TPTP Problem File: SWW483_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW483_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 50
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_50 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 188 ( 85 unt; 42 typ; 0 def)
% Number of atoms : 253 ( 120 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 121 ( 14 ~; 0 |; 22 &)
% ( 28 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 3 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 24 ( 15 >; 9 *; 0 +; 0 <<)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-3 aty)
% Number of functors : 20 ( 20 usr; 6 con; 0-4 aty)
% Number of variables : 227 ( 201 !; 0 ?; 227 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:22
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
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_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (36)
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_Fields_Ofield,type,
field:
!>[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_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__semiring,type,
number_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__field,type,
real_field:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[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_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[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_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( 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_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > A ) ).
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_x____,type,
x: real ).
tff(sy_v_y____,type,
y: real ).
tff(sy_v_z,type,
z: complex ).
%----Relevant facts (97)
tff(fact_0_y0,axiom,
y != zero_zero(real) ).
tff(fact_1_xy,axiom,
z = complex1(x,y) ).
tff(fact_2_sqrt4,axiom,
sqrt(number_number_of(real,bit0(bit0(bit1(pls))))) = number_number_of(real,bit0(bit1(pls))) ).
tff(fact_3__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Az_A_061_AComplex_Ax_Ay_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [X1: real,Y1: real] : z != complex1(X1,Y1) ).
tff(fact_4_th1,axiom,
minus_minus(real,sqrt(inverse_divide(real,times_times(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x)),number_number_of(real,bit0(bit0(bit1(pls)))))),sqrt(inverse_divide(real,times_times(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x)),number_number_of(real,bit0(bit0(bit1(pls))))))) = x ).
tff(fact_5_th2,axiom,
inverse_divide(real,times_times(real,number_number_of(real,bit0(bit1(pls))),times_times(real,y,sqrt(inverse_divide(real,times_times(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x)),number_number_of(real,bit0(bit0(bit1(pls)))))))),abs_abs(real,y)) = y ).
tff(fact_6_real__average__minus__first,axiom,
! [B1: real,A2: real] : minus_minus(real,inverse_divide(real,plus_plus(real,A2,B1),number_number_of(real,bit0(bit1(pls)))),A2) = inverse_divide(real,minus_minus(real,B1,A2),number_number_of(real,bit0(bit1(pls)))) ).
tff(fact_7_real__average__minus__second,axiom,
! [A2: real,B1: real] : minus_minus(real,inverse_divide(real,plus_plus(real,B1,A2),number_number_of(real,bit0(bit1(pls)))),A2) = inverse_divide(real,minus_minus(real,B1,A2),number_number_of(real,bit0(bit1(pls)))) ).
tff(fact_8__0960_A_060_061_A_Isqrt_A_Ix_A_K_Ax_A_L_Ay_A_K_Ay_J_A_N_Ax_J_A_P_A2_096,axiom,
ord_less_eq(real,zero_zero(real),inverse_divide(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls))))) ).
tff(fact_9_eq__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [W: int,B: A,A1: A] :
( ( A1 = inverse_divide(A,B,number_number_of(A,W)) )
<=> ( ( ( number_number_of(A,W) != zero_zero(A) )
=> ( times_times(A,A1,number_number_of(A,W)) = B ) )
& ( ( number_number_of(A,W) = zero_zero(A) )
=> ( A1 = zero_zero(A) ) ) ) ) ) ).
tff(fact_10_divide__eq__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [A1: A,W: int,B: A] :
( ( inverse_divide(A,B,number_number_of(A,W)) = A1 )
<=> ( ( ( number_number_of(A,W) != zero_zero(A) )
=> ( B = times_times(A,A1,number_number_of(A,W)) ) )
& ( ( number_number_of(A,W) = zero_zero(A) )
=> ( A1 = zero_zero(A) ) ) ) ) ) ).
tff(fact_11__0960_A_060_061_A_Isqrt_A_Ix_A_K_Ax_A_L_Ay_A_K_Ay_J_A_L_Ax_J_A_P_A2_096,axiom,
ord_less_eq(real,zero_zero(real),inverse_divide(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls))))) ).
tff(fact_12_real__sqrt__abs2,axiom,
! [X: real] : sqrt(times_times(real,X,X)) = abs_abs(real,X) ).
tff(fact_13_th_I2_J,axiom,
! [Ya: real,Xa: real] : ord_less_eq(real,zero_zero(real),inverse_divide(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,Xa,Xa),times_times(real,Ya,Ya))),Xa),number_number_of(real,bit0(bit1(pls))))) ).
tff(fact_14_real__two__squares__add__zero__iff,axiom,
! [Ya: real,Xa: real] :
( ( plus_plus(real,times_times(real,Xa,Xa),times_times(real,Ya,Ya)) = zero_zero(real) )
<=> ( ( Xa = zero_zero(real) )
& ( Ya = zero_zero(real) ) ) ) ).
tff(fact_15_right__diff__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& ring(B2) )
=> ! [C1: B2,B1: B2,V: int] : times_times(B2,number_number_of(B2,V),minus_minus(B2,B1,C1)) = minus_minus(B2,times_times(B2,number_number_of(B2,V),B1),times_times(B2,number_number_of(B2,V),C1)) ) ).
tff(fact_16_left__diff__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& ring(B2) )
=> ! [V: int,B1: B2,A2: B2] : times_times(B2,minus_minus(B2,A2,B1),number_number_of(B2,V)) = minus_minus(B2,times_times(B2,A2,number_number_of(B2,V)),times_times(B2,B1,number_number_of(B2,V))) ) ).
tff(fact_17_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_18_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Ya: int,Xa: int] :
( ( number_number_of(A,Xa) = number_number_of(A,Ya) )
<=> ( Xa = Ya ) ) ) ).
tff(fact_19_rel__simps_I51_J,axiom,
! [L: int,K1: int] :
( ( bit1(K1) = bit1(L) )
<=> ( K1 = L ) ) ).
tff(fact_20_mult__Pls,axiom,
! [W1: int] : times_times(int,pls,W1) = pls ).
tff(fact_21_diff__bin__simps_I7_J,axiom,
! [L1: int,K: int] : minus_minus(int,bit0(K),bit0(L1)) = bit0(minus_minus(int,K,L1)) ).
tff(fact_22_mult__Bit0,axiom,
! [L1: int,K: int] : times_times(int,bit0(K),L1) = bit0(times_times(int,K,L1)) ).
tff(fact_23_rel__simps_I48_J,axiom,
! [L: int,K1: int] :
( ( bit0(K1) = bit0(L) )
<=> ( K1 = L ) ) ).
tff(fact_24_real__sqrt__eq__iff,axiom,
! [Ya: real,Xa: real] :
( ( sqrt(Xa) = sqrt(Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_25_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( plus_plus(A,A1,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_26_le__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int,Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),number_number_of(A,Ya))
<=> ord_less_eq(int,Xa,Ya) ) ) ).
tff(fact_27_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W1: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W1),Z)) = times_times(A,number_number_of(A,times_times(int,V,W1)),Z) ) ).
tff(fact_28_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : times_times(A,number_number_of(A,V),number_number_of(A,W1)) = number_number_of(A,times_times(int,V,W1)) ) ).
tff(fact_29_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W1: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W1),Z)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W1)),Z) ) ).
tff(fact_30_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : plus_plus(A,number_number_of(A,V),number_number_of(A,W1)) = number_number_of(A,plus_plus(int,V,W1)) ) ).
tff(fact_31_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_32_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_33_diff__bin__simps_I10_J,axiom,
! [L1: int,K: int] : minus_minus(int,bit1(K),bit1(L1)) = bit0(minus_minus(int,K,L1)) ).
tff(fact_34_diff__bin__simps_I9_J,axiom,
! [L1: int,K: int] : minus_minus(int,bit1(K),bit0(L1)) = bit1(minus_minus(int,K,L1)) ).
tff(fact_35_rel__simps_I50_J,axiom,
! [L1: int,K: int] : bit1(K) != bit0(L1) ).
tff(fact_36_rel__simps_I49_J,axiom,
! [L1: int,K: int] : bit0(K) != bit1(L1) ).
tff(fact_37_diff__bin__simps_I3_J,axiom,
! [L1: int] : minus_minus(int,pls,bit0(L1)) = bit0(minus_minus(int,pls,L1)) ).
tff(fact_38_rel__simps_I44_J,axiom,
! [K1: int] :
( ( bit0(K1) = pls )
<=> ( K1 = pls ) ) ).
tff(fact_39_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_40_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_41_real__divide__square__eq,axiom,
! [A2: real,R: real] : inverse_divide(real,times_times(real,R,A2),times_times(real,R,R)) = inverse_divide(real,A2,R) ).
tff(fact_42_real__sqrt__le__iff,axiom,
! [Ya: real,Xa: real] :
( ord_less_eq(real,sqrt(Xa),sqrt(Ya))
<=> ord_less_eq(real,Xa,Ya) ) ).
tff(fact_43_real__sqrt__eq__0__iff,axiom,
! [Xa: real] :
( ( sqrt(Xa) = zero_zero(real) )
<=> ( Xa = zero_zero(real) ) ) ).
tff(fact_44_real__sqrt__zero,axiom,
sqrt(zero_zero(real)) = zero_zero(real) ).
tff(fact_45_th_I1_J,axiom,
! [Ya: real,Xa: real] : 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))))) ).
tff(fact_46_left__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [V: int,B1: B2,A2: B2] : times_times(B2,plus_plus(B2,A2,B1),number_number_of(B2,V)) = plus_plus(B2,times_times(B2,A2,number_number_of(B2,V)),times_times(B2,B1,number_number_of(B2,V))) ) ).
tff(fact_47_right__distrib__number__of,axiom,
! [B2: $tType] :
( ( number(B2)
& semiring(B2) )
=> ! [C1: B2,B1: B2,V: int] : times_times(B2,number_number_of(B2,V),plus_plus(B2,B1,C1)) = plus_plus(B2,times_times(B2,number_number_of(B2,V),B1),times_times(B2,number_number_of(B2,V),C1)) ) ).
tff(fact_48_real__sqrt__le__0__iff,axiom,
! [Xa: real] :
( ord_less_eq(real,sqrt(Xa),zero_zero(real))
<=> ord_less_eq(real,Xa,zero_zero(real)) ) ).
tff(fact_49_real__sqrt__ge__0__iff,axiom,
! [Ya: real] :
( ord_less_eq(real,zero_zero(real),sqrt(Ya))
<=> ord_less_eq(real,zero_zero(real),Ya) ) ).
tff(fact_50_le__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),zero_zero(A))
<=> ord_less_eq(int,Xa,pls) ) ) ).
tff(fact_51_le__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int] :
( ord_less_eq(A,zero_zero(A),number_number_of(A,Ya))
<=> ord_less_eq(int,pls,Ya) ) ) ).
tff(fact_52_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_53_diff__bin__simps_I1_J,axiom,
! [K: int] : minus_minus(int,K,pls) = K ).
tff(fact_54_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_55_real__sqrt__le__mono,axiom,
! [Y: real,X: real] :
( ord_less_eq(real,X,Y)
=> ord_less_eq(real,sqrt(X),sqrt(Y)) ) ).
tff(fact_56_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ord_less_eq(real,zero_zero(real),X)
=> ( ( sqrt(X) = zero_zero(real) )
=> ( X = zero_zero(real) ) ) ) ).
tff(fact_57_real__sqrt__ge__zero,axiom,
! [X: real] :
( ord_less_eq(real,zero_zero(real),X)
=> ord_less_eq(real,zero_zero(real),sqrt(X)) ) ).
tff(fact_58_le__real__sqrt__sumsq,axiom,
! [Y: real,X: real] : ord_less_eq(real,X,sqrt(plus_plus(real,times_times(real,X,X),times_times(real,Y,Y)))) ).
tff(fact_59_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [Xa: A,W: int] :
( ( number_number_of(A,W) = Xa )
<=> ( Xa = number_number_of(A,W) ) ) ) ).
tff(fact_60_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : number_number_of(A,times_times(int,V,W1)) = times_times(A,number_number_of(A,V),number_number_of(A,W1)) ) ).
tff(fact_61_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : number_number_of(A,plus_plus(int,V,W1)) = plus_plus(A,number_number_of(A,V),number_number_of(A,W1)) ) ).
tff(fact_62_add__diff__add,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [D: A,B1: A,C1: A,A2: A] : minus_minus(A,plus_plus(A,A2,C1),plus_plus(A,B1,D)) = plus_plus(A,minus_minus(A,A2,B1),minus_minus(A,C1,D)) ) ).
tff(fact_63_number__of__diff,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : number_number_of(A,minus_minus(int,V,W1)) = minus_minus(A,number_number_of(A,V),number_number_of(A,W1)) ) ).
tff(fact_64_real__mult__left__cancel,axiom,
! [B: real,A1: real,C: real] :
( ( C != zero_zero(real) )
=> ( ( times_times(real,C,A1) = times_times(real,C,B) )
<=> ( A1 = B ) ) ) ).
tff(fact_65_real__mult__right__cancel,axiom,
! [B: real,A1: real,C: real] :
( ( C != zero_zero(real) )
=> ( ( times_times(real,A1,C) = times_times(real,B,C) )
<=> ( A1 = B ) ) ) ).
tff(fact_66_real__sqrt__mult,axiom,
! [Y: real,X: real] : sqrt(times_times(real,X,Y)) = times_times(real,sqrt(X),sqrt(Y)) ).
tff(fact_67_real__sqrt__divide,axiom,
! [Y: real,X: real] : sqrt(inverse_divide(real,X,Y)) = inverse_divide(real,sqrt(X),sqrt(Y)) ).
tff(fact_68_mult__diff__mult,axiom,
! [A: $tType] :
( ring(A)
=> ! [B1: A,A2: A,Y: A,X: A] : minus_minus(A,times_times(A,X,Y),times_times(A,A2,B1)) = plus_plus(A,times_times(A,X,minus_minus(A,Y,B1)),times_times(A,minus_minus(A,X,A2),B1)) ) ).
tff(fact_69_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_70_add__number__of__diff1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [C1: A,W1: int,V: int] : plus_plus(A,number_number_of(A,V),minus_minus(A,number_number_of(A,W1),C1)) = minus_minus(A,number_number_of(A,plus_plus(int,V,W1)),C1) ) ).
tff(fact_71_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A2: A] : plus_plus(A,number_number_of(A,pls),A2) = A2 ) ).
tff(fact_72_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A2: A] : plus_plus(A,A2,number_number_of(A,pls)) = A2 ) ).
tff(fact_73_lemma__MVT,axiom,
! [B: real,A1: real,F: fun(real,real)] : minus_minus(real,aa(real,real,F,A1),times_times(real,inverse_divide(real,minus_minus(real,aa(real,real,F,B),aa(real,real,F,A1)),minus_minus(real,B,A1)),A1)) = minus_minus(real,aa(real,real,F,B),times_times(real,inverse_divide(real,minus_minus(real,aa(real,real,F,B),aa(real,real,F,A1)),minus_minus(real,B,A1)),B)) ).
tff(fact_74_number__of__Bit0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : number_number_of(A,bit0(W1)) = plus_plus(A,plus_plus(A,zero_zero(A),number_number_of(A,W1)),number_number_of(A,W1)) ) ).
tff(fact_75_divide__eq__eq__number__of,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [W: int,C: A,B: A] :
( ( inverse_divide(A,B,C) = number_number_of(A,W) )
<=> ( ( ( C != zero_zero(A) )
=> ( B = times_times(A,number_number_of(A,W),C) ) )
& ( ( C = zero_zero(A) )
=> ( number_number_of(A,W) = zero_zero(A) ) ) ) ) ) ).
tff(fact_76_eq__divide__eq__number__of,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [C: A,B: A,W: int] :
( ( number_number_of(A,W) = inverse_divide(A,B,C) )
<=> ( ( ( C != zero_zero(A) )
=> ( times_times(A,number_number_of(A,W),C) = B ) )
& ( ( C = zero_zero(A) )
=> ( number_number_of(A,W) = zero_zero(A) ) ) ) ) ) ).
tff(fact_77_DERIV__mult__lemma,axiom,
! [A: $tType] :
( real_field(A)
=> ! [H: A,D: A,C1: A,B1: A,A2: A] : inverse_divide(A,minus_minus(A,times_times(A,A2,B1),times_times(A,C1,D)),H) = plus_plus(A,times_times(A,A2,inverse_divide(A,minus_minus(A,B1,D),H)),times_times(A,inverse_divide(A,minus_minus(A,A2,C1),H),D)) ) ).
tff(fact_78_divide__Numeral0,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number_ring(A) )
=> ! [X: A] : inverse_divide(A,X,number_number_of(A,pls)) = zero_zero(A) ) ).
tff(fact_79_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A2: A] : times_times(A,number_number_of(A,bit1(pls)),A2) = A2 ) ).
tff(fact_80_mult__numeral__1__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A2: A] : times_times(A,A2,number_number_of(A,bit1(pls))) = A2 ) ).
tff(fact_81_divide__Numeral1,axiom,
! [A: $tType] :
( ( field(A)
& number_ring(A) )
=> ! [X: A] : inverse_divide(A,X,number_number_of(A,bit1(pls))) = X ) ).
tff(fact_82_divide__numeral__1,axiom,
! [A: $tType] :
( ( field(A)
& number_ring(A) )
=> ! [A2: A] : inverse_divide(A,A2,number_number_of(A,bit1(pls))) = A2 ) ).
tff(fact_83_semiring__mult__2,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [Z: A] : times_times(A,number_number_of(A,bit0(bit1(pls))),Z) = plus_plus(A,Z,Z) ) ).
tff(fact_84_mult__2,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A] : times_times(A,number_number_of(A,bit0(bit1(pls))),Z) = plus_plus(A,Z,Z) ) ).
tff(fact_85_semiring__mult__2__right,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [Z: A] : times_times(A,Z,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,Z,Z) ) ).
tff(fact_86_mult__2__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A] : times_times(A,Z,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,Z,Z) ) ).
tff(fact_87_eq__divide__2__times__iff,axiom,
! [Za: real,Ya: real,Xa: real] :
( ( Xa = inverse_divide(real,Ya,times_times(real,number_number_of(real,bit0(bit1(pls))),Za)) )
<=> ( times_times(real,number_number_of(real,bit0(bit1(pls))),Xa) = inverse_divide(real,Ya,Za) ) ) ).
tff(fact_88_complex__mult,axiom,
! [D: real,C1: real,B1: real,A2: real] : times_times(complex,complex1(A2,B1),complex1(C1,D)) = complex1(minus_minus(real,times_times(real,A2,C1),times_times(real,B1,D)),plus_plus(real,times_times(real,A2,D),times_times(real,B1,C1))) ).
tff(fact_89_sq4_H_I2_J,axiom,
sqrt(inverse_divide(real,power_power(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(nat,bit0(bit1(pls)))),number_number_of(real,bit0(bit0(bit1(pls)))))) = inverse_divide(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls)))) ).
tff(fact_90_sq4_H_I1_J,axiom,
sqrt(inverse_divide(real,power_power(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(nat,bit0(bit1(pls)))),number_number_of(real,bit0(bit0(bit1(pls)))))) = inverse_divide(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls)))) ).
tff(fact_91_Complex__eq__number__of,axiom,
! [W: int,B: real,A1: real] :
( ( complex1(A1,B) = number_number_of(complex,W) )
<=> ( ( A1 = number_number_of(real,W) )
& ( B = zero_zero(real) ) ) ) ).
tff(fact_92_abs__le__zero__iff,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] :
( ord_less_eq(A,abs_abs(A,A1),zero_zero(A))
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_93_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ord_less_eq(A,plus_plus(A,A1,A1),zero_zero(A))
<=> ord_less_eq(A,A1,zero_zero(A)) ) ) ).
tff(fact_94_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ord_less_eq(A,zero_zero(A),plus_plus(A,A1,A1))
<=> ord_less_eq(A,zero_zero(A),A1) ) ) ).
tff(fact_95_real__sum__of__halves,axiom,
! [X: real] : plus_plus(real,inverse_divide(real,X,number_number_of(real,bit0(bit1(pls)))),inverse_divide(real,X,number_number_of(real,bit0(bit1(pls))))) = X ).
tff(fact_96_add__left__cancel,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 ) ) ) ).
%----Arities (44)
tff(arity_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(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__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
tff(arity_Int_Oint___Groups_Oab__group__add,axiom,
ab_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_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(real) ).
tff(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_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___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__field,axiom,
real_field(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__group__add,axiom,
ab_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___Fields_Ofield,axiom,
field(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__semigroup__add,axiom,
cancel_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__field,axiom,
real_field(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__semiring,axiom,
number_semiring(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__group__add,axiom,
ab_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___Fields_Ofield,axiom,
field(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 (3)
tff(conj_0,hypothesis,
$true ).
tff(conj_1,hypothesis,
$true ).
tff(conj_2,conjecture,
( ( minus_minus(real,times_times(real,sqrt(inverse_divide(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls))))),sqrt(inverse_divide(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls)))))),times_times(real,sqrt(inverse_divide(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls))))),sqrt(inverse_divide(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls))))))) = x )
& ( inverse_divide(real,times_times(real,number_number_of(real,bit0(bit1(pls))),times_times(real,y,times_times(real,sqrt(inverse_divide(real,minus_minus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls))))),sqrt(inverse_divide(real,plus_plus(real,sqrt(plus_plus(real,times_times(real,x,x),times_times(real,y,y))),x),number_number_of(real,bit0(bit1(pls)))))))),abs_abs(real,y)) = y ) ) ).
%------------------------------------------------------------------------------