TPTP Problem File: SCT169_5.p
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%------------------------------------------------------------------------------
% File : SCT169_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 29
% 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_29 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.4.0, 0.50 v7.1.0, 1.00 v6.4.0
% Syntax : Number of formulae : 137 ( 41 unt; 32 typ; 0 def)
% Number of atoms : 212 ( 160 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 143 ( 36 ~; 3 |; 23 &)
% ( 32 <=>; 49 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 42 ( 24 >; 18 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-3 aty)
% Number of functors : 24 ( 24 usr; 4 con; 0-5 aty)
% Number of variables : 362 ( 324 !; 9 ?; 362 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:13:51
%------------------------------------------------------------------------------
%----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_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (28)
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_c_HOL_Oequal__class_Oequal,type,
equal_equal:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Ohd,type,
hd:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * list(B) ) > list(B) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
tff(sy_c_List_Olist_Olist__rec,type,
list_rec:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),fun(T,T))) * list(A) ) > T ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( fun(A,list(B)) * list(A) ) > list(B) ) ).
tff(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
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_pp,type,
pp: bool > $o ).
tff(sy_v_a,type,
a: arrow_411405190le_alt ).
tff(sy_v_b,type,
b: arrow_411405190le_alt ).
%----Relevant facts (100)
tff(fact_0_alt3,axiom,
? [A1: arrow_411405190le_alt,B1: arrow_411405190le_alt,C1: arrow_411405190le_alt] : distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,A1,cons(arrow_411405190le_alt,B1,cons(arrow_411405190le_alt,C1,nil(arrow_411405190le_alt))))) ).
tff(fact_1_distinct_Osimps_I1_J,axiom,
! [A: $tType] : distinct(A,nil(A)) ).
tff(fact_2_distinct__singleton,axiom,
! [A: $tType,X: A] : distinct(A,cons(A,X,nil(A))) ).
tff(fact_3_list_Oinject,axiom,
! [A: $tType,List3: list(A),A3: A,List: list(A),Aa: A] :
( ( cons(A,Aa,List) = cons(A,A3,List3) )
<=> ( ( Aa = A3 )
& ( List = List3 ) ) ) ).
tff(fact_4_distinct__length__2__or__more,axiom,
! [A: $tType,Xs1: list(A),Ba: A,Aa: A] :
( distinct(A,cons(A,Aa,cons(A,Ba,Xs1)))
<=> ( ( Aa != Ba )
& distinct(A,cons(A,Aa,Xs1))
& distinct(A,cons(A,Ba,Xs1)) ) ) ).
tff(fact_5_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A2: A] : nil(A) != cons(A,A2,List2) ).
tff(fact_6_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A2: A] : cons(A,A2,List2) != nil(A) ).
tff(fact_7_neq__Nil__conv,axiom,
! [A: $tType,Xs1: list(A)] :
( ( Xs1 != nil(A) )
<=> ? [Y3: A,Ys4: list(A)] : Xs1 = cons(A,Y3,Ys4) ) ).
tff(fact_8_list_Oexhaust,axiom,
! [A: $tType,Y2: list(A)] :
( ( Y2 != nil(A) )
=> ~ ! [A1: A,List1: list(A)] : Y2 != cons(A,A1,List1) ) ).
tff(fact_9_not__Cons__self,axiom,
! [A: $tType,X: A,Xs: list(A)] : Xs != cons(A,X,Xs) ).
tff(fact_10_not__Cons__self2,axiom,
! [A: $tType,Xs: list(A),X: A] : cons(A,X,Xs) != Xs ).
tff(fact_11_insert__Nil,axiom,
! [A: $tType,X: A] : insert(A,X,nil(A)) = cons(A,X,nil(A)) ).
tff(fact_12_splice_Osimps_I2_J,axiom,
! [A: $tType,Va: list(A),V: A] : splice(A,cons(A,V,Va),nil(A)) = cons(A,V,Va) ).
tff(fact_13_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys: list(A),Y2: A,Xs: list(A),X: A] : splice(A,cons(A,X,Xs),cons(A,Y2,Ys)) = cons(A,X,cons(A,Y2,splice(A,Xs,Ys))) ).
tff(fact_14_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( distinct(A,Xs)
=> distinct(A,insert(A,X,Xs)) ) ).
tff(fact_15_splice__Nil2,axiom,
! [A: $tType,Xs: list(A)] : splice(A,Xs,nil(A)) = Xs ).
tff(fact_16_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).
tff(fact_17_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,nil(B)) = F1 ).
tff(fact_18_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),Aa: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,cons(B,Aa,List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,Aa),List) ).
tff(fact_19_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : ~ list_ex1(A,P,nil(A)) ).
tff(fact_20_list__nonempty__induct,axiom,
! [A: $tType,P: fun(list(A),bool),Xs1: list(A)] :
( ( Xs1 != nil(A) )
=> ( ! [X2: A] : pp(aa(list(A),bool,P,cons(A,X2,nil(A))))
=> ( ! [X2: A,Xs2: list(A)] :
( ( Xs2 != nil(A) )
=> ( pp(aa(list(A),bool,P,Xs2))
=> pp(aa(list(A),bool,P,cons(A,X2,Xs2))) ) )
=> pp(aa(list(A),bool,P,Xs1)) ) ) ) ).
tff(fact_21_list_Orecs_I1_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,nil(B)) = F1 ).
tff(fact_22_list_Orecs_I2_J,axiom,
! [A: $tType,B: $tType,List: list(B),Aa: B,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,cons(B,Aa,List)) = aa(A,A,aa(list(B),fun(A,A),aa(B,fun(list(B),fun(A,A)),F2,Aa),List),list_rec(A,B,F1,F2,List)) ).
tff(fact_23_last_Osimps,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( ( Xs = nil(A) )
=> ( last(A,cons(A,X,Xs)) = X ) )
& ( ( Xs != nil(A) )
=> ( last(A,cons(A,X,Xs)) = last(A,Xs) ) ) ) ).
tff(fact_24_butlast_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( ( Xs = nil(A) )
=> ( butlast(A,cons(A,X,Xs)) = nil(A) ) )
& ( ( Xs != nil(A) )
=> ( butlast(A,cons(A,X,Xs)) = cons(A,X,butlast(A,Xs)) ) ) ) ).
tff(fact_25_insort__key_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X1: A,F: fun(A,B)] : linorder_insort_key(A,B,F,X1,nil(A)) = cons(A,X1,nil(A)) ) ).
tff(fact_26_butlast_Osimps_I1_J,axiom,
! [A: $tType] : butlast(A,nil(A)) = nil(A) ).
tff(fact_27_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xs1: list(B),Y: B,X1: B,F: fun(B,A)] :
( ( aa(B,A,F,X1) != aa(B,A,F,Y) )
=> ( linorder_insort_key(B,A,F,Y,linorder_insort_key(B,A,F,X1,Xs1)) = linorder_insort_key(B,A,F,X1,linorder_insort_key(B,A,F,Y,Xs1)) ) ) ) ).
tff(fact_28_insort__not__Nil,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs1: list(A),Aa: A,F: fun(A,B)] : linorder_insort_key(A,B,F,Aa,Xs1) != nil(A) ) ).
tff(fact_29_last__ConsR,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( last(A,cons(A,X,Xs)) = last(A,Xs) ) ) ).
tff(fact_30_last__ConsL,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( Xs = nil(A) )
=> ( last(A,cons(A,X,Xs)) = X ) ) ).
tff(fact_31_distinct__butlast,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( distinct(A,Xs)
=> distinct(A,butlast(A,Xs)) ) ) ).
tff(fact_32_sort__key__simps_I2_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs1: list(A),X1: A,F: fun(A,B)] : linorder_sort_key(A,B,F,cons(A,X1,Xs1)) = linorder_insort_key(A,B,F,X1,linorder_sort_key(A,B,F,Xs1)) ) ).
tff(fact_33_snoc__eq__iff__butlast,axiom,
! [A: $tType,Ys1: list(A),X1: A,Xs1: list(A)] :
( ( append(A,Xs1,cons(A,X1,nil(A))) = Ys1 )
<=> ( ( Ys1 != nil(A) )
& ( butlast(A,Ys1) = Xs1 )
& ( last(A,Ys1) = X1 ) ) ) ).
tff(fact_34_append__butlast__last__id,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( append(A,butlast(A,Xs),cons(A,last(A,Xs),nil(A))) = Xs ) ) ).
tff(fact_35_null__rec_I2_J,axiom,
! [B: $tType] : null(B,nil(B)) ).
tff(fact_36_List_Onull__def,axiom,
! [A: $tType,Xs1: list(A)] :
( null(A,Xs1)
<=> ( Xs1 = nil(A) ) ) ).
tff(fact_37_eq__Nil__null,axiom,
! [A: $tType,Xs1: list(A)] :
( ( Xs1 = nil(A) )
<=> null(A,Xs1) ) ).
tff(fact_38_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),X1: A,P: fun(A,bool)] :
( ( pp(aa(A,bool,P,X1))
=> ( takeWhile(A,P,cons(A,X1,Xs1)) = cons(A,X1,takeWhile(A,P,Xs1)) ) )
& ( ~ pp(aa(A,bool,P,X1))
=> ( takeWhile(A,P,cons(A,X1,Xs1)) = nil(A) ) ) ) ).
tff(fact_39_append__same__eq,axiom,
! [A: $tType,Zs: list(A),Xs1: list(A),Ys1: list(A)] :
( ( append(A,Ys1,Xs1) = append(A,Zs,Xs1) )
<=> ( Ys1 = Zs ) ) ).
tff(fact_40_same__append__eq,axiom,
! [A: $tType,Zs: list(A),Ys1: list(A),Xs1: list(A)] :
( ( append(A,Xs1,Ys1) = append(A,Xs1,Zs) )
<=> ( Ys1 = Zs ) ) ).
tff(fact_41_append__assoc,axiom,
! [A: $tType,Zs1: list(A),Ys: list(A),Xs: list(A)] : append(A,append(A,Xs,Ys),Zs1) = append(A,Xs,append(A,Ys,Zs1)) ).
tff(fact_42_append__Cons,axiom,
! [A: $tType,Ys: list(A),Xs: list(A),X: A] : append(A,cons(A,X,Xs),Ys) = cons(A,X,append(A,Xs,Ys)) ).
tff(fact_43_append__self__conv2,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( append(A,Xs1,Ys1) = Ys1 )
<=> ( Xs1 = nil(A) ) ) ).
tff(fact_44_append__self__conv,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( append(A,Xs1,Ys1) = Xs1 )
<=> ( Ys1 = nil(A) ) ) ).
tff(fact_45_append__is__Nil__conv,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( append(A,Xs1,Ys1) = nil(A) )
<=> ( ( Xs1 = nil(A) )
& ( Ys1 = nil(A) ) ) ) ).
tff(fact_46_self__append__conv2,axiom,
! [A: $tType,Xs1: list(A),Ys1: list(A)] :
( ( Ys1 = append(A,Xs1,Ys1) )
<=> ( Xs1 = nil(A) ) ) ).
tff(fact_47_self__append__conv,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( Xs1 = append(A,Xs1,Ys1) )
<=> ( Ys1 = nil(A) ) ) ).
tff(fact_48_Nil__is__append__conv,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( nil(A) = append(A,Xs1,Ys1) )
<=> ( ( Xs1 = nil(A) )
& ( Ys1 = nil(A) ) ) ) ).
tff(fact_49_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : takeWhile(A,P,nil(A)) = nil(A) ).
tff(fact_50_sort__key__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B)] : linorder_sort_key(A,B,F,nil(A)) = nil(A) ) ).
tff(fact_51_distinct__sort,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs1: list(A),F: fun(A,B)] :
( distinct(A,linorder_sort_key(A,B,F,Xs1))
<=> distinct(A,Xs1) ) ) ).
tff(fact_52_append1__eq__conv,axiom,
! [A: $tType,Y: A,Ys1: list(A),X1: A,Xs1: list(A)] :
( ( append(A,Xs1,cons(A,X1,nil(A))) = append(A,Ys1,cons(A,Y,nil(A))) )
<=> ( ( Xs1 = Ys1 )
& ( X1 = Y ) ) ) ).
tff(fact_53_append__eq__appendI,axiom,
! [A: $tType,Us1: list(A),Ys: list(A),Zs1: list(A),Xs11: list(A),Xs: list(A)] :
( ( append(A,Xs,Xs11) = Zs1 )
=> ( ( Ys = append(A,Xs11,Us1) )
=> ( append(A,Xs,Ys) = append(A,Zs1,Us1) ) ) ) ).
tff(fact_54_takeWhile__tail,axiom,
! [A: $tType,L: list(A),Xs1: list(A),X1: A,P: fun(A,bool)] :
( ~ pp(aa(A,bool,P,X1))
=> ( takeWhile(A,P,append(A,Xs1,cons(A,X1,L))) = takeWhile(A,P,Xs1) ) ) ).
tff(fact_55_append__eq__append__conv2,axiom,
! [A: $tType,Ts: list(A),Zs: list(A),Ys1: list(A),Xs1: list(A)] :
( ( append(A,Xs1,Ys1) = append(A,Zs,Ts) )
<=> ? [Us: list(A)] :
( ( ( Xs1 = append(A,Zs,Us) )
& ( append(A,Us,Ys1) = Ts ) )
| ( ( append(A,Xs1,Us) = Zs )
& ( Ys1 = append(A,Us,Ts) ) ) ) ) ).
tff(fact_56_Cons__eq__appendI,axiom,
! [A: $tType,Zs1: list(A),Xs: list(A),Ys: list(A),Xs11: list(A),X: A] :
( ( cons(A,X,Xs11) = Ys )
=> ( ( Xs = append(A,Xs11,Zs1) )
=> ( cons(A,X,Xs) = append(A,Ys,Zs1) ) ) ) ).
tff(fact_57_eq__Nil__appendI,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( Xs = Ys )
=> ( Xs = append(A,nil(A),Ys) ) ) ).
tff(fact_58_append__Nil2,axiom,
! [A: $tType,Xs: list(A)] : append(A,Xs,nil(A)) = Xs ).
tff(fact_59_append__Nil,axiom,
! [A: $tType,Ys: list(A)] : append(A,nil(A),Ys) = Ys ).
tff(fact_60_distinct__takeWhile,axiom,
! [A: $tType,P: fun(A,bool),Xs1: list(A)] :
( distinct(A,Xs1)
=> distinct(A,takeWhile(A,P,Xs1)) ) ).
tff(fact_61_Cons__eq__append__conv,axiom,
! [A: $tType,Zs: list(A),Ys1: list(A),Xs1: list(A),X1: A] :
( ( cons(A,X1,Xs1) = append(A,Ys1,Zs) )
<=> ( ( ( Ys1 = nil(A) )
& ( cons(A,X1,Xs1) = Zs ) )
| ? [Ys3: list(A)] :
( ( cons(A,X1,Ys3) = Ys1 )
& ( Xs1 = append(A,Ys3,Zs) ) ) ) ) ).
tff(fact_62_append__eq__Cons__conv,axiom,
! [A: $tType,Xs1: list(A),X1: A,Zs: list(A),Ys1: list(A)] :
( ( append(A,Ys1,Zs) = cons(A,X1,Xs1) )
<=> ( ( ( Ys1 = nil(A) )
& ( Zs = cons(A,X1,Xs1) ) )
| ? [Ys3: list(A)] :
( ( Ys1 = cons(A,X1,Ys3) )
& ( append(A,Ys3,Zs) = Xs1 ) ) ) ) ).
tff(fact_63_butlast__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( ( Ys = nil(A) )
=> ( butlast(A,append(A,Xs,Ys)) = butlast(A,Xs) ) )
& ( ( Ys != nil(A) )
=> ( butlast(A,append(A,Xs,Ys)) = append(A,Xs,butlast(A,Ys)) ) ) ) ).
tff(fact_64_last__appendL,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Ys = nil(A) )
=> ( last(A,append(A,Xs,Ys)) = last(A,Xs) ) ) ).
tff(fact_65_last__appendR,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Ys != nil(A) )
=> ( last(A,append(A,Xs,Ys)) = last(A,Ys) ) ) ).
tff(fact_66_last__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( ( Ys = nil(A) )
=> ( last(A,append(A,Xs,Ys)) = last(A,Xs) ) )
& ( ( Ys != nil(A) )
=> ( last(A,append(A,Xs,Ys)) = last(A,Ys) ) ) ) ).
tff(fact_67_butlast__snoc,axiom,
! [A: $tType,X: A,Xs: list(A)] : butlast(A,append(A,Xs,cons(A,X,nil(A)))) = Xs ).
tff(fact_68_null__rec_I1_J,axiom,
! [A: $tType,Xs: list(A),X: A] : ~ null(A,cons(A,X,Xs)) ).
tff(fact_69_last__snoc,axiom,
! [A: $tType,X: A,Xs: list(A)] : last(A,append(A,Xs,cons(A,X,nil(A)))) = X ).
tff(fact_70_rev__cases,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ~ ! [Ys2: list(A),Y1: A] : Xs != append(A,Ys2,cons(A,Y1,nil(A))) ) ).
tff(fact_71_rev__induct,axiom,
! [A: $tType,Xs1: list(A),P: fun(list(A),bool)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [X2: A,Xs2: list(A)] :
( pp(aa(list(A),bool,P,Xs2))
=> pp(aa(list(A),bool,P,append(A,Xs2,cons(A,X2,nil(A))))) )
=> pp(aa(list(A),bool,P,Xs1)) ) ) ).
tff(fact_72_equal__Nil__null,axiom,
! [A: $tType,Xs1: list(A)] :
( equal_equal(list(A),Xs1,nil(A))
<=> null(A,Xs1) ) ).
tff(fact_73_rotate__simps,axiom,
! [A: $tType,B: $tType,Xs: list(B),X: B] :
( ( rotate1(A,nil(A)) = nil(A) )
& ( rotate1(B,cons(B,X,Xs)) = append(B,Xs,cons(B,X,nil(B))) ) ) ).
tff(fact_74_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,Xs1: list(B),X1: B,F: fun(B,list(A))] : maps(B,A,F,cons(B,X1,Xs1)) = append(A,aa(B,list(A),F,X1),maps(B,A,F,Xs1)) ).
tff(fact_75_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs1: list(A)] :
( ( rotate1(A,Xs1) = nil(A) )
<=> ( Xs1 = nil(A) ) ) ).
tff(fact_76_distinct1__rotate,axiom,
! [A: $tType,Xs1: list(A)] :
( distinct(A,rotate1(A,Xs1))
<=> distinct(A,Xs1) ) ).
tff(fact_77_equal__list__def,axiom,
! [A: $tType,Y: list(A),X1: list(A)] :
( equal_equal(list(A),X1,Y)
<=> ( X1 = Y ) ) ).
tff(fact_78_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,list(A))] : maps(B,A,F,nil(B)) = nil(A) ).
tff(fact_79_dropWhile__eq__Cons__conv,axiom,
! [A: $tType,Ys1: list(A),Y: A,Xs1: list(A),P: fun(A,bool)] :
( ( dropWhile(A,P,Xs1) = cons(A,Y,Ys1) )
<=> ( ( Xs1 = append(A,takeWhile(A,P,Xs1),cons(A,Y,Ys1)) )
& ~ pp(aa(A,bool,P,Y)) ) ) ).
tff(fact_80_rev_Osimps_I2_J,axiom,
! [A: $tType,Xs: list(A),X: A] : rev(A,cons(A,X,Xs)) = append(A,rev(A,Xs),cons(A,X,nil(A))) ).
tff(fact_81_rev__is__rev__conv,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( rev(A,Xs1) = rev(A,Ys1) )
<=> ( Xs1 = Ys1 ) ) ).
tff(fact_82_rev_Osimps_I1_J,axiom,
! [A: $tType] : rev(A,nil(A)) = nil(A) ).
tff(fact_83_Nil__is__rev__conv,axiom,
! [A: $tType,Xs1: list(A)] :
( ( nil(A) = rev(A,Xs1) )
<=> ( Xs1 = nil(A) ) ) ).
tff(fact_84_rev__is__Nil__conv,axiom,
! [A: $tType,Xs1: list(A)] :
( ( rev(A,Xs1) = nil(A) )
<=> ( Xs1 = nil(A) ) ) ).
tff(fact_85_rev__append,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] : rev(A,append(A,Xs,Ys)) = append(A,rev(A,Ys),rev(A,Xs)) ).
tff(fact_86_distinct__rev,axiom,
! [A: $tType,Xs1: list(A)] :
( distinct(A,rev(A,Xs1))
<=> distinct(A,Xs1) ) ).
tff(fact_87_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),X1: A,P: fun(A,bool)] :
( ( pp(aa(A,bool,P,X1))
=> ( dropWhile(A,P,cons(A,X1,Xs1)) = dropWhile(A,P,Xs1) ) )
& ( ~ pp(aa(A,bool,P,X1))
=> ( dropWhile(A,P,cons(A,X1,Xs1)) = cons(A,X1,Xs1) ) ) ) ).
tff(fact_88_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : dropWhile(A,P,nil(A)) = nil(A) ).
tff(fact_89_singleton__rev__conv,axiom,
! [A: $tType,Xs1: list(A),X1: A] :
( ( cons(A,X1,nil(A)) = rev(A,Xs1) )
<=> ( Xs1 = cons(A,X1,nil(A)) ) ) ).
tff(fact_90_rev__singleton__conv,axiom,
! [A: $tType,X1: A,Xs1: list(A)] :
( ( rev(A,Xs1) = cons(A,X1,nil(A)) )
<=> ( Xs1 = cons(A,X1,nil(A)) ) ) ).
tff(fact_91_rev__eq__Cons__iff,axiom,
! [A: $tType,Ys1: list(A),Y: A,Xs1: list(A)] :
( ( rev(A,Xs1) = cons(A,Y,Ys1) )
<=> ( Xs1 = append(A,rev(A,Ys1),cons(A,Y,nil(A))) ) ) ).
tff(fact_92_distinct__dropWhile,axiom,
! [A: $tType,P: fun(A,bool),Xs1: list(A)] :
( distinct(A,Xs1)
=> distinct(A,dropWhile(A,P,Xs1)) ) ).
tff(fact_93_rev__swap,axiom,
! [A: $tType,Ys1: list(A),Xs1: list(A)] :
( ( rev(A,Xs1) = Ys1 )
<=> ( Xs1 = rev(A,Ys1) ) ) ).
tff(fact_94_rev__rev__ident,axiom,
! [A: $tType,Xs: list(A)] : rev(A,rev(A,Xs)) = Xs ).
tff(fact_95_takeWhile__dropWhile__id,axiom,
! [A: $tType,Xs1: list(A),P: fun(A,bool)] : append(A,takeWhile(A,P,Xs1),dropWhile(A,P,Xs1)) = Xs1 ).
tff(fact_96_hd__rev,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( hd(A,rev(A,Xs)) = last(A,Xs) ) ) ).
tff(fact_97_last__rev,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( last(A,rev(A,Xs)) = hd(A,Xs) ) ) ).
tff(fact_98_hd_Osimps,axiom,
! [A: $tType,Xs: list(A),X: A] : hd(A,cons(A,X,Xs)) = X ).
tff(fact_99_hd__append,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( ( ( Xs = nil(A) )
=> ( hd(A,append(A,Xs,Ys)) = hd(A,Ys) ) )
& ( ( Xs != nil(A) )
=> ( hd(A,append(A,Xs,Ys)) = hd(A,Xs) ) ) ) ).
%----Arities (1)
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
a != b ).
tff(conj_1,conjecture,
? [C: arrow_411405190le_alt] : distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,a,cons(arrow_411405190le_alt,b,cons(arrow_411405190le_alt,C,nil(arrow_411405190le_alt))))) ).
%------------------------------------------------------------------------------