SET007 Axioms: SET007+194.ax
%------------------------------------------------------------------------------
% File : SET007+194 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of the Trigonometric Function
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : sin_cos2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 68 ( 25 unt; 0 def)
% Number of atoms : 170 ( 61 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 106 ( 4 ~; 0 |; 49 &)
% ( 3 <=>; 50 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 36 ( 36 usr; 13 con; 0-4 aty)
% Number of variables : 61 ( 61 !; 0 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_sin_cos2,axiom,
! [A] :
( v1_xcmplx_0(k2_sin_cos2(A))
& v1_xreal_0(k2_sin_cos2(A)) ) ).
fof(fc2_sin_cos2,axiom,
! [A] :
( v1_xcmplx_0(k5_sin_cos2(A))
& v1_xreal_0(k5_sin_cos2(A)) ) ).
fof(fc3_sin_cos2,axiom,
! [A] :
( v1_xcmplx_0(k8_sin_cos2(A))
& v1_xreal_0(k8_sin_cos2(A)) ) ).
fof(fc4_sin_cos2,axiom,
( v1_relat_1(k1_sin_cos2)
& v1_funct_1(k1_sin_cos2)
& v1_seq_1(k1_sin_cos2)
& v1_partfun1(k1_sin_cos2,k1_numbers,k1_numbers) ) ).
fof(fc5_sin_cos2,axiom,
( v1_relat_1(k4_sin_cos2)
& v1_funct_1(k4_sin_cos2)
& v1_seq_1(k4_sin_cos2)
& v1_partfun1(k4_sin_cos2,k1_numbers,k1_numbers) ) ).
fof(fc6_sin_cos2,axiom,
( v1_relat_1(k7_sin_cos2)
& v1_funct_1(k7_sin_cos2)
& v1_seq_1(k7_sin_cos2)
& v1_partfun1(k7_sin_cos2,k1_numbers,k1_numbers) ) ).
fof(t1_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> r1_xreal_0(k3_xcmplx_0(np__2,k8_square_1(k3_xcmplx_0(A,B))),k2_xcmplx_0(A,B)) ) ) ) ).
fof(t2_sin_cos2,axiom,
r1_rfunct_2(k18_sin_cos,k2_rcomp_1(np__0,k12_binop_2(k32_sin_cos,np__2))) ).
fof(t3_sin_cos2,axiom,
r2_rfunct_2(k18_sin_cos,k2_rcomp_1(k12_binop_2(k32_sin_cos,np__2),k32_sin_cos)) ).
fof(t4_sin_cos2,axiom,
r2_rfunct_2(k21_sin_cos,k2_rcomp_1(np__0,k12_binop_2(k32_sin_cos,np__2))) ).
fof(t5_sin_cos2,axiom,
r2_rfunct_2(k21_sin_cos,k2_rcomp_1(k12_binop_2(k32_sin_cos,np__2),k32_sin_cos)) ).
fof(t6_sin_cos2,axiom,
r2_rfunct_2(k18_sin_cos,k2_rcomp_1(k32_sin_cos,k11_binop_2(k12_binop_2(np__3,np__2),k32_sin_cos))) ).
fof(t7_sin_cos2,axiom,
r1_rfunct_2(k18_sin_cos,k2_rcomp_1(k11_binop_2(k12_binop_2(np__3,np__2),k32_sin_cos),k11_binop_2(np__2,k32_sin_cos))) ).
fof(t8_sin_cos2,axiom,
r1_rfunct_2(k21_sin_cos,k2_rcomp_1(k32_sin_cos,k11_binop_2(k12_binop_2(np__3,np__2),k32_sin_cos))) ).
fof(t9_sin_cos2,axiom,
r1_rfunct_2(k21_sin_cos,k2_rcomp_1(k11_binop_2(k12_binop_2(np__3,np__2),k32_sin_cos),k11_binop_2(np__2,k32_sin_cos))) ).
fof(t10_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A) = k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,k2_xcmplx_0(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)) ) ) ).
fof(t11_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A) = k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,k2_xcmplx_0(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)) ) ) ).
fof(d1_sin_cos2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k1_sin_cos2
<=> ( k1_relat_1(A) = k1_numbers
& ! [B] :
( v1_xreal_0(B)
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k12_binop_2(k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,B),k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k4_xcmplx_0(B))),np__2) ) ) ) ) ).
fof(d2_sin_cos2,axiom,
! [A] : k2_sin_cos2(A) = k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A) ).
fof(d3_sin_cos2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k4_sin_cos2
<=> ( k1_relat_1(A) = k1_numbers
& ! [B] :
( v1_xreal_0(B)
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k12_binop_2(k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,B),k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k4_xcmplx_0(B))),np__2) ) ) ) ) ).
fof(d4_sin_cos2,axiom,
! [A] : k5_sin_cos2(A) = k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A) ).
fof(d5_sin_cos2,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ( A = k7_sin_cos2
<=> ( k1_relat_1(A) = k1_numbers
& ! [B] :
( v1_xreal_0(B)
=> k2_seq_1(k1_numbers,k1_numbers,A,B) = k12_binop_2(k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,B),k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k4_xcmplx_0(B))),k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,B),k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k4_xcmplx_0(B)))) ) ) ) ) ).
fof(d6_sin_cos2,axiom,
! [A] : k8_sin_cos2(A) = k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A) ).
fof(t12_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,k2_xcmplx_0(A,B)) = k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,A),k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,B)) ) ) ).
fof(t13_sin_cos2,axiom,
k2_seq_1(k1_numbers,k1_numbers,k26_sin_cos,np__0) = np__1 ).
fof(t14_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k10_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A))) = np__1
& k10_binop_2(k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A))) = np__1 ) ) ).
fof(t15_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A) != np__0
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),np__0)
& k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,np__0) = np__1 ) ) ).
fof(t16_sin_cos2,axiom,
k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,np__0) = np__0 ).
fof(t17_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A) = k12_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)) ) ).
fof(t18_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A)) = k11_binop_2(k12_binop_2(np__1,np__2),k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k3_xcmplx_0(np__2,A)),np__1))
& k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)) = k11_binop_2(k12_binop_2(np__1,np__2),k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k3_xcmplx_0(np__2,A)),np__1)) ) ) ).
fof(t19_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k4_xcmplx_0(A)) = k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)
& k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k4_xcmplx_0(A)) = k7_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A))
& k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,k4_xcmplx_0(A)) = k7_binop_2(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A)) ) ) ).
fof(t20_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k2_xcmplx_0(A,B)) = k9_binop_2(k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B)))
& k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k6_xcmplx_0(A,B)) = k10_binop_2(k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B))) ) ) ) ).
fof(t21_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k2_xcmplx_0(A,B)) = k9_binop_2(k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B)))
& k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k6_xcmplx_0(A,B)) = k10_binop_2(k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B))) ) ) ) ).
fof(t22_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,k2_xcmplx_0(A,B)) = k12_binop_2(k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,B)),k9_binop_2(np__1,k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,B))))
& k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,k6_xcmplx_0(A,B)) = k12_binop_2(k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,B)),k10_binop_2(np__1,k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,B)))) ) ) ) ).
fof(t23_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k3_xcmplx_0(np__2,A)) = k11_binop_2(k11_binop_2(np__2,k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A)),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A))
& k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k3_xcmplx_0(np__2,A)) = k10_binop_2(k11_binop_2(np__2,k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A))),np__1)
& k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,k3_xcmplx_0(np__2,A)) = k12_binop_2(k11_binop_2(np__2,k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A)),k9_binop_2(np__1,k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A)))) ) ) ).
fof(t24_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k10_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B))) = k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k2_xcmplx_0(A,B)),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k6_xcmplx_0(A,B)))
& k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k2_xcmplx_0(A,B)),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k6_xcmplx_0(A,B))) = k10_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)))
& k10_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B))) = k10_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B))) ) ) ) ).
fof(t25_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k9_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B))) = k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k2_xcmplx_0(A,B)),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k6_xcmplx_0(A,B)))
& k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k2_xcmplx_0(A,B)),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k6_xcmplx_0(A,B))) = k9_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B)))
& k9_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B))) = k9_binop_2(k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)),k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B))) ) ) ) ).
fof(t26_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B)) = k11_binop_2(k11_binop_2(np__2,k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k2_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2)))),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k6_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2))))
& k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,B)) = k11_binop_2(k11_binop_2(np__2,k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k6_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2)))),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k2_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2)))) ) ) ) ).
fof(t27_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)) = k11_binop_2(k11_binop_2(np__2,k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k2_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2)))),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k6_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2))))
& k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)) = k11_binop_2(k11_binop_2(np__2,k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k6_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2)))),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k2_xcmplx_0(k7_xcmplx_0(A,np__2),k7_xcmplx_0(B,np__2)))) ) ) ) ).
fof(t28_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,B)) = k12_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k2_xcmplx_0(A,B)),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B)))
& k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,B)) = k12_binop_2(k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k6_xcmplx_0(A,B)),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,B))) ) ) ) ).
fof(t29_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k3_prepower(k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A)),B) = k9_binop_2(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,k3_xcmplx_0(B,A)),k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,k3_xcmplx_0(B,A))) ) ) ).
fof(t30_sin_cos2,axiom,
( k1_relat_1(k1_sin_cos2) = k1_numbers
& k1_relat_1(k4_sin_cos2) = k1_numbers
& k1_relat_1(k7_sin_cos2) = k1_numbers ) ).
fof(t31_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_fdiff_1(k1_sin_cos2,A)
& k1_fdiff_1(k1_sin_cos2,A) = k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A) ) ) ).
fof(t32_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_fdiff_1(k4_sin_cos2,A)
& k1_fdiff_1(k4_sin_cos2,A) = k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A) ) ) ).
fof(t33_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_fdiff_1(k7_sin_cos2,A)
& k1_fdiff_1(k7_sin_cos2,A) = k12_binop_2(np__1,k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A))) ) ) ).
fof(t34_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_fdiff_1(k1_sin_cos2,k1_numbers)
& k1_fdiff_1(k1_sin_cos2,A) = k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A) ) ) ).
fof(t35_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_fdiff_1(k4_sin_cos2,k1_numbers)
& k1_fdiff_1(k4_sin_cos2,A) = k2_seq_1(k1_numbers,k1_numbers,k1_sin_cos2,A) ) ) ).
fof(t36_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_fdiff_1(k7_sin_cos2,k1_numbers)
& k1_fdiff_1(k7_sin_cos2,A) = k12_binop_2(np__1,k7_square_1(k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A))) ) ) ).
fof(t37_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> r1_xreal_0(np__1,k2_seq_1(k1_numbers,k1_numbers,k4_sin_cos2,A)) ) ).
fof(t38_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> r1_fcont_1(k1_sin_cos2,A) ) ).
fof(t39_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> r1_fcont_1(k4_sin_cos2,A) ) ).
fof(t40_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> r1_fcont_1(k7_sin_cos2,A) ) ).
fof(t41_sin_cos2,axiom,
r2_fcont_1(k1_sin_cos2,k1_numbers) ).
fof(t42_sin_cos2,axiom,
r2_fcont_1(k4_sin_cos2,k1_numbers) ).
fof(t43_sin_cos2,axiom,
r2_fcont_1(k7_sin_cos2,k1_numbers) ).
fof(t44_sin_cos2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(np__1,k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A))
& ~ r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k7_sin_cos2,A),k7_binop_2(np__1)) ) ) ).
fof(dt_k1_sin_cos2,axiom,
( v1_funct_1(k1_sin_cos2)
& m2_relset_1(k1_sin_cos2,k1_numbers,k1_numbers) ) ).
fof(dt_k2_sin_cos2,axiom,
$true ).
fof(dt_k3_sin_cos2,axiom,
! [A] : m1_subset_1(k3_sin_cos2(A),k1_numbers) ).
fof(redefinition_k3_sin_cos2,axiom,
! [A] : k3_sin_cos2(A) = k2_sin_cos2(A) ).
fof(dt_k4_sin_cos2,axiom,
( v1_funct_1(k4_sin_cos2)
& m2_relset_1(k4_sin_cos2,k1_numbers,k1_numbers) ) ).
fof(dt_k5_sin_cos2,axiom,
$true ).
fof(dt_k6_sin_cos2,axiom,
! [A] : m1_subset_1(k6_sin_cos2(A),k1_numbers) ).
fof(redefinition_k6_sin_cos2,axiom,
! [A] : k6_sin_cos2(A) = k5_sin_cos2(A) ).
fof(dt_k7_sin_cos2,axiom,
( v1_funct_1(k7_sin_cos2)
& m2_relset_1(k7_sin_cos2,k1_numbers,k1_numbers) ) ).
fof(dt_k8_sin_cos2,axiom,
$true ).
fof(dt_k9_sin_cos2,axiom,
! [A] : m1_subset_1(k9_sin_cos2(A),k1_numbers) ).
fof(redefinition_k9_sin_cos2,axiom,
! [A] : k9_sin_cos2(A) = k8_sin_cos2(A) ).
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