TPTP Problem File: LCL780_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL780_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 94
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_94 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 167 ( 52 unt; 47 typ; 0 def)
% Number of atoms : 235 ( 89 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 147 ( 32 ~; 2 |; 8 &)
% ( 34 <=>; 71 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 51 ( 26 >; 25 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-3 aty)
% Number of functors : 30 ( 30 usr; 8 con; 0-5 aty)
% Number of variables : 261 ( 239 !; 2 ?; 261 :)
% ( 20 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:31
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
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_Type_Otype,type,
type: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (41)
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_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[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_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_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:
!>[T4: $tType] : ( ( fun(nat,T4) * fun(dB,fun(dB,T4)) * fun(dB,T4) * dB ) > T4 ) ).
tff(sy_c_Lambda_OdB_OdB__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: ( dB * nat ) > dB ).
tff(sy_c_Lambda_Oliftn,type,
liftn: ( nat * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: ( dB * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubstn,type,
substn: ( 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_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Type_Oshift,type,
shift:
!>[A: $tType] : ( ( fun(nat,A) * nat * A ) > fun(nat,A) ) ).
tff(sy_c_Type_Otype_OAtom,type,
atom: nat > type ).
tff(sy_c_Type_Otype_OFun,type,
fun1: ( type * type ) > type ).
tff(sy_c_Type_Otype_Otype__case,type,
type_case:
!>[T4: $tType] : ( ( fun(nat,T4) * fun(type,fun(type,T4)) * type ) > T4 ) ).
tff(sy_c_Type_Otype_Otype__rec,type,
type_rec:
!>[T4: $tType] : ( ( fun(nat,T4) * fun(type,fun(type,fun(T4,fun(T4,T4)))) * type ) > T4 ) ).
tff(sy_c_Type_Otype_Otype__size,type,
type_size: type > nat ).
tff(sy_c_Type_Otyping,type,
typing: ( fun(nat,type) * dB * type ) > $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_T,type,
t1: type ).
tff(sy_v_U,type,
u1: type ).
tff(sy_v_e,type,
e: fun(nat,type) ).
tff(sy_v_i,type,
i: nat ).
tff(sy_v_t,type,
t: dB ).
tff(sy_v_u,type,
u: dB ).
%----Relevant facts (99)
tff(fact_0_subst__lemma,axiom,
! [I: nat,Ua: type,U2: dB,E1: fun(nat,type),T2: type,T1: dB,E: fun(nat,type)] :
( typing(E,T1,T2)
=> ( typing(E1,U2,Ua)
=> ( ( E = shift(type,E1,I,Ua) )
=> typing(E1,subst(T1,U2,I),T2) ) ) ) ).
tff(fact_1_shift__eq,axiom,
! [A: $tType,T2: A,E: fun(nat,A),J1: nat,I: nat] :
( ( I = J1 )
=> ( aa(nat,A,shift(A,E,I,T2),J1) = T2 ) ) ).
tff(fact_2_subst__Var__IT,axiom,
! [J: nat,I1: nat,R: dB] :
( it(R)
=> it(subst(R,var(I1),J)) ) ).
tff(fact_3_lift__type,axiom,
! [Ua: type,I: nat,T2: type,T1: dB,E: fun(nat,type)] :
( typing(E,T1,T2)
=> typing(shift(type,E,I,Ua),lift(T1,I),T2) ) ).
tff(fact_4_lift__IT,axiom,
! [I1: nat,T: dB] :
( it(T)
=> it(lift(T,I1)) ) ).
tff(fact_5_Var__IT,axiom,
! [N2: nat] : it(var(N2)) ).
tff(fact_6_Lambda,axiom,
! [R: dB] :
( it(R)
=> it(abs(R)) ) ).
tff(fact_7_typing_OVar,axiom,
! [T2: type,X1: nat,Env: fun(nat,type)] :
( ( aa(nat,type,Env,X1) = T2 )
=> typing(Env,var(X1),T2) ) ).
tff(fact_8_typing__elims_I1_J,axiom,
! [T2: type,I: nat,E: fun(nat,type)] :
( typing(E,var(I),T2)
=> ( aa(nat,type,E,I) = T2 ) ) ).
tff(fact_9_abs__typeE,axiom,
! [T2: type,T1: dB,E: fun(nat,type)] :
( typing(E,abs(T1),T2)
=> ~ ! [U: type,V: type] : ~ typing(shift(type,E,zero_zero(nat),U),T1,V) ) ).
tff(fact_10_subst__lift,axiom,
! [S: dB,K: nat,T: dB] : subst(lift(T,K),S,K) = T ).
tff(fact_11_shift__gt,axiom,
! [A: $tType,T2: A,E: fun(nat,A),I: nat,J1: nat] :
( ord_less(nat,J1,I)
=> ( aa(nat,A,shift(A,E,I,T2),J1) = aa(nat,A,E,J1) ) ) ).
tff(fact_12_dB_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( var(Nat2) = var(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_13_dB_Osimps_I3_J,axiom,
! [DB2: dB,DB: dB] :
( ( abs(DB) = abs(DB2) )
<=> ( DB = DB2 ) ) ).
tff(fact_14_dB_Osimps_I7_J,axiom,
! [Nat1: nat,DB1: dB] : abs(DB1) != var(Nat1) ).
tff(fact_15_dB_Osimps_I6_J,axiom,
! [DB1: dB,Nat1: nat] : var(Nat1) != abs(DB1) ).
tff(fact_16_subst__lt,axiom,
! [U1: dB,I1: nat,J: nat] :
( ord_less(nat,J,I1)
=> ( subst(var(J),U1,I1) = var(J) ) ) ).
tff(fact_17_subst__eq,axiom,
! [U1: dB,K: nat] : subst(var(K),U1,K) = U1 ).
tff(fact_18_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_19_less__nat__zero__code,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_20_less__zeroE,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_21_Abs,axiom,
! [Ua: type,T1: dB,T2: type,Env: fun(nat,type)] :
( typing(shift(type,Env,zero_zero(nat),T2),T1,Ua)
=> typing(Env,abs(T1),fun1(T2,Ua)) ) ).
tff(fact_22_dB_Osize_I1_J,axiom,
! [Nat1: nat] : dB_size(var(Nat1)) = zero_zero(nat) ).
tff(fact_23_not__less0,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_24_gr__implies__not0,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ( N2 != zero_zero(nat) ) ) ).
tff(fact_25_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N2) ) ).
tff(fact_26_type_Osimps_I2_J,axiom,
! [Type23: type,Type13: type,Type22: type,Type12: type] :
( ( fun1(Type12,Type22) = fun1(Type13,Type23) )
<=> ( ( Type12 = Type13 )
& ( Type22 = Type23 ) ) ) ).
tff(fact_27_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_28_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N: nat,M: nat] :
( ( ord_less(nat,M,N)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M)) )
=> ( ( ( M = N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M)) )
=> ( ( ord_less(nat,N,M)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M)) ) ) ) ).
tff(fact_29_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_30_less__not__refl2,axiom,
! [M2: nat,N2: nat] :
( ord_less(nat,N2,M2)
=> ( M2 != N2 ) ) ).
tff(fact_31_less__irrefl__nat,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_32_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_33_nat__neq__iff,axiom,
! [N: nat,M: nat] :
( ( M != N )
<=> ( ord_less(nat,M,N)
| ord_less(nat,N,M) ) ) ).
tff(fact_34_less__not__refl,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_35_typing__elims_I3_J,axiom,
! [T2: type,T1: dB,E: fun(nat,type)] :
( typing(E,abs(T1),T2)
=> ~ ! [T3: type,U: type] :
( ( T2 = fun1(T3,U) )
=> ~ typing(shift(type,E,zero_zero(nat),T3),T1,U) ) ) ).
tff(fact_36_type_Osimps_I6_J,axiom,
! [A: $tType,Type22: type,Type12: type,F2: fun(type,fun(type,A)),F1: fun(nat,A)] : type_case(A,F1,F2,fun1(Type12,Type22)) = aa(type,A,aa(type,fun(type,A),F2,Type12),Type22) ).
tff(fact_37_dB_Osize_I4_J,axiom,
! [Nat1: nat] : size_size(dB,var(Nat1)) = zero_zero(nat) ).
tff(fact_38_of__nat__0__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] :
( ord_less(A,zero_zero(A),semiring_1_of_nat(A,N))
<=> ord_less(nat,zero_zero(nat),N) ) ) ).
tff(fact_39_substn__subst__0,axiom,
! [S: dB,T: dB] : substn(T,S,zero_zero(nat)) = subst(T,S,zero_zero(nat)) ).
tff(fact_40_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_41_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat,M: nat] :
( ( semiring_1_of_nat(A,M) = semiring_1_of_nat(A,N) )
<=> ( M = N ) ) ) ).
tff(fact_42_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_43_of__nat__less__iff,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_44_of__nat__less__0__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M2: nat] : ~ ord_less(A,semiring_1_of_nat(A,M2),zero_zero(A)) ) ).
tff(fact_45_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ord_less(A,semiring_1_of_nat(A,M2),semiring_1_of_nat(A,N2)) ) ) ).
tff(fact_46_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,M2: nat] :
( ord_less(A,semiring_1_of_nat(A,M2),semiring_1_of_nat(A,N2))
=> ord_less(nat,M2,N2) ) ) ).
tff(fact_47_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat2: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_48_substn__subst__n,axiom,
! [N2: nat,S: dB,T: dB] : substn(T,S,N2) = subst(T,liftn(N2,S,zero_zero(nat)),N2) ).
tff(fact_49_zero__less__int__conv,axiom,
! [N: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,N))
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_50_type_Osimps_I5_J,axiom,
! [A: $tType,Nat2: nat,F2: fun(type,fun(type,A)),F1: fun(nat,A)] : type_case(A,F1,F2,atom(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_51_infinite__descent0,axiom,
! [N: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( ! [N3: nat] :
( ord_less(nat,zero_zero(nat),N3)
=> ( ~ pp(aa(nat,bool,P1,N3))
=> ? [M1: nat] :
( ord_less(nat,M1,N3)
& ~ pp(aa(nat,bool,P1,M1)) ) ) )
=> pp(aa(nat,bool,P1,N)) ) ) ).
tff(fact_52_type_Orecs_I2_J,axiom,
! [A: $tType,Type22: type,Type12: type,F2: fun(type,fun(type,fun(A,fun(A,A)))),F1: fun(nat,A)] : type_rec(A,F1,F2,fun1(Type12,Type22)) = aa(A,A,aa(A,fun(A,A),aa(type,fun(A,fun(A,A)),aa(type,fun(type,fun(A,fun(A,A))),F2,Type12),Type22),type_rec(A,F1,F2,Type12)),type_rec(A,F1,F2,Type22)) ).
tff(fact_53_type_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( atom(Nat2) = atom(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_54_int__less__0__conv,axiom,
! [K: nat] : ~ ord_less(int,semiring_1_of_nat(int,K),zero_zero(int)) ).
tff(fact_55_int__int__eq,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat(int,M) = semiring_1_of_nat(int,N) )
<=> ( M = N ) ) ).
tff(fact_56_type_Orecs_I1_J,axiom,
! [A: $tType,Nat2: nat,F2: fun(type,fun(type,fun(A,fun(A,A)))),F1: fun(nat,A)] : type_rec(A,F1,F2,atom(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_57_liftn__0,axiom,
! [K: nat,T: dB] : liftn(zero_zero(nat),T,K) = T ).
tff(fact_58_type_Osimps_I3_J,axiom,
! [Type21: type,Type11: type,Nat1: nat] : atom(Nat1) != fun1(Type11,Type21) ).
tff(fact_59_type_Osimps_I4_J,axiom,
! [Nat1: nat,Type21: type,Type11: type] : fun1(Type11,Type21) != atom(Nat1) ).
tff(fact_60_int__eq__0__conv,axiom,
! [N: nat] :
( ( semiring_1_of_nat(int,N) = zero_zero(int) )
<=> ( N = zero_zero(nat) ) ) ).
tff(fact_61_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_62_zless__int,axiom,
! [N: nat,M: nat] :
( ord_less(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N))
<=> ord_less(nat,M,N) ) ).
tff(fact_63_zero__less__imp__eq__int,axiom,
! [K: int] :
( ord_less(int,zero_zero(int),K)
=> ? [N3: nat] :
( ord_less(nat,zero_zero(nat),N3)
& ( K = semiring_1_of_nat(int,N3) ) ) ) ).
tff(fact_64_type_Osize_I1_J,axiom,
! [Nat1: nat] : type_size(atom(Nat1)) = zero_zero(nat) ).
tff(fact_65_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_66_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = semiring_1_of_nat(int,zero_zero(nat)) ).
tff(fact_67_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_68_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_69_type_Osize_I3_J,axiom,
! [Nat1: nat] : size_size(type,atom(Nat1)) = zero_zero(nat) ).
tff(fact_70_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_71_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_72_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int,W: int] :
( ( ring_1_of_int(A,W) = ring_1_of_int(A,Z) )
<=> ( W = Z ) ) ) ).
tff(fact_73_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_75_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_76_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_77_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W: int] :
( ord_less(A,ring_1_of_int(A,W),ring_1_of_int(A,Z))
<=> ord_less(int,W,Z) ) ) ).
tff(fact_78_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N2: nat] : ring_1_of_int(A,semiring_1_of_nat(int,N2)) = semiring_1_of_nat(A,N2) ) ).
tff(fact_79_of__int__int__eq,axiom,
! [N2: nat] : ring_1_of_int(int,semiring_1_of_nat(int,N2)) = semiring_1_of_nat(int,N2) ).
tff(fact_80_type_Oexhaust,axiom,
! [Y: type] :
( ! [Nat: nat] : Y != atom(Nat)
=> ~ ! [Type1: type,Type2: type] : Y != fun1(Type1,Type2) ) ).
tff(fact_81_zero__less__nat__eq,axiom,
! [Z: int] :
( ord_less(nat,zero_zero(nat),nat1(Z))
<=> ord_less(int,zero_zero(int),Z) ) ).
tff(fact_82_nat__0,axiom,
nat1(zero_zero(int)) = zero_zero(nat) ).
tff(fact_83_zless__nat__conj,axiom,
! [Z: int,W: int] :
( ord_less(nat,nat1(W),nat1(Z))
<=> ( ord_less(int,zero_zero(int),Z)
& ord_less(int,W,Z) ) ) ).
tff(fact_84_nat__int,axiom,
! [N2: nat] : nat1(semiring_1_of_nat(int,N2)) = N2 ).
tff(fact_85_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_86_transfer__nat__int__numerals_I1_J,axiom,
zero_zero(nat) = nat1(zero_zero(int)) ).
tff(fact_87_nat__mono__iff,axiom,
! [W: int,Z: int] :
( ord_less(int,zero_zero(int),Z)
=> ( ord_less(nat,nat1(W),nat1(Z))
<=> ord_less(int,W,Z) ) ) ).
tff(fact_88_zless__nat__eq__int__zless,axiom,
! [Z: int,M: nat] :
( ord_less(nat,M,nat1(Z))
<=> ord_less(int,semiring_1_of_nat(int,M),Z) ) ).
tff(fact_89_split__nat,axiom,
! [I: int,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,nat1(I)))
<=> ( ! [N1: nat] :
( ( I = semiring_1_of_nat(int,N1) )
=> pp(aa(nat,bool,P1,N1)) )
& ( ord_less(int,I,zero_zero(int))
=> pp(aa(nat,bool,P1,zero_zero(nat))) ) ) ) ).
tff(fact_90_nat__less__iff,axiom,
! [M: nat,W: int] :
( ord_less_eq(int,zero_zero(int),W)
=> ( ord_less(nat,nat1(W),M)
<=> ord_less(int,W,semiring_1_of_nat(int,M)) ) ) ).
tff(fact_91_of__nat__nat,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: int] :
( ord_less_eq(int,zero_zero(int),Z2)
=> ( semiring_1_of_nat(A,nat1(Z2)) = ring_1_of_int(A,Z2) ) ) ) ).
tff(fact_92_of__nat__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less_eq(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N))
<=> ord_less_eq(nat,M,N) ) ) ).
tff(fact_93_of__int__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W: int] :
( ord_less_eq(A,ring_1_of_int(A,W),ring_1_of_int(A,Z))
<=> ord_less_eq(int,W,Z) ) ) ).
tff(fact_94_nat__0__iff,axiom,
! [I: int] :
( ( nat1(I) = zero_zero(nat) )
<=> ord_less_eq(int,I,zero_zero(int)) ) ).
tff(fact_95_int__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_eq(int,zero_zero(int),Z2)
=> ( semiring_1_of_nat(int,nat1(Z2)) = Z2 ) )
& ( ~ ord_less_eq(int,zero_zero(int),Z2)
=> ( semiring_1_of_nat(int,nat1(Z2)) = zero_zero(int) ) ) ) ).
tff(fact_96_of__int__0__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,zero_zero(A),ring_1_of_int(A,Z))
<=> ord_less_eq(int,zero_zero(int),Z) ) ) ).
tff(fact_97_of__int__le__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,ring_1_of_int(A,Z),zero_zero(A))
<=> ord_less_eq(int,Z,zero_zero(int)) ) ) ).
tff(fact_98_eq__nat__nat__iff,axiom,
! [Z1: int,Z: int] :
( ord_less_eq(int,zero_zero(int),Z)
=> ( ord_less_eq(int,zero_zero(int),Z1)
=> ( ( nat1(Z) = nat1(Z1) )
<=> ( Z = Z1 ) ) ) ) ).
%----Arities (11)
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___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(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___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----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 (5)
tff(conj_0,hypothesis,
it(t) ).
tff(conj_1,hypothesis,
typing(shift(type,e,i,u1),t,t1) ).
tff(conj_2,hypothesis,
it(u) ).
tff(conj_3,hypothesis,
typing(e,u,u1) ).
tff(conj_4,conjecture,
it(subst(t,u,i)) ).
%------------------------------------------------------------------------------