TPTP Problem File: SCT159+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SCT159+1 : TPTP v8.2.0. Released v5.2.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem 433377, 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_433377.500.p [Bla11]
% Status : ContradictoryAxioms
% Rating : 0.50 v8.2.0, 0.47 v8.1.0, 0.53 v7.5.0, 0.56 v7.4.0, 0.50 v7.3.0, 0.52 v7.2.0, 0.48 v7.1.0, 0.35 v7.0.0, 0.37 v6.4.0, 0.38 v6.2.0, 0.48 v6.1.0, 0.60 v6.0.0, 0.57 v5.5.0, 0.63 v5.4.0, 0.68 v5.3.0, 0.78 v5.2.0
% Syntax : Number of formulae : 533 ( 113 unt; 0 def)
% Number of atoms : 1401 ( 356 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 1002 ( 134 ~; 30 |; 63 &)
% ( 130 <=>; 645 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 39 ( 38 usr; 1 prp; 0-6 aty)
% Number of functors : 42 ( 42 usr; 8 con; 0-5 aty)
% Number of variables : 1533 (1509 !; 24 ?)
% SPC : FOF_CAX_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-03-01 11:24:27
% : Renamed from SWW161+1
%------------------------------------------------------------------------------
%----Relevant facts (491)
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_card__image,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Finite__Set_Ocard(T_a,V_A_2) ) ).
fof(fact_inj__image__eq__iff,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_Set_Oimage(T_a,T_b,V_f_2,V_A_2) = c_Set_Oimage(T_a,T_b,V_f_2,V_B_2)
<=> V_A_2 = V_B_2 ) ) ).
fof(fact_top1I,axiom,
! [V_x_2,T_a] : hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_x_2)) ).
fof(fact_inj__eq,axiom,
! [V_y_2,V_x_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ).
fof(fact_injD,axiom,
! [V_y_2,V_x_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)
=> V_x_2 = V_y_2 ) ) ).
fof(fact_top__apply,axiom,
! [V_x_2,T_b,T_a] :
( class_Orderings_Otop(T_a)
=> hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_b,T_a)),V_x_2) = c_Orderings_Otop__class_Otop(T_a) ) ).
fof(fact_inj__on__def,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
<=> ! [B_x] :
( c_member(T_a,B_x,V_A_2)
=> ! [B_xa] :
( c_member(T_a,B_xa,V_A_2)
=> ( hAPP(V_f_2,B_x) = hAPP(V_f_2,B_xa)
=> B_x = B_xa ) ) ) ) ).
fof(fact_zero__reorient,axiom,
! [V_x_2,T_a] :
( class_Groups_Ozero(T_a)
=> ( c_Groups_Ozero__class_Ozero(T_a) = V_x_2
<=> V_x_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ).
fof(fact_assms_I3_J,axiom,
c_Arrow__Order__Mirabelle_OIIA(v_F) ).
fof(fact_u,axiom,
c_Arrow__Order__Mirabelle_Ounanimity(v_F) ).
fof(fact_finite__indi,axiom,
c_Finite__Set_Ofinite(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) ).
fof(fact_finite__imageI,axiom,
! [V_h_2,T_b,V_Fa_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_Fa_2)
=> c_Finite__Set_Ofinite(T_b,c_Set_Oimage(T_a,T_b,V_h_2,V_Fa_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_finite__option__UNIV,axiom,
! [T_a] :
( c_Finite__Set_Ofinite(tc_Option_Ooption(T_a),c_Orderings_Otop__class_Otop(tc_fun(tc_Option_Ooption(T_a),tc_HOL_Obool)))
<=> c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ) ).
fof(fact_finite__Plus__UNIV__iff,axiom,
! [T_b,T_a] :
( c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_a,T_b),tc_HOL_Obool)))
<=> ( c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
& c_Finite__Set_Ofinite(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) ) ) ).
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__fun__UNIVD2,axiom,
! [T_b,T_a] :
( c_Finite__Set_Ofinite(tc_fun(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool)))
=> c_Finite__Set_Ofinite(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) ) ).
fof(fact_finite__Prod__UNIV,axiom,
! [T_b,T_a] :
( c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_Finite__Set_Ofinite(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))
=> c_Finite__Set_Ofinite(tc_prod(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool))) ) ) ).
fof(fact_finite__UNIV,axiom,
! [T_a] :
( class_Finite__Set_Ofinite(T_a)
=> c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ) ).
fof(fact_card__infinite,axiom,
! [V_A_2,T_a] :
( ~ c_Finite__Set_Ofinite(T_a,V_A_2)
=> c_Finite__Set_Ocard(T_a,V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ).
fof(fact_finite__imageD,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Finite__Set_Ofinite(T_a,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2))
=> ( c_Fun_Oinj__on(T_b,T_a,V_f_2,V_A_2)
=> c_Finite__Set_Ofinite(T_b,V_A_2) ) ) ).
fof(fact_card__eq__UNIV__imp__eq__UNIV,axiom,
! [V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_Finite__Set_Ocard(T_a,V_A_2) = c_Finite__Set_Ocard(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> V_A_2 = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) ) ) ).
fof(fact_finite__UNIV__surj__inj,axiom,
! [V_f_2,T_a] :
( c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_Set_Oimage(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))
=> c_Fun_Oinj__on(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ) ) ).
fof(fact_finite__UNIV__inj__surj,axiom,
! [V_f_2,T_a] :
( c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_Fun_Oinj__on(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> c_Set_Oimage(T_a,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) ) ) ).
fof(fact_eq__card__imp__inj__on,axiom,
! [V_f_2,T_b,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Finite__Set_Ocard(T_a,V_A_2)
=> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) ) ) ).
fof(fact_inj__on__iff__eq__card,axiom,
! [V_f_2,T_b,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
<=> c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Finite__Set_Ocard(T_a,V_A_2) ) ) ).
fof(fact_card__bij__eq,axiom,
! [V_g_2,V_B_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),V_B_2)
=> ( c_Fun_Oinj__on(T_b,T_a,V_g_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oimage(T_b,T_a,V_g_2,V_B_2),V_A_2)
=> ( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Finite__Set_Ofinite(T_b,V_B_2)
=> c_Finite__Set_Ocard(T_a,V_A_2) = c_Finite__Set_Ocard(T_b,V_B_2) ) ) ) ) ) ) ).
fof(fact_finite__UNIV__card__ge__0,axiom,
! [T_a] :
( c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))) ) ).
fof(fact_the__inv__into__onto,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> c_Set_Oimage(T_b,T_a,c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = V_A_2 ) ).
fof(fact_inj__on__the__inv__into,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> c_Fun_Oinj__on(T_b,T_a,c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) ) ).
fof(fact_the__inv__f__f,axiom,
! [V_x_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> hAPP(c_Fun_Othe__inv__into(T_a,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2 ) ).
fof(fact_card__eq__0__iff,axiom,
! [V_A_2,T_a] :
( c_Finite__Set_Ocard(T_a,V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
| ~ c_Finite__Set_Ofinite(T_a,V_A_2) ) ) ).
fof(fact_pigeonhole,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)),c_Finite__Set_Ocard(T_b,V_A_2))
=> ~ c_Fun_Oinj__on(T_b,T_a,V_f_2,V_A_2) ) ).
fof(fact_inj__image__subset__iff,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_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_inj__image__mem__iff,axiom,
! [V_A_2,V_a_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_member(T_b,hAPP(V_f_2,V_a_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))
<=> c_member(T_a,V_a_2,V_A_2) ) ) ).
fof(fact_range__ex1__eq,axiom,
! [V_b_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)))
=> ( c_member(T_b,V_b_2,c_Set_Oimage(T_a,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))))
<=> ? [B_x] :
( V_b_2 = hAPP(V_f_2,B_x)
& ! [B_y] :
( V_b_2 = hAPP(V_f_2,B_y)
=> B_y = B_x ) ) ) ) ).
fof(fact_card__image__le,axiom,
! [V_f_2,T_b,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_b,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)),c_Finite__Set_Ocard(T_a,V_A_2)) ) ).
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_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_inj__on__empty,axiom,
! [V_f_2,T_b,T_a] : c_Fun_Oinj__on(T_a,T_b,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ).
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_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_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_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_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_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_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__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__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__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_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__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__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_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_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_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_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_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__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_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_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_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_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_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_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_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_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_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_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_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__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__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_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_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_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_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_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__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__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_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_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_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_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__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_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_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_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_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_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_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_the__inv__into__f__eq,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hAPP(V_f_2,V_x_2) = V_y_2
=> ( c_member(T_a,V_x_2,V_A_2)
=> hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_y_2) = V_x_2 ) ) ) ).
fof(fact_the__inv__into__f__f,axiom,
! [V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_member(T_a,V_x_2,V_A_2)
=> hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2 ) ) ).
fof(fact_the__inv__into__into,axiom,
! [V_B_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_member(T_b,V_x_2,c_Set_Oimage(T_a,T_b,V_f_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,hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_x_2),V_B_2) ) ) ) ).
fof(fact_card__seteq,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_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_B_2),c_Finite__Set_Ocard(T_a,V_A_2))
=> V_A_2 = V_B_2 ) ) ) ).
fof(fact_card__mono,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_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,V_B_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_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_top__greatest,axiom,
! [V_x,T_a] :
( class_Orderings_Otop(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Orderings_Otop__class_Otop(T_a)) ) ).
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_inj__on__contraD,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_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)
=> hAPP(V_f_2,V_x_2) != hAPP(V_f_2,V_y_2) ) ) ) ) ).
fof(fact_inj__on__iff,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_member(T_a,V_x_2,V_A_2)
=> ( c_member(T_a,V_y_2,V_A_2)
=> ( hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)
<=> V_x_2 = V_y_2 ) ) ) ) ).
fof(fact_inj__onD,axiom,
! [V_y_2,V_x_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)
=> ( c_member(T_a,V_x_2,V_A_2)
=> ( c_member(T_a,V_y_2,V_A_2)
=> V_x_2 = V_y_2 ) ) ) ) ).
fof(fact_subset__inj__on,axiom,
! [V_A_2,V_B_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_B_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2) ) ) ).
fof(fact_f__the__inv__into__f,axiom,
! [V_y_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_member(T_b,V_y_2,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))
=> hAPP(V_f_2,hAPP(c_Fun_Othe__inv__into(T_a,T_b,V_A_2,V_f_2),V_y_2)) = V_y_2 ) ) ).
fof(fact_card__gt__0__iff,axiom,
! [V_A_2,T_a] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,V_A_2))
<=> ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
& c_Finite__Set_Ofinite(T_a,V_A_2) ) ) ).
fof(fact_card__inj__on__le,axiom,
! [V_B_2,V_A_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),V_B_2)
=> ( c_Finite__Set_Ofinite(T_b,V_B_2)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2)) ) ) ) ).
fof(fact_finite__surj,axiom,
! [V_f_2,V_B_2,T_b,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),V_B_2,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2))
=> c_Finite__Set_Ofinite(T_b,V_B_2) ) ) ).
fof(fact_card_Oempty,axiom,
! [T_a] : c_Finite__Set_Ocard(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_card__ge__0__finite,axiom,
! [V_A_2,T_a] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,V_A_2))
=> c_Finite__Set_Ofinite(T_a,V_A_2) ) ).
fof(fact_finite__surj__inj,axiom,
! [V_f_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oimage(T_a,T_a,V_f_2,V_A_2))
=> c_Fun_Oinj__on(T_a,T_a,V_f_2,V_A_2) ) ) ).
fof(fact_endo__inj__surj,axiom,
! [V_f_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oimage(T_a,T_a,V_f_2,V_A_2),V_A_2)
=> ( c_Fun_Oinj__on(T_a,T_a,V_f_2,V_A_2)
=> c_Set_Oimage(T_a,T_a,V_f_2,V_A_2) = V_A_2 ) ) ) ).
fof(fact_inj__on__iff__card__le,axiom,
! [V_B_2,T_b,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Finite__Set_Ofinite(T_b,V_B_2)
=> ( ? [B_f] :
( c_Fun_Oinj__on(T_a,T_b,B_f,V_A_2)
& c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,B_f,V_A_2),V_B_2) )
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2)) ) ) ) ).
fof(fact_le0,axiom,
! [V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) ).
fof(fact_less__zeroE,axiom,
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ).
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_image__eqI,axiom,
! [T_a,V_A_2,T_b,V_x_2,V_f_2,V_b_2] :
( V_b_2 = hAPP(V_f_2,V_x_2)
=> ( c_member(T_b,V_x_2,V_A_2)
=> c_member(T_a,V_b_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)) ) ) ).
fof(fact_UNIV__I,axiom,
! [V_x_2,T_a] : c_member(T_a,V_x_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ).
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_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_inj__on__iff__surj,axiom,
! [V_A_H_2,T_b,T_a,V_A_2] :
( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> ( ? [B_f] :
( c_Fun_Oinj__on(T_a,T_b,B_f,V_A_2)
& c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,B_f,V_A_2),V_A_H_2) )
<=> ? [B_g] : c_Set_Oimage(T_b,T_a,B_g,V_A_H_2) = V_A_2 ) ) ).
fof(fact_ex__nat__less__eq,axiom,
! [V_P_2,V_n_2] :
( ? [B_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_m,V_n_2)
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ? [B_x] :
( c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))
& hBOOL(hAPP(V_P_2,B_x)) ) ) ).
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_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_finite__atLeastLessThan,axiom,
! [V_u_2,V_l_2] : c_Finite__Set_Ofinite(tc_Nat_Onat,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2,V_u_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_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__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_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__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_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_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_bot__nat__def,axiom,
c_Orderings_Obot__class_Obot(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
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_inj__on__strict__subset,axiom,
! [V_A_2,V_B_2,V_f_2,T_b,T_a] :
( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_B_2)
=> ( 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_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_2,V_B_2)) ) ) ).
fof(fact_psubset__card__mono,axiom,
! [V_A_2,V_B_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_B_2)
=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,V_B_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_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_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_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__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_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_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_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_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_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_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_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_card__psubset,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_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,V_B_2))
=> c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ) ).
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_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_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_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_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_rev__image__eqI,axiom,
! [T_b,V_f_2,V_b_2,V_A_2,V_x_2,T_a] :
( c_member(T_a,V_x_2,V_A_2)
=> ( V_b_2 = hAPP(V_f_2,V_x_2)
=> c_member(T_b,V_b_2,c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) ) ) ).
fof(fact_imageI,axiom,
! [V_f_2,T_b,V_A_2,V_x_2,T_a] :
( c_member(T_a,V_x_2,V_A_2)
=> c_member(T_b,hAPP(V_f_2,V_x_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) ) ).
fof(fact_image__iff,axiom,
! [V_A_2,V_f_2,T_b,V_z_2,T_a] :
( c_member(T_a,V_z_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2))
<=> ? [B_x] :
( c_member(T_b,B_x,V_A_2)
& V_z_2 = hAPP(V_f_2,B_x) ) ) ).
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_UNIV__not__empty,axiom,
! [T_a] : c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).
fof(fact_image__is__empty,axiom,
! [V_A_2,V_f_2,T_a,T_b] :
( c_Set_Oimage(T_b,T_a,V_f_2,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_b,tc_HOL_Obool)) ) ).
fof(fact_image__empty,axiom,
! [V_f_2,T_a,T_b] : c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ).
fof(fact_empty__is__image,axiom,
! [V_A_2,V_f_2,T_b,T_a] :
( c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_Set_Oimage(T_b,T_a,V_f_2,V_A_2)
<=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ).
fof(fact_gr0I,axiom,
! [V_n] :
( V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) ) ).
fof(fact_gr__implies__not0,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ).
fof(fact_less__nat__zero__code,axiom,
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ).
fof(fact_neq0__conv,axiom,
! [V_n_2] :
( V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) ) ).
fof(fact_not__less0,axiom,
! [V_n] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ).
fof(fact_subset__UNIV,axiom,
! [V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_image__mono,axiom,
! [V_f_2,T_b,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_b,tc_HOL_Obool),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2),c_Set_Oimage(T_a,T_b,V_f_2,V_B_2)) ) ).
fof(fact_subset__image__iff,axiom,
! [V_A_2,V_f_2,T_b,V_B_2,T_a] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Set_Oimage(T_b,T_a,V_f_2,V_A_2))
<=> ? [B_AA] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),B_AA,V_A_2)
& V_B_2 = c_Set_Oimage(T_b,T_a,V_f_2,B_AA) ) ) ).
fof(fact_le__0__eq,axiom,
! [V_n_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
<=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ).
fof(fact_less__eq__nat_Osimps_I1_J,axiom,
! [V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) ).
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_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_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__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__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_atLeastLessThan__inj_I2_J,axiom,
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2)
=> V_b_2 = V_d_2 ) ) ) ) ).
fof(fact_atLeastLessThan__inj_I1_J,axiom,
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2)
=> V_a_2 = V_c_2 ) ) ) ) ).
fof(fact_atLeastLessThan__eq__iff,axiom,
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c_2,V_d_2)
=> ( c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2)
<=> ( V_a_2 = V_c_2
& V_b_2 = V_d_2 ) ) ) ) ) ).
fof(fact_atLeastLessThan0,axiom,
! [V_m_2] : c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) ).
fof(fact_range__eqI,axiom,
! [T_b,T_a,V_x_2,V_f_2,V_b_2] :
( V_b_2 = hAPP(V_f_2,V_x_2)
=> c_member(T_a,V_b_2,c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) ) ).
fof(fact_rangeI,axiom,
! [T_b,V_x_2,V_f_2,T_a] : c_member(T_a,hAPP(V_f_2,V_x_2),c_Set_Oimage(T_b,T_a,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) ).
fof(fact_atLeastLessThan__empty,axiom,
! [V_a_2,V_b_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_a_2)
=> c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).
fof(fact_atLeastLessThan__empty__iff2,axiom,
! [V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2)
<=> ~ c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_atLeastLessThan__empty__iff,axiom,
! [V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
<=> ~ c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_atLeastLessThan__subset__iff,axiom,
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2),c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_a_2)
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_a_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_d_2) ) ) ) ) ).
fof(fact_all__nat__less__eq,axiom,
! [V_P_2,V_n_2] :
( ! [B_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_m,V_n_2)
=> hBOOL(hAPP(V_P_2,B_m)) )
<=> ! [B_x] :
( c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))
=> hBOOL(hAPP(V_P_2,B_x)) ) ) ).
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_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_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_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_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_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_infinite__UNIV__nat,axiom,
~ c_Finite__Set_Ofinite(tc_Nat_Onat,c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) ).
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_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_iso__tuple__UNIV__I,axiom,
! [V_x_2,T_a] : c_member(T_a,V_x_2,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ).
fof(fact_finite__fun__UNIVD1,axiom,
! [T_b,T_a] :
( c_Finite__Set_Ofinite(tc_fun(T_a,T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_a,T_b),tc_HOL_Obool)))
=> ( c_Finite__Set_Ocard(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) != hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
=> c_Finite__Set_Ofinite(T_a,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool))) ) ) ).
fof(fact_folding__one__idem_Osubset__idem,axiom,
! [V_B_2,V_A_2,V_Fa_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2)
=> ( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)
=> hAPP(hAPP(V_f_2,hAPP(V_Fa_2,V_B_2)),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2) ) ) ) ) ).
fof(fact_folding__image__simple__idem_Osubset__idem,axiom,
! [V_B_2,V_A_2,V_Fa_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_Fa_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_Fa_2,V_B_2)),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2) ) ) ) ).
fof(fact_folding__image__simple__idem_Oin__idem,axiom,
! [V_x_2,V_A_2,V_Fa_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_Fa_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_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2) ) ) ) ).
fof(fact_Suc__mono,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),hAPP(c_Nat_OSuc,V_n)) ) ).
fof(fact_lessI,axiom,
! [V_n] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,hAPP(c_Nat_OSuc,V_n)) ).
fof(fact_zero__less__Suc,axiom,
! [V_n] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n)) ).
fof(fact_n__not__Suc__n,axiom,
! [V_n] : V_n != hAPP(c_Nat_OSuc,V_n) ).
fof(fact_Suc__n__not__n,axiom,
! [V_n] : hAPP(c_Nat_OSuc,V_n) != V_n ).
fof(fact_nat_Oinject,axiom,
! [V_nat_H_2,V_nat_2] :
( hAPP(c_Nat_OSuc,V_nat_2) = hAPP(c_Nat_OSuc,V_nat_H_2)
<=> V_nat_2 = V_nat_H_2 ) ).
fof(fact_Suc__inject,axiom,
! [V_y,V_x] :
( hAPP(c_Nat_OSuc,V_x) = hAPP(c_Nat_OSuc,V_y)
=> V_x = V_y ) ).
fof(fact_inj__Suc,axiom,
! [V_N_2] : c_Fun_Oinj__on(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,V_N_2) ).
fof(fact_folding__one__idem_Oidem,axiom,
! [V_x_2,V_Fa_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2)
=> hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2 ) ).
fof(fact_folding__image__simple__idem_Oidem,axiom,
! [V_x_2,V_Fa_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_Fa_2)
=> hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2 ) ).
fof(fact_image__Suc__atLeastLessThan,axiom,
! [V_j_2,V_i_2] : c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2)) ).
fof(fact_Suc__n__not__le__n,axiom,
! [V_n] : ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n),V_n) ).
fof(fact_not__less__eq__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__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2) ) ).
fof(fact_le__Suc__eq,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))
<=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
| V_m_2 = hAPP(c_Nat_OSuc,V_n_2) ) ) ).
fof(fact_Suc__le__mono,axiom,
! [V_m_2,V_n_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),hAPP(c_Nat_OSuc,V_m_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_m_2) ) ).
fof(fact_le__SucI,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_m,hAPP(c_Nat_OSuc,V_n)) ) ).
fof(fact_le__SucE,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n))
=> ( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> V_m = hAPP(c_Nat_OSuc,V_n) ) ) ).
fof(fact_Suc__leD,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_not__less__eq,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(tc_Nat_Onat,V_n_2,hAPP(c_Nat_OSuc,V_m_2)) ) ).
fof(fact_less__Suc__eq,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,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_Suc__less__eq,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),hAPP(c_Nat_OSuc,V_n_2))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) ).
fof(fact_not__less__less__Suc__eq,axiom,
! [V_m_2,V_n_2] :
( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,hAPP(c_Nat_OSuc,V_m_2))
<=> V_n_2 = V_m_2 ) ) ).
fof(fact_less__antisym,axiom,
! [V_m,V_n] :
( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,V_m)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n,hAPP(c_Nat_OSuc,V_m))
=> V_m = V_n ) ) ).
fof(fact_less__SucI,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)) ) ).
fof(fact_Suc__lessI,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> ( hAPP(c_Nat_OSuc,V_m) != V_n
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) ) ) ).
fof(fact_less__trans__Suc,axiom,
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_j,V_k)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i),V_k) ) ) ).
fof(fact_less__SucE,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n))
=> ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n)
=> V_m = V_n ) ) ).
fof(fact_Suc__lessD,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_Suc__less__SucD,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),hAPP(c_Nat_OSuc,V_n))
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_Suc__neq__Zero,axiom,
! [V_m] : hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_Zero__neq__Suc,axiom,
! [V_m] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m) ).
fof(fact_nat_Osimps_I3_J,axiom,
! [V_nat_H_1] : hAPP(c_Nat_OSuc,V_nat_H_1) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_Suc__not__Zero,axiom,
! [V_m] : hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_nat_Osimps_I2_J,axiom,
! [V_nat_H] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H) ).
fof(fact_Zero__not__Suc,axiom,
! [V_m] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m) ).
fof(fact_gr0__conv__Suc,axiom,
! [V_n_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)
<=> ? [B_m] : V_n_2 = hAPP(c_Nat_OSuc,B_m) ) ).
fof(fact_less__Suc0,axiom,
! [V_n_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))
<=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ).
fof(fact_less__Suc__eq__0__disj,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))
<=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
| ? [B_j] :
( V_m_2 = hAPP(c_Nat_OSuc,B_j)
& c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_j,V_n_2) ) ) ) ).
fof(fact_Suc__le__lessD,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_le__less__Suc__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_n_2,hAPP(c_Nat_OSuc,V_m_2))
<=> V_n_2 = V_m_2 ) ) ).
fof(fact_Suc__leI,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,hAPP(c_Nat_OSuc,V_m),V_n) ) ).
fof(fact_le__imp__less__Suc,axiom,
! [V_n,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)) ) ).
fof(fact_Suc__le__eq,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2)
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) ).
fof(fact_less__Suc__eq__le,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) ).
fof(fact_less__eq__Suc__le,axiom,
! [V_m_2,V_n_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2) ) ).
fof(fact_folding__one__idem_Oin__idem,axiom,
! [V_x_2,V_A_2,V_Fa_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2)
=> ( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_member(T_a,V_x_2,V_A_2)
=> hAPP(hAPP(V_f_2,V_x_2),hAPP(V_Fa_2,V_A_2)) = hAPP(V_Fa_2,V_A_2) ) ) ) ).
fof(fact_folding__one__idem_Ohom__commute,axiom,
! [V_N_2,V_h_2,V_Fa_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_2)
=> ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(V_f_2,B_x),B_y)) = hAPP(hAPP(V_f_2,hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
=> ( c_Finite__Set_Ofinite(T_a,V_N_2)
=> ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
=> hAPP(V_h_2,hAPP(V_Fa_2,V_N_2)) = hAPP(V_Fa_2,c_Set_Oimage(T_a,T_a,V_h_2,V_N_2)) ) ) ) ) ).
fof(fact_greaterThan__0,axiom,
c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) ).
fof(fact_finite__PlusD_I2_J,axiom,
! [V_B_2,V_A_2,T_b,T_a] :
( c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2))
=> c_Finite__Set_Ofinite(T_b,V_B_2) ) ).
fof(fact_greaterThan__eq__iff,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_SetInterval_Oord__class_OgreaterThan(T_a,V_x_2) = c_SetInterval_Oord__class_OgreaterThan(T_a,V_y_2)
<=> V_x_2 = V_y_2 ) ) ).
fof(fact_greaterThan__iff,axiom,
! [V_k_2,V_i_2,T_a] :
( class_Orderings_Oord(T_a)
=> ( c_member(T_a,V_i_2,c_SetInterval_Oord__class_OgreaterThan(T_a,V_k_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_k_2,V_i_2) ) ) ).
fof(fact_greaterThan__subset__iff,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Orderings_Olinorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OgreaterThan(T_a,V_x_2),c_SetInterval_Oord__class_OgreaterThan(T_a,V_y_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_y_2,V_x_2) ) ) ).
fof(fact_finite__Plus__iff,axiom,
! [V_B_2,V_A_2,T_b,T_a] :
( c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2))
<=> ( c_Finite__Set_Ofinite(T_a,V_A_2)
& c_Finite__Set_Ofinite(T_b,V_B_2) ) ) ).
fof(fact_finite__Plus,axiom,
! [V_B_2,T_b,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Finite__Set_Ofinite(T_b,V_B_2)
=> c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) ) ) ).
fof(fact_finite__PlusD_I1_J,axiom,
! [V_B_2,V_A_2,T_b,T_a] :
( c_Finite__Set_Ofinite(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2))
=> c_Finite__Set_Ofinite(T_a,V_A_2) ) ).
fof(fact_UNIV__Plus__UNIV,axiom,
! [T_b,T_a] : c_Sum__Type_OPlus(T_a,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_a,T_b),tc_HOL_Obool)) ).
fof(fact_Plus__eq__empty__conv,axiom,
! [V_B_2,V_A_2,T_b,T_a] :
( c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_sum(T_a,T_b),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_b,tc_HOL_Obool)) ) ) ).
fof(fact_card__Plus__conv__if,axiom,
! [V_B_2,T_b,V_A_2,T_a] :
( ( ( c_Finite__Set_Ofinite(T_a,V_A_2)
& c_Finite__Set_Ofinite(T_b,V_B_2) )
=> c_Finite__Set_Ocard(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2)) )
& ( ~ ( c_Finite__Set_Ofinite(T_a,V_A_2)
& c_Finite__Set_Ofinite(T_b,V_B_2) )
=> c_Finite__Set_Ocard(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ).
fof(fact_termination__basic__simps_I1_J,axiom,
! [V_z,V_y,V_x] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_y)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) ) ).
fof(fact_termination__basic__simps_I2_J,axiom,
! [V_y,V_z,V_x] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,V_z)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) ) ).
fof(fact_termination__basic__simps_I4_J,axiom,
! [V_y,V_z,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_z)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) ) ).
fof(fact_termination__basic__simps_I3_J,axiom,
! [V_z,V_y,V_x] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,V_y)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) ) ).
fof(fact_nat__add__commute,axiom,
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m) ).
fof(fact_nat__add__left__commute,axiom,
! [V_z,V_y,V_x] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,V_z)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_y,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_x,V_z)) ).
fof(fact_nat__add__assoc,axiom,
! [V_k,V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n),V_k) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_k)) ).
fof(fact_nat__add__left__cancel,axiom,
! [V_n_2,V_m_2,V_k_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2)
<=> V_m_2 = V_n_2 ) ).
fof(fact_nat__add__right__cancel,axiom,
! [V_n_2,V_k_2,V_m_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_k_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_k_2)
<=> V_m_2 = V_n_2 ) ).
fof(fact_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oab__semigroup__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) ).
fof(fact_add__left__cancel,axiom,
! [V_c_2,V_b_2,V_a_2,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2)
<=> V_b_2 = V_c_2 ) ) ).
fof(fact_add__right__cancel,axiom,
! [V_c_2,V_a_2,V_b_2,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2) = c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2)
<=> V_b_2 = V_c_2 ) ) ).
fof(fact_add__left__imp__eq,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)
=> V_b = V_c ) ) ).
fof(fact_add__imp__eq,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Ocancel__ab__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a,V_b) = c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)
=> V_b = V_c ) ) ).
fof(fact_add__right__imp__eq,axiom,
! [V_c,V_a,V_b,T_a] :
( class_Groups_Ocancel__semigroup__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_b,V_a) = c_Groups_Oplus__class_Oplus(T_a,V_c,V_a)
=> V_b = V_c ) ) ).
fof(fact_subset__card__intvl__is__intvl,axiom,
! [V_k_2,V_A_2] :
( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool),V_A_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,c_Finite__Set_Ocard(tc_Nat_Onat,V_A_2))))
=> V_A_2 = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,c_Finite__Set_Ocard(tc_Nat_Onat,V_A_2))) ) ).
fof(fact_le__add2,axiom,
! [V_m,V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) ).
fof(fact_le__add1,axiom,
! [V_m,V_n] : c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,V_m)) ).
fof(fact_le__iff__add,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
<=> ? [B_k] : V_n_2 = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k) ) ).
fof(fact_nat__add__left__cancel__le,axiom,
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2))
<=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) ).
fof(fact_trans__le__add1,axiom,
! [V_m,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_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)) ) ).
fof(fact_trans__le__add2,axiom,
! [V_m,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_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)) ) ).
fof(fact_add__le__mono1,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,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)) ) ).
fof(fact_add__le__mono,axiom,
! [V_l,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_k,V_l)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l)) ) ) ).
fof(fact_add__leD2,axiom,
! [V_n,V_k,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n) ) ).
fof(fact_add__leD1,axiom,
! [V_n,V_k,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n) ) ).
fof(fact_add__leE,axiom,
! [V_n,V_k,V_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_k),V_n)
=> ~ ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m,V_n)
=> ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_k,V_n) ) ) ).
fof(fact_not__add__less1,axiom,
! [V_j,V_i] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_i) ).
fof(fact_not__add__less2,axiom,
! [V_i,V_j] : ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_i),V_i) ).
fof(fact_nat__add__left__cancel__less,axiom,
! [V_n_2,V_m_2,V_k_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_m_2),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k_2,V_n_2))
<=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) ).
fof(fact_trans__less__add1,axiom,
! [V_m,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_m)) ) ).
fof(fact_trans__less__add2,axiom,
! [V_m,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_j)) ) ).
fof(fact_add__less__mono1,axiom,
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_k)) ) ).
fof(fact_add__less__mono,axiom,
! [V_l,V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_j)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_k),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_j,V_l)) ) ) ).
fof(fact_less__add__eq__less,axiom,
! [V_n,V_m,V_l,V_k] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_k,V_l)
=> ( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_l) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_k,V_n)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m,V_n) ) ) ).
fof(fact_add__lessD1,axiom,
! [V_k,V_j,V_i] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_j),V_k)
=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,V_k) ) ).
fof(fact_add__eq__self__zero,axiom,
! [V_n,V_m] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n) = V_m
=> V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ).
fof(fact_add__is__0,axiom,
! [V_n_2,V_m_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
& V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ).
fof(fact_Nat_Oadd__0__right,axiom,
! [V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m ).
fof(fact_plus__nat_Oadd__0,axiom,
! [V_n] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n) = V_n ).
fof(fact_add__less__imp__less__left,axiom,
! [V_b,V_a,V_c,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ).
fof(fact_add__less__imp__less__right,axiom,
! [V_b,V_c,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))
=> c_Orderings_Oord__class_Oless(T_a,V_a,V_b) ) ) ).
fof(fact_add__strict__mono,axiom,
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) ).
fof(fact_add__strict__left__mono,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) ) ) ).
fof(fact_add__strict__right__mono,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) ) ).
fof(fact_add__less__cancel__left,axiom,
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_add__less__cancel__right,axiom,
! [V_b_2,V_c_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_add_Ocomm__neutral,axiom,
! [V_a,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) ).
fof(fact_add__0__right,axiom,
! [V_a,T_a] :
( class_Groups_Omonoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) ).
fof(fact_double__zero__sym,axiom,
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ).
fof(fact_add__0,axiom,
! [V_a,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a ) ).
fof(fact_add__0__left,axiom,
! [V_a,T_a] :
( class_Groups_Omonoid__add(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a ) ).
fof(fact_add__le__imp__le__left,axiom,
! [V_b,V_a,V_c,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) ).
fof(fact_add__le__imp__le__right,axiom,
! [V_b,V_c,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c))
=> c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b) ) ) ).
fof(fact_add__mono,axiom,
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) ).
fof(fact_add__left__mono,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_a),c_Groups_Oplus__class_Oplus(T_a,V_c,V_b)) ) ) ).
fof(fact_add__right__mono,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) ) ).
fof(fact_add__le__cancel__left,axiom,
! [V_b_2,V_a_2,V_c_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_a_2),c_Groups_Oplus__class_Oplus(T_a,V_c_2,V_b_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_add__le__cancel__right,axiom,
! [V_b_2,V_c_2,V_a_2,T_a] :
( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_c_2),c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_c_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_add__Suc__shift,axiom,
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)) ).
fof(fact_add__Suc,axiom,
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) ).
fof(fact_add__Suc__right,axiom,
! [V_n,V_m] : c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_n)) ).
fof(fact_add__nonpos__nonpos,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ).
fof(fact_add__increasing2,axiom,
! [V_a,V_b,V_c,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_c)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) ).
fof(fact_add__increasing,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) ).
fof(fact_add__nonneg__eq__0__iff,axiom,
! [V_y_2,V_x_2,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_x_2)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_y_2)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_x_2,V_y_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> ( V_x_2 = c_Groups_Ozero__class_Ozero(T_a)
& V_y_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ) ) ).
fof(fact_add__nonneg__nonneg,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) ).
fof(fact_double__add__le__zero__iff__single__add__le__zero,axiom,
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ).
fof(fact_zero__le__double__add__iff__zero__le__single__add,axiom,
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2))
<=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) ).
fof(fact_add__less__le__mono,axiom,
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) ).
fof(fact_add__le__less__mono,axiom,
! [V_d,V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
=> ( c_Orderings_Oord__class_Oless(T_a,V_c,V_d)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ) ) ).
fof(fact_add__neg__neg,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ).
fof(fact_add__pos__pos,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) ).
fof(fact_double__add__less__zero__iff__single__add__less__zero,axiom,
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ).
fof(fact_zero__less__double__add__iff__zero__less__single__add,axiom,
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2))
<=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a_2) ) ) ).
fof(fact_add__is__1,axiom,
! [V_n_2,V_m_2] :
( c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
<=> ( ( V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
& V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
| ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
& V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) ) ).
fof(fact_one__is__add,axiom,
! [V_n_2,V_m_2] :
( hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2)
<=> ( ( V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
& V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
| ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
& V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) ) ).
fof(fact_add__gr__0,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,V_n_2))
<=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_m_2)
| c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) ) ) ).
fof(fact_less__add__Suc1,axiom,
! [V_m,V_i] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_i,V_m))) ).
fof(fact_less__add__Suc2,axiom,
! [V_m,V_i] : c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_i,hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m,V_i))) ).
fof(fact_less__iff__Suc__add,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
<=> ? [B_k] : V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_m_2,B_k)) ) ).
fof(fact_add__pos__nonneg,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) ).
fof(fact_add__nonneg__pos,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_b)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,V_a,V_b)) ) ) ) ).
fof(fact_add__strict__increasing,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) ).
fof(fact_add__strict__increasing2,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) ).
fof(fact_add__neg__nonpos,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ).
fof(fact_add__nonpos__neg,axiom,
! [V_b,V_a,T_a] :
( class_Groups_Oordered__comm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Ozero__class_Ozero(T_a))
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ).
fof(fact_card__Plus,axiom,
! [V_B_2,T_b,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Finite__Set_Ofinite(T_b,V_B_2)
=> c_Finite__Set_Ocard(tc_sum(T_a,T_b),c_Sum__Type_OPlus(T_a,T_b,V_A_2,V_B_2)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_b,V_B_2)) ) ) ).
fof(fact_pos__add__strict,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b,V_c)
=> c_Orderings_Oord__class_Oless(T_a,V_b,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c)) ) ) ) ).
fof(fact_even__less__0__iff,axiom,
! [V_a_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ).
fof(fact_double__eq__0__iff,axiom,
! [V_a_2,T_a] :
( class_Groups_Olinordered__ab__group__add(T_a)
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ).
fof(fact_linorder__neqE__linordered__idom,axiom,
! [V_y,V_x,T_a] :
( class_Rings_Olinordered__idom(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_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = V_a ) ).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) ).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [V_d,V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),c_Groups_Oplus__class_Oplus(T_a,V_b,V_d)) ) ).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_b) ) ).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [V_c,V_b,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_b),V_c) = c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_b,V_c)) ) ).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [V_d,V_c,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a,V_c),V_d) ) ).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [V_d,V_c,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_c,V_d)) = c_Groups_Oplus__class_Oplus(T_a,V_c,c_Groups_Oplus__class_Oplus(T_a,V_a,V_d)) ) ).
fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [V_c,V_a,T_a] :
( class_Rings_Ocomm__semiring__1(T_a)
=> c_Groups_Oplus__class_Oplus(T_a,V_a,V_c) = c_Groups_Oplus__class_Oplus(T_a,V_c,V_a) ) ).
fof(fact_add__0__iff,axiom,
! [V_a_2,V_b_2,T_a] :
( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a)
=> ( V_b_2 = c_Groups_Oplus__class_Oplus(T_a,V_b_2,V_a_2)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ).
fof(fact_nat_Osize_I2_J,axiom,
! [V_nat] : c_Nat_Onat_Onat__size(hAPP(c_Nat_OSuc,V_nat)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_Onat_Onat__size(V_nat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ).
fof(fact_setsum__op__ivl__Suc,axiom,
! [V_f_2,V_m_2,V_n_2,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Ozero__class_Ozero(T_a) )
& ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_n_2,V_m_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2)),hAPP(V_f_2,V_n_2)) ) ) ) ).
fof(fact_nat_Osize_I1_J,axiom,
c_Nat_Onat_Onat__size(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_setsum__add__nat__ivl,axiom,
! [V_f_2,V_p_2,V_n_2,V_m_2,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_n_2,V_p_2)
=> c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_n_2,V_p_2))) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_p_2)) ) ) ) ).
fof(fact_setsum__shift__lb__Suc0__0__upt,axiom,
! [V_k_2,V_f_2,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(T_a)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)) ) ) ).
fof(fact_setsum__head__upt__Suc,axiom,
! [V_f_2,V_n_2,V_m_2,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(V_f_2,V_m_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2))) ) ) ).
fof(fact_setsum_Oinfinite,axiom,
! [V_g_2,V_A_2,T_a,T_b] :
( class_Groups_Ocomm__monoid__add(T_b)
=> ( ~ c_Finite__Set_Ofinite(T_a,V_A_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,V_A_2) = c_Groups_Ozero__class_Ozero(T_b) ) ) ).
fof(fact_setsum__infinite,axiom,
! [V_f_2,V_A_2,T_a,T_b] :
( class_Groups_Ocomm__monoid__add(T_b)
=> ( ~ c_Finite__Set_Ofinite(T_a,V_A_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,V_A_2) = c_Groups_Ozero__class_Ozero(T_b) ) ) ).
fof(fact_setsum__eq__0__iff,axiom,
! [V_f_2,V_Fa_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_Fa_2)
=> ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_Fa_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
<=> ! [B_x] :
( c_member(T_a,B_x,V_Fa_2)
=> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) ).
fof(fact_setsum__eq__Suc0__iff,axiom,
! [V_f_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_A_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
<=> ? [B_x] :
( c_member(T_a,B_x,V_A_2)
& hAPP(V_f_2,B_x) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
& ! [B_xa] :
( c_member(T_a,B_xa,V_A_2)
=> ( B_x != B_xa
=> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) ) ) ).
fof(fact_setsum__empty,axiom,
! [V_f_2,T_b,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_a,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_a) ) ).
fof(fact_setsum_Oempty,axiom,
! [V_g_2,T_b,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_a,V_g_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_a) ) ).
fof(fact_nat_Osize_I4_J,axiom,
! [V_nat] : c_Nat_Osize__class_Osize(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_nat)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Nat_Osize__class_Osize(tc_Nat_Onat,V_nat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ).
fof(fact_setsum__strict__mono,axiom,
! [V_g_2,V_f_2,V_A_2,T_a,T_b] :
( ( class_Groups_Ocomm__monoid__add(T_b)
& class_Groups_Oordered__cancel__ab__semigroup__add(T_b) )
=> ( 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] :
( c_member(T_a,B_x,V_A_2)
=> c_Orderings_Oord__class_Oless(T_b,hAPP(V_f_2,B_x),hAPP(V_g_2,B_x)) )
=> c_Orderings_Oord__class_Oless(T_b,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,V_A_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,V_A_2)) ) ) ) ) ).
fof(fact_nat__size,axiom,
! [V_n] : c_Nat_Osize__class_Osize(tc_Nat_Onat,V_n) = V_n ).
fof(fact_nat_Osize_I3_J,axiom,
c_Nat_Osize__class_Osize(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_setsum__SucD,axiom,
! [V_n_2,V_A_2,V_f_2,T_a] :
( c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_A_2) = hAPP(c_Nat_OSuc,V_n_2)
=> ? [B_x] :
( c_member(T_a,B_x,V_A_2)
& c_Orderings_Oord__class_Oless(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(V_f_2,B_x)) ) ) ).
fof(fact_setsum__reindex__cong,axiom,
! [V_h_2,V_g_2,V_B_2,V_A_2,V_f_2,T_b,T_a,T_c] :
( class_Groups_Ocomm__monoid__add(T_c)
=> ( c_Fun_Oinj__on(T_a,T_b,V_f_2,V_A_2)
=> ( V_B_2 = c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)
=> ( ! [B_a] :
( c_member(T_a,B_a,V_A_2)
=> hAPP(V_g_2,B_a) = hAPP(V_h_2,hAPP(V_f_2,B_a)) )
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c,V_h_2,V_B_2) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_c,V_g_2,V_A_2) ) ) ) ) ).
fof(fact_setsum__ivl__cong,axiom,
! [V_g_2,V_f_2,V_d_2,V_b_2,V_c_2,V_a_2,T_a,T_b] :
( ( class_Groups_Ocomm__monoid__add(T_b)
& class_Orderings_Oord(T_a) )
=> ( V_a_2 = V_c_2
=> ( V_b_2 = V_d_2
=> ( ! [B_x] :
( c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,B_x)
=> ( c_Orderings_Oord__class_Oless(T_a,B_x,V_d_2)
=> hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x) ) )
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_a_2,V_b_2)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,c_SetInterval_Oord__class_OatLeastLessThan(T_a,V_c_2,V_d_2)) ) ) ) ) ).
fof(fact_setsum__cl__ivl__Suc,axiom,
! [V_f_2,V_m_2,V_n_2,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Ozero__class_Ozero(T_a) )
& ( ~ c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),V_m_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)),hAPP(V_f_2,hAPP(c_Nat_OSuc,V_n_2))) ) ) ) ).
fof(fact_finite__atLeastAtMost,axiom,
! [V_u_2,V_l_2] : c_Finite__Set_Ofinite(tc_Nat_Onat,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)) ).
fof(fact_atLeastatMost__subset__iff,axiom,
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_c_2,V_d_2))
<=> ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_a_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_d_2) ) ) ) ) ).
fof(fact_atLeastatMost__empty,axiom,
! [V_a_2,V_b_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(T_a,V_b_2,V_a_2)
=> c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ).
fof(fact_atLeastatMost__empty__iff,axiom,
! [V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
<=> ~ c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_atLeastatMost__empty__iff2,axiom,
! [V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2)
<=> ~ c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2) ) ) ).
fof(fact_atLeastLessThanSuc__atLeastAtMost,axiom,
! [V_u_2,V_l_2] : c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2,hAPP(c_Nat_OSuc,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2) ).
fof(fact_ex__nat__less,axiom,
! [V_P_2,V_n_2] :
( ? [B_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m,V_n_2)
& hBOOL(hAPP(V_P_2,B_m)) )
<=> ? [B_x] :
( c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))
& hBOOL(hAPP(V_P_2,B_x)) ) ) ).
fof(fact_all__nat__less,axiom,
! [V_P_2,V_n_2] :
( ! [B_m] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_m,V_n_2)
=> hBOOL(hAPP(V_P_2,B_m)) )
<=> ! [B_x] :
( c_member(tc_Nat_Onat,B_x,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))
=> hBOOL(hAPP(V_P_2,B_x)) ) ) ).
fof(fact_image__Suc__atLeastAtMost,axiom,
! [V_j_2,V_i_2] : c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2)) ).
fof(fact_setsum__shift__lb__Suc0__0,axiom,
! [V_k_2,V_f_2] :
( hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) = c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)) ) ).
fof(fact_atLeastatMost__psubset__iff,axiom,
! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2),c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_c_2,V_d_2))
<=> ( ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_a_2,V_b_2)
| ( c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_a_2)
& c_Orderings_Oord__class_Oless__eq(T_a,V_b_2,V_d_2)
& ( c_Orderings_Oord__class_Oless(T_a,V_c_2,V_a_2)
| c_Orderings_Oord__class_Oless(T_a,V_b_2,V_d_2) ) ) )
& c_Orderings_Oord__class_Oless__eq(T_a,V_c_2,V_d_2) ) ) ) ).
fof(fact_setsum__head__Suc,axiom,
! [V_f_2,V_n_2,V_m_2,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = c_Groups_Oplus__class_Oplus(T_a,hAPP(V_f_2,V_m_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2))) ) ) ).
fof(fact_setsum__ub__add__nat,axiom,
! [V_p_2,V_f_2,V_n_2,V_m_2,T_a] :
( class_Groups_Ocomm__monoid__add(T_a)
=> ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,c_Groups_Oone__class_Oone(tc_Nat_Onat)))
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_p_2))) = c_Groups_Oplus__class_Oplus(T_a,c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_a,V_f_2,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n_2,V_p_2)))) ) ) ).
fof(fact_bool_Osize_I1_J,axiom,
c_HOL_Obool_Obool__size(c_fTrue) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_card__UNIV__unit,axiom,
c_Finite__Set_Ocard(tc_Product__Type_Ounit,c_Orderings_Otop__class_Otop(tc_fun(tc_Product__Type_Ounit,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(tc_Nat_Onat) ).
fof(fact_less__add__one,axiom,
! [V_a,T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless(T_a,V_a,c_Groups_Oplus__class_Oplus(T_a,V_a,c_Groups_Oone__class_Oone(T_a))) ) ).
fof(fact_not__one__le__zero,axiom,
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ~ c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Ozero__class_Ozero(T_a)) ) ).
fof(fact_zero__le__one,axiom,
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless__eq(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oone__class_Oone(T_a)) ) ).
fof(fact_not__one__less__zero,axiom,
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> ~ c_Orderings_Oord__class_Oless(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Ozero__class_Ozero(T_a)) ) ).
fof(fact_zero__less__one,axiom,
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oone__class_Oone(T_a)) ) ).
fof(fact_one__neq__zero,axiom,
! [T_a] :
( class_Rings_Ozero__neq__one(T_a)
=> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a) ) ).
fof(fact_zero__neq__one,axiom,
! [T_a] :
( class_Rings_Ozero__neq__one(T_a)
=> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a) ) ).
fof(fact_One__nat__def,axiom,
c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ).
fof(fact_Suc__eq__plus1__left,axiom,
! [V_n] : hAPP(c_Nat_OSuc,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),V_n) ).
fof(fact_Suc__eq__plus1,axiom,
! [V_n] : hAPP(c_Nat_OSuc,V_n) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,V_n,c_Groups_Oone__class_Oone(tc_Nat_Onat)) ).
fof(fact_one__reorient,axiom,
! [V_x_2,T_a] :
( class_Groups_Oone(T_a)
=> ( c_Groups_Oone__class_Oone(T_a) = V_x_2
<=> V_x_2 = c_Groups_Oone__class_Oone(T_a) ) ) ).
fof(fact_zero__less__two,axiom,
! [T_a] :
( class_Rings_Olinordered__semidom(T_a)
=> c_Orderings_Oord__class_Oless(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),c_Groups_Oone__class_Oone(T_a))) ) ).
fof(fact_setsum__eq__1__iff,axiom,
! [V_f_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,tc_Nat_Onat,V_f_2,V_A_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat)
<=> ? [B_x] :
( c_member(T_a,B_x,V_A_2)
& hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(tc_Nat_Onat)
& ! [B_xa] :
( c_member(T_a,B_xa,V_A_2)
=> ( B_x != B_xa
=> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) ) ) ).
fof(fact_bool_Osize_I2_J,axiom,
c_HOL_Obool_Obool__size(c_fFalse) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ).
fof(fact_Ints__odd__less__0,axiom,
! [V_a_2,T_a] :
( class_Rings_Olinordered__idom(T_a)
=> ( c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a))
=> ( c_Orderings_Oord__class_Oless(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),V_a_2),V_a_2),c_Groups_Ozero__class_Ozero(T_a))
<=> c_Orderings_Oord__class_Oless(T_a,V_a_2,c_Groups_Ozero__class_Ozero(T_a)) ) ) ) ).
fof(fact_ex__least__nat__less,axiom,
! [V_n_2,V_P_2] :
( ~ hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))
=> ( hBOOL(hAPP(V_P_2,V_n_2))
=> ? [B_k] :
( c_Orderings_Oord__class_Oless(tc_Nat_Onat,B_k,V_n_2)
& ! [B_i] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,B_i,B_k)
=> ~ hBOOL(hAPP(V_P_2,B_i)) )
& hBOOL(hAPP(V_P_2,c_Groups_Oplus__class_Oplus(tc_Nat_Onat,B_k,c_Groups_Oone__class_Oone(tc_Nat_Onat)))) ) ) ) ).
fof(fact_Ints__0,axiom,
! [T_a] :
( class_Rings_Oring__1(T_a)
=> c_member(T_a,c_Groups_Ozero__class_Ozero(T_a),c_Int_Oring__1__class_OInts(T_a)) ) ).
fof(fact_Ints__add,axiom,
! [V_b_2,V_a_2,T_a] :
( class_Rings_Oring__1(T_a)
=> ( c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a))
=> ( c_member(T_a,V_b_2,c_Int_Oring__1__class_OInts(T_a))
=> c_member(T_a,c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_b_2),c_Int_Oring__1__class_OInts(T_a)) ) ) ) ).
fof(fact_Ints__1,axiom,
! [T_a] :
( class_Rings_Oring__1(T_a)
=> c_member(T_a,c_Groups_Oone__class_Oone(T_a),c_Int_Oring__1__class_OInts(T_a)) ) ).
fof(fact_Ints__double__eq__0__iff,axiom,
! [V_a_2,T_a] :
( class_Int_Oring__char__0(T_a)
=> ( c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a))
=> ( c_Groups_Oplus__class_Oplus(T_a,V_a_2,V_a_2) = c_Groups_Ozero__class_Ozero(T_a)
<=> V_a_2 = c_Groups_Ozero__class_Ozero(T_a) ) ) ) ).
fof(fact_Ints__odd__nonzero,axiom,
! [V_a_2,T_a] :
( class_Int_Oring__char__0(T_a)
=> ( c_member(T_a,V_a_2,c_Int_Oring__1__class_OInts(T_a))
=> c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oplus__class_Oplus(T_a,c_Groups_Oone__class_Oone(T_a),V_a_2),V_a_2) != c_Groups_Ozero__class_Ozero(T_a) ) ) ).
fof(fact_card_Oneutral,axiom,
! [V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( ( ? [B_x] : c_member(T_a,B_x,V_A_2)
=> c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
=> c_Finite__Set_Ocard(T_a,V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ).
fof(fact_card_Oinsert,axiom,
! [V_x_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( ~ c_member(T_a,V_x_2,V_A_2)
=> c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,V_A_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_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_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_folding__image__simple__idem_Oinsert__idem,axiom,
! [V_x_2,V_A_2,V_Fa_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_Fa_2)
=> ( c_Finite__Set_Ofinite(T_b,V_A_2)
=> hAPP(V_Fa_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_Fa_2,V_A_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_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_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_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__image,axiom,
! [V_f_2,T_b,V_A_2,V_x_2,T_a] :
( c_member(T_a,V_x_2,V_A_2)
=> c_Set_Oinsert(T_b,hAPP(V_f_2,V_x_2),c_Set_Oimage(T_a,T_b,V_f_2,V_A_2)) = c_Set_Oimage(T_a,T_b,V_f_2,V_A_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_image__insert,axiom,
! [V_B_2,V_a_2,V_f_2,T_a,T_b] : c_Set_Oimage(T_b,T_a,V_f_2,c_Set_Oinsert(T_b,V_a_2,V_B_2)) = c_Set_Oinsert(T_a,hAPP(V_f_2,V_a_2),c_Set_Oimage(T_b,T_a,V_f_2,V_B_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_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__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_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_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_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_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_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__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_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_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_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_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_atLeastAtMost__singleton_H,axiom,
! [V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( V_a_2 = V_b_2
=> c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) = c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ) ) ).
fof(fact_atLeastAtMost__singleton__iff,axiom,
! [V_c_2,V_b_2,V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> ( c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_b_2) = c_Set_Oinsert(T_a,V_c_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))
<=> ( V_a_2 = V_b_2
& V_b_2 = V_c_2 ) ) ) ).
fof(fact_atLeastAtMost__singleton,axiom,
! [V_a_2,T_a] :
( class_Orderings_Oorder(T_a)
=> c_SetInterval_Oord__class_OatLeastAtMost(T_a,V_a_2,V_a_2) = c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ) ).
fof(fact_card__insert__le,axiom,
! [V_x_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,c_Finite__Set_Ocard(T_a,V_A_2),c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2))) ) ).
fof(fact_atLeastAtMostSuc__conv,axiom,
! [V_n_2,V_m_2] :
( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))
=> c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) = c_Set_Oinsert(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) ) ).
fof(fact_atLeastLessThan__singleton,axiom,
! [V_m_2] : c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_m_2)) = c_Set_Oinsert(tc_Nat_Onat,V_m_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) ).
fof(fact_folding__one__idem_Oinsert__idem,axiom,
! [V_x_2,V_A_2,V_Fa_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one__idem(T_a,V_f_2,V_Fa_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))
=> hAPP(V_Fa_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_Fa_2,V_A_2)) ) ) ) ).
fof(fact_setsum_Oinsert,axiom,
! [V_g_2,V_x_2,V_A_2,T_a,T_b] :
( class_Groups_Ocomm__monoid__add(T_b)
=> ( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( ~ c_member(T_a,V_x_2,V_A_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Groups_Oplus__class_Oplus(T_b,hAPP(V_g_2,V_x_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_g_2,V_A_2)) ) ) ) ).
fof(fact_setsum__insert,axiom,
! [V_f_2,V_a_2,V_Fa_2,T_a,T_b] :
( class_Groups_Ocomm__monoid__add(T_b)
=> ( c_Finite__Set_Ofinite(T_a,V_Fa_2)
=> ( ~ c_member(T_a,V_a_2,V_Fa_2)
=> c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,c_Set_Oinsert(T_a,V_a_2,V_Fa_2)) = c_Groups_Oplus__class_Oplus(T_b,hAPP(V_f_2,V_a_2),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_a,T_b,V_f_2,V_Fa_2)) ) ) ) ).
fof(fact_card__insert__disjoint,axiom,
! [V_x_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( ~ c_member(T_a,V_x_2,V_A_2)
=> c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = hAPP(c_Nat_OSuc,c_Finite__Set_Ocard(T_a,V_A_2)) ) ) ).
fof(fact_card__insert__if,axiom,
! [V_x_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( ( c_member(T_a,V_x_2,V_A_2)
=> c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = c_Finite__Set_Ocard(T_a,V_A_2) )
& ( ~ c_member(T_a,V_x_2,V_A_2)
=> c_Finite__Set_Ocard(T_a,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = hAPP(c_Nat_OSuc,c_Finite__Set_Ocard(T_a,V_A_2)) ) ) ) ).
fof(fact_atLeastLessThanSuc,axiom,
! [V_n_2,V_m_2] :
( ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
=> c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) = c_Set_Oinsert(tc_Nat_Onat,V_n_2,c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,V_n_2)) )
& ( ~ c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2)
=> c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) ) ) ).
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_card__Suc__eq,axiom,
! [V_k_2,V_A_2,T_a] :
( c_Finite__Set_Ocard(T_a,V_A_2) = hAPP(c_Nat_OSuc,V_k_2)
<=> ? [B_b,B_B] :
( V_A_2 = c_Set_Oinsert(T_a,B_b,B_B)
& ~ c_member(T_a,B_b,B_B)
& c_Finite__Set_Ocard(T_a,B_B) = V_k_2
& ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
=> B_B = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) ) ).
fof(fact_folding__one_Oinsert,axiom,
! [V_x_2,V_A_2,V_Fa_2,V_f_2,T_a] :
( c_Finite__Set_Ofolding__one(T_a,V_f_2,V_Fa_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_Fa_2,c_Set_Oinsert(T_a,V_x_2,V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_Fa_2,V_A_2)) ) ) ) ) ).
fof(fact_card_Oremove,axiom,
! [V_x_2,V_A_2,T_a] :
( c_Finite__Set_Ofinite(T_a,V_A_2)
=> ( c_member(T_a,V_x_2,V_A_2)
=> c_Finite__Set_Ocard(T_a,V_A_2) = c_Groups_Oplus__class_Oplus(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Finite__Set_Ocard(T_a,c_Groups_Ominus__class_Ominus(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)))))) ) ) ).
%----Arity declarations (36)
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__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_Otop,axiom,
! [T_2,T_1] :
( class_Orderings_Otop(T_1)
=> class_Orderings_Otop(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__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add,axiom,
class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le,axiom,
class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add,axiom,
class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Oordered__comm__monoid__add,axiom,
class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add,axiom,
class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Ocancel__semigroup__add,axiom,
class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Rings_Olinordered__semidom,axiom,
class_Rings_Olinordered__semidom(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Oab__semigroup__add,axiom,
class_Groups_Oab__semigroup__add(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Ocomm__monoid__add,axiom,
class_Groups_Ocomm__monoid__add(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Rings_Ocomm__semiring__1,axiom,
class_Rings_Ocomm__semiring__1(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Rings_Ozero__neq__one,axiom,
class_Rings_Ozero__neq__one(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__Groups_Omonoid__add,axiom,
class_Groups_Omonoid__add(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_Nat__Onat__Groups_Ozero,axiom,
class_Groups_Ozero(tc_Nat_Onat) ).
fof(arity_Nat__Onat__Groups_Oone,axiom,
class_Groups_Oone(tc_Nat_Onat) ).
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__Orderings_Oorder,axiom,
class_Orderings_Oorder(tc_HOL_Obool) ).
fof(arity_HOL__Obool__Orderings_Otop,axiom,
class_Orderings_Otop(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_sum__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_sum(T_2,T_1)) ) ).
fof(arity_Option__Ooption__Finite__Set_Ofinite,axiom,
! [T_1] :
( class_Finite__Set_Ofinite(T_1)
=> class_Finite__Set_Ofinite(tc_Option_Ooption(T_1)) ) ).
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)) ) ).
fof(arity_Product____Type__Ounit__Finite__Set_Ofinite,axiom,
class_Finite__Set_Ofinite(tc_Product__Type_Ounit) ).
%----Helper facts (4)
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)) ) ).
fof(help_c__fFalse__1,axiom,
~ hBOOL(c_fFalse) ).
fof(help_c__fTrue__1,axiom,
hBOOL(c_fTrue) ).
%----Conjectures (2)
fof(conj_0,hypothesis,
! [B_h] :
( c_Fun_Oinj__on(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,B_h,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool)))
=> ( c_Set_Oimage(tc_Arrow__Order__Mirabelle_Oindi,tc_Nat_Onat,B_h,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_Finite__Set_Ocard(tc_Arrow__Order__Mirabelle_Oindi,c_Orderings_Otop__class_Otop(tc_fun(tc_Arrow__Order__Mirabelle_Oindi,tc_HOL_Obool))))
=> v_thesis____ ) ) ).
fof(conj_1,conjecture,
v_thesis____ ).
%------------------------------------------------------------------------------