TPTP Problem File: SCT265_5.p
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%------------------------------------------------------------------------------
% File : SCT265_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 307
% Version : Especial.
% English : Formalization of two proofs of Arrow's impossibility theorem. One
% formalization is based on utility functions, the other one on
% strict partial orders.
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : arrow_307 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 160 ( 42 unt; 47 typ; 0 def)
% Number of atoms : 296 ( 79 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 247 ( 64 ~; 8 |; 19 &)
% ( 39 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 33 ( 19 >; 14 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-4 aty)
% Number of functors : 36 ( 36 usr; 18 con; 0-5 aty)
% Number of variables : 305 ( 268 !; 5 ?; 305 :)
% ( 32 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:27:57
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oalt,type,
arrow_411405190le_alt: $tType ).
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oindi,type,
arrow_159774573e_indi: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (41)
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OIIA,type,
arrow_1958449194le_IIA: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OLin,type,
arrow_1985332922le_Lin: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OProf,type,
arrow_610318064e_Prof: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Oabove,type,
arrow_1158827142_above: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)))) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omkbot,type,
arrow_276188178_mkbot: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omktop,type,
arrow_424895264_mktop: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Ounanimity,type,
arrow_2069624013nimity: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[A: $tType] : ( fun(A,bool) > nat ) ).
tff(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > $o ) ).
tff(sy_c_FuncSet_OPi,type,
pi:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,fun(B,bool)) ) > fun(fun(A,B),bool) ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,B) * B ) > A ) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : fun(bool,fun(A,fun(A,A))) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( ( A * B ) > product_prod(A,B) ) ).
tff(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( fun(A,nat) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_F,type,
f: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) ).
tff(sy_v_Lab____,type,
lab: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_v_Lba____,type,
lba: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_v_P____,type,
p: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) ).
tff(sy_v_a____,type,
a: arrow_411405190le_alt ).
tff(sy_v_b____,type,
b: arrow_411405190le_alt ).
tff(sy_v_c____,type,
c: arrow_411405190le_alt ).
tff(sy_v_d____,type,
d: arrow_411405190le_alt ).
tff(sy_v_e____,type,
e: arrow_411405190le_alt ).
tff(sy_v_h____,type,
h: fun(arrow_159774573e_indi,nat) ).
tff(sy_v_n____,type,
n: nat ).
%----Relevant facts (100)
tff(fact_0__096P_A_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),p,arrow_610318064e_Prof) ).
tff(fact_1_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_2_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_3__096c_A_126_061_Ad_096,axiom,
c != d ).
tff(fact_4__096_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e))),arrow_610318064e_Prof) ).
tff(fact_5__096c_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ae_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,e),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e))))) ).
tff(fact_6__096e_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ad_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,e,d),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e))))) ).
tff(fact_7__C2_C,axiom,
! [P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B4: arrow_411405190le_alt,A5: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A5 != B4 )
=> ( ( Aa != B4 )
=> ( ( Ba != A5 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( ! [I3: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A5,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P1,I3)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A5,B4),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P1)) ) ) ) ) ) ) ) ) ).
tff(fact_8__C1_C,axiom,
! [P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B4: arrow_411405190le_alt,A5: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A5 != B4 )
=> ( ( Aa != B4 )
=> ( ( Ba != A5 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( ! [I3: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A5,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P1,I3)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A5,B4),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P1)) ) ) ) ) ) ) ) ) ).
tff(fact_9__C4_C,axiom,
! [P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Ca: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( Ba != Ca )
=> ( ( Aa != Ca )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( ! [I3: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Ca),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P1,I3)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Ca),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P1)) ) ) ) ) ) ) ) ).
tff(fact_10_pairwise__neutrality,axiom,
! [P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B4: arrow_411405190le_alt,A5: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A5 != B4 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( ! [I3: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A5,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P1,I3)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A5,B4),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P1)) ) ) ) ) ) ) ).
tff(fact_11__C3_C,axiom,
! [P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( ! [I3: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P1,I3)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P1)) ) ) ) ) ) ).
tff(fact_12__096ALL_Ai_O_A_Ic_A_060_092_060_094bsub_062P_Ai_092_060_094esub_062_Ad_J_A_061_A_Ic_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ad_J_096,axiom,
! [I1: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),e)) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> ( ( ( aa(arrow_159774573e_indi,nat,h,I1) = n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_1158827142_above,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),c),e)) )
& ( ( aa(arrow_159774573e_indi,nat,h,I1) != n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),e)) ) ) ) ) ) ).
tff(fact_13_PW,axiom,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,p))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e))))) ) ).
tff(fact_14_in__mkbot,axiom,
! [Z1: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,L1),Z1))
<=> ( ( Y1 != Z1 )
& ( ( X1 = Z1 )
=> ( X1 != Y1 ) )
& ( ( X1 != Z1 )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1) ) ) ) ).
tff(fact_15_in__mktop,axiom,
! [Z1: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,L1),Z1))
<=> ( ( X1 != Z1 )
& ( ( Y1 = Z1 )
=> ( X1 != Y1 ) )
& ( ( Y1 != Z1 )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1) ) ) ) ).
tff(fact_16__096c_A_060_092_060_094bsub_062P_A_Iinv_Ah_An_J_092_060_094esub_062_Ad_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,hilbert_inv_into(arrow_159774573e_indi,nat,top_top(fun(arrow_159774573e_indi,bool)),h,n))) ).
tff(fact_17_split__paired__All,axiom,
! [A: $tType,B: $tType,Pa: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,Pa,X11))
<=> ! [A2: A,B1: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A2,B1))) ) ).
tff(fact_18_Pair__eq,axiom,
! [A: $tType,B: $tType,B4: B,A5: A,Ba: B,Aa: A] :
( ( product_Pair(A,B,Aa,Ba) = product_Pair(A,B,A5,B4) )
<=> ( ( Aa = A5 )
& ( Ba = B4 ) ) ) ).
tff(fact_19__096_Ie_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ad_J_A_061_Ib_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_J_096,axiom,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,e,d),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e)))))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),lab),lba))) ) ).
tff(fact_20__096ALL_Ai_O_A_Ie_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ad_J_A_061_A_Ib_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_J_096,axiom,
! [I1: arrow_159774573e_indi] :
( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,e,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),e)) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> ( ( ( aa(arrow_159774573e_indi,nat,h,I1) = n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,e,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_1158827142_above,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),c),e)) )
& ( ( aa(arrow_159774573e_indi,nat,h,I1) != n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,e,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),e)) ) ) ) )
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),lab) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),lba) ) ) ) ).
tff(fact_21_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S1: fun(product_prod(A,B),bool),R1: fun(product_prod(A,B),bool)] :
( ! [X3: A,Xa: B] :
( member(product_prod(A,B),product_Pair(A,B,X3,Xa),R1)
<=> member(product_prod(A,B),product_Pair(A,B,X3,Xa),S1) )
<=> ( R1 = S1 ) ) ).
tff(fact_22_in__measure,axiom,
! [A: $tType,F: fun(A,nat),Y1: A,X1: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y1),measure(A,F))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X1)),aa(A,nat,F,Y1))) ) ).
tff(fact_23_in__above,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_1158827142_above,L1),Aa),Ba))
<=> ( ( X1 != Y1 )
& ( ( X1 = Ba )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Y1),L1) )
& ( ( X1 != Ba )
=> ( ( ( Y1 = Ba )
=> ( ( X1 = Aa )
| member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Aa),L1) ) )
& ( ( Y1 != Ba )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1) ) ) ) ) ) ) ) ).
tff(fact_24_less__not__refl,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
tff(fact_25_nat__neq__iff,axiom,
! [Na: nat,M1: nat] :
( ( M1 != Na )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M1)) ) ) ).
tff(fact_26_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
tff(fact_27_less__irrefl__nat,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
tff(fact_28__096Lba_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lba,arrow_1985332922le_Lin) ).
tff(fact_29__096Lab_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lab,arrow_1985332922le_Lin) ).
tff(fact_30__096a_A_126_061_Ab_096,axiom,
a != b ).
tff(fact_31__096b_A_060_092_060_094bsub_062Lba_092_060_094esub_062_Aa_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),lba) ).
tff(fact_32__096a_A_060_092_060_094bsub_062Lab_092_060_094esub_062_Ab_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),lab) ).
tff(fact_33_top1I,axiom,
! [A: $tType,X1: A] : pp(aa(A,bool,top_top(fun(A,bool)),X1)) ).
tff(fact_34__096_Ib_M_Aa_J_A_126_058_ALab_096,axiom,
~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),lab) ).
tff(fact_35__096_Ia_M_Ab_J_A_126_058_ALba_096,axiom,
~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),lba) ).
tff(fact_36_PiProf,axiom,
! [Na: nat] : member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),Na)),lab),lba),arrow_610318064e_Prof) ).
tff(fact_37__096_B_Bthesis_O_A_I_B_BLba_O_A_091_124_Ab_A_060_092_060_094bsub_062Lba_092_060_094esub_062_Aa_059_ALba_A_058_ALin_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [Lba: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),Lba)
=> ~ member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Lba,arrow_1985332922le_Lin) ) ).
tff(fact_38__096_B_Bthesis_O_A_I_B_BLab_O_A_091_124_Aa_A_060_092_060_094bsub_062Lab_092_060_094esub_062_Ab_059_ALab_A_058_ALin_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [Lab: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),Lab)
=> ~ member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Lab,arrow_1985332922le_Lin) ) ).
tff(fact_39_assms_I1_J,axiom,
member(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),f,pi(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_610318064e_Prof,combk(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_1985332922le_Lin))) ).
tff(fact_40_n_I1_J,axiom,
pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),n),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool))))) ).
tff(fact_41_n_I2_J,axiom,
! [M2: nat] :
( ord_less_eq(nat,M2,n)
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),M2)),lab),lba))) ) ).
tff(fact_42_injh,axiom,
inj_on(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))) ).
tff(fact_43_const__Lin__Prof,axiom,
! [L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combk(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_159774573e_indi,L1),arrow_610318064e_Prof) ) ).
tff(fact_44_Lin__irrefl,axiom,
! [Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),L1)
=> ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),L1) ) ) ).
tff(fact_45_notin__Lin__iff,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> ( ( X1 != Y1 )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Y1,X1),L1) ) ) ) ).
tff(fact_46_mktop__Lin,axiom,
! [X1: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,L1),X1),arrow_1985332922le_Lin) ) ).
tff(fact_47_mkbot__Lin,axiom,
! [X1: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,L1),X1),arrow_1985332922le_Lin) ) ).
tff(fact_48_above__Lin,axiom,
! [L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y1 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_1158827142_above,L1),X1),Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_49_Pair__inject,axiom,
! [A: $tType,B: $tType,B3: B,A4: A,B2: B,A3: A] :
( ( product_Pair(A,B,A3,B2) = product_Pair(A,B,A4,B3) )
=> ~ ( ( A3 = A4 )
=> ( B2 != B3 ) ) ) ).
tff(fact_50_nat__less__cases,axiom,
! [Pa: fun(nat,fun(nat,bool)),Na: nat,M1: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> ( ( ( M1 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) ) ) ) ).
tff(fact_51_less__not__refl3,axiom,
! [T: nat,S: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T))
=> ( S != T ) ) ).
tff(fact_52_less__not__refl2,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( M != N ) ) ).
tff(fact_53_n_I3_J,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),plus_plus(nat,n,one_one(nat)))),lab),lba))) ).
tff(fact_54__096_Ic_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ae_J_A_061_Ia_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_J_096,axiom,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,e),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e)))))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),plus_plus(nat,n,one_one(nat)))),lab),lba))) ) ).
tff(fact_55_IIA__def,axiom,
! [Fa: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_1958449194le_IIA(Fa)
<=> ! [X3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X3,arrow_610318064e_Prof)
=> ! [Xa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Xa,arrow_610318064e_Prof)
=> ! [A2: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ! [I2: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A2,B1),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A2,B1),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa,I2)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A2,B1),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Fa,X3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A2,B1),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Fa,Xa)) ) ) ) ) ) ).
tff(fact_56_unanimity__def,axiom,
! [Fa: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_2069624013nimity(Fa)
<=> ! [X3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X3,arrow_610318064e_Prof)
=> ! [A2: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ! [I2: arrow_159774573e_indi] : member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A2,B1),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I2))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A2,B1),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Fa,X3)) ) ) ) ).
tff(fact_57__096ALL_Ai_O_A_Ic_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_092_060_094esub_062_Ae_J_A_061_A_Ia_A_060_092_060_094bsub_062_Iif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_J_096,axiom,
! [I1: arrow_159774573e_indi] :
( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,e),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),e)) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),n))
=> ( ( ( aa(arrow_159774573e_indi,nat,h,I1) = n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,e),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_1158827142_above,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),c),e)) )
& ( ( aa(arrow_159774573e_indi,nat,h,I1) != n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,e),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I1)),e)) ) ) ) )
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),plus_plus(nat,n,one_one(nat))))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),lab) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I1)),plus_plus(nat,n,one_one(nat))))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),lba) ) ) ) ).
tff(fact_58_split__paired__Ex,axiom,
! [A: $tType,B: $tType,Pa: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,Pa,X11))
<=> ? [A2: A,B1: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A2,B1))) ) ).
tff(fact_59_nat__add__left__cancel,axiom,
! [Na: nat,M1: nat,K2: nat] :
( ( plus_plus(nat,K2,M1) = plus_plus(nat,K2,Na) )
<=> ( M1 = Na ) ) ).
tff(fact_60_nat__add__right__cancel,axiom,
! [Na: nat,K2: nat,M1: nat] :
( ( plus_plus(nat,M1,K2) = plus_plus(nat,Na,K2) )
<=> ( M1 = Na ) ) ).
tff(fact_61_nat__add__left__cancel__less,axiom,
! [Na: nat,M1: nat,K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,K2,M1)),plus_plus(nat,K2,Na)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na)) ) ).
tff(fact_62_nat__add__left__cancel__le,axiom,
! [Na: nat,M1: nat,K2: nat] :
( ord_less_eq(nat,plus_plus(nat,K2,M1),plus_plus(nat,K2,Na))
<=> ord_less_eq(nat,M1,Na) ) ).
tff(fact_63__096EX_An_060N_O_A_IALL_Am_060_061n_O_Ab_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_Am_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_J_A_G_Aa_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_096,axiom,
? [N1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))))
& ! [M2: nat] :
( ord_less_eq(nat,M2,N1)
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),M2)),lab),lba))) )
& member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),plus_plus(nat,N1,one_one(nat)))),lab),lba))) ) ).
tff(fact_64__096_B_Bthesis_O_A_I_B_Bn_O_A_091_124_An_A_060_AN_059_AALL_Am_060_061n_O_Ab_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_Am_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_059_Aa_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [N1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))))
=> ( ! [M2: nat] :
( ord_less_eq(nat,M2,N1)
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),M2)),lab),lba))) )
=> ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),plus_plus(nat,N1,one_one(nat)))),lab),lba))) ) ) ).
tff(fact_65_le__refl,axiom,
! [N: nat] : ord_less_eq(nat,N,N) ).
tff(fact_66_le__add2,axiom,
! [M: nat,N: nat] : ord_less_eq(nat,N,plus_plus(nat,M,N)) ).
tff(fact_67_le__add1,axiom,
! [M: nat,N: nat] : ord_less_eq(nat,N,plus_plus(nat,N,M)) ).
tff(fact_68_le__iff__add,axiom,
! [Na: nat,M1: nat] :
( ord_less_eq(nat,M1,Na)
<=> ? [K1: nat] : Na = plus_plus(nat,M1,K1) ) ).
tff(fact_69_nat__add__commute,axiom,
! [N: nat,M: nat] : plus_plus(nat,M,N) = plus_plus(nat,N,M) ).
tff(fact_70_nat__le__linear,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
| ord_less_eq(nat,N,M) ) ).
tff(fact_71_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X: nat] : plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z)) ).
tff(fact_72_nat__add__assoc,axiom,
! [K: nat,N: nat,M: nat] : plus_plus(nat,plus_plus(nat,M,N),K) = plus_plus(nat,M,plus_plus(nat,N,K)) ).
tff(fact_73_eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( M = N )
=> ord_less_eq(nat,M,N) ) ).
tff(fact_74_trans__le__add1,axiom,
! [M: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ord_less_eq(nat,I,plus_plus(nat,J,M)) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X1: A] :
( member(A,X1,A1)
<=> pp(aa(A,bool,A1,X1)) ) ).
tff(fact_77_trans__le__add2,axiom,
! [M: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ord_less_eq(nat,I,plus_plus(nat,M,J)) ) ).
tff(fact_78_add__le__mono1,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ord_less_eq(nat,plus_plus(nat,I,K),plus_plus(nat,J,K)) ) ).
tff(fact_79_le__trans,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,J,K)
=> ord_less_eq(nat,I,K) ) ) ).
tff(fact_80_le__antisym,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ( ord_less_eq(nat,N,M)
=> ( M = N ) ) ) ).
tff(fact_81_add__le__mono,axiom,
! [L: nat,K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,K,L)
=> ord_less_eq(nat,plus_plus(nat,I,K),plus_plus(nat,J,L)) ) ) ).
tff(fact_82_add__leD2,axiom,
! [N: nat,K: nat,M: nat] :
( ord_less_eq(nat,plus_plus(nat,M,K),N)
=> ord_less_eq(nat,K,N) ) ).
tff(fact_83_add__leD1,axiom,
! [N: nat,K: nat,M: nat] :
( ord_less_eq(nat,plus_plus(nat,M,K),N)
=> ord_less_eq(nat,M,N) ) ).
tff(fact_84_add__leE,axiom,
! [N: nat,K: nat,M: nat] :
( ord_less_eq(nat,plus_plus(nat,M,K),N)
=> ~ ( ord_less_eq(nat,M,N)
=> ~ ord_less_eq(nat,K,N) ) ) ).
tff(fact_85_not__add__less1,axiom,
! [J: nat,I: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I,J)),I)) ).
tff(fact_86_not__add__less2,axiom,
! [I: nat,J: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,J,I)),I)) ).
tff(fact_87_trans__less__add1,axiom,
! [M: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),plus_plus(nat,J,M))) ) ).
tff(fact_88_trans__less__add2,axiom,
! [M: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),plus_plus(nat,M,J))) ) ).
tff(fact_89_add__less__mono1,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I,K)),plus_plus(nat,J,K))) ) ).
tff(fact_90_add__less__mono,axiom,
! [L: nat,K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I,K)),plus_plus(nat,J,L))) ) ) ).
tff(fact_91_less__add__eq__less,axiom,
! [N: nat,M: nat,L: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
=> ( ( plus_plus(nat,M,L) = plus_plus(nat,K,N) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
tff(fact_92_add__lessD1,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I,J)),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K)) ) ).
tff(fact_93_nat__less__le,axiom,
! [Na: nat,M1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
<=> ( ord_less_eq(nat,M1,Na)
& ( M1 != Na ) ) ) ).
tff(fact_94_le__eq__less__or__eq,axiom,
! [Na: nat,M1: nat] :
( ord_less_eq(nat,M1,Na)
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
| ( M1 = Na ) ) ) ).
tff(fact_95_less__imp__le__nat,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ord_less_eq(nat,M,N) ) ).
tff(fact_96_le__neq__implies__less,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ( ( M != N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
tff(fact_97_less__or__eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| ( M = N ) )
=> ord_less_eq(nat,M,N) ) ).
tff(fact_98_Pi__UNIV,axiom,
! [B: $tType,A: $tType,A1: fun(A,bool)] : pi(A,B,A1,combk(fun(B,bool),A,top_top(fun(B,bool)))) = top_top(fun(fun(A,B),bool)) ).
tff(fact_99_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,Aa: A,Ca: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,Ca,Aa)),plus_plus(A,Ca,Ba)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),Ba)) ) ) ).
%----Arities (1)
tff(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
%----Helper facts (11)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),fTrue),X),Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),fFalse),X),Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e))))) ).
%------------------------------------------------------------------------------