--- a/src/HOL/Finite.ML Tue Jan 09 15:18:07 2001 +0100
+++ b/src/HOL/Finite.ML Tue Jan 09 15:22:13 2001 +0100
@@ -74,7 +74,7 @@
Addsimps[finite_insert];
(*The image of a finite set is finite *)
-Goal "finite F ==> finite(h``F)";
+Goal "finite F ==> finite(h`F)";
by (etac finite_induct 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
@@ -127,11 +127,11 @@
val lemma = result();
(*Lemma for proving finite_imageD*)
-Goal "finite B ==> ALL A. f``A = B --> inj_on f A --> finite A";
+Goal "finite B ==> ALL A. f`A = B --> inj_on f A --> finite A";
by (etac finite_induct 1);
by (ALLGOALS Asm_simp_tac);
by (Clarify_tac 1);
-by (subgoal_tac "EX y:A. f y = x & F = f``(A-{y})" 1);
+by (subgoal_tac "EX y:A. f y = x & F = f`(A-{y})" 1);
by (Clarify_tac 1);
by (full_simp_tac (simpset() addsimps [inj_on_def]) 1);
by (blast_tac (claset() addSDs [lemma RS iffD1]) 1);
@@ -143,7 +143,7 @@
(asm_full_simp_tac (simpset() addsimps [inj_on_image_set_diff])));
val lemma = result();
-Goal "[| finite(f``A); inj_on f A |] ==> finite A";
+Goal "[| finite(f`A); inj_on f A |] ==> finite A";
by (dtac lemma 1);
by (Blast_tac 1);
qed "finite_imageD";
@@ -195,7 +195,7 @@
(** The powerset of a finite set **)
Goal "finite(Pow A) ==> finite A";
-by (subgoal_tac "finite ((%x.{x})``A)" 1);
+by (subgoal_tac "finite ((%x.{x})`A)" 1);
by (rtac finite_subset 2);
by (assume_tac 3);
by (ALLGOALS
@@ -214,7 +214,7 @@
AddIffs [finite_Pow_iff];
Goal "finite(r^-1) = finite r";
-by (subgoal_tac "r^-1 = (%(x,y).(y,x))``r" 1);
+by (subgoal_tac "r^-1 = (%(x,y).(y,x))`r" 1);
by (Asm_simp_tac 1);
by (rtac iffI 1);
by (etac (rewrite_rule [inj_on_def] finite_imageD) 1);
@@ -467,14 +467,14 @@
(*** Cardinality of image ***)
-Goal "finite A ==> card (f `` A) <= card A";
+Goal "finite A ==> card (f ` A) <= card A";
by (etac finite_induct 1);
by (Simp_tac 1);
by (asm_simp_tac (simpset() addsimps [le_SucI, finite_imageI,
card_insert_if]) 1);
qed "card_image_le";
-Goal "finite(A) ==> inj_on f A --> card (f `` A) = card A";
+Goal "finite(A) ==> inj_on f A --> card (f ` A) = card A";
by (etac finite_induct 1);
by (ALLGOALS Asm_simp_tac);
by Safe_tac;
@@ -486,7 +486,7 @@
by (Blast_tac 1);
qed_spec_mp "card_image";
-Goal "[| finite A; f``A <= A; inj_on f A |] ==> f``A = A";
+Goal "[| finite A; f`A <= A; inj_on f A |] ==> f`A = A";
by (asm_simp_tac (simpset() addsimps [card_seteq, card_image]) 1);
qed "endo_inj_surj";
@@ -853,14 +853,14 @@
by (auto_tac (claset() addIs [finite_subset], simpset()));
qed "choose_deconstruct";
-Goal "[| finite(A); finite(B); f``A <= B; inj_on f A |] \
+Goal "[| finite(A); finite(B); f`A <= B; inj_on f A |] \
\ ==> card A <= card B";
by (auto_tac (claset() addIs [card_mono],
simpset() addsimps [card_image RS sym]));
qed "card_inj_on_le";
Goal "[| finite A; finite B; \
-\ f``A <= B; inj_on f A; g``B <= A; inj_on g B |] \
+\ f`A <= B; inj_on f A; g`B <= A; inj_on g B |] \
\ ==> card(A) = card(B)";
by (auto_tac (claset() addIs [le_anti_sym,card_inj_on_le], simpset()));
qed "card_bij_eq";