(* Title: ZF/AC/AC10_AC15.ML
ID: $Id$
Author: Krzysztof Grabczewski
The proofs needed to state that AC10, ..., AC15 are equivalent to the rest.
We need the following:
WO1 ==> AC10(n) ==> AC11 ==> AC12 ==> AC15 ==> WO6
In order to add the formulations AC13 and AC14 we need:
AC10(succ(n)) ==> AC13(n) ==> AC14 ==> AC15
or
AC1 ==> AC13(1); AC13(m) ==> AC13(n) ==> AC14 ==> AC15 (m le n)
So we don't have to prove all implications of both cases.
Moreover we don't need to prove AC13(1) ==> AC1 and AC11 ==> AC14 as
Rubin & Rubin do.
*)
(* ********************************************************************** *)
(* Lemmas used in the proofs in which the conclusion is AC13, AC14 *)
(* or AC15 *)
(* - cons_times_nat_not_Finite *)
(* - ex_fun_AC13_AC15 *)
(* ********************************************************************** *)
Goalw [lepoll_def] "A~=0 ==> B lepoll A*B";
by (etac not_emptyE 1);
by (res_inst_tac [("x","lam z:B. <x,z>")] exI 1);
by (fast_tac (claset() addSIs [snd_conv, lam_injective]) 1);
qed "lepoll_Sigma";
Goal "0~:A ==> ALL B:{cons(0,x*nat). x:A}. ~Finite(B)";
by (rtac ballI 1);
by (etac RepFunE 1);
by (hyp_subst_tac 1);
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));
by (Fast_tac 1);
qed "cons_times_nat_not_Finite";
Goal "[| Union(C)=A; a:A |] ==> EX B:C. a:B & B <= A";
by (Fast_tac 1);
val lemma1 = result();
Goalw [pairwise_disjoint_def]
"[| 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 1);
val lemma2 = result();
Goalw [sets_of_size_between_def]
"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 2);
by (REPEAT (etac conjE 1));
by (dresolve_tac [consI1 RSN (2, lemma1)] 1);
by (etac bexE 1);
by (rtac ex1I 1);
by (Fast_tac 1);
by (REPEAT (etac conjE 1));
by (rtac lemma2 1 THEN (REPEAT (assume_tac 1)));
val lemma3 = result();
Goalw [lepoll_def] "[| 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);
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 (claset() addIs [LeastI2]
addSEs [Ord_in_Ord, inj_is_fun RS apply_type]) 1);
by (etac RepFunE 1);
by (rtac LeastI2 1);
by (Fast_tac 1);
by (fast_tac (claset() addSEs [Ord_in_Ord, inj_is_fun RS apply_type]) 1);
by (fast_tac (claset() addEs [sym, left_inverse RS ssubst]) 1);
val lemma4 = result();
Goal "[| 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";
by (Asm_simp_tac 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);
by (rtac conjI 1);
by (fast_tac (claset() addSIs [fst_type] addSEs [consE]) 1);
by (fast_tac (claset() addSEs [equalityE,
Diff_lepoll RS (nat_into_Ord RSN (2, lemma4))]) 1);
val lemma5 = result();
Goal "[| 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";
by (fast_tac (empty_cs addSDs [lemma3, theI] addDs [bspec]
addSEs [exE, conjE]
addIs [exI, ballI, lemma5]) 1);
qed "ex_fun_AC13_AC15";
(* ********************************************************************** *)
(* The target proofs *)
(* ********************************************************************** *)
(* ********************************************************************** *)
(* AC10(n) ==> AC11 *)
(* ********************************************************************** *)
Goalw AC_defs "[| n:nat; 1 le n; AC10(n) |] ==> AC11";
by (rtac bexI 1 THEN (assume_tac 2));
by (Fast_tac 1);
qed "AC10_AC11";
(* ********************************************************************** *)
(* AC11 ==> AC12 *)
(* ********************************************************************** *)
Goalw AC_defs "AC11 ==> AC12";
by (fast_tac (FOL_cs addSEs [bexE] addIs [bexI]) 1);
qed "AC11_AC12";
(* ********************************************************************** *)
(* AC12 ==> AC15 *)
(* ********************************************************************** *)
Goalw AC_defs "AC12 ==> AC15";
by Safe_tac;
by (etac allE 1);
by (etac impE 1);
by (etac cons_times_nat_not_Finite 1);
by (fast_tac (claset() addSIs [ex_fun_AC13_AC15]) 1);
qed "AC12_AC15";
(* ********************************************************************** *)
(* AC15 ==> WO6 *)
(* ********************************************************************** *)
(* in a separate file *)
(* ********************************************************************** *)
(* 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) *)
(* ********************************************************************** *)
Goalw AC_defs "[| n:nat; 1 le n; AC10(n) |] ==> AC13(n)";
by Safe_tac;
by (fast_tac (empty_cs addSEs [allE, cons_times_nat_not_Finite RSN (2, impE),
ex_fun_AC13_AC15]) 1);
qed "AC10_AC13";
(* ********************************************************************** *)
(* The proofs needed in the second case, not in the first *)
(* ********************************************************************** *)
(* ********************************************************************** *)
(* AC1 ==> AC13(1) *)
(* ********************************************************************** *)
Goalw AC_defs "AC1 ==> AC13(1)";
by (rtac allI 1);
by (etac allE 1);
by (rtac impI 1);
by (mp_tac 1);
by (etac exE 1);
by (res_inst_tac [("x","lam x:A. {f`x}")] exI 1);
by (asm_simp_tac (simpset() addsimps
[singleton_eqpoll_1 RS eqpoll_imp_lepoll,
singletonI RS not_emptyI]) 1);
qed "AC1_AC13";
(* ********************************************************************** *)
(* AC13(m) ==> AC13(n) for m <= n *)
(* ********************************************************************** *)
Goalw AC_defs "[| m le n; AC13(m) |] ==> AC13(n)";
by (dtac le_imp_lepoll 1);
by (fast_tac (claset() addSEs [lepoll_trans]) 1);
qed "AC13_mono";
(* ********************************************************************** *)
(* The proofs necessary for both cases *)
(* ********************************************************************** *)
(* ********************************************************************** *)
(* AC13(n) ==> AC14 if 1 <= n *)
(* ********************************************************************** *)
Goalw AC_defs "[| n:nat; 1 le n; AC13(n) |] ==> AC14";
by (fast_tac (FOL_cs addIs [bexI]) 1);
qed "AC13_AC14";
(* ********************************************************************** *)
(* AC14 ==> AC15 *)
(* ********************************************************************** *)
Goalw AC_defs "AC14 ==> AC15";
by (Fast_tac 1);
qed "AC14_AC15";
(* ********************************************************************** *)
(* The redundant proofs; however cited by Rubin & Rubin *)
(* ********************************************************************** *)
(* ********************************************************************** *)
(* AC13(1) ==> AC1 *)
(* ********************************************************************** *)
Goal "[| A~=0; A lepoll 1 |] ==> EX a. A={a}";
by (fast_tac (claset() addSEs [not_emptyE, lepoll_1_is_sing]) 1);
qed "lemma_aux";
Goal "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)";
by (rtac lam_type 1);
by (dtac bspec 1 THEN (assume_tac 1));
by (REPEAT (etac conjE 1));
by (eresolve_tac [lemma_aux RS exE] 1 THEN (assume_tac 1));
by (asm_full_simp_tac (simpset() addsimps [the_element]) 1);
val lemma = result();
Goalw AC_defs "AC13(1) ==> AC1";
by (fast_tac (claset() addSEs [lemma]) 1);
qed "AC13_AC1";
(* ********************************************************************** *)
(* AC11 ==> AC14 *)
(* ********************************************************************** *)
Goalw [AC11_def, AC14_def] "AC11 ==> AC14";
by (fast_tac (claset() addSIs [AC10_AC13]) 1);
qed "AC11_AC14";