(* Title: ZF/AC/AC16_WO4.ML
ID: $Id$
Author: Krzysztof Grabczewski
The proof of AC16(n, k) ==> WO4(n-k)
*)
(* ********************************************************************** *)
(* The case of finite set *)
(* ********************************************************************** *)
Goalw [Finite_def] "[| 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 (etac bexE 1);
by (dresolve_tac [eqpoll_sym RS (eqpoll_def RS def_imp_iff RS iffD1)] 1);
by (etac 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 1);
by (rewrite_goals_tac [bij_def, surj_def]);
by (fast_tac (claset() 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 [apply_type, ltE]) 1);
val lemma1 = result();
(* ********************************************************************** *)
(* The case of infinite set *)
(* ********************************************************************** *)
(* well_ord(x,r) ==> well_ord({{y,z}. y:x}, Something(x,z)) **)
bind_thm ("well_ord_paired", (paired_bij RS bij_is_inj RS well_ord_rvimage));
Goal "[| A lepoll B; ~ A lepoll C |] ==> ~ B lepoll C";
by (fast_tac (claset() addEs [notE, lepoll_trans]) 1);
qed "lepoll_trans1";
(* ********************************************************************** *)
(* There exists a well ordered set y such that ... *)
(* ********************************************************************** *)
val lepoll_paired = paired_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll;
Goal "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 [transfer thy Ord_nat RS well_ord_Memrel RS (Ord_Hartog RS
well_ord_Memrel RSN (2, well_ord_Un)) RS exE] 1);
by (fast_tac
(claset() addSIs [Ord_Hartog, well_ord_Memrel, well_ord_paired,
HartogI RSN (2, lepoll_trans1),
subset_imp_lepoll RS (lepoll_paired RSN (2, lepoll_trans))]
addSEs [nat_not_Finite RS notE] addEs [mem_asym]
addSDs [Un_upper1 RS subset_imp_lepoll RS lepoll_Finite,
lepoll_paired RS lepoll_Finite]) 1);
val lemma2 = result();
val [prem] = goal thy "~Finite(B) ==> ~Finite(A Un B)";
by (fast_tac (claset()
addSIs [subset_imp_lepoll RS (prem RSN (2, lepoll_infinite))]) 1);
qed "infinite_Un";
(* ********************************************************************** *)
(* 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} *)
(* ********************************************************************** *)
(*Proof simplified by LCP*)
Goal "[| ~(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 (auto_tac (claset(), simpset() addsimps [inj_is_fun RS apply_type]));
qed "succ_not_lepoll_lemma";
Goalw [lepoll_def, eqpoll_def, bij_def, surj_def]
"[| ~A eqpoll B; A lepoll B |] ==> succ(A) lepoll B";
by (fast_tac (claset() addSEs [succ_not_lepoll_lemma, inj_is_fun]) 1);
qed "succ_not_lepoll_imp_eqpoll";
(* ********************************************************************** *)
(* There is a k-2 element subset of y *)
(* ********************************************************************** *)
Goalw [lepoll_def, eqpoll_def]
"[| n:nat; nat lepoll X |] ==> EX Y. Y<=X & n eqpoll Y";
by (fast_tac (subset_cs addSDs [Ord_nat RSN (2, OrdmemD) RSN (2, restrict_inj)]
addSEs [restrict_bij, inj_is_fun RS fun_is_rel RS image_subset]) 1);
qed "nat_lepoll_imp_ex_eqpoll_n";
val ordertype_eqpoll =
ordermap_bij RS (exI RS (eqpoll_def RS def_imp_iff RS iffD2));
Goalw [lesspoll_def] "n: nat ==> n lesspoll nat";
by (fast_tac (claset() 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);
qed "n_lesspoll_nat";
Goal "[| a<=y; b:y-a; u:x |] ==> cons(b, cons(u, a)) : Pow(x Un y)";
by (Fast_tac 1);
qed "cons_cons_subset";
Goal "[| 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 (claset() addSIs [cons_eqpoll_succ]) 1);
qed "cons_cons_eqpoll";
Goal "[| succ(k) eqpoll A; k eqpoll B; B <= A; a : A-B; k:nat \
\ |] ==> A = cons(a, B)";
by (rtac equalityI 1);
by (Fast_tac 2);
by (resolve_tac [Diff_eq_0_iff RS iffD1] 1);
by (rtac 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 1);
by (dtac cons_eqpoll_succ 1);
by (Fast_tac 1);
by (fast_tac
(claset()
addSEs [[eqpoll_sym RS eqpoll_imp_lepoll, subset_imp_lepoll] MRS
(lepoll_trans RS lepoll_trans) RS succ_lepoll_natE]) 1);
qed "set_eq_cons";
Goal "[| cons(x,a) = cons(y,a); x~: a |] ==> x = y ";
by (fast_tac (claset() addSEs [equalityE]) 1);
qed "cons_eqE";
Goal "A = B ==> A Int C = B Int C";
by (Asm_simp_tac 1);
qed "eq_imp_Int_eq";
(* ********************************************************************** *)
(* some arithmetic *)
(* ********************************************************************** *)
Goal "[| k:nat; m:nat |] ==> \
\ ALL A B. A eqpoll k #+ m & k lepoll B & B<=A --> A-B lepoll m";
by (induct_tac "k" 1);
by (simp_tac (simpset() addsimps [add_0]) 1);
by (fast_tac (claset() 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 1);
by (eres_inst_tac [("x","A - {xa}")] allE 1);
by (eres_inst_tac [("x","B - {xa}")] allE 1);
by (etac impE 1);
by (asm_full_simp_tac (simpset() addsimps [add_succ]) 1);
by (fast_tac (claset() 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 1);
qed "eqpoll_sum_imp_Diff_lepoll_lemma";
Goal "[| 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 1);
qed "eqpoll_sum_imp_Diff_lepoll";
(* ********************************************************************** *)
(* similar properties for eqpoll *)
(* ********************************************************************** *)
Goal "[| k:nat; m:nat |] ==> \
\ ALL A B. A eqpoll k #+ m & k eqpoll B & B<=A --> A-B eqpoll m";
by (induct_tac "k" 1);
by (fast_tac (claset() addSDs [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_0_is_0]
addss (simpset() 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 (claset() addSEs [eqpoll_imp_lepoll]) 1);
by (eres_inst_tac [("x","A - {xa}")] allE 1);
by (eres_inst_tac [("x","B - {xa}")] allE 1);
by (etac impE 1);
by (fast_tac (claset() addSIs [Diff_sing_eqpoll,
eqpoll_sym RSN (2, Diff_sing_eqpoll) RS eqpoll_sym]
addss (simpset() addsimps [add_succ])) 1);
by (res_inst_tac [("P","%z. z eqpoll m")] subst 1 THEN (assume_tac 2));
by (Fast_tac 1);
qed "eqpoll_sum_imp_Diff_eqpoll_lemma";
Goal "[| 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 1);
qed "eqpoll_sum_imp_Diff_eqpoll";
(* ********************************************************************** *)
(* LL can be well ordered *)
(* ********************************************************************** *)
Goal "{x:Pow(X). x lepoll 0} = {0}";
by (fast_tac (claset() addSDs [lepoll_0_is_0] addSIs [lepoll_refl]) 1);
qed "subsets_lepoll_0_eq_unit";
Goal "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 (claset() addIs [le_refl, leI, le_imp_lepoll]
addSDs [lepoll_succ_disj]
addSEs [nat_into_Ord, lepoll_trans, eqpoll_imp_lepoll]) 1);
qed "subsets_lepoll_succ";
Goal "n:nat ==> {z:Pow(y). z lepoll n} Int {z:Pow(y). z eqpoll succ(n)} = 0";
by (fast_tac (claset() addSEs [eqpoll_sym RS eqpoll_imp_lepoll
RS lepoll_trans RS succ_lepoll_natE]
addSIs [equals0I]) 1);
qed "Int_empty";
Open_locale "AC16";
val all_ex = thm "all_ex";
val disjoint = thm "disjoint";
val includes = thm "includes";
val WO_R = thm "WO_R";
val k_def = thm "k_def";
val lnat = thm "lnat";
val mnat = thm "mnat";
val mpos = thm "mpos";
val Infinite = thm "Infinite";
val noLepoll = thm "noLepoll";
val LL_def = thm "LL_def";
val MM_def = thm "MM_def";
val GG_def = thm "GG_def";
val s_def = thm "s_def";
Addsimps [disjoint, WO_R, lnat, mnat, mpos, Infinite];
AddSIs [disjoint, WO_R, lnat, mnat, mpos];
Goalw [k_def] "k: nat";
by Auto_tac;
qed "knat";
Addsimps [knat]; AddSIs [knat];
AddSIs [Infinite]; (*if notI is removed!*)
AddSEs [Infinite RS notE];
AddEs [[disjoint, IntI] MRS (equals0D RS notE)];
(*use k = succ(l) *)
val includes_l = simplify (FOL_ss addsimps [k_def]) includes;
val all_ex_l = simplify (FOL_ss addsimps [k_def]) all_ex;
(* ********************************************************************** *)
(* 1. y is less than or equipollent to {v:s(u). a <= v} *)
(* where a is certain k-2 element subset of y *)
(* ********************************************************************** *)
Goal "[| l eqpoll a; a <= y |] ==> y - a eqpoll y";
by (cut_facts_tac [WO_R, Infinite, lnat] 1);
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);
qed "Diff_Finite_eqpoll";
Goalw [s_def] "s(u) <= t_n";
by (Fast_tac 1);
qed "s_subset";
Goalw [s_def, succ_def, k_def]
"[| w:t_n; cons(b,cons(u,a)) <= w; a <= y; b : y-a; l eqpoll a \
\ |] ==> w: s(u)";
by (fast_tac (claset() addDs [eqpoll_imp_lepoll RS cons_lepoll_cong]
addSEs [subset_imp_lepoll RSN (2, lepoll_trans)]) 1);
qed "sI";
Goalw [s_def] "v : s(u) ==> u : v";
by (Fast_tac 1);
qed "in_s_imp_u_in";
Goal "[| l eqpoll a; a <= y; b : y - a; u : x |] \
\ ==> EX! c. c: s(u) & a <= c & b:c";
by (rtac (all_ex_l RS ballE) 1);
by (blast_tac (claset() delrules [PowI]
addSIs [cons_cons_subset,
eqpoll_sym RS cons_cons_eqpoll]) 2);
by (etac ex1E 1);
by (res_inst_tac [("a","w")] ex1I 1);
by (blast_tac (claset() addIs [sI]) 1);
by (etac allE 1);
by (etac impE 1);
by (assume_tac 2);
by (fast_tac (claset() addSEs [s_subset RS subsetD, in_s_imp_u_in]) 1);
qed "ex1_superset_a";
Goal "[| ALL v:s(u). succ(l) eqpoll v Int y; \
\ l eqpoll a; a <= y; b : y - a; u : x |] \
\ ==> (THE c. c: s(u) & a <= c & b:c) \
\ Int y = cons(b, a)";
by (forward_tac [ex1_superset_a RS theI] 1 THEN REPEAT (assume_tac 1));
by (rtac set_eq_cons 1);
by (REPEAT (Fast_tac 1));
qed "the_eq_cons";
Goal "[| ALL v:s(u). succ(l) eqpoll v Int y; \
\ l eqpoll a; a <= y; u:x |] \
\ ==> y lepoll {v:s(u). a <= v}";
by (resolve_tac [Diff_Finite_eqpoll RS eqpoll_sym RS
eqpoll_imp_lepoll RS lepoll_trans] 1
THEN REPEAT (Fast_tac 1));
by (res_inst_tac
[("f3", "lam b:y-a. THE c. c: s(u) & a <= c & b:c")]
(exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
by (simp_tac (simpset() addsimps [inj_def]) 1);
by (rtac conjI 1);
by (rtac lam_type 1);
by (forward_tac [ex1_superset_a RS theI] 1 THEN REPEAT (Fast_tac 1));
by (Asm_simp_tac 1);
by (Clarify_tac 1);
by (rtac cons_eqE 1);
by (Fast_tac 2);
by (dres_inst_tac [("A","THE c. ?P(c)"), ("C","y")] eq_imp_Int_eq 1);
by (asm_full_simp_tac (simpset() addsimps [the_eq_cons]) 1);
qed "y_lepoll_subset_s";
(* ********************************************************************** *)
(* back to the second part *)
(* ********************************************************************** *)
Goal "w <= x Un y ==> w Int (x - {u}) = w - cons(u, w Int y)";
by (Fast_tac 1);
qed "w_Int_eq_w_Diff";
Goal "[| w:{v:s(u). a <= v}; \
\ l eqpoll a; u : x; \
\ ALL v:s(u). succ(l) eqpoll v Int y \
\ |] ==> w Int (x - {u}) eqpoll m";
by (etac CollectE 1);
by (stac w_Int_eq_w_Diff 1);
by (fast_tac (claset() addSDs [s_subset RS subsetD,
includes_l RS subsetD]) 1);
by (fast_tac (claset() addSDs [bspec]
addDs [s_subset RS subsetD, includes_l RS subsetD]
addSEs [eqpoll_sym RS cons_eqpoll_succ RS eqpoll_sym, in_s_imp_u_in]
addSIs [eqpoll_sum_imp_Diff_eqpoll]) 1);
qed "w_Int_eqpoll_m";
(* ********************************************************************** *)
(* 2. {v:s(u). a <= v} is less than or equipollent *)
(* to {v:Pow(x). v eqpoll n-k} *)
(* ********************************************************************** *)
Goal "x eqpoll m ==> x ~= 0";
by (cut_facts_tac [mpos] 1);
by (fast_tac (claset() addSEs [zero_lt_natE]
addSDs [eqpoll_succ_imp_not_empty]) 1);
qed "eqpoll_m_not_empty";
Goal "[| z : xa Int (x - {u}); l eqpoll a; a <= y; u:x |] \
\ ==> EX! w. w : t_n & cons(z, cons(u, a)) <= w";
by (rtac (all_ex RS bspec) 1);
by (rewtac k_def);
by (fast_tac (claset() addSIs [cons_eqpoll_succ] addEs [eqpoll_sym]) 1);
qed "cons_cons_in";
Goal "[| ALL v:s(u). succ(l) eqpoll v Int y; \
\ a <= y; l eqpoll a; u : x |] \
\ ==> {v:s(u). a <= v} lepoll {v:Pow(x). v eqpoll m}";
by (res_inst_tac [("f3","lam w:{v:s(u). a <= v}. w Int (x - {u})")]
(exI RS (lepoll_def RS def_imp_iff RS iffD2)) 1);
by (simp_tac (simpset() addsimps [inj_def]) 1);
by (rtac conjI 1);
by (rtac lam_type 1);
by (rtac CollectI 1);
by (Fast_tac 1);
by (rtac w_Int_eqpoll_m 1 THEN REPEAT (assume_tac 1));
by (REPEAT (resolve_tac [ballI, impI] 1));
(** LEVEL 8 **)
by (resolve_tac [w_Int_eqpoll_m RS eqpoll_m_not_empty RS not_emptyE] 1);
by (EVERY (map Blast_tac [4,3,2,1]));
by (dresolve_tac [equalityD1 RS subsetD] 1 THEN (assume_tac 1));
by (ftac cons_cons_in 1 THEN REPEAT (assume_tac 1));
by (etac ex1_two_eq 1);
by (REPEAT (blast_tac
(claset() addDs [s_subset RS subsetD, in_s_imp_u_in]) 1));
qed "subset_s_lepoll_w";
(* ********************************************************************** *)
(* well_ord_subsets_lepoll_n *)
(* ********************************************************************** *)
Goal "n:nat ==> EX S. well_ord({z: Pow(y) . z eqpoll succ(n)}, S)";
by (resolve_tac [WO_R RS well_ord_infinite_subsets_eqpoll_X
RS (eqpoll_def RS def_imp_iff RS iffD1) RS exE] 1);
by (REPEAT (fast_tac (claset() addIs [bij_is_inj RS well_ord_rvimage]) 1));
qed "well_ord_subsets_eqpoll_n";
Goal "n:nat ==> EX R. well_ord({z:Pow(y). z lepoll n}, R)";
by (induct_tac "n" 1);
by (force_tac (claset() addSIs [well_ord_unit],
simpset() addsimps [subsets_lepoll_0_eq_unit]) 1);
by (etac exE 1);
by (resolve_tac [well_ord_subsets_eqpoll_n RS exE] 1 THEN assume_tac 1);
by (asm_simp_tac (simpset() addsimps [subsets_lepoll_succ]) 1);
by (dtac 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 (claset() addSEs [bij_is_inj RS well_ord_rvimage]) 1);
qed "well_ord_subsets_lepoll_n";
Goalw [LL_def, MM_def] "LL <= {z:Pow(y). z lepoll succ(k #+ m)}";
by (cut_facts_tac [includes] 1);
by (fast_tac (claset() addIs [subset_imp_lepoll
RS (eqpoll_imp_lepoll
RSN (2, lepoll_trans))]) 1);
qed "LL_subset";
Goal "EX S. well_ord(LL,S)";
by (rtac (well_ord_subsets_lepoll_n RS exE) 1);
by (blast_tac (claset() addIs [LL_subset RSN (2, well_ord_subset)]) 2);
by Auto_tac;
qed "well_ord_LL";
(* ********************************************************************** *)
(* every element of LL is a contained in exactly one element of MM *)
(* ********************************************************************** *)
Goalw [MM_def, LL_def] "v:LL ==> EX! w. w:MM & v<=w";
by Safe_tac;
by (Fast_tac 1);
by (resolve_tac [lepoll_imp_eqpoll_subset RS exE] 1 THEN (assume_tac 1));
by (res_inst_tac [("x","x")] (all_ex RS ballE) 1);
by (fast_tac (claset() addSEs [eqpoll_sym]) 2);
by (Blast_tac 1);
qed "unique_superset_in_MM";
val unique_superset1 = unique_superset_in_MM RS theI RS conjunct1;
val unique_superset2 = unique_superset_in_MM RS the_equality2;
(* ********************************************************************** *)
(* The function GG satisfies the conditions of WO4 *)
(* ********************************************************************** *)
(* ********************************************************************** *)
(* The union of appropriate values is the whole x *)
(* ********************************************************************** *)
Goalw [LL_def] "w : MM ==> w Int y : LL";
by (Fast_tac 1);
qed "Int_in_LL";
Goalw [LL_def]
"v : LL ==> v = (THE x. x : MM & v <= x) Int y";
by (Clarify_tac 1);
by (stac unique_superset2 1);
by (auto_tac (claset(), simpset() addsimps [Int_in_LL]));
qed "in_LL_eq_Int";
Goal "v : LL ==> (THE x. x : MM & v <= x) <= x Un y";
by (dtac unique_superset1 1);
by (rewtac MM_def);
by (fast_tac (claset() addSDs [unique_superset1, includes RS subsetD]) 1);
qed "the_in_MM_subset";
Goalw [GG_def] "v : LL ==> GG ` v <= x";
by (ftac the_in_MM_subset 1);
by (ftac in_LL_eq_Int 1);
by (force_tac (claset() addEs [equalityE], simpset()) 1);
qed "GG_subset";
Goal "n:nat ==> EX z. z<=y & n eqpoll z";
by (etac nat_lepoll_imp_ex_eqpoll_n 1);
by (resolve_tac [ordertype_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll
RSN (2, lepoll_trans)] 1);
by (rtac WO_R 2);
by (fast_tac
(claset() delrules [notI]
addSIs [nat_le_infinite_Ord RS le_imp_lepoll]
addIs [Ord_ordertype,
ordertype_eqpoll RS eqpoll_imp_lepoll
RS lepoll_infinite]) 1);
qed "ex_subset_eqpoll_n";
Goal "u:x ==> EX v : s(u). succ(k) lepoll v Int y";
by (rtac ccontr 1);
by (subgoal_tac "ALL v:s(u). k eqpoll v Int y" 1);
by (full_simp_tac (simpset() addsimps [s_def]) 2);
by (blast_tac (claset() addIs [succ_not_lepoll_imp_eqpoll]) 2);
by (rewtac k_def);
by (cut_facts_tac [all_ex, includes, lnat] 1);
by (rtac (ex_subset_eqpoll_n RS exE) 1 THEN assume_tac 1);
by (rtac (noLepoll RS notE) 1);
by (blast_tac (claset() addIs
[[y_lepoll_subset_s, subset_s_lepoll_w] MRS lepoll_trans]) 1);
qed "exists_proper_in_s";
Goal "u:x ==> EX w:MM. u:w";
by (eresolve_tac [exists_proper_in_s RS bexE] 1);
by (rewrite_goals_tac [MM_def, s_def]);
by (Fast_tac 1);
qed "exists_in_MM";
Goal "u:x ==> EX w:LL. u:GG`w";
by (rtac (exists_in_MM RS bexE) 1);
by (assume_tac 1);
by (rtac bexI 1);
by (etac Int_in_LL 2);
by (rewtac GG_def);
by (asm_simp_tac (simpset() addsimps [Int_in_LL]) 1);
by (stac unique_superset2 1);
by (REPEAT (fast_tac (claset() addSEs [Int_in_LL]) 1));
qed "exists_in_LL";
Goal "well_ord(LL,S) ==> \
\ (UN b<ordertype(LL,S). GG ` (converse(ordermap(LL,S)) ` b)) = x";
by (rtac equalityI 1);
by (rtac subsetI 1);
by (etac OUN_E 1);
by (resolve_tac [GG_subset RS subsetD] 1);
by (assume_tac 2);
by (blast_tac (claset() addIs [ordermap_bij RS bij_converse_bij RS
bij_is_fun RS apply_type, ltD]) 1);
by (rtac subsetI 1);
by (eresolve_tac [exists_in_LL RS bexE] 1);
by (force_tac (claset() addIs [Ord_ordertype RSN (2, ltI),
ordermap_type RS apply_type],
simpset() addsimps [ordermap_bij RS bij_is_inj RS left_inverse]) 1);
qed "OUN_eq_x";
(* ********************************************************************** *)
(* Every element of the family is less than or equipollent to n-k (m) *)
(* ********************************************************************** *)
Goalw [MM_def] "w : MM ==> w eqpoll succ(k #+ m)";
by (fast_tac (claset() addDs [includes RS subsetD]) 1);
qed "in_MM_eqpoll_n";
Goalw [LL_def, MM_def] "w : LL ==> succ(k) lepoll w";
by (Fast_tac 1);
qed "in_LL_eqpoll_n";
val in_LL = in_LL_eq_Int RS equalityD1 RS (Int_lower1 RSN (2, subset_trans));
Goalw [GG_def]
"well_ord(LL,S) ==> \
\ ALL b<ordertype(LL,S). GG ` (converse(ordermap(LL,S)) ` b) lepoll m";
by (rtac oallI 1);
by (asm_simp_tac
(simpset() addsimps [ltD,
ordermap_bij RS bij_converse_bij RS
bij_is_fun RS apply_type]) 1);
by (cut_facts_tac [includes] 1);
by (rtac eqpoll_sum_imp_Diff_lepoll 1);
by (REPEAT
(fast_tac (claset() delrules [subsetI]
addSDs [ltD]
addSIs [eqpoll_sum_imp_Diff_lepoll, in_LL_eqpoll_n]
addIs [unique_superset1 RS in_MM_eqpoll_n, in_LL,
ordermap_bij RS bij_converse_bij RS
bij_is_fun RS apply_type]) 1 ));
qed "all_in_lepoll_m";
Goal "EX a f. Ord(a) & domain(f) = a & \
\ (UN b<a. f ` b) = x & (ALL b<a. f ` b lepoll m)";
by (resolve_tac [well_ord_LL RS exE] 1 THEN REPEAT (assume_tac 1));
by (rename_tac "S" 1);
by (res_inst_tac [("x","ordertype(LL,S)")] exI 1);
by (res_inst_tac [("x",
"lam b:ordertype(LL,S). GG ` (converse(ordermap(LL,S)) ` b)")]
exI 1);
by (Simp_tac 1);
by (REPEAT (ares_tac [conjI, lam_funtype RS domain_of_fun,
Ord_ordertype,
all_in_lepoll_m, OUN_eq_x] 1));
qed "conclusion";
Close_locale "AC16";
(* ********************************************************************** *)
(* The main theorem AC16(n, k) ==> WO4(n-k) *)
(* ********************************************************************** *)
Goalw [AC16_def,WO4_def]
"[| AC16(k #+ m, k); 0 < k; 0 < m; k:nat; m:nat |] ==> WO4(m)";
by (rtac allI 1);
by (case_tac "Finite(A)" 1);
by (rtac lemma1 1 THEN REPEAT (assume_tac 1));
by (cut_facts_tac [lemma2] 1);
by (REPEAT (eresolve_tac [exE, conjE] 1));
by (eres_inst_tac [("x","A Un y")] allE 1);
by (ftac infinite_Un 1 THEN (mp_tac 1));
by (etac zero_lt_natE 1); by (assume_tac 1);
by (Clarify_tac 1);
by (DEPTH_SOLVE (ares_tac [export conclusion] 1));
qed "AC16_WO4";