--- a/src/ZF/Arith.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/Arith.ML Thu Jun 23 17:38:12 1994 +0200
@@ -156,19 +156,17 @@
(ALLGOALS
(asm_simp_tac (arith_ss addsimps [add_0_right, add_succ_right]))) ]);
+(*for a/c rewriting*)
val add_left_commute = prove_goal Arith.thy
- "!!m n k. [| m:nat; n:nat; k:nat |] ==> m#+(n#+k)=n#+(m#+k)"
- (fn _ => [rtac (add_commute RS trans) 1,
- rtac (add_assoc RS trans) 3,
- rtac (add_commute RS subst_context) 4,
- REPEAT (ares_tac [add_type] 1)]);
+ "!!m n k. [| m:nat; n:nat |] ==> m#+(n#+k)=n#+(m#+k)"
+ (fn _ => [asm_simp_tac (ZF_ss addsimps [add_assoc RS sym, add_commute]) 1]);
(*Addition is an AC-operator*)
val add_ac = [add_assoc, add_commute, add_left_commute];
(*Cancellation law on the left*)
-val [knat,eqn] = goal Arith.thy
- "[| k:nat; k #+ m = k #+ n |] ==> m=n";
+val [eqn,knat] = goal Arith.thy
+ "[| k #+ m = k #+ n; k:nat |] ==> m=n";
by (rtac (eqn RS rev_mp) 1);
by (nat_ind_tac "k" [knat] 1);
by (ALLGOALS (simp_tac arith_ss));
@@ -221,6 +219,16 @@
[ (etac nat_induct 1),
(ALLGOALS (asm_simp_tac (arith_ss addsimps [add_mult_distrib]))) ]);
+(*for a/c rewriting*)
+val mult_left_commute = prove_goal Arith.thy
+ "!!m n k. [| m:nat; n:nat; k:nat |] ==> m #* (n #* k) = n #* (m #* k)"
+ (fn _ => [rtac (mult_commute RS trans) 1,
+ rtac (mult_assoc RS trans) 3,
+ rtac (mult_commute RS subst_context) 6,
+ REPEAT (ares_tac [mult_type] 1)]);
+
+val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
+
(*** Difference ***)
@@ -241,11 +249,17 @@
(*Subtraction is the inverse of addition. *)
val [mnat,nnat] = goal Arith.thy
- "[| m:nat; n:nat |] ==> (n#+m) #-n = m";
+ "[| m:nat; n:nat |] ==> (n#+m) #- n = m";
by (rtac (nnat RS nat_induct) 1);
by (ALLGOALS (asm_simp_tac (arith_ss addsimps [mnat])));
val diff_add_inverse = result();
+goal Arith.thy
+ "!!m n. [| m:nat; n:nat |] ==> (m#+n) #- n = m";
+by (res_inst_tac [("m1","m")] (add_commute RS ssubst) 1);
+by (REPEAT (ares_tac [diff_add_inverse] 1));
+val diff_add_inverse2 = result();
+
val [mnat,nnat] = goal Arith.thy
"[| m:nat; n:nat |] ==> n #- (n#+m) = 0";
by (rtac (nnat RS nat_induct) 1);
@@ -311,7 +325,7 @@
goal Arith.thy
"!!m n. [| 0<n; m:nat; n:nat |] ==> (m div n)#*n #+ m mod n = m";
by (etac complete_induct 1);
-by (res_inst_tac [("Q","x<n")] (excluded_middle RS disjE) 1);
+by (excluded_middle_tac "x<n" 1);
(*case x<n*)
by (asm_simp_tac (arith_ss addsimps [mod_less, div_less]) 2);
(*case n le x*)
--- a/src/ZF/Cardinal.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/Cardinal.ML Thu Jun 23 17:38:12 1994 +0200
@@ -55,8 +55,8 @@
(** Equipollence is an equivalence relation **)
goalw Cardinal.thy [eqpoll_def] "X eqpoll X";
-br exI 1;
-br id_bij 1;
+by (rtac exI 1);
+by (rtac id_bij 1);
val eqpoll_refl = result();
goalw Cardinal.thy [eqpoll_def] "!!X Y. X eqpoll Y ==> Y eqpoll X";
@@ -71,8 +71,8 @@
(** Le-pollence is a partial ordering **)
goalw Cardinal.thy [lepoll_def] "!!X Y. X<=Y ==> X lepoll Y";
-br exI 1;
-be id_subset_inj 1;
+by (rtac exI 1);
+by (etac id_subset_inj 1);
val subset_imp_lepoll = result();
val lepoll_refl = subset_refl RS subset_imp_lepoll;
@@ -97,7 +97,7 @@
val [major,minor] = goal Cardinal.thy
"[| X eqpoll Y; [| X lepoll Y; Y lepoll X |] ==> P |] ==> P";
-br minor 1;
+by (rtac minor 1);
by (REPEAT (resolve_tac [major, eqpoll_imp_lepoll, eqpoll_sym] 1));
val eqpollE = result();
@@ -113,7 +113,7 @@
by (rtac the_equality 1);
by (fast_tac (ZF_cs addSIs [premP,premOrd,premNot]) 1);
by (REPEAT (etac conjE 1));
-be (premOrd RS Ord_linear_lt) 1;
+by (etac (premOrd RS Ord_linear_lt) 1);
by (ALLGOALS (fast_tac (ZF_cs addSIs [premP] addSDs [premNot])));
val Least_equality = result();
@@ -140,18 +140,24 @@
(*LEAST really is the smallest*)
goal Cardinal.thy "!!i. [| P(i); i < (LEAST x.P(x)) |] ==> Q";
-br (Least_le RSN (2,lt_trans2) RS lt_anti_refl) 1;
+by (rtac (Least_le RSN (2,lt_trans2) RS lt_irrefl) 1);
by (REPEAT (eresolve_tac [asm_rl, ltE] 1));
val less_LeastE = result();
+(*If there is no such P then LEAST is vacuously 0*)
+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);
+val Least_0 = result();
+
goal Cardinal.thy "Ord(LEAST x.P(x))";
-by (res_inst_tac [("Q","EX i. Ord(i) & P(i)")] (excluded_middle RS disjE) 1);
+by (excluded_middle_tac "EX i. Ord(i) & P(i)" 1);
by (safe_tac ZF_cs);
-br (Least_le RS ltE) 2;
+by (rtac (Least_le RS ltE) 2);
by (REPEAT_SOME assume_tac);
-bw Least_def;
-by (rtac (the_0 RS ssubst) 1 THEN rtac Ord_0 2);
-by (fast_tac FOL_cs 1);
+by (etac (Least_0 RS ssubst) 1);
+by (rtac Ord_0 1);
val Ord_Least = result();
@@ -165,17 +171,17 @@
(*Need AC to prove X lepoll Y ==> |X| le |Y| ; see well_ord_lepoll_imp_le *)
goalw Cardinal.thy [eqpoll_def,cardinal_def] "!!X Y. X eqpoll Y ==> |X| = |Y|";
-br Least_cong 1;
+by (rtac Least_cong 1);
by (fast_tac (ZF_cs addEs [comp_bij,bij_converse_bij]) 1);
val cardinal_cong = result();
(*Under AC, the premise becomes trivial; one consequence is ||A|| = |A|*)
goalw Cardinal.thy [eqpoll_def, cardinal_def]
"!!A. well_ord(A,r) ==> |A| eqpoll A";
-br LeastI 1;
-be Ord_ordertype 2;
-br exI 1;
-be (ordertype_bij RS bij_converse_bij) 1;
+by (rtac LeastI 1);
+by (etac Ord_ordertype 2);
+by (rtac exI 1);
+by (etac (ordertype_bij RS bij_converse_bij) 1);
val well_ord_cardinal_eqpoll = result();
val Ord_cardinal_eqpoll = well_ord_Memrel RS well_ord_cardinal_eqpoll
@@ -183,8 +189,8 @@
goal Cardinal.thy
"!!X Y. [| well_ord(X,r); well_ord(Y,s); |X| = |Y| |] ==> X eqpoll Y";
-br (eqpoll_sym RS eqpoll_trans) 1;
-be well_ord_cardinal_eqpoll 1;
+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);
val well_ord_cardinal_eqE = result();
@@ -192,55 +198,71 @@
(** Observations from Kunen, page 28 **)
goalw Cardinal.thy [cardinal_def] "!!i. Ord(i) ==> |i| le i";
-be (eqpoll_refl RS Least_le) 1;
+by (etac (eqpoll_refl RS Least_le) 1);
val Ord_cardinal_le = result();
goalw Cardinal.thy [Card_def] "!!i. Card(i) ==> |i| = i";
-be sym 1;
+by (etac sym 1);
val Card_cardinal_eq = result();
val prems = goalw Cardinal.thy [Card_def,cardinal_def]
"[| Ord(i); !!j. j<i ==> ~(j eqpoll i) |] ==> Card(i)";
-br (Least_equality RS ssubst) 1;
+by (rtac (Least_equality RS ssubst) 1);
by (REPEAT (ares_tac ([refl,eqpoll_refl]@prems) 1));
val CardI = result();
goalw Cardinal.thy [Card_def, cardinal_def] "!!i. Card(i) ==> Ord(i)";
-be ssubst 1;
-br Ord_Least 1;
+by (etac ssubst 1);
+by (rtac Ord_Least 1);
val Card_is_Ord = result();
-goalw Cardinal.thy [cardinal_def] "Ord( |i| )";
-br Ord_Least 1;
+goalw Cardinal.thy [cardinal_def] "Ord( |A| )";
+by (rtac Ord_Least 1);
val Ord_cardinal = result();
+goal Cardinal.thy "Card(0)";
+by (rtac (Ord_0 RS CardI) 1);
+by (fast_tac (ZF_cs addSEs [ltE]) 1);
+val Card_0 = result();
+
+goalw Cardinal.thy [cardinal_def] "Card( |A| )";
+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 (rtac less_LeastE 1);
+by (assume_tac 2);
+by (etac eqpoll_trans 1);
+by (REPEAT (ares_tac [LeastI] 1));
+val Card_cardinal = result();
+
(*Kunen's Lemma 10.5*)
goal Cardinal.thy "!!i j. [| |i| le j; j le i |] ==> |j| = |i|";
-br (eqpollI RS cardinal_cong) 1;
-be (le_imp_subset RS subset_imp_lepoll) 1;
-br lepoll_trans 1;
-be (le_imp_subset RS subset_imp_lepoll) 2;
-br (eqpoll_sym RS eqpoll_imp_lepoll) 1;
-br Ord_cardinal_eqpoll 1;
+by (rtac (eqpollI RS cardinal_cong) 1);
+by (etac (le_imp_subset RS subset_imp_lepoll) 1);
+by (rtac lepoll_trans 1);
+by (etac (le_imp_subset RS subset_imp_lepoll) 2);
+by (rtac (eqpoll_sym RS eqpoll_imp_lepoll) 1);
+by (rtac Ord_cardinal_eqpoll 1);
by (REPEAT (eresolve_tac [ltE, Ord_succD] 1));
val cardinal_eq_lemma = result();
goal Cardinal.thy "!!i j. 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]));
-br cardinal_eq_lemma 1;
-ba 2;
-be le_trans 1;
-be ltE 1;
-be Ord_cardinal_le 1;
+by (rtac cardinal_eq_lemma 1);
+by (assume_tac 2);
+by (etac le_trans 1);
+by (etac ltE 1);
+by (etac Ord_cardinal_le 1);
val cardinal_mono = result();
(*Since we have |succ(nat)| le |nat|, the converse of cardinal_mono fails!*)
goal Cardinal.thy "!!i j. [| |i| < |j|; Ord(i); Ord(j) |] ==> i < j";
-br Ord_linear2 1;
+by (rtac Ord_linear2 1);
by (REPEAT_SOME assume_tac);
-be (lt_trans2 RS lt_anti_refl) 1;
-be cardinal_mono 1;
+by (etac (lt_trans2 RS lt_irrefl) 1);
+by (etac cardinal_mono 1);
val cardinal_lt_imp_lt = result();
goal Cardinal.thy "!!i j. [| |i| < k; Ord(i); Card(k) |] ==> i < k";
@@ -262,7 +284,7 @@
val swap_swap_identity = result();
goal Cardinal.thy "!!A. [| x:A; y:A |] ==> swap(A,x,y) : bij(A,A)";
-br nilpotent_imp_bijective 1;
+by (rtac nilpotent_imp_bijective 1);
by (REPEAT (ares_tac [swap_type, comp_eq_id_iff RS iffD2,
ballI, swap_swap_identity] 1));
val swap_bij = result();
@@ -272,24 +294,24 @@
(*Lemma suggested by Mike Fourman*)
val [prem] = goalw Cardinal.thy [inj_def]
"f: inj(succ(m), succ(n)) ==> (lam x:m. if(f`x=n, f`m, f`x)) : inj(m,n)";
-br CollectI 1;
+by (rtac CollectI 1);
(*Proving it's in the function space m->n*)
by (cut_facts_tac [prem] 1);
-br (if_type RS lam_type) 1;
-by (fast_tac (ZF_cs addSEs [mem_anti_refl] addEs [apply_funtype RS succE]) 1);
-by (fast_tac (ZF_cs addSEs [mem_anti_refl] addEs [apply_funtype RS succE]) 1);
+by (rtac (if_type RS lam_type) 1);
+by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [apply_funtype RS succE]) 1);
+by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [apply_funtype RS succE]) 1);
(*Proving it's injective*)
by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1);
(*Adding prem earlier would cause the simplifier to loop*)
by (cut_facts_tac [prem] 1);
-by (fast_tac (ZF_cs addSEs [mem_anti_refl]) 1);
+by (fast_tac (ZF_cs addSEs [mem_irrefl]) 1);
val inj_succ_succD = result();
val [prem] = goalw Cardinal.thy [lepoll_def]
"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);
-br ballI 1;
+by (rtac ballI 1);
by (eres_inst_tac [("n","n")] natE 1);
by (asm_simp_tac (ZF_ss addsimps [inj_def, succI1 RS Pi_empty2]) 1);
by (fast_tac (ZF_cs addSIs [succ_leI] addSDs [inj_succ_succD]) 1);
@@ -298,24 +320,23 @@
goal Cardinal.thy
"!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n";
-br iffI 1;
+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_asym] addSEs [eqpollE]) 1);
+by (fast_tac (ZF_cs addIs [nat_lepoll_imp_le, le_anti_sym]
+ addSEs [eqpollE]) 1);
val nat_eqpoll_iff = result();
goalw Cardinal.thy [Card_def,cardinal_def]
"!!n. n: nat ==> Card(n)";
-br (Least_equality RS ssubst) 1;
+by (rtac (Least_equality RS ssubst) 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_anti_refl]) 1);
+by (fast_tac (ZF_cs addSEs [lt_irrefl]) 1);
val nat_into_Card = result();
-val Card_0 = nat_0I RS nat_into_Card;
-
(*Part of Kunen's Lemma 10.6*)
goal Cardinal.thy "!!n. [| succ(n) lepoll n; n:nat |] ==> P";
-br (nat_lepoll_imp_le RS lt_anti_refl) 1;
+by (rtac (nat_lepoll_imp_le RS lt_irrefl) 1);
by (REPEAT (ares_tac [nat_succI] 1));
val succ_lepoll_natE = result();
@@ -324,29 +345,34 @@
(*This implies Kunen's Lemma 10.6*)
goal Cardinal.thy "!!n. [| n<i; n:nat |] ==> ~ i lepoll n";
-br notI 1;
+by (rtac notI 1);
by (rtac succ_lepoll_natE 1 THEN assume_tac 2);
by (rtac lepoll_trans 1 THEN assume_tac 2);
-be ltE 1;
+by (etac ltE 1);
by (REPEAT (ares_tac [Ord_succ_subsetI RS subset_imp_lepoll] 1));
val lt_not_lepoll = result();
-goalw Cardinal.thy [Card_def,cardinal_def] "Card(nat)";
-br (Least_equality RS ssubst) 1;
-by (REPEAT_FIRST (ares_tac [eqpoll_refl, Ord_nat, refl]));
-be ltE 1;
-by (asm_simp_tac (ZF_ss addsimps [eqpoll_iff, lt_not_lepoll, ltI]) 1);
-val Card_nat = result();
-
goal Cardinal.thy "!!i n. [| Ord(i); n:nat |] ==> i eqpoll n <-> i=n";
-br iffI 1;
+by (rtac iffI 1);
by (asm_simp_tac (ZF_ss 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
REPEAT (assume_tac 1));
by (rtac (lt_not_lepoll RS notE) 1 THEN (REPEAT (assume_tac 1)));
-be eqpoll_imp_lepoll 1;
+by (etac eqpoll_imp_lepoll 1);
val Ord_nat_eqpoll_iff = result();
+goalw Cardinal.thy [Card_def,cardinal_def] "Card(nat)";
+by (rtac (Least_equality RS ssubst) 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);
+val Card_nat = result();
+(*Allows showing that |i| is a limit cardinal*)
+goal Cardinal.thy "!!i. nat le i ==> nat le |i|";
+by (rtac (Card_nat RS Card_cardinal_eq RS subst) 1);
+by (etac cardinal_mono 1);
+val nat_le_cardinal = result();
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/CardinalArith.ML Thu Jun 23 17:38:12 1994 +0200
@@ -0,0 +1,508 @@
+(* Title: ZF/CardinalArith.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+Cardinal arithmetic -- WITHOUT the Axiom of Choice
+*)
+
+open CardinalArith;
+
+(*Use AC to discharge first premise*)
+goal CardinalArith.thy
+ "!!A 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);
+by (rtac lepoll_trans 1);
+by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
+by (assume_tac 1);
+by (etac (le_imp_subset RS subset_imp_lepoll RS lepoll_trans) 1);
+by (rtac eqpoll_imp_lepoll 1);
+by (rewtac lepoll_def);
+by (etac exE 1);
+by (rtac well_ord_cardinal_eqpoll 1);
+by (etac well_ord_rvimage 1);
+by (assume_tac 1);
+val well_ord_lepoll_imp_le = result();
+
+val case_ss = ZF_ss addsimps [Inl_iff, Inl_Inr_iff, Inr_iff, Inr_Inl_iff,
+ case_Inl, case_Inr, InlI, InrI];
+
+
+(** Congruence laws for successor, cardinal addition and multiplication **)
+
+val bij_inverse_ss =
+ case_ss addsimps [bij_is_fun RS apply_type,
+ bij_converse_bij RS bij_is_fun RS apply_type,
+ left_inverse_bij, right_inverse_bij];
+
+
+(*Congruence law for succ under equipollence*)
+goalw CardinalArith.thy [eqpoll_def]
+ "!!A B. A eqpoll B ==> succ(A) eqpoll succ(B)";
+by (safe_tac ZF_cs);
+by (rtac exI 1);
+by (res_inst_tac [("c", "%z.if(z=A,B,f`z)"),
+ ("d", "%z.if(z=B,A,converse(f)`z)")] lam_bijective 1);
+by (ALLGOALS
+ (asm_simp_tac (bij_inverse_ss addsimps [succI2, mem_imp_not_eq]
+ setloop etac succE )));
+val succ_eqpoll_cong = result();
+
+(*Congruence law for + under equipollence*)
+goalw CardinalArith.thy [eqpoll_def]
+ "!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D";
+by (safe_tac ZF_cs);
+by (rtac exI 1);
+by (res_inst_tac [("c", "case(%x. Inl(f`x), %y. Inr(fa`y))"),
+ ("d", "case(%x. Inl(converse(f)`x), %y. Inr(converse(fa)`y))")]
+ lam_bijective 1);
+by (safe_tac (ZF_cs addSEs [sumE]));
+by (ALLGOALS (asm_simp_tac bij_inverse_ss));
+val sum_eqpoll_cong = result();
+
+(*Congruence law for * under equipollence*)
+goalw CardinalArith.thy [eqpoll_def]
+ "!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D";
+by (safe_tac ZF_cs);
+by (rtac exI 1);
+by (res_inst_tac [("c", "split(%x y. <f`x, fa`y>)"),
+ ("d", "split(%x y. <converse(f)`x, converse(fa)`y>)")]
+ lam_bijective 1);
+by (safe_tac ZF_cs);
+by (ALLGOALS (asm_simp_tac bij_inverse_ss));
+val prod_eqpoll_cong = result();
+
+
+(*** Cardinal addition ***)
+
+(** Cardinal addition is commutative **)
+
+(*Easier to prove the two directions separately*)
+goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A";
+by (rtac exI 1);
+by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")]
+ lam_bijective 1);
+by (safe_tac (ZF_cs addSEs [sumE]));
+by (ALLGOALS (asm_simp_tac case_ss));
+val sum_commute_eqpoll = result();
+
+goalw CardinalArith.thy [cadd_def] "i |+| j = j |+| i";
+by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
+val cadd_commute = result();
+
+(** Cardinal addition is associative **)
+
+goalw CardinalArith.thy [eqpoll_def] "(A+B)+C eqpoll A+(B+C)";
+by (rtac exI 1);
+by (res_inst_tac [("c", "case(case(Inl, %y.Inr(Inl(y))), %y. Inr(Inr(y)))"),
+ ("d", "case(%x.Inl(Inl(x)), case(%x.Inl(Inr(x)), Inr))")]
+ lam_bijective 1);
+by (ALLGOALS (asm_simp_tac (case_ss setloop etac sumE)));
+val sum_assoc_eqpoll = result();
+
+(*Unconditional version requires AC*)
+goalw CardinalArith.thy [cadd_def]
+ "!!i j k. [| Ord(i); Ord(j); Ord(k) |] ==> \
+\ (i |+| j) |+| k = i |+| (j |+| k)";
+by (rtac cardinal_cong 1);
+br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS
+ eqpoll_trans) 1;
+by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2);
+br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS
+ eqpoll_sym) 2;
+by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
+val Ord_cadd_assoc = result();
+
+(** 0 is the identity for addition **)
+
+goalw CardinalArith.thy [eqpoll_def] "0+A eqpoll A";
+by (rtac exI 1);
+by (res_inst_tac [("c", "case(%x.x, %y.y)"), ("d", "Inr")]
+ lam_bijective 1);
+by (ALLGOALS (asm_simp_tac (case_ss setloop eresolve_tac [sumE,emptyE])));
+val sum_0_eqpoll = result();
+
+goalw CardinalArith.thy [cadd_def] "!!i. Card(i) ==> 0 |+| i = i";
+by (asm_simp_tac (ZF_ss addsimps [sum_0_eqpoll RS cardinal_cong,
+ Card_cardinal_eq]) 1);
+val cadd_0 = result();
+
+(** Addition of finite cardinals is "ordinary" addition **)
+
+goalw CardinalArith.thy [eqpoll_def] "succ(A)+B eqpoll succ(A+B)";
+by (rtac exI 1);
+by (res_inst_tac [("c", "%z.if(z=Inl(A),A+B,z)"),
+ ("d", "%z.if(z=A+B,Inl(A),z)")]
+ lam_bijective 1);
+by (ALLGOALS
+ (asm_simp_tac (case_ss addsimps [succI2, mem_imp_not_eq]
+ setloop eresolve_tac [sumE,succE])));
+val sum_succ_eqpoll = result();
+
+(*Pulling the succ(...) outside the |...| requires m, n: nat *)
+(*Unconditional version requires AC*)
+goalw CardinalArith.thy [cadd_def]
+ "!!m n. [| Ord(m); Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|";
+by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1);
+by (rtac (succ_eqpoll_cong RS cardinal_cong) 1);
+by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1);
+by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
+val cadd_succ_lemma = result();
+
+val [mnat,nnat] = goal CardinalArith.thy
+ "[| m: nat; n: nat |] ==> m |+| n = m#+n";
+by (cut_facts_tac [nnat] 1);
+by (nat_ind_tac "m" [mnat] 1);
+by (asm_simp_tac (arith_ss addsimps [nat_into_Card RS cadd_0]) 1);
+by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma,
+ nat_into_Card RS Card_cardinal_eq]) 1);
+val nat_cadd_eq_add = result();
+
+
+(*** Cardinal multiplication ***)
+
+(** Cardinal multiplication is commutative **)
+
+(*Easier to prove the two directions separately*)
+goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A";
+by (rtac exI 1);
+by (res_inst_tac [("c", "split(%x y.<y,x>)"), ("d", "split(%x y.<y,x>)")]
+ lam_bijective 1);
+by (safe_tac ZF_cs);
+by (ALLGOALS (asm_simp_tac ZF_ss));
+val prod_commute_eqpoll = result();
+
+goalw CardinalArith.thy [cmult_def] "i |*| j = j |*| i";
+by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
+val cmult_commute = result();
+
+(** Cardinal multiplication is associative **)
+
+goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)";
+by (rtac exI 1);
+by (res_inst_tac [("c", "split(%w z. split(%x y. <x,<y,z>>, w))"),
+ ("d", "split(%x. split(%y z. <<x,y>, z>))")]
+ lam_bijective 1);
+by (safe_tac ZF_cs);
+by (ALLGOALS (asm_simp_tac ZF_ss));
+val prod_assoc_eqpoll = result();
+
+(*Unconditional version requires AC*)
+goalw CardinalArith.thy [cmult_def]
+ "!!i j k. [| Ord(i); Ord(j); Ord(k) |] ==> \
+\ (i |*| j) |*| k = i |*| (j |*| k)";
+by (rtac cardinal_cong 1);
+br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS
+ eqpoll_trans) 1;
+by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2);
+br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS
+ eqpoll_sym) 2;
+by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
+val Ord_cmult_assoc = result();
+
+(** Cardinal multiplication distributes over addition **)
+
+goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)";
+by (rtac exI 1);
+by (res_inst_tac
+ [("c", "split(%x z. case(%y.Inl(<y,z>), %y.Inr(<y,z>), x))"),
+ ("d", "case(split(%x y.<Inl(x),y>), split(%x y.<Inr(x),y>))")]
+ lam_bijective 1);
+by (safe_tac (ZF_cs addSEs [sumE]));
+by (ALLGOALS (asm_simp_tac case_ss));
+val sum_prod_distrib_eqpoll = result();
+
+goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A";
+by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
+by (simp_tac (ZF_ss addsimps [lam_type]) 1);
+val prod_square_lepoll = result();
+
+goalw CardinalArith.thy [cmult_def] "!!k. Card(k) ==> k le k |*| k";
+by (rtac le_trans 1);
+by (rtac well_ord_lepoll_imp_le 2);
+by (rtac prod_square_lepoll 3);
+by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
+by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
+val cmult_square_le = result();
+
+(** Multiplication by 0 yields 0 **)
+
+goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0";
+by (rtac exI 1);
+by (rtac lam_bijective 1);
+by (safe_tac ZF_cs);
+val prod_0_eqpoll = result();
+
+goalw CardinalArith.thy [cmult_def] "0 |*| i = 0";
+by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong,
+ Card_0 RS Card_cardinal_eq]) 1);
+val cmult_0 = result();
+
+(** 1 is the identity for multiplication **)
+
+goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A";
+by (rtac exI 1);
+by (res_inst_tac [("c", "snd"), ("d", "%z.<x,z>")] lam_bijective 1);
+by (safe_tac ZF_cs);
+by (ALLGOALS (asm_simp_tac ZF_ss));
+val prod_singleton_eqpoll = result();
+
+goalw CardinalArith.thy [cmult_def, succ_def] "!!i. Card(i) ==> 1 |*| i = i";
+by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong,
+ Card_cardinal_eq]) 1);
+val cmult_1 = result();
+
+(** Multiplication of finite cardinals is "ordinary" multiplication **)
+
+goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B";
+by (rtac exI 1);
+by (res_inst_tac [("c", "split(%x y. if(x=A, Inl(y), Inr(<x,y>)))"),
+ ("d", "case(%y. <A,y>, %z.z)")]
+ lam_bijective 1);
+by (safe_tac (ZF_cs addSEs [sumE]));
+by (ALLGOALS
+ (asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq])));
+val prod_succ_eqpoll = result();
+
+
+(*Unconditional version requires AC*)
+goalw CardinalArith.thy [cmult_def, cadd_def]
+ "!!m n. [| Ord(m); Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";
+by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1);
+by (rtac (cardinal_cong RS sym) 1);
+by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1);
+by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
+val cmult_succ_lemma = result();
+
+val [mnat,nnat] = goal CardinalArith.thy
+ "[| m: nat; n: nat |] ==> m |*| n = m#*n";
+by (cut_facts_tac [nnat] 1);
+by (nat_ind_tac "m" [mnat] 1);
+by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1);
+by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma,
+ nat_cadd_eq_add]) 1);
+val nat_cmult_eq_mult = result();
+
+
+(*** Infinite Cardinals are Limit Ordinals ***)
+
+goalw CardinalArith.thy [lepoll_def, inj_def]
+ "!!i. nat <= A ==> succ(A) lepoll A";
+by (res_inst_tac [("x",
+ "lam z:succ(A). if(z=A, 0, if(z:nat, succ(z), z))")] exI 1);
+by (rtac (lam_type RS CollectI) 1);
+by (rtac if_type 1);
+by (etac ([asm_rl, nat_0I] MRS subsetD) 1);
+by (etac succE 1);
+by (contr_tac 1);
+by (rtac if_type 1);
+by (assume_tac 2);
+by (etac ([asm_rl, nat_succI] MRS subsetD) 1 THEN assume_tac 1);
+by (REPEAT (rtac ballI 1));
+by (asm_simp_tac
+ (ZF_ss addsimps [succ_inject_iff, succ_not_0, succ_not_0 RS not_sym]
+ setloop split_tac [expand_if]) 1);
+by (safe_tac (ZF_cs addSIs [nat_0I, nat_succI]));
+val nat_succ_lepoll = result();
+
+goal CardinalArith.thy "!!i. nat <= A ==> succ(A) eqpoll A";
+by (etac (nat_succ_lepoll RS eqpollI) 1);
+by (rtac (subset_succI RS subset_imp_lepoll) 1);
+val nat_succ_eqpoll = result();
+
+goalw CardinalArith.thy [InfCard_def] "!!i. InfCard(i) ==> Card(i)";
+by (etac conjunct1 1);
+val InfCard_is_Card = result();
+
+(*Kunen's Lemma 10.11*)
+goalw CardinalArith.thy [InfCard_def] "!!i. InfCard(i) ==> Limit(i)";
+by (etac conjE 1);
+by (rtac (ltI RS non_succ_LimitI) 1);
+by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
+by (etac Card_is_Ord 1);
+by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD]));
+by (forward_tac [Card_is_Ord RS Ord_succD] 1);
+by (rewtac Card_def);
+by (res_inst_tac [("i", "succ(y)")] lt_irrefl 1);
+by (dtac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
+(*Tricky combination of substitutions; backtracking needed*)
+by (etac ssubst 1 THEN etac ssubst 1 THEN rtac Ord_cardinal_le 1);
+by (assume_tac 1);
+val InfCard_is_Limit = result();
+
+
+
+(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
+
+(*A general fact about ordermap*)
+goalw Cardinal.thy [eqpoll_def]
+ "!!A. [| well_ord(A,r); x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
+by (rtac exI 1);
+by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
+by (etac (ordertype_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
+by (rtac pred_subset 1);
+val ordermap_eqpoll_pred = result();
+
+(** Establishing the well-ordering **)
+
+goalw CardinalArith.thy [inj_def]
+ "!!k. Ord(k) ==> \
+\ (lam z:k*k. split(%x y. <x Un y, <x, y>>, z)) : inj(k*k, k*k*k)";
+by (safe_tac ZF_cs);
+by (fast_tac (ZF_cs addIs [lam_type, Un_least_lt RS ltD, ltI]
+ addSEs [split_type]) 1);
+by (asm_full_simp_tac ZF_ss 1);
+val csquare_lam_inj = result();
+
+goalw CardinalArith.thy [csquare_rel_def]
+ "!!k. Ord(k) ==> well_ord(k*k, csquare_rel(k))";
+by (rtac (csquare_lam_inj RS well_ord_rvimage) 1);
+by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
+val well_ord_csquare = result();
+
+(** Characterising initial segments of the well-ordering **)
+
+goalw CardinalArith.thy [csquare_rel_def]
+ "!!k. [| x<k; y<k; z<k |] ==> \
+\ <<x,y>, <z,z>> : csquare_rel(k) --> x le z & y le z";
+by (REPEAT (etac ltE 1));
+by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
+ Un_absorb, Un_least_mem_iff, ltD]) 1);
+by (safe_tac (ZF_cs addSEs [mem_irrefl]
+ addSIs [Un_upper1_le, Un_upper2_le]));
+by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ])));
+val csquareD_lemma = result();
+val csquareD = csquareD_lemma RS mp |> standard;
+
+goalw CardinalArith.thy [pred_def]
+ "!!k. z<k ==> pred(k*k, <z,z>, csquare_rel(k)) <= succ(z)*succ(z)";
+by (safe_tac (lemmas_cs addSEs [SigmaE])); (*avoids using succCI*)
+by (rtac (csquareD RS conjE) 1);
+by (rewtac lt_def);
+by (assume_tac 4);
+by (ALLGOALS (fast_tac ZF_cs));
+val pred_csquare_subset = result();
+
+goalw CardinalArith.thy [csquare_rel_def]
+ "!!k. [| x<z; y<z; z<k |] ==> \
+\ <<x,y>, <z,z>> : csquare_rel(k)";
+by (subgoals_tac ["x<k", "y<k"] 1);
+by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
+by (REPEAT (etac ltE 1));
+by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
+ Un_absorb, Un_least_mem_iff, ltD]) 1);
+val csquare_ltI = result();
+
+(*Part of the traditional proof. UNUSED since it's harder to prove & apply *)
+goalw CardinalArith.thy [csquare_rel_def]
+ "!!k. [| x le z; y le z; z<k |] ==> \
+\ <<x,y>, <z,z>> : csquare_rel(k) | x=z & y=z";
+by (subgoals_tac ["x<k", "y<k"] 1);
+by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
+by (REPEAT (etac ltE 1));
+by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
+ Un_absorb, Un_least_mem_iff, ltD]) 1);
+by (REPEAT_FIRST (etac succE));
+by (ALLGOALS
+ (asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym,
+ subset_Un_iff2 RS iff_sym, OrdmemD])));
+val csquare_or_eqI = result();
+
+(** The cardinality of initial segments **)
+
+goal CardinalArith.thy
+ "!!k. [| InfCard(k); x<k; y<k; z=succ(x Un y) |] ==> \
+\ ordermap(k*k, csquare_rel(k)) ` <x,y> lepoll \
+\ ordermap(k*k, csquare_rel(k)) ` <z,z>";
+by (subgoals_tac ["z<k", "well_ord(k*k, csquare_rel(k))"] 1);
+by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 2);
+by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, Limit_has_succ]) 2);
+by (rtac (OrdmemD RS subset_imp_lepoll) 1);
+by (res_inst_tac [("z1","z")] (csquare_ltI RS less_imp_ordermap_in) 1);
+by (etac well_ord_is_wf 4);
+by (ALLGOALS
+ (fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap]
+ addSEs [ltE])));
+val ordermap_z_lepoll = result();
+
+(*Kunen: "each <x,y>: k*k has no more than z*z predecessors..." (page 29) *)
+goalw CardinalArith.thy [cmult_def]
+ "!!k. [| InfCard(k); x<k; y<k; z=succ(x Un y) |] ==> \
+\ | ordermap(k*k, csquare_rel(k)) ` <x,y> | le |succ(z)| |*| |succ(z)|";
+by (rtac (well_ord_rmult RS well_ord_lepoll_imp_le) 1);
+by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
+by (subgoals_tac ["z<k"] 1);
+by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit,
+ Limit_has_succ]) 2);
+by (rtac (ordermap_z_lepoll RS lepoll_trans) 1);
+by (REPEAT_SOME assume_tac);
+by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
+by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 1);
+by (fast_tac (ZF_cs addIs [ltD]) 1);
+by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
+ assume_tac 1);
+by (REPEAT_FIRST (etac ltE));
+by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
+by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));
+val ordermap_csquare_le = result();
+
+(*Kunen: "... so the order type <= k" *)
+goal CardinalArith.thy
+ "!!k. [| InfCard(k); ALL y:k. InfCard(y) --> y |*| y = y |] ==> \
+\ ordertype(k*k, csquare_rel(k)) le k";
+by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
+by (rtac all_lt_imp_le 1);
+by (assume_tac 1);
+by (etac (well_ord_csquare RS Ord_ordertype) 1);
+by (rtac Card_lt_imp_lt 1);
+by (etac InfCard_is_Card 3);
+by (etac ltE 2 THEN assume_tac 2);
+by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1);
+by (safe_tac (ZF_cs addSEs [ltE]));
+by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
+by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
+by (rtac (ordermap_csquare_le RS lt_trans1) 1 THEN
+ REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
+by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1 THEN
+ REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
+(*the finite case: xb Un y < nat *)
+by (res_inst_tac [("j", "nat")] lt_trans2 1);
+by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2);
+by (asm_full_simp_tac
+ (ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
+ nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
+(*case nat le (xb Un y), equivalently InfCard(xb Un y) *)
+by (asm_full_simp_tac
+ (ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
+ le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt,
+ Ord_Un, ltI, nat_le_cardinal,
+ Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
+val ordertype_csquare_le = result();
+
+(*This lemma can easily be generalized to premise well_ord(A*A,r) *)
+goalw CardinalArith.thy [cmult_def]
+ "!!k. Ord(k) ==> k |*| k = |ordertype(k*k, csquare_rel(k))|";
+by (rtac cardinal_cong 1);
+by (rewtac eqpoll_def);
+by (rtac exI 1);
+by (etac (well_ord_csquare RS ordertype_bij) 1);
+val csquare_eq_ordertype = result();
+
+(*Main result: Kunen's Theorem 10.12*)
+goal CardinalArith.thy
+ "!!k. InfCard(k) ==> k |*| k = k";
+by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
+by (etac rev_mp 1);
+by (trans_ind_tac "k" [] 1);
+by (rtac impI 1);
+by (rtac le_anti_sym 1);
+by (etac (InfCard_is_Card RS cmult_square_le) 2);
+by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
+by (assume_tac 2);
+by (assume_tac 2);
+by (asm_simp_tac
+ (ZF_ss addsimps [csquare_eq_ordertype, Ord_cardinal_le,
+ well_ord_csquare RS Ord_ordertype]) 1);
+val InfCard_csquare_eq = result();
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/CardinalArith.thy Thu Jun 23 17:38:12 1994 +0200
@@ -0,0 +1,29 @@
+(* Title: ZF/CardinalArith.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+Cardinal Arithmetic
+*)
+
+CardinalArith = Cardinal + OrderArith + Arith +
+consts
+ InfCard :: "i=>o"
+ "|*|" :: "[i,i]=>i" (infixl 70)
+ "|+|" :: "[i,i]=>i" (infixl 65)
+ csquare_rel :: "i=>i"
+
+rules
+
+ InfCard_def "InfCard(i) == Card(i) & nat le i"
+
+ cadd_def "i |+| j == | i+j |"
+
+ cmult_def "i |*| j == | i*j |"
+
+ csquare_rel_def
+ "csquare_rel(k) == rvimage(k*k, lam z:k*k. split(%x y. <x Un y, <x,y>>, z), \
+\ rmult(k,Memrel(k), k*k, \
+\ rmult(k,Memrel(k), k,Memrel(k))))"
+
+end
--- a/src/ZF/Epsilon.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/Epsilon.ML Thu Jun 23 17:38:12 1994 +0200
@@ -219,7 +219,7 @@
goal Epsilon.thy "rank(Pow(a)) = succ(rank(a))";
by (rtac (rank RS trans) 1);
-by (rtac le_asym 1);
+by (rtac le_anti_sym 1);
by (DO_GOAL [rtac (Ord_rank RS Ord_succ RS UN_least_le),
etac (PowD RS rank_mono RS succ_leI)] 1);
by (DO_GOAL [rtac ([Pow_top, le_refl] MRS UN_upper_le),
@@ -252,7 +252,7 @@
val rank_Union = result();
goal Epsilon.thy "rank(eclose(a)) = rank(a)";
-by (rtac le_asym 1);
+by (rtac le_anti_sym 1);
by (rtac (arg_subset_eclose RS rank_mono) 2);
by (res_inst_tac [("a1","eclose(a)")] (rank RS ssubst) 1);
by (rtac (Ord_rank RS UN_least_le) 1);
--- a/src/ZF/Fin.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/Fin.ML Thu Jun 23 17:38:12 1994 +0200
@@ -48,7 +48,7 @@
\ !!x y. [| x: A; y: Fin(A); x~:y; P(y) |] ==> P(cons(x,y)) \
\ |] ==> P(b)";
by (rtac (major RS Fin.induct) 1);
-by (res_inst_tac [("Q","a:b")] (excluded_middle RS disjE) 2);
+by (excluded_middle_tac "a:b" 2);
by (etac (cons_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*)
by (REPEAT (ares_tac prems 1));
val Fin_induct = result();
--- a/src/ZF/Order.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/Order.ML Thu Jun 23 17:38:12 1994 +0200
@@ -6,9 +6,6 @@
For Order.thy. Orders in Zermelo-Fraenkel Set Theory
*)
-(*TO PURE/TACTIC.ML*)
-fun subgoals_tac sprops = EVERY' (map subgoal_tac sprops);
-
open Order;
@@ -44,8 +41,8 @@
goalw Order.thy [ord_iso_def]
"!!f. [| f: ord_iso(A,r,B,s); <x,y>: s; x:B; y:B |] ==> \
\ <converse(f) ` x, converse(f) ` y> : r";
-be CollectE 1;
-be (bspec RS bspec RS iffD2) 1;
+by (etac CollectE 1);
+by (etac (bspec RS bspec RS iffD2) 1);
by (REPEAT (eresolve_tac [asm_rl,
bij_converse_bij RS bij_is_fun RS apply_type] 1));
by (asm_simp_tac (ZF_ss addsimps [right_inverse_bij]) 1);
@@ -142,10 +139,10 @@
goal Order.thy
"!!r. [| well_ord(A,r); well_ord(B,s); \
\ f: ord_iso(A,r, B,s); g: ord_iso(A,r, B,s) |] ==> f = g";
-br fun_extension 1;
-be (ord_iso_is_bij RS bij_is_fun) 1;
-be (ord_iso_is_bij RS bij_is_fun) 1;
-br well_ord_iso_unique_lemma 1;
+by (rtac fun_extension 1);
+by (etac (ord_iso_is_bij RS bij_is_fun) 1);
+by (etac (ord_iso_is_bij RS bij_is_fun) 1);
+by (rtac well_ord_iso_unique_lemma 1);
by (REPEAT_SOME assume_tac);
val well_ord_iso_unique = result();
@@ -157,7 +154,7 @@
by (safe_tac ZF_cs);
by (linear_case_tac 1);
(*case x=xb*)
-by (fast_tac (ZF_cs addSEs [wf_on_anti_sym]) 1);
+by (fast_tac (ZF_cs addSEs [wf_on_asym]) 1);
(*case x<xb*)
by (fast_tac (ZF_cs addSEs [wf_on_chain3]) 1);
val well_ordI = result();
@@ -172,7 +169,7 @@
val [major,minor] = goalw Order.thy [pred_def]
"[| y: pred(A,x,r); [| y:A; <y,x>:r |] ==> P |] ==> P";
-br (major RS CollectE) 1;
+by (rtac (major RS CollectE) 1);
by (REPEAT (ares_tac [minor] 1));
val predE = result();
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/OrderArith.ML Thu Jun 23 17:38:12 1994 +0200
@@ -0,0 +1,246 @@
+(* Title: ZF/OrderArith.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+Towards ordinal arithmetic
+*)
+
+(*for deleting an unwanted assumption*)
+val thin = prove_goal pure_thy "[| PROP P; PROP Q |] ==> PROP Q"
+ (fn prems => [resolve_tac prems 1]);
+
+open OrderArith;
+
+
+(**** Addition of relations -- disjoint sum ****)
+
+(** Rewrite rules. Can be used to obtain introduction rules **)
+
+goalw OrderArith.thy [radd_def]
+ "<Inl(a), Inr(b)> : radd(A,r,B,s) <-> a:A & b:B";
+by (fast_tac sum_cs 1);
+val radd_Inl_Inr_iff = result();
+
+goalw OrderArith.thy [radd_def]
+ "<Inl(a'), Inl(a)> : radd(A,r,B,s) <-> <a',a>:r & a':A & a:A";
+by (fast_tac sum_cs 1);
+val radd_Inl_iff = result();
+
+goalw OrderArith.thy [radd_def]
+ "<Inr(b'), Inr(b)> : radd(A,r,B,s) <-> <b',b>:s & b':B & b:B";
+by (fast_tac sum_cs 1);
+val radd_Inr_iff = result();
+
+goalw OrderArith.thy [radd_def]
+ "<Inr(b), Inl(a)> : radd(A,r,B,s) <-> False";
+by (fast_tac sum_cs 1);
+val radd_Inr_Inl_iff = result();
+
+(** Elimination Rule **)
+
+val major::prems = goalw OrderArith.thy [radd_def]
+ "[| <p',p> : radd(A,r,B,s); \
+\ !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q; \
+\ !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q; \
+\ !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q \
+\ |] ==> Q";
+by (cut_facts_tac [major] 1);
+(*Split into the three cases*)
+by (REPEAT_FIRST
+ (eresolve_tac [CollectE, Pair_inject, conjE, exE, SigmaE, disjE]));
+(*Apply each premise to correct subgoal; can't just use fast_tac
+ because hyp_subst_tac would delete equalities too quickly*)
+by (EVERY (map (fn prem =>
+ EVERY1 [rtac prem, assume_tac, REPEAT o fast_tac sum_cs])
+ prems));
+val raddE = result();
+
+(** Type checking **)
+
+goalw OrderArith.thy [radd_def] "radd(A,r,B,s) <= (A+B) * (A+B)";
+by (rtac Collect_subset 1);
+val radd_type = result();
+
+val field_radd = standard (radd_type RS field_rel_subset);
+
+(** Linearity **)
+
+val sum_ss = ZF_ss addsimps [Pair_iff, InlI, InrI, Inl_iff, Inr_iff,
+ Inl_Inr_iff, Inr_Inl_iff];
+
+val radd_ss = sum_ss addsimps [radd_Inl_iff, radd_Inr_iff,
+ radd_Inl_Inr_iff, radd_Inr_Inl_iff];
+
+goalw OrderArith.thy [linear_def]
+ "!!r s. [| linear(A,r); linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))";
+by (REPEAT_FIRST (ares_tac [ballI] ORELSE' etac sumE));
+by (ALLGOALS (asm_simp_tac radd_ss));
+val linear_radd = result();
+
+
+(** Well-foundedness **)
+
+goal OrderArith.thy
+ "!!r s. [| wf[A](r); wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))";
+by (rtac wf_onI2 1);
+by (subgoal_tac "ALL x:A. Inl(x): Ba" 1);
+(*Proving the lemma, which is needed twice!*)
+by (eres_inst_tac [("P", "y : A + B")] thin 2);
+by (rtac ballI 2);
+by (eres_inst_tac [("r","r"),("a","x")] wf_on_induct 2 THEN assume_tac 2);
+by (etac (bspec RS mp) 2);
+by (fast_tac sum_cs 2);
+by (best_tac (sum_cs addSEs [raddE]) 2);
+(*Returning to main part of proof*)
+by (REPEAT_FIRST (eresolve_tac [sumE, ssubst]));
+by (best_tac sum_cs 1);
+by (eres_inst_tac [("r","s"),("a","ya")] wf_on_induct 1 THEN assume_tac 1);
+by (etac (bspec RS mp) 1);
+by (fast_tac sum_cs 1);
+by (best_tac (sum_cs addSEs [raddE]) 1);
+val wf_on_radd = result();
+
+goal OrderArith.thy
+ "!!r s. [| wf(r); wf(s) |] ==> wf(radd(field(r),r,field(s),s))";
+by (asm_full_simp_tac (ZF_ss addsimps [wf_iff_wf_on_field]) 1);
+by (rtac (field_radd RSN (2, wf_on_subset_A)) 1);
+by (REPEAT (ares_tac [wf_on_radd] 1));
+val wf_radd = result();
+
+goal OrderArith.thy
+ "!!r s. [| well_ord(A,r); well_ord(B,s) |] ==> \
+\ well_ord(A+B, radd(A,r,B,s))";
+by (rtac well_ordI 1);
+by (asm_full_simp_tac (ZF_ss addsimps [well_ord_def, wf_on_radd]) 1);
+by (asm_full_simp_tac
+ (ZF_ss addsimps [well_ord_def, tot_ord_def, linear_radd]) 1);
+val well_ord_radd = result();
+
+
+(**** Multiplication of relations -- lexicographic product ****)
+
+(** Rewrite rule. Can be used to obtain introduction rules **)
+
+goalw OrderArith.thy [rmult_def]
+ "!!r s. <<a',b'>, <a,b>> : rmult(A,r,B,s) <-> \
+\ (<a',a>: r & a':A & a:A & b': B & b: B) | \
+\ (<b',b>: s & a'=a & a:A & b': B & b: B)";
+by (fast_tac ZF_cs 1);
+val rmult_iff = result();
+
+val major::prems = goal OrderArith.thy
+ "[| <<a',b'>, <a,b>> : rmult(A,r,B,s); \
+\ [| <a',a>: r; a':A; a:A; b':B; b:B |] ==> Q; \
+\ [| <b',b>: s; a:A; a'=a; b':B; b:B |] ==> Q \
+\ |] ==> Q";
+by (rtac (major RS (rmult_iff RS iffD1) RS disjE) 1);
+by (DEPTH_SOLVE (eresolve_tac ([asm_rl, conjE] @ prems) 1));
+val rmultE = result();
+
+(** Type checking **)
+
+goalw OrderArith.thy [rmult_def] "rmult(A,r,B,s) <= (A*B) * (A*B)";
+by (rtac Collect_subset 1);
+val rmult_type = result();
+
+val field_rmult = standard (rmult_type RS field_rel_subset);
+
+(** Linearity **)
+
+val [lina,linb] = goal OrderArith.thy
+ "[| linear(A,r); linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))";
+by (rewtac linear_def); (*Note! the premises are NOT rewritten*)
+by (REPEAT_FIRST (ares_tac [ballI] ORELSE' etac SigmaE));
+by (asm_simp_tac (ZF_ss addsimps [rmult_iff]) 1);
+by (res_inst_tac [("x","xa"), ("y","xb")] (lina RS linearE) 1);
+by (res_inst_tac [("x","ya"), ("y","yb")] (linb RS linearE) 4);
+by (REPEAT_SOME (fast_tac ZF_cs));
+val linear_rmult = result();
+
+
+(** Well-foundedness **)
+
+goal OrderArith.thy
+ "!!r s. [| wf[A](r); wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))";
+by (rtac wf_onI2 1);
+by (etac SigmaE 1);
+by (etac ssubst 1);
+by (subgoal_tac "ALL b:B. <x,b>: Ba" 1);
+by (fast_tac ZF_cs 1);
+by (eres_inst_tac [("a","x")] wf_on_induct 1 THEN assume_tac 1);
+by (rtac ballI 1);
+by (eres_inst_tac [("a","b")] wf_on_induct 1 THEN assume_tac 1);
+by (etac (bspec RS mp) 1);
+by (fast_tac ZF_cs 1);
+by (best_tac (ZF_cs addSEs [rmultE]) 1);
+val wf_on_rmult = result();
+
+
+goal OrderArith.thy
+ "!!r s. [| wf(r); wf(s) |] ==> wf(rmult(field(r),r,field(s),s))";
+by (asm_full_simp_tac (ZF_ss addsimps [wf_iff_wf_on_field]) 1);
+by (rtac (field_rmult RSN (2, wf_on_subset_A)) 1);
+by (REPEAT (ares_tac [wf_on_rmult] 1));
+val wf_rmult = result();
+
+goal OrderArith.thy
+ "!!r s. [| well_ord(A,r); well_ord(B,s) |] ==> \
+\ well_ord(A*B, rmult(A,r,B,s))";
+by (rtac well_ordI 1);
+by (asm_full_simp_tac (ZF_ss addsimps [well_ord_def, wf_on_rmult]) 1);
+by (asm_full_simp_tac
+ (ZF_ss addsimps [well_ord_def, tot_ord_def, linear_rmult]) 1);
+val well_ord_rmult = result();
+
+
+(**** Inverse image of a relation ****)
+
+(** Rewrite rule **)
+
+goalw OrderArith.thy [rvimage_def]
+ "<a,b> : rvimage(A,f,r) <-> <f`a,f`b>: r & a:A & b:A";
+by (fast_tac ZF_cs 1);
+val rvimage_iff = result();
+
+(** Type checking **)
+
+goalw OrderArith.thy [rvimage_def] "rvimage(A,f,r) <= A*A";
+by (rtac Collect_subset 1);
+val rvimage_type = result();
+
+val field_rvimage = standard (rvimage_type RS field_rel_subset);
+
+
+(** Linearity **)
+
+val [finj,lin] = goalw OrderArith.thy [inj_def]
+ "[| f: inj(A,B); linear(B,r) |] ==> linear(A,rvimage(A,f,r))";
+by (rewtac linear_def); (*Note! the premises are NOT rewritten*)
+by (REPEAT_FIRST (ares_tac [ballI]));
+by (asm_simp_tac (ZF_ss addsimps [rvimage_iff]) 1);
+by (cut_facts_tac [finj] 1);
+by (res_inst_tac [("x","f`x"), ("y","f`y")] (lin RS linearE) 1);
+by (REPEAT_SOME (fast_tac (ZF_cs addSIs [apply_type])));
+val linear_rvimage = result();
+
+
+(** Well-foundedness **)
+
+goal OrderArith.thy
+ "!!r. [| f: A->B; wf[B](r) |] ==> wf[A](rvimage(A,f,r))";
+by (rtac wf_onI2 1);
+by (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba" 1);
+by (fast_tac ZF_cs 1);
+by (eres_inst_tac [("a","f`y")] wf_on_induct 1);
+by (fast_tac (ZF_cs addSIs [apply_type]) 1);
+by (best_tac (ZF_cs addSIs [apply_type] addSDs [rvimage_iff RS iffD1]) 1);
+val wf_on_rvimage = result();
+
+goal OrderArith.thy
+ "!!r. [| f: inj(A,B); well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))";
+by (rtac well_ordI 1);
+by (rewrite_goals_tac [well_ord_def, tot_ord_def]);
+by (fast_tac (ZF_cs addSIs [wf_on_rvimage, inj_is_fun]) 1);
+by (fast_tac (ZF_cs addSIs [linear_rvimage]) 1);
+val well_ord_rvimage = result();
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/OrderArith.thy Thu Jun 23 17:38:12 1994 +0200
@@ -0,0 +1,31 @@
+(* Title: ZF/OrderArith.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+Towards ordinal arithmetic
+*)
+
+OrderArith = Order + Sum +
+consts
+ radd, rmult :: "[i,i,i,i]=>i"
+ rvimage :: "[i,i,i]=>i"
+
+rules
+ (*disjoint sum of two relations; underlies ordinal addition*)
+ radd_def "radd(A,r,B,s) == \
+\ {z: (A+B) * (A+B). \
+\ (EX x y. z = <Inl(x), Inr(y)>) | \
+\ (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r) | \
+\ (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
+
+ (*lexicographic product of two relations; underlies ordinal multiplication*)
+ rmult_def "rmult(A,r,B,s) == \
+\ {z: (A*B) * (A*B). \
+\ EX x' y' x y. z = <<x',y'>, <x,y>> & \
+\ (<x',x>: r | (x'=x & <y',y>: s))}"
+
+ (*inverse image of a relation*)
+ rvimage_def "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
+
+end
--- a/src/ZF/OrderType.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/OrderType.ML Thu Jun 23 17:38:12 1994 +0200
@@ -9,8 +9,8 @@
(*Requires Ordinal.thy as parent; otherwise could be in Order.ML*)
goal OrderType.thy "!!i. Ord(i) ==> well_ord(i, Memrel(i))";
-br well_ordI 1;
-br (wf_Memrel RS wf_imp_wf_on) 1;
+by (rtac well_ordI 1);
+by (rtac (wf_Memrel RS wf_imp_wf_on) 1);
by (asm_simp_tac (ZF_ss addsimps [linear_def, Memrel_iff]) 1);
by (REPEAT (resolve_tac [ballI, Ord_linear] 1));;
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));;
@@ -18,28 +18,36 @@
open OrderType;
+goalw OrderType.thy [ordermap_def,ordertype_def]
+ "ordermap(A,r) : A -> ordertype(A,r)";
+by (rtac lam_type 1);
+by (rtac (lamI RS imageI) 1);
+by (REPEAT (assume_tac 1));
+val ordermap_type = result();
+
(** Unfolding of ordermap **)
+(*Useful for cardinality reasoning; see CardinalArith.ML*)
goalw OrderType.thy [ordermap_def, pred_def]
"!!r. [| wf[A](r); x:A |] ==> \
+\ ordermap(A,r) ` x = ordermap(A,r) `` pred(A,x,r)";
+by (asm_simp_tac ZF_ss 1);
+by (etac (wfrec_on RS trans) 1);
+by (assume_tac 1);
+by (asm_simp_tac (ZF_ss addsimps [subset_iff, image_lam,
+ vimage_singleton_iff]) 1);
+val ordermap_eq_image = result();
+
+goal OrderType.thy
+ "!!r. [| wf[A](r); x:A |] ==> \
\ ordermap(A,r) ` x = {ordermap(A,r)`y . y : pred(A,x,r)}";
-by (asm_simp_tac ZF_ss 1);
-be (wfrec_on RS trans) 1;
-ba 1;
-by (asm_simp_tac (ZF_ss addsimps [subset_iff, image_lam,
- vimage_singleton_iff]) 1);
+by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, pred_subset,
+ ordermap_type RS image_fun]) 1);
val ordermap_pred_unfold = result();
(*pred-unfolded version. NOT suitable for rewriting -- loops!*)
val ordermap_unfold = rewrite_rule [pred_def] ordermap_pred_unfold;
-goalw OrderType.thy [ordermap_def,ordertype_def]
- "ordermap(A,r) : A -> ordertype(A,r)";
-br lam_type 1;
-by (rtac (lamI RS imageI) 1);
-by (REPEAT (assume_tac 1));
-val ordermap_type = result();
-
(** Showing that ordermap, ordertype yield ordinals **)
fun ordermap_elim_tac i =
@@ -53,7 +61,7 @@
by (wf_on_ind_tac "x" [] 1);
by (asm_simp_tac (ZF_ss addsimps [ordermap_pred_unfold]) 1);
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
-bws [pred_def,Transset_def];
+by (rewrite_goals_tac [pred_def,Transset_def]);
by (fast_tac ZF_cs 2);
by (safe_tac ZF_cs);
by (ordermap_elim_tac 1);
@@ -65,7 +73,7 @@
by (rtac ([ordermap_type, subset_refl] MRS image_fun RS ssubst) 1);
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
by (fast_tac (ZF_cs addIs [Ord_ordermap]) 2);
-bws [Transset_def,well_ord_def];
+by (rewrite_goals_tac [Transset_def,well_ord_def]);
by (safe_tac ZF_cs);
by (ordermap_elim_tac 1);
by (fast_tac ZF_cs 1);
@@ -77,7 +85,7 @@
"!!r. [| <w,x>: r; wf[A](r); w: A; x: A |] ==> \
\ ordermap(A,r)`w : ordermap(A,r)`x";
by (eres_inst_tac [("x1", "x")] (ordermap_unfold RS ssubst) 1);
-ba 1;
+by (assume_tac 1);
by (fast_tac ZF_cs 1);
val less_imp_ordermap_in = result();
@@ -88,10 +96,10 @@
by (safe_tac ZF_cs);
by (linear_case_tac 1);
by (fast_tac (ZF_cs addSEs [mem_not_refl RS notE]) 1);
-bd less_imp_ordermap_in 1;
+by (dtac less_imp_ordermap_in 1);
by (REPEAT_SOME assume_tac);
-be mem_anti_sym 1;
-ba 1;
+by (etac mem_asym 1);
+by (assume_tac 1);
val ordermap_in_imp_less = result();
val ordermap_surj =
@@ -102,8 +110,8 @@
goalw OrderType.thy [well_ord_def, tot_ord_def, bij_def, inj_def]
"!!r. well_ord(A,r) ==> ordermap(A,r) : bij(A, ordertype(A,r))";
by (safe_tac ZF_cs);
-br ordermap_type 1;
-br ordermap_surj 2;
+by (rtac ordermap_type 1);
+by (rtac ordermap_surj 2);
by (linear_case_tac 1);
(*The two cases yield similar contradictions*)
by (ALLGOALS (dtac less_imp_ordermap_in));
@@ -115,10 +123,10 @@
"!!r. well_ord(A,r) ==> \
\ ordermap(A,r) : ord_iso(A,r, ordertype(A,r), Memrel(ordertype(A,r)))";
by (safe_tac ZF_cs);
-br ordertype_bij 1;
-ba 1;
+by (rtac ordertype_bij 1);
+by (assume_tac 1);
by (fast_tac (ZF_cs addSEs [MemrelE, ordermap_in_imp_less]) 2);
-bw well_ord_def;
+by (rewtac well_ord_def);
by (fast_tac (ZF_cs addSIs [MemrelI, less_imp_ordermap_in,
ordermap_type RS apply_type]) 1);
val ordertype_ord_iso = result();
--- a/src/ZF/Ordinal.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/Ordinal.ML Thu Jun 23 17:38:12 1994 +0200
@@ -185,10 +185,10 @@
goal Ordinal.thy "~ (ALL i. i:X <-> Ord(i))";
by (rtac notI 1);
by (forw_inst_tac [("x", "X")] spec 1);
-by (safe_tac (ZF_cs addSEs [mem_anti_refl]));
+by (safe_tac (ZF_cs addSEs [mem_irrefl]));
by (swap_res_tac [Ord_is_Transset RSN (2,OrdI)] 1);
by (fast_tac ZF_cs 2);
-bw Transset_def;
+by (rewtac Transset_def);
by (safe_tac ZF_cs);
by (asm_full_simp_tac ZF_ss 1);
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
@@ -223,14 +223,14 @@
val lt_trans = result();
goalw Ordinal.thy [lt_def] "!!i j. [| i<j; j<i |] ==> P";
-by (REPEAT (eresolve_tac [asm_rl, conjE, mem_anti_sym] 1));
-val lt_anti_sym = result();
+by (REPEAT (eresolve_tac [asm_rl, conjE, mem_asym] 1));
+val lt_asym = result();
-val lt_anti_refl = prove_goal Ordinal.thy "i<i ==> P"
- (fn [major]=> [ (rtac (major RS (major RS lt_anti_sym)) 1) ]);
+val lt_irrefl = prove_goal Ordinal.thy "i<i ==> P"
+ (fn [major]=> [ (rtac (major RS (major RS lt_asym)) 1) ]);
val lt_not_refl = prove_goal Ordinal.thy "~ i<i"
- (fn _=> [ (rtac notI 1), (etac lt_anti_refl 1) ]);
+ (fn _=> [ (rtac notI 1), (etac lt_irrefl 1) ]);
(** le is less than or equals; recall i le j abbrevs i<succ(j) !! **)
@@ -261,8 +261,8 @@
goal Ordinal.thy "!!i j. [| i le j; j le i |] ==> i=j";
by (asm_full_simp_tac (ZF_ss addsimps [le_iff]) 1);
-by (fast_tac (ZF_cs addEs [lt_anti_sym]) 1);
-val le_asym = result();
+by (fast_tac (ZF_cs addEs [lt_asym]) 1);
+val le_anti_sym = result();
goal Ordinal.thy "i le 0 <-> i=0";
by (fast_tac (ZF_cs addSIs [Ord_0 RS le_refl] addSEs [leE, lt0E]) 1);
@@ -273,7 +273,7 @@
val lt_cs =
ZF_cs addSIs [le_refl, leCI]
addSDs [le0D]
- addSEs [lt_anti_refl, lt0E, leE];
+ addSEs [lt_irrefl, lt0E, leE];
(*** Natural Deduction rules for Memrel ***)
@@ -394,13 +394,13 @@
val Ord_linear_le = result();
goal Ordinal.thy "!!i j. j le i ==> ~ i<j";
-by (fast_tac (lt_cs addEs [lt_anti_sym]) 1);
+by (fast_tac (lt_cs addEs [lt_asym]) 1);
val le_imp_not_lt = result();
goal Ordinal.thy "!!i j. [| ~ i<j; Ord(i); Ord(j) |] ==> j le i";
by (res_inst_tac [("i","i"),("j","j")] Ord_linear2 1);
by (REPEAT (SOMEGOAL assume_tac));
-by (fast_tac (lt_cs addEs [lt_anti_sym]) 1);
+by (fast_tac (lt_cs addEs [lt_asym]) 1);
val not_lt_imp_le = result();
goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> ~ i<j <-> j le i";
@@ -430,7 +430,7 @@
by (rtac (not_lt_iff_le RS iffD1) 1);
by (assume_tac 1);
by (assume_tac 1);
-by (fast_tac (ZF_cs addEs [ltE, mem_anti_refl]) 1);
+by (fast_tac (ZF_cs addEs [ltE, mem_irrefl]) 1);
val subset_imp_le = result();
goal Ordinal.thy "!!i j. i le j ==> i<=j";
@@ -453,7 +453,7 @@
val [ordi,ordj,minor] = goal Ordinal.thy
"[| Ord(i); Ord(j); !!x. x<j ==> x<i |] ==> j le i";
by (REPEAT_FIRST (ares_tac [notI RS not_lt_imp_le, ordi, ordj]));
-be (minor RS lt_anti_refl) 1;
+by (etac (minor RS lt_irrefl) 1);
val all_lt_imp_le = result();
(** Transitive laws **)
@@ -472,13 +472,13 @@
goal Ordinal.thy "!!i j. i<j ==> succ(i) le j";
by (rtac (not_lt_iff_le RS iffD1) 1);
-by (fast_tac (lt_cs addEs [lt_anti_sym]) 3);
+by (fast_tac (lt_cs addEs [lt_asym]) 3);
by (ALLGOALS (fast_tac (ZF_cs addEs [ltE] addIs [Ord_succ])));
val succ_leI = result();
goal Ordinal.thy "!!i j. succ(i) le j ==> i<j";
by (rtac (not_le_iff_lt RS iffD1) 1);
-by (fast_tac (lt_cs addEs [lt_anti_sym]) 3);
+by (fast_tac (lt_cs addEs [lt_asym]) 3);
by (ALLGOALS (fast_tac (ZF_cs addEs [ltE, make_elim Ord_succD])));
val succ_leE = result();
@@ -509,8 +509,8 @@
goal Ordinal.thy "!!i j. [| Ord(i); Ord(j) |] ==> i Un j < k <-> i<k & j<k";
by (safe_tac (ZF_cs addSIs [Un_least_lt]));
-br (Un_upper2_le RS lt_trans1) 2;
-br (Un_upper1_le RS lt_trans1) 1;
+by (rtac (Un_upper2_le RS lt_trans1) 2);
+by (rtac (Un_upper1_le RS lt_trans1) 1);
by (REPEAT_SOME assume_tac);
val Un_least_lt_iff = result();
@@ -601,15 +601,15 @@
"!!i. [| 0<i; ALL y. succ(y) ~= i |] ==> Limit(i)";
by (safe_tac subset_cs);
by (rtac (not_le_iff_lt RS iffD1) 2);
-by (fast_tac (lt_cs addEs [lt_anti_sym]) 4);
+by (fast_tac (lt_cs addEs [lt_asym]) 4);
by (REPEAT (eresolve_tac [asm_rl, ltE, Ord_succ] 1));
val non_succ_LimitI = result();
goal Ordinal.thy "!!i. Limit(succ(i)) ==> P";
-br lt_anti_refl 1;
-br Limit_has_succ 1;
-ba 1;
-be (Limit_is_Ord RS Ord_succD RS le_refl) 1;
+by (rtac lt_irrefl 1);
+by (rtac Limit_has_succ 1);
+by (assume_tac 1);
+by (etac (Limit_is_Ord RS Ord_succD RS le_refl) 1);
val succ_LimitE = result();
goal Ordinal.thy "!!i. [| Limit(i); i le succ(j) |] ==> i le j";
--- a/src/ZF/Perm.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/Perm.ML Thu Jun 23 17:38:12 1994 +0200
@@ -142,7 +142,7 @@
val prems = goal Perm.thy
"f: inj(A,B) ==> converse(f): inj(range(f), A)";
-bw inj_def; (*rewrite subgoal but not prems!!*)
+by (rewtac inj_def); (*rewrite subgoal but not prems!!*)
by (cut_facts_tac prems 1);
by (safe_tac ZF_cs);
(*apply f to both sides and simplify using right_inverse
@@ -359,7 +359,7 @@
by (safe_tac ZF_cs);
by (dres_inst_tac [("t", "%h.h`y ")] subst_context 1);
by (asm_full_simp_tac (ZF_ss addsimps [comp_fun_apply]) 1);
-br fun_extension 1;
+by (rtac fun_extension 1);
by (REPEAT (ares_tac [comp_fun, lam_type] 1));
by (asm_simp_tac (ZF_ss addsimps [comp_fun_apply]) 1);
val comp_eq_id_iff = result();
@@ -475,7 +475,7 @@
by (assume_tac 1);
by (dtac apply_equality 1);
by (assume_tac 1);
-by (res_inst_tac [("a","m")] mem_anti_refl 1);
+by (res_inst_tac [("a","m")] mem_irrefl 1);
by (fast_tac ZF_cs 1);
val inj_succ_restrict = result();
--- a/src/ZF/WF.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/WF.ML Thu Jun 23 17:38:12 1994 +0200
@@ -62,7 +62,7 @@
"[| !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A \
\ |] ==> y:B |] \
\ ==> wf[A](r)";
-br wf_onI 1;
+by (rtac wf_onI 1);
by (res_inst_tac [ ("c", "u") ] (prem RS DiffE) 1);
by (contr_tac 3);
by (fast_tac ZF_cs 2);
@@ -131,7 +131,7 @@
\ !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A \
\ |] ==> y:B |] \
\ ==> wf(r)";
-br ([wf_onI2, subs] MRS (wf_on_subset_A RS wf_on_field_imp_wf)) 1;
+by (rtac ([wf_onI2, subs] MRS (wf_on_subset_A RS wf_on_field_imp_wf)) 1);
by (REPEAT (ares_tac [indhyp] 1));
val wfI2 = result();
@@ -148,7 +148,7 @@
by (wf_ind_tac "a" [] 2);
by (fast_tac ZF_cs 2);
by (fast_tac FOL_cs 1);
-val wf_anti_sym = result();
+val wf_asym = result();
goal WF.thy "!!r. [| wf[A](r); a: A |] ==> <a,a> ~: r";
by (wf_on_ind_tac "a" [] 1);
@@ -160,7 +160,7 @@
by (wf_on_ind_tac "a" [] 2);
by (fast_tac ZF_cs 2);
by (fast_tac ZF_cs 1);
-val wf_on_anti_sym = result();
+val wf_on_asym = result();
(*Needed to prove well_ordI. Could also reason that wf[A](r) means
wf(r Int A*A); thus wf( (r Int A*A)^+ ) and use wf_not_refl *)
@@ -180,7 +180,7 @@
(*transitive closure of a WF relation is WF provided A is downwards closed*)
val [wfr,subs] = goal WF.thy
"[| wf[A](r); r-``A <= A |] ==> wf[A](r^+)";
-br wf_onI2 1;
+by (rtac wf_onI2 1);
by (bchain_tac 1);
by (eres_inst_tac [("a","y")] (wfr RS wf_on_induct) 1);
by (rtac (impI RS ballI) 1);
@@ -194,8 +194,8 @@
goal WF.thy "!!r. wf(r) ==> wf(r^+)";
by (asm_full_simp_tac (ZF_ss addsimps [wf_iff_wf_on_field]) 1);
-br (trancl_type RS field_rel_subset RSN (2, wf_on_subset_A)) 1;
-be wf_on_trancl 1;
+by (rtac (trancl_type RS field_rel_subset RSN (2, wf_on_subset_A)) 1);
+by (etac wf_on_trancl 1);
by (fast_tac ZF_cs 1);
val wf_trancl = result();
@@ -342,7 +342,7 @@
goalw WF.thy [wf_on_def, wfrec_on_def]
"!!A r. [| wf[A](r); a: A |] ==> \
\ wfrec[A](r,a,H) = H(a, lam x: (r-``{a}) Int A. wfrec[A](r,x,H))";
-be (wfrec RS trans) 1;
+by (etac (wfrec RS trans) 1);
by (asm_simp_tac (ZF_ss addsimps [vimage_Int_square, cons_subset_iff]) 1);
val wfrec_on = result();
--- a/src/ZF/func.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/func.ML Thu Jun 23 17:38:12 1994 +0200
@@ -230,7 +230,7 @@
val image_lam = result();
goal ZF.thy "!!C A. [| f : Pi(A,B); C <= A |] ==> f``C = {f`x. x:C}";
-be (eta RS subst) 1;
+by (etac (eta RS subst) 1);
by (asm_full_simp_tac (FOL_ss addsimps [beta, image_lam, subset_iff]
addcongs [RepFun_cong]) 1);
val image_fun = result();
--- a/src/ZF/pair.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/pair.ML Thu Jun 23 17:38:12 1994 +0200
@@ -35,13 +35,13 @@
val Pair_neq_fst = prove_goalw ZF.thy [Pair_def] "<a,b>=a ==> P"
(fn [major]=>
- [ (rtac (consI1 RS mem_anti_sym RS FalseE) 1),
+ [ (rtac (consI1 RS mem_asym RS FalseE) 1),
(rtac (major RS subst) 1),
(rtac consI1 1) ]);
val Pair_neq_snd = prove_goalw ZF.thy [Pair_def] "<a,b>=b ==> P"
(fn [major]=>
- [ (rtac (consI1 RS consI2 RS mem_anti_sym RS FalseE) 1),
+ [ (rtac (consI1 RS consI2 RS mem_asym RS FalseE) 1),
(rtac (major RS subst) 1),
(rtac (consI1 RS consI2) 1) ]);
--- a/src/ZF/upair.ML Thu Jun 23 16:44:57 1994 +0200
+++ b/src/ZF/upair.ML Thu Jun 23 17:38:12 1994 +0200
@@ -207,20 +207,21 @@
val expand_if = prove_goal ZF.thy
"P(if(Q,x,y)) <-> ((Q --> P(x)) & (~Q --> P(y)))"
- (fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
+ (fn _=> [ (excluded_middle_tac "Q" 1),
(asm_simp_tac if_ss 1),
(asm_simp_tac if_ss 1) ]);
val prems = goal ZF.thy
"[| P ==> a: A; ~P ==> b: A |] ==> if(P,a,b): A";
-by (res_inst_tac [("Q","P")] (excluded_middle RS disjE) 1);
+by (excluded_middle_tac "P" 1);
by (ALLGOALS (asm_simp_tac (if_ss addsimps prems)));
val if_type = result();
(*** Foundation lemmas ***)
-val mem_anti_sym = prove_goal ZF.thy "[| a:b; b:a |] ==> P"
+(*was called mem_anti_sym*)
+val mem_asym = prove_goal ZF.thy "[| a:b; b:a |] ==> P"
(fn prems=>
[ (rtac disjE 1),
(res_inst_tac [("A","{a,b}")] foundation 1),
@@ -229,15 +230,16 @@
(fast_tac (lemmas_cs addIs (prems@[consI1,consI2])
addSEs [consE,equalityE]) 1) ]);
-val mem_anti_refl = prove_goal ZF.thy "a:a ==> P"
- (fn [major]=> [ (rtac (major RS (major RS mem_anti_sym)) 1) ]);
+(*was called mem_anti_refl*)
+val mem_irrefl = prove_goal ZF.thy "a:a ==> P"
+ (fn [major]=> [ (rtac (major RS (major RS mem_asym)) 1) ]);
-val mem_not_refl = prove_goal ZF.thy "a~:a"
- (K [ (rtac notI 1), (etac mem_anti_refl 1) ]);
+val mem_not_refl = prove_goal ZF.thy "a ~: a"
+ (K [ (rtac notI 1), (etac mem_irrefl 1) ]);
(*Good for proving inequalities by rewriting*)
val mem_imp_not_eq = prove_goal ZF.thy "!!a A. a:A ==> a ~= A"
- (fn _=> [ fast_tac (lemmas_cs addSEs [mem_anti_refl]) 1 ]);
+ (fn _=> [ fast_tac (lemmas_cs addSEs [mem_irrefl]) 1 ]);
(*** Rules for succ ***)
@@ -279,12 +281,12 @@
(fn [major]=>
[ (rtac (major RS equalityE) 1),
(REPEAT (eresolve_tac [asm_rl, sym, succE, make_elim succ_subsetD,
- mem_anti_sym] 1)) ]);
+ mem_asym] 1)) ]);
val succ_inject_iff = prove_goal ZF.thy "succ(m) = succ(n) <-> m=n"
(fn _=> [ (fast_tac (FOL_cs addSEs [succ_inject]) 1) ]);
-(*UpairI1/2 should become UpairCI; mem_anti_refl as a hazE? *)
+(*UpairI1/2 should become UpairCI; mem_irrefl as a hazE? *)
val upair_cs = lemmas_cs
addSIs [singletonI, DiffI, IntI, UnCI, consCI, succCI, UpairI1,UpairI2]
addSEs [singletonE, DiffE, IntE, UnE, consE, succE, UpairE];