TPTP Problem File: SCT175_5.p
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%------------------------------------------------------------------------------
% File : SCT175_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 46
% 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_46 [Bla13]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v7.4.0, 0.75 v7.1.0, 1.00 v6.4.0
% Syntax : Number of formulae : 142 ( 30 unt; 35 typ; 0 def)
% Number of atoms : 247 ( 41 equ)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 173 ( 33 ~; 2 |; 7 &)
% ( 31 <=>; 100 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 45 ( 27 >; 18 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-6 aty)
% Number of functors : 19 ( 19 usr; 4 con; 0-6 aty)
% Number of variables : 484 ( 442 !; 4 ?; 484 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:03
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oalt,type,
arrow_411405190le_alt: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (31)
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $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_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_Order__Relation_Olinear__order__on,type,
order_1409979114der_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Order__Relation_Opartial__order__on,type,
order_915043626der_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Order__Relation_Ostrict__linear__order__on,type,
order_215145569der_on:
!>[A: $tType] : ( fun(A,bool) > fun(fun(product_prod(A,A),bool),bool) ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $o ) ).
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,C2: $tType] : ( ( fun(product_prod(A,B),C2) * A * B ) > C2 ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C2: $tType] : ( ( fun(A,fun(B,C2)) * product_prod(A,B) ) > C2 ) ).
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_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( fun(A,bool) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( fun(product_prod(A,B),bool) > fun(product_prod(B,A),bool) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( fun(product_prod(B,B),bool) * fun(A,B) ) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Relation_Osym,type,
sym:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > 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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_L,type,
l: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
%----Relevant facts (99)
tff(fact_0_converse__converse,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool)] : converse(B,A,converse(A,B,R)) = R ).
tff(fact_1_notin__Lin__iff,axiom,
! [Y1: arrow_411405190le_alt,X: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),La,arrow_1985332922le_Lin)
=> ( ( X != Y1 )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X,Y1),La)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Y1,X),La) ) ) ) ).
tff(fact_2_linear__order__on__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( order_1409979114der_on(A,A2,converse(A,A,R))
<=> order_1409979114der_on(A,A2,R) ) ).
tff(fact_3_Lin__def,axiom,
arrow_1985332922le_Lin = collect(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),order_215145569der_on(arrow_411405190le_alt,top_top(fun(arrow_411405190le_alt,bool)))) ).
tff(fact_4_preorder__on__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( order_preorder_on(A,A2,converse(A,A,R))
<=> order_preorder_on(A,A2,R) ) ).
tff(fact_5_partial__order__on__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( order_915043626der_on(A,A2,converse(A,A,R))
<=> order_915043626der_on(A,A2,R) ) ).
tff(fact_6_total__on__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( total_on(A,A2,converse(A,A,R))
<=> total_on(A,A2,R) ) ).
tff(fact_7_converse__inv__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(product_prod(B,B),bool)] : converse(A,A,inv_image(B,A,R1,F)) = inv_image(B,A,converse(B,B,R1),F) ).
tff(fact_8_antisym__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( antisym(A,converse(A,A,R))
<=> antisym(A,R) ) ).
tff(fact_9_acyclic__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( transitive_acyclic(A,converse(A,A,R))
<=> transitive_acyclic(A,R) ) ).
tff(fact_10_converse__Id__on,axiom,
! [A: $tType,A2: fun(A,bool)] : converse(A,A,id_on(A,A2)) = id_on(A,A2) ).
tff(fact_11_sym__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( sym(A,converse(A,A,R))
<=> sym(A,R) ) ).
tff(fact_12_converse__iff,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),B1: B,A1: A] :
( member(product_prod(A,B),product_Pair(A,B,A1,B1),converse(B,A,R))
<=> member(product_prod(B,A),product_Pair(B,A,B1,A1),R) ) ).
tff(fact_13_in__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(A,B),R: fun(product_prod(B,B),bool),Y1: A,X: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Y1),inv_image(B,A,R,F))
<=> member(product_prod(B,B),product_Pair(B,B,aa(A,B,F,X),aa(A,B,F,Y1)),R) ) ).
tff(fact_14_sym__Id__on,axiom,
! [A: $tType,A2: fun(A,bool)] : sym(A,id_on(A,A2)) ).
tff(fact_15_antisym__Id__on,axiom,
! [A: $tType,A2: fun(A,bool)] : antisym(A,id_on(A,A2)) ).
tff(fact_16_sym__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( sym(A,R)
<=> ! [X2: A,Y2: A] :
( member(product_prod(A,A),product_Pair(A,A,X2,Y2),R)
=> member(product_prod(A,A),product_Pair(A,A,Y2,X2),R) ) ) ).
tff(fact_17_antisym__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( antisym(A,R)
<=> ! [X2: A,Y2: A] :
( member(product_prod(A,A),product_Pair(A,A,X2,Y2),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Y2,X2),R)
=> ( X2 = Y2 ) ) ) ) ).
tff(fact_18_total__on__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( total_on(A,A2,R)
<=> ! [X2: A] :
( member(A,X2,A2)
=> ! [Xa: A] :
( member(A,Xa,A2)
=> ( ( X2 != Xa )
=> ( member(product_prod(A,A),product_Pair(A,A,X2,Xa),R)
| member(product_prod(A,A),product_Pair(A,A,Xa,X2),R) ) ) ) ) ) ).
tff(fact_19_Id__on__iff,axiom,
! [A: $tType,A2: fun(A,bool),Y1: A,X: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Y1),id_on(A,A2))
<=> ( ( X = Y1 )
& member(A,X,A2) ) ) ).
tff(fact_20_sym__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),R: fun(product_prod(A,A),bool)] :
( sym(A,R)
=> sym(B,inv_image(A,B,R,F)) ) ).
tff(fact_21_symD,axiom,
! [A: $tType,B1: A,A1: A,R: fun(product_prod(A,A),bool)] :
( sym(A,R)
=> ( member(product_prod(A,A),product_Pair(A,A,A1,B1),R)
=> member(product_prod(A,A),product_Pair(A,A,B1,A1),R) ) ) ).
tff(fact_22_antisymD,axiom,
! [A: $tType,B1: A,A1: A,R: fun(product_prod(A,A),bool)] :
( antisym(A,R)
=> ( member(product_prod(A,A),product_Pair(A,A,A1,B1),R)
=> ( member(product_prod(A,A),product_Pair(A,A,B1,A1),R)
=> ( A1 = B1 ) ) ) ) ).
tff(fact_23_Id__on__eqI,axiom,
! [A: $tType,A2: fun(A,bool),B1: A,A1: A] :
( ( A1 = B1 )
=> ( member(A,A1,A2)
=> member(product_prod(A,A),product_Pair(A,A,A1,B1),id_on(A,A2)) ) ) ).
tff(fact_24_partial__order__on__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( order_915043626der_on(A,A2,R)
<=> ( order_preorder_on(A,A2,R)
& antisym(A,R) ) ) ).
tff(fact_25_linear__order__on__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( order_1409979114der_on(A,A2,R)
<=> ( order_915043626der_on(A,A2,R)
& total_on(A,A2,R) ) ) ).
tff(fact_26_converseD,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),B1: B,A1: A] :
( member(product_prod(A,B),product_Pair(A,B,A1,B1),converse(B,A,R))
=> member(product_prod(B,A),product_Pair(B,A,B1,A1),R) ) ).
tff(fact_27_converseI,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool),B1: B,A1: A] :
( member(product_prod(A,B),product_Pair(A,B,A1,B1),R)
=> member(product_prod(B,A),product_Pair(B,A,B1,A1),converse(A,B,R)) ) ).
tff(fact_28_sym__conv__converse__eq,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( sym(A,R)
<=> ( converse(A,A,R) = R ) ) ).
tff(fact_29_Id__onE,axiom,
! [A: $tType,A2: fun(A,bool),C1: product_prod(A,A)] :
( member(product_prod(A,A),C1,id_on(A,A2))
=> ~ ! [X1: A] :
( member(A,X1,A2)
=> ( C1 != product_Pair(A,A,X1,X1) ) ) ) ).
tff(fact_30_converseE,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),Yx: product_prod(A,B)] :
( member(product_prod(A,B),Yx,converse(B,A,R))
=> ~ ! [X1: B,Y: A] :
( ( Yx = product_Pair(A,B,Y,X1) )
=> ~ member(product_prod(B,A),product_Pair(B,A,X1,Y),R) ) ) ).
tff(fact_31_antisymI,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( ! [X1: A,Y: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Y,X1),R)
=> ( X1 = Y ) ) )
=> antisym(A,R) ) ).
tff(fact_32_symI,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( ! [A3: A,B2: A] :
( member(product_prod(A,A),product_Pair(A,A,A3,B2),R)
=> member(product_prod(A,A),product_Pair(A,A,B2,A3),R) )
=> sym(A,R) ) ).
tff(fact_33_top1I,axiom,
! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).
tff(fact_34_UNIV__I,axiom,
! [A: $tType,X: A] : member(A,X,top_top(fun(A,bool))) ).
tff(fact_35_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : member(A,X,top_top(fun(A,bool))) ).
tff(fact_36_split__paired__All,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ! [A4: A,B3: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A4,B3))) ) ).
tff(fact_37_Pair__eq,axiom,
! [A: $tType,B: $tType,B6: B,A7: A,B1: B,A1: A] :
( ( product_Pair(A,B,A1,B1) = product_Pair(A,B,A7,B6) )
<=> ( ( A1 = A7 )
& ( B1 = B6 ) ) ) ).
tff(fact_38_Pair__inject,axiom,
! [A: $tType,B: $tType,B5: B,A6: A,B4: B,A5: A] :
( ( product_Pair(A,B,A5,B4) = product_Pair(A,B,A6,B5) )
=> ~ ( ( A5 = A6 )
=> ( B4 != B5 ) ) ) ).
tff(fact_39_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_40_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ? [A4: A,B3: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A4,B3))) ) ).
tff(fact_41_prod_Orecs,axiom,
! [B: $tType,A: $tType,C2: $tType,B1: C2,A1: B,F11: fun(B,fun(C2,A))] : product_prod_rec(B,C2,A,F11,product_Pair(B,C2,A1,B1)) = aa(C2,A,aa(B,fun(C2,A),F11,A1),B1) ).
tff(fact_42_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_43_UNIV__eq__I,axiom,
! [A: $tType,A2: fun(A,bool)] :
( ! [X1: A] : member(A,X1,A2)
=> ( top_top(fun(A,bool)) = A2 ) ) ).
tff(fact_44_UNIV__witness,axiom,
! [A: $tType] :
? [X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_45_prod__induct6,axiom,
! [F1: $tType,E: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,product_prod(C2,product_prod(D,product_prod(E,F1))))),P1: fun(product_prod(A,product_prod(B,product_prod(C2,product_prod(D,product_prod(E,F1))))),bool)] :
( ! [A3: A,B2: B,C: C2,D1: D,E1: E,F2: F1] : pp(aa(product_prod(A,product_prod(B,product_prod(C2,product_prod(D,product_prod(E,F1))))),bool,P1,product_Pair(A,product_prod(B,product_prod(C2,product_prod(D,product_prod(E,F1)))),A3,product_Pair(B,product_prod(C2,product_prod(D,product_prod(E,F1))),B2,product_Pair(C2,product_prod(D,product_prod(E,F1)),C,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C2,product_prod(D,product_prod(E,F1))))),bool,P1,X)) ) ).
tff(fact_46_prod__cases6,axiom,
! [A: $tType,B: $tType,C2: $tType,D: $tType,E: $tType,F1: $tType,Y3: product_prod(A,product_prod(B,product_prod(C2,product_prod(D,product_prod(E,F1)))))] :
~ ! [A3: A,B2: B,C: C2,D1: D,E1: E,F2: F1] : Y3 != product_Pair(A,product_prod(B,product_prod(C2,product_prod(D,product_prod(E,F1)))),A3,product_Pair(B,product_prod(C2,product_prod(D,product_prod(E,F1))),B2,product_Pair(C2,product_prod(D,product_prod(E,F1)),C,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2))))) ).
tff(fact_47_prod__induct5,axiom,
! [E: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,product_prod(C2,product_prod(D,E)))),P1: fun(product_prod(A,product_prod(B,product_prod(C2,product_prod(D,E)))),bool)] :
( ! [A3: A,B2: B,C: C2,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C2,product_prod(D,E)))),bool,P1,product_Pair(A,product_prod(B,product_prod(C2,product_prod(D,E))),A3,product_Pair(B,product_prod(C2,product_prod(D,E)),B2,product_Pair(C2,product_prod(D,E),C,product_Pair(D,E,D1,E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C2,product_prod(D,E)))),bool,P1,X)) ) ).
tff(fact_48_prod__cases5,axiom,
! [A: $tType,B: $tType,C2: $tType,D: $tType,E: $tType,Y3: product_prod(A,product_prod(B,product_prod(C2,product_prod(D,E))))] :
~ ! [A3: A,B2: B,C: C2,D1: D,E1: E] : Y3 != product_Pair(A,product_prod(B,product_prod(C2,product_prod(D,E))),A3,product_Pair(B,product_prod(C2,product_prod(D,E)),B2,product_Pair(C2,product_prod(D,E),C,product_Pair(D,E,D1,E1)))) ).
tff(fact_49_prod__cases4,axiom,
! [A: $tType,B: $tType,C2: $tType,D: $tType,Y3: product_prod(A,product_prod(B,product_prod(C2,D)))] :
~ ! [A3: A,B2: B,C: C2,D1: D] : Y3 != product_Pair(A,product_prod(B,product_prod(C2,D)),A3,product_Pair(B,product_prod(C2,D),B2,product_Pair(C2,D,C,D1))) ).
tff(fact_50_prod__induct4,axiom,
! [D: $tType,C2: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,product_prod(C2,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C2,D))),bool)] :
( ! [A3: A,B2: B,C: C2,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C2,D))),bool,P1,product_Pair(A,product_prod(B,product_prod(C2,D)),A3,product_Pair(B,product_prod(C2,D),B2,product_Pair(C2,D,C,D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C2,D))),bool,P1,X)) ) ).
tff(fact_51_prod__cases3,axiom,
! [A: $tType,B: $tType,C2: $tType,Y3: product_prod(A,product_prod(B,C2))] :
~ ! [A3: A,B2: B,C: C2] : Y3 != product_Pair(A,product_prod(B,C2),A3,product_Pair(B,C2,B2,C)) ).
tff(fact_52_prod__induct3,axiom,
! [C2: $tType,B: $tType,A: $tType,X: product_prod(A,product_prod(B,C2)),P1: fun(product_prod(A,product_prod(B,C2)),bool)] :
( ! [A3: A,B2: B,C: C2] : pp(aa(product_prod(A,product_prod(B,C2)),bool,P1,product_Pair(A,product_prod(B,C2),A3,product_Pair(B,C2,B2,C))))
=> pp(aa(product_prod(A,product_prod(B,C2)),bool,P1,X)) ) ).
tff(fact_53_PairE,axiom,
! [A: $tType,B: $tType,P3: product_prod(A,B)] :
~ ! [X1: A,Y: B] : P3 != product_Pair(A,B,X1,Y) ).
tff(fact_54_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod(A,B)] :
~ ! [A3: A,B2: B] : Y3 != product_Pair(A,B,A3,B2) ).
tff(fact_55_internal__split__conv,axiom,
! [B: $tType,A: $tType,C2: $tType,B1: C2,A1: B,C1: fun(B,fun(C2,A))] : produc1605651328_split(B,C2,A,C1,product_Pair(B,C2,A1,B1)) = aa(C2,A,aa(B,fun(C2,A),C1,A1),B1) ).
tff(fact_56_curry__conv,axiom,
! [A: $tType,B: $tType,C2: $tType,B1: C2,A1: B,F: fun(product_prod(B,C2),A)] : product_curry(B,C2,A,F,A1,B1) = aa(product_prod(B,C2),A,F,product_Pair(B,C2,A1,B1)) ).
tff(fact_57_curryI,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,A1,B1)))
=> pp(product_curry(A,B,bool,F,A1,B1)) ) ).
tff(fact_58_curryE,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,A1,B1))
=> pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,A1,B1))) ) ).
tff(fact_59_curryD,axiom,
! [A: $tType,B: $tType,B1: B,A1: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,A1,B1))
=> pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,A1,B1))) ) ).
tff(fact_60_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y1: B,X: B,F: fun(B,A),R: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R,F,X,Y1)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,F,X)),aa(B,A,F,Y1))) ) ).
tff(fact_61_irrefl__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( irrefl(A,R)
<=> ! [X2: A] : ~ member(product_prod(A,A),product_Pair(A,A,X2,X2),R) ) ).
tff(fact_62_strict__linear__order__on__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( pp(aa(fun(product_prod(A,A),bool),bool,order_215145569der_on(A,A2),R))
<=> ( trans(A,R)
& irrefl(A,R)
& total_on(A,A2,R) ) ) ).
tff(fact_63_trans__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( trans(A,converse(A,A,R))
<=> trans(A,R) ) ).
tff(fact_64_transD,axiom,
! [A: $tType,C1: A,B1: A,A1: A,R: fun(product_prod(A,A),bool)] :
( trans(A,R)
=> ( member(product_prod(A,A),product_Pair(A,A,A1,B1),R)
=> ( member(product_prod(A,A),product_Pair(A,A,B1,C1),R)
=> member(product_prod(A,A),product_Pair(A,A,A1,C1),R) ) ) ) ).
tff(fact_65_trans__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( trans(A,R)
<=> ! [X2: A,Y2: A,Z2: A] :
( member(product_prod(A,A),product_Pair(A,A,X2,Y2),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Y2,Z2),R)
=> member(product_prod(A,A),product_Pair(A,A,X2,Z2),R) ) ) ) ).
tff(fact_66_trans__Id__on,axiom,
! [A: $tType,A2: fun(A,bool)] : trans(A,id_on(A,A2)) ).
tff(fact_67_trans__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),R: fun(product_prod(A,A),bool)] :
( trans(A,R)
=> trans(B,inv_image(A,B,R,F)) ) ).
tff(fact_68_transI,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( ! [X1: A,Y: A,Z: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Y,Z),R)
=> member(product_prod(A,A),product_Pair(A,A,X1,Z),R) ) )
=> trans(A,R) ) ).
tff(fact_69_preorder__on__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( order_preorder_on(A,A2,R)
<=> ( refl_on(A,A2,R)
& trans(A,R) ) ) ).
tff(fact_70_acyclic__irrefl,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( transitive_acyclic(A,R)
<=> irrefl(A,transitive_trancl(A,R)) ) ).
tff(fact_71_trancl_Or__into__trancl,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),R)
=> member(product_prod(A,A),product_Pair(A,A,A1,B1),transitive_trancl(A,R)) ) ).
tff(fact_72_refl__on__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( refl_on(A,A2,converse(A,A,R))
<=> refl_on(A,A2,R) ) ).
tff(fact_73_trancl__id,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( trans(A,R)
=> ( transitive_trancl(A,R) = R ) ) ).
tff(fact_74_trans__trancl,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] : trans(A,transitive_trancl(A,R)) ).
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,A2: fun(A,bool),X: A] :
( member(A,X,A2)
<=> pp(aa(A,bool,A2,X)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_refl__on__Id__on,axiom,
! [A: $tType,A2: fun(A,bool)] : refl_on(A,A2,id_on(A,A2)) ).
tff(fact_79_trancl__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] : transitive_trancl(A,converse(A,A,R)) = converse(A,A,transitive_trancl(A,R)) ).
tff(fact_80_sym__trancl,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( sym(A,R)
=> sym(A,transitive_trancl(A,R)) ) ).
tff(fact_81_r__into__trancl_H,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),P2: product_prod(A,A)] :
( member(product_prod(A,A),P2,R)
=> member(product_prod(A,A),P2,transitive_trancl(A,R)) ) ).
tff(fact_82_refl__onD,axiom,
! [A: $tType,A1: A,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( refl_on(A,A2,R)
=> ( member(A,A1,A2)
=> member(product_prod(A,A),product_Pair(A,A,A1,A1),R) ) ) ).
tff(fact_83_refl__onD1,axiom,
! [A: $tType,Y1: A,X: A,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( refl_on(A,A2,R)
=> ( member(product_prod(A,A),product_Pair(A,A,X,Y1),R)
=> member(A,X,A2) ) ) ).
tff(fact_84_refl__onD2,axiom,
! [A: $tType,Y1: A,X: A,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( refl_on(A,A2,R)
=> ( member(product_prod(A,A),product_Pair(A,A,X,Y1),R)
=> member(A,Y1,A2) ) ) ).
tff(fact_85_trancl__trans,axiom,
! [A: $tType,C1: A,R: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),transitive_trancl(A,R))
=> ( member(product_prod(A,A),product_Pair(A,A,B1,C1),transitive_trancl(A,R))
=> member(product_prod(A,A),product_Pair(A,A,A1,C1),transitive_trancl(A,R)) ) ) ).
tff(fact_86_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,C1: A,R: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),transitive_trancl(A,R))
=> ( member(product_prod(A,A),product_Pair(A,A,B1,C1),R)
=> member(product_prod(A,A),product_Pair(A,A,A1,C1),transitive_trancl(A,R)) ) ) ).
tff(fact_87_trancl__into__trancl2,axiom,
! [A: $tType,C1: A,R: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),R)
=> ( member(product_prod(A,A),product_Pair(A,A,B1,C1),transitive_trancl(A,R))
=> member(product_prod(A,A),product_Pair(A,A,A1,C1),transitive_trancl(A,R)) ) ) ).
tff(fact_88_r__r__into__trancl,axiom,
! [A: $tType,C1: A,R1: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),R1)
=> ( member(product_prod(A,A),product_Pair(A,A,B1,C1),R1)
=> member(product_prod(A,A),product_Pair(A,A,A1,C1),transitive_trancl(A,R1)) ) ) ).
tff(fact_89_trancl__converseD,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),Y1: A,X: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Y1),transitive_trancl(A,converse(A,A,R)))
=> member(product_prod(A,A),product_Pair(A,A,X,Y1),converse(A,A,transitive_trancl(A,R))) ) ).
tff(fact_90_trancl__converseI,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),Y1: A,X: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Y1),converse(A,A,transitive_trancl(A,R)))
=> member(product_prod(A,A),product_Pair(A,A,X,Y1),transitive_trancl(A,converse(A,A,R))) ) ).
tff(fact_91_acyclic__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( transitive_acyclic(A,R)
<=> ! [X2: A] : ~ member(product_prod(A,A),product_Pair(A,A,X2,X2),transitive_trancl(A,R)) ) ).
tff(fact_92_acyclicI,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( ! [X1: A] : ~ member(product_prod(A,A),product_Pair(A,A,X1,X1),transitive_trancl(A,R))
=> transitive_acyclic(A,R) ) ).
tff(fact_93_tranclE,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),transitive_trancl(A,R))
=> ( ~ member(product_prod(A,A),product_Pair(A,A,A1,B1),R)
=> ~ ! [C: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,C),transitive_trancl(A,R))
=> ~ member(product_prod(A,A),product_Pair(A,A,C,B1),R) ) ) ) ).
tff(fact_94_converse__tranclE,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),Z1: A,X: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Z1),transitive_trancl(A,R))
=> ( ~ member(product_prod(A,A),product_Pair(A,A,X,Z1),R)
=> ~ ! [Y: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Y),R)
=> ~ member(product_prod(A,A),product_Pair(A,A,Y,Z1),transitive_trancl(A,R)) ) ) ) ).
tff(fact_95_irrefl__trancl__rD,axiom,
! [A: $tType,Y1: A,X: A,R: fun(product_prod(A,A),bool)] :
( ! [X1: A] : ~ member(product_prod(A,A),product_Pair(A,A,X1,X1),transitive_trancl(A,R))
=> ( member(product_prod(A,A),product_Pair(A,A,X,Y1),R)
=> ( X != Y1 ) ) ) ).
tff(fact_96_trancl__trans__induct,axiom,
! [A: $tType,P1: fun(A,fun(A,bool)),R: fun(product_prod(A,A),bool),Y1: A,X: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Y1),transitive_trancl(A,R))
=> ( ! [X1: A,Y: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y),R)
=> pp(aa(A,bool,aa(A,fun(A,bool),P1,X1),Y)) )
=> ( ! [X1: A,Y: A,Z: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y),transitive_trancl(A,R))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P1,X1),Y))
=> ( member(product_prod(A,A),product_Pair(A,A,Y,Z),transitive_trancl(A,R))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P1,Y),Z))
=> pp(aa(A,bool,aa(A,fun(A,bool),P1,X1),Z)) ) ) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),P1,X),Y1)) ) ) ) ).
tff(fact_97_converse__trancl__induct,axiom,
! [A: $tType,P1: fun(A,bool),R: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),transitive_trancl(A,R))
=> ( ! [Y: A] :
( member(product_prod(A,A),product_Pair(A,A,Y,B1),R)
=> pp(aa(A,bool,P1,Y)) )
=> ( ! [Y: A,Z: A] :
( member(product_prod(A,A),product_Pair(A,A,Y,Z),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Z,B1),transitive_trancl(A,R))
=> ( pp(aa(A,bool,P1,Z))
=> pp(aa(A,bool,P1,Y)) ) ) )
=> pp(aa(A,bool,P1,A1)) ) ) ) ).
tff(fact_98_trancl__induct,axiom,
! [A: $tType,P1: fun(A,bool),R: fun(product_prod(A,A),bool),B1: A,A1: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,B1),transitive_trancl(A,R))
=> ( ! [Y: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,Y),R)
=> pp(aa(A,bool,P1,Y)) )
=> ( ! [Y: A,Z: A] :
( member(product_prod(A,A),product_Pair(A,A,A1,Y),transitive_trancl(A,R))
=> ( member(product_prod(A,A),product_Pair(A,A,Y,Z),R)
=> ( pp(aa(A,bool,P1,Y))
=> pp(aa(A,bool,P1,Z)) ) ) )
=> pp(aa(A,bool,P1,B1)) ) ) ) ).
%----Arities (2)
tff(arity_fun___Orderings_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Otop,axiom,
top(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,
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),converse(arrow_411405190le_alt,arrow_411405190le_alt,l),arrow_1985332922le_Lin)
<=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),l,arrow_1985332922le_Lin) ) ).
%------------------------------------------------------------------------------