TPTP Problem File: SCT254_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SCT254_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 280
% 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_280 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 163 ( 42 unt; 42 typ; 0 def)
% Number of atoms : 336 ( 98 equ)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 272 ( 57 ~; 5 |; 12 &)
% ( 30 <=>; 168 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 37 ( 22 >; 15 *; 0 +; 0 <<)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-4 aty)
% Number of functors : 23 ( 23 usr; 6 con; 0-6 aty)
% Number of variables : 501 ( 457 !; 9 ?; 501 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:27
%------------------------------------------------------------------------------
%----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 (36)
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[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_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_Odictator,type,
arrow_987702531ctator: ( 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_159774573e_indi ) > $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omkbot,type,
arrow_276188178_mkbot: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * 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(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * 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_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[A: $tType] : ( fun(A,bool) > nat ) ).
tff(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > $o ) ).
tff(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,B) ) > fun(B,A) ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[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_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(product_prod(A,B),C) * A * B ) > C ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( fun(A,bool) > fun(fun(A,bool),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_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_h____,type,
h: fun(arrow_159774573e_indi,nat) ).
tff(sy_v_n____,type,
n: nat ).
%----Relevant facts (100)
tff(fact_0_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_1_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_2_injh,axiom,
inj_on(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))) ).
tff(fact_3_top1I,axiom,
! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).
tff(fact_4_UNIV__I,axiom,
! [A: $tType,X: A] : member(A,X,top_top(fun(A,bool))) ).
tff(fact_5_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : member(A,X,top_top(fun(A,bool))) ).
tff(fact_6_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [X: B] : aa(B,A,top_top(fun(B,A)),X) = top_top(A) ) ).
tff(fact_7_n_I1_J,axiom,
ord_less(nat,n,finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))) ).
tff(fact_8_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_9_dictator__def,axiom,
! [I2: arrow_159774573e_indi,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_987702531ctator(Fa,I2)
<=> ! [X4: 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)),X4,arrow_610318064e_Prof)
=> ( 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,X4) = aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I2) ) ) ) ).
tff(fact_10_inv__equality,axiom,
! [A: $tType,B: $tType,F: fun(A,B),G: fun(B,A)] :
( ! [X1: A] : aa(B,A,G,aa(A,B,F,X1)) = X1
=> ( ! [Y1: B] : aa(A,B,F,aa(B,A,G,Y1)) = Y1
=> ( hilbert_inv_into(A,B,top_top(fun(A,bool)),F) = G ) ) ) ).
tff(fact_11_Sup__UNIV,axiom,
! [A: $tType] :
( comple187826305attice(A)
=> ( complete_Sup_Sup(A,top_top(fun(A,bool))) = top_top(A) ) ) ).
tff(fact_12_inv__id,axiom,
! [A: $tType] : hilbert_inv_into(A,A,top_top(fun(A,bool)),id(A)) = id(A) ).
tff(fact_13_UNIV__witness,axiom,
! [A: $tType] :
? [X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_14_UNIV__eq__I,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X1: A] : member(A,X1,A1)
=> ( top_top(fun(A,bool)) = A1 ) ) ).
tff(fact_15_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ~ ord_less(A,X2,X2) ) ).
tff(fact_16_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ( X != Y )
<=> ( ord_less(A,X,Y)
| ord_less(A,Y,X) ) ) ) ).
tff(fact_17_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
<=> ( ord_less(A,Y,X)
| ( X = Y ) ) ) ) ).
tff(fact_18_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
| ( X2 = Y2 )
| ord_less(A,Y2,X2) ) ) ).
tff(fact_19_less__Sup__iff,axiom,
! [A: $tType] :
( comple1035589618norder(A)
=> ! [S: fun(A,bool),Aa: A] :
( ord_less(A,Aa,complete_Sup_Sup(A,S))
<=> ? [X4: A] :
( member(A,X4,S)
& ord_less(A,Aa,X4) ) ) ) ).
tff(fact_20_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ ord_less(A,Y,X)
=> ( ~ ord_less(A,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_21_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ( X2 != Y2 )
=> ( ~ ord_less(A,X2,Y2)
=> ord_less(A,Y2,X2) ) ) ) ).
tff(fact_22_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( X2 != Y2 ) ) ) ).
tff(fact_23_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ~ ord_less(A,Y2,X2) ) ) ).
tff(fact_24_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ~ ord_less(A,Y2,X2) ) ) ).
tff(fact_25_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( X2 != Y2 ) ) ) ).
tff(fact_26_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( Y2 != X2 ) ) ) ).
tff(fact_27_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P1: bool,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ord_less(A,Y,X)
=> pp(P1) ) ) ) ).
tff(fact_28_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A4: A] :
( ord_less(A,A4,B3)
=> ~ ord_less(A,B3,A4) ) ) ).
tff(fact_29_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C4: A,B3: A,A4: A] :
( ( A4 = B3 )
=> ( ord_less(A,B3,C4)
=> ord_less(A,A4,C4) ) ) ) ).
tff(fact_30_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C4: A,B3: A,A4: A] :
( ord_less(A,A4,B3)
=> ( ( B3 = C4 )
=> ord_less(A,A4,C4) ) ) ) ).
tff(fact_31_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( ord_less(A,Y2,Z2)
=> ord_less(A,X2,Z2) ) ) ) ).
tff(fact_32_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ~ ord_less(A,Y2,X2) ) ) ).
tff(fact_33_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less(A,X2,Y2)
=> ( ( X2 != Y2 )
=> ord_less(A,Y2,X2) ) ) ) ).
tff(fact_34_Union__UNIV,axiom,
! [A: $tType] : complete_Sup_Sup(fun(A,bool),top_top(fun(fun(A,bool),bool))) = top_top(fun(A,bool)) ).
tff(fact_35_less__top,axiom,
! [A: $tType] :
( top(A)
=> ! [Aa: A] :
( ( Aa != top_top(A) )
<=> ord_less(A,Aa,top_top(A)) ) ) ).
tff(fact_36_not__top__less,axiom,
! [A: $tType] :
( top(A)
=> ! [A4: A] : ~ ord_less(A,top_top(A),A4) ) ).
tff(fact_37_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,Y: B,X: A,A1: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A1)
=> ( member(A,X,A1)
=> ( ( aa(A,B,F,X) = Y )
=> ( aa(B,A,hilbert_inv_into(A,B,A1,F),Y) = X ) ) ) ) ).
tff(fact_38_inv__into__f__f,axiom,
! [B: $tType,A: $tType,X: A,A1: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A1)
=> ( member(A,X,A1)
=> ( aa(B,A,hilbert_inv_into(A,B,A1,F),aa(A,B,F,X)) = X ) ) ) ).
tff(fact_39_inv__f__eq,axiom,
! [B: $tType,A: $tType,Y: B,X: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ( aa(A,B,F,X) = Y )
=> ( aa(B,A,hilbert_inv_into(A,B,top_top(fun(A,bool)),F),Y) = X ) ) ) ).
tff(fact_40_inv__f__f,axiom,
! [B: $tType,A: $tType,X: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( aa(B,A,hilbert_inv_into(A,B,top_top(fun(A,bool)),F),aa(A,B,F,X)) = X ) ) ).
tff(fact_41_Sup__eq__top__iff,axiom,
! [A: $tType] :
( comple1035589618norder(A)
=> ! [A1: fun(A,bool)] :
( ( complete_Sup_Sup(A,A1) = top_top(A) )
<=> ! [X4: A] :
( ord_less(A,X4,top_top(A))
=> ? [Xa: A] :
( member(A,Xa,A1)
& ord_less(A,X4,Xa) ) ) ) ) ).
tff(fact_42_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ! [X1: B] : aa(A,B,F,aa(B,A,G,X1)) = X1
=> ( hilbert_inv_into(A,B,top_top(fun(A,bool)),F) = G ) ) ) ).
tff(fact_43_inj__on__id,axiom,
! [A: $tType,A1: fun(A,bool)] : inj_on(A,A,id(A),A1) ).
tff(fact_44_injD,axiom,
! [B: $tType,A: $tType,Y: A,X: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
=> ( X = Y ) ) ) ).
tff(fact_45_inj__eq,axiom,
! [B: $tType,A: $tType,Y: A,X: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_46__C3_C,axiom,
! [P3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: 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)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P3,arrow_610318064e_Prof)
=> ( ! [I1: 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),P1,I1))
<=> 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),P3,I1)) )
=> ( 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,P1))
<=> 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,P3)) ) ) ) ) ) ).
tff(fact_47_pairwise__neutrality,axiom,
! [P3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B5: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B5 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P3,arrow_610318064e_Prof)
=> ( ! [I1: 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),P1,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P3,I1)) )
=> ( 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,P1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),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,P3)) ) ) ) ) ) ) ).
tff(fact_48__C4_C,axiom,
! [P3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),C1: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( Ba != C1 )
=> ( ( Aa != C1 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P3,arrow_610318064e_Prof)
=> ( ! [I1: 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),P1,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,C1),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P3,I1)) )
=> ( 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,P1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,C1),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,P3)) ) ) ) ) ) ) ) ).
tff(fact_49_UnionI,axiom,
! [A: $tType,A1: A,C3: fun(fun(A,bool),bool),X6: fun(A,bool)] :
( member(fun(A,bool),X6,C3)
=> ( member(A,A1,X6)
=> member(A,A1,complete_Sup_Sup(fun(A,bool),C3)) ) ) ).
tff(fact_50__C2_C,axiom,
! [P3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B5: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B5 )
=> ( ( Aa != B5 )
=> ( ( Ba != A6 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P3,arrow_610318064e_Prof)
=> ( ! [I1: 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),P1,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P3,I1)) )
=> ( 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,P1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),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,P3)) ) ) ) ) ) ) ) ) ).
tff(fact_51__C1_C,axiom,
! [P3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B5: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B5 )
=> ( ( Aa != B5 )
=> ( ( Ba != A6 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P3,arrow_610318064e_Prof)
=> ( ! [I1: 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),P1,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P3,I1)) )
=> ( 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,P1))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),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,P3)) ) ) ) ) ) ) ) ) ).
tff(fact_52_inj__onD,axiom,
! [B: $tType,A: $tType,Y: A,X: A,A1: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A1)
=> ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
=> ( member(A,X,A1)
=> ( member(A,Y,A1)
=> ( X = Y ) ) ) ) ) ).
tff(fact_53_inj__on__iff,axiom,
! [B: $tType,A: $tType,Y: A,X: A,A1: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A1)
=> ( member(A,X,A1)
=> ( member(A,Y,A1)
=> ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
<=> ( X = Y ) ) ) ) ) ).
tff(fact_54_inj__on__contraD,axiom,
! [B: $tType,A: $tType,Y: A,X: A,A1: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A1)
=> ( ( X != Y )
=> ( member(A,X,A1)
=> ( member(A,Y,A1)
=> ( aa(A,B,F,X) != aa(A,B,F,Y) ) ) ) ) ) ).
tff(fact_55_inj__on__def,axiom,
! [B: $tType,A: $tType,A1: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A1)
<=> ! [X4: A] :
( member(A,X4,A1)
=> ! [Xa: A] :
( member(A,Xa,A1)
=> ( ( aa(A,B,F,X4) = aa(A,B,F,Xa) )
=> ( X4 = Xa ) ) ) ) ) ).
tff(fact_56_id__apply,axiom,
! [A: $tType,X2: A] : aa(A,A,id(A),X2) = X2 ).
tff(fact_57_id__def,axiom,
! [A: $tType,X5: A] : aa(A,A,id(A),X5) = X5 ).
tff(fact_58_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)
<=> ! [X4: 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)),X4,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)
=> ! [A3: arrow_411405190le_alt,B2: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa,I)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),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,X4))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),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_59_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)
<=> ! [X4: 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)),X4,arrow_610318064e_Prof)
=> ! [A3: arrow_411405190le_alt,B2: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] : member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),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,X4)) ) ) ) ).
tff(fact_60_UnionE,axiom,
! [A: $tType,C3: fun(fun(A,bool),bool),A1: A] :
( member(A,A1,complete_Sup_Sup(fun(A,bool),C3))
=> ~ ! [X3: fun(A,bool)] :
( member(A,A1,X3)
=> ~ member(fun(A,bool),X3,C3) ) ) ).
tff(fact_61_injI,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( ! [X1: A,Y1: A] :
( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
=> ( X1 = Y1 ) )
=> inj_on(A,B,F,top_top(fun(A,bool))) ) ).
tff(fact_62_split__paired__All,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ! [X12: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X12))
<=> ! [A3: A,B2: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A3,B2))) ) ).
tff(fact_63_Pair__eq,axiom,
! [A: $tType,B: $tType,B5: B,A6: A,Ba: B,Aa: A] :
( ( product_Pair(A,B,Aa,Ba) = product_Pair(A,B,A6,B5) )
<=> ( ( Aa = A6 )
& ( Ba = B5 ) ) ) ).
tff(fact_64_inj__onI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A1: fun(A,bool)] :
( ! [X1: A,Y1: A] :
( member(A,X1,A1)
=> ( member(A,Y1,A1)
=> ( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
=> ( X1 = Y1 ) ) ) )
=> inj_on(A,B,F,A1) ) ).
tff(fact_65_Pair__inject,axiom,
! [A: $tType,B: $tType,B4: B,A5: A,B3: B,A4: A] :
( ( product_Pair(A,B,A4,B3) = product_Pair(A,B,A5,B4) )
=> ~ ( ( A4 = A5 )
=> ( B3 != B4 ) ) ) ).
tff(fact_66_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ? [X12: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X12))
<=> ? [A3: A,B2: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A3,B2))) ) ).
tff(fact_67_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,F11: fun(B,fun(C,A))] : product_prod_rec(B,C,A,F11,product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),F11,Aa),Ba) ).
tff(fact_68_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(B,A),A1: fun(A,bool)] :
( ! [X1: A] :
( member(A,X1,A1)
=> ( aa(B,A,G,aa(A,B,F,X1)) = X1 ) )
=> inj_on(A,B,F,A1) ) ).
tff(fact_69_inj__on__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),A1: fun(A,bool)] :
( ! [A2: A] :
( member(A,A2,A1)
=> ( aa(A,B,F,A2) = aa(A,B,G,A2) ) )
=> ( inj_on(A,B,F,A1)
<=> inj_on(A,B,G,A1) ) ) ).
tff(fact_70_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod(A,B)] :
~ ! [A2: A,B1: B] : Y2 != product_Pair(A,B,A2,B1) ).
tff(fact_71_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X1: A,Y1: B] : P2 != product_Pair(A,B,X1,Y1) ).
tff(fact_72_prod__induct6,axiom,
! [F1: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool)] :
( ! [A2: A,B1: B,C2: C,D1: D,E1: E,F2: F1] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),A2,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1))),B1,product_Pair(C,product_prod(D,product_prod(E,F1)),C2,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,X)) ) ).
tff(fact_73_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F1: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))] :
~ ! [A2: A,B1: B,C2: C,D1: D,E1: E,F2: F1] : Y2 != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),A2,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1))),B1,product_Pair(C,product_prod(D,product_prod(E,F1)),C2,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2))))) ).
tff(fact_74_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A2: A,B1: B,C2: C,D1: D,E1: E] : Y2 != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A2,product_Pair(B,product_prod(C,product_prod(D,E)),B1,product_Pair(C,product_prod(D,E),C2,product_Pair(D,E,D1,E1)))) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X: A] :
( member(A,X,A1)
<=> pp(aa(A,bool,A1,X)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A2: A,B1: B,C2: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A2,product_Pair(B,product_prod(C,product_prod(D,E)),B1,product_Pair(C,product_prod(D,E),C2,product_Pair(D,E,D1,E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,X)) ) ).
tff(fact_79_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A2: A,B1: B,C2: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,product_Pair(A,product_prod(B,product_prod(C,D)),A2,product_Pair(B,product_prod(C,D),B1,product_Pair(C,D,C2,D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,X)) ) ).
tff(fact_80_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A2: A,B1: B,C2: C,D1: D] : Y2 != product_Pair(A,product_prod(B,product_prod(C,D)),A2,product_Pair(B,product_prod(C,D),B1,product_Pair(C,D,C2,D1))) ).
tff(fact_81_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y2: product_prod(A,product_prod(B,C))] :
~ ! [A2: A,B1: B,C2: C] : Y2 != product_Pair(A,product_prod(B,C),A2,product_Pair(B,C,B1,C2)) ).
tff(fact_82_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A2: A,B1: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P1,product_Pair(A,product_prod(B,C),A2,product_Pair(B,C,B1,C2))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,X)) ) ).
tff(fact_83_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,C1: fun(B,fun(C,A))] : produc1605651328_split(B,C,A,C1,product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),C1,Aa),Ba) ).
tff(fact_84_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,Ba: B,F: fun(B,A),Aa: A] :
( ord_less(A,Aa,aa(B,A,F,Ba))
=> ( ord_less(B,Ba,C1)
=> ( ! [X1: B,Y1: B] :
( ord_less(B,X1,Y1)
=> ord_less(A,aa(B,A,F,X1),aa(B,A,F,Y1)) )
=> ord_less(A,Aa,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_85_gt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A] :
? [X11: A] : ord_less(A,X,X11) ) ).
tff(fact_86_lt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X2: A] :
? [Y1: A] : ord_less(A,Y1,X2) ) ).
tff(fact_87_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,Ba: B,F: fun(B,A),Aa: A] :
( ( Aa = aa(B,A,F,Ba) )
=> ( ord_less(B,Ba,C1)
=> ( ! [X1: B,Y1: B] :
( ord_less(B,X1,Y1)
=> ord_less(A,aa(B,A,F,X1),aa(B,A,F,Y1)) )
=> ord_less(A,Aa,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_88_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F: fun(A,B),Ba: A,Aa: A] :
( ord_less(A,Aa,Ba)
=> ( ord_less(B,aa(A,B,F,Ba),C1)
=> ( ! [X1: A,Y1: A] :
( ord_less(A,X1,Y1)
=> ord_less(B,aa(A,B,F,X1),aa(A,B,F,Y1)) )
=> ord_less(B,aa(A,B,F,Aa),C1) ) ) ) ) ).
tff(fact_89_dense,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ? [Z1: A] :
( ord_less(A,X2,Z1)
& ord_less(A,Z1,Y2) ) ) ) ).
tff(fact_90_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,F: fun(A,B),Ba: A,Aa: A] :
( ord_less(A,Aa,Ba)
=> ( ( aa(A,B,F,Ba) = C1 )
=> ( ! [X1: A,Y1: A] :
( ord_less(A,X1,Y1)
=> ord_less(B,aa(A,B,F,X1),aa(A,B,F,Y1)) )
=> ord_less(B,aa(A,B,F,Aa),C1) ) ) ) ) ).
tff(fact_91_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,Ba: C,Aa: B,F: fun(product_prod(B,C),A)] : product_curry(B,C,A,F,Aa,Ba) = aa(product_prod(B,C),A,F,product_Pair(B,C,Aa,Ba)) ).
tff(fact_92_in__mktop,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y: arrow_411405190le_alt,X: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X,Y),arrow_424895264_mktop(L,Z))
<=> ( ( X != Z )
& ( ( Y = Z )
=> ( X != Y ) )
& ( ( Y != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X,Y),L) ) ) ) ).
tff(fact_93_curryI,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,Aa,Ba)))
=> pp(product_curry(A,B,bool,F,Aa,Ba)) ) ).
tff(fact_94_curryD,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,Aa,Ba))
=> pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_95_curryE,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,Aa,Ba))
=> pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_96_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y: arrow_411405190le_alt,X: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X,Y),arrow_276188178_mkbot(L,Z))
<=> ( ( Y != Z )
& ( ( X = Z )
=> ( X != Y ) )
& ( ( X != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X,Y),L) ) ) ) ).
tff(fact_97_Pow__UNIV,axiom,
! [A: $tType] : pow(A,top_top(fun(A,bool))) = top_top(fun(fun(A,bool),bool)) ).
tff(fact_98_Union__Pow__eq,axiom,
! [A: $tType,A1: fun(A,bool)] : complete_Sup_Sup(fun(A,bool),pow(A,A1)) = A1 ).
tff(fact_99_Pow__top,axiom,
! [A: $tType,A1: fun(A,bool)] : member(fun(A,bool),A1,pow(A,A1)) ).
%----Arities (15)
tff(arity_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [T_1: $tType,T_2: $tType] :
( comple187826305attice(T_2)
=> comple187826305attice(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_HOL_Obool___Complete__Lattices_Ocomplete__lattice,axiom,
comple187826305attice(bool) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Otop,axiom,
top(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (5)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
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_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
arrow_987702531ctator(f,aa(nat,arrow_159774573e_indi,hilbert_inv_into(arrow_159774573e_indi,nat,top_top(fun(arrow_159774573e_indi,bool)),h),n)) ).
%------------------------------------------------------------------------------