TPTP Problem File: NUM988_5.p
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%------------------------------------------------------------------------------
% File : NUM988_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 149
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_149 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 147 ( 51 unt; 27 typ; 0 def)
% Number of atoms : 225 ( 67 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 109 ( 4 ~; 1 |; 1 &)
% ( 32 <=>; 71 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 10 ( 6 >; 4 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-3 aty)
% Number of functors : 10 ( 10 usr; 5 con; 0-3 aty)
% Number of variables : 247 ( 228 !; 1 ?; 247 :)
% ( 18 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:22
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
%----Explicit typings (24)
tff(sy_cl_Groups_Oone,type,
one:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[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__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_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_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
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_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_m1____,type,
m1: int ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (98)
tff(fact_0__096_126_Am1_A_060_A1_A_L_Aint_An_096,axiom,
~ ord_less(int,m1,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_1_t1,axiom,
ord_less(int,one_one(int),t) ).
tff(fact_2_n1pos,axiom,
ord_less(int,zero_zero(int),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_3_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less_eq(A,zero_zero(A),plus_plus(A,A2,A2))
<=> ord_less_eq(A,zero_zero(A),A2) ) ) ).
tff(fact_4_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less_eq(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less_eq(A,A2,zero_zero(A)) ) ) ).
tff(fact_5_of__nat__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,one_one(nat)) = one_one(A) ) ) ).
tff(fact_6_of__nat__add,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,M: nat] : semiring_1_of_nat(A,plus_plus(nat,M,N)) = plus_plus(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ).
tff(fact_7_of__nat__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,Ma: nat] :
( ord_less_eq(A,semiring_1_of_nat(A,Ma),semiring_1_of_nat(A,Na))
<=> ord_less_eq(nat,Ma,Na) ) ) ).
tff(fact_8_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_9_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A1: A] : minus_minus(A,A1,A1) = zero_zero(A) ) ).
tff(fact_10_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,C1: A,A2: A] :
( ord_less_eq(A,plus_plus(A,A2,C1),plus_plus(A,B1,C1))
<=> ord_less_eq(A,A2,B1) ) ) ).
tff(fact_11_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,A2: A,C1: A] :
( ord_less_eq(A,plus_plus(A,C1,A2),plus_plus(A,C1,B1))
<=> ord_less_eq(A,A2,B1) ) ) ).
tff(fact_12_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( zero_zero(A) = plus_plus(A,A2,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_13_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( plus_plus(A,A2,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_14_tpos,axiom,
ord_less_eq(int,one_one(int),t) ).
tff(fact_15_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A2: A,B1: A] :
( ( plus_plus(A,B1,A2) = plus_plus(A,C1,A2) )
<=> ( B1 = C1 ) ) ) ).
tff(fact_16_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B1: A,A2: A] :
( ( plus_plus(A,A2,B1) = plus_plus(A,A2,C1) )
<=> ( B1 = C1 ) ) ) ).
tff(fact_17_diff__0__eq__0,axiom,
! [N: nat] : minus_minus(nat,zero_zero(nat),N) = zero_zero(nat) ).
tff(fact_18_diff__self__eq__0,axiom,
! [M: nat] : minus_minus(nat,M,M) = zero_zero(nat) ).
tff(fact_19_diff__diff__left,axiom,
! [K: nat,J: nat,I: nat] : minus_minus(nat,minus_minus(nat,I,J),K) = minus_minus(nat,I,plus_plus(nat,J,K)) ).
tff(fact_20_nat__add__left__cancel,axiom,
! [Na: nat,Ma: nat,K1: nat] :
( ( plus_plus(nat,K1,Ma) = plus_plus(nat,K1,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_21_nat__add__right__cancel,axiom,
! [Na: nat,K1: nat,Ma: nat] :
( ( plus_plus(nat,Ma,K1) = plus_plus(nat,Na,K1) )
<=> ( Ma = Na ) ) ).
tff(fact_22_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [Na: nat,Ma: nat] :
( ( semiring_1_of_nat(A,Ma) = semiring_1_of_nat(A,Na) )
<=> ( Ma = Na ) ) ) ).
tff(fact_23_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,A2: A,C1: A] :
( ord_less(A,plus_plus(A,C1,A2),plus_plus(A,C1,B1))
<=> ord_less(A,A2,B1) ) ) ).
tff(fact_24_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,C1: A,A2: A] :
( ord_less(A,plus_plus(A,A2,C1),plus_plus(A,B1,C1))
<=> ord_less(A,A2,B1) ) ) ).
tff(fact_25_of__nat__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,Ma: nat] :
( ord_less(A,semiring_1_of_nat(A,Ma),semiring_1_of_nat(A,Na))
<=> ord_less(nat,Ma,Na) ) ) ).
tff(fact_26_add__is__0,axiom,
! [Na: nat,Ma: nat] :
( ( plus_plus(nat,Ma,Na) = zero_zero(nat) )
<=> ( ( Ma = zero_zero(nat) )
& ( Na = zero_zero(nat) ) ) ) ).
tff(fact_27_le0,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_28_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_29_le__0__eq,axiom,
! [Na: nat] :
( ord_less_eq(nat,Na,zero_zero(nat))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_30_diff__is__0__eq,axiom,
! [Na: nat,Ma: nat] :
( ( minus_minus(nat,Ma,Na) = zero_zero(nat) )
<=> ord_less_eq(nat,Ma,Na) ) ).
tff(fact_31_diff__diff__right,axiom,
! [I: nat,J: nat,K: nat] :
( ord_less_eq(nat,K,J)
=> ( minus_minus(nat,I,minus_minus(nat,J,K)) = minus_minus(nat,plus_plus(nat,I,K),J) ) ) ).
tff(fact_32_nat__add__left__cancel__le,axiom,
! [Na: nat,Ma: nat,K1: nat] :
( ord_less_eq(nat,plus_plus(nat,K1,Ma),plus_plus(nat,K1,Na))
<=> ord_less_eq(nat,Ma,Na) ) ).
tff(fact_33_n0,axiom,
ord_less(nat,zero_zero(nat),n) ).
tff(fact_34_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_35_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),plus_plus(A,A2,A2))
<=> ord_less(A,zero_zero(A),A2) ) ) ).
tff(fact_36_of__nat__0__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat] :
( ord_less(A,zero_zero(A),semiring_1_of_nat(A,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_37_zle__add1__eq__le,axiom,
! [Z1: int,Wa: int] :
( ord_less(int,Wa,plus_plus(int,Z1,one_one(int)))
<=> ord_less_eq(int,Wa,Z1) ) ).
tff(fact_38_zle__diff1__eq,axiom,
! [Z1: int,Wa: int] :
( ord_less_eq(int,Wa,minus_minus(int,Z1,one_one(int)))
<=> ord_less(int,Wa,Z1) ) ).
tff(fact_39_le__refl,axiom,
! [N: nat] : ord_less_eq(nat,N,N) ).
tff(fact_40_le__add2,axiom,
! [M: nat,N: nat] : ord_less_eq(nat,N,plus_plus(nat,M,N)) ).
tff(fact_41_le__add1,axiom,
! [M: nat,N: nat] : ord_less_eq(nat,N,plus_plus(nat,N,M)) ).
tff(fact_42_plus__nat_Oadd__0,axiom,
! [N: nat] : plus_plus(nat,zero_zero(nat),N) = N ).
tff(fact_43_Nat_Oadd__0__right,axiom,
! [M: nat] : plus_plus(nat,M,zero_zero(nat)) = M ).
tff(fact_44_minus__nat_Odiff__0,axiom,
! [M: nat] : minus_minus(nat,M,zero_zero(nat)) = M ).
tff(fact_45_Nat_Odiff__le__self,axiom,
! [N: nat,M: nat] : ord_less_eq(nat,minus_minus(nat,M,N),M) ).
tff(fact_46_le__iff__add,axiom,
! [Na: nat,Ma: nat] :
( ord_less_eq(nat,Ma,Na)
<=> ? [K2: nat] : Na = plus_plus(nat,Ma,K2) ) ).
tff(fact_47_nat__add__commute,axiom,
! [N: nat,M: nat] : plus_plus(nat,M,N) = plus_plus(nat,N,M) ).
tff(fact_48_nat__le__linear,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
| ord_less_eq(nat,N,M) ) ).
tff(fact_49_diff__add__0,axiom,
! [M: nat,N: nat] : minus_minus(nat,N,plus_plus(nat,N,M)) = zero_zero(nat) ).
tff(fact_50_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X: nat] : plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z)) ).
tff(fact_51_diff__add__inverse2,axiom,
! [N: nat,M: nat] : minus_minus(nat,plus_plus(nat,M,N),N) = M ).
tff(fact_52_diff__add__inverse,axiom,
! [M: nat,N: nat] : minus_minus(nat,plus_plus(nat,N,M),N) = M ).
tff(fact_53_nat__add__assoc,axiom,
! [K: nat,N: nat,M: nat] : plus_plus(nat,plus_plus(nat,M,N),K) = plus_plus(nat,M,plus_plus(nat,N,K)) ).
tff(fact_54_le__diff__conv,axiom,
! [I1: nat,K1: nat,J1: nat] :
( ord_less_eq(nat,minus_minus(nat,J1,K1),I1)
<=> ord_less_eq(nat,J1,plus_plus(nat,I1,K1)) ) ).
tff(fact_55_Nat_Odiff__cancel,axiom,
! [N: nat,M: nat,K: nat] : minus_minus(nat,plus_plus(nat,K,M),plus_plus(nat,K,N)) = minus_minus(nat,M,N) ).
tff(fact_56_diff__cancel2,axiom,
! [N: nat,K: nat,M: nat] : minus_minus(nat,plus_plus(nat,M,K),plus_plus(nat,N,K)) = minus_minus(nat,M,N) ).
tff(fact_57_eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( M = N )
=> ord_less_eq(nat,M,N) ) ).
tff(fact_58_trans__le__add1,axiom,
! [M: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ord_less_eq(nat,I,plus_plus(nat,J,M)) ) ).
tff(fact_59_trans__le__add2,axiom,
! [M: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ord_less_eq(nat,I,plus_plus(nat,M,J)) ) ).
tff(fact_60_le__add__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ord_less_eq(nat,K,N)
=> ord_less_eq(nat,M,minus_minus(nat,plus_plus(nat,N,M),K)) ) ).
tff(fact_61_diff__is__0__eq_H,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ( minus_minus(nat,M,N) = zero_zero(nat) ) ) ).
tff(fact_62_add__le__mono1,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ord_less_eq(nat,plus_plus(nat,I,K),plus_plus(nat,J,K)) ) ).
tff(fact_63_diff__le__mono2,axiom,
! [L: nat,N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ord_less_eq(nat,minus_minus(nat,L,N),minus_minus(nat,L,M)) ) ).
tff(fact_64_diff__le__mono,axiom,
! [L: nat,N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ord_less_eq(nat,minus_minus(nat,M,L),minus_minus(nat,N,L)) ) ).
tff(fact_65_diff__diff__cancel,axiom,
! [N: nat,I: nat] :
( ord_less_eq(nat,I,N)
=> ( minus_minus(nat,N,minus_minus(nat,N,I)) = I ) ) ).
tff(fact_66_le__add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ord_less_eq(nat,N,M)
=> ( plus_plus(nat,N,minus_minus(nat,M,N)) = M ) ) ).
tff(fact_67_add__diff__assoc,axiom,
! [I: nat,J: nat,K: nat] :
( ord_less_eq(nat,K,J)
=> ( plus_plus(nat,I,minus_minus(nat,J,K)) = minus_minus(nat,plus_plus(nat,I,J),K) ) ) ).
tff(fact_68_le__diff__conv2,axiom,
! [I1: nat,J1: nat,K1: nat] :
( ord_less_eq(nat,K1,J1)
=> ( ord_less_eq(nat,I1,minus_minus(nat,J1,K1))
<=> ord_less_eq(nat,plus_plus(nat,I1,K1),J1) ) ) ).
tff(fact_69_le__add__diff__inverse2,axiom,
! [M: nat,N: nat] :
( ord_less_eq(nat,N,M)
=> ( plus_plus(nat,minus_minus(nat,M,N),N) = M ) ) ).
tff(fact_70_le__imp__diff__is__add,axiom,
! [K1: nat,J1: nat,I1: nat] :
( ord_less_eq(nat,I1,J1)
=> ( ( minus_minus(nat,J1,I1) = K1 )
<=> ( J1 = plus_plus(nat,K1,I1) ) ) ) ).
tff(fact_71_diff__add__assoc,axiom,
! [I: nat,J: nat,K: nat] :
( ord_less_eq(nat,K,J)
=> ( minus_minus(nat,plus_plus(nat,I,J),K) = plus_plus(nat,I,minus_minus(nat,J,K)) ) ) ).
tff(fact_72_add__diff__assoc2,axiom,
! [I: nat,J: nat,K: nat] :
( ord_less_eq(nat,K,J)
=> ( plus_plus(nat,minus_minus(nat,J,K),I) = minus_minus(nat,plus_plus(nat,J,I),K) ) ) ).
tff(fact_73_diff__add__assoc2,axiom,
! [I: nat,J: nat,K: nat] :
( ord_less_eq(nat,K,J)
=> ( minus_minus(nat,plus_plus(nat,J,I),K) = plus_plus(nat,minus_minus(nat,J,K),I) ) ) ).
tff(fact_74_le__trans,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,J,K)
=> ord_less_eq(nat,I,K) ) ) ).
tff(fact_75_le__antisym,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ( ord_less_eq(nat,N,M)
=> ( M = N ) ) ) ).
tff(fact_76_add__le__mono,axiom,
! [L: nat,K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,K,L)
=> ord_less_eq(nat,plus_plus(nat,I,K),plus_plus(nat,J,L)) ) ) ).
tff(fact_77_eq__diff__iff,axiom,
! [Na: nat,Ma: nat,K1: nat] :
( ord_less_eq(nat,K1,Ma)
=> ( ord_less_eq(nat,K1,Na)
=> ( ( minus_minus(nat,Ma,K1) = minus_minus(nat,Na,K1) )
<=> ( Ma = Na ) ) ) ) ).
tff(fact_78_Nat_Odiff__diff__eq,axiom,
! [N: nat,M: nat,K: nat] :
( ord_less_eq(nat,K,M)
=> ( ord_less_eq(nat,K,N)
=> ( minus_minus(nat,minus_minus(nat,M,K),minus_minus(nat,N,K)) = minus_minus(nat,M,N) ) ) ) ).
tff(fact_79_le__diff__iff,axiom,
! [Na: nat,Ma: nat,K1: nat] :
( ord_less_eq(nat,K1,Ma)
=> ( ord_less_eq(nat,K1,Na)
=> ( ord_less_eq(nat,minus_minus(nat,Ma,K1),minus_minus(nat,Na,K1))
<=> ord_less_eq(nat,Ma,Na) ) ) ) ).
tff(fact_80_diffs0__imp__equal,axiom,
! [N: nat,M: nat] :
( ( minus_minus(nat,M,N) = zero_zero(nat) )
=> ( ( minus_minus(nat,N,M) = zero_zero(nat) )
=> ( M = N ) ) ) ).
tff(fact_81_add__eq__self__zero,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = M )
=> ( N = zero_zero(nat) ) ) ).
tff(fact_82_add__leD2,axiom,
! [N: nat,K: nat,M: nat] :
( ord_less_eq(nat,plus_plus(nat,M,K),N)
=> ord_less_eq(nat,K,N) ) ).
tff(fact_83_add__leD1,axiom,
! [N: nat,K: nat,M: nat] :
( ord_less_eq(nat,plus_plus(nat,M,K),N)
=> ord_less_eq(nat,M,N) ) ).
tff(fact_84_add__leE,axiom,
! [N: nat,K: nat,M: nat] :
( ord_less_eq(nat,plus_plus(nat,M,K),N)
=> ~ ( ord_less_eq(nat,M,N)
=> ~ ord_less_eq(nat,K,N) ) ) ).
tff(fact_85_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_86_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_87_zdiff__int,axiom,
! [M: nat,N: nat] :
( ord_less_eq(nat,N,M)
=> ( minus_minus(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N)) = semiring_1_of_nat(int,minus_minus(nat,M,N)) ) ) ).
tff(fact_88_of__nat__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: nat,N: nat] :
( ord_less_eq(nat,N,M)
=> ( semiring_1_of_nat(A,minus_minus(nat,M,N)) = minus_minus(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ) ) ).
tff(fact_89_zadd__int__left,axiom,
! [Z: int,N: nat,M: nat] : plus_plus(int,semiring_1_of_nat(int,M),plus_plus(int,semiring_1_of_nat(int,N),Z)) = plus_plus(int,semiring_1_of_nat(int,plus_plus(nat,M,N)),Z) ).
tff(fact_90_zadd__int,axiom,
! [N: nat,M: nat] : plus_plus(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N)) = semiring_1_of_nat(int,plus_plus(nat,M,N)) ).
tff(fact_91_zle__int,axiom,
! [Na: nat,Ma: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,Ma),semiring_1_of_nat(int,Na))
<=> ord_less_eq(nat,Ma,Na) ) ).
tff(fact_92_int__1,axiom,
semiring_1_of_nat(int,one_one(nat)) = one_one(int) ).
tff(fact_93_int__eq__0__conv,axiom,
! [Na: nat] :
( ( semiring_1_of_nat(int,Na) = zero_zero(int) )
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_94_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_95_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C: A] :
( ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B))
=> ord_less(A,A1,B) ) ) ).
tff(fact_96_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C: A,A1: A] :
( ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,C))
=> ord_less(A,A1,B) ) ) ).
tff(fact_97_add__strict__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,C,D)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,D)) ) ) ) ).
%----Arities (19)
tff(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(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___Groups_Ogroup__add,axiom,
group_add(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___Groups_Oone,axiom,
one(int) ).
tff(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
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) ).
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 (1)
tff(conj_0,conjecture,
ord_less_eq(int,minus_minus(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),m1),zero_zero(int)) ).
%------------------------------------------------------------------------------