--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/AC/AC16_WO4.ML Tue Jul 25 17:31:53 1995 +0200
@@ -0,0 +1,697 @@
+
+open AC16_WO4;
+
+(* ********************************************************************** *)
+(* The case of finite set *)
+(* ********************************************************************** *)
+
+goalw thy [Finite_def] "!!A. [| Finite(A); 0<m; m:nat |] ==> \
+\ EX a f. Ord(a) & domain(f) = a & \
+\ (UN b<a. f`b) = A & (ALL b<a. f`b lepoll m)";
+by (eresolve_tac [bexE] 1);
+by (dresolve_tac [eqpoll_sym RS (eqpoll_def RS def_imp_iff RS iffD1)] 1);
+by (eresolve_tac [exE] 1);
+by (res_inst_tac [("x","n")] exI 1);
+by (res_inst_tac [("x","lam i:n. {f`i}")] exI 1);
+by (asm_full_simp_tac AC_ss 1);
+by (rewrite_goals_tac [bij_def, surj_def]);
+by (fast_tac (AC_cs addSIs [ltI, nat_into_Ord, lam_funtype RS domain_of_fun,
+ equalityI, singleton_eqpoll_1 RS eqpoll_imp_lepoll RS lepoll_trans,
+ nat_1_lepoll_iff RS iffD2]
+ addSEs [singletonE, apply_type, ltE]) 1);
+val lemma1 = result();
+
+(* ********************************************************************** *)
+(* The case of infinite set *)
+(* ********************************************************************** *)
+
+goal thy "!!x. well_ord(x,r) ==> well_ord({{y,z}. y:x},?s(x,z))";
+by (eresolve_tac [paired_bij RS bij_is_inj RS well_ord_rvimage] 1);
+val well_ord_paired = uresult();
+
+goal thy "!!A. [| A lepoll B; ~ A lepoll C |] ==> ~ B lepoll C";
+by (fast_tac (FOL_cs addEs [notE, lepoll_trans]) 1);
+val lepoll_trans1 = result();
+
+goalw thy [lepoll_def]
+ "!!X.[| Y lepoll X; well_ord(X, R) |] ==> EX S. well_ord(Y, S)";
+by (fast_tac (AC_cs addSEs [well_ord_rvimage]) 1);
+val well_ord_lepoll = result();
+
+goal thy "!!X. [| well_ord(X,R); well_ord(Y,S) \
+\ |] ==> EX T. well_ord(X Un Y, T)";
+by (eresolve_tac [well_ord_radd RS (Un_lepoll_sum RS well_ord_lepoll)] 1);
+by (assume_tac 1);
+val well_ord_Un = result();
+
+(* ********************************************************************** *)
+(* There exists a well ordered set y such that ... *)
+(* ********************************************************************** *)
+
+goal thy "EX y R. well_ord(y,R) & x Int y = 0 & ~y lepoll z & ~Finite(y)";
+by (res_inst_tac [("x","{{a,x}. a:nat Un Hartog(z)}")] exI 1);
+by (resolve_tac [Ord_nat RS well_ord_Memrel RS (Ord_Hartog RS
+ well_ord_Memrel RSN (2, well_ord_Un)) RS exE] 1);
+by (fast_tac (AC_cs addSIs [Ord_Hartog, well_ord_Memrel, well_ord_paired,
+ equals0I, HartogI RSN (2, lepoll_trans1),
+ subset_imp_lepoll RS (paired_eqpoll RS eqpoll_sym RS
+ eqpoll_imp_lepoll RSN (2, lepoll_trans))]
+ addSEs [RepFunE, nat_not_Finite RS notE] addEs [mem_asym]
+ addSDs [Un_upper1 RS subset_imp_lepoll RS lepoll_Finite,
+ paired_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
+ RS lepoll_Finite]) 1);
+val lemma2 = result();
+
+val [prem] = goal thy "~Finite(B) ==> ~Finite(A Un B)";
+by (fast_tac (AC_cs
+ addSIs [subset_imp_lepoll RS (prem RSN (2, lepoll_infinite))]) 1);
+val infinite_Un = result();
+
+(* ********************************************************************** *)
+(* There is a v : s_u such that k lepoll x Int y (in our case succ(k)) *)
+(* The idea of the proof is the following : *)
+(* Suppose not, i.e. every element of s_u has exactly k-1 elements of y *)
+(* Thence y is less than or equipollent to {v:Pow(x). v eqpoll n#-k} *)
+(* We have obtained this result in two steps : *)
+(* 1. y is less than or equipollent to {v:s_u. a <= v} *)
+(* where a is certain k-2 element subset of y *)
+(* 2. {v:s_u. a <= v} is less than or equipollent *)
+(* to {v:Pow(x). v eqpoll n-k} *)
+(* ********************************************************************** *)
+
+goal thy "!!A. [| ~(EX x:A. f`x=y); f : inj(A, B); y:B |] \
+\ ==> (lam a:succ(A). if(a=A, y, f`a)) : inj(succ(A), B)";
+by (res_inst_tac [("d","%z. if(z=y, A, converse(f)`z)")] lam_injective 1);
+by (fast_tac (AC_cs addSEs [inj_is_fun RS apply_type]
+ addIs [expand_if RS iffD2]) 1);
+by (REPEAT (split_tac [expand_if] 1));
+by (fast_tac (AC_cs addSEs [left_inverse]) 1);
+val succ_not_lepoll_lemma = result();
+
+goalw thy [lepoll_def, eqpoll_def, bij_def, surj_def]
+ "!!A. [| ~A eqpoll B; A lepoll B |] ==> succ(A) lepoll B";
+by (fast_tac (AC_cs addSEs [succ_not_lepoll_lemma, inj_is_fun]) 1);
+val succ_not_lepoll_imp_eqpoll = result();
+
+val [prem] = goalw thy [s_u_def]
+ "(ALL v:s_u(u, t_n, k, y). k eqpoll v Int y ==> False) \
+\ ==> EX v : s_u(u, t_n, k, y). succ(k) lepoll v Int y";
+by (excluded_middle_tac "?P" 1 THEN (assume_tac 2));
+by (resolve_tac [prem RS FalseE] 1);
+by (resolve_tac [ballI] 1);
+by (eresolve_tac [CollectE] 1);
+by (eresolve_tac [conjE] 1);
+by (eresolve_tac [swap] 1);
+by (fast_tac (AC_cs addSEs [succ_not_lepoll_imp_eqpoll]) 1);
+val suppose_not = result();
+
+(* ********************************************************************** *)
+(* There is a k-2 element subset of y *)
+(* ********************************************************************** *)
+
+goalw thy [lepoll_def, eqpoll_def]
+ "!!X. [| n:nat; nat lepoll X |] ==> EX Y. Y<=X & n eqpoll Y";
+by (fast_tac (FOL_cs addSDs [Ord_nat RSN (2, OrdmemD) RSN (2, restrict_inj)]
+ addSIs [subset_refl]
+ addSEs [restrict_bij, inj_is_fun RS fun_is_rel RS image_subset]) 1);
+val nat_lepoll_imp_ex_eqpoll_n = result();
+
+val ordertype_eqpoll =
+ ordermap_bij RS (exI RS (eqpoll_def RS def_imp_iff RS iffD2));
+
+goal thy "!!y. [| well_ord(y,R); ~Finite(y); n:nat \
+\ |] ==> EX z. z<=y & n eqpoll z";
+by (eresolve_tac [nat_lepoll_imp_ex_eqpoll_n] 1);
+by (resolve_tac [ordertype_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
+ RSN (2, lepoll_trans)] 1 THEN (assume_tac 2));
+by (fast_tac (AC_cs addSIs [nat_le_infinite_Ord RS le_imp_lepoll]
+ addSEs [Ord_ordertype, ordertype_eqpoll RS eqpoll_imp_lepoll
+ RS lepoll_infinite]) 1);
+val ex_subset_eqpoll_n = result();
+
+goalw thy [lesspoll_def] "!!n. n: nat ==> n lesspoll nat";
+by (fast_tac (AC_cs addSEs [Ord_nat RSN (2, ltI) RS leI RS le_imp_lepoll,
+ eqpoll_sym RS eqpoll_imp_lepoll]
+ addIs [Ord_nat RSN (2, nat_succI RS ltI) RS leI
+ RS le_imp_lepoll RS lepoll_trans RS succ_lepoll_natE]) 1);
+val n_lesspoll_nat = result();
+
+goal thy "!!y. [| well_ord(y,R); ~Finite(y); k eqpoll a; a <= y; k: nat |] \
+\ ==> y - a eqpoll y";
+by (fast_tac (empty_cs addIs [lepoll_lesspoll_lesspoll]
+ addSIs [Card_cardinal, Diff_lesspoll_eqpoll_Card RS eqpoll_trans,
+ Card_cardinal RS Card_is_Ord RS nat_le_infinite_Ord
+ RS le_imp_lepoll]
+ addSEs [well_ord_cardinal_eqpoll,
+ well_ord_cardinal_eqpoll RS eqpoll_sym,
+ eqpoll_sym RS eqpoll_imp_lepoll,
+ n_lesspoll_nat RS lesspoll_lepoll_lesspoll,
+ well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
+ RS lepoll_infinite]) 1);
+val Diff_Finite_eqpoll = result();
+
+goal thy "!!x. [| a<=y; b:y-a; u:x |] ==> cons(b, cons(u, a)) : Pow(x Un y)";
+by (fast_tac AC_cs 1);
+val cons_cons_subset = result();
+
+goal thy "!!x. [| a eqpoll k; a<=y; b:y-a; u:x; x Int y = 0 \
+\ |] ==> cons(b, cons(u, a)) eqpoll succ(succ(k))";
+by (fast_tac (AC_cs addSIs [cons_eqpoll_succ] addEs [equals0D]) 1);
+val cons_cons_eqpoll = result();
+
+goalw thy [s_u_def] "s_u(u, t_n, k, y) <= t_n";
+by (fast_tac AC_cs 1);
+val s_u_subset = result();
+
+goalw thy [s_u_def, succ_def]
+ "!!w. [| w:t_n; cons(b,cons(u,a)) <= w; a <= y; b : y-a; k eqpoll a \
+\ |] ==> w: s_u(u, t_n, succ(k), y)";
+by (fast_tac (AC_cs addDs [eqpoll_imp_lepoll RS cons_lepoll_cong]
+ addSEs [subset_imp_lepoll RSN (2, lepoll_trans)]) 1);
+val s_uI = result();
+
+goalw thy [s_u_def] "!!v. v : s_u(u, t_n, k, y) ==> u : v";
+by (fast_tac AC_cs 1);
+val in_s_u_imp_u_in = result();
+
+goal thy
+ "!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}. \
+\ EX! w. w:t_n & z <= w; \
+\ k eqpoll a; a <= y; b : y - a; u : x; x Int y = 0 |] \
+\ ==> EX! c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c";
+by (eresolve_tac [ballE] 1);
+by (fast_tac (FOL_cs addSIs [CollectI, cons_cons_subset,
+ eqpoll_sym RS cons_cons_eqpoll]) 2);
+by (eresolve_tac [ex1E] 1);
+by (res_inst_tac [("a","w")] ex1I 1);
+by (resolve_tac [conjI] 1);
+by (resolve_tac [CollectI] 1);
+by (fast_tac (FOL_cs addSEs [s_uI]) 1);
+by (fast_tac AC_cs 1);
+by (fast_tac AC_cs 1);
+by (eresolve_tac [allE] 1);
+by (eresolve_tac [impE] 1);
+by (assume_tac 2);
+by (fast_tac (AC_cs addSEs [s_u_subset RS subsetD, in_s_u_imp_u_in]) 1);
+val ex1_superset_a = result();
+
+goal thy
+ "!!A. [| succ(k) eqpoll A; k eqpoll B; B <= A; a : A-B; k:nat \
+\ |] ==> A = cons(a, B)";
+by (resolve_tac [equalityI] 1);
+by (fast_tac AC_cs 2);
+by (resolve_tac [Diff_eq_0_iff RS iffD1] 1);
+by (resolve_tac [equals0I] 1);
+by (dresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll] 1);
+by (dresolve_tac [eqpoll_sym RS cons_eqpoll_succ] 1);
+by (fast_tac AC_cs 1);
+by (dresolve_tac [cons_eqpoll_succ] 1);
+by (fast_tac AC_cs 1);
+by (fast_tac (AC_cs addSIs [nat_succI]
+ addSEs [[eqpoll_sym RS eqpoll_imp_lepoll, subset_imp_lepoll] MRS
+ (lepoll_trans RS lepoll_trans) RS succ_lepoll_natE]) 1);
+val set_eq_cons = result();
+
+goal thy
+ "!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}. \
+\ EX! w. w:t_n & z <= w; \
+\ ALL v:s_u(u, t_n, succ(k), y). succ(k) eqpoll v Int y; \
+\ k eqpoll a; a <= y; b : y - a; u : x; x Int y = 0; k:nat \
+\ |] ==> (THE c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c) \
+\ Int y = cons(b, a)";
+by (dresolve_tac [ex1_superset_a RS theI] 1 THEN REPEAT (assume_tac 1));
+by (resolve_tac [set_eq_cons] 1);
+by (REPEAT (fast_tac AC_cs 1));
+val the_eq_cons = result();
+
+goal thy "!!a. [| cons(x,a) = cons(y,a); x~: a |] ==> x = y ";
+by (fast_tac (AC_cs addSEs [equalityE]) 1);
+val cons_eqE = result();
+
+goal thy "!!A. A = B ==> A Int C = B Int C";
+by (asm_simp_tac AC_ss 1);
+val eq_imp_Int_eq = result();
+
+goal thy "!!a. [| a=b; a=c; b=d |] ==> c=d";
+by (asm_full_simp_tac AC_ss 1);
+val msubst = result();
+
+(* ********************************************************************** *)
+(* 1. y is less than or equipollent to {v:s_u. a <= v} *)
+(* where a is certain k-2 element subset of y *)
+(* ********************************************************************** *)
+
+goal thy
+ "!!y. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(k))}. \
+\ EX! w. w:t_n & z <= w; \
+\ ALL v:s_u(u, t_n, succ(k), y). succ(k) eqpoll v Int y; \
+\ well_ord(y,R); ~Finite(y); k eqpoll a; a <= y; \
+\ k:nat; u:x; x Int y = 0 |] \
+\ ==> y lepoll {v:s_u(u, t_n, succ(k), y). a <= v}";
+by (resolve_tac [Diff_Finite_eqpoll RS eqpoll_sym RS
+ eqpoll_imp_lepoll RS lepoll_trans] 1
+ THEN REPEAT (assume_tac 1));
+by (res_inst_tac [("f3","lam b:y-a. \
+\ THE c. c:{v:s_u(u, t_n, succ(k), y). a <= v} & b:c")]
+ (exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
+by (resolve_tac [inj_def RS def_imp_eq RS ssubst] 1);
+by (resolve_tac [CollectI] 1);
+by (resolve_tac [lam_type] 1);
+by (resolve_tac [ex1_superset_a RS theI RS conjunct1] 1
+ THEN REPEAT (assume_tac 1));
+by (resolve_tac [ballI] 1);
+by (resolve_tac [ballI] 1);
+by (resolve_tac [beta RS ssubst] 1 THEN (assume_tac 1));
+by (resolve_tac [beta RS ssubst] 1 THEN (assume_tac 1));
+by (resolve_tac [impI] 1);
+by (resolve_tac [cons_eqE] 1);
+by (fast_tac AC_cs 2);
+by (dres_inst_tac [("A","THE c. ?P(c)"), ("C","y")] eq_imp_Int_eq 1);
+by (eresolve_tac [[asm_rl, the_eq_cons, the_eq_cons] MRS msubst] 1
+ THEN REPEAT (assume_tac 1));
+val y_lepoll_subset_s_u = result();
+
+(* ********************************************************************** *)
+(* some arithmetic *)
+(* ********************************************************************** *)
+
+goal thy
+ "!!k. [| k:nat; m:nat |] ==> \
+\ ALL A B. A eqpoll k #+ m & k lepoll B & B<=A --> A-B lepoll m";
+by (eres_inst_tac [("n","k")] nat_induct 1);
+by (simp_tac (AC_ss addsimps [add_0]) 1);
+by (fast_tac (AC_cs addIs [eqpoll_imp_lepoll RS
+ (Diff_subset RS subset_imp_lepoll RS lepoll_trans)]) 1);
+by (REPEAT (resolve_tac [allI,impI] 1));
+by (resolve_tac [succ_lepoll_imp_not_empty RS not_emptyE] 1);
+by (fast_tac AC_cs 1);
+by (eres_inst_tac [("x","A - {xa}")] allE 1);
+by (eres_inst_tac [("x","B - {xa}")] allE 1);
+by (eresolve_tac [impE] 1);
+by (asm_full_simp_tac (AC_ss addsimps [add_succ]) 1);
+by (fast_tac (AC_cs addSIs [Diff_sing_eqpoll, lepoll_Diff_sing]) 1);
+by (res_inst_tac [("P","%z. z lepoll m")] subst 1 THEN (assume_tac 2));
+by (fast_tac (AC_cs addSIs [equalityI]) 1);
+val eqpoll_sum_imp_Diff_lepoll_lemma = result();
+
+goal thy "!!k. [| A eqpoll succ(k #+ m); B<=A; succ(k) lepoll B; \
+\ k:nat; m:nat |] \
+\ ==> A-B lepoll m";
+by (dresolve_tac [add_succ RS ssubst] 1);
+by (dresolve_tac [nat_succI RS eqpoll_sum_imp_Diff_lepoll_lemma] 1
+ THEN (REPEAT (assume_tac 1)));
+by (fast_tac AC_cs 1);
+val eqpoll_sum_imp_Diff_lepoll = result();
+
+(* ********************************************************************** *)
+(* similar properties for eqpoll *)
+(* ********************************************************************** *)
+
+goal thy
+ "!!k. [| k:nat; m:nat |] ==> \
+\ ALL A B. A eqpoll k #+ m & k eqpoll B & B<=A --> A-B eqpoll m";
+by (eres_inst_tac [("n","k")] nat_induct 1);
+by (fast_tac (AC_cs addSDs [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_0_is_0]
+ addss (AC_ss addsimps [add_0])) 1);
+by (REPEAT (resolve_tac [allI,impI] 1));
+by (resolve_tac [succ_lepoll_imp_not_empty RS not_emptyE] 1);
+by (fast_tac (AC_cs addSEs [eqpoll_imp_lepoll]) 1);
+by (eres_inst_tac [("x","A - {xa}")] allE 1);
+by (eres_inst_tac [("x","B - {xa}")] allE 1);
+by (eresolve_tac [impE] 1);
+by (fast_tac (AC_cs addSIs [Diff_sing_eqpoll,
+ eqpoll_sym RSN (2, Diff_sing_eqpoll) RS eqpoll_sym]
+ addss (AC_ss addsimps [add_succ])) 1);
+by (res_inst_tac [("P","%z. z eqpoll m")] subst 1 THEN (assume_tac 2));
+by (fast_tac (AC_cs addSIs [equalityI]) 1);
+val eqpoll_sum_imp_Diff_eqpoll_lemma = result();
+
+goal thy "!!k. [| A eqpoll succ(k #+ m); B<=A; succ(k) eqpoll B; \
+\ k:nat; m:nat |] \
+\ ==> A-B eqpoll m";
+by (dresolve_tac [add_succ RS ssubst] 1);
+by (dresolve_tac [nat_succI RS eqpoll_sum_imp_Diff_eqpoll_lemma] 1
+ THEN (REPEAT (assume_tac 1)));
+by (fast_tac AC_cs 1);
+val eqpoll_sum_imp_Diff_eqpoll = result();
+
+(* ********************************************************************** *)
+(* back to the second part *)
+(* ********************************************************************** *)
+
+goal thy "!!w. [| x Int y = 0; w <= x Un y |] \
+\ ==> w Int (x - {u}) = w - cons(u, w Int y)";
+by (fast_tac (AC_cs addSIs [equalityI] addEs [equals0D]) 1);
+val w_Int_eq_w_Diff = result();
+
+goal thy "!!w. [| w:{v:s_u(u, t_n, succ(l), y). a <= v}; \
+\ l eqpoll a; t_n <= {v:Pow(x Un y). v eqpoll succ(succ(l) #+ m)}; \
+\ m:nat; l:nat; x Int y = 0; u : x; \
+\ ALL v:s_u(u, t_n, succ(l), y). succ(l) eqpoll v Int y \
+\ |] ==> w Int (x - {u}) eqpoll m";
+by (eresolve_tac [CollectE] 1);
+by (resolve_tac [w_Int_eq_w_Diff RS ssubst] 1 THEN (assume_tac 1));
+by (fast_tac (AC_cs addSDs [s_u_subset RS subsetD]) 1);
+by (fast_tac (AC_cs addEs [equals0D] addSDs [bspec]
+ addDs [s_u_subset RS subsetD]
+ addSEs [eqpoll_sym RS cons_eqpoll_succ RS eqpoll_sym, in_s_u_imp_u_in]
+ addSIs [nat_succI, eqpoll_sum_imp_Diff_eqpoll]) 1);
+val w_Int_eqpoll_m = result();
+
+goal thy "!!m. [| 0<m; x eqpoll m; m:nat |] ==> x ~= 0";
+by (fast_tac (empty_cs
+ addSEs [mem_irrefl, ltE, eqpoll_succ_imp_not_empty, natE]) 1);
+val eqpoll_m_not_empty = result();
+
+goal thy
+ "!!z. [| z : xa Int (x - {u}); l eqpoll a; a <= y; x Int y = 0; u:x \
+\ |] ==> cons(z, cons(u, a)) : {v: Pow(x Un y). v eqpoll succ(succ(l))}";
+by (fast_tac (AC_cs addSIs [cons_eqpoll_succ] addEs [equals0D, eqpoll_sym]) 1);
+val cons_cons_in = result();
+
+(* ********************************************************************** *)
+(* 2. {v:s_u. a <= v} is less than or equipollent *)
+(* to {v:Pow(x). v eqpoll n-k} *)
+(* ********************************************************************** *)
+
+goal thy
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(succ(l))}. \
+\ EX! w. w:t_n & z <= w; \
+\ t_n <= {v:Pow(x Un y). v eqpoll succ(succ(l) #+ m)}; \
+\ 0<m; m:nat; l:nat; \
+\ ALL v:s_u(u, t_n, succ(l), y). succ(l) eqpoll v Int y; \
+\ a <= y; l eqpoll a; x Int y = 0; u : x \
+\ |] ==> {v:s_u(u, t_n, succ(l), y). a <= v} \
+\ lepoll {v:Pow(x). v eqpoll m}";
+by (res_inst_tac [("f3","lam w:{v:s_u(u, t_n, succ(l), y). a <= v}. \
+\ w Int (x - {u})")]
+ (exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
+by (resolve_tac [inj_def RS def_imp_eq RS ssubst] 1);
+by (resolve_tac [CollectI] 1);
+by (resolve_tac [lam_type] 1);
+by (resolve_tac [CollectI] 1);
+by (fast_tac AC_cs 1);
+by (resolve_tac [w_Int_eqpoll_m] 1 THEN REPEAT (assume_tac 1));
+by (simp_tac AC_ss 1);
+by (REPEAT (resolve_tac [ballI, impI] 1));
+by (eresolve_tac [w_Int_eqpoll_m RSN (2, eqpoll_m_not_empty) RS not_emptyE] 1
+ THEN REPEAT (assume_tac 1));
+by (dresolve_tac [equalityD1 RS subsetD] 1 THEN (assume_tac 1));
+by (dresolve_tac [cons_cons_in RSN (2, bspec)] 1 THEN REPEAT (assume_tac 1));
+by (eresolve_tac [ex1_two_eq] 1);
+by (fast_tac (AC_cs addSEs [s_u_subset RS subsetD, in_s_u_imp_u_in]) 1);
+by (fast_tac (AC_cs addSEs [s_u_subset RS subsetD, in_s_u_imp_u_in]) 1);
+val subset_s_u_lepoll_w = result();
+
+goal thy "!!k. [| 0<k; k:nat |] ==> EX l:nat. k = succ(l)";
+by (eresolve_tac [natE] 1);
+by (fast_tac (empty_cs addSEs [ltE, mem_irrefl]) 1);
+by (fast_tac (empty_cs addSIs [refl, bexI]) 1);
+val ex_eq_succ = result();
+
+goal thy
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. \
+\ EX! w. w:t_n & z <= w; \
+\ well_ord(y,R); ~Finite(y); u:x; x Int y = 0; \
+\ t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)}; \
+\ ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat \
+\ |] ==> EX v : s_u(u, t_n, k, y). succ(k) lepoll v Int y";
+by (resolve_tac [suppose_not] 1);
+by (eresolve_tac [ex_eq_succ RS bexE] 1 THEN (assume_tac 1));
+by (hyp_subst_tac 1);
+by (res_inst_tac [("n1","xa")] (ex_subset_eqpoll_n RS exE) 1
+ THEN REPEAT (assume_tac 1));
+by (eresolve_tac [conjE] 1);
+by (forward_tac [[y_lepoll_subset_s_u, subset_s_u_lepoll_w] MRS lepoll_trans] 1
+ THEN REPEAT (assume_tac 1));
+by (contr_tac 1);
+val exists_proper_in_s_u = result();
+
+(* ********************************************************************** *)
+(* LL can be well ordered *)
+(* ********************************************************************** *)
+
+goal thy "{x:Pow(X). x lepoll 0} = {0}";
+by (fast_tac (AC_cs addSDs [lepoll_0_is_0]
+ addSIs [singletonI, lepoll_refl, equalityI]
+ addSEs [singletonE]) 1);
+val subsets_lepoll_0_eq_unit = result();
+
+goal thy "!!X. [| well_ord(X, R); ~Finite(X); n:nat |] \
+\ ==> EX S. well_ord({Y: Pow(X) . Y eqpoll succ(n)}, S)";
+by (resolve_tac [well_ord_infinite_subsets_eqpoll_X
+ RS (eqpoll_def RS def_imp_iff RS iffD1) RS exE] 1
+ THEN (REPEAT (assume_tac 1)));
+by (fast_tac (ZF_cs addSEs [bij_is_inj RS well_ord_rvimage]) 1);
+val well_ord_subsets_eqpoll_n = result();
+
+goal thy "!!n. n:nat ==> {z:Pow(y). z lepoll succ(n)} = \
+\ {z:Pow(y). z lepoll n} Un {z:Pow(y). z eqpoll succ(n)}";
+by (fast_tac (ZF_cs addSIs [le_refl, leI,
+ le_imp_lepoll, equalityI]
+ addSDs [lepoll_succ_disj]
+ addSEs [nat_into_Ord, lepoll_trans, eqpoll_imp_lepoll]) 1);
+val subsets_lepoll_succ = result();
+
+goal thy "!!n. n:nat ==> \
+\ {z:Pow(y). z lepoll n} Int {z:Pow(y). z eqpoll succ(n)} = 0";
+by (fast_tac (ZF_cs addSEs [eqpoll_sym RS eqpoll_imp_lepoll
+ RS lepoll_trans RS succ_lepoll_natE]
+ addSIs [equals0I]) 1);
+val Int_empty = result();
+
+(* ********************************************************************** *)
+(* unit set is well-ordered by the empty relation *)
+(* ********************************************************************** *)
+
+goalw thy [irrefl_def, trans_on_def, part_ord_def, linear_def, tot_ord_def]
+ "tot_ord({a},0)";
+by (simp_tac ZF_ss 1);
+val tot_ord_unit = result();
+
+goalw thy [wf_on_def, wf_def] "wf[{a}](0)";
+by (fast_tac (ZF_cs addSIs [equalityI]) 1);
+val wf_on_unit = result();
+
+goalw thy [well_ord_def] "well_ord({a},0)";
+by (simp_tac (ZF_ss addsimps [tot_ord_unit, wf_on_unit]) 1);
+val well_ord_unit = result();
+
+(* ********************************************************************** *)
+(* well_ord_subsets_lepoll_n *)
+(* ********************************************************************** *)
+
+goal thy "!!y r. [| well_ord(y,r); ~Finite(y); n:nat |] ==> \
+\ EX R. well_ord({z:Pow(y). z lepoll n}, R)";
+by (eresolve_tac [nat_induct] 1);
+by (fast_tac (ZF_cs addSIs [well_ord_unit]
+ addss (ZF_ss addsimps [subsets_lepoll_0_eq_unit])) 1);
+by (eresolve_tac [exE] 1);
+by (eresolve_tac [well_ord_subsets_eqpoll_n RS exE] 1
+ THEN REPEAT (assume_tac 1));
+by (asm_simp_tac (ZF_ss addsimps [subsets_lepoll_succ]) 1);
+by (dresolve_tac [well_ord_radd] 1 THEN (assume_tac 1));
+by (eresolve_tac [Int_empty RS disj_Un_eqpoll_sum RS
+ (eqpoll_def RS def_imp_iff RS iffD1) RS exE] 1);
+by (fast_tac (ZF_cs addSEs [bij_is_inj RS well_ord_rvimage]) 1);
+val well_ord_subsets_lepoll_n = result();
+
+goalw thy [LL_def, MM_def]
+ "!!x. t_n <= {v:Pow(x Un y). v eqpoll n} \
+\ ==> LL(t_n, k, y) <= {z:Pow(y). z lepoll n}";
+by (fast_tac (AC_cs addSEs [RepFunE]
+ addIs [subset_imp_lepoll RS (eqpoll_imp_lepoll
+ RSN (2, lepoll_trans))]) 1);
+val LL_subset = result();
+
+goal thy "!!x. [| t_n <= {v:Pow(x Un y). v eqpoll n}; \
+\ well_ord(y, R); ~Finite(y); n:nat \
+\ |] ==> EX S. well_ord(LL(t_n, k, y), S)";
+by (fast_tac (FOL_cs addIs [exI]
+ addSEs [LL_subset RSN (2, well_ord_subset)]
+ addEs [well_ord_subsets_lepoll_n RS exE]) 1);
+val well_ord_LL = result();
+
+(* ********************************************************************** *)
+(* every element of LL is a contained in exactly one element of MM *)
+(* ********************************************************************** *)
+
+goalw thy [MM_def, LL_def]
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\ t_n <= {v:Pow(x Un y). v eqpoll n}; \
+\ v:LL(t_n, k, y) \
+\ |] ==> EX! w. w:MM(t_n, k, y) & v<=w";
+by (step_tac (AC_cs addSEs [RepFunE]) 1);
+by (resolve_tac [lepoll_imp_eqpoll_subset RS exE] 1 THEN (assume_tac 1));
+by (eres_inst_tac [("x","xa")] ballE 1);
+by (fast_tac (AC_cs addSEs [eqpoll_sym]) 2);
+by (res_inst_tac [("a","v")] ex1I 1);
+by (fast_tac AC_cs 1);
+by (eresolve_tac [ex1E] 1);
+by (res_inst_tac [("x","v")] allE 1 THEN (assume_tac 1));
+by (eres_inst_tac [("x","xb")] allE 1);
+by (fast_tac AC_cs 1);
+val unique_superset_in_MM = result();
+
+(* ********************************************************************** *)
+(* The function GG satisfies the conditions of WO4 *)
+(* ********************************************************************** *)
+
+(* ********************************************************************** *)
+(* The union of appropriate values is the whole x *)
+(* ********************************************************************** *)
+
+goal thy
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. \
+\ EX! w. w:t_n & z <= w; \
+\ well_ord(y,R); ~Finite(y); u:x; x Int y = 0; \
+\ t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)}; \
+\ ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat \
+\ |] ==> EX w:MM(t_n, succ(k), y). u:w";
+by (eresolve_tac [exists_proper_in_s_u RS bexE] 1
+ THEN REPEAT (assume_tac 1));
+by (rewrite_goals_tac [MM_def, s_u_def]);
+by (fast_tac AC_cs 1);
+val exists_in_MM = result();
+
+goalw thy [LL_def] "!!w. w : MM(t_n, k, y) ==> w Int y : LL(t_n, k, y)";
+by (fast_tac AC_cs 1);
+val Int_in_LL = result();
+
+goalw thy [MM_def] "MM(t_n, k, y) <= t_n";
+by (fast_tac AC_cs 1);
+val MM_subset = result();
+
+goal thy
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. \
+\ EX! w. w:t_n & z <= w; \
+\ well_ord(y,R); ~Finite(y); u:x; x Int y = 0; \
+\ t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)}; \
+\ ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat \
+\ |] ==> EX w:LL(t_n, succ(k), y). u:GG(t_n, succ(k), y)`w";
+by (forward_tac [exists_in_MM] 1 THEN REPEAT (assume_tac 1));
+by (eresolve_tac [bexE] 1);
+by (res_inst_tac [("x","w Int y")] bexI 1);
+by (eresolve_tac [Int_in_LL] 2);
+by (rewrite_goals_tac [GG_def]);
+by (asm_full_simp_tac (AC_ss addsimps [Int_in_LL]) 1);
+by (eresolve_tac [unique_superset_in_MM RS the_equality2 RS ssubst] 1
+ THEN (assume_tac 1));
+by (REPEAT (fast_tac (AC_cs addEs [equals0D] addSEs [Int_in_LL]) 1));
+val exists_in_LL = result();
+
+goalw thy [LL_def]
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\ t_n <= {v:Pow(x Un y). v eqpoll n}; \
+\ v : LL(t_n, k, y) |] \
+\ ==> v = (THE x. x : MM(t_n, k, y) & v <= x) Int y";
+by (fast_tac (AC_cs addSEs [Int_in_LL,
+ unique_superset_in_MM RS the_equality2 RS ssubst]) 1);
+val in_LL_eq_Int = result();
+
+goal thy
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\ t_n <= {v:Pow(x Un y). v eqpoll n}; \
+\ v : LL(t_n, k, y) |] \
+\ ==> (THE x. x : MM(t_n, k, y) & v <= x) <= x Un y";
+by (fast_tac (AC_cs addSDs [unique_superset_in_MM RS theI RS conjunct1 RS
+ (MM_subset RS subsetD)]) 1);
+val the_in_MM_subset = result();
+
+goalw thy [GG_def]
+ "!!x. [| ALL z:{z: Pow(x Un y) . z eqpoll k}. EX! w. w:t_n & z <= w; \
+\ t_n <= {v:Pow(x Un y). v eqpoll n}; \
+\ v : LL(t_n, k, y) |] \
+\ ==> GG(t_n, k, y) ` v <= x";
+by (asm_full_simp_tac AC_ss 1);
+by (forward_tac [the_in_MM_subset] 1 THEN REPEAT (assume_tac 1));
+by (dresolve_tac [in_LL_eq_Int] 1 THEN REPEAT (assume_tac 1));
+by (resolve_tac [subsetI] 1);
+by (eresolve_tac [DiffE] 1);
+by (eresolve_tac [swap] 1);
+by (fast_tac (AC_cs addEs [ssubst]) 1);
+val GG_subset = result();
+
+goal thy
+ "!!x. [| well_ord(LL(t_n, succ(k), y), S); \
+\ ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. EX! w. w:t_n & z <= w; \
+\ well_ord(y,R); ~Finite(y); x Int y = 0; \
+\ t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)}; \
+\ ~ y lepoll {v:Pow(x). v eqpoll m}; 0<k; 0<m; k:nat; m:nat \
+\ |] ==> (UN b<ordertype(LL(t_n, succ(k), y), S). \
+\ (GG(t_n, succ(k), y)) ` \
+\ (converse(ordermap(LL(t_n, succ(k), y), S)) ` b)) = x";
+by (resolve_tac [equalityI] 1);
+by (resolve_tac [subsetI] 1);
+by (eresolve_tac [OUN_E] 1);
+by (eresolve_tac [GG_subset RS subsetD] 1 THEN TRYALL assume_tac);
+by (eresolve_tac [ordermap_bij RS bij_converse_bij RS
+ bij_is_fun RS apply_type] 1);
+by (eresolve_tac [ltD] 1);
+by (resolve_tac [subsetI] 1);
+by (eresolve_tac [exists_in_LL RS bexE] 1 THEN REPEAT (assume_tac 1));
+by (resolve_tac [OUN_I] 1);
+by (resolve_tac [Ord_ordertype RSN (2, ltI)] 1 THEN (assume_tac 2));
+by (eresolve_tac [ordermap_type RS apply_type] 1);
+by (eresolve_tac [ordermap_bij RS bij_is_inj RS left_inverse RS ssubst] 1
+ THEN REPEAT (assume_tac 1));
+val OUN_eq_x = result();
+
+(* ********************************************************************** *)
+(* Every element of the family is less than or equipollent to n-k (m) *)
+(* ********************************************************************** *)
+
+goalw thy [MM_def]
+ "!!w. [| w : MM(t_n, k, y); t_n <= {v:Pow(x Un y). v eqpoll n} \
+\ |] ==> w eqpoll n";
+by (fast_tac AC_cs 1);
+val in_MM_eqpoll_n = result();
+
+goalw thy [LL_def, MM_def]
+ "!!w. w : LL(t_n, k, y) ==> k lepoll w";
+by (fast_tac AC_cs 1);
+val in_LL_eqpoll_n = result();
+
+goalw thy [GG_def]
+ "!!x. [| \
+\ ALL z:{z: Pow(x Un y) . z eqpoll succ(k)}. EX! w. w:t_n & z <= w; \
+\ t_n <= {v:Pow(x Un y). v eqpoll succ(k #+ m)}; \
+\ well_ord(LL(t_n, succ(k), y), S); k:nat; m:nat |] \
+\ ==> ALL b<ordertype(LL(t_n, succ(k), y), S). \
+\ (GG(t_n, succ(k), y)) ` \
+\ (converse(ordermap(LL(t_n, succ(k), y), S)) ` b) lepoll m";
+by (resolve_tac [oallI] 1);
+by (asm_full_simp_tac (AC_ss addsimps [ltD,
+ ordermap_bij RS bij_converse_bij RS bij_is_fun RS apply_type]) 1);
+by (resolve_tac [eqpoll_sum_imp_Diff_lepoll] 1);
+by (REPEAT (fast_tac (FOL_cs addSDs [ltD]
+ addSIs [eqpoll_sum_imp_Diff_lepoll, in_LL_eqpoll_n]
+ addEs [unique_superset_in_MM RS theI RS conjunct1 RS in_MM_eqpoll_n,
+ in_LL_eq_Int RS equalityD1 RS (Int_lower1 RSN (2, subset_trans)),
+ ordermap_bij RS bij_converse_bij RS bij_is_fun RS apply_type]) 1));
+val all_in_lepoll_m = result();
+
+(* ********************************************************************** *)
+(* The main theorem AC16(n, k) ==> WO4(n-k) *)
+(* ********************************************************************** *)
+
+goalw thy [AC16_def,WO4_def]
+ "!!n k. [| AC16(k #+ m, k); 0 < k; 0 < m; k:nat; m:nat |] ==> WO4(m)";
+by (resolve_tac [allI] 1);
+by (excluded_middle_tac "Finite(A)" 1);
+by (resolve_tac [lemma1] 2 THEN REPEAT (assume_tac 2));
+by (resolve_tac [lemma2 RS revcut_rl] 1);
+by (REPEAT (eresolve_tac [exE, conjE] 1));
+by (eres_inst_tac [("x","A Un y")] allE 1);
+by (forward_tac [infinite_Un] 1 THEN (mp_tac 1));
+by (REPEAT (eresolve_tac [exE, conjE] 1));
+by (resolve_tac [well_ord_LL RS exE] 1 THEN REPEAT (assume_tac 1));
+by (fast_tac (AC_cs addSIs [nat_succI, add_type]) 1);
+by (res_inst_tac [("x","ordertype(LL(T, succ(k), y), x)")] exI 1);
+by (res_inst_tac [("x","lam b:ordertype(LL(T, succ(k), y), x). \
+\ (GG(T, succ(k), y)) ` \
+\ (converse(ordermap(LL(T, succ(k), y), x)) ` b)")] exI 1);
+by (simp_tac AC_ss 1);
+by (fast_tac (empty_cs addSIs [conjI, lam_funtype RS domain_of_fun]
+ addSEs [Ord_ordertype, all_in_lepoll_m, OUN_eq_x]) 1);
+qed "AC16_WO4";