--- a/src/ZF/AC/AC10_AC15.ML Mon Jan 29 14:16:13 1996 +0100
+++ b/src/ZF/AC/AC10_AC15.ML Tue Jan 30 13:42:57 1996 +0100
@@ -1,6 +1,6 @@
-(* Title: ZF/AC/AC10_AC15.ML
+(* Title: ZF/AC/AC10_AC15.ML
ID: $Id$
- Author: Krzysztof Grabczewski
+ Author: Krzysztof Grabczewski
The proofs needed to state that AC10, ..., AC15 are equivalent to the rest.
We need the following:
@@ -21,10 +21,10 @@
*)
(* ********************************************************************** *)
-(* Lemmas used in the proofs in which the conclusion is AC13, AC14 *)
-(* or AC15 *)
-(* - cons_times_nat_not_Finite *)
-(* - ex_fun_AC13_AC15 *)
+(* Lemmas used in the proofs in which the conclusion is AC13, AC14 *)
+(* or AC15 *)
+(* - cons_times_nat_not_Finite *)
+(* - ex_fun_AC13_AC15 *)
(* ********************************************************************** *)
goalw thy [lepoll_def] "!!A. A~=0 ==> B lepoll A*B";
@@ -40,7 +40,7 @@
by (rtac notI 1);
by (dresolve_tac [subset_consI RS subset_imp_lepoll RS lepoll_Finite] 1);
by (resolve_tac [lepoll_Sigma RS lepoll_Finite RS (nat_not_Finite RS notE)] 1
- THEN (assume_tac 2));
+ THEN (assume_tac 2));
by (fast_tac AC_cs 1);
val cons_times_nat_not_Finite = result();
@@ -49,17 +49,17 @@
val lemma1 = result();
goalw thy [pairwise_disjoint_def]
- "!!A. [| pairwise_disjoint(A); B:A; C:A; a:B; a:C |] ==> B=C";
+ "!!A. [| pairwise_disjoint(A); B:A; C:A; a:B; a:C |] ==> B=C";
by (dtac IntI 1 THEN (assume_tac 1));
by (dres_inst_tac [("A","B Int C")] not_emptyI 1);
by (fast_tac ZF_cs 1);
val lemma2 = result();
goalw thy [sets_of_size_between_def]
- "!!A. ALL B:{cons(0, x*nat). x:A}. pairwise_disjoint(f`B) & \
-\ sets_of_size_between(f`B, 2, n) & Union(f`B)=B \
-\ ==> ALL B:A. EX! u. u:f`cons(0, B*nat) & u <= cons(0, B*nat) & \
-\ 0:u & 2 lepoll u & u lepoll n";
+ "!!A. ALL B:{cons(0, x*nat). x:A}. pairwise_disjoint(f`B) & \
+\ sets_of_size_between(f`B, 2, n) & Union(f`B)=B \
+\ ==> ALL B:A. EX! u. u:f`cons(0, B*nat) & u <= cons(0, B*nat) & \
+\ 0:u & 2 lepoll u & u lepoll n";
by (rtac ballI 1);
by (etac ballE 1);
by (fast_tac ZF_cs 2);
@@ -75,12 +75,12 @@
goalw thy [lepoll_def] "!!A. [| A lepoll i; Ord(i) |] ==> {P(a). a:A} lepoll i";
by (etac exE 1);
by (res_inst_tac
- [("x", "lam x:RepFun(A, P). LEAST j. EX a:A. x=P(a) & f`a=j")] exI 1);
+ [("x", "lam x:RepFun(A, P). LEAST j. EX a:A. x=P(a) & f`a=j")] exI 1);
by (res_inst_tac [("d", "%y. P(converse(f)`y)")] lam_injective 1);
by (etac RepFunE 1);
by (forward_tac [inj_is_fun RS apply_type] 1 THEN (assume_tac 1));
by (fast_tac (AC_cs addIs [LeastI2]
- addSEs [Ord_in_Ord, inj_is_fun RS apply_type]) 1);
+ addSEs [Ord_in_Ord, inj_is_fun RS apply_type]) 1);
by (etac RepFunE 1);
by (rtac LeastI2 1);
by (fast_tac AC_cs 1);
@@ -89,38 +89,38 @@
val lemma4 = result();
goal thy "!!A. [| n:nat; B:A; u(B) <= cons(0, B*nat); 0:u(B); 2 lepoll u(B); \
-\ u(B) lepoll succ(n) |] \
-\ ==> (lam x:A. {fst(x). x:u(x)-{0}})`B ~= 0 & \
-\ (lam x:A. {fst(x). x:u(x)-{0}})`B <= B & \
-\ (lam x:A. {fst(x). x:u(x)-{0}})`B lepoll n";
+\ u(B) lepoll succ(n) |] \
+\ ==> (lam x:A. {fst(x). x:u(x)-{0}})`B ~= 0 & \
+\ (lam x:A. {fst(x). x:u(x)-{0}})`B <= B & \
+\ (lam x:A. {fst(x). x:u(x)-{0}})`B lepoll n";
by (asm_simp_tac AC_ss 1);
by (rtac conjI 1);
by (fast_tac (empty_cs addSDs [RepFun_eq_0_iff RS iffD1]
- addDs [lepoll_Diff_sing]
- addEs [lepoll_trans RS succ_lepoll_natE, ssubst]
- addSIs [notI, lepoll_refl, nat_0I]) 1);
+ addDs [lepoll_Diff_sing]
+ addEs [lepoll_trans RS succ_lepoll_natE, ssubst]
+ addSIs [notI, lepoll_refl, nat_0I]) 1);
by (rtac conjI 1);
by (fast_tac (ZF_cs addSIs [fst_type] addSEs [consE]) 1);
by (fast_tac (ZF_cs addSEs [equalityE,
- Diff_lepoll RS (nat_into_Ord RSN (2, lemma4))]) 1);
+ Diff_lepoll RS (nat_into_Ord RSN (2, lemma4))]) 1);
val lemma5 = result();
goal thy "!!A. [| EX f. ALL B:{cons(0, x*nat). x:A}. \
-\ pairwise_disjoint(f`B) & \
-\ sets_of_size_between(f`B, 2, succ(n)) & \
-\ Union(f`B)=B; n:nat |] \
-\ ==> EX f. ALL B:A. f`B ~= 0 & f`B <= B & f`B lepoll n";
+\ pairwise_disjoint(f`B) & \
+\ sets_of_size_between(f`B, 2, succ(n)) & \
+\ Union(f`B)=B; n:nat |] \
+\ ==> EX f. ALL B:A. f`B ~= 0 & f`B <= B & f`B lepoll n";
by (fast_tac (empty_cs addSDs [lemma3, theI] addDs [bspec]
- addSEs [exE, conjE]
- addIs [exI, ballI, lemma5]) 1);
+ addSEs [exE, conjE]
+ addIs [exI, ballI, lemma5]) 1);
val ex_fun_AC13_AC15 = result();
(* ********************************************************************** *)
-(* The target proofs *)
+(* The target proofs *)
(* ********************************************************************** *)
(* ********************************************************************** *)
-(* AC10(n) ==> AC11 *)
+(* AC10(n) ==> AC11 *)
(* ********************************************************************** *)
goalw thy AC_defs "!!Z. [| n:nat; 1 le n; AC10(n) |] ==> AC11";
@@ -129,7 +129,7 @@
qed "AC10_AC11";
(* ********************************************************************** *)
-(* AC11 ==> AC12 *)
+(* AC11 ==> AC12 *)
(* ********************************************************************** *)
goalw thy AC_defs "!! Z. AC11 ==> AC12";
@@ -137,7 +137,7 @@
qed "AC11_AC12";
(* ********************************************************************** *)
-(* AC12 ==> AC15 *)
+(* AC12 ==> AC15 *)
(* ********************************************************************** *)
goalw thy AC_defs "!!Z. AC12 ==> AC15";
@@ -149,35 +149,35 @@
qed "AC12_AC15";
(* ********************************************************************** *)
-(* AC15 ==> WO6 *)
+(* AC15 ==> WO6 *)
(* ********************************************************************** *)
(* in a separate file *)
(* ********************************************************************** *)
-(* The proof needed in the first case, not in the second *)
+(* The proof needed in the first case, not in the second *)
(* ********************************************************************** *)
(* ********************************************************************** *)
-(* AC10(n) ==> AC13(n-1) if 2 le n *)
-(* *)
-(* Because of the change to the formal definition of AC10(n) we prove *)
-(* the following obviously equivalent theorem : *)
-(* AC10(n) implies AC13(n) for (1 le n) *)
+(* AC10(n) ==> AC13(n-1) if 2 le n *)
+(* *)
+(* Because of the change to the formal definition of AC10(n) we prove *)
+(* the following obviously equivalent theorem : *)
+(* AC10(n) implies AC13(n) for (1 le n) *)
(* ********************************************************************** *)
goalw thy AC_defs "!!n. [| n:nat; 1 le n; AC10(n) |] ==> AC13(n)";
by (safe_tac ZF_cs);
by (fast_tac (empty_cs addSEs [allE, cons_times_nat_not_Finite RSN (2, impE),
- ex_fun_AC13_AC15]) 1);
+ ex_fun_AC13_AC15]) 1);
qed "AC10_AC13";
(* ********************************************************************** *)
-(* The proofs needed in the second case, not in the first *)
+(* The proofs needed in the second case, not in the first *)
(* ********************************************************************** *)
(* ********************************************************************** *)
-(* AC1 ==> AC13(1) *)
+(* AC1 ==> AC13(1) *)
(* ********************************************************************** *)
goalw thy AC_defs "!!Z. AC1 ==> AC13(1)";
@@ -188,13 +188,13 @@
by (etac exE 1);
by (res_inst_tac [("x","lam x:A. {f`x}")] exI 1);
by (asm_full_simp_tac (AC_ss addsimps
- [singleton_eqpoll_1 RS eqpoll_imp_lepoll,
- singletonI RS not_emptyI]) 1);
+ [singleton_eqpoll_1 RS eqpoll_imp_lepoll,
+ singletonI RS not_emptyI]) 1);
by (fast_tac (AC_cs addSEs [singletonE, apply_type]) 1);
qed "AC1_AC13";
(* ********************************************************************** *)
-(* AC13(m) ==> AC13(n) for m <= n *)
+(* AC13(m) ==> AC13(n) for m <= n *)
(* ********************************************************************** *)
goalw thy AC_defs "!!m n. [| m:nat; n:nat; m le n; AC13(m) |] ==> AC13(n)";
@@ -203,11 +203,11 @@
qed "AC13_mono";
(* ********************************************************************** *)
-(* The proofs necessary for both cases *)
+(* The proofs necessary for both cases *)
(* ********************************************************************** *)
(* ********************************************************************** *)
-(* AC13(n) ==> AC14 if 1 <= n *)
+(* AC13(n) ==> AC14 if 1 <= n *)
(* ********************************************************************** *)
goalw thy AC_defs "!!n. [| n:nat; 1 le n; AC13(n) |] ==> AC14";
@@ -215,7 +215,7 @@
qed "AC13_AC14";
(* ********************************************************************** *)
-(* AC14 ==> AC15 *)
+(* AC14 ==> AC15 *)
(* ********************************************************************** *)
goalw thy AC_defs "!!Z. AC14 ==> AC15";
@@ -223,11 +223,11 @@
qed "AC14_AC15";
(* ********************************************************************** *)
-(* The redundant proofs; however cited by Rubin & Rubin *)
+(* The redundant proofs; however cited by Rubin & Rubin *)
(* ********************************************************************** *)
(* ********************************************************************** *)
-(* AC13(1) ==> AC1 *)
+(* AC13(1) ==> AC1 *)
(* ********************************************************************** *)
goal thy "!!A. [| A~=0; A lepoll 1 |] ==> EX a. A={a}";
@@ -235,7 +235,7 @@
val lemma_aux = result();
goal thy "!!f. ALL B:A. f(B)~=0 & f(B)<=B & f(B) lepoll 1 \
-\ ==> (lam x:A. THE y. f(x)={y}) : (PROD X:A. X)";
+\ ==> (lam x:A. THE y. f(x)={y}) : (PROD X:A. X)";
by (rtac lam_type 1);
by (dtac bspec 1 THEN (assume_tac 1));
by (REPEAT (etac conjE 1));
@@ -249,7 +249,7 @@
qed "AC13_AC1";
(* ********************************************************************** *)
-(* AC11 ==> AC14 *)
+(* AC11 ==> AC14 *)
(* ********************************************************************** *)
goalw thy [AC11_def, AC14_def] "!!Z. AC11 ==> AC14";