TPTP Problem File: SCT104+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SCT104+1 : TPTP v8.2.0. Released v5.2.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem 432372, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : arrow_432372.500.p [Bla11]
% Status : Theorem
% Rating : 1.00 v7.0.0, 0.93 v6.4.0, 1.00 v6.2.0, 0.96 v6.1.0, 0.97 v6.0.0, 1.00 v5.2.0
% Syntax : Number of formulae : 523 ( 123 unt; 0 def)
% Number of atoms : 1277 ( 251 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 882 ( 128 ~; 28 |; 39 &)
% ( 101 <=>; 586 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-6 aty)
% Number of functors : 58 ( 58 usr; 13 con; 0-6 aty)
% Number of variables : 1791 (1777 !; 14 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-03-01 11:06:37
% : Renamed from SWW106+1
%------------------------------------------------------------------------------
%----Relevant facts (494)
fof(fact_ext,axiom,
! [V_g_2,V_f_2] :
( ! [B_x] : hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)
=> V_f_2 = V_g_2 ) ).
fof(fact_equalityCE,axiom,
! [V_c_2,T_a,V_B_2,V_A_2] :
( V_A_2 = V_B_2
=> ( ( c_member(T_a,V_c_2,V_A_2)
=> ~ c_member(T_a,V_c_2,V_B_2) )
=> ~ ( ~ c_member(T_a,V_c_2,V_A_2)
=> c_member(T_a,V_c_2,V_B_2) ) ) ) ).
fof(fact_converse__converse,axiom,
! [V_r_2,T_a,T_b] : c_Relation_Oconverse(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2)) = V_r_2 ).
fof(fact_mem__def,axiom,
! [V_A_2,V_x_2,T_a] :
( c_member(T_a,V_x_2,V_A_2)
<=> hBOOL(hAPP(V_A_2,V_x_2)) ) ).
fof(fact_eq__mem,axiom,
! [V_y_2,V_x_2,T_a] :
( c_member(T_a,V_x_2,c_fequal(V_y_2))
<=> V_x_2 = V_y_2 ) ).
fof(fact_eqset__imp__iff,axiom,
! [V_x_2,T_a,V_B_2,V_A_2] :
( V_A_2 = V_B_2
=> ( c_member(T_a,V_x_2,V_A_2)
<=> c_member(T_a,V_x_2,V_B_2) ) ) ).
fof(fact_eqelem__imp__iff,axiom,
! [V_A_2,T_a,V_y_2,V_x_2] :
( V_x_2 = V_y_2
=> ( c_member(T_a,V_x_2,V_A_2)
<=> c_member(T_a,V_y_2,V_A_2) ) ) ).
fof(fact_eq__mem__trans,axiom,
! [V_A_2,T_a,V_b_2,V_a_2] :
( V_a_2 = V_b_2
=> ( c_member(T_a,V_b_2,V_A_2)
=> c_member(T_a,V_a_2,V_A_2) ) ) ).
fof(fact_notin__Lin__iff,axiom,
! [V_y_2,V_x_2,V_La_2] :
( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin)
=> ( V_x_2 != V_y_2
=> ( ~ c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_x_2,V_y_2),V_La_2)
<=> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_y_2,V_x_2),V_La_2) ) ) ) ).
fof(fact_total__on__converse,axiom,
! [V_r_2,V_A_2,T_a] :
( c_Relation_Ototal__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
<=> c_Relation_Ototal__on(T_a,V_A_2,V_r_2) ) ).
fof(fact_converse__inv__image,axiom,
! [V_f_2,V_R_2,T_b,T_a] : c_Relation_Oconverse(T_a,T_a,hAPP(c_Relation_Oinv__image(T_b,T_a,V_R_2),V_f_2)) = hAPP(c_Relation_Oinv__image(T_b,T_a,c_Relation_Oconverse(T_b,T_b,V_R_2)),V_f_2) ).
fof(fact_antisym__converse,axiom,
! [V_r_2,T_a] :
( c_Relation_Oantisym(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
<=> c_Relation_Oantisym(T_a,V_r_2) ) ).
fof(fact_converse__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_Oconverse(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = c_Relation_OId__on(T_a,V_A_2) ).
fof(fact_ord_OgreaterThanAtMost__iff,axiom,
! [V_u_2,V_l_2,V_less_2,V_less__eq_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OgreaterThanAtMost(T_a,V_less__eq_2,V_less_2,V_l_2,V_u_2))
<=> ( hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2))
& hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) ) ) ).
fof(fact_ord_OatLeastLessThan__iff,axiom,
! [V_u_2,V_l_2,V_less_2,V_less__eq_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OatLeastLessThan(T_a,V_less__eq_2,V_less_2,V_l_2,V_u_2))
<=> ( hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2))
& hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) ) ) ).
fof(fact_ord_OgreaterThanLessThan__iff,axiom,
! [V_u_2,V_l_2,V_less_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OgreaterThanLessThan(T_a,V_less_2,V_l_2,V_u_2))
<=> ( hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2))
& hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) ) ) ).
fof(fact_ord_OatLeastAtMost__iff,axiom,
! [V_u_2,V_l_2,V_less__eq_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OatLeastAtMost(T_a,V_less__eq_2,V_l_2,V_u_2))
<=> ( hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2))
& hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) ) ) ).
fof(fact_ord_OgreaterThan__iff,axiom,
! [V_k_2,V_less_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OgreaterThan(T_a,V_less_2,V_k_2))
<=> hBOOL(hAPP(hAPP(V_less_2,V_k_2),V_i_2)) ) ).
fof(fact_ord_OlessThan__iff,axiom,
! [V_k_2,V_less_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OlessThan(T_a,V_less_2,V_k_2))
<=> hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_k_2)) ) ).
fof(fact_antisym__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_Oantisym(T_a,c_Relation_OId__on(T_a,V_A_2)) ).
fof(fact_in__inv__image,axiom,
! [V_f_2,V_r_2,T_b,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),hAPP(c_Relation_Oinv__image(T_b,T_a,V_r_2),V_f_2))
<=> c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,hAPP(V_f_2,V_x_2),hAPP(V_f_2,V_y_2)),V_r_2) ) ).
fof(fact_antisym__def,axiom,
! [V_r_2,T_a] :
( c_Relation_Oantisym(T_a,V_r_2)
<=> ! [B_x,B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2)
=> B_x = B_y ) ) ) ).
fof(fact_antisymD,axiom,
! [V_b_2,V_a_2,V_r_2,T_a] :
( c_Relation_Oantisym(T_a,V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),V_r_2)
=> V_a_2 = V_b_2 ) ) ) ).
fof(fact_total__on__def,axiom,
! [V_r_2,V_A_2,T_a] :
( c_Relation_Ototal__on(T_a,V_A_2,V_r_2)
<=> ! [B_x] :
( c_member(T_a,B_x,V_A_2)
=> ! [B_xa] :
( c_member(T_a,B_xa,V_A_2)
=> ( B_x != B_xa
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_xa),V_r_2)
| c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_xa,B_x),V_r_2) ) ) ) ) ) ).
fof(fact_Id__on__eqI,axiom,
! [V_A_2,T_a,V_b_2,V_a_2] :
( V_a_2 = V_b_2
=> ( c_member(T_a,V_a_2,V_A_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Relation_OId__on(T_a,V_A_2)) ) ) ).
fof(fact_Id__on__iff,axiom,
! [V_A_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Relation_OId__on(T_a,V_A_2))
<=> ( V_x_2 = V_y_2
& c_member(T_a,V_x_2,V_A_2) ) ) ).
fof(fact_converseD,axiom,
! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Relation_Oconverse(T_b,T_a,V_r_2))
=> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),V_r_2) ) ).
fof(fact_converseI,axiom,
! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
=> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),c_Relation_Oconverse(T_a,T_b,V_r_2)) ) ).
fof(fact_converse__iff,axiom,
! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Relation_Oconverse(T_b,T_a,V_r_2))
<=> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),V_r_2) ) ).
fof(fact_ord_OatMost__iff,axiom,
! [V_k_2,V_less__eq_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OatMost(T_a,V_less__eq_2,V_k_2))
<=> hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_k_2)) ) ).
fof(fact_ord_OatLeast__iff,axiom,
! [V_k_2,V_less__eq_2,V_i_2,T_a] :
( c_member(T_a,V_i_2,c_SetInterval_Oord_OatLeast(T_a,V_less__eq_2,V_k_2))
<=> hBOOL(hAPP(hAPP(V_less__eq_2,V_k_2),V_i_2)) ) ).
fof(fact_irrefl__def,axiom,
! [V_r_2,T_a] :
( c_Relation_Oirrefl(T_a,V_r_2)
<=> ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),V_r_2) ) ).
fof(fact_Id__onE,axiom,
! [V_A_2,V_c_2,T_a] :
( c_member(tc_prod(T_a,T_a),V_c_2,c_Relation_OId__on(T_a,V_A_2))
=> ~ ! [B_x] :
( c_member(T_a,B_x,V_A_2)
=> V_c_2 != c_Product__Type_OPair(T_a,T_a,B_x,B_x) ) ) ).
fof(fact_in__rel__def,axiom,
! [V_y_2,V_x_2,V_R_2,T_b,T_a] :
( hBOOL(hAPP(hAPP(c_FunDef_Oin__rel(T_a,T_b,V_R_2),V_x_2),V_y_2))
<=> c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_x_2,V_y_2),V_R_2) ) ).
fof(fact_converseE,axiom,
! [V_r_2,V_yx_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),V_yx_2,c_Relation_Oconverse(T_b,T_a,V_r_2))
=> ~ ! [B_x,B_y] :
( V_yx_2 = c_Product__Type_OPair(T_a,T_b,B_y,B_x)
=> ~ c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,B_y),V_r_2) ) ) ).
fof(fact_same__fstI,axiom,
! [T_a,V_R_2,V_y_2,V_y_H_2,T_b,V_x_2,V_P_2] :
( hBOOL(hAPP(V_P_2,V_x_2))
=> ( c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,V_y_H_2,V_y_2),hAPP(V_R_2,V_x_2))
=> c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_x_2,V_y_H_2),c_Product__Type_OPair(T_a,T_b,V_x_2,V_y_2)),c_Recdef_Osame__fst(T_a,T_b,V_P_2,V_R_2)) ) ) ).
fof(fact_RangeI,axiom,
! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
=> c_member(T_b,V_b_2,c_Relation_ORange(T_a,T_b,V_r_2)) ) ).
fof(fact_DomainI,axiom,
! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
=> c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)) ) ).
fof(fact_Nitpick_Orefl_H__def,axiom,
! [V_r_2,T_a] :
( c_Nitpick_Orefl_H(T_a,V_r_2)
<=> ! [B_x] : c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),V_r_2) ) ).
fof(fact_antisymI,axiom,
! [V_r_2,T_a] :
( ! [B_x,B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2)
=> B_x = B_y ) )
=> c_Relation_Oantisym(T_a,V_r_2) ) ).
fof(fact_Range__converse,axiom,
! [V_r_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2)) = c_Relation_ODomain(T_a,T_b,V_r_2) ).
fof(fact_Domain__converse,axiom,
! [V_r_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) = c_Relation_ORange(T_b,T_a,V_r_2) ).
fof(fact_Range__def,axiom,
! [V_r_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,V_r_2) = c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) ).
fof(fact_Domain__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_ODomain(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2 ).
fof(fact_Range__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_ORange(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2 ).
fof(fact_Domain__iff,axiom,
! [V_r_2,T_b,V_a_2,T_a] :
( c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2))
<=> ? [B_y] : c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,B_y),V_r_2) ) ).
fof(fact_Range__iff,axiom,
! [V_r_2,T_b,V_a_2,T_a] :
( c_member(T_a,V_a_2,c_Relation_ORange(T_b,T_a,V_r_2))
<=> ? [B_y] : c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_y,V_a_2),V_r_2) ) ).
fof(fact_in__lex__prod,axiom,
! [V_s_2,V_r_2,V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OPair(T_a,T_b,V_a_H_2,V_b_H_2)),c_Wellfounded_Olex__prod(T_a,T_b,V_r_2,V_s_2))
<=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_H_2),V_r_2)
| ( V_a_2 = V_a_H_2
& c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,V_b_2,V_b_H_2),V_s_2) ) ) ) ).
fof(fact_RangeE,axiom,
! [V_r_2,T_b,V_b_2,T_a] :
( c_member(T_a,V_b_2,c_Relation_ORange(T_b,T_a,V_r_2))
=> ~ ! [B_x] : ~ c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2) ) ).
fof(fact_DomainE,axiom,
! [V_r_2,T_b,V_a_2,T_a] :
( c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2))
=> ~ ! [B_y] : ~ c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,B_y),V_r_2) ) ).
fof(fact_refl__onD,axiom,
! [V_a_2,V_r_2,V_A_2,T_a] :
( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
=> ( c_member(T_a,V_a_2,V_A_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2) ) ) ).
fof(fact_refl__onD1,axiom,
! [V_y_2,V_x_2,V_r_2,V_A_2,T_a] :
( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),V_r_2)
=> c_member(T_a,V_x_2,V_A_2) ) ) ).
fof(fact_refl__onD2,axiom,
! [V_y_2,V_x_2,V_r_2,V_A_2,T_a] :
( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),V_r_2)
=> c_member(T_a,V_y_2,V_A_2) ) ) ).
fof(fact_wfrec__rel_Oequations,axiom,
! [V_a2_2,V_a1_2,V_F_2,V_R_2,T_b,T_a] :
( c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_a1_2,hAPP(hAPP(V_F_2,V_a2_2),V_a1_2))
<=> ? [B_g] :
( hAPP(hAPP(V_F_2,V_a2_2),V_a1_2) = hAPP(hAPP(V_F_2,B_g),V_a1_2)
& ! [B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2)
=> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(B_g,B_z)) ) ) ) ).
fof(fact_single__valuedD,axiom,
! [V_z_2,V_y_2,V_x_2,V_r_2,T_b,T_a] :
( c_Relation_Osingle__valued(T_a,T_b,V_r_2)
=> ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_x_2,V_y_2),V_r_2)
=> ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_x_2,V_z_2),V_r_2)
=> V_y_2 = V_z_2 ) ) ) ).
fof(fact_single__valued__def,axiom,
! [V_r_2,T_b,T_a] :
( c_Relation_Osingle__valued(T_a,T_b,V_r_2)
<=> ! [B_x,B_y] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_y),V_r_2)
=> ! [B_z] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_z),V_r_2)
=> B_y = B_z ) ) ) ).
fof(fact_cut__apply,axiom,
! [V_f_2,T_b,V_r_2,V_a_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_a_2),V_r_2)
=> hAPP(c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_a_2),V_x_2) = hAPP(V_f_2,V_x_2) ) ).
fof(fact_refl__on__converse,axiom,
! [V_r_2,V_A_2,T_a] :
( c_Relation_Orefl__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
<=> c_Relation_Orefl__on(T_a,V_A_2,V_r_2) ) ).
fof(fact_refl__on__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_Orefl__on(T_a,V_A_2,c_Relation_OId__on(T_a,V_A_2)) ).
fof(fact_single__valued__Id__on,axiom,
! [V_A_2,T_a] : c_Relation_Osingle__valued(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) ).
fof(fact_tfl__cut__apply,axiom,
! [T_b,V_a_2,V_x_2,T_a,B_f,B_R] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_a_2),B_R)
=> hAPP(c_Recdef_Ocut(T_a,T_b,B_f,B_R,V_a_2),V_x_2) = hAPP(B_f,V_x_2) ) ).
fof(fact_cuts__eq,axiom,
! [V_g_2,V_x_2,V_r_2,V_f_2,T_b,T_a] :
( c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_x_2) = c_Recdef_Ocut(T_a,T_b,V_g_2,V_r_2,V_x_2)
<=> ! [B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_x_2),V_r_2)
=> hAPP(V_f_2,B_y) = hAPP(V_g_2,B_y) ) ) ).
fof(fact_wfrec__rel_Osimps,axiom,
! [V_a2_2,V_a1_2,V_F_2,V_R_2,T_b,T_a] :
( c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_a1_2,V_a2_2)
<=> ? [B_g] :
( V_a2_2 = hAPP(hAPP(V_F_2,B_g),V_a1_2)
& ! [B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2)
=> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(B_g,B_z)) ) ) ) ).
fof(fact_wfrec__rel_Ointros,axiom,
! [V_g_2,V_F_2,T_b,V_R_2,V_x_2,T_a] :
( ! [B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_x_2),V_R_2)
=> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(V_g_2,B_z)) )
=> c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_x_2,hAPP(hAPP(V_F_2,V_g_2),V_x_2)) ) ).
fof(fact_single__valuedI,axiom,
! [V_r_2,T_b,T_a] :
( ! [B_x,B_y] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_y),V_r_2)
=> ! [B_z] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_z),V_r_2)
=> B_y = B_z ) )
=> c_Relation_Osingle__valued(T_a,T_b,V_r_2) ) ).
fof(fact_acc_Osimps,axiom,
! [V_r_2,V_a_2,T_a] :
( c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2))
<=> ! [B_x] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,V_a_2),V_r_2)
=> c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) ).
fof(fact_acc__downward,axiom,
! [V_a_2,V_r_2,V_b_2,T_a] :
( c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
=> c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) ).
fof(fact_Nitpick_Owf__wfrec_H__def,axiom,
! [V_x_2,V_F_2,V_R_2,T_a,T_b] : c_Nitpick_Owf__wfrec_H(T_b,T_a,V_R_2,V_F_2,V_x_2) = hAPP(hAPP(V_F_2,c_Recdef_Ocut(T_b,T_a,c_Nitpick_Owf__wfrec(T_b,T_a,V_R_2,V_F_2),V_R_2,V_x_2)),V_x_2) ).
fof(fact_IdI,axiom,
! [V_a_2,T_a] : c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),c_Relation_OId(T_a)) ).
fof(fact_Pair__inject,axiom,
! [V_b_H,V_a_H,V_b,V_a,T_b,T_a] :
( c_Product__Type_OPair(T_a,T_b,V_a,V_b) = c_Product__Type_OPair(T_a,T_b,V_a_H,V_b_H)
=> ~ ( V_a = V_a_H
=> V_b != V_b_H ) ) ).
fof(fact_Pair__eq,axiom,
! [V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_b,T_a] :
( c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2) = c_Product__Type_OPair(T_a,T_b,V_a_H_2,V_b_H_2)
<=> ( V_a_2 = V_a_H_2
& V_b_2 = V_b_H_2 ) ) ).
fof(fact_converse__Id,axiom,
! [T_a] : c_Relation_Oconverse(T_a,T_a,c_Relation_OId(T_a)) = c_Relation_OId(T_a) ).
fof(fact_antisym__Id,axiom,
! [T_a] : c_Relation_Oantisym(T_a,c_Relation_OId(T_a)) ).
fof(fact_single__valued__Id,axiom,
! [T_a] : c_Relation_Osingle__valued(T_a,T_a,c_Relation_OId(T_a)) ).
fof(fact_pair__in__Id__conv,axiom,
! [V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Relation_OId(T_a))
<=> V_a_2 = V_b_2 ) ).
fof(fact_split__paired__All,axiom,
! [T_b,T_a,V_P_2] :
( ! [B_x1] : hBOOL(hAPP(V_P_2,B_x1))
<=> ! [B_a,B_b] : hBOOL(hAPP(V_P_2,c_Product__Type_OPair(T_a,T_b,B_a,B_b))) ) ).
fof(fact_prod_Orecs,axiom,
! [V_b_2,V_a_2,V_f1_2,T_a,T_c,T_b] : c_Product__Type_Oprod_Oprod__rec(T_b,T_c,T_a,V_f1_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2) ).
fof(fact_not__acc__down,axiom,
! [V_R_2,V_x_2,T_a] :
( ~ c_member(T_a,V_x_2,c_Wellfounded_Oacc(T_a,V_R_2))
=> ~ ! [B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_x_2),V_R_2)
=> c_member(T_a,B_z,c_Wellfounded_Oacc(T_a,V_R_2)) ) ) ).
fof(fact_acc_OaccI,axiom,
! [V_r_2,V_x_2,T_a] :
( ! [B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_x_2),V_r_2)
=> c_member(T_a,B_y,c_Wellfounded_Oacc(T_a,V_r_2)) )
=> c_member(T_a,V_x_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ).
fof(fact_IdE,axiom,
! [V_p_2,T_a] :
( c_member(tc_prod(T_a,T_a),V_p_2,c_Relation_OId(T_a))
=> ~ ! [B_x] : V_p_2 != c_Product__Type_OPair(T_a,T_a,B_x,B_x) ) ).
fof(fact_internal__split__conv,axiom,
! [V_b_2,V_a_2,V_c_2,T_a,T_c,T_b] : c_Product__Type_Ointernal__split(T_b,T_c,T_a,V_c_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(V_c_2,V_a_2),V_b_2) ).
fof(fact_acc__downwards__aux,axiom,
! [V_r_2,V_a_2,V_b_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2))
=> c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) ).
fof(fact_acc__downwards,axiom,
! [V_b_2,V_r_2,V_a_2,T_a] :
( c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) ).
fof(fact_curryI,axiom,
! [V_b_2,V_a_2,T_b,T_a,V_f_2] :
( hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2)))
=> hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2)) ) ).
fof(fact_curryE,axiom,
! [V_b_2,V_a_2,V_f_2,T_b,T_a] :
( hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2))
=> hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))) ) ).
fof(fact_curryD,axiom,
! [V_b_2,V_a_2,V_f_2,T_b,T_a] :
( hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2))
=> hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))) ) ).
fof(fact_curry__conv,axiom,
! [V_b_2,V_a_2,V_f_2,T_a,T_c,T_b] : c_Product__Type_Ocurry(T_b,T_c,T_a,V_f_2,V_a_2,V_b_2) = hAPP(V_f_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) ).
fof(fact_rtrancl_Ortrancl__refl,axiom,
! [V_r_2,V_a_2,T_a] : c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ).
fof(fact_r__into__rtrancl,axiom,
! [V_r_2,V_p_2,T_a] :
( c_member(tc_prod(T_a,T_a),V_p_2,V_r_2)
=> c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ).
fof(fact_Not__Domain__rtrancl,axiom,
! [V_y_2,V_R_2,V_x_2,T_a] :
( ~ c_member(T_a,V_x_2,c_Relation_ODomain(T_a,T_a,V_R_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
<=> V_x_2 = V_y_2 ) ) ).
fof(fact_single__valued__confluent,axiom,
! [V_z_2,V_y_2,V_x_2,V_r_2,T_a] :
( c_Relation_Osingle__valued(T_a,T_a,V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_y_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
| c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_z_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ) ) ).
fof(fact_rtrancl__converseI,axiom,
! [V_r_2,V_x_2,V_y_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_y_2,V_x_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) ) ).
fof(fact_rtrancl__converseD,axiom,
! [V_r_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_y_2,V_x_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ).
fof(fact_converse__rtrancl__into__rtrancl,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ).
fof(fact_rtrancl_Ortrancl__into__rtrancl,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ).
fof(fact_rtrancl__idemp,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) ).
fof(fact_rtrancl__converse,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) = c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ).
fof(fact_rtrancl__trans,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ).
fof(fact_rtrancl__induct2,axiom,
! [V_P_2,V_r_2,V_by_2,V_bx_2,V_ay_2,V_ax_2,T_b,T_a] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
=> ( hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2))
=> ( ! [B_a,B_b] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,B_a,B_b)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
=> ! [B_aa,B_ba] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba)),V_r_2)
=> ( hBOOL(hAPP(hAPP(V_P_2,B_a),B_b))
=> hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba)) ) ) )
=> hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2)) ) ) ) ).
fof(fact_converse__rtrancl__induct2,axiom,
! [V_P_2,V_r_2,V_by_2,V_bx_2,V_ay_2,V_ax_2,T_b,T_a] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
=> ( hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2))
=> ( ! [B_a,B_b,B_aa,B_ba] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba)),V_r_2)
=> ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
=> ( hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba))
=> hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)) ) ) )
=> hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2)) ) ) ) ).
fof(fact_converse__rtranclE2,axiom,
! [V_r_2,V_zb_2,V_za_2,V_xb_2,V_xa_2,T_b,T_a] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2),c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
=> ( c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2) != c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)
=> ~ ! [B_a,B_b] :
( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2),c_Product__Type_OPair(T_a,T_b,B_a,B_b)),V_r_2)
=> ~ c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)) ) ) ) ).
fof(fact_rtranclE,axiom,
! [V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( V_a_2 != V_b_2
=> ~ ! [B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_y),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_b_2),V_r_2) ) ) ) ).
fof(fact_converse__rtranclE,axiom,
! [V_r_2,V_z_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( V_x_2 != V_z_2
=> ~ ! [B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,B_y),V_r_2)
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ) ).
fof(fact_converse__rtrancl__induct,axiom,
! [V_P_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( hBOOL(hAPP(V_P_2,V_b_2))
=> ( ! [B_y,B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( hBOOL(hAPP(V_P_2,B_z))
=> hBOOL(hAPP(V_P_2,B_y)) ) ) )
=> hBOOL(hAPP(V_P_2,V_a_2)) ) ) ) ).
fof(fact_rtrancl__induct,axiom,
! [V_P_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( hBOOL(hAPP(V_P_2,V_a_2))
=> ( ! [B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_y),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ! [B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2)
=> ( hBOOL(hAPP(V_P_2,B_y))
=> hBOOL(hAPP(V_P_2,B_z)) ) ) )
=> hBOOL(hAPP(V_P_2,V_b_2)) ) ) ) ).
fof(fact_acyclic__impl__antisym__rtrancl,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,V_r_2)
=> c_Relation_Oantisym(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ).
fof(fact_acyclic__converse,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
<=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ).
fof(fact_acyclic__insert,axiom,
! [V_r_2,V_x_2,V_y_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_y_2,V_x_2),V_r_2))
<=> ( c_Wellfounded_Oacyclic(T_a,V_r_2)
& ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ).
fof(fact_trancl__into__rtrancl,axiom,
! [V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ).
fof(fact_acyclic__def,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,V_r_2)
<=> ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ).
fof(fact_trancl__converseI,axiom,
! [V_r_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) ) ).
fof(fact_trancl__converseD,axiom,
! [V_r_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ).
fof(fact_SigmaI,axiom,
! [V_B_2,V_b_2,T_b,V_A_2,V_a_2,T_a] :
( c_member(T_a,V_a_2,V_A_2)
=> ( c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2))
=> c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2)) ) ) ).
fof(fact_rtrancl__trancl__trancl,axiom,
! [V_z_2,V_r_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_y_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ).
fof(fact_insertCI,axiom,
! [V_b_2,V_B_2,V_a_2,T_a] :
( ( ~ c_member(T_a,V_a_2,V_B_2)
=> V_a_2 = V_b_2 )
=> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)) ) ).
fof(fact_insertE,axiom,
! [V_A_2,V_b_2,V_a_2,T_a] :
( c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2))
=> ( V_a_2 != V_b_2
=> c_member(T_a,V_a_2,V_A_2) ) ) ).
fof(fact_trancl_Or__into__trancl,axiom,
! [V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ).
fof(fact_insert__code,axiom,
! [V_x_2,V_A_2,V_y_2,T_a] :
( hBOOL(hAPP(c_Set_Oinsert(T_a,V_y_2,V_A_2),V_x_2))
<=> ( V_y_2 = V_x_2
| hBOOL(hAPP(V_A_2,V_x_2)) ) ) ).
fof(fact_insert__commute,axiom,
! [V_A_2,V_y_2,V_x_2,T_a] : c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_y_2,V_A_2)) = c_Set_Oinsert(T_a,V_y_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) ).
fof(fact_insert__absorb2,axiom,
! [V_A_2,V_x_2,T_a] : c_Set_Oinsert(T_a,V_x_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Set_Oinsert(T_a,V_x_2,V_A_2) ).
fof(fact_Domain__insert,axiom,
! [V_r_2,V_b_2,V_a_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Set_Oinsert(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)) = c_Set_Oinsert(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)) ).
fof(fact_Range__insert,axiom,
! [V_r_2,V_b_2,V_a_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Set_Oinsert(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_a_2,V_b_2),V_r_2)) = c_Set_Oinsert(T_a,V_b_2,c_Relation_ORange(T_b,T_a,V_r_2)) ).
fof(fact_insert__absorb,axiom,
! [V_A_2,V_a_2,T_a] :
( c_member(T_a,V_a_2,V_A_2)
=> c_Set_Oinsert(T_a,V_a_2,V_A_2) = V_A_2 ) ).
fof(fact_insertI2,axiom,
! [V_b_2,V_B_2,V_a_2,T_a] :
( c_member(T_a,V_a_2,V_B_2)
=> c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)) ) ).
fof(fact_insert__ident,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( ~ c_member(T_a,V_x_2,V_A_2)
=> ( ~ c_member(T_a,V_x_2,V_B_2)
=> ( c_Set_Oinsert(T_a,V_x_2,V_A_2) = c_Set_Oinsert(T_a,V_x_2,V_B_2)
<=> V_A_2 = V_B_2 ) ) ) ).
fof(fact_insert__iff,axiom,
! [V_A_2,V_b_2,V_a_2,T_a] :
( c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,V_A_2))
<=> ( V_a_2 = V_b_2
| c_member(T_a,V_a_2,V_A_2) ) ) ).
fof(fact_insertI1,axiom,
! [V_B_2,V_a_2,T_a] : c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) ).
fof(fact_r__into__trancl_H,axiom,
! [V_r_2,V_p_2,T_a] :
( c_member(tc_prod(T_a,T_a),V_p_2,V_r_2)
=> c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ).
fof(fact_trancl__rtrancl__absorb,axiom,
! [V_R_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Otrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ).
fof(fact_rtrancl__trancl__absorb,axiom,
! [V_R_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ).
fof(fact_trancl__converse,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) = c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) ).
fof(fact_trancl__domain,axiom,
! [V_r_2,T_a] : c_Relation_ODomain(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Relation_ODomain(T_a,T_a,V_r_2) ).
fof(fact_trancl__range,axiom,
! [V_r_2,T_a] : c_Relation_ORange(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Relation_ORange(T_a,T_a,V_r_2) ).
fof(fact_trancl__trans,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ).
fof(fact_Transitive__Closure_Otrancl__into__trancl,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ).
fof(fact_trancl__into__trancl2,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ).
fof(fact_r__r__into__trancl,axiom,
! [V_c_2,V_R_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_R_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_R_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) ) ) ).
fof(fact_SigmaE2,axiom,
! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
=> ~ ( c_member(T_a,V_a_2,V_A_2)
=> ~ c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) ) ) ).
fof(fact_SigmaD2,axiom,
! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
=> c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) ) ).
fof(fact_SigmaD1,axiom,
! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
=> c_member(T_a,V_a_2,V_A_2) ) ).
fof(fact_mem__Sigma__iff,axiom,
! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
<=> ( c_member(T_a,V_a_2,V_A_2)
& c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) ) ) ).
fof(fact_rtrancl__eq__or__trancl,axiom,
! [V_R_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
<=> ( V_x_2 = V_y_2
| ( V_x_2 != V_y_2
& c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) ) ) ) ).
fof(fact_rtrancl__into__trancl2,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ).
fof(fact_rtranclD,axiom,
! [V_R_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
=> ( V_a_2 = V_b_2
| ( V_a_2 != V_b_2
& c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) ) ) ) ).
fof(fact_rtrancl__into__trancl1,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ).
fof(fact_trancl__rtrancl__trancl,axiom,
! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ).
fof(fact_SigmaE,axiom,
! [V_B_2,V_A_2,V_c_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),V_c_2,c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
=> ~ ! [B_x] :
( c_member(T_a,B_x,V_A_2)
=> ! [B_y] :
( c_member(T_b,B_y,hAPP(V_B_2,B_x))
=> V_c_2 != c_Product__Type_OPair(T_a,T_b,B_x,B_y) ) ) ) ).
fof(fact_acyclicI,axiom,
! [V_r_2,T_a] :
( ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ).
fof(fact_tranclD2,axiom,
! [V_R_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Otrancl(T_a,V_R_2))
=> ? [B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,B_z),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
& c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_y_2),V_R_2) ) ) ).
fof(fact_tranclD,axiom,
! [V_R_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Otrancl(T_a,V_R_2))
=> ? [B_z] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,B_z),V_R_2)
& c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) ) ) ).
fof(fact_wf__insert,axiom,
! [V_r_2,V_x_2,V_y_2,T_a] :
( c_Wellfounded_Owf(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_y_2,V_x_2),V_r_2))
<=> ( c_Wellfounded_Owf(T_a,V_r_2)
& ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ).
fof(fact_tranclE,axiom,
! [V_r_2,V_b_2,V_a_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ( ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
=> ~ ! [B_c] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_c),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_c,V_b_2),V_r_2) ) ) ) ).
fof(fact_converse__tranclE,axiom,
! [V_r_2,V_z_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ( ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_z_2),V_r_2)
=> ~ ! [B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,B_y),V_r_2)
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ) ).
fof(fact_wf__inv__image,axiom,
! [V_f_2,T_b,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> c_Wellfounded_Owf(T_b,hAPP(c_Relation_Oinv__image(T_a,T_b,V_r_2),V_f_2)) ) ).
fof(fact_wf__lex__prod,axiom,
! [V_rb_2,T_b,V_ra_2,T_a] :
( c_Wellfounded_Owf(T_a,V_ra_2)
=> ( c_Wellfounded_Owf(T_b,V_rb_2)
=> c_Wellfounded_Owf(tc_prod(T_a,T_b),c_Wellfounded_Olex__prod(T_a,T_b,V_ra_2,V_rb_2)) ) ) ).
fof(fact_wf__trancl,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> c_Wellfounded_Owf(T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ).
fof(fact_wf__acyclic,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ).
fof(fact_wf__not__refl,axiom,
! [V_a_2,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2) ) ).
fof(fact_wf__not__sym,axiom,
! [V_x_2,V_a_2,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_x_2),V_r_2)
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_a_2),V_r_2) ) ) ).
fof(fact_wf__asym,axiom,
! [V_x_2,V_a_2,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_x_2),V_r_2)
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_a_2),V_r_2) ) ) ).
fof(fact_wf__irrefl,axiom,
! [V_a_2,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2) ) ).
fof(fact_acc__wfD,axiom,
! [V_x_2,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> c_member(T_a,V_x_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ).
fof(fact_wf__acc__iff,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
<=> ! [B_x] : c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2)) ) ).
fof(fact_wf__converse__trancl,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
=> c_Wellfounded_Owf(T_a,c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) ).
fof(fact_wf__measure,axiom,
! [V_f_2,T_a] : c_Wellfounded_Owf(T_a,hAPP(c_Wellfounded_Omeasure(T_a),V_f_2)) ).
fof(fact_tfl__wfrec,axiom,
! [T_b,T_a,B_M,B_R] :
( c_Wellfounded_Owf(T_a,B_R)
=> ! [B_x] : hAPP(c_Recdef_Owfrec(T_a,T_b,B_R,B_M),B_x) = hAPP(hAPP(B_M,c_Recdef_Ocut(T_a,T_b,c_Recdef_Owfrec(T_a,T_b,B_R,B_M),B_R,B_x)),B_x) ) ).
fof(fact_wfrec,axiom,
! [V_a_2,V_H_2,T_b,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> hAPP(c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2),V_a_2) = hAPP(hAPP(V_H_2,c_Recdef_Ocut(T_a,T_b,c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2),V_r_2,V_a_2)),V_a_2) ) ).
fof(fact_def__wfrec,axiom,
! [V_a_2,V_H_2,V_r_2,T_b,T_a,V_f_2] :
( V_f_2 = c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2)
=> ( c_Wellfounded_Owf(T_a,V_r_2)
=> hAPP(V_f_2,V_a_2) = hAPP(hAPP(V_H_2,c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_a_2)),V_a_2) ) ) ).
fof(fact_wf__eq__minimal,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
<=> ! [B_Q] :
( ? [B_x] : c_member(T_a,B_x,B_Q)
=> ? [B_x] :
( c_member(T_a,B_x,B_Q)
& ! [B_y] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2)
=> ~ c_member(T_a,B_y,B_Q) ) ) ) ) ).
fof(fact_wf__in__rel,axiom,
! [V_R_2,T_a] :
( c_Wellfounded_Owf(T_a,V_R_2)
=> c_Wellfounded_OwfP(T_a,c_FunDef_Oin__rel(T_a,T_a,V_R_2)) ) ).
fof(fact_wf__same__fst,axiom,
! [T_a,V_R_2,T_b,V_P_2] :
( ! [B_x] :
( hBOOL(hAPP(V_P_2,B_x))
=> c_Wellfounded_Owf(T_b,hAPP(V_R_2,B_x)) )
=> c_Wellfounded_Owf(tc_prod(T_a,T_b),c_Recdef_Osame__fst(T_a,T_b,V_P_2,V_R_2)) ) ).
fof(fact_acc__wfI,axiom,
! [V_r_2,T_a] :
( ! [B_x] : c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2))
=> c_Wellfounded_Owf(T_a,V_r_2) ) ).
fof(fact_in__measure,axiom,
! [V_f_2,V_y_2,V_x_2,T_a] :
( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),hAPP(c_Wellfounded_Omeasure(T_a),V_f_2))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(V_f_2,V_x_2),hAPP(V_f_2,V_y_2)) ) ).
fof(fact_irrefl__trancl__rD,axiom,
! [V_y_2,V_x_2,V_r_2,T_a] :
( ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),V_r_2)
=> V_x_2 != V_y_2 ) ) ).
fof(fact_measure__def,axiom,
! [T_a] : c_Wellfounded_Omeasure(T_a) = c_Relation_Oinv__image(tc_Nat_Onat,T_a,c_Wellfounded_Oless__than) ).
fof(fact_wf__less__than,axiom,
c_Wellfounded_Owf(tc_Nat_Onat,c_Wellfounded_Oless__than) ).
fof(fact_less__than__iff,axiom,
! [V_y_2,V_x_2] :
( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_x_2,V_y_2),c_Wellfounded_Oless__than)
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x_2,V_y_2) ) ).
fof(fact_mlex__less,axiom,
! [V_R_2,T_a,V_y_2,V_x_2,V_f_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(V_f_2,V_x_2),hAPP(V_f_2,V_y_2))
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)) ) ).
fof(fact_less__than__def,axiom,
c_Wellfounded_Oless__than = c_Transitive__Closure_Otrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat) ).
fof(fact_finite__acyclic__wf__converse,axiom,
! [V_r_2,T_a] :
( c_Finite__Set_Ofinite(tc_prod(T_a,T_a),V_r_2)
=> ( c_Wellfounded_Oacyclic(T_a,V_r_2)
=> c_Wellfounded_Owf(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) ) ) ).
fof(fact_min__ext__wf,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> c_Wellfounded_Owf(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Omin__ext(T_a,V_r_2)) ) ).
fof(fact_nat__less__cases,axiom,
! [V_P_2,V_n_2,V_m_2] :
( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
=> ( ( V_m_2 = V_n_2
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
=> ( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
=> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) ) ) ) ).
fof(fact_finite__Range,axiom,
! [V_r_2,T_b,T_a] :
( c_Finite__Set_Ofinite(tc_prod(T_a,T_b),V_r_2)
=> c_Finite__Set_Ofinite(T_b,c_Relation_ORange(T_a,T_b,V_r_2)) ) ).
fof(fact_finite__Domain,axiom,
! [V_r_2,T_b,T_a] :
( c_Finite__Set_Ofinite(tc_prod(T_a,T_b),V_r_2)
=> c_Finite__Set_Ofinite(T_a,c_Relation_ODomain(T_a,T_b,V_r_2)) ) ).
fof(fact_wf__pred__nat,axiom,
c_Wellfounded_Owf(tc_Nat_Onat,c_Wellfounded_Opred__nat) ).
fof(fact_finite__trancl,axiom,
! [V_r_2,T_a] :
( c_Finite__Set_Ofinite(tc_prod(T_a,T_a),c_Transitive__Closure_Otrancl(T_a,V_r_2))
<=> c_Finite__Set_Ofinite(tc_prod(T_a,T_a),V_r_2) ) ).
fof(fact_finite__converse,axiom,
! [V_r_2,T_b,T_a] :
( c_Finite__Set_Ofinite(tc_prod(T_a,T_b),c_Relation_Oconverse(T_b,T_a,V_r_2))
<=> c_Finite__Set_Ofinite(tc_prod(T_b,T_a),V_r_2) ) ).
fof(fact_wf__iff__acyclic__if__finite,axiom,
! [V_r_2,T_a] :
( c_Finite__Set_Ofinite(tc_prod(T_a,T_a),V_r_2)
=> ( c_Wellfounded_Owf(T_a,V_r_2)
<=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ) ).
fof(fact_finite__acyclic__wf,axiom,
! [V_r_2,T_a] :
( c_Finite__Set_Ofinite(tc_prod(T_a,T_a),V_r_2)
=> ( c_Wellfounded_Oacyclic(T_a,V_r_2)
=> c_Wellfounded_Owf(T_a,V_r_2) ) ) ).
fof(fact_less__eq,axiom,
! [V_n_2,V_m_2] :
( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_m_2,V_n_2),c_Transitive__Closure_Otrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) ).
fof(fact_wf__mlex,axiom,
! [V_f_2,V_R_2,T_a] :
( c_Wellfounded_Owf(T_a,V_R_2)
=> c_Wellfounded_Owf(T_a,c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)) ) ).
fof(fact_less__not__refl,axiom,
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n) ).
fof(fact_nat__neq__iff,axiom,
! [V_n_2,V_m_2] :
( V_m_2 != V_n_2
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2) ) ) ).
fof(fact_linorder__neqE__nat,axiom,
! [V_y,V_x] :
( V_x != V_y
=> ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_y,V_x) ) ) ).
fof(fact_less__irrefl__nat,axiom,
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_n) ).
fof(fact_less__not__refl2,axiom,
! [V_m,V_n] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m)
=> V_m != V_n ) ).
fof(fact_less__not__refl3,axiom,
! [V_t,V_s] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_s,V_t)
=> V_s != V_t ) ).
fof(fact_finite_OinsertI,axiom,
! [V_a_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> c_Finite__Set_Ofinite(T_a,c_Set_Oinsert(T_a,V_a_2,V_A_2)) ) ).
fof(fact_Nitpick_Owf_H__def,axiom,
! [V_r_2,T_a] :
( c_Nitpick_Owf_H(T_a,V_r_2)
<=> ( c_Wellfounded_Oacyclic(T_a,V_r_2)
& ( c_Finite__Set_Ofinite(tc_prod(T_a,T_a),V_r_2)
| c_Nitpick_Ounknown(tc_HOL_Obool) ) ) ) ).
fof(fact_finite__insert,axiom,
! [V_A_2,V_a_2,T_a] :
( c_Finite__Set_Ofinite(T_a,c_Set_Oinsert(T_a,V_a_2,V_A_2))
<=> c_Finite__Set_Ofinite(T_a,V_A_2) ) ).
fof(fact_wf__finite__psubset,axiom,
! [T_a] : c_Wellfounded_Owf(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Ofinite__psubset(T_a)) ).
fof(fact_max__ext__wf,axiom,
! [V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> c_Wellfounded_Owf(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Omax__ext(T_a,V_r_2)) ) ).
fof(fact_finite__nat__set__iff__bounded,axiom,
! [V_N_2] :
( c_Finite__Set_Ofinite(tc_Nat_Onat,V_N_2)
<=> ? [B_m] :
! [B_x] :
( c_member(tc_Nat_Onat,B_x,V_N_2)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_x,B_m) ) ) ).
fof(fact_finite__code,axiom,
! [V_A_2,T_a] :
( class_Finite__Set_Ofinite(T_a)
=> c_Finite__Set_Ofinite(T_a,V_A_2) ) ).
fof(fact_finite,axiom,
! [V_A_2,T_a] :
( class_Finite__Set_Ofinite(T_a)
=> c_Finite__Set_Ofinite(T_a,V_A_2) ) ).
fof(fact_folding__image__simple_Oinsert,axiom,
! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofolding__image__simple(T_a,T_b,V_f_2,V_z_2,V_g_2,V_F_2)
=> ( c_Finite__Set_Ofinite(T_b,V_A_2)
=> ( ~ c_member(T_b,V_x_2,V_A_2)
=> hAPP(V_F_2,c_Set_Oinsert(T_b,V_x_2,V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) ) ) ) ).
fof(fact_pair__leqI1,axiom,
! [V_t_2,V_s_2,V_b_2,V_a_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_a_2,V_b_2)
=> c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_a_2,V_s_2),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_b_2,V_t_2)),c_FunDef_Opair__leq) ) ).
fof(fact_folding__image__simple__idem_Oinsert__idem,axiom,
! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_F_2)
=> ( c_Finite__Set_Ofinite(T_b,V_A_2)
=> hAPP(V_F_2,c_Set_Oinsert(T_b,V_x_2,V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) ) ) ).
fof(fact_folding__image__simple__idem_Oin__idem,axiom,
! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_F_2)
=> ( c_Finite__Set_Ofinite(T_b,V_A_2)
=> ( c_member(T_b,V_x_2,V_A_2)
=> hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2) ) ) ) ).
fof(fact_pair__lessI1,axiom,
! [V_t_2,V_s_2,V_b_2,V_a_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_a_2,V_b_2)
=> c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_a_2,V_s_2),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_b_2,V_t_2)),c_FunDef_Opair__less) ) ).
fof(fact_folding__image__simple__idem_Oidem,axiom,
! [V_x_2,V_F_2,V_g_2,V_z_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_F_2)
=> hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2 ) ).
fof(fact_wf__pair__less,axiom,
c_Wellfounded_Owf(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__less) ).
fof(fact_pair__less__def,axiom,
c_FunDef_Opair__less = c_Wellfounded_Olex__prod(tc_Nat_Onat,tc_Nat_Onat,c_Wellfounded_Oless__than,c_Wellfounded_Oless__than) ).
fof(fact_wmax__insertI,axiom,
! [V_XS_2,V_x_2,V_YS_2,V_y_2] :
( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_y_2,V_YS_2)
=> ( c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_y_2),c_FunDef_Opair__leq)
=> ( c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_XS_2,V_YS_2),c_FunDef_Omax__weak)
=> c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_XS_2),V_YS_2),c_FunDef_Omax__weak) ) ) ) ).
fof(fact_wmin__insertI,axiom,
! [V_YS_2,V_y_2,V_XS_2,V_x_2] :
( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_XS_2)
=> ( c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_y_2),c_FunDef_Opair__leq)
=> ( c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_XS_2,V_YS_2),c_FunDef_Omin__weak)
=> c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_XS_2,c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_y_2,V_YS_2)),c_FunDef_Omin__weak) ) ) ) ).
fof(fact_smax__insertI,axiom,
! [V_X_2,V_x_2,V_Y_2,V_y_2] :
( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_y_2,V_Y_2)
=> ( c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_y_2),c_FunDef_Opair__less)
=> ( c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_X_2,V_Y_2),c_FunDef_Omax__strict)
=> c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_X_2),V_Y_2),c_FunDef_Omax__strict) ) ) ) ).
fof(fact_smin__insertI,axiom,
! [V_YS_2,V_y_2,V_XS_2,V_x_2] :
( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_XS_2)
=> ( c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_x_2,V_y_2),c_FunDef_Opair__less)
=> ( c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_XS_2,V_YS_2),c_FunDef_Omin__strict)
=> c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_XS_2,c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_y_2,V_YS_2)),c_FunDef_Omin__strict) ) ) ) ).
fof(fact_pair__lessI2,axiom,
! [V_t_2,V_s_2,V_b_2,V_a_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_a_2,V_b_2)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_s_2,V_t_2)
=> c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_a_2,V_s_2),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_b_2,V_t_2)),c_FunDef_Opair__less) ) ) ).
fof(fact_le__refl,axiom,
! [V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_n) ).
fof(fact_nat__le__linear,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
| c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m) ) ).
fof(fact_eq__imp__le,axiom,
! [V_n,V_m] :
( V_m = V_n
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_le__trans,axiom,
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_j,V_k)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_i,V_k) ) ) ).
fof(fact_le__antisym,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,V_m)
=> V_m = V_n ) ) ).
fof(fact_less__or__eq__imp__le,axiom,
! [V_n,V_m] :
( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
| V_m = V_n )
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_le__neq__implies__less,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> ( V_m != V_n
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ) ).
fof(fact_less__imp__le__nat,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_le__eq__less__or__eq,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
| V_m_2 = V_n_2 ) ) ).
fof(fact_nat__less__le,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
<=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
& V_m_2 != V_n_2 ) ) ).
fof(fact_termination__basic__simps_I5_J,axiom,
! [V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_y) ) ).
fof(fact_finite__nat__set__iff__bounded__le,axiom,
! [V_N_2] :
( c_Finite__Set_Ofinite(tc_Nat_Onat,V_N_2)
<=> ? [B_m] :
! [B_x] :
( c_member(tc_Nat_Onat,B_x,V_N_2)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_x,B_m) ) ) ).
fof(fact_mlex__leq,axiom,
! [V_R_2,T_a,V_y_2,V_x_2,V_f_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(V_f_2,V_x_2),hAPP(V_f_2,V_y_2))
=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),V_R_2)
=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_y_2),c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)) ) ) ).
fof(fact_min__strict__def,axiom,
c_FunDef_Omin__strict = c_Wellfounded_Omin__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__less) ).
fof(fact_max__strict__def,axiom,
c_FunDef_Omax__strict = c_Wellfounded_Omax__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__less) ).
fof(fact_pred__nat__trancl__eq__le,axiom,
! [V_n_2,V_m_2] :
( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_m_2,V_n_2),c_Transitive__Closure_Ortrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) ).
fof(fact_pair__leqI2,axiom,
! [V_t_2,V_s_2,V_b_2,V_a_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_a_2,V_b_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_s_2,V_t_2)
=> c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_a_2,V_s_2),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_b_2,V_t_2)),c_FunDef_Opair__leq) ) ) ).
fof(fact_order__refl,axiom,
! [V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_x) ) ).
fof(fact_min__rpair__set,axiom,
c_FunDef_Oreduction__pair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Product__Type_OPair(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),c_FunDef_Omin__strict,c_FunDef_Omin__weak)) ).
fof(fact_max__rpair__set,axiom,
c_FunDef_Oreduction__pair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Product__Type_OPair(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),c_FunDef_Omax__strict,c_FunDef_Omax__weak)) ).
fof(fact_linorder__not__less,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) ) ) ).
fof(fact_subsetD,axiom,
! [V_c_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_member(T_a,V_c_2,V_A_2)
=> c_member(T_a,V_c_2,V_B_2) ) ) ).
fof(fact_rev__finite__subset,axiom,
! [V_A_2,V_B_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_B_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_Finite__Set_Ofinite(T_a,V_A_2) ) ) ).
fof(fact_finite__subset,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_Finite__Set_Ofinite(T_a,V_B_2)
=> c_Finite__Set_Ofinite(T_a,V_A_2) ) ) ).
fof(fact_in__mono,axiom,
! [V_x_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_member(T_a,V_x_2,V_A_2)
=> c_member(T_a,V_x_2,V_B_2) ) ) ).
fof(fact_set__rev__mp,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( c_member(T_a,V_x_2,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_member(T_a,V_x_2,V_B_2) ) ) ).
fof(fact_set__mp,axiom,
! [V_x_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_member(T_a,V_x_2,V_A_2)
=> c_member(T_a,V_x_2,V_B_2) ) ) ).
fof(fact_psubsetD,axiom,
! [V_c_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_member(T_a,V_c_2,V_A_2)
=> c_member(T_a,V_c_2,V_B_2) ) ) ).
fof(fact_rtrancl__subset__rtrancl,axiom,
! [V_s_2,V_r_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Transitive__Closure_Ortrancl(T_a,V_s_2))
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_2)) ) ).
fof(fact_rtrancl__subset,axiom,
! [V_S_2,V_R_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_S_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2))
=> c_Transitive__Closure_Ortrancl(T_a,V_S_2) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ) ) ).
fof(fact_rtrancl__mono,axiom,
! [V_s_2,V_r_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_2)) ) ).
fof(fact_wf__subset,axiom,
! [V_p_2,V_r_2,T_a] :
( c_Wellfounded_Owf(T_a,V_r_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_p_2,V_r_2)
=> c_Wellfounded_Owf(T_a,V_p_2) ) ) ).
fof(fact_acc__subset,axiom,
! [V_R2_2,V_R1_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R1_2,V_R2_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Oacc(T_a,V_R2_2),c_Wellfounded_Oacc(T_a,V_R1_2)) ) ).
fof(fact_Domain__mono,axiom,
! [V_s_2,V_r_2,T_b,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_2,V_s_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_r_2),c_Relation_ODomain(T_a,T_b,V_s_2)) ) ).
fof(fact_insert__mono,axiom,
! [V_a_2,V_D_2,V_C_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_C_2),c_Set_Oinsert(T_a,V_a_2,V_D_2)) ) ).
fof(fact_subset__insertI2,axiom,
! [V_b_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_b_2,V_B_2)) ) ).
fof(fact_subset__insertI,axiom,
! [V_a_2,V_B_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) ).
fof(fact_wfP__subset,axiom,
! [V_p_2,V_r_2,T_a] :
( c_Wellfounded_OwfP(T_a,V_r_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool)),V_p_2,V_r_2)
=> c_Wellfounded_OwfP(T_a,V_p_2) ) ) ).
fof(fact_antisym__subset,axiom,
! [V_s_2,V_r_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
=> ( c_Relation_Oantisym(T_a,V_s_2)
=> c_Relation_Oantisym(T_a,V_r_2) ) ) ).
fof(fact_acyclic__subset,axiom,
! [V_r_2,V_s_2,T_a] :
( c_Wellfounded_Oacyclic(T_a,V_s_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) ) ).
fof(fact_single__valued__subset,axiom,
! [V_s_2,V_r_2,T_b,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_2,V_s_2)
=> ( c_Relation_Osingle__valued(T_a,T_b,V_s_2)
=> c_Relation_Osingle__valued(T_a,T_b,V_r_2) ) ) ).
fof(fact_subset__insert,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( ~ c_member(T_a,V_x_2,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_x_2,V_B_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ).
fof(fact_insert__subset,axiom,
! [V_B_2,V_A_2,V_x_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_x_2,V_A_2),V_B_2)
<=> ( c_member(T_a,V_x_2,V_B_2)
& c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ).
fof(fact_trancl__mono,axiom,
! [V_s_2,V_r_2,V_p_2,T_a] :
( c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_r_2))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
=> c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_s_2)) ) ) ).
fof(fact_folding__image__simple__idem_Osubset__idem,axiom,
! [V_B_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_F_2)
=> ( c_Finite__Set_Ofinite(T_b,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),V_B_2,V_A_2)
=> hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2) ) ) ) ).
fof(fact_le__fun__def,axiom,
! [V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
<=> ! [B_x] : c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x)) ) ) ).
fof(fact_linorder__linear,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
| c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) ).
fof(fact_order__eq__iff,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_x_2 = V_y_2
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) ) ) ) ).
fof(fact_order__eq__refl,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( V_x = V_y
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) ) ) ).
fof(fact_le__funD,axiom,
! [V_x_2,V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
=> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)) ) ) ).
fof(fact_order__antisym__conv,axiom,
! [V_x_2,V_y_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ) ).
fof(fact_ord__eq__le__trans,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c) ) ) ) ).
fof(fact_xt1_I3_J,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a) ) ) ) ).
fof(fact_ord__le__eq__trans,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c) ) ) ) ).
fof(fact_xt1_I4_J,axiom,
! [V_c,V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a) ) ) ) ).
fof(fact_order__antisym,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> V_x = V_y ) ) ) ).
fof(fact_order__trans,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_z) ) ) ) ).
fof(fact_xt1_I5_J,axiom,
! [V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> V_x = V_y ) ) ) ).
fof(fact_xt1_I6_J,axiom,
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x) ) ) ) ).
fof(fact_le__funE,axiom,
! [V_x_2,V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
=> c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)) ) ) ).
fof(fact_linorder__le__cases,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) ).
fof(fact_order__less__irrefl,axiom,
! [V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_x) ) ).
fof(fact_linorder__neq__iff,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( V_x_2 != V_y_2
<=> ( c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
| c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) ) ) ) ).
fof(fact_not__less__iff__gr__or__eq,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
<=> ( c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2)
| V_x_2 = V_y_2 ) ) ) ).
fof(fact_linorder__less__linear,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
| V_x = V_y
| c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ).
fof(fact_linorder__antisym__conv3,axiom,
! [V_x_2,V_y_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ) ).
fof(fact_linorder__neqE,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( V_x != V_y
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ) ).
fof(fact_less__imp__neq,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> V_x != V_y ) ) ).
fof(fact_order__less__not__sym,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ).
fof(fact_order__less__imp__not__less,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ).
fof(fact_order__less__imp__not__eq,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> V_x != V_y ) ) ).
fof(fact_order__less__imp__not__eq2,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> V_y != V_x ) ) ).
fof(fact_order__less__asym_H,axiom,
! [V_b,V_a,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_b,V_a) ) ) ).
fof(fact_xt1_I9_J,axiom,
! [V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ).
fof(fact_ord__eq__less__trans,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_c) ) ) ) ).
fof(fact_xt1_I1_J,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a = V_b
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_b)
=> c_Orderings_Oord__class_Oless(T_a,V_c,V_a) ) ) ) ).
fof(fact_ord__less__eq__trans,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Orderings_Oord(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_c) ) ) ) ).
fof(fact_xt1_I2_J,axiom,
! [V_c,V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_a)
=> ( V_b = V_c
=> c_Orderings_Oord__class_Oless(T_a,V_c,V_a) ) ) ) ).
fof(fact_order__less__trans,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_z) ) ) ) ).
fof(fact_xt1_I10_J,axiom,
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_z,V_x) ) ) ) ).
fof(fact_order__less__asym,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ).
fof(fact_linorder__cases,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ( V_x != V_y
=> c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ) ).
fof(fact_in__finite__psubset,axiom,
! [V_B_2,V_A_2,T_a] :
( c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Wellfounded_Ofinite__psubset(T_a))
<=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
& c_Finite__Set_Ofinite(T_a,V_B_2) ) ) ).
fof(fact_xt1_I8_J,axiom,
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_z,V_x) ) ) ) ).
fof(fact_order__le__less__trans,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_z) ) ) ) ).
fof(fact_xt1_I7_J,axiom,
! [V_z,V_x,V_y,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y)
=> c_Orderings_Oord__class_Oless(T_a,V_z,V_x) ) ) ) ).
fof(fact_order__less__le__trans,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_z) ) ) ) ).
fof(fact_xt1_I11_J,axiom,
! [V_a,V_b,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> ( V_a != V_b
=> c_Orderings_Oord__class_Oless(T_a,V_b,V_a) ) ) ) ).
fof(fact_order__le__neq__trans,axiom,
! [V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( V_a != V_b
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) ).
fof(fact_order__le__imp__less__or__eq,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
| V_x = V_y ) ) ) ).
fof(fact_linorder__antisym__conv2,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ) ).
fof(fact_order__less__imp__le,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y) ) ) ).
fof(fact_leD,axiom,
! [V_x,V_y,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y) ) ) ).
fof(fact_xt1_I12_J,axiom,
! [V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a != V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> c_Orderings_Oord__class_Oless(T_a,V_b,V_a) ) ) ) ).
fof(fact_order__neq__le__trans,axiom,
! [V_b,V_a,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a != V_b
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ) ).
fof(fact_linorder__antisym__conv1,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ) ).
fof(fact_not__leE,axiom,
! [V_x,V_y,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> c_Orderings_Oord__class_Oless(T_a,V_x,V_y) ) ) ).
fof(fact_leI,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless(T_a,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x) ) ) ).
fof(fact_order__le__less,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
<=> ( c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
| V_x_2 = V_y_2 ) ) ) ).
fof(fact_less__le__not__le,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Opreorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
& ~ c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) ) ) ) ).
fof(fact_order__less__le,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x_2,V_y_2)
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
& V_x_2 != V_y_2 ) ) ) ).
fof(fact_linorder__le__less__linear,axiom,
! [V_y,V_x,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
| c_Orderings_Oord__class_Oless(T_a,V_y,V_x) ) ) ).
fof(fact_linorder__not__le,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
<=> c_Orderings_Oord__class_Oless(T_a,V_y_2,V_x_2) ) ) ).
fof(fact_rp__inv__image__rp,axiom,
! [V_f_2,T_b,V_P_2,T_a] :
( c_FunDef_Oreduction__pair(T_a,V_P_2)
=> c_FunDef_Oreduction__pair(T_b,c_FunDef_Orp__inv__image(T_a,T_b,V_P_2,V_f_2)) ) ).
fof(fact_reduction__pairI,axiom,
! [V_S_2,V_R_2,T_a] :
( c_Wellfounded_Owf(T_a,V_R_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_a,T_a,V_R_2,V_S_2),V_R_2)
=> c_FunDef_Oreduction__pair(T_a,c_Product__Type_OPair(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)) ) ) ).
fof(fact_smin__emptyI,axiom,
! [V_X_2] :
( V_X_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))
=> c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_X_2,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),c_FunDef_Omin__strict) ) ).
fof(fact_wmin__emptyI,axiom,
! [V_X_2] : c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),V_X_2,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),c_FunDef_Omin__weak) ).
fof(fact_equalityI,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)
=> V_A_2 = V_B_2 ) ) ).
fof(fact_emptyE,axiom,
! [V_a_2,T_a] : ~ c_member(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_finite_OemptyI,axiom,
! [T_a] : c_Finite__Set_Ofinite(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_empty__subsetI,axiom,
! [V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_A_2) ).
fof(fact_wf__empty,axiom,
! [T_a] : c_Wellfounded_Owf(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).
fof(fact_rel__compI,axiom,
! [V_s_2,V_c_2,T_c,V_r_2,V_b_2,V_a_2,T_b,T_a] :
( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
=> ( c_member(tc_prod(T_b,T_c),c_Product__Type_OPair(T_b,T_c,V_b_2,V_c_2),V_s_2)
=> c_member(tc_prod(T_a,T_c),c_Product__Type_OPair(T_a,T_c,V_a_2,V_c_2),c_Relation_Orel__comp(T_a,T_b,T_c,V_r_2,V_s_2)) ) ) ).
fof(fact_less__by__empty,axiom,
! [V_B_2,T_a,V_A_2] :
( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_A_2,V_B_2) ) ).
fof(fact_subset__refl,axiom,
! [V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) ).
fof(fact_subset__empty,axiom,
! [V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))
<=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).
fof(fact_set__eq__subset,axiom,
! [T_a,V_B_2,V_A_2] :
( V_A_2 = V_B_2
<=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
& c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ) ) ).
fof(fact_rev__predicate1D,axiom,
! [V_Q_2,T_a,V_x_2,V_P_2] :
( hBOOL(hAPP(V_P_2,V_x_2))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2)
=> hBOOL(hAPP(V_Q_2,V_x_2)) ) ) ).
fof(fact_equalityD1,axiom,
! [T_a,V_B_2,V_A_2] :
( V_A_2 = V_B_2
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ).
fof(fact_equalityD2,axiom,
! [T_a,V_B_2,V_A_2] :
( V_A_2 = V_B_2
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ) ).
fof(fact_predicate1D,axiom,
! [V_x_2,V_Q_2,V_P_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2)
=> ( hBOOL(hAPP(V_P_2,V_x_2))
=> hBOOL(hAPP(V_Q_2,V_x_2)) ) ) ).
fof(fact_subset__trans,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) ) ) ).
fof(fact_rel__comp__mono,axiom,
! [V_s_2,V_s_H_2,T_c,V_r_2,V_r_H_2,T_b,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_H_2,V_r_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),V_s_H_2,V_s_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_b,T_c,V_r_H_2,V_s_H_2),c_Relation_Orel__comp(T_a,T_b,T_c,V_r_2,V_s_2)) ) ) ).
fof(fact_predicate2D,axiom,
! [V_y_2,V_x_2,V_Q_2,V_P_2,T_b,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),V_P_2,V_Q_2)
=> ( hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2))
=> hBOOL(hAPP(hAPP(V_Q_2,V_x_2),V_y_2)) ) ) ).
fof(fact_equalityE,axiom,
! [T_a,V_B_2,V_A_2] :
( V_A_2 = V_B_2
=> ~ ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ~ c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ) ) ).
fof(fact_rev__predicate2D,axiom,
! [V_Q_2,T_b,T_a,V_y_2,V_x_2,V_P_2] :
( hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_b,tc_HOL_Obool)),V_P_2,V_Q_2)
=> hBOOL(hAPP(hAPP(V_Q_2,V_x_2),V_y_2)) ) ) ).
fof(fact_subset__singletonD,axiom,
! [V_x_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
| V_A_2 = c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ) ) ).
fof(fact_less__fun__def,axiom,
! [V_g_2,V_f_2,T_a,T_b] :
( class_Orderings_Oord(T_b)
=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,T_b),V_f_2,V_g_2)
<=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
& ~ c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_g_2,V_f_2) ) ) ) ).
fof(fact_subset__psubset__trans,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
=> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) ) ) ).
fof(fact_psubset__subset__trans,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
=> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) ) ) ).
fof(fact_psubset__imp__subset,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ).
fof(fact_subset__iff__psubset__eq,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
<=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
| V_A_2 = V_B_2 ) ) ).
fof(fact_psubset__eq,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
<=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
& V_A_2 != V_B_2 ) ) ).
fof(fact_not__psubset__empty,axiom,
! [V_A_2,T_a] : ~ c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_psubset__trans,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
=> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2) ) ) ).
fof(fact_bot__apply,axiom,
! [V_x_2,T_b,T_a] :
( class_Orderings_Obot(T_a)
=> hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_x_2) = c_Orderings_Obot__class_Obot(T_a) ) ).
fof(fact_r__comp__rtrancl__eq,axiom,
! [V_r_2,T_a] : c_Relation_Orel__comp(T_a,T_a,T_a,V_r_2,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Relation_Orel__comp(T_a,T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2),V_r_2) ).
fof(fact_rtrancl__idemp__self__comp,axiom,
! [V_R_2,T_a] : c_Relation_Orel__comp(T_a,T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ).
fof(fact_trancl__empty,axiom,
! [T_a] : c_Transitive__Closure_Otrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) ).
fof(fact_R__O__Id,axiom,
! [V_R_2,T_b,T_a] : c_Relation_Orel__comp(T_a,T_b,T_b,V_R_2,c_Relation_OId(T_b)) = V_R_2 ).
fof(fact_Id__O__R,axiom,
! [V_R_2,T_b,T_a] : c_Relation_Orel__comp(T_a,T_a,T_b,c_Relation_OId(T_a),V_R_2) = V_R_2 ).
fof(fact_finite_Oequations_I1_J,axiom,
! [T_a] : c_Finite__Set_Ofinite(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_Sigma__empty1,axiom,
! [V_B_2,T_b,T_a] : c_Product__Type_OSigma(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ).
fof(fact_Domain__empty__iff,axiom,
! [V_r_2,T_b,T_a] :
( c_Relation_ODomain(T_a,T_b,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
<=> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ) ).
fof(fact_Domain__empty,axiom,
! [T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).
fof(fact_Range__empty__iff,axiom,
! [V_r_2,T_a,T_b] :
( c_Relation_ORange(T_b,T_a,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
<=> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)) ) ).
fof(fact_Range__empty,axiom,
! [T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).
fof(fact_refl__on__empty,axiom,
! [T_a] : c_Relation_Orefl__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).
fof(fact_rel__comp__empty1,axiom,
! [V_R_2,T_b,T_c,T_a] : c_Relation_Orel__comp(T_a,T_c,T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool)),V_R_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ).
fof(fact_rel__comp__empty2,axiom,
! [V_R_2,T_b,T_c,T_a] : c_Relation_Orel__comp(T_a,T_c,T_b,V_R_2,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ).
fof(fact_O__assoc,axiom,
! [V_T_2,V_S_2,V_R_2,T_d,T_b,T_c,T_a] : c_Relation_Orel__comp(T_a,T_c,T_b,c_Relation_Orel__comp(T_a,T_d,T_c,V_R_2,V_S_2),V_T_2) = c_Relation_Orel__comp(T_a,T_d,T_b,V_R_2,c_Relation_Orel__comp(T_d,T_c,T_b,V_S_2,V_T_2)) ).
fof(fact_Id__on__empty,axiom,
! [T_a] : c_Relation_OId__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) ).
fof(fact_converse__rel__comp,axiom,
! [V_s_2,V_r_2,T_c,T_a,T_b] : c_Relation_Oconverse(T_b,T_a,c_Relation_Orel__comp(T_b,T_c,T_a,V_r_2,V_s_2)) = c_Relation_Orel__comp(T_a,T_c,T_b,c_Relation_Oconverse(T_c,T_a,V_s_2),c_Relation_Oconverse(T_b,T_c,V_r_2)) ).
fof(fact_empty__not__insert,axiom,
! [V_A_2,V_a_2,T_a] : c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) != c_Set_Oinsert(T_a,V_a_2,V_A_2) ).
fof(fact_insert__not__empty,axiom,
! [V_A_2,V_a_2,T_a] : c_Set_Oinsert(T_a,V_a_2,V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).
fof(fact_doubleton__eq__iff,axiom,
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( c_Set_Oinsert(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Set_Oinsert(T_a,V_c_2,c_Set_Oinsert(T_a,V_d_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
<=> ( ( V_a_2 = V_c_2
& V_b_2 = V_d_2 )
| ( V_a_2 = V_d_2
& V_b_2 = V_c_2 ) ) ) ).
fof(fact_singleton__inject,axiom,
! [V_b_2,V_a_2,T_a] :
( c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))
=> V_a_2 = V_b_2 ) ).
fof(fact_wf__no__loop,axiom,
! [V_R_2,T_a] :
( c_Relation_Orel__comp(T_a,T_a,T_a,V_R_2,V_R_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
=> c_Wellfounded_Owf(T_a,V_R_2) ) ).
fof(fact_wf__comp__self,axiom,
! [V_R_2,T_a] :
( c_Wellfounded_Owf(T_a,V_R_2)
<=> c_Wellfounded_Owf(T_a,c_Relation_Orel__comp(T_a,T_a,T_a,V_R_2,V_R_2)) ) ).
fof(fact_equals0D,axiom,
! [V_a_2,T_a,V_A_2] :
( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> ~ c_member(T_a,V_a_2,V_A_2) ) ).
fof(fact_empty__iff,axiom,
! [V_c_2,T_a] : ~ c_member(T_a,V_c_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_ex__in__conv,axiom,
! [V_A_2,T_a] :
( ? [B_x] : c_member(T_a,B_x,V_A_2)
<=> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).
fof(fact_all__not__in__conv,axiom,
! [V_A_2,T_a] :
( ! [B_x] : ~ c_member(T_a,B_x,V_A_2)
<=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ).
fof(fact_single__valued__rel__comp,axiom,
! [V_s_2,T_c,V_r_2,T_b,T_a] :
( c_Relation_Osingle__valued(T_a,T_b,V_r_2)
=> ( c_Relation_Osingle__valued(T_b,T_c,V_s_2)
=> c_Relation_Osingle__valued(T_a,T_c,c_Relation_Orel__comp(T_a,T_b,T_c,V_r_2,V_s_2)) ) ) ).
fof(fact_antisym__empty,axiom,
! [T_a] : c_Relation_Oantisym(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) ).
fof(fact_total__on__empty,axiom,
! [V_r_2,T_a] : c_Relation_Ototal__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_r_2) ).
fof(fact_bot__least,axiom,
! [V_x,T_a] :
( class_Orderings_Obot(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Orderings_Obot__class_Obot(T_a),V_x) ) ).
fof(fact_folding__image__simple_Oempty,axiom,
! [V_F_2,V_g_2,V_z_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofolding__image__simple(T_a,T_b,V_f_2,V_z_2,V_g_2,V_F_2)
=> hAPP(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = V_z_2 ) ).
fof(fact_singleton__iff,axiom,
! [V_a_2,V_b_2,T_a] :
( c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
<=> V_b_2 = V_a_2 ) ).
fof(fact_singletonE,axiom,
! [V_a_2,V_b_2,T_a] :
( c_member(T_a,V_b_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
=> V_b_2 = V_a_2 ) ).
fof(fact_trancl__unfold__left,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,V_r_2) = c_Relation_Orel__comp(T_a,T_a,T_a,V_r_2,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ).
fof(fact_trancl__unfold__right,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,V_r_2) = c_Relation_Orel__comp(T_a,T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2),V_r_2) ).
fof(fact_rtrancl__empty,axiom,
! [T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Relation_OId(T_a) ).
fof(fact_smax__emptyI,axiom,
! [V_Y_2] :
( c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_Y_2)
=> ( V_Y_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))
=> c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_Y_2),c_FunDef_Omax__strict) ) ) ).
fof(fact_wmax__emptyI,axiom,
! [V_X_2] :
( c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat),V_X_2)
=> c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),c_FunDef_Omax__weak) ) ).
fof(fact_refl__on__comp__subset,axiom,
! [V_r_2,V_A_2,T_a] :
( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_Orel__comp(T_a,T_a,T_a,c_Relation_Oconverse(T_a,T_a,V_r_2),V_r_2)) ) ).
fof(fact_the__elem__eq,axiom,
! [V_x_2,T_a] : c_Set_Othe__elem(T_a,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_x_2 ).
fof(fact_folding__one_Oinsert,axiom,
! [V_x_2,V_A_2,V_F_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one(T_a,V_f_2,V_F_2)
=> ( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( ~ c_member(T_a,V_x_2,V_A_2)
=> ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> hAPP(V_F_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2)) ) ) ) ) ).
fof(fact_folding__one_Osingleton,axiom,
! [V_x_2,V_F_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one(T_a,V_f_2,V_F_2)
=> hAPP(V_F_2,c_Set_Oinsert(T_a,V_x_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_x_2 ) ).
fof(fact_min__weak__def,axiom,
c_FunDef_Omin__weak = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),c_Wellfounded_Omin__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__leq),c_Set_Oinsert(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)))) ).
fof(fact_max__weak__def,axiom,
c_FunDef_Omax__weak = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),c_Wellfounded_Omax__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__leq),c_Set_Oinsert(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)))) ).
fof(fact_sup1E,axiom,
! [V_x_2,V_B_2,V_A_2,T_a] :
( hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2))
=> ( ~ hBOOL(hAPP(V_A_2,V_x_2))
=> hBOOL(hAPP(V_B_2,V_x_2)) ) ) ).
fof(fact_sup1CI,axiom,
! [T_a,V_A_2,V_x_2,V_B_2] :
( ( ~ hBOOL(hAPP(V_B_2,V_x_2))
=> hBOOL(hAPP(V_A_2,V_x_2)) )
=> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2)) ) ).
fof(fact_UnE,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
=> ( ~ c_member(T_a,V_c_2,V_A_2)
=> c_member(T_a,V_c_2,V_B_2) ) ) ).
fof(fact_UnCI,axiom,
! [V_A_2,V_B_2,V_c_2,T_a] :
( ( ~ c_member(T_a,V_c_2,V_B_2)
=> c_member(T_a,V_c_2,V_A_2) )
=> c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ).
fof(fact_Un__empty__left,axiom,
! [V_B_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2) = V_B_2 ).
fof(fact_Un__empty__right,axiom,
! [V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = V_A_2 ).
fof(fact_Un__empty,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
<=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
& V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).
fof(fact_finite__UnI,axiom,
! [V_G_2,V_F_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_F_2)
=> ( c_Finite__Set_Ofinite(T_a,V_G_2)
=> c_Finite__Set_Ofinite(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_F_2,V_G_2)) ) ) ).
fof(fact_finite__Un,axiom,
! [V_G_2,V_F_2,T_a] :
( c_Finite__Set_Ofinite(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_F_2,V_G_2))
<=> ( c_Finite__Set_Ofinite(T_a,V_F_2)
& c_Finite__Set_Ofinite(T_a,V_G_2) ) ) ).
fof(fact_insert__is__Un,axiom,
! [V_A_2,V_a_2,T_a] : c_Set_Oinsert(T_a,V_a_2,V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_A_2) ).
fof(fact_in__rtrancl__UnI,axiom,
! [V_S_2,V_R_2,V_x_2,T_a] :
( ( c_member(tc_prod(T_a,T_a),V_x_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2))
| c_member(tc_prod(T_a,T_a),V_x_2,c_Transitive__Closure_Ortrancl(T_a,V_S_2)) )
=> c_member(tc_prod(T_a,T_a),V_x_2,c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2))) ) ).
fof(fact_rtrancl__Un__subset,axiom,
! [V_S_2,V_R_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Transitive__Closure_Ortrancl(T_a,V_S_2)),c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2))) ).
fof(fact_wf__union__merge,axiom,
! [V_S_2,V_R_2,T_a] :
( c_Wellfounded_Owf(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2))
<=> c_Wellfounded_Owf(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_a,T_a,V_R_2,V_R_2),c_Relation_Orel__comp(T_a,T_a,T_a,V_S_2,V_R_2)),V_S_2)) ) ).
fof(fact_trancl__unfold,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,V_r_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_Orel__comp(T_a,T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2),V_r_2)) ).
fof(fact_union__comp__emptyL,axiom,
! [V_B_2,V_C_2,V_A_2,T_a] :
( c_Relation_Orel__comp(T_a,T_a,T_a,V_A_2,V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
=> ( c_Relation_Orel__comp(T_a,T_a,T_a,V_B_2,V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
=> c_Relation_Orel__comp(T_a,T_a,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) ) ) ).
fof(fact_union__comp__emptyR,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] :
( c_Relation_Orel__comp(T_a,T_a,T_a,V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
=> ( c_Relation_Orel__comp(T_a,T_a,T_a,V_A_2,V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
=> c_Relation_Orel__comp(T_a,T_a,T_a,V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_B_2,V_C_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) ) ) ).
fof(fact_rtrancl__reflcl__absorb,axiom,
! [V_R_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Relation_OId(T_a)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ).
fof(fact_rtrancl__reflcl,axiom,
! [V_R_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,c_Relation_OId(T_a))) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ).
fof(fact_antisym__reflcl,axiom,
! [V_r_2,T_a] :
( c_Relation_Oantisym(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a)))
<=> c_Relation_Oantisym(T_a,V_r_2) ) ).
fof(fact_rtrancl__Un__rtrancl,axiom,
! [V_S_2,V_R_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Transitive__Closure_Ortrancl(T_a,V_S_2))) = c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)) ).
fof(fact_converse__Un,axiom,
! [V_s_2,V_r_2,T_a,T_b] : c_Relation_Oconverse(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_r_2,V_s_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),c_Relation_Oconverse(T_b,T_a,V_r_2),c_Relation_Oconverse(T_b,T_a,V_s_2)) ).
fof(fact_Un__absorb,axiom,
! [V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) = V_A_2 ).
fof(fact_Un__commute,axiom,
! [V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) ).
fof(fact_Un__left__absorb,axiom,
! [V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ).
fof(fact_Un__left__commute,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)) ).
fof(fact_Un__assoc,axiom,
! [V_C_2,V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) ).
fof(fact_bex__Un,axiom,
! [V_P_2,V_B_2,V_A_2,T_a] :
( ? [B_x] :
( c_member(T_a,B_x,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
& hBOOL(hAPP(V_P_2,B_x)) )
<=> ( ? [B_x] :
( c_member(T_a,B_x,V_A_2)
& hBOOL(hAPP(V_P_2,B_x)) )
| ? [B_x] :
( c_member(T_a,B_x,V_B_2)
& hBOOL(hAPP(V_P_2,B_x)) ) ) ) ).
fof(fact_ball__Un,axiom,
! [V_P_2,V_B_2,V_A_2,T_a] :
( ! [B_x] :
( c_member(T_a,B_x,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
=> hBOOL(hAPP(V_P_2,B_x)) )
<=> ( ! [B_x] :
( c_member(T_a,B_x,V_A_2)
=> hBOOL(hAPP(V_P_2,B_x)) )
& ! [B_x] :
( c_member(T_a,B_x,V_B_2)
=> hBOOL(hAPP(V_P_2,B_x)) ) ) ) ).
fof(fact_sup1I1,axiom,
! [V_B_2,T_a,V_x_2,V_A_2] :
( hBOOL(hAPP(V_A_2,V_x_2))
=> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2)) ) ).
fof(fact_sup1I2,axiom,
! [V_A_2,T_a,V_x_2,V_B_2] :
( hBOOL(hAPP(V_B_2,V_x_2))
=> hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2)) ) ).
fof(fact_Range__Un__eq,axiom,
! [V_B_2,V_A_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ORange(T_b,T_a,V_A_2),c_Relation_ORange(T_b,T_a,V_B_2)) ).
fof(fact_Domain__Un__eq,axiom,
! [V_B_2,V_A_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_A_2),c_Relation_ODomain(T_a,T_b,V_B_2)) ).
fof(fact_refl__on__Un,axiom,
! [V_s_2,V_B_2,V_r_2,V_A_2,T_a] :
( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
=> ( c_Relation_Orefl__on(T_a,V_B_2,V_s_2)
=> c_Relation_Orefl__on(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)) ) ) ).
fof(fact_Un__insert__right,axiom,
! [V_B_2,V_a_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) = c_Set_Oinsert(T_a,V_a_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ).
fof(fact_Un__insert__left,axiom,
! [V_C_2,V_B_2,V_a_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_B_2),V_C_2) = c_Set_Oinsert(T_a,V_a_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) ).
fof(fact_UnI2,axiom,
! [V_A_2,V_B_2,V_c_2,T_a] :
( c_member(T_a,V_c_2,V_B_2)
=> c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ).
fof(fact_UnI1,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( c_member(T_a,V_c_2,V_A_2)
=> c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ).
fof(fact_Un__iff,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
<=> ( c_member(T_a,V_c_2,V_A_2)
| c_member(T_a,V_c_2,V_B_2) ) ) ).
fof(fact_rel__comp__distrib2,axiom,
! [V_R_2,V_T_2,V_S_2,T_b,T_c,T_a] : c_Relation_Orel__comp(T_a,T_c,T_b,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_c),tc_HOL_Obool),V_S_2,V_T_2),V_R_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_c,T_b,V_S_2,V_R_2),c_Relation_Orel__comp(T_a,T_c,T_b,V_T_2,V_R_2)) ).
fof(fact_rel__comp__distrib,axiom,
! [V_T_2,V_S_2,V_R_2,T_b,T_c,T_a] : c_Relation_Orel__comp(T_a,T_c,T_b,V_R_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),V_S_2,V_T_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_c,T_b,V_R_2,V_S_2),c_Relation_Orel__comp(T_a,T_c,T_b,V_R_2,V_T_2)) ).
fof(fact_Un__mono,axiom,
! [V_D_2,V_B_2,V_C_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_D_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2)) ) ) ).
fof(fact_Un__least,axiom,
! [V_B_2,V_C_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) ) ) ).
fof(fact_Un__absorb2,axiom,
! [V_A_2,V_B_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)
=> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_A_2 ) ).
fof(fact_Un__absorb1,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_B_2 ) ).
fof(fact_subset__Un__eq,axiom,
! [V_B_2,V_A_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
<=> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_B_2 ) ).
fof(fact_Un__upper2,axiom,
! [V_A_2,V_B_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ).
fof(fact_Un__upper1,axiom,
! [V_B_2,V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ).
fof(fact_folding__image__simple__idem_Ounion__idem,axiom,
! [V_B_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofolding__image__simple__idem(T_a,T_b,V_f_2,V_z_2,V_g_2,V_F_2)
=> ( c_Finite__Set_Ofinite(T_b,V_A_2)
=> ( c_Finite__Set_Ofinite(T_b,V_B_2)
=> hAPP(V_F_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) ) ) ) ).
fof(fact_wf__union__compatible,axiom,
! [V_S_2,V_R_2,T_a] :
( c_Wellfounded_Owf(T_a,V_R_2)
=> ( c_Wellfounded_Owf(T_a,V_S_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_a,T_a,V_R_2,V_S_2),V_R_2)
=> c_Wellfounded_Owf(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)) ) ) ) ).
fof(fact_trancl__reflcl,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) ).
fof(fact_reflcl__trancl,axiom,
! [V_r_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Otrancl(T_a,V_r_2),c_Relation_OId(T_a)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) ).
fof(fact_rtrancl__unfold,axiom,
! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,V_r_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_OId(T_a),c_Relation_Orel__comp(T_a,T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2),V_r_2)) ).
fof(fact_max__ext__additive,axiom,
! [V_D_2,V_C_2,V_R_2,V_B_2,V_A_2,T_a] :
( c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Wellfounded_Omax__ext(T_a,V_R_2))
=> ( c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2),c_Wellfounded_Omax__ext(T_a,V_R_2))
=> c_member(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_D_2)),c_Wellfounded_Omax__ext(T_a,V_R_2)) ) ) ).
fof(fact_max__ext__compat,axiom,
! [V_S_2,V_R_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_a,T_a,V_R_2,V_S_2),V_R_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool),c_Relation_Orel__comp(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Omax__ext(T_a,V_R_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool),c_Wellfounded_Omax__ext(T_a,V_S_2),c_Set_Oinsert(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool))))),c_Wellfounded_Omax__ext(T_a,V_R_2)) ) ).
fof(fact_min__ext__compat,axiom,
! [V_S_2,V_R_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Orel__comp(T_a,T_a,T_a,V_R_2,V_S_2),V_R_2)
=> c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool),c_Relation_Orel__comp(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Omin__ext(T_a,V_R_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool),c_Wellfounded_Omin__ext(T_a,V_S_2),c_Set_Oinsert(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),c_Product__Type_OPair(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_a,tc_HOL_Obool),tc_fun(T_a,tc_HOL_Obool)),tc_HOL_Obool))))),c_Wellfounded_Omin__ext(T_a,V_R_2)) ) ).
fof(fact_sup__apply,axiom,
! [V_x_2,V_g_2,V_f_2,T_b,T_a] :
( class_Lattices_Olattice(T_a)
=> hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,T_a),V_f_2,V_g_2),V_x_2) = c_Lattices_Osemilattice__sup__class_Osup(T_a,hAPP(V_f_2,V_x_2),hAPP(V_g_2,V_x_2)) ) ).
fof(fact_sup__eq__bot__iff,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Lattices_Obounded__lattice__bot(T_a)
=> ( c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x_2,V_y_2) = c_Orderings_Obot__class_Obot(T_a)
<=> ( V_x_2 = c_Orderings_Obot__class_Obot(T_a)
& V_y_2 = c_Orderings_Obot__class_Obot(T_a) ) ) ) ).
fof(fact_pair__leq__def,axiom,
c_FunDef_Opair__leq = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),tc_HOL_Obool),c_FunDef_Opair__less,c_Relation_OId(tc_prod(tc_Nat_Onat,tc_Nat_Onat))) ).
fof(fact_sup_Oidem,axiom,
! [V_a,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_a) = V_a ) ).
fof(fact_sup__idem,axiom,
! [V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_x) = V_x ) ).
fof(fact_sup_Ocommute,axiom,
! [V_b,V_a,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_a) ) ).
fof(fact_inf__sup__aci_I5_J,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Olattice(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_x) ) ).
fof(fact_sup__commute,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_x) ) ).
fof(fact_sup_Oleft__idem,axiom,
! [V_b,V_a,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b) ) ).
fof(fact_inf__sup__aci_I8_J,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Olattice(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) ) ).
fof(fact_sup__left__idem,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) ) ).
fof(fact_sup_Oleft__commute,axiom,
! [V_c,V_a,V_b,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_c)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_c)) ) ).
fof(fact_inf__sup__aci_I7_J,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Lattices_Olattice(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z)) ) ).
fof(fact_sup__left__commute,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z)) ) ).
fof(fact_sup_Oassoc,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_c) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,V_c)) ) ).
fof(fact_inf__sup__aci_I6_J,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Lattices_Olattice(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) ) ).
fof(fact_sup__assoc,axiom,
! [V_z,V_y,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) ) ).
fof(fact_le__supE,axiom,
! [V_x,V_b,V_a,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_x)
=> ~ ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x)
=> ~ c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x) ) ) ) ).
fof(fact_sup__mono,axiom,
! [V_d,V_b,V_c,V_a,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_d)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),c_Lattices_Osemilattice__sup__class_Osup(T_a,V_c,V_d)) ) ) ) ).
fof(fact_sup__least,axiom,
! [V_z,V_x,V_y,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z),V_x) ) ) ) ).
fof(fact_le__supI,axiom,
! [V_b,V_x,V_a,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_x) ) ) ) ).
fof(fact_sup__absorb1,axiom,
! [V_x,V_y,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = V_x ) ) ).
fof(fact_sup__absorb2,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = V_y ) ) ).
fof(fact_le__supI2,axiom,
! [V_a,V_b,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) ) ) ).
fof(fact_le__supI1,axiom,
! [V_b,V_a,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) ) ) ).
fof(fact_le__sup__iff,axiom,
! [V_z_2,V_y_2,V_x_2,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x_2,V_y_2),V_z_2)
<=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_z_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_z_2) ) ) ) ).
fof(fact_le__iff__sup,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_x_2,V_y_2)
<=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x_2,V_y_2) = V_y_2 ) ) ).
fof(fact_sup__ge2,axiom,
! [V_x,V_y,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) ) ).
fof(fact_inf__sup__ord_I4_J,axiom,
! [V_x,V_y,T_a] :
( class_Lattices_Olattice(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_y,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) ) ).
fof(fact_sup__ge1,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) ) ).
fof(fact_inf__sup__ord_I3_J,axiom,
! [V_y,V_x,T_a] :
( class_Lattices_Olattice(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) ) ).
fof(fact_less__supI1,axiom,
! [V_b,V_a,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_a)
=> c_Orderings_Oord__class_Oless(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) ) ) ).
fof(fact_less__supI2,axiom,
! [V_a,V_b,V_x,T_a] :
( class_Lattices_Osemilattice__sup(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_x,V_b)
=> c_Orderings_Oord__class_Oless(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) ) ) ).
fof(fact_sup__bot__left,axiom,
! [V_x,T_a] :
( class_Lattices_Obounded__lattice__bot(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Orderings_Obot__class_Obot(T_a),V_x) = V_x ) ).
fof(fact_sup__bot__right,axiom,
! [V_x,T_a] :
( class_Lattices_Obounded__lattice__bot(T_a)
=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Orderings_Obot__class_Obot(T_a)) = V_x ) ).
fof(fact_folding__one_Oclosed,axiom,
! [V_A_2,V_F_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one(T_a,V_f_2,V_F_2)
=> ( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> ( ! [B_x,B_y] : c_member(T_a,hAPP(hAPP(V_f_2,B_x),B_y),c_Set_Oinsert(T_a,B_x,c_Set_Oinsert(T_a,B_y,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
=> c_member(T_a,hAPP(V_F_2,V_A_2),V_A_2) ) ) ) ) ).
fof(fact_irrefl__tranclI,axiom,
! [V_x_2,V_r_2,T_a] :
( c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Oconverse(T_a,T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
=> ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_x_2,V_x_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ).
fof(fact_inf1I,axiom,
! [T_a,V_B_2,V_x_2,V_A_2] :
( hBOOL(hAPP(V_A_2,V_x_2))
=> ( hBOOL(hAPP(V_B_2,V_x_2))
=> hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2)) ) ) ).
fof(fact_inf1E,axiom,
! [V_x_2,V_B_2,V_A_2,T_a] :
( hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_x_2))
=> ~ ( hBOOL(hAPP(V_A_2,V_x_2))
=> ~ hBOOL(hAPP(V_B_2,V_x_2)) ) ) ).
fof(fact_IntI,axiom,
! [V_B_2,V_A_2,V_c_2,T_a] :
( c_member(T_a,V_c_2,V_A_2)
=> ( c_member(T_a,V_c_2,V_B_2)
=> c_member(T_a,V_c_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ) ).
%----Arity declarations (26)
fof(arity_HOL__Obool__Lattices_Obounded__lattice,axiom,
class_Lattices_Obounded__lattice(tc_HOL_Obool) ).
fof(arity_fun__Lattices_Obounded__lattice,axiom,
! [T_2,T_1] :
( class_Lattices_Obounded__lattice(T_1)
=> class_Lattices_Obounded__lattice(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Lattices_Obounded__lattice__bot,axiom,
! [T_2,T_1] :
( class_Lattices_Obounded__lattice(T_1)
=> class_Lattices_Obounded__lattice__bot(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Lattices_Osemilattice__sup,axiom,
! [T_2,T_1] :
( class_Lattices_Olattice(T_1)
=> class_Lattices_Osemilattice__sup(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Orderings_Opreorder,axiom,
! [T_2,T_1] :
( class_Orderings_Opreorder(T_1)
=> class_Orderings_Opreorder(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Finite__Set_Ofinite,axiom,
! [T_1,T_2] :
( ( class_Finite__Set_Ofinite(T_2)
& class_Finite__Set_Ofinite(T_1) )
=> class_Finite__Set_Ofinite(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Lattices_Olattice,axiom,
! [T_2,T_1] :
( class_Lattices_Olattice(T_1)
=> class_Lattices_Olattice(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Orderings_Oorder,axiom,
! [T_2,T_1] :
( class_Orderings_Oorder(T_1)
=> class_Orderings_Oorder(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Orderings_Oord,axiom,
! [T_2,T_1] :
( class_Orderings_Oord(T_1)
=> class_Orderings_Oord(tc_fun(T_2,T_1)) ) ).
fof(arity_fun__Orderings_Obot,axiom,
! [T_2,T_1] :
( class_Orderings_Obot(T_1)
=> class_Orderings_Obot(tc_fun(T_2,T_1)) ) ).
fof(arity_Nat__Onat__Lattices_Osemilattice__sup,axiom,
class_Lattices_Osemilattice__sup(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Orderings_Opreorder,axiom,
class_Orderings_Opreorder(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Orderings_Olinorder,axiom,
class_Orderings_Olinorder(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Lattices_Olattice,axiom,
class_Lattices_Olattice(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Orderings_Oorder,axiom,
class_Orderings_Oorder(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Orderings_Oord,axiom,
class_Orderings_Oord(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Orderings_Obot,axiom,
class_Orderings_Obot(tc_Nat_Onat) ).
fof(arity_HOL__Obool__Lattices_Obounded__lattice__bot,axiom,
class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Lattices_Osemilattice__sup,axiom,
class_Lattices_Osemilattice__sup(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Orderings_Opreorder,axiom,
class_Orderings_Opreorder(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Finite__Set_Ofinite,axiom,
class_Finite__Set_Ofinite(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Lattices_Olattice,axiom,
class_Lattices_Olattice(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Orderings_Oorder,axiom,
class_Orderings_Oorder(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Orderings_Oord,axiom,
class_Orderings_Oord(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Orderings_Obot,axiom,
class_Orderings_Obot(tc_HOL_Obool) ).
fof(arity_prod__Finite__Set_Ofinite,axiom,
! [T_1,T_2] :
( ( class_Finite__Set_Ofinite(T_2)
& class_Finite__Set_Ofinite(T_1) )
=> class_Finite__Set_Ofinite(tc_prod(T_2,T_1)) ) ).
%----Helper facts (2)
fof(help_c__fequal__1,axiom,
! [V_y_2,V_x_2] :
( ~ hBOOL(hAPP(c_fequal(V_x_2),V_y_2))
| V_x_2 = V_y_2 ) ).
fof(help_c__fequal__2,axiom,
! [V_y_2,V_x_2] :
( V_x_2 != V_y_2
| hBOOL(hAPP(c_fequal(V_x_2),V_y_2)) ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Relation_Oconverse(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_L),c_Arrow__Order__Mirabelle_OLin)
<=> c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),v_L,c_Arrow__Order__Mirabelle_OLin) ) ).
%------------------------------------------------------------------------------