TPTP Problem File: LCL754_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL754_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 52
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_52 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 177 ( 57 unt; 43 typ; 0 def)
% Number of atoms : 305 ( 73 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 211 ( 40 ~; 6 |; 15 &)
% ( 30 <=>; 120 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 33 ( 18 >; 15 *; 0 +; 0 <<)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-4 aty)
% Number of functors : 19 ( 19 usr; 5 con; 0-5 aty)
% Number of variables : 311 ( 278 !; 4 ?; 311 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:13
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Lambda_OdB,type,
dB: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (38)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__ring,type,
number_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[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_Int_Onumber__semiring,type,
number_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_InductTermi_OIT,type,
it: dB > $o ).
tff(sy_c_Int_Onat,type,
nat1: int > nat ).
tff(sy_c_Int_Onumber__class_Onumber__of,type,
number_number_of:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_Lambda_OdB_OAbs,type,
abs: dB > dB ).
tff(sy_c_Lambda_OdB_OVar,type,
var: nat > dB ).
tff(sy_c_Lambda_OdB_OdB__case,type,
dB_case:
!>[T1: $tType] : ( ( fun(nat,T1) * fun(dB,fun(dB,T1)) * fun(dB,T1) * dB ) > T1 ) ).
tff(sy_c_Lambda_OdB_OdB__rec,type,
dB_rec:
!>[T1: $tType] : ( ( fun(nat,T1) * fun(dB,fun(dB,fun(T1,fun(T1,T1)))) * fun(dB,fun(T1,T1)) * dB ) > T1 ) ).
tff(sy_c_Lambda_Olift,type,
lift: ( dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: ( dB * dB * nat ) > dB ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Nat__Transfer_Otransfer__morphism,type,
nat_tr1645093318rphism:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > $o ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $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_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_ia,type,
ia: nat ).
tff(sy_v_ja,type,
ja: nat ).
tff(sy_v_ra,type,
ra: dB ).
%----Relevant facts (98)
tff(fact_0_Lambda,axiom,
! [R: dB] :
( it(R)
=> it(abs(R)) ) ).
tff(fact_1_dB_Osimps_I3_J,axiom,
! [DB2: dB,DB: dB] :
( ( abs(DB) = abs(DB2) )
<=> ( DB = DB2 ) ) ).
tff(fact_2_dB_Osimps_I1_J,axiom,
! [Nat2: nat,Nat: nat] :
( ( var(Nat) = var(Nat2) )
<=> ( Nat = Nat2 ) ) ).
tff(fact_3_subst__eq,axiom,
! [U: dB,K: nat] : subst(var(K),U,K) = U ).
tff(fact_4_dB_Osimps_I7_J,axiom,
! [Nat1: nat,DB1: dB] : abs(DB1) != var(Nat1) ).
tff(fact_5_dB_Osimps_I6_J,axiom,
! [DB1: dB,Nat1: nat] : var(Nat1) != abs(DB1) ).
tff(fact_6_lift__IT,axiom,
! [I1: nat,T: dB] :
( it(T)
=> it(lift(T,I1)) ) ).
tff(fact_7_dB_Osimps_I12_J,axiom,
! [A: $tType,DB: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,abs(DB)) = aa(dB,A,F3,DB) ).
tff(fact_8_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat)) = aa(nat,A,F1,Nat) ).
tff(fact_9_subst__lt,axiom,
! [U: dB,I1: nat,J1: nat] :
( ord_less(nat,J1,I1)
=> ( subst(var(J1),U,I1) = var(J1) ) ) ).
tff(fact_10_dB_Orecs_I1_J,axiom,
! [A: $tType,Nat: nat,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,var(Nat)) = aa(nat,A,F1,Nat) ).
tff(fact_11_dB_Orecs_I3_J,axiom,
! [A: $tType,DB: dB,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,abs(DB)) = aa(A,A,aa(dB,fun(A,A),F3,DB),dB_rec(A,F1,F2,F3,DB)) ).
tff(fact_12_subst__lift,axiom,
! [S: dB,K: nat,T: dB] : subst(lift(T,K),S,K) = T ).
tff(fact_13_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_14_nat__neq__iff,axiom,
! [N1: nat,M1: nat] :
( ( M1 != N1 )
<=> ( ord_less(nat,M1,N1)
| ord_less(nat,N1,M1) ) ) ).
tff(fact_15_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_16_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_17_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_18_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_19_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N1: nat,M1: nat] :
( ( ord_less(nat,M1,N1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( ( M1 = N1 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( ord_less(nat,N1,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) ) ) ) ).
tff(fact_20_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ ord_less(A,X,X) ) ).
tff(fact_21_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 != Y1 )
<=> ( ord_less(A,X1,Y1)
| ord_less(A,Y1,X1) ) ) ) ).
tff(fact_22_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
=> ( ( X != Y )
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_23_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_24_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ord_less(A,Y,Z2)
=> ord_less(A,X,Z2) ) ) ) ).
tff(fact_25_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A3: A] :
( ord_less(A,A3,B2)
=> ( ( B2 = C1 )
=> ord_less(A,A3,C1) ) ) ) ).
tff(fact_26_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A3: A] :
( ( A3 = B2 )
=> ( ord_less(A,B2,C1)
=> ord_less(A,A3,C1) ) ) ) ).
tff(fact_27_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( ord_less(A,A3,B2)
=> ~ ord_less(A,B2,A3) ) ) ).
tff(fact_28_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P1: bool,Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( ord_less(A,Y1,X1)
=> pp(P1) ) ) ) ).
tff(fact_29_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( Y != X ) ) ) ).
tff(fact_30_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_31_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_32_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_33_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_34_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_35_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ~ ord_less(A,Y1,X1)
=> ( ~ ord_less(A,X1,Y1)
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_36_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
| ( X = Y )
| ord_less(A,Y,X) ) ) ).
tff(fact_37_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
<=> ( ord_less(A,Y1,X1)
| ( X1 = Y1 ) ) ) ) ).
tff(fact_38_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),A2: A] :
( ord_less(A,A2,aa(B,A,F,B1))
=> ( ord_less(B,B1,C)
=> ( ! [X2: B,Y3: B] :
( ord_less(B,X2,Y3)
=> ord_less(A,aa(B,A,F,X2),aa(B,A,F,Y3)) )
=> ord_less(A,A2,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_39_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C: B,B1: B,F: fun(B,A),A2: A] :
( ( A2 = aa(B,A,F,B1) )
=> ( ord_less(B,B1,C)
=> ( ! [X2: B,Y3: B] :
( ord_less(B,X2,Y3)
=> ord_less(A,aa(B,A,F,X2),aa(B,A,F,Y3)) )
=> ord_less(A,A2,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_40_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,F: fun(A,B),B1: A,A2: A] :
( ord_less(A,A2,B1)
=> ( ord_less(B,aa(A,B,F,B1),C)
=> ( ! [X2: A,Y3: A] :
( ord_less(A,X2,Y3)
=> ord_less(B,aa(A,B,F,X2),aa(A,B,F,Y3)) )
=> ord_less(B,aa(A,B,F,A2),C) ) ) ) ) ).
tff(fact_41_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C: B,F: fun(A,B),B1: A,A2: A] :
( ord_less(A,A2,B1)
=> ( ( aa(A,B,F,B1) = C )
=> ( ! [X2: A,Y3: A] :
( ord_less(A,X2,Y3)
=> ord_less(B,aa(A,B,F,X2),aa(A,B,F,Y3)) )
=> ord_less(B,aa(A,B,F,A2),C) ) ) ) ) ).
tff(fact_42_dense,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ? [Z1: A] :
( ord_less(A,X,Z1)
& ord_less(A,Z1,Y) ) ) ) ).
tff(fact_43_lt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A] :
? [Y3: A] : ord_less(A,Y3,X) ) ).
tff(fact_44_gt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X1: A] :
? [X11: A] : ord_less(A,X1,X11) ) ).
tff(fact_45_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_46_of__nat__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat,M1: nat] :
( ord_less(A,semiring_1_of_nat(A,M1),semiring_1_of_nat(A,N1))
<=> ord_less(nat,M1,N1) ) ) ).
tff(fact_47_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,X1: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ord_less(B,aa(A,B,F,X1),aa(A,B,F,Y1))
<=> ord_less(A,X1,Y1) ) ) ) ).
tff(fact_48_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Y1: A,X1: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ord_less(A,X1,Y1)
=> ord_less(B,aa(A,B,F,X1),aa(A,B,F,Y1)) ) ) ) ).
tff(fact_49_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N1: nat,M1: nat] :
( ( semiring_1_of_nat(A,M1) = semiring_1_of_nat(A,N1) )
<=> ( M1 = N1 ) ) ) ).
tff(fact_50_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,X1: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_51_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N))
=> ord_less(nat,M,N) ) ) ).
tff(fact_52_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ) ).
tff(fact_53_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X2: A,Y3: A] :
( ord_less(A,X2,Y3)
=> ord_less(B,aa(A,B,F,X2),aa(A,B,F,Y3)) )
=> order_strict_mono(A,B,F) ) ) ).
tff(fact_54_transfer__int__nat__relations_I2_J,axiom,
! [Y1: nat,X1: nat] :
( ord_less(int,semiring_1_of_nat(int,X1),semiring_1_of_nat(int,Y1))
<=> ord_less(nat,X1,Y1) ) ).
tff(fact_55_zless__int,axiom,
! [N1: nat,M1: nat] :
( ord_less(int,semiring_1_of_nat(int,M1),semiring_1_of_nat(int,N1))
<=> ord_less(nat,M1,N1) ) ).
tff(fact_56_infinite__descent__measure,axiom,
! [A: $tType,X1: A,V1: fun(A,nat),P1: fun(A,bool)] :
( ! [X2: A] :
( ~ pp(aa(A,bool,P1,X2))
=> ? [Y2: A] :
( ord_less(nat,aa(A,nat,V1,Y2),aa(A,nat,V1,X2))
& ~ pp(aa(A,bool,P1,Y2)) ) )
=> pp(aa(A,bool,P1,X1)) ) ).
tff(fact_57_int__if__cong,axiom,
! [Y1: nat,X1: nat,P1: bool] :
( ( pp(P1)
=> ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) )
& ( ~ pp(P1)
=> ( semiring_1_of_nat(int,Y1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) ) ) ).
tff(fact_58_transfer__int__nat__relations_I1_J,axiom,
! [Y1: nat,X1: nat] :
( ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,Y1) )
<=> ( X1 = Y1 ) ) ).
tff(fact_59_int__int__eq,axiom,
! [N1: nat,M1: nat] :
( ( semiring_1_of_nat(int,M1) = semiring_1_of_nat(int,N1) )
<=> ( M1 = N1 ) ) ).
tff(fact_60_transfer__morphismI,axiom,
! [B: $tType,A: $tType,A1: fun(A,bool),F: fun(A,B)] : nat_tr1645093318rphism(A,B,F,A1) ).
tff(fact_61_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] : ring_1_of_int(A,semiring_1_of_nat(int,N)) = semiring_1_of_nat(A,N) ) ).
tff(fact_62_zless__nat__eq__int__zless,axiom,
! [Z: int,M1: nat] :
( ord_less(nat,M1,nat1(Z))
<=> ord_less(int,semiring_1_of_nat(int,M1),Z) ) ).
tff(fact_63_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int,W1: int] :
( ( ring_1_of_int(A,W1) = ring_1_of_int(A,Z) )
<=> ( W1 = Z ) ) ) ).
tff(fact_64_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W1: int] :
( ord_less(A,ring_1_of_int(A,W1),ring_1_of_int(A,Z))
<=> ord_less(int,W1,Z) ) ) ).
tff(fact_65_of__int__int__eq,axiom,
! [N: nat] : ring_1_of_int(int,semiring_1_of_nat(int,N)) = semiring_1_of_nat(int,N) ).
tff(fact_66_nat__int,axiom,
! [N: nat] : nat1(semiring_1_of_nat(int,N)) = N ).
tff(fact_67_transfer__morphism__def,axiom,
! [B: $tType,A: $tType,A1: fun(A,bool),F: fun(A,B)] : nat_tr1645093318rphism(A,B,F,A1) ).
tff(fact_68_nat__if__cong,axiom,
! [Y1: int,X1: int,P1: bool] :
( ( pp(P1)
=> ( nat1(X1) = nat1(if(int,P1,X1,Y1)) ) )
& ( ~ pp(P1)
=> ( nat1(Y1) = nat1(if(int,P1,X1,Y1)) ) ) ) ).
tff(fact_69_zless__nat__conj,axiom,
! [Z: int,W1: int] :
( ord_less(nat,nat1(W1),nat1(Z))
<=> ( ord_less(int,zero_zero(int),Z)
& ord_less(int,W1,Z) ) ) ).
tff(fact_70_nat__mono__iff,axiom,
! [W1: int,Z: int] :
( ord_less(int,zero_zero(int),Z)
=> ( ord_less(nat,nat1(W1),nat1(Z))
<=> ord_less(int,W1,Z) ) ) ).
tff(fact_71_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int,X1: int] :
( ord_less(A,number_number_of(A,X1),number_number_of(A,Y1))
<=> ord_less(int,X1,Y1) ) ) ).
tff(fact_72_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y1: int,X1: int] :
( ( number_number_of(A,X1) = number_number_of(A,Y1) )
<=> ( X1 = Y1 ) ) ) ).
tff(fact_73_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_74_of__int__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [V: int] : ring_1_of_int(A,number_number_of(int,V)) = number_number_of(A,V) ) ).
tff(fact_75_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_76_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_77_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( ring_1_of_int(A,Z) = zero_zero(A) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_78_of__nat__0__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat] :
( ord_less(A,zero_zero(A),semiring_1_of_nat(A,N1))
<=> ord_less(nat,zero_zero(nat),N1) ) ) ).
tff(fact_79_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less(A,zero_zero(A),ring_1_of_int(A,Z))
<=> ord_less(int,zero_zero(int),Z) ) ) ).
tff(fact_80_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less(A,ring_1_of_int(A,Z),zero_zero(A))
<=> ord_less(int,Z,zero_zero(int)) ) ) ).
tff(fact_81_number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [K: int] : number_number_of(A,K) = ring_1_of_int(A,K) ) ).
tff(fact_82_int__number__of__def,axiom,
! [W: int] : number_number_of(int,W) = ring_1_of_int(int,W) ).
tff(fact_83_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X1: A,W1: int] :
( ( number_number_of(A,W1) = X1 )
<=> ( X1 = number_number_of(A,W1) ) ) ) ).
tff(fact_84_less__number__of__int__code,axiom,
! [L: int,K1: int] :
( ord_less(int,number_number_of(int,K1),number_number_of(int,L))
<=> ord_less(int,K1,L) ) ).
tff(fact_85_number__of__int,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [N: nat] : number_number_of(A,semiring_1_of_nat(int,N)) = semiring_1_of_nat(A,N) ) ).
tff(fact_86_of__nat__less__0__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat] : ~ ord_less(A,semiring_1_of_nat(A,M),zero_zero(A)) ) ).
tff(fact_87_int__less__0__conv,axiom,
! [K: nat] : ~ ord_less(int,semiring_1_of_nat(int,K),zero_zero(int)) ).
tff(fact_88_nat__number__of,axiom,
! [W: int] : nat1(number_number_of(int,W)) = number_number_of(nat,W) ).
tff(fact_89_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_90_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_91_neq0__conv,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_92_nat__0,axiom,
nat1(zero_zero(int)) = zero_zero(nat) ).
tff(fact_93_zero__less__nat__eq,axiom,
! [Z: int] :
( ord_less(nat,zero_zero(nat),nat1(Z))
<=> ord_less(int,zero_zero(int),Z) ) ).
tff(fact_94_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_95_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_96_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_97_number__of__is__id,axiom,
! [K: int] : number_number_of(int,K) = K ).
%----Arities (28)
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
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_Oord,axiom,
ord(bool) ).
%----Helper facts (5)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fTrue,X,Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
it(ra) ).
tff(conj_1,hypothesis,
! [I: nat,J: nat] : it(subst(ra,var(I),J)) ).
tff(conj_2,conjecture,
it(subst(abs(ra),var(ia),ja)) ).
%------------------------------------------------------------------------------