TPTP Problem File: SCT179_5.p
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%------------------------------------------------------------------------------
% File : SCT179_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 56
% 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_56 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.5.0, 0.00 v7.4.0, 0.50 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 173 ( 59 unt; 47 typ; 0 def)
% Number of atoms : 224 ( 57 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 119 ( 21 ~; 8 |; 8 &)
% ( 22 <=>; 60 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 46 ( 28 >; 18 *; 0 +; 0 <<)
% Number of predicates : 20 ( 19 usr; 0 prp; 1-6 aty)
% Number of functors : 25 ( 25 usr; 6 con; 0-5 aty)
% Number of variables : 354 ( 303 !; 4 ?; 354 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:10
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oalt,type,
arrow_411405190le_alt: $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 (42)
tff(sy_cl_Groups_Ominus,type,
cl_Groups_Ominus:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[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_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_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > 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_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
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_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $o ) ).
tff(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( fun(product_prod(A,B),bool) > fun(A,bool) ) ).
tff(sy_c_Relation_OId,type,
id:
!>[A: $tType] : fun(product_prod(A,A),bool) ).
tff(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( fun(A,bool) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Relation_ORange,type,
range:
!>[A: $tType,B: $tType] : ( fun(product_prod(A,B),bool) > fun(B,bool) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
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_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_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,A),bool) * fun(product_prod(B,B),bool) ) > fun(product_prod(product_prod(A,B),product_prod(A,B)),bool) ) ).
tff(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( fun(A,nat) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
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_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : fun(A,fun(fun(A,bool),bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_r,type,
r: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
%----Relevant facts (100)
tff(fact_0_strict__linear__order__on__diff__Id,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( order_1409979114der_on(A,A1,R1)
=> pp(aa(fun(product_prod(A,A),bool),bool,order_215145569der_on(A,A1),minus_minus(fun(product_prod(A,A),bool),R1,id(A)))) ) ).
tff(fact_1_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_2_Diff__iff,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),minus_minus(fun(A,bool),A1,B1)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1))
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B1)) ) ) ).
tff(fact_3_Diff__idemp,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] : minus_minus(fun(A,bool),minus_minus(fun(A,bool),A1,B1),B1) = minus_minus(fun(A,bool),A1,B1) ).
tff(fact_4_DiffI,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),minus_minus(fun(A,bool),A1,B1))) ) ) ).
tff(fact_5_DiffE,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),minus_minus(fun(A,bool),A1,B1)))
=> ~ ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B1)) ) ) ).
tff(fact_6_top1I,axiom,
! [A: $tType,X: A] : pp(aa(A,bool,top_top(fun(A,bool)),X)) ).
tff(fact_7_UNIV__I,axiom,
! [A: $tType,X: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),top_top(fun(A,bool)))) ).
tff(fact_8_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),top_top(fun(A,bool)))) ).
tff(fact_9_well__order__on__def,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( order_well_order_on(A,A1,R1)
<=> ( order_1409979114der_on(A,A1,R1)
& wf(A,minus_minus(fun(product_prod(A,A),bool),R1,id(A))) ) ) ).
tff(fact_10_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B1: fun(A,B),A1: fun(A,B),X3: A] : aa(A,B,minus_minus(fun(A,B),A1,B1),X3) = minus_minus(B,aa(A,B,A1,X3),aa(A,B,B1,X3)) ) ).
tff(fact_11_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [X: B,B1: fun(B,A),A1: fun(B,A)] : aa(B,A,minus_minus(fun(B,A),A1,B1),X) = minus_minus(A,aa(B,A,A1,X),aa(B,A,B1,X)) ) ).
tff(fact_12_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_13_DiffD2,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),minus_minus(fun(A,bool),A1,B1)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B1)) ) ).
tff(fact_14_DiffD1,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),minus_minus(fun(A,bool),A1,B1)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1)) ) ).
tff(fact_15_set__diff__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] : minus_minus(fun(A,bool),A1,B1) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,combc(A,fun(A,bool),bool,member(A),A1)),combb(bool,bool,A,fNot,combc(A,fun(A,bool),bool,member(A),B1)))) ).
tff(fact_16_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_17_UNIV__eq__I,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X1: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A1))
=> ( top_top(fun(A,bool)) = A1 ) ) ).
tff(fact_18_UNIV__witness,axiom,
! [A: $tType] :
? [X1: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),top_top(fun(A,bool)))) ).
tff(fact_19_irrefl__diff__Id,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool)] : irrefl(A,minus_minus(fun(product_prod(A,A),bool),R1,id(A))) ).
tff(fact_20_total__on__diff__Id,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( total_on(A,A1,minus_minus(fun(product_prod(A,A),bool),R1,id(A)))
<=> total_on(A,A1,R1) ) ).
tff(fact_21_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D1: A,C2: A,B2: A,A3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C2,D1) )
=> ( ( A3 = B2 )
<=> ( C2 = D1 ) ) ) ) ).
tff(fact_22_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y1: B,X: B,F: fun(B,A),R1: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R1,F,X,Y1)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R1,aa(B,A,F,X)),aa(B,A,F,Y1))) ) ).
tff(fact_23_well__order__on,axiom,
! [A: $tType,A1: fun(A,bool)] :
? [X11: fun(product_prod(A,A),bool)] : order_well_order_on(A,A1,X11) ).
tff(fact_24_Diff__UNIV,axiom,
! [A: $tType,A1: fun(A,bool)] : minus_minus(fun(A,bool),A1,top_top(fun(A,bool))) = bot_bot(fun(A,bool)) ).
tff(fact_25_all__not__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X4: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A1))
<=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_26_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X4: A] : ~ pp(aa(A,bool,P1,X4)) ) ).
tff(fact_27_empty__iff,axiom,
! [A: $tType,C2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),bot_bot(fun(A,bool)))) ).
tff(fact_28_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X4: A] : ~ pp(aa(A,bool,P1,X4)) ) ).
tff(fact_29_emptyE,axiom,
! [A: $tType,A3: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),bot_bot(fun(A,bool)))) ).
tff(fact_30_empty__Diff,axiom,
! [A: $tType,A1: fun(A,bool)] : minus_minus(fun(A,bool),bot_bot(fun(A,bool)),A1) = bot_bot(fun(A,bool)) ).
tff(fact_31_Diff__cancel,axiom,
! [A: $tType,A1: fun(A,bool)] : minus_minus(fun(A,bool),A1,A1) = bot_bot(fun(A,bool)) ).
tff(fact_32_bot__empty__eq,axiom,
! [A: $tType,X3: A] :
( pp(aa(A,bool,bot_bot(fun(A,bool)),X3))
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),bot_bot(fun(A,bool)))) ) ).
tff(fact_33_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,combk(bool,A,fFalse)) ).
tff(fact_34_ex__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ? [X4: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A1))
<=> ( A1 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_35_equals0D,axiom,
! [A: $tType,A3: A,A1: fun(A,bool)] :
( ( A1 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A1)) ) ).
tff(fact_36_bot__apply,axiom,
! [B: $tType,A: $tType] :
( bot(A)
=> ! [X: B] : aa(B,A,bot_bot(fun(B,A)),X) = bot_bot(A) ) ).
tff(fact_37_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X3: A] : aa(A,B,bot_bot(fun(A,B)),X3) = bot_bot(B) ) ).
tff(fact_38_total__on__empty,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool)] : total_on(A,bot_bot(fun(A,bool)),R1) ).
tff(fact_39_lnear__order__on__empty,axiom,
! [A: $tType] : order_1409979114der_on(A,bot_bot(fun(A,bool)),bot_bot(fun(product_prod(A,A),bool))) ).
tff(fact_40_well__order__on__empty,axiom,
! [A: $tType] : order_well_order_on(A,bot_bot(fun(A,bool)),bot_bot(fun(product_prod(A,A),bool))) ).
tff(fact_41_UNIV__not__empty,axiom,
! [A: $tType] : top_top(fun(A,bool)) != bot_bot(fun(A,bool)) ).
tff(fact_42_Diff__empty,axiom,
! [A: $tType,A1: fun(A,bool)] : minus_minus(fun(A,bool),A1,bot_bot(fun(A,bool))) = A1 ).
tff(fact_43_wf__empty,axiom,
! [A: $tType] : wf(A,bot_bot(fun(product_prod(A,A),bool))) ).
tff(fact_44_linear__order__on__def,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( order_1409979114der_on(A,A1,R1)
<=> ( order_915043626der_on(A,A1,R1)
& total_on(A,A1,R1) ) ) ).
tff(fact_45_strict__linear__order__on__def,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( pp(aa(fun(product_prod(A,A),bool),bool,order_215145569der_on(A,A1),R1))
<=> ( trans(A,R1)
& irrefl(A,R1)
& total_on(A,A1,R1) ) ) ).
tff(fact_46_equals0I,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [Y: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y),A1))
=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_47_Range__Id,axiom,
! [A: $tType] : range(A,A,id(A)) = top_top(fun(A,bool)) ).
tff(fact_48_Domain__Id,axiom,
! [A: $tType] : domain(A,A,id(A)) = top_top(fun(A,bool)) ).
tff(fact_49_refl__Id,axiom,
! [A: $tType] : refl_on(A,top_top(fun(A,bool)),id(A)) ).
tff(fact_50_Domain__empty,axiom,
! [B: $tType,A: $tType] : domain(A,B,bot_bot(fun(product_prod(A,B),bool))) = bot_bot(fun(A,bool)) ).
tff(fact_51_Range__empty,axiom,
! [B: $tType,A: $tType] : range(B,A,bot_bot(fun(product_prod(B,A),bool))) = bot_bot(fun(A,bool)) ).
tff(fact_52_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R1: fun(product_prod(A,B),bool)] :
( ( domain(A,B,R1) = bot_bot(fun(A,bool)) )
<=> ( R1 = bot_bot(fun(product_prod(A,B),bool)) ) ) ).
tff(fact_53_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R1: fun(product_prod(B,A),bool)] :
( ( range(B,A,R1) = bot_bot(fun(A,bool)) )
<=> ( R1 = bot_bot(fun(product_prod(B,A),bool)) ) ) ).
tff(fact_54_refl__on__empty,axiom,
! [A: $tType] : refl_on(A,bot_bot(fun(A,bool)),bot_bot(fun(product_prod(A,A),bool))) ).
tff(fact_55_trans__Id,axiom,
! [A: $tType] : trans(A,id(A)) ).
tff(fact_56_partial__order__on__empty,axiom,
! [A: $tType] : order_915043626der_on(A,bot_bot(fun(A,bool)),bot_bot(fun(product_prod(A,A),bool))) ).
tff(fact_57_trans__lex__prod,axiom,
! [A: $tType,B: $tType,R2: fun(product_prod(B,B),bool),R11: fun(product_prod(A,A),bool)] :
( trans(A,R11)
=> ( trans(B,R2)
=> trans(product_prod(A,B),lex_prod(A,B,R11,R2)) ) ) ).
tff(fact_58_wf__lex__prod,axiom,
! [A: $tType,B: $tType,Rb: fun(product_prod(B,B),bool),Ra: fun(product_prod(A,A),bool)] :
( wf(A,Ra)
=> ( wf(B,Rb)
=> wf(product_prod(A,B),lex_prod(A,B,Ra,Rb)) ) ) ).
tff(fact_59_wf__measure,axiom,
! [A: $tType,F: fun(A,nat)] : wf(A,measure(A,F)) ).
tff(fact_60_trans__diff__Id,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool)] :
( trans(A,R1)
=> ( antisym(A,R1)
=> trans(A,minus_minus(fun(product_prod(A,A),bool),R1,id(A))) ) ) ).
tff(fact_61_antisym__empty,axiom,
! [A: $tType] : antisym(A,bot_bot(fun(product_prod(A,A),bool))) ).
tff(fact_62_antisym__Id,axiom,
! [A: $tType] : antisym(A,id(A)) ).
tff(fact_63_partial__order__on__def,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( order_915043626der_on(A,A1,R1)
<=> ( order_preorder_on(A,A1,R1)
& antisym(A,R1) ) ) ).
tff(fact_64_preorder__on__def,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( order_preorder_on(A,A1,R1)
<=> ( refl_on(A,A1,R1)
& trans(A,R1) ) ) ).
tff(fact_65_preorder__on__empty,axiom,
! [A: $tType] : order_preorder_on(A,bot_bot(fun(A,bool)),bot_bot(fun(product_prod(A,A),bool))) ).
tff(fact_66_Id__on__empty,axiom,
! [A: $tType] : id_on(A,bot_bot(fun(A,bool))) = bot_bot(fun(product_prod(A,A),bool)) ).
tff(fact_67_Range__Diff__subset,axiom,
! [A: $tType,B: $tType,B1: fun(product_prod(B,A),bool),A1: fun(product_prod(B,A),bool)] : ord_less_eq(fun(A,bool),minus_minus(fun(A,bool),range(B,A,A1),range(B,A,B1)),range(B,A,minus_minus(fun(product_prod(B,A),bool),A1,B1))) ).
tff(fact_68_Domain__Diff__subset,axiom,
! [B: $tType,A: $tType,B1: fun(product_prod(A,B),bool),A1: fun(product_prod(A,B),bool)] : ord_less_eq(fun(A,bool),minus_minus(fun(A,bool),domain(A,B,A1),domain(A,B,B1)),domain(A,B,minus_minus(fun(product_prod(A,B),bool),A1,B1))) ).
tff(fact_69_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ord_less_eq(A,X2,X2) ) ).
tff(fact_70_subsetD,axiom,
! [A: $tType,C2: A,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B1)) ) ) ).
tff(fact_71_equalityI,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( ord_less_eq(fun(A,bool),B1,A1)
=> ( A1 = B1 ) ) ) ).
tff(fact_72_subset__empty,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,bot_bot(fun(A,bool)))
<=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_73_empty__subsetI,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),bot_bot(fun(A,bool)),A1) ).
tff(fact_74_Domain__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : domain(A,A,id_on(A,A1)) = A1 ).
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] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),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_trans__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : trans(A,id_on(A,A1)) ).
tff(fact_79_Domain__mono,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R1: fun(product_prod(A,B),bool)] :
( ord_less_eq(fun(product_prod(A,B),bool),R1,S)
=> ord_less_eq(fun(A,bool),domain(A,B,R1),domain(A,B,S)) ) ).
tff(fact_80_bot__least,axiom,
! [A: $tType] :
( bot(A)
=> ! [A2: A] : ord_less_eq(A,bot_bot(A),A2) ) ).
tff(fact_81_bot__unique,axiom,
! [A: $tType] :
( bot(A)
=> ! [A3: A] :
( ord_less_eq(A,A3,bot_bot(A))
<=> ( A3 = bot_bot(A) ) ) ) ).
tff(fact_82_le__bot,axiom,
! [A: $tType] :
( bot(A)
=> ! [A2: A] :
( ord_less_eq(A,A2,bot_bot(A))
=> ( A2 = bot_bot(A) ) ) ) ).
tff(fact_83_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D1: A,C2: A,B2: A,A3: A] :
( ( minus_minus(A,A3,B2) = minus_minus(A,C2,D1) )
=> ( ord_less_eq(A,A3,B2)
<=> ord_less_eq(A,C2,D1) ) ) ) ).
tff(fact_84_top__le,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] :
( ord_less_eq(A,top_top(A),A2)
=> ( A2 = top_top(A) ) ) ) ).
tff(fact_85_top__unique,axiom,
! [A: $tType] :
( top(A)
=> ! [A3: A] :
( ord_less_eq(A,top_top(A),A3)
<=> ( A3 = top_top(A) ) ) ) ).
tff(fact_86_top__greatest,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] : ord_less_eq(A,A2,top_top(A)) ) ).
tff(fact_87_subset__UNIV,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),A1,top_top(fun(A,bool))) ).
tff(fact_88_double__diff,axiom,
! [A: $tType,C1: fun(A,bool),B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( ord_less_eq(fun(A,bool),B1,C1)
=> ( minus_minus(fun(A,bool),B1,minus_minus(fun(A,bool),C1,A1)) = A1 ) ) ) ).
tff(fact_89_Diff__mono,axiom,
! [A: $tType,B1: fun(A,bool),D: fun(A,bool),C1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,C1)
=> ( ord_less_eq(fun(A,bool),D,B1)
=> ord_less_eq(fun(A,bool),minus_minus(fun(A,bool),A1,B1),minus_minus(fun(A,bool),C1,D)) ) ) ).
tff(fact_90_Diff__subset,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] : ord_less_eq(fun(A,bool),minus_minus(fun(A,bool),A1,B1),A1) ).
tff(fact_91_subset__refl,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),A1,A1) ).
tff(fact_92_set__eq__subset,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
<=> ( ord_less_eq(fun(A,bool),A1,B1)
& ord_less_eq(fun(A,bool),B1,A1) ) ) ).
tff(fact_93_rev__predicate1D,axiom,
! [A: $tType,Q1: fun(A,bool),X: A,P1: fun(A,bool)] :
( pp(aa(A,bool,P1,X))
=> ( ord_less_eq(fun(A,bool),P1,Q1)
=> pp(aa(A,bool,Q1,X)) ) ) ).
tff(fact_94_equalityD1,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
=> ord_less_eq(fun(A,bool),A1,B1) ) ).
tff(fact_95_equalityD2,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
=> ord_less_eq(fun(A,bool),B1,A1) ) ).
tff(fact_96_in__mono,axiom,
! [A: $tType,X: A,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),B1)) ) ) ).
tff(fact_97_predicate1D,axiom,
! [A: $tType,X: A,Q1: fun(A,bool),P1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),P1,Q1)
=> ( pp(aa(A,bool,P1,X))
=> pp(aa(A,bool,Q1,X)) ) ) ).
tff(fact_98_set__rev__mp,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),A1))
=> ( ord_less_eq(fun(A,bool),A1,B1)
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),B1)) ) ) ).
tff(fact_99_set__mp,axiom,
! [A: $tType,X: A,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),B1)) ) ) ).
%----Arities (8)
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_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(fun(T_1,T_2)) ) ).
tff(arity_fun___Groups_Ominus,axiom,
! [T_1: $tType,T_2: $tType] :
( cl_Groups_Ominus(T_2)
=> cl_Groups_Ominus(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Otop,axiom,
top(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
tff(arity_HOL_Obool___Groups_Ominus,axiom,
cl_Groups_Ominus(bool) ).
%----Helper facts (15)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
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_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
order_1409979114der_on(arrow_411405190le_alt,top_top(fun(arrow_411405190le_alt,bool)),r) ).
tff(conj_1,hypothesis,
wf(arrow_411405190le_alt,minus_minus(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),r,id(arrow_411405190le_alt))) ).
tff(conj_2,conjecture,
? [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : pp(aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool,order_215145569der_on(arrow_411405190le_alt,top_top(fun(arrow_411405190le_alt,bool))),L)) ).
%------------------------------------------------------------------------------