TPTP Problem File: NUM974_5.p
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%------------------------------------------------------------------------------
% File : NUM974_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 112
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_112 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.4.0, 0.50 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 143 ( 42 unt; 26 typ; 0 def)
% Number of atoms : 255 ( 70 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 165 ( 27 ~; 2 |; 9 &)
% ( 42 <=>; 85 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 13 ( 8 >; 5 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 1 prp; 0-3 aty)
% Number of functors : 10 ( 10 usr; 3 con; 0-4 aty)
% Number of variables : 190 ( 169 !; 4 ?; 190 :)
% ( 17 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:24:38
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (22)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[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_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_Onat,type,
nat1: int > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
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_aa,type,
aa:
!>[A: $tType,B2: $tType] : ( ( fun(A,B2) * A ) > B2 ) ).
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,
t: int ).
tff(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (98)
tff(fact_0_t1,axiom,
ord_less(int,one_one(int),t) ).
tff(fact_1_zero__less__diff,axiom,
! [Ma: nat,N1: nat] :
( ord_less(nat,zero_zero(nat),minus_minus(nat,N1,Ma))
<=> ord_less(nat,Ma,N1) ) ).
tff(fact_2_diff__0__eq__0,axiom,
! [N2: nat] : ( minus_minus(nat,zero_zero(nat),N2) = zero_zero(nat) ) ).
tff(fact_3_diff__self__eq__0,axiom,
! [M: nat] : ( minus_minus(nat,M,M) = zero_zero(nat) ) ).
tff(fact_4_neq0__conv,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_5_less__nat__zero__code,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_6_less__zeroE,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_7_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] : ( minus_minus(A,A2,A2) = zero_zero(A) ) ) ).
tff(fact_8_diff__less,axiom,
! [M: nat,N2: nat] :
( ord_less(nat,zero_zero(nat),N2)
=> ( ord_less(nat,zero_zero(nat),M)
=> ord_less(nat,minus_minus(nat,M,N2),M) ) ) ).
tff(fact_9_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,B)
<=> ord_less(A,minus_minus(A,A1,B),zero_zero(A)) ) ) ).
tff(fact_10_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_11_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_12_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_13_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y1: A,X2: A] :
( ( X2 != Y1 )
=> ( ~ ord_less(A,X2,Y1)
=> ord_less(A,Y1,X2) ) ) ) ).
tff(fact_14_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
tff(fact_15_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ( minus_minus(A,A1,B) = minus_minus(A,C,D) )
=> ( ( A1 = B )
<=> ( C = D ) ) ) ) ).
tff(fact_16_nat__less__cases,axiom,
! [P: fun(nat,fun(nat,bool)),N1: nat,Ma: nat] :
( ( ord_less(nat,Ma,N1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),Ma)) )
=> ( ( ( Ma = N1 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),Ma)) )
=> ( ( ord_less(nat,N1,Ma)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),Ma)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),Ma)) ) ) ) ).
tff(fact_17_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_18_less__not__refl2,axiom,
! [M: nat,N2: nat] :
( ord_less(nat,N2,M)
=> ( M != N2 ) ) ).
tff(fact_19_less__irrefl__nat,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_20_linorder__neqE__nat,axiom,
! [Y1: nat,X2: nat] :
( ( X2 != Y1 )
=> ( ~ ord_less(nat,X2,Y1)
=> ord_less(nat,Y1,X2) ) ) ).
tff(fact_21_nat__neq__iff,axiom,
! [N1: nat,Ma: nat] :
( ( Ma != N1 )
<=> ( ord_less(nat,Ma,N1)
| ord_less(nat,N1,Ma) ) ) ).
tff(fact_22_less__not__refl,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_23_diff__commute,axiom,
! [K: nat,J: nat,I1: nat] : ( minus_minus(nat,minus_minus(nat,I1,J),K) = minus_minus(nat,minus_minus(nat,I1,K),J) ) ).
tff(fact_24_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_25_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_26_right__minus__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B: A,A1: A] :
( ( minus_minus(A,A1,B) = zero_zero(A) )
<=> ( A1 = B ) ) ) ).
tff(fact_27_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B: A,A1: A] :
( ( A1 = B )
<=> ( minus_minus(A,A1,B) = zero_zero(A) ) ) ) ).
tff(fact_28_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] : ( minus_minus(A,A2,zero_zero(A)) = A2 ) ) ).
tff(fact_29_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ( minus_minus(A,A1,B) = minus_minus(A,C,D) )
=> ( ord_less(A,A1,B)
<=> ord_less(A,C,D) ) ) ) ).
tff(fact_30_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N2) ) ).
tff(fact_31_gr__implies__not0,axiom,
! [N2: nat,M: nat] :
( ord_less(nat,M,N2)
=> ( N2 != zero_zero(nat) ) ) ).
tff(fact_32_not__less0,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_33_diffs0__imp__equal,axiom,
! [N2: nat,M: nat] :
( ( minus_minus(nat,M,N2) = zero_zero(nat) )
=> ( ( minus_minus(nat,N2,M) = zero_zero(nat) )
=> ( M = N2 ) ) ) ).
tff(fact_34_minus__nat_Odiff__0,axiom,
! [M: nat] : ( minus_minus(nat,M,zero_zero(nat)) = M ) ).
tff(fact_35_diff__less__mono2,axiom,
! [L: nat,N2: nat,M: nat] :
( ord_less(nat,M,N2)
=> ( ord_less(nat,M,L)
=> ord_less(nat,minus_minus(nat,L,N2),minus_minus(nat,L,M)) ) ) ).
tff(fact_36_less__imp__diff__less,axiom,
! [N2: nat,K: nat,J: nat] :
( ord_less(nat,J,K)
=> ord_less(nat,minus_minus(nat,J,N2),K) ) ).
tff(fact_37_zero__less__nat__eq,axiom,
! [Z: int] :
( ord_less(nat,zero_zero(nat),nat1(Z))
<=> ord_less(int,zero_zero(int),Z) ) ).
tff(fact_38_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_39_nat__0,axiom,
nat1(zero_zero(int)) = zero_zero(nat) ).
tff(fact_40_transfer__nat__int__numerals_I2_J,axiom,
one_one(nat) = nat1(one_one(int)) ).
tff(fact_41_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_42_tpos,axiom,
ord_less_eq(int,one_one(int),t) ).
tff(fact_43_transfer__nat__int__numerals_I1_J,axiom,
zero_zero(nat) = nat1(zero_zero(int)) ).
tff(fact_44_Euler_Oaux1,axiom,
! [A2: int,X2: int] :
( ord_less(int,zero_zero(int),X2)
=> ( ord_less(int,X2,A2)
=> ( ( X2 != minus_minus(int,A2,one_one(int)) )
=> ord_less(int,X2,minus_minus(int,A2,one_one(int))) ) ) ) ).
tff(fact_45_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_46_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N1: nat,Ma: nat] :
( ( semiring_1_of_nat(A,Ma) = semiring_1_of_nat(A,N1) )
<=> ( Ma = N1 ) ) ) ).
tff(fact_47_of__nat__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat,Ma: nat] :
( ord_less_eq(A,semiring_1_of_nat(A,Ma),semiring_1_of_nat(A,N1))
<=> ord_less_eq(nat,Ma,N1) ) ) ).
tff(fact_48_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_49_of__nat__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat,Ma: nat] :
( ord_less(A,semiring_1_of_nat(A,Ma),semiring_1_of_nat(A,N1))
<=> ord_less(nat,Ma,N1) ) ) ).
tff(fact_50_of__nat__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,one_one(nat)) = one_one(A) ) ) ).
tff(fact_51_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_52_nat__0__iff,axiom,
! [I: int] :
( ( nat1(I) = zero_zero(nat) )
<=> ord_less_eq(int,I,zero_zero(int)) ) ).
tff(fact_53_zle__diff1__eq,axiom,
! [Z: int,W: int] :
( ord_less_eq(int,W,minus_minus(int,Z,one_one(int)))
<=> ord_less(int,W,Z) ) ).
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_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_56_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = semiring_1_of_nat(int,zero_zero(nat)) ).
tff(fact_57_zero__le__imp__of__nat,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,M)) ) ).
tff(fact_58_of__nat__0__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,N2)) ) ).
tff(fact_59_zero__zle__int,axiom,
! [N2: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,N2)) ).
tff(fact_60_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z2: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,Z2)) ).
tff(fact_61_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_62_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: fun(int,bool)] :
( ? [X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
& pp(aa(int,bool,P,X1)) )
<=> ? [X1: nat] : pp(aa(int,bool,P,semiring_1_of_nat(int,X1))) ) ).
tff(fact_63_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: fun(int,bool)] :
( ! [X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
=> pp(aa(int,bool,P,X1)) )
<=> ! [X1: nat] : pp(aa(int,bool,P,semiring_1_of_nat(int,X1))) ) ).
tff(fact_64_int__eq__0__conv,axiom,
! [N1: nat] :
( ( semiring_1_of_nat(int,N1) = zero_zero(int) )
<=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_65_int__le__0__conv,axiom,
! [N1: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,N1),zero_zero(int))
<=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_66_int__eq__iff,axiom,
! [Z: int,Ma: nat] :
( ( semiring_1_of_nat(int,Ma) = Z )
<=> ( ( Ma = nat1(Z) )
& ord_less_eq(int,zero_zero(int),Z) ) ) ).
tff(fact_67_nat__0__le,axiom,
! [Z2: int] :
( ord_less_eq(int,zero_zero(int),Z2)
=> ( semiring_1_of_nat(int,nat1(Z2)) = Z2 ) ) ).
tff(fact_68_nat__int,axiom,
! [N2: nat] : ( nat1(semiring_1_of_nat(int,N2)) = N2 ) ).
tff(fact_69_Euler_Oaux2,axiom,
! [B1: int,C1: int,A2: int] :
( ord_less(int,A2,C1)
=> ( ord_less(int,B1,C1)
=> ( ord_less_eq(int,A2,B1)
| ord_less_eq(int,B1,A2) ) ) ) ).
tff(fact_70_less__int__def,axiom,
! [W: int,Z: int] :
( ord_less(int,Z,W)
<=> ( ord_less_eq(int,Z,W)
& ( Z != W ) ) ) ).
tff(fact_71_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq(int,zero_zero(int),one_one(int)) ).
tff(fact_72_eq__nat__nat__iff,axiom,
! [Z3: int,Z: int] :
( ord_less_eq(int,zero_zero(int),Z)
=> ( ord_less_eq(int,zero_zero(int),Z3)
=> ( ( nat1(Z) = nat1(Z3) )
<=> ( Z = Z3 ) ) ) ) ).
tff(fact_73_transfer__nat__int__relations_I1_J,axiom,
! [Y: int,X: int] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less_eq(int,zero_zero(int),Y)
=> ( ( nat1(X) = nat1(Y) )
<=> ( X = Y ) ) ) ) ).
tff(fact_74_all__nat,axiom,
! [P: fun(nat,bool)] :
( ! [X11: nat] : pp(aa(nat,bool,P,X11))
<=> ! [X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
=> pp(aa(nat,bool,P,nat1(X1))) ) ) ).
tff(fact_75_ex__nat,axiom,
! [P: fun(nat,bool)] :
( ? [X11: nat] : pp(aa(nat,bool,P,X11))
<=> ? [X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
& pp(aa(nat,bool,P,nat1(X1))) ) ) ).
tff(fact_76_nat__eq__iff2,axiom,
! [W: int,Ma: nat] :
( ( Ma = nat1(W) )
<=> ( ( ord_less_eq(int,zero_zero(int),W)
=> ( W = semiring_1_of_nat(int,Ma) ) )
& ( ~ ord_less_eq(int,zero_zero(int),W)
=> ( Ma = zero_zero(nat) ) ) ) ) ).
tff(fact_77_nat__eq__iff,axiom,
! [Ma: nat,W: int] :
( ( nat1(W) = Ma )
<=> ( ( ord_less_eq(int,zero_zero(int),W)
=> ( W = semiring_1_of_nat(int,Ma) ) )
& ( ~ ord_less_eq(int,zero_zero(int),W)
=> ( Ma = zero_zero(nat) ) ) ) ) ).
tff(fact_78_nat__less__iff,axiom,
! [Ma: nat,W: int] :
( ord_less_eq(int,zero_zero(int),W)
=> ( ord_less(nat,nat1(W),Ma)
<=> ord_less(int,W,semiring_1_of_nat(int,Ma)) ) ) ).
tff(fact_79_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ( minus_minus(A,A1,B) = minus_minus(A,C,D) )
=> ( ord_less_eq(A,A1,B)
<=> ord_less_eq(A,C,D) ) ) ) ).
tff(fact_80_zless__int,axiom,
! [N1: nat,Ma: nat] :
( ord_less(int,semiring_1_of_nat(int,Ma),semiring_1_of_nat(int,N1))
<=> ord_less(nat,Ma,N1) ) ).
tff(fact_81_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [Y: nat,X: nat] :
( ord_less(int,semiring_1_of_nat(int,X),semiring_1_of_nat(int,Y))
<=> ord_less(nat,X,Y) ) ).
tff(fact_82_transfer__int__nat__numerals_I2_J,axiom,
one_one(int) = semiring_1_of_nat(int,one_one(nat)) ).
tff(fact_83_int__1,axiom,
semiring_1_of_nat(int,one_one(nat)) = one_one(int) ).
tff(fact_84_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ord_less_eq(int,one_one(int),Z)
<=> ord_less(int,zero_zero(int),Z) ) ).
tff(fact_85_nat__le__0,axiom,
! [Z2: int] :
( ord_less_eq(int,Z2,zero_zero(int))
=> ( nat1(Z2) = zero_zero(nat) ) ) ).
tff(fact_86_nat__diff__distrib,axiom,
! [Z2: int,Z1: int] :
( ord_less_eq(int,zero_zero(int),Z1)
=> ( ord_less_eq(int,Z1,Z2)
=> ( nat1(minus_minus(int,Z2,Z1)) = minus_minus(nat,nat1(Z2),nat1(Z1)) ) ) ) ).
tff(fact_87_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_88_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,M: nat] :
( ord_less(nat,M,N2)
=> ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N2)) ) ) ).
tff(fact_89_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N2: nat,M: nat] :
( ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N2))
=> ord_less(nat,M,N2) ) ) ).
tff(fact_90_zero__less__int__conv,axiom,
! [N1: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,N1))
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_91_split__nat,axiom,
! [I: int,P: fun(nat,bool)] :
( pp(aa(nat,bool,P,nat1(I)))
<=> ( ! [N: nat] :
( ( I = semiring_1_of_nat(int,N) )
=> pp(aa(nat,bool,P,N)) )
& ( ord_less(int,I,zero_zero(int))
=> pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).
tff(fact_92_not__one__le__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less_eq(A,one_one(A),zero_zero(A)) ) ).
tff(fact_93_zero__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less_eq(A,zero_zero(A),one_one(A)) ) ).
tff(fact_94_zless__nat__eq__int__zless,axiom,
! [Z: int,Ma: nat] :
( ord_less(nat,Ma,nat1(Z))
<=> ord_less(int,semiring_1_of_nat(int,Ma),Z) ) ).
tff(fact_95_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,A1,B)
<=> ord_less_eq(A,minus_minus(A,A1,B),zero_zero(A)) ) ) ).
tff(fact_96_nat__less__eq__zless,axiom,
! [Z: int,W: int] :
( ord_less_eq(int,zero_zero(int),W)
=> ( ord_less(nat,nat1(W),nat1(Z))
<=> ord_less(int,W,Z) ) ) ).
tff(fact_97_transfer__nat__int__relations_I2_J,axiom,
! [Y: int,X: int] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less_eq(int,zero_zero(int),Y)
=> ( ord_less(nat,nat1(X),nat1(Y))
<=> ord_less(int,X,Y) ) ) ) ).
%----Arities (15)
tff(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
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_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(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_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
%----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,
! [Tn: nat] :
( ( Tn = minus_minus(nat,nat1(t),one_one(nat)) )
=> ( ord_less(nat,zero_zero(nat),Tn)
=> thesis ) ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------