TPTP Problem File: SWW481_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW481_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 35
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_35 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 199 ( 80 unt; 45 typ; 0 def)
% Number of atoms : 277 ( 98 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 131 ( 8 ~; 0 |; 13 &)
% ( 41 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 18 ( 12 >; 6 *; 0 +; 0 <<)
% Number of predicates : 23 ( 22 usr; 0 prp; 1-3 aty)
% Number of functors : 19 ( 19 usr; 8 con; 0-3 aty)
% Number of variables : 261 ( 233 !; 0 ?; 261 :)
% ( 28 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:04
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
%----Explicit typings (41)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield__inverse__zero,type,
field_inverse_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_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,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_c_Complex_Ocomplex_OComplex,type,
complex1: ( real * real ) > complex ).
tff(sy_c_Fields_Oinverse__class_Odivide,type,
inverse_divide:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_OBit0,type,
bit0: int > int ).
tff(sy_c_Int_OBit1,type,
bit1: int > int ).
tff(sy_c_Int_OPls,type,
pls: int ).
tff(sy_c_Int_Onumber__class_Onumber__of,type,
number_number_of:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_RealVector_Onorm__class_Onorm,type,
norm_norm:
!>[A: $tType] : ( A > real ) ).
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_x____,type,
x: real ).
tff(sy_v_xa____,type,
xa: real ).
tff(sy_v_y____,type,
y: real ).
tff(sy_v_ya____,type,
ya: real ).
tff(sy_v_z,type,
z: complex ).
%----Relevant facts (97)
tff(fact_0_y0,axiom,
y != zero_zero(real) ).
tff(fact_1__0960_A_060_061_Acmod_A_IComplex_Ax_Ay_J_A_L_Ax_096,axiom,
ord_less_eq(real,zero_zero(real),plus_plus(real,norm_norm(complex,complex1(xa,ya)),xa)) ).
tff(fact_2__0960_A_060_061_Acmod_A_IComplex_Ax_Ay_J_A_N_Ax_096,axiom,
ord_less_eq(real,zero_zero(real),minus_minus(real,norm_norm(complex,complex1(xa,ya)),xa)) ).
tff(fact_3_eq__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [W1: int,B1: A,A2: A] :
( ( A2 = inverse_divide(A,B1,number_number_of(A,W1)) )
<=> ( ( ( number_number_of(A,W1) != zero_zero(A) )
=> ( times_times(A,A2,number_number_of(A,W1)) = B1 ) )
& ( ( number_number_of(A,W1) = zero_zero(A) )
=> ( A2 = zero_zero(A) ) ) ) ) ) ).
tff(fact_4_divide__eq__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [A2: A,W1: int,B1: A] :
( ( inverse_divide(A,B1,number_number_of(A,W1)) = A2 )
<=> ( ( ( number_number_of(A,W1) != zero_zero(A) )
=> ( B1 = times_times(A,A2,number_number_of(A,W1)) ) )
& ( ( number_number_of(A,W1) = zero_zero(A) )
=> ( A2 = zero_zero(A) ) ) ) ) ) ).
tff(fact_5_le__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Yb: int] :
( ord_less_eq(A,zero_zero(A),number_number_of(A,Yb))
<=> ord_less_eq(int,pls,Yb) ) ) ).
tff(fact_6_le__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xb: int] :
( ord_less_eq(A,number_number_of(A,Xb),zero_zero(A))
<=> ord_less_eq(int,Xb,pls) ) ) ).
tff(fact_7_real__sqrt__ge__0__iff,axiom,
! [Yb: real] :
( ord_less_eq(real,zero_zero(real),sqrt(Yb))
<=> ord_less_eq(real,zero_zero(real),Yb) ) ).
tff(fact_8_real__sqrt__le__0__iff,axiom,
! [Xb: real] :
( ord_less_eq(real,sqrt(Xb),zero_zero(real))
<=> ord_less_eq(real,Xb,zero_zero(real)) ) ).
tff(fact_9_real__two__squares__add__zero__iff,axiom,
! [Yb: real,Xb: real] :
( ( plus_plus(real,times_times(real,Xb,Xb),times_times(real,Yb,Yb)) = zero_zero(real) )
<=> ( ( Xb = zero_zero(real) )
& ( Yb = zero_zero(real) ) ) ) ).
tff(fact_10_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_11_right__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [C: B2,B: B2,V: int] : times_times(B2,number_number_of(B2,V),plus_plus(B2,B,C)) = plus_plus(B2,times_times(B2,number_number_of(B2,V),B),times_times(B2,number_number_of(B2,V),C)) ) ).
tff(fact_12_left__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [V: int,B: B2,A1: B2] : times_times(B2,plus_plus(B2,A1,B),number_number_of(B2,V)) = plus_plus(B2,times_times(B2,A1,number_number_of(B2,V)),times_times(B2,B,number_number_of(B2,V))) ) ).
tff(fact_13_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_14_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_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_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Yb: int,Xb: int] :
( ( number_number_of(A,Xb) = number_number_of(A,Yb) )
<=> ( Xb = Yb ) ) ) ).
tff(fact_18_rel__simps_I51_J,axiom,
! [L: int,K3: int] :
( ( bit1(K3) = bit1(L) )
<=> ( K3 = L ) ) ).
tff(fact_19_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_20_add__Bit0__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit0(L1)) = bit0(plus_plus(int,K,L1)) ).
tff(fact_21_mult__Bit0,axiom,
! [L1: int,K: int] : times_times(int,bit0(K),L1) = bit0(times_times(int,K,L1)) ).
tff(fact_22_rel__simps_I48_J,axiom,
! [L: int,K3: int] :
( ( bit0(K3) = bit0(L) )
<=> ( K3 = L ) ) ).
tff(fact_23_real__sqrt__eq__iff,axiom,
! [Yb: real,Xb: real] :
( ( sqrt(Xb) = sqrt(Yb) )
<=> ( Xb = Yb ) ) ).
tff(fact_24_xy,axiom,
z = complex1(x,y) ).
tff(fact_25_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_26_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_27_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_28_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_29_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A1: A] : minus_minus(A,A1,A1) = zero_zero(A) ) ).
tff(fact_30_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W),Z)) = times_times(A,number_number_of(A,times_times(int,V,W)),Z) ) ).
tff(fact_31_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : times_times(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,times_times(int,V,W)) ) ).
tff(fact_32_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W),Z)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W)),Z) ) ).
tff(fact_33_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : plus_plus(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,plus_plus(int,V,W)) ) ).
tff(fact_34_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_35_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_36_mult__Bit1,axiom,
! [L1: int,K: int] : times_times(int,bit1(K),L1) = plus_plus(int,bit0(times_times(int,K,L1)),L1) ).
tff(fact_37_add__Bit1__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit1(K),bit0(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_38_add__Bit0__Bit1,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit1(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_39_rel__simps_I50_J,axiom,
! [L1: int,K: int] : bit1(K) != bit0(L1) ).
tff(fact_40_rel__simps_I49_J,axiom,
! [L1: int,K: int] : bit0(K) != bit1(L1) ).
tff(fact_41_rel__simps_I44_J,axiom,
! [K3: int] :
( ( bit0(K3) = pls )
<=> ( K3 = pls ) ) ).
tff(fact_42_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_43_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_44_rel__simps_I34_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit1(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_45_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_46_rel__simps_I31_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit0(K3),bit0(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_47_real__divide__square__eq,axiom,
! [A1: real,R: real] : inverse_divide(real,times_times(real,R,A1),times_times(real,R,R)) = inverse_divide(real,A1,R) ).
tff(fact_48_real__sqrt__le__iff,axiom,
! [Yb: real,Xb: real] :
( ord_less_eq(real,sqrt(Xb),sqrt(Yb))
<=> ord_less_eq(real,Xb,Yb) ) ).
tff(fact_49_real__sqrt__eq__0__iff,axiom,
! [Xb: real] :
( ( sqrt(Xb) = zero_zero(real) )
<=> ( Xb = zero_zero(real) ) ) ).
tff(fact_50_real__sqrt__zero,axiom,
sqrt(zero_zero(real)) = zero_zero(real) ).
tff(fact_51_left__diff__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& ring(B2) )
=> ! [V: int,B: B2,A1: B2] : times_times(B2,minus_minus(B2,A1,B),number_number_of(B2,V)) = minus_minus(B2,times_times(B2,A1,number_number_of(B2,V)),times_times(B2,B,number_number_of(B2,V))) ) ).
tff(fact_52_right__diff__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& ring(B2) )
=> ! [C: B2,B: B2,V: int] : times_times(B2,number_number_of(B2,V),minus_minus(B2,B,C)) = minus_minus(B2,times_times(B2,number_number_of(B2,V),B),times_times(B2,number_number_of(B2,V),C)) ) ).
tff(fact_53_le__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Yb: int,Xb: int] :
( ord_less_eq(A,number_number_of(A,Xb),number_number_of(A,Yb))
<=> ord_less_eq(int,Xb,Yb) ) ) ).
tff(fact_54_rel__simps_I22_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit1(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_55_rel__simps_I32_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit0(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_56_rel__simps_I27_J,axiom,
! [K3: int] :
( ord_less_eq(int,bit0(K3),pls)
<=> ord_less_eq(int,K3,pls) ) ).
tff(fact_57_rel__simps_I21_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit0(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_58_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_59_less__eq__number__of__int__code,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,number_number_of(int,K3),number_number_of(int,L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_60_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D1: A,C1: A,B1: A,A2: A] :
( ( minus_minus(A,A2,B1) = minus_minus(A,C1,D1) )
=> ( ( A2 = B1 )
<=> ( C1 = D1 ) ) ) ) ).
tff(fact_61_number__of__diff,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,minus_minus(int,V,W)) = minus_minus(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_62_add__Pls__right,axiom,
! [K: int] : plus_plus(int,K,pls) = K ).
tff(fact_63_add__Pls,axiom,
! [K: int] : plus_plus(int,pls,K) = K ).
tff(fact_64_Bit0__def,axiom,
! [K: int] : bit0(K) = plus_plus(int,K,K) ).
tff(fact_65_less__eq__int__code_I16_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit1(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_66_less__eq__int__code_I13_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit0(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_67_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_68_less__eq__int__code_I14_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_69_right__minus__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A2: A] :
( ( minus_minus(A,A2,B1) = zero_zero(A) )
<=> ( A2 = B1 ) ) ) ).
tff(fact_70_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A2: A] :
( ( A2 = B1 )
<=> ( minus_minus(A,A2,B1) = zero_zero(A) ) ) ) ).
tff(fact_71_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A1: A] : minus_minus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_72_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D1: A,C1: A,B1: A,A2: A] :
( ( minus_minus(A,A2,B1) = minus_minus(A,C1,D1) )
=> ( ord_less_eq(A,A2,B1)
<=> ord_less_eq(A,C1,D1) ) ) ) ).
tff(fact_73_add__diff__add,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [D: A,B: A,C: A,A1: A] : minus_minus(A,plus_plus(A,A1,C),plus_plus(A,B,D)) = plus_plus(A,minus_minus(A,A1,B),minus_minus(A,C,D)) ) ).
tff(fact_74_add__diff__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B: A,A1: A] : minus_minus(A,plus_plus(A,A1,B),B) = A1 ) ).
tff(fact_75_diff__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B: A,A1: A] : plus_plus(A,minus_minus(A,A1,B),B) = A1 ) ).
tff(fact_76_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xb: A] :
( ( zero_zero(A) = Xb )
<=> ( Xb = zero_zero(A) ) ) ) ).
tff(fact_77_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C: A,B: A,A1: A] : times_times(A,times_times(A,A1,B),C) = times_times(A,A1,times_times(A,B,C)) ) ).
tff(fact_78_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B: A] :
( ( plus_plus(A,B,A1) = plus_plus(A,C,A1) )
=> ( B = C ) ) ) ).
tff(fact_79_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_80_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_81_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,A1,plus_plus(A,B,C)) ) ).
tff(fact_82_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B1: A,A2: A] :
( ord_less_eq(A,A2,B1)
<=> ord_less_eq(A,minus_minus(A,A2,B1),zero_zero(A)) ) ) ).
tff(fact_83_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [Xb: A,W1: int] :
( ( number_number_of(A,W1) = Xb )
<=> ( Xb = number_number_of(A,W1) ) ) ) ).
tff(fact_84_mult__diff__mult,axiom,
! [A: $tType] :
( ring(A)
=> ! [B: A,A1: A,Y: A,X: A] : minus_minus(A,times_times(A,X,Y),times_times(A,A1,B)) = plus_plus(A,times_times(A,X,minus_minus(A,Y,B)),times_times(A,minus_minus(A,X,A1),B)) ) ).
tff(fact_85_add__number__of__diff1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [C: A,W: int,V: int] : plus_plus(A,number_number_of(A,V),minus_minus(A,number_number_of(A,W),C)) = minus_minus(A,number_number_of(A,plus_plus(int,V,W)),C) ) ).
tff(fact_86_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_87_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_88_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_89_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_90_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C: A] :
( ord_less_eq(A,plus_plus(A,C,A1),plus_plus(A,C,B))
=> ord_less_eq(A,A1,B) ) ) ).
tff(fact_91_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C: A,A1: A] :
( ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B,C))
=> ord_less_eq(A,A1,B) ) ) ).
tff(fact_92_add__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,C,D)
=> ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B,D)) ) ) ) ).
tff(fact_93_add__left__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ord_less_eq(A,plus_plus(A,C,A1),plus_plus(A,C,B)) ) ) ).
tff(fact_94_add__right__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B,C)) ) ) ).
tff(fact_95_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,times_times(int,V,W)) = times_times(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_96_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,plus_plus(int,V,W)) = plus_plus(A,number_number_of(A,V),number_number_of(A,W)) ) ).
%----Arities (54)
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(int) ).
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
tff(arity_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(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___Rings_Osemiring,axiom,
semiring(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(real) ).
tff(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(real) ).
tff(arity_RealDef_Oreal___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(real) ).
tff(arity_RealDef_Oreal___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__group__add,axiom,
ab_group_add(real) ).
tff(arity_RealDef_Oreal___Groups_Omonoid__add,axiom,
monoid_add(real) ).
tff(arity_RealDef_Oreal___Groups_Ogroup__add,axiom,
group_add(real) ).
tff(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring,axiom,
semiring(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Rings_Oring,axiom,
ring(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
tff(arity_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__group__add,axiom,
ab_group_add(complex) ).
tff(arity_Complex_Ocomplex___Groups_Omonoid__add,axiom,
monoid_add(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ogroup__add,axiom,
group_add(complex) ).
tff(arity_Complex_Ocomplex___Int_Oring__char__0,axiom,
ring_char_0(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__ring,axiom,
number_ring(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring,axiom,
semiring(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oring,axiom,
ring(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber,axiom,
number(complex) ).
%----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(real,zero_zero(real),inverse_divide(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,xa,xa),times_times(real,ya,ya))),xa),number_number_of(real,bit0(bit1(pls))))) ).
%------------------------------------------------------------------------------