addSIs [if_type RS lam_type]
addEs [apply_funtype RS succE]) 1);
(*Proving it's injective*)
-by (asm_simp_tac (simpset() addsplits [split_if]) 1);
+by (Asm_simp_tac 1);
by (blast_tac (claset() delrules [equalityI]) 1);
qed "inj_not_surj_succ";
(** Variations on transitivity **)
Goalw [lesspoll_def]
- "!!X. [| X lesspoll Y; Y lesspoll Z |] ==> X lesspoll Z";
+ "[| X lesspoll Y; Y lesspoll Z |] ==> X lesspoll Z";
by (blast_tac (claset() addSEs [eqpollE] addIs [eqpollI, lepoll_trans]) 1);
qed "lesspoll_trans";
Goalw [lesspoll_def]
- "!!X. [| X lesspoll Y; Y lepoll Z |] ==> X lesspoll Z";
+ "[| X lesspoll Y; Y lepoll Z |] ==> X lesspoll Z";
by (blast_tac (claset() addSEs [eqpollE] addIs [eqpollI, lepoll_trans]) 1);
qed "lesspoll_lepoll_lesspoll";
Goalw [lesspoll_def]
- "!!X. [| X lesspoll Y; Z lepoll X |] ==> Z lesspoll Y";
+ "[| X lesspoll Y; Z lepoll X |] ==> Z lesspoll Y";
by (blast_tac (claset() addSEs [eqpollE] addIs [eqpollI, lepoll_trans]) 1);
qed "lepoll_lesspoll_lesspoll";
@@ -195,7 +195,7 @@
by (ALLGOALS (blast_tac (claset() addSIs [premP] addSDs [premNot])));
qed "Least_equality";
-Goal "!!i. [| P(i); Ord(i) |] ==> P(LEAST x. P(x))";
+Goal "[| P(i); Ord(i) |] ==> P(LEAST x. P(x))";
by (etac rev_mp 1);
by (trans_ind_tac "i" [] 1);
by (rtac impI 1);
@@ -206,7 +206,7 @@
qed "LeastI";
(*Proof is almost identical to the one above!*)
-Goal "!!i. [| P(i); Ord(i) |] ==> (LEAST x. P(x)) le i";
+Goal "[| P(i); Ord(i) |] ==> (LEAST x. P(x)) le i";
by (etac rev_mp 1);
by (trans_ind_tac "i" [] 1);
by (rtac impI 1);
@@ -217,7 +217,7 @@
qed "Least_le";
(*LEAST really is the smallest*)
-Goal "!!i. [| P(i); i < (LEAST x. P(x)) |] ==> Q";
+Goal "[| P(i); i < (LEAST x. P(x)) |] ==> Q";
by (rtac (Least_le RSN (2,lt_trans2) RS lt_irrefl) 1);
by (REPEAT (eresolve_tac [asm_rl, ltE] 1));
qed "less_LeastE";
@@ -232,7 +232,7 @@
(*If there is no such P then LEAST is vacuously 0*)
Goalw [Least_def]
- "!!P. [| ~ (EX i. Ord(i) & P(i)) |] ==> (LEAST x. P(x)) = 0";
+ "[| ~ (EX i. Ord(i) & P(i)) |] ==> (LEAST x. P(x)) = 0";
by (rtac the_0 1);
by (Blast_tac 1);
qed "Least_0";
@@ -264,7 +264,7 @@
(*Under AC, the premise becomes trivial; one consequence is ||A|| = |A|*)
Goalw [cardinal_def]
- "!!A. well_ord(A,r) ==> |A| eqpoll A";
+ "well_ord(A,r) ==> |A| eqpoll A";
by (rtac LeastI 1);
by (etac Ord_ordertype 2);
by (etac (ordermap_bij RS bij_converse_bij RS bij_imp_eqpoll) 1);
@@ -274,25 +274,25 @@
bind_thm ("Ord_cardinal_eqpoll", well_ord_Memrel RS well_ord_cardinal_eqpoll);
Goal
- "!!X Y. [| well_ord(X,r); well_ord(Y,s); |X| = |Y| |] ==> X eqpoll 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 (simpset() addsimps [well_ord_cardinal_eqpoll]) 1);
qed "well_ord_cardinal_eqE";
Goal
- "!!X Y. [| well_ord(X,r); well_ord(Y,s) |] ==> |X| = |Y| <-> X eqpoll Y";
+ "[| well_ord(X,r); well_ord(Y,s) |] ==> |X| = |Y| <-> X eqpoll Y";
by (blast_tac (claset() addIs [cardinal_cong, well_ord_cardinal_eqE]) 1);
qed "well_ord_cardinal_eqpoll_iff";
(** Observations from Kunen, page 28 **)
-Goalw [cardinal_def] "!!i. Ord(i) ==> |i| le i";
+Goalw [cardinal_def] "Ord(i) ==> |i| le i";
by (etac (eqpoll_refl RS Least_le) 1);
qed "Ord_cardinal_le";
-Goalw [Card_def] "!!K. Card(K) ==> |K| = K";
+Goalw [Card_def] "Card(K) ==> |K| = K";
by (etac sym 1);
qed "Card_cardinal_eq";
@@ -308,7 +308,7 @@
by (rtac Ord_Least 1);
qed "Card_is_Ord";
-Goal "!!K. Card(K) ==> K le |K|";
+Goal "Card(K) ==> K le |K|";
by (asm_simp_tac (simpset() addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
qed "Card_cardinal_le";
@@ -326,7 +326,7 @@
by (ALLGOALS assume_tac);
qed "Card_iff_initial";
-Goalw [lesspoll_def] "!!a. [| Card(a); i<a |] ==> i lesspoll a";
+Goalw [lesspoll_def] "[| Card(a); i<a |] ==> i lesspoll a";
by (dresolve_tac [Card_iff_initial RS iffD1] 1);
by (blast_tac (claset() addSIs [leI RS le_imp_lepoll]) 1);
qed "lt_Card_imp_lesspoll";
@@ -359,7 +359,7 @@
qed "Card_cardinal";
(*Kunen's Lemma 10.5*)
-Goal "!!i j. [| |i| le j; j le i |] ==> |j| = |i|";
+Goal "[| |i| le j; j le i |] ==> |j| = |i|";
by (rtac (eqpollI RS cardinal_cong) 1);
by (etac le_imp_lepoll 1);
by (rtac lepoll_trans 1);
@@ -369,7 +369,7 @@
by (REPEAT (eresolve_tac [ltE, Ord_succD] 1));
qed "cardinal_eq_lemma";
-Goal "!!i j. i le j ==> |i| le |j|";
+Goal "i le j ==> |i| le |j|";
by (res_inst_tac [("i","|i|"),("j","|j|")] Ord_linear_le 1);
by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI]));
by (rtac cardinal_eq_lemma 1);
@@ -380,23 +380,23 @@
qed "cardinal_mono";
(*Since we have |succ(nat)| le |nat|, the converse of cardinal_mono fails!*)
-Goal "!!i j. [| |i| < |j|; Ord(i); Ord(j) |] ==> i < j";
+Goal "[| |i| < |j|; Ord(i); Ord(j) |] ==> i < j";
by (rtac Ord_linear2 1);
by (REPEAT_SOME assume_tac);
by (etac (lt_trans2 RS lt_irrefl) 1);
by (etac cardinal_mono 1);
qed "cardinal_lt_imp_lt";
-Goal "!!i j. [| |i| < K; Ord(i); Card(K) |] ==> i < K";
+Goal "[| |i| < K; Ord(i); Card(K) |] ==> i < K";
by (asm_simp_tac (simpset() addsimps
[cardinal_lt_imp_lt, Card_is_Ord, Card_cardinal_eq]) 1);
qed "Card_lt_imp_lt";
-Goal "!!i j. [| Ord(i); Card(K) |] ==> (|i| < K) <-> (i < K)";
+Goal "[| Ord(i); Card(K) |] ==> (|i| < K) <-> (i < K)";
by (blast_tac (claset() addIs [Card_lt_imp_lt, Ord_cardinal_le RS lt_trans1]) 1);
qed "Card_lt_iff";
-Goal "!!i j. [| Ord(i); Card(K) |] ==> (K le |i|) <-> (K le i)";
+Goal "[| Ord(i); Card(K) |] ==> (K le |i|) <-> (K le i)";
by (asm_simp_tac (simpset() addsimps
[Card_lt_iff, Card_is_Ord, Ord_cardinal,
not_lt_iff_le RS iff_sym]) 1);
@@ -404,7 +404,7 @@
(*Can use AC or finiteness to discharge first premise*)
Goal
- "!!A B. [| well_ord(B,r); A lepoll B |] ==> |A| le |B|";
+ "[| well_ord(B,r); A lepoll B |] ==> |A| le |B|";
by (res_inst_tac [("i","|A|"),("j","|B|")] Ord_linear_le 1);
by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI]));
by (rtac (eqpollI RS cardinal_cong) 1 THEN assume_tac 1);
@@ -421,7 +421,7 @@
qed "well_ord_lepoll_imp_Card_le";
-Goal "!!A. [| A lepoll i; Ord(i) |] ==> |A| le i";
+Goal "[| A lepoll i; Ord(i) |] ==> |A| le i";
by (rtac le_trans 1);
by (etac (well_ord_Memrel RS well_ord_lepoll_imp_Card_le) 1);
by (assume_tac 1);
@@ -432,7 +432,7 @@
(*** The finite cardinals ***)
Goalw [lepoll_def, inj_def]
- "!!A B. [| cons(u,A) lepoll cons(v,B); u~:A; v~:B |] ==> A lepoll B";
+ "[| cons(u,A) lepoll cons(v,B); u~:A; v~:B |] ==> A lepoll B";
by Safe_tac;
by (res_inst_tac [("x", "lam x:A. if(f`x=v, f`u, f`x)")] exI 1);
by (rtac CollectI 1);
@@ -441,18 +441,18 @@
by (blast_tac (claset() addEs [apply_funtype RS consE]) 1);
by (blast_tac (claset() addSEs [mem_irrefl] addEs [apply_funtype RS consE]) 1);
(*Proving it's injective*)
-by (asm_simp_tac (simpset() addsplits [split_if]) 1);
+by (Asm_simp_tac 1);
by (Blast_tac 1);
qed "cons_lepoll_consD";
Goal
- "!!A B. [| cons(u,A) eqpoll cons(v,B); u~:A; v~:B |] ==> A eqpoll B";
+ "[| cons(u,A) eqpoll cons(v,B); u~:A; v~:B |] ==> A eqpoll B";
by (asm_full_simp_tac (simpset() addsimps [eqpoll_iff]) 1);
by (blast_tac (claset() addIs [cons_lepoll_consD]) 1);
qed "cons_eqpoll_consD";
(*Lemma suggested by Mike Fourman*)
-Goalw [succ_def] "!!m n. succ(m) lepoll succ(n) ==> m lepoll n";
+Goalw [succ_def] "succ(m) lepoll succ(n) ==> m lepoll n";
by (etac cons_lepoll_consD 1);
by (REPEAT (rtac mem_not_refl 1));
qed "succ_lepoll_succD";
@@ -471,7 +471,7 @@
bind_thm ("nat_lepoll_imp_le", nat_lepoll_imp_le_lemma RS bspec RS mp);
Goal
- "!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n";
+ "[| m:nat; n: nat |] ==> m eqpoll n <-> m = n";
by (rtac iffI 1);
by (asm_simp_tac (simpset() addsimps [eqpoll_refl]) 2);
by (blast_tac (claset() addIs [nat_lepoll_imp_le, le_anti_sym]
@@ -480,7 +480,7 @@
(*The object of all this work: every natural number is a (finite) cardinal*)
Goalw [Card_def,cardinal_def]
- "!!n. n: nat ==> Card(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 (simpset() addsimps [lt_nat_in_nat RS nat_eqpoll_iff]) 1);
@@ -488,7 +488,7 @@
qed "nat_into_Card";
(*Part of Kunen's Lemma 10.6*)
-Goal "!!n. [| succ(n) lepoll n; n:nat |] ==> P";
+Goal "[| succ(n) lepoll n; n:nat |] ==> P";
by (rtac (nat_lepoll_imp_le RS lt_irrefl) 1);
by (REPEAT (ares_tac [nat_succI] 1));
qed "succ_lepoll_natE";
@@ -497,7 +497,7 @@
(** lepoll, lesspoll and natural numbers **)
Goalw [lesspoll_def]
- "!!m. [| A lepoll m; m:nat |] ==> A lesspoll succ(m)";
+ "[| A lepoll m; m:nat |] ==> A lesspoll succ(m)";
by (rtac conjI 1);
by (blast_tac (claset() addIs [subset_imp_lepoll RSN (2,lepoll_trans)]) 1);
by (rtac notI 1);
@@ -507,17 +507,17 @@
qed "lepoll_imp_lesspoll_succ";
Goalw [lesspoll_def, lepoll_def, eqpoll_def, bij_def]
- "!!m. [| A lesspoll succ(m); m:nat |] ==> A lepoll m";
+ "[| A lesspoll succ(m); m:nat |] ==> A lepoll m";
by (Clarify_tac 1);
by (blast_tac (claset() addSIs [inj_not_surj_succ]) 1);
qed "lesspoll_succ_imp_lepoll";
-Goal "!!m. m:nat ==> A lesspoll succ(m) <-> A lepoll m";
+Goal "m:nat ==> A lesspoll succ(m) <-> A lepoll m";
by (blast_tac (claset() addSIs [lepoll_imp_lesspoll_succ,
lesspoll_succ_imp_lepoll]) 1);
qed "lesspoll_succ_iff";
-Goal "!!A m. [| A lepoll succ(m); m:nat |] ==> \
+Goal "[| A lepoll succ(m); m:nat |] ==> \
\ A lepoll m | A eqpoll succ(m)";
by (rtac disjCI 1);
by (rtac lesspoll_succ_imp_lepoll 1);
@@ -529,7 +529,7 @@
(*** The first infinite cardinal: Omega, or nat ***)
(*This implies Kunen's Lemma 10.6*)
-Goal "!!n. [| n<i; n:nat |] ==> ~ i lepoll n";
+Goal "[| n<i; n:nat |] ==> ~ i lepoll n";
by (rtac notI 1);
by (rtac succ_lepoll_natE 1 THEN assume_tac 2);
by (rtac lepoll_trans 1 THEN assume_tac 2);
@@ -537,7 +537,7 @@
by (REPEAT (ares_tac [Ord_succ_subsetI RS subset_imp_lepoll] 1));
qed "lt_not_lepoll";
-Goal "!!i n. [| Ord(i); n:nat |] ==> i eqpoll n <-> i=n";
+Goal "[| Ord(i); n:nat |] ==> i eqpoll n <-> i=n";
by (rtac iffI 1);
by (asm_simp_tac (simpset() addsimps [eqpoll_refl]) 2);
by (rtac Ord_linear_lt 1);
@@ -556,7 +556,7 @@
qed "Card_nat";
(*Allows showing that |i| is a limit cardinal*)
-Goal "!!i. nat le i ==> nat le |i|";
+Goal "nat le i ==> nat le |i|";
by (rtac (Card_nat RS Card_cardinal_eq RS subst) 1);
by (etac cardinal_mono 1);
qed "nat_le_cardinal";
@@ -567,7 +567,7 @@
(*Congruence law for cons under equipollence*)
Goalw [lepoll_def]
- "!!A B. [| A lepoll B; b ~: B |] ==> cons(a,A) lepoll cons(b,B)";
+ "[| A lepoll B; b ~: B |] ==> cons(a,A) lepoll cons(b,B)";
by Safe_tac;
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)")]
@@ -579,18 +579,18 @@
qed "cons_lepoll_cong";
Goal
- "!!A B. [| A eqpoll B; a ~: A; b ~: B |] ==> cons(a,A) eqpoll cons(b,B)";
+ "[| A eqpoll B; a ~: A; b ~: B |] ==> cons(a,A) eqpoll cons(b,B)";
by (asm_full_simp_tac (simpset() addsimps [eqpoll_iff, cons_lepoll_cong]) 1);
qed "cons_eqpoll_cong";
Goal
- "!!A B. [| a ~: A; b ~: B |] ==> \
+ "[| a ~: A; b ~: B |] ==> \
\ cons(a,A) lepoll cons(b,B) <-> A lepoll B";
by (blast_tac (claset() addIs [cons_lepoll_cong, cons_lepoll_consD]) 1);
qed "cons_lepoll_cons_iff";
Goal
- "!!A B. [| a ~: A; b ~: B |] ==> \
+ "[| a ~: A; b ~: B |] ==> \
\ cons(a,A) eqpoll cons(b,B) <-> A eqpoll B";
by (blast_tac (claset() addIs [cons_eqpoll_cong, cons_eqpoll_consD]) 1);
qed "cons_eqpoll_cons_iff";
@@ -606,37 +606,35 @@
(*Congruence law for succ under equipollence*)
Goalw [succ_def]
- "!!A B. A eqpoll B ==> succ(A) eqpoll succ(B)";
+ "A eqpoll B ==> succ(A) eqpoll succ(B)";
by (REPEAT (ares_tac [cons_eqpoll_cong, mem_not_refl] 1));
qed "succ_eqpoll_cong";
(*Congruence law for + under equipollence*)
Goalw [eqpoll_def]
- "!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D";
+ "[| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D";
by (blast_tac (claset() addSIs [sum_bij]) 1);
qed "sum_eqpoll_cong";
(*Congruence law for * under equipollence*)
Goalw [eqpoll_def]
- "!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D";
+ "[| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D";
by (blast_tac (claset() addSIs [prod_bij]) 1);
qed "prod_eqpoll_cong";
Goalw [eqpoll_def]
- "!!f. [| f: inj(A,B); A Int B = 0 |] ==> A Un (B - range(f)) eqpoll B";
+ "[| f: inj(A,B); A Int B = 0 |] ==> A Un (B - range(f)) eqpoll B";
by (rtac exI 1);
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 (blast_tac (claset() addSIs [if_type, inj_is_fun RS apply_type]) 1);
by (asm_simp_tac
- (simpset() addsimps [inj_converse_fun RS apply_funtype]
- addsplits [split_if]) 1);
+ (simpset() addsimps [inj_converse_fun RS apply_funtype]) 1);
by (asm_simp_tac (simpset() addsimps [inj_is_fun RS apply_rangeI, left_inverse]
setloop etac UnE') 1);
by (asm_simp_tac
- (simpset() addsimps [inj_converse_fun RS apply_funtype, right_inverse]
- addsplits [split_if]) 1);
+ (simpset() addsimps [inj_converse_fun RS apply_funtype, right_inverse]) 1);
by (blast_tac (claset() addEs [equals0D]) 1);
qed "inj_disjoint_eqpoll";
@@ -646,7 +644,7 @@
(*If A has at most n+1 elements and a:A then A-{a} has at most n.*)
Goalw [succ_def]
- "!!A a n. [| a:A; A lepoll succ(n) |] ==> A - {a} lepoll n";
+ "[| a:A; A lepoll succ(n) |] ==> A - {a} lepoll n";
by (rtac cons_lepoll_consD 1);
by (rtac mem_not_refl 3);
by (eresolve_tac [cons_Diff RS ssubst] 1);
@@ -655,19 +653,19 @@
(*If A has at least n+1 elements then A-{a} has at least n.*)
Goalw [succ_def]
- "!!A a n. [| succ(n) lepoll A |] ==> n lepoll A - {a}";
+ "[| succ(n) lepoll A |] ==> n lepoll A - {a}";
by (rtac cons_lepoll_consD 1);
by (rtac mem_not_refl 2);
by (Blast_tac 2);
by (blast_tac (claset() addIs [subset_imp_lepoll RSN (2, lepoll_trans)]) 1);
qed "lepoll_Diff_sing";
-Goal "!!A a n. [| a:A; A eqpoll succ(n) |] ==> A - {a} eqpoll n";
+Goal "[| a:A; A eqpoll succ(n) |] ==> A - {a} eqpoll n";
by (blast_tac (claset() addSIs [eqpollI] addSEs [eqpollE]
addIs [Diff_sing_lepoll,lepoll_Diff_sing]) 1);
qed "Diff_sing_eqpoll";
-Goal "!!A. [| A lepoll 1; a:A |] ==> A = {a}";
+Goal "[| A lepoll 1; a:A |] ==> A = {a}";
by (forward_tac [Diff_sing_lepoll] 1);
by (assume_tac 1);
by (dtac lepoll_0_is_0 1);
@@ -679,8 +677,7 @@
by (res_inst_tac [("d","%z. snd(z)")] lam_injective 1);
by (split_tac [split_if] 1);
by (blast_tac (claset() addSIs [InlI, InrI]) 1);
-by (asm_full_simp_tac (simpset() addsimps [Inl_def, Inr_def]
- addsplits [split_if]) 1);
+by (asm_full_simp_tac (simpset() addsimps [Inl_def, Inr_def]) 1);
qed "Un_lepoll_sum";
@@ -691,15 +688,15 @@
qed "Finite_0";
Goalw [Finite_def]
- "!!A. [| A lepoll n; n:nat |] ==> Finite(A)";
+ "[| A lepoll n; n:nat |] ==> Finite(A)";
by (etac rev_mp 1);
by (etac nat_induct 1);
by (blast_tac (claset() addSDs [lepoll_0_is_0] addSIs [eqpoll_refl,nat_0I]) 1);
-by (blast_tac (claset() addSDs [lepoll_succ_disj] addSIs [nat_succI]) 1);
+by (blast_tac (claset() addSDs [lepoll_succ_disj]) 1);
qed "lepoll_nat_imp_Finite";
Goalw [Finite_def]
- "!!X. [| Y lepoll X; Finite(X) |] ==> Finite(Y)";
+ "[| Y lepoll X; Finite(X) |] ==> Finite(Y)";
by (blast_tac
(claset() addSEs [eqpollE]
addIs [lepoll_trans RS
@@ -710,7 +707,7 @@
bind_thm ("Finite_Diff", Diff_subset RS subset_Finite);
-Goalw [Finite_def] "!!x. Finite(x) ==> Finite(cons(y,x))";
+Goalw [Finite_def] "Finite(x) ==> Finite(cons(y,x))";
by (excluded_middle_tac "y:x" 1);
by (asm_simp_tac (simpset() addsimps [cons_absorb]) 2);
by (etac bexE 1);
@@ -720,19 +717,19 @@
(simpset() addsimps [succ_def, cons_eqpoll_cong, mem_not_refl]) 1);
qed "Finite_cons";
-Goalw [succ_def] "!!x. Finite(x) ==> Finite(succ(x))";
+Goalw [succ_def] "Finite(x) ==> Finite(succ(x))";
by (etac Finite_cons 1);
qed "Finite_succ";
Goalw [Finite_def]
- "!!i. [| Ord(i); ~ Finite(i) |] ==> nat le i";
+ "[| Ord(i); ~ Finite(i) |] ==> nat le i";
by (eresolve_tac [Ord_nat RSN (2,Ord_linear2)] 1);
by (assume_tac 2);
by (blast_tac (claset() addSIs [eqpoll_refl] addSEs [ltE]) 1);
qed "nat_le_infinite_Ord";
Goalw [Finite_def, eqpoll_def]
- "!!A. Finite(A) ==> EX r. well_ord(A,r)";
+ "Finite(A) ==> EX r. well_ord(A,r)";
by (blast_tac (claset() addIs [well_ord_rvimage, bij_is_inj, well_ord_Memrel,
nat_into_Ord]) 1);
qed "Finite_imp_well_ord";
@@ -753,7 +750,7 @@
by (Blast.depth_tac (claset()) 4 1);
qed "nat_wf_on_converse_Memrel";
-Goal "!!n. n:nat ==> well_ord(n,converse(Memrel(n)))";
+Goal "n:nat ==> well_ord(n,converse(Memrel(n)))";
by (forward_tac [transfer thy Ord_nat RS Ord_in_Ord RS well_ord_Memrel] 1);
by (rewtac well_ord_def);
by (blast_tac (claset() addSIs [tot_ord_converse,
@@ -761,7 +758,7 @@
qed "nat_well_ord_converse_Memrel";
Goal
- "!!A. [| well_ord(A,r); \
+ "[| well_ord(A,r); \
\ well_ord(ordertype(A,r), converse(Memrel(ordertype(A, r)))) \
\ |] ==> well_ord(A,converse(r))";
by (resolve_tac [well_ord_Int_iff RS iffD1] 1);
@@ -773,7 +770,7 @@
qed "well_ord_converse";
Goal
- "!!A. [| well_ord(A,r); A eqpoll n; n:nat |] ==> ordertype(A,r)=n";
+ "[| well_ord(A,r); A eqpoll n; n:nat |] ==> ordertype(A,r)=n";
by (rtac (Ord_ordertype RS Ord_nat_eqpoll_iff RS iffD1) 1 THEN
REPEAT (assume_tac 1));
by (rtac eqpoll_trans 1 THEN assume_tac 2);
@@ -782,7 +779,7 @@
qed "ordertype_eq_n";
Goalw [Finite_def]
- "!!A. [| Finite(A); well_ord(A,r) |] ==> well_ord(A,converse(r))";
+ "[| Finite(A); well_ord(A,r) |] ==> well_ord(A,converse(r))";
by (rtac well_ord_converse 1 THEN assume_tac 1);
by (blast_tac (claset() addDs [ordertype_eq_n]
addSIs [nat_well_ord_converse_Memrel]) 1);