Removal of needless "addIs [equality]", etc.
(* Title: ZF/AC/DC.ML
ID: $Id$
Author: Krzysztof Grabczewski
The proofs concerning the Axiom of Dependent Choice
*)
open DC;
(* ********************************************************************** *)
(* DC ==> DC(omega) *)
(* *)
(* The scheme of the proof: *)
(* *)
(* Assume DC. Let R and x satisfy the premise of DC(omega). *)
(* *)
(* Define XX and RR as follows: *)
(* *)
(* XX = (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}) *)
(* f RR g iff domain(g)=succ(domain(f)) & *)
(* restrict(g, domain(f)) = f *)
(* *)
(* Then RR satisfies the hypotheses of DC. *)
(* So applying DC: *)
(* *)
(* EX f:nat->XX. ALL n:nat. f`n RR f`succ(n) *)
(* *)
(* Thence *)
(* *)
(* ff = {<n, f`succ(n)`n>. n:nat} *)
(* *)
(* is the desired function. *)
(* *)
(* ********************************************************************** *)
goal thy "{<z1,z2>:XX*XX. domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1} <= XX*XX";
by (Fast_tac 1);
val lemma1_1 = result();
goal thy "!!X. ALL Y:Pow(X). Y lesspoll nat --> (EX x:X. <Y, x> : R) \
\ ==> {<z1,z2>: (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}) * \
\ (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}). \
\ domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1} ~= 0";
by (etac ballE 1);
by (eresolve_tac [empty_subsetI RS PowI RSN (2, notE)] 2);
by (eresolve_tac [nat_0I RS n_lesspoll_nat RSN (2, impE)] 1);
by (etac bexE 1);
by (res_inst_tac [("a","<0, {<0, x>}>")] not_emptyI 1);
by (rtac CollectI 1);
by (rtac SigmaI 1);
by (fast_tac (!claset addSIs [nat_0I RS UN_I, empty_fun]) 1);
by (rtac (nat_1I RS UN_I) 1);
by (fast_tac (!claset addSIs [singleton_fun RS Pi_type]
addss (!simpset addsimps [singleton_0 RS sym])) 1);
by (asm_full_simp_tac (!simpset addsimps [domain_0, domain_cons,
singleton_0]) 1);
val lemma1_2 = result();
goal thy "!!X. ALL Y:Pow(X). Y lesspoll nat --> (EX x:X. <Y, x> : R) \
\ ==> range({<z1,z2>: (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}) * \
\ (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}). \
\ domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1}) \
\ <= domain({<z1,z2>: (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}) * \
\ (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}). \
\ domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1})";
by (rtac range_subset_domain 1);
by (rtac subsetI 2);
by (etac CollectD1 2);
by (etac UN_E 1);
by (etac CollectE 1);
by (dresolve_tac [fun_is_rel RS image_subset RS PowI RSN (2, bspec)] 1
THEN (assume_tac 1));
by (eresolve_tac [[n_lesspoll_nat, nat_into_Ord RSN (2, image_Ord_lepoll)]
MRS lepoll_lesspoll_lesspoll RSN (2, impE)] 1
THEN REPEAT (assume_tac 1));
by (etac bexE 1);
by (res_inst_tac [("x","cons(<n,x>, g)")] exI 1);
by (rtac CollectI 1);
by (rtac SigmaI 1);
by (Fast_tac 1);
by (rtac UN_I 1);
by (etac nat_succI 1);
by (rtac CollectI 1);
by (etac cons_fun_type2 1 THEN (assume_tac 1));
by (fast_tac (!claset addSEs [succE] addss (!simpset
addsimps [cons_image_n, cons_val_n, cons_image_k, cons_val_k])) 1);
by (asm_full_simp_tac (!simpset
addsimps [domain_cons, domain_of_fun, succ_def, restrict_cons_eq]) 1);
val lemma1_3 = result();
goal thy "!!X. [| XX = (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}); \
\ RR = {<z1,z2>:XX*XX. domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1}; \
\ ALL Y:Pow(X). Y lesspoll nat --> (EX x:X. <Y, x> : R) \
\ |] ==> RR <= XX*XX & RR ~= 0 & range(RR) <= domain(RR)";
by (fast_tac (!claset addSIs [lemma1_1] addSEs [lemma1_2, lemma1_3]) 1);
val lemma1 = result();
goal thy
"!!X.[| XX = (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}); \
\ ALL n:nat. <f`n, f`succ(n)> : \
\ {<z1,z2>:XX*XX. domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1}; \
\ f: nat -> XX; n:nat \
\ |] ==> EX k:nat. f`succ(n) : k -> X & n:k \
\ & <f`succ(n)``n, f`succ(n)`n> : R";
by (etac nat_induct 1);
by (dresolve_tac [nat_1I RSN (2, apply_type)] 1);
by (dresolve_tac [nat_0I RSN (2, bspec)] 1);
by (Asm_full_simp_tac 1);
by (Step_tac 1);
by (rtac bexI 1 THEN (assume_tac 2));
by (best_tac (!claset addIs [ltD]
addSEs [nat_0_le RS leE]
addEs [sym RS trans RS succ_neq_0, domain_of_fun]
addss (!simpset)) 1);
(** LEVEL 7 **)
by (dresolve_tac [nat_succI RSN (2, bspec)] 1 THEN (assume_tac 1));
by (subgoal_tac "f ` succ(succ(x)) : succ(k)->X" 1);
by (dresolve_tac [nat_succI RS nat_succI RSN (2, apply_type)] 1
THEN (assume_tac 1));
by (Full_simp_tac 1);
by (Step_tac 1);
by (forw_inst_tac [("a","succ(k)")] (domain_of_fun RS sym RS trans) 1 THEN
(assume_tac 1));
by (forw_inst_tac [("a","xa")] (domain_of_fun RS sym RS trans) 1 THEN
(assume_tac 1));
by (fast_tac (!claset addSEs [nat_succI, nat_into_Ord RS succ_in_succ]
addSDs [nat_into_Ord RS succ_in_succ RSN (2, bspec)]) 1);
by (dtac domain_of_fun 1);
by (Full_simp_tac 1);
by (deepen_tac (!claset addDs [domain_of_fun RS sym RS trans]) 0 1);
val lemma2 = result();
goal thy
"!!X.[| XX = (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}); \
\ ALL n:nat. <f`n, f`succ(n)> : \
\ {<z1,z2>:XX*XX. domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1}; \
\ f: nat -> XX; n:nat \
\ |] ==> ALL x:n. f`succ(n)`x = f`succ(x)`x";
by (etac nat_induct 1);
by (Fast_tac 1);
by (rtac ballI 1);
by (etac succE 1);
by (rtac restrict_eq_imp_val_eq 1);
by (dresolve_tac [nat_succI RSN (2, bspec)] 1 THEN (assume_tac 1));
by (Asm_full_simp_tac 1);
by (dtac lemma2 1 THEN REPEAT (assume_tac 1));
by (fast_tac (!claset addSDs [domain_of_fun]) 1);
by (dres_inst_tac [("x","xa")] bspec 1 THEN (assume_tac 1));
by (eresolve_tac [sym RS trans RS sym] 1);
by (resolve_tac [restrict_eq_imp_val_eq RS sym] 1);
by (dresolve_tac [nat_succI RSN (2, bspec)] 1 THEN (assume_tac 1));
by (Asm_full_simp_tac 1);
by (dtac lemma2 1 THEN REPEAT (assume_tac 1));
by (fast_tac (FOL_cs addSDs [domain_of_fun]
addSEs [bexE, nat_into_Ord RSN (2, OrdmemD) RS subsetD]) 1);
val lemma3_1 = result();
goal thy "!!n. ALL x:n. f`succ(n)`x = f`succ(x)`x \
\ ==> {f`succ(x)`x. x:n} = {f`succ(n)`x. x:n}";
by (Asm_full_simp_tac 1);
val lemma3_2 = result();
goal thy "!!X. [| XX = (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R}); \
\ ALL n:nat. <f`n, f`succ(n)> : \
\ {<z1,z2>:XX*XX. domain(z2)=succ(domain(z1)) \
\ & restrict(z2, domain(z1)) = z1}; \
\ f: nat -> XX; n:nat \
\ |] ==> (lam x:nat. f`succ(x)`x) `` n = f`succ(n)``n";
by (etac natE 1);
by (asm_full_simp_tac (!simpset addsimps [image_0]) 1);
by (resolve_tac [image_lam RS ssubst] 1);
by (fast_tac (!claset addSEs [[nat_succI, Ord_nat] MRS OrdmemD]) 1);
by (resolve_tac [lemma3_1 RS lemma3_2 RS ssubst] 1
THEN REPEAT (assume_tac 1));
by (fast_tac (!claset addSEs [nat_succI]) 1);
by (dresolve_tac [nat_succI RSN (4, lemma2)] 1
THEN REPEAT (assume_tac 1));
by (fast_tac (!claset addSEs [nat_into_Ord RSN (2, OrdmemD) RSN
(2, image_fun RS sym)]) 1);
val lemma3 = result();
goal thy "!!f. [| f:A->B; B<=C |] ==> f:A->C";
by (rtac Pi_type 1 THEN (assume_tac 1));
by (fast_tac (!claset addSEs [apply_type]) 1);
val fun_type_gen = result();
goalw thy [DC_def, DC0_def] "!!Z. DC0 ==> DC(nat)";
by (REPEAT (resolve_tac [allI, impI] 1));
by (REPEAT (eresolve_tac [conjE, allE] 1));
by (eresolve_tac [[refl, refl] MRS lemma1 RSN (2, impE)] 1
THEN (assume_tac 1));
by (etac bexE 1);
by (res_inst_tac [("x","lam n:nat. f`succ(n)`n")] bexI 1);
by (fast_tac (!claset addSIs [lam_type] addSDs [refl RS lemma2]
addSEs [fun_type_gen, apply_type]) 2);
by (rtac oallI 1);
by (forward_tac [ltD RSN (3, refl RS lemma2)] 1
THEN assume_tac 2);
by (fast_tac (!claset addSEs [fun_type_gen]) 1);
by (eresolve_tac [ltD RSN (3, refl RS lemma3) RS ssubst] 1
THEN assume_tac 2);
by (fast_tac (!claset addSEs [fun_type_gen]) 1);
by (fast_tac (!claset addss (!simpset)) 1);
qed "DC0_DC_nat";
(* ********************************************************************** *)
(* DC(omega) ==> DC *)
(* *)
(* The scheme of the proof: *)
(* *)
(* Assume DC(omega). Let R and x satisfy the premise of DC. *)
(* *)
(* Define XX and RR as follows: *)
(* *)
(* XX = (UN n:nat. *)
(* {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}) *)
(* RR = {<z1,z2>:Fin(XX)*XX. (domain(z2)=succ(UN f:z1. domain(f)) *)
(* & (ALL f:z1. restrict(z2, domain(f)) = f)) | *)
(* (~ (EX g:XX. domain(g)=succ(UN f:z1. domain(f)) *)
(* & (ALL f:z1. restrict(g, domain(f)) = f)) & *)
(* z2={<0,x>})} *)
(* *)
(* Then XX and RR satisfy the hypotheses of DC(omega). *)
(* So applying DC: *)
(* *)
(* EX f:nat->XX. ALL n:nat. f``n RR f`n *)
(* *)
(* Thence *)
(* *)
(* ff = {<n, f`n`n>. n:nat} *)
(* *)
(* is the desired function. *)
(* *)
(* ********************************************************************** *)
goalw thy [lesspoll_def, Finite_def]
"!!A. A lesspoll nat ==> Finite(A)";
by (fast_tac (!claset addSDs [ltD, lepoll_imp_ex_le_eqpoll]
addSIs [Ord_nat]) 1);
val lesspoll_nat_is_Finite = result();
goal thy "!!n. n:nat ==> ALL A. (A eqpoll n & A <= X) --> A : Fin(X)";
by (etac nat_induct 1);
by (rtac allI 1);
by (fast_tac (!claset addSIs [Fin.emptyI]
addSDs [eqpoll_imp_lepoll RS lepoll_0_is_0]) 1);
by (rtac allI 1);
by (rtac impI 1);
by (etac conjE 1);
by (resolve_tac [eqpoll_succ_imp_not_empty RS not_emptyE] 1
THEN (assume_tac 1));
by (forward_tac [Diff_sing_eqpoll] 1 THEN (assume_tac 1));
by (etac allE 1);
by (etac impE 1);
by (Fast_tac 1);
by (dtac subsetD 1 THEN (assume_tac 1));
by (dresolve_tac [Fin.consI] 1 THEN (assume_tac 1));
by (asm_full_simp_tac (!simpset addsimps [cons_Diff]) 1);
val Finite_Fin_lemma = result();
goalw thy [Finite_def] "!!A. [| Finite(A); A <= X |] ==> A : Fin(X)";
by (etac bexE 1);
by (dtac Finite_Fin_lemma 1);
by (etac allE 1);
by (etac impE 1);
by (assume_tac 2);
by (Fast_tac 1);
val Finite_Fin = result();
goal thy "!!x. x: X \
\ ==> {<0,x>}: (UN n:nat. {f:succ(n)->X. ALL k:n. <f`k, f`succ(k)> : R})";
by (rtac (nat_0I RS UN_I) 1);
by (fast_tac (!claset addSIs [singleton_fun RS Pi_type]
addss (!simpset addsimps [singleton_0 RS sym])) 1);
val singleton_in_funs = result();
goal thy
"!!X. [| XX = (UN n:nat. \
\ {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}); \
\ RR = {<z1,z2>:Fin(XX)*XX. (domain(z2)=succ(UN f:z1. domain(f)) \
\ & (ALL f:z1. restrict(z2, domain(f)) = f)) | \
\ (~ (EX g:XX. domain(g)=succ(UN f:z1. domain(f)) \
\ & (ALL f:z1. restrict(g, domain(f)) = f)) & z2={<0,x>})}; \
\ range(R) <= domain(R); x:domain(R) \
\ |] ==> RR <= Pow(XX)*XX & \
\ (ALL Y:Pow(XX). Y lesspoll nat --> (EX x:XX. <Y,x>:RR))";
by (rtac conjI 1);
by (deepen_tac (!claset addSEs [FinD RS PowI]) 0 1);
by (rtac (impI RS ballI) 1);
by (dresolve_tac [[lesspoll_nat_is_Finite, PowD] MRS Finite_Fin] 1
THEN (assume_tac 1));
by (excluded_middle_tac "EX g:XX. domain(g)=succ(UN f:Y. domain(f)) \
\ & (ALL f:Y. restrict(g, domain(f)) = f)" 1);
by (etac subst 2 THEN (*elimination equation for greater speed*)
fast_tac (!claset addss (!simpset)) 2);
by (step_tac (!claset delrules [domainE]) 1);
by (swap_res_tac [bexI] 1 THEN etac singleton_in_funs 2);
by (asm_full_simp_tac (!simpset addsimps [nat_0I RSN (2, bexI),
cons_fun_type2, empty_fun]) 1);
val lemma1 = result();
goal thy "!!f. [| f:nat->X; n:nat |] ==> \
\ (UN x:f``succ(n). P(x)) = P(f`n) Un (UN x:f``n. P(x))";
by (asm_full_simp_tac (!simpset
addsimps [Ord_nat RSN (2, OrdmemD) RSN (2, image_fun),
[nat_succI, Ord_nat] MRS OrdmemD RSN (2, image_fun)]) 1);
val UN_image_succ_eq = result();
goal thy "!!f. [| (UN x:f``n. P(x)) = y; P(f`n) = succ(y); \
\ f:nat -> X; n:nat |] ==> (UN x:f``succ(n). P(x)) = succ(y)";
by (asm_full_simp_tac (!simpset addsimps [UN_image_succ_eq]) 1);
by (Fast_tac 1);
val UN_image_succ_eq_succ = result();
goal thy "!!f. [| f:succ(n) -> D; n:nat; \
\ domain(f)=succ(x); x=y |] ==> f`y : D";
by (fast_tac (!claset addEs [apply_type]
addSDs [[sym, domain_of_fun] MRS trans]) 1);
val apply_domain_type = result();
goal thy "!!f. [| f : nat -> X; n:nat |] ==> f``succ(n) = cons(f`n, f``n)";
by (asm_full_simp_tac (!simpset
addsimps [nat_succI, Ord_nat RSN (2, OrdmemD), image_fun]) 1);
val image_fun_succ = result();
goal thy "!!f. [| domain(f`n) = succ(u); f : nat -> (UN n:nat. \
\ {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}); \
\ u=k; n:nat \
\ |] ==> f`n : succ(k) -> domain(R)";
by (dtac apply_type 1 THEN (assume_tac 1));
by (fast_tac (!claset addEs [UN_E, domain_eq_imp_fun_type]) 1);
val f_n_type = result();
goal thy "!!f. [| f : nat -> (UN n:nat. \
\ {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}); \
\ domain(f`n) = succ(k); n:nat \
\ |] ==> ALL i:k. <f`n`i, f`n`succ(i)> : R";
by (dtac apply_type 1 THEN (assume_tac 1));
by (etac UN_E 1);
by (etac CollectE 1);
by (dresolve_tac [domain_of_fun RS sym RS trans] 1 THEN (assume_tac 1));
by (Asm_full_simp_tac 1);
val f_n_pairs_in_R = result();
goalw thy [restrict_def]
"!!f. [| restrict(f, domain(x))=x; f:n->X; domain(x) <= n \
\ |] ==> restrict(cons(<n, y>, f), domain(x)) = x";
by (eresolve_tac [sym RS trans RS sym] 1);
by (rtac fun_extension 1);
by (fast_tac (!claset addSIs [lam_type]) 1);
by (fast_tac (!claset addSIs [lam_type]) 1);
by (asm_full_simp_tac (!simpset addsimps [subsetD RS cons_val_k]) 1);
val restrict_cons_eq_restrict = result();
goal thy "!!f. [| ALL x:f``n. restrict(f`n, domain(x))=x; \
\ f : nat -> (UN n:nat. \
\ {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}); \
\ n:nat; domain(f`n) = succ(n); \
\ (UN x:f``n. domain(x)) <= n |] \
\ ==> ALL x:f``succ(n). restrict(cons(<succ(n),y>, f`n), domain(x))=x";
by (rtac ballI 1);
by (asm_full_simp_tac (!simpset addsimps [image_fun_succ]) 1);
by (dtac f_n_type 1 THEN REPEAT (ares_tac [refl] 1));
by (etac disjE 1);
by (asm_full_simp_tac (!simpset addsimps [domain_of_fun, restrict_cons_eq]) 1);
by (dtac bspec 1 THEN (assume_tac 1));
by (fast_tac (!claset addSEs [restrict_cons_eq_restrict]) 1);
val all_in_image_restrict_eq = result();
goal thy
"!!X.[| ALL b<nat. <f``b, f`b> : \
\ {<z1,z2>:Fin(XX)*XX. (domain(z2)=succ(UN f:z1. domain(f)) \
\ & (ALL f:z1. restrict(z2, domain(f)) = f)) | \
\ (~ (EX g:XX. domain(g)=succ(UN f:z1. domain(f)) \
\ & (ALL f:z1. restrict(g, domain(f)) = f)) & z2={<0,x>})}; \
\ XX = (UN n:nat. \
\ {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}); \
\ f: nat -> XX; range(R) <= domain(R); x:domain(R) \
\ |] ==> ALL b<nat. <f``b, f`b> : \
\ {<z1,z2>:Fin(XX)*XX. (domain(z2)=succ(UN f:z1. domain(f)) \
\ & (UN f:z1. domain(f)) = b \
\ & (ALL f:z1. restrict(z2, domain(f)) = f))}";
by (rtac oallI 1);
by (dtac ltD 1);
by (etac nat_induct 1);
by (dresolve_tac [[nat_0I, Ord_nat] MRS ltI RSN (2, ospec)] 1);
by (fast_tac (FOL_cs addss
(!simpset addsimps [image_0, singleton_fun RS domain_of_fun,
singleton_0, singleton_in_funs])) 1);
(*induction step*) (** LEVEL 5 **)
by (Full_simp_tac 1);
by (dresolve_tac [[nat_succI, Ord_nat] MRS ltI RSN (2, ospec)] 1
THEN (assume_tac 1));
by (REPEAT (eresolve_tac [conjE, disjE] 1));
by (fast_tac (FOL_cs addSEs [trans, subst_context]
addSIs [UN_image_succ_eq_succ] addss (!simpset)) 1);
by (etac conjE 1);
by (etac notE 1);
by (asm_full_simp_tac (!simpset addsimps [UN_image_succ_eq_succ]) 1);
(** LEVEL 12 **)
by (REPEAT (eresolve_tac [conjE, bexE] 1));
by (dtac apply_domain_type 1 THEN REPEAT (assume_tac 1));
by (etac domainE 1);
by (etac domainE 1);
by (forward_tac [f_n_type] 1 THEN REPEAT (assume_tac 1));
by (rtac bexI 1);
by (etac nat_succI 2);
by (res_inst_tac [("x","cons(<succ(xa), ya>, f`xa)")] bexI 1);
by (rtac conjI 1);
by (fast_tac (FOL_cs
addEs [subst_context RSN (2, trans) RS domain_cons_eq_succ,
subst_context, all_in_image_restrict_eq, trans, equalityD1]) 2);
by (eresolve_tac [rangeI RSN (2, subsetD) RSN (2, cons_fun_type2)] 2
THEN REPEAT (assume_tac 2));
by (rtac ballI 1);
by (etac succE 1);
(** LEVEL 25 **)
by (dresolve_tac [domain_of_fun RSN (2, f_n_pairs_in_R)] 2
THEN REPEAT (assume_tac 2));
by (dtac bspec 2 THEN (assume_tac 2));
by (asm_full_simp_tac (!simpset
addsimps [nat_into_Ord RS succ_in_succ, succI2, cons_val_k]) 2);
by (asm_full_simp_tac (!simpset addsimps [cons_val_n, cons_val_k]) 1);
val simplify_recursion = result();
goal thy "!!X. [| XX = (UN n:nat. \
\ {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}); \
\ ALL b<nat. <f``b, f`b> : \
\ {<z1,z2>:Fin(XX)*XX. (domain(z2)=succ(UN f:z1. domain(f)) \
\ & (UN f:z1. domain(f)) = b \
\ & (ALL f:z1. restrict(z2, domain(f)) = f))}; \
\ f: nat -> XX; range(R) <= domain(R); x:domain(R); n:nat \
\ |] ==> f`n : succ(n) -> domain(R) \
\ & (ALL i:n. <f`n`i, f`n`succ(i)>:R)";
by (dtac ospec 1);
by (eresolve_tac [Ord_nat RSN (2, ltI)] 1);
by (etac CollectE 1);
by (Asm_full_simp_tac 1);
by (rtac conjI 1);
by (fast_tac (!claset
addSEs [trans RS domain_eq_imp_fun_type, subst_context]) 1);
by (fast_tac (FOL_cs addSEs [conjE, f_n_pairs_in_R, trans, subst_context]) 1);
val lemma2 = result();
goal thy "!!n. [| XX = (UN n:nat. \
\ {f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R}); \
\ ALL b<nat. <f``b, f`b> : \
\ {<z1,z2>:Fin(XX)*XX. (domain(z2)=succ(UN f:z1. domain(f)) \
\ & (UN f:z1. domain(f)) = b \
\ & (ALL f:z1. restrict(z2, domain(f)) = f))}; \
\ f : nat -> XX; n:nat; range(R) <= domain(R); x:domain(R) \
\ |] ==> f`n`n = f`succ(n)`n";
by (forward_tac [lemma2 RS conjunct1 RS domain_of_fun] 1
THEN REPEAT (assume_tac 1));
by (dresolve_tac [[nat_succI, Ord_nat] MRS ltI RSN (2, ospec)] 1
THEN (assume_tac 1));
by (Asm_full_simp_tac 1);
by (REPEAT (etac conjE 1));
by (etac ballE 1);
by (eresolve_tac [restrict_eq_imp_val_eq RS sym] 1);
by (fast_tac (!claset addSEs [ssubst]) 1);
by (asm_full_simp_tac (!simpset
addsimps [[nat_succI, Ord_nat] MRS OrdmemD RSN (2, image_fun)]) 1);
val lemma3 = result();
goalw thy [DC_def, DC0_def] "!!Z. DC(nat) ==> DC0";
by (REPEAT (resolve_tac [allI, impI] 1));
by (REPEAT (eresolve_tac [asm_rl, conjE, ex_in_domain RS exE, allE] 1));
by (eresolve_tac [[refl, refl] MRS lemma1 RSN (2, impE)] 1
THEN REPEAT (assume_tac 1));
by (etac bexE 1);
by (dresolve_tac [refl RSN (2, simplify_recursion)] 1
THEN REPEAT (assume_tac 1));
by (res_inst_tac [("x","lam n:nat. f`n`n")] bexI 1);
by (rtac lam_type 2);
by (eresolve_tac [[refl RS lemma2 RS conjunct1, succI1] MRS apply_type] 2
THEN REPEAT (assume_tac 2));
by (rtac ballI 1);
by (forward_tac [refl RS (nat_succI RSN (6, lemma2)) RS conjunct2] 1
THEN REPEAT (assume_tac 1));
by (dresolve_tac [refl RS lemma3] 1 THEN REPEAT (assume_tac 1));
by (asm_full_simp_tac (!simpset addsimps [nat_succI]) 1);
qed "DC_nat_DC0";
(* ********************************************************************** *)
(* ALL K. Card(K) --> DC(K) ==> WO3 *)
(* ********************************************************************** *)
goalw thy [lesspoll_def]
"!!A. [| ~ A lesspoll B; C lesspoll B |] ==> A - C ~= 0";
by (fast_tac (!claset addSDs [Diff_eq_0_iff RS iffD1 RS subset_imp_lepoll]
addSIs [eqpollI] addEs [notE] addSEs [eqpollE, lepoll_trans]) 1);
val lemma1 = result();
val [f_type, Ord_a, not_eq] = goalw thy [inj_def]
"[| f:a->X; Ord(a); (!!b c. [| b<c; c:a |] ==> f`b~=f`c) \
\ |] ==> f:inj(a, X)";
by (resolve_tac [f_type RS CollectI] 1);
by (REPEAT (resolve_tac [ballI,impI] 1));
by (resolve_tac [Ord_a RS Ord_in_Ord RS Ord_linear_lt] 1
THEN (assume_tac 1));
by (eres_inst_tac [("j","x")] (Ord_a RS Ord_in_Ord) 1);
by (REPEAT (fast_tac (!claset addDs [not_eq, not_eq RS not_sym]) 1));
val fun_Ord_inj = result();
goal thy "!!a. [| f:X->Y; A<=X; a:A |] ==> f`a : f``A";
by (fast_tac (!claset addSEs [image_fun RS ssubst]) 1);
val value_in_image = result();
goalw thy [DC_def, WO3_def]
"!!Z. ALL K. Card(K) --> DC(K) ==> WO3";
by (rtac allI 1);
by (excluded_middle_tac "A lesspoll Hartog(A)" 1);
by (fast_tac (!claset addSDs [lesspoll_imp_ex_lt_eqpoll]
addSIs [Ord_Hartog, leI RS le_imp_subset]) 2);
by (REPEAT (eresolve_tac [allE, impE] 1));
by (rtac Card_Hartog 1);
by (eres_inst_tac [("x","A")] allE 1);
by (eres_inst_tac [("x","{<z1,z2>:Pow(A)*A . z1 \
\ lesspoll Hartog(A) & z2 ~: z1}")] allE 1);
by (Asm_full_simp_tac 1);
by (etac impE 1);
by (fast_tac (!claset addEs [lemma1 RS not_emptyE]) 1);
by (etac bexE 1);
by (resolve_tac [exI RS (lepoll_def RS (def_imp_iff RS iffD2))
RS (HartogI RS notE)] 1);
by (resolve_tac [Ord_Hartog RSN (2, fun_Ord_inj)] 1 THEN (assume_tac 1));
by (dresolve_tac [Ord_Hartog RSN (2, OrdmemD) RSN (2,
ltD RSN (3, value_in_image))] 1
THEN REPEAT (assume_tac 1));
by (fast_tac (!claset addSDs [Ord_Hartog RSN (2, ltI) RSN (2, ospec)]
addEs [subst]) 1);
qed "DC_WO3";
(* ********************************************************************** *)
(* WO1 ==> ALL K. Card(K) --> DC(K) *)
(* ********************************************************************** *)
goal thy
"!!a. [| Ord(a); b:a |] ==> (lam x:a. P(x))``b = (lam x:b. P(x))``b";
by (rtac images_eq 1);
by (REPEAT (fast_tac (!claset addSEs [RepFunI, OrdmemD]
addSIs [lam_type]) 2));
by (rtac ballI 1);
by (dresolve_tac [OrdmemD RS subsetD] 1
THEN REPEAT (assume_tac 1));
by (Asm_full_simp_tac 1);
val lam_images_eq = result();
goalw thy [lesspoll_def] "!!K. [| Card(K); b:K |] ==> b lesspoll K";
by (asm_full_simp_tac (!simpset addsimps [Card_iff_initial]) 1);
by (fast_tac (!claset addSIs [le_imp_lepoll, ltI, leI]) 1);
val in_Card_imp_lesspoll = result();
goal thy "(lam b:a. P(b)) : a -> {P(b). b:a}";
by (fast_tac (!claset addSIs [lam_type, RepFunI]) 1);
val lam_type_RepFun = result();
goal thy "!!Z. [| ALL Y:Pow(X). Y lesspoll a --> (EX x:X. <Y, x> : R); \
\ b:a; Z:Pow(X); Z lesspoll a |] \
\ ==> {x:X. <Z,x> : R} ~= 0";
by (fast_tac (FOL_cs addEs [bexE, equals0D]
addSDs [bspec] addIs [CollectI]) 1);
val lemma_ = result();
goal thy "!!K. [| Card(K); well_ord(X,Q); \
\ ALL Y:Pow(X). Y lesspoll K --> (EX x:X. <Y, x> : R); b:K |] \
\ ==> ff(b, X, Q, R) : {x:X. <(lam c:b. ff(c, X, Q, R))``b, x> : R}";
by (res_inst_tac [("P","b:K")] impE 1 THEN TRYALL assume_tac);
by (res_inst_tac [("i","b")] (Card_is_Ord RS Ord_in_Ord RS trans_induct) 1
THEN REPEAT (assume_tac 1));
by (rtac impI 1);
by (resolve_tac [ff_def RS def_transrec RS ssubst] 1);
by (etac the_first_in 1);
by (Fast_tac 1);
by (asm_full_simp_tac (!simpset
addsimps [[lam_type_RepFun, subset_refl] MRS image_fun]) 1);
by (etac lemma_ 1 THEN (assume_tac 1));
by (fast_tac (!claset addSEs [RepFunE, impE, notE]
addEs [Card_is_Ord RSN (2, OrdmemD) RS subsetD]) 1);
by (fast_tac (!claset addSEs [[in_Card_imp_lesspoll, RepFun_lepoll]
MRS lepoll_lesspoll_lesspoll]) 1);
val lemma = result();
goalw thy [DC_def, WO1_def]
"!!Z. WO1 ==> ALL K. Card(K) --> DC(K)";
by (REPEAT (resolve_tac [allI,impI] 1));
by (REPEAT (eresolve_tac [allE,exE,conjE] 1));
by (res_inst_tac [("x","lam b:K. ff(b, X, Ra, R)")] bexI 1);
by (rtac lam_type 2);
by (resolve_tac [lemma RS CollectD1] 2 THEN REPEAT (assume_tac 2));
by (asm_full_simp_tac (!simpset
addsimps [[Card_is_Ord, ltD] MRS lam_images_eq]) 1);
by (fast_tac (!claset addSEs [ltE, lemma RS CollectD2]) 1);
qed" WO1_DC_Card";