(* Title: ZF/AC/WO1-WO6.ML
ID: $Id$
Author: Krzysztof Gr`abczewski
Proofs needed to state that formulations WO1,...,WO6 are all equivalent.
All but one WO6 ==> WO1 (placed in separate file WO6_WO1.ML)
Every proofs (exept one) presented in this file are referred as clear
by Rubin & Rubin (page 2).
They refer reader to a book by G"odel to see the proof WO1 ==> WO2.
Fortunately order types made this proof also very easy.
*)
(* ********************************************************************** *)
goalw thy WO_defs "!!Z. WO2 ==> WO3";
by (fast_tac ZF_cs 1);
result();
(* ********************************************************************** *)
goalw thy (eqpoll_def::WO_defs) "!!Z. WO3 ==> WO1";
by (fast_tac (ZF_cs addSEs [bij_is_inj RS well_ord_rvimage,
well_ord_Memrel RS well_ord_subset]) 1);
result();
(* ********************************************************************** *)
goalw thy (eqpoll_def::WO_defs) "!!Z. WO1 ==> WO2";
by (fast_tac (ZF_cs addSIs [Ord_ordertype, ordermap_bij]) 1);
result();
(* ********************************************************************** *)
goal thy "!!f. f: A->B ==> (lam x:A. {f`x}): A -> {{b}. b:B}";
by (fast_tac (ZF_cs addSIs [lam_type, RepFunI, apply_type]) 1);
val lam_sets = result();
goalw thy [surj_def] "!!f. f:surj(A,B) ==> (UN a:A. {f`a}) = B";
by (fast_tac (ZF_cs addSIs [equalityI] addSEs [singletonE, apply_type]) 1);
val surj_imp_eq_ = result();
goal thy "!!f. [| f:surj(A,B); Ord(A) |] ==> (UN a<A. {f`a}) = B";
by (fast_tac (AC_cs addSDs [surj_imp_eq_]
addSIs [equalityI, ltI] addSEs [ltE]) 1);
val surj_imp_eq = result();
goalw thy WO_defs "!!Z. WO1 ==> WO4(1)";
by (resolve_tac [allI] 1);
by (eres_inst_tac [("x","A")] allE 1);
by (eresolve_tac [exE] 1);
by (REPEAT (resolve_tac [exI, conjI] 1));
by (eresolve_tac [Ord_ordertype] 1);
by (resolve_tac [conjI] 1);
by (eresolve_tac [ordermap_bij RS bij_converse_bij RS bij_is_fun RS
lam_sets RS domain_of_fun] 1);
by (asm_simp_tac (AC_ss addsimps [singleton_eqpoll_1 RS eqpoll_imp_lepoll,
Ord_ordertype RSN (2, ordermap_bij RS bij_converse_bij RS
bij_is_surj RS surj_imp_eq)]) 1);
result();
(* ********************************************************************** *)
goalw thy WO_defs "!!Z. [| m:nat; n:nat; m le n; WO4(m) |] ==> WO4(n)";
by (fast_tac (AC_cs addIs [nat_le_imp_lepoll RSN (2, lepoll_trans)]) 1);
result();
(* ********************************************************************** *)
val prems = goalw thy WO_defs "!!Z. [| m:nat; 1 le m; WO4(m) |] ==> WO5";
by (fast_tac ZF_cs 1);
result();
(* ********************************************************************** *)
val prems = goalw thy WO_defs "!!Z. WO5 ==> WO6";
by (fast_tac ZF_cs 1);
result();