SET007 Axioms: SET007+762.ax
%------------------------------------------------------------------------------
% File : SET007+762 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Solving Roots of Polynomial Equation of Degree 4
% Version : [Urb08] axioms.
% English : Solving Roots of Polynomial Equation of Degree 4 with Real
% Coefficients},
% 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 : polyeq_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 19 ( 2 unt; 0 def)
% Number of atoms : 182 ( 60 equ)
% Maximal formula atoms : 18 ( 9 avg)
% Number of connectives : 178 ( 15 ~; 10 |; 36 &)
% ( 0 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 15 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 18 ( 18 usr; 7 con; 0-6 aty)
% Number of variables : 114 ( 114 !; 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_polyeq_2,axiom,
! [A,B,C,D,E,F] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C)
& v1_xreal_0(D)
& v1_xreal_0(E)
& v1_xreal_0(F) )
=> ( v1_xcmplx_0(k1_polyeq_2(A,B,C,D,E,F))
& v1_xreal_0(k1_polyeq_2(A,B,C,D,E,F)) ) ) ).
fof(fc2_polyeq_2,axiom,
! [A,B,C,D,E,F] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_xreal_0(C)
& v1_xreal_0(D)
& v1_xreal_0(E)
& v1_xreal_0(F) )
=> ( v1_xcmplx_0(k2_polyeq_2(A,B,C,D,E,F))
& v1_xreal_0(k2_polyeq_2(A,B,C,D,E,F)) ) ) ).
fof(d1_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> k1_polyeq_2(A,B,C,D,E,F) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k2_newton(F,np__4)),k3_xcmplx_0(B,k2_newton(F,np__3))),k3_xcmplx_0(C,k5_square_1(F))),k3_xcmplx_0(D,F)),E) ) ) ) ) ) ) ).
fof(t1_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( k1_polyeq_2(A,np__0,B,np__0,C,D) = np__0
=> ( A = np__0
| C = np__0
| r1_xreal_0(k6_xcmplx_0(k5_square_1(B),k3_xcmplx_0(k3_xcmplx_0(np__4,A),C)),np__0)
| ( D != np__0
& ~ ( D != k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k1_quin_1(A,B,C))),k3_xcmplx_0(np__2,A)))
& D != k8_square_1(k7_xcmplx_0(k6_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k1_quin_1(A,B,C))),k3_xcmplx_0(np__2,A)))
& D != k4_xcmplx_0(k8_square_1(k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k1_quin_1(A,B,C))),k3_xcmplx_0(np__2,A))))
& D != k4_xcmplx_0(k8_square_1(k7_xcmplx_0(k6_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k1_quin_1(A,B,C))),k3_xcmplx_0(np__2,A)))) ) ) ) ) ) ) ) ) ).
fof(t2_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ( ( E = k2_xcmplx_0(D,k7_xcmplx_0(np__1,D))
& k1_polyeq_2(A,B,C,B,A,D) = np__0 )
=> ( A = np__0
| ( D != np__0
& k6_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k5_square_1(E)),k3_xcmplx_0(B,E)),C),k3_xcmplx_0(np__2,A)) = np__0 ) ) ) ) ) ) ) ) ).
fof(t3_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ( ( E = k2_xcmplx_0(D,k7_xcmplx_0(np__1,D))
& k1_polyeq_2(A,B,C,B,A,D) = np__0 )
=> ( A = np__0
| r1_xreal_0(k2_xcmplx_0(k6_xcmplx_0(k5_square_1(B),k3_xcmplx_0(k3_xcmplx_0(np__4,A),C)),k3_xcmplx_0(np__8,k5_square_1(A))),np__0)
| ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ( ( F = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k2_xcmplx_0(k6_xcmplx_0(k5_square_1(B),k3_xcmplx_0(k3_xcmplx_0(np__4,A),C)),k3_xcmplx_0(np__8,k5_square_1(A))))),k3_xcmplx_0(np__2,A))
& G = k7_xcmplx_0(k6_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k2_xcmplx_0(k6_xcmplx_0(k5_square_1(B),k3_xcmplx_0(k3_xcmplx_0(np__4,A),C)),k3_xcmplx_0(np__8,k5_square_1(A))))),k3_xcmplx_0(np__2,A)) )
=> ( D != np__0
& ~ ( D != k7_xcmplx_0(k2_xcmplx_0(F,k8_square_1(k1_quin_1(np__1,k4_xcmplx_0(F),np__1))),np__2)
& D != k7_xcmplx_0(k2_xcmplx_0(G,k8_square_1(k1_quin_1(np__1,k4_xcmplx_0(G),np__1))),np__2)
& D != k7_xcmplx_0(k6_xcmplx_0(F,k8_square_1(k1_quin_1(np__1,k4_xcmplx_0(F),np__1))),np__2)
& D != k7_xcmplx_0(k6_xcmplx_0(G,k8_square_1(k1_quin_1(np__1,k4_xcmplx_0(G),np__1))),np__2) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_newton(A,np__3) = k3_xcmplx_0(k5_square_1(A),A)
& k3_xcmplx_0(k2_newton(A,np__3),A) = k2_newton(A,np__4)
& k3_xcmplx_0(k5_square_1(A),k5_square_1(A)) = k2_newton(A,np__4) ) ) ).
fof(t5_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_xcmplx_0(A,B) != np__0
=> k2_newton(k2_xcmplx_0(A,B),np__4) = k2_xcmplx_0(k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__3,B),k5_square_1(A)),k3_xcmplx_0(k3_xcmplx_0(np__3,k5_square_1(B)),A))),k2_newton(B,np__3)),A),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__3),k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__3,B),k5_square_1(A)),k3_xcmplx_0(k3_xcmplx_0(np__3,k5_square_1(B)),A))),k2_newton(B,np__3)),B)) ) ) ) ).
fof(t6_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_xcmplx_0(A,B) != np__0
=> k2_newton(k2_xcmplx_0(A,B),np__4) = k2_xcmplx_0(k2_xcmplx_0(k2_newton(A,np__4),k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__4,B),k2_newton(A,np__3)),k3_xcmplx_0(k3_xcmplx_0(np__6,k5_square_1(B)),k5_square_1(A))),k3_xcmplx_0(k3_xcmplx_0(np__4,k2_newton(B,np__3)),A))),k2_newton(B,np__4)) ) ) ) ).
fof(t7_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ! [J] :
( v1_xreal_0(J)
=> ( ! [K] :
( v1_xreal_0(K)
=> k1_polyeq_2(A,B,C,D,E,K) = k1_polyeq_2(F,G,H,I,J,K) )
=> ( E = J
& k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(A,B),C),D) = k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(F,G),H),I)
& k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(A,B),C),D) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(F,G),H),I) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ! [J] :
( v1_xreal_0(J)
=> ( ! [K] :
( v1_xreal_0(K)
=> k1_polyeq_2(A,B,C,D,E,K) = k1_polyeq_2(F,G,H,I,J,K) )
=> ( k6_xcmplx_0(A,F) = k6_xcmplx_0(H,C)
& k6_xcmplx_0(B,G) = k6_xcmplx_0(I,D) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t9_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ! [J] :
( v1_xreal_0(J)
=> ( ! [K] :
( v1_xreal_0(K)
=> k1_polyeq_2(A,B,C,D,E,K) = k1_polyeq_2(F,G,H,I,J,K) )
=> ( A = F
& B = G
& C = H
& D = I
& E = J ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> k2_polyeq_2(A,B,C,D,E,F) = k3_xcmplx_0(A,k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(F,B),k6_xcmplx_0(F,C)),k6_xcmplx_0(F,D)),k6_xcmplx_0(F,E))) ) ) ) ) ) ) ).
fof(t10_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ! [J] :
( v1_xreal_0(J)
=> ( ! [K] :
( v1_xreal_0(K)
=> k1_polyeq_2(A,B,C,D,E,K) = k2_polyeq_2(A,G,H,I,J,K) )
=> ( A = np__0
| k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k2_newton(F,np__4)),k3_xcmplx_0(B,k2_newton(F,np__3))),k3_xcmplx_0(C,k5_square_1(F))),k3_xcmplx_0(D,F)),E),A) = k6_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k5_square_1(F),k5_square_1(F)),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(G,H),I),k3_xcmplx_0(k5_square_1(F),F))),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(G,I),k3_xcmplx_0(H,I)),k3_xcmplx_0(G,H)),k5_square_1(F))),k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(G,H),I),F)),k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(F,G),k6_xcmplx_0(F,H)),k6_xcmplx_0(F,I)),J)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t11_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ! [J] :
( v1_xreal_0(J)
=> ( ! [K] :
( v1_xreal_0(K)
=> k1_polyeq_2(A,B,C,D,E,K) = k2_polyeq_2(A,G,H,I,J,K) )
=> ( A = np__0
| k7_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k2_newton(F,np__4)),k3_xcmplx_0(B,k2_newton(F,np__3))),k3_xcmplx_0(C,k5_square_1(F))),k3_xcmplx_0(D,F)),E),A) = k2_xcmplx_0(k6_xcmplx_0(k2_xcmplx_0(k6_xcmplx_0(k2_newton(F,np__4),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(G,H),I),J),k2_newton(F,np__3))),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(G,H),k3_xcmplx_0(G,I)),k3_xcmplx_0(G,J)),k2_xcmplx_0(k3_xcmplx_0(H,I),k3_xcmplx_0(H,J))),k3_xcmplx_0(I,J)),k5_square_1(F))),k3_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(G,H),I),k3_xcmplx_0(k3_xcmplx_0(G,H),J)),k3_xcmplx_0(k3_xcmplx_0(G,I),J)),k3_xcmplx_0(k3_xcmplx_0(H,I),J)),F)),k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(G,H),I),J)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ! [H] :
( v1_xreal_0(H)
=> ! [I] :
( v1_xreal_0(I)
=> ( ! [J] :
( v1_xreal_0(J)
=> k1_polyeq_2(A,B,C,D,E,J) = k2_polyeq_2(A,F,G,H,I,J) )
=> ( A = np__0
| ( k7_xcmplx_0(B,A) = k4_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(F,G),H),I))
& k7_xcmplx_0(C,A) = k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(F,G),k3_xcmplx_0(F,H)),k3_xcmplx_0(F,I)),k2_xcmplx_0(k3_xcmplx_0(G,H),k3_xcmplx_0(G,I))),k3_xcmplx_0(H,I))
& k7_xcmplx_0(D,A) = k4_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(F,G),H),k3_xcmplx_0(k3_xcmplx_0(F,G),I)),k3_xcmplx_0(k3_xcmplx_0(F,H),I)),k3_xcmplx_0(k3_xcmplx_0(G,H),I)))
& k7_xcmplx_0(E,A) = k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(F,G),H),I) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_polyeq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ! [D] :
( v1_xreal_0(D)
=> k2_xcmplx_0(k2_newton(D,np__4),k2_newton(A,np__4)) = k3_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(B,A),D),k2_xcmplx_0(k5_square_1(D),k5_square_1(A))) )
=> ( A = np__0
| k2_xcmplx_0(k6_xcmplx_0(k6_xcmplx_0(k2_newton(C,np__4),k3_xcmplx_0(B,k2_newton(C,np__3))),k3_xcmplx_0(B,C)),np__1) = np__0 ) ) ) ) ) ).
fof(dt_k1_polyeq_2,axiom,
$true ).
fof(dt_k2_polyeq_2,axiom,
$true ).
%------------------------------------------------------------------------------