SET007 Axioms: SET007+197.ax
%------------------------------------------------------------------------------
% File : SET007+197 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Chinese Remainder Theorem
% 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 : wsierp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 75 ( 6 unt; 0 def)
% Number of atoms : 384 ( 90 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 357 ( 48 ~; 9 |; 110 &)
% ( 6 <=>; 184 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-3 aty)
% Number of functors : 51 ( 51 usr; 9 con; 0-4 aty)
% Number of variables : 191 ( 183 !; 8 ?)
% 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_wsierp_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_xcmplx_0(k2_newton(A,B))
& v1_xreal_0(k2_newton(A,B))
& v1_int_1(k2_newton(A,B))
& v1_rat_1(k2_newton(A,B)) ) ) ).
fof(fc2_wsierp_1,axiom,
! [A,B] :
( m1_finseq_1(A,k4_numbers)
=> ( v1_xcmplx_0(k1_funct_1(A,B))
& v1_xreal_0(k1_funct_1(A,B))
& v1_int_1(k1_funct_1(A,B))
& v1_rat_1(k1_funct_1(A,B)) ) ) ).
fof(t1_wsierp_1,axiom,
$true ).
fof(t2_wsierp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ( k2_newton(A,np__2) = k3_xcmplx_0(A,A)
& k2_newton(k4_xcmplx_0(A),np__2) = k2_newton(A,np__2) ) ) ).
fof(t3_wsierp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k2_newton(k4_xcmplx_0(A),k2_nat_1(np__2,B)) = k2_newton(A,k2_nat_1(np__2,B))
& k2_newton(k4_xcmplx_0(A),k1_nat_1(k2_nat_1(np__2,B),np__1)) = k4_xcmplx_0(k2_newton(A,k1_nat_1(k2_nat_1(np__2,B),np__1))) ) ) ) ).
fof(t4_wsierp_1,axiom,
$true ).
fof(t5_wsierp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B)
& k2_newton(A,C) = k2_newton(B,C) )
=> ( r1_xreal_0(C,np__0)
| A = B ) ) ) ) ) ).
fof(t6_wsierp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ~ r1_xreal_0(A,k2_square_1(B,C))
<=> ( ~ r1_xreal_0(A,B)
& ~ r1_xreal_0(A,C) ) ) ) ) ) ).
fof(t7_wsierp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,np__0)
& r1_xreal_0(C,B) )
=> ( r1_xreal_0(C,k6_xcmplx_0(B,A))
& r1_xreal_0(k2_xcmplx_0(C,A),B) ) ) ) ) ) ).
fof(t8_wsierp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( ( r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,C) )
| ( ~ r1_xreal_0(np__0,A)
& r1_xreal_0(C,B) ) )
=> ( ~ r1_xreal_0(B,k2_xcmplx_0(C,A))
& ~ r1_xreal_0(k6_xcmplx_0(B,A),C) ) ) ) ) ) ).
fof(t9_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r2_int_1(A,B)
& r2_int_1(A,C) )
=> r2_int_1(A,k2_xcmplx_0(B,C)) ) ) ) ) ).
fof(t10_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ! [E] :
( v1_int_1(E)
=> ( ( r2_int_1(A,B)
& r2_int_1(A,C) )
=> r2_int_1(A,k2_xcmplx_0(k3_xcmplx_0(B,D),k3_xcmplx_0(C,E))) ) ) ) ) ) ) ).
fof(t11_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( k3_int_2(A,B) = np__1
& k3_int_2(C,B) = np__1 )
=> k3_int_2(k3_xcmplx_0(A,C),B) = np__1 ) ) ) ) ).
fof(t12_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k6_nat_1(A,B) = np__1
& k6_nat_1(C,B) = np__1 )
=> k6_nat_1(k2_nat_1(A,C),B) = np__1 ) ) ) ) ).
fof(t13_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ( k3_int_2(np__0,A) = k1_prepower(A)
& k3_int_2(np__1,A) = np__1 ) ) ).
fof(t14_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> r1_int_2(np__1,A) ) ).
fof(t15_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_2(B,C)
=> r1_int_2(k2_newton(B,A),C) ) ) ) ) ).
fof(t16_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ( r1_int_2(C,D)
=> r1_int_2(k2_newton(C,A),k2_newton(D,B)) ) ) ) ) ) ).
fof(t17_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ( k3_int_2(C,D) = np__1
=> ( k3_int_2(C,k2_newton(D,A)) = np__1
& k3_int_2(k2_newton(C,B),k2_newton(D,A)) = np__1 ) ) ) ) ) ) ).
fof(t18_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r2_int_1(k1_prepower(A),B)
<=> r2_int_1(A,B) ) ) ) ).
fof(t19_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,B)
=> r1_nat_1(k2_wsierp_1(A,C),k2_wsierp_1(B,C)) ) ) ) ) ).
fof(t20_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_nat_1(A,np__1)
=> A = np__1 ) ) ).
fof(t21_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(A,B)
& k6_nat_1(B,C) = np__1 )
=> k6_nat_1(A,C) = np__1 ) ) ) ) ).
fof(t22_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( A != np__0
=> ( r2_int_1(A,B)
<=> v1_int_1(k7_xcmplx_0(B,A)) ) ) ) ) ).
fof(t23_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,k5_real_1(B,C))
=> ( r1_xreal_0(A,B)
& r1_xreal_0(C,B) ) ) ) ) ) ).
fof(d1_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( ~ r2_hidden(A,k4_finseq_1(B))
=> ( C = k2_finseq_3(A,B)
<=> C = B ) )
& ( r2_hidden(A,k4_finseq_1(B))
=> ( C = k2_finseq_3(A,B)
<=> ( k1_nat_1(k3_finseq_1(C),np__1) = k3_finseq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( ~ r1_xreal_0(A,D)
=> k1_funct_1(C,D) = k1_funct_1(B,D) )
& ( r1_xreal_0(A,D)
=> k1_funct_1(C,D) = k1_funct_1(B,k1_nat_1(D,np__1)) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_wsierp_1,axiom,
$true ).
fof(t25_wsierp_1,axiom,
$true ).
fof(t26_wsierp_1,axiom,
! [A,B,C] :
( k2_finseq_3(np__1,k9_finseq_1(A)) = k1_xboole_0
& k2_finseq_3(np__1,k10_finseq_1(A,B)) = k9_finseq_1(B)
& k2_finseq_3(np__2,k10_finseq_1(A,B)) = k9_finseq_1(A)
& k2_finseq_3(np__1,k11_finseq_1(A,B,C)) = k10_finseq_1(B,C)
& k2_finseq_3(np__2,k11_finseq_1(A,B,C)) = k10_finseq_1(A,C)
& k2_finseq_3(np__3,k11_finseq_1(A,B,C)) = k10_finseq_1(A,B) ) ).
fof(t27_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( r2_hidden(A,k4_finseq_1(B))
=> k3_real_1(k15_rvsum_1(k11_wsierp_1(k1_numbers,A,B)),k1_wsierp_1(B,A)) = k15_rvsum_1(B) ) ) ) ).
fof(t28_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_trees_4(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k4_finseq_1(B))
=> m2_subset_1(k6_real_1(k10_wsierp_1(B),k3_wsierp_1(B,A)),k1_numbers,k5_numbers) ) ) ) ).
fof(t29_wsierp_1,axiom,
! [A] :
( v1_rat_1(A)
=> r1_int_2(k2_rat_1(A),k1_rat_1(A)) ) ).
fof(t30_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_rat_1(C)
=> ( ( C = k7_xcmplx_0(B,A)
& r1_int_2(B,A) )
=> ( C = np__0
| A = np__0
| ( B = k2_rat_1(C)
& A = k1_rat_1(C) ) ) ) ) ) ) ).
fof(t31_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ? [C] :
( v1_rat_1(C)
& A = k2_newton(C,B) )
& ! [C] :
( v1_int_1(C)
=> A != k2_newton(C,B) ) ) ) ) ).
fof(t32_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ? [C] :
( v1_rat_1(C)
& A = k2_newton(C,B) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> A != k2_wsierp_1(C,B) ) ) ) ) ).
fof(t33_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_nat_1(k2_wsierp_1(B,A),k2_wsierp_1(C,A))
=> ( r1_xreal_0(A,np__0)
| r1_nat_1(B,C) ) ) ) ) ) ).
fof(t34_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_int_1(C)
& ? [D] :
( v1_int_1(D)
& k6_nat_1(A,B) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)) ) ) ) ) ).
fof(t35_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ? [C] :
( v1_int_1(C)
& ? [D] :
( v1_int_1(D)
& k3_int_2(A,B) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)) ) ) ) ) ).
fof(t36_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r2_int_1(A,k3_xcmplx_0(B,C))
& k3_int_2(A,B) = np__1 )
=> r2_int_1(A,C) ) ) ) ) ).
fof(t37_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k6_nat_1(A,B) = np__1
& r1_nat_1(A,k2_nat_1(B,C)) )
=> r1_nat_1(A,C) ) ) ) ) ).
fof(t38_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& B != np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k6_nat_1(A,B) != k5_real_1(k2_nat_1(A,C),k2_nat_1(B,D)) ) ) ) ) ) ).
fof(t39_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& k6_nat_1(A,B) = np__1
& k2_wsierp_1(C,A) = k2_wsierp_1(D,B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( C = k2_wsierp_1(E,B)
& D = k2_wsierp_1(E,A) ) ) ) ) ) ) ) ).
fof(t40_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ? [D] :
( v1_int_1(D)
& ? [E] :
( v1_int_1(E)
& k2_xcmplx_0(k3_xcmplx_0(A,D),k3_xcmplx_0(B,E)) = C ) )
<=> r2_int_1(k3_int_2(A,B),C) ) ) ) ) ).
fof(t41_wsierp_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ! [E] :
( v1_int_1(E)
=> ( k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)) = E
=> ( A = np__0
| B = np__0
| ! [F] :
( v1_int_1(F)
=> ! [G] :
( v1_int_1(G)
=> ~ ( k2_xcmplx_0(k3_xcmplx_0(A,F),k3_xcmplx_0(B,G)) = E
& ! [H] :
( v1_int_1(H)
=> ~ ( F = k2_xcmplx_0(C,k3_xcmplx_0(H,k7_xcmplx_0(B,k3_int_2(A,B))))
& G = k6_xcmplx_0(D,k3_xcmplx_0(H,k7_xcmplx_0(A,k3_int_2(A,B)))) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t42_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( k6_nat_1(A,B) = np__1
& k2_nat_1(A,B) = k2_wsierp_1(C,D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( A = k2_wsierp_1(E,D)
& B = k2_wsierp_1(F,D) ) ) ) ) ) ) ) ) ).
fof(t43_wsierp_1,axiom,
! [A] :
( m1_trees_4(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(A))
=> k6_nat_1(k3_wsierp_1(A,C),B) = np__1 ) )
=> k6_nat_1(k10_wsierp_1(A),B) = np__1 ) ) ) ).
fof(t44_wsierp_1,axiom,
! [A] :
( m1_trees_4(A,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,k3_finseq_1(A))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& r2_hidden(C,k4_finseq_1(A)) )
=> ( B = C
| k6_nat_1(k3_wsierp_1(A,B),k3_wsierp_1(A,C)) = np__1 ) ) ) ) )
=> ! [B] :
( m1_trees_4(B,k1_numbers,k6_wsierp_1)
=> ~ ( k3_finseq_1(B) = k3_finseq_1(A)
& ! [C] :
( m1_trees_4(C,k1_numbers,k6_wsierp_1)
=> ~ ( k3_finseq_1(C) = k3_finseq_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(A))
=> k3_real_1(k4_real_1(k3_wsierp_1(A,D),k1_wsierp_1(C,D)),k1_wsierp_1(B,D)) = k3_real_1(k4_real_1(k3_wsierp_1(A,np__1),k1_wsierp_1(C,np__1)),k1_wsierp_1(B,np__1)) ) ) ) ) ) ) ) ) ).
fof(t45_wsierp_1,axiom,
$true ).
fof(t46_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ~ ( A != np__0
& k3_int_2(A,B) = np__1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( np__0 != C
& np__0 != D
& r1_xreal_0(C,k9_square_1(A))
& r1_xreal_0(D,k9_square_1(A))
& ( r2_int_1(A,k2_xcmplx_0(k3_xcmplx_0(B,C),D))
| r2_int_1(A,k6_xcmplx_0(k3_xcmplx_0(B,C),D)) ) ) ) ) ) ) ) ).
fof(t47_wsierp_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> r1_tarski(k4_finseq_1(k2_finseq_3(A,B)),k4_finseq_1(B)) ) ) ).
fof(t48_wsierp_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( k2_finseq_3(np__1,k7_finseq_1(k9_finseq_1(B),A)) = A
& k2_finseq_3(k1_nat_1(k3_finseq_1(A),np__1),k7_finseq_1(A,k9_finseq_1(B))) = A ) ) ).
fof(dt_k1_wsierp_1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k1_wsierp_1(A,B),k1_numbers) ) ).
fof(redefinition_k1_wsierp_1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k1_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_wsierp_1(A,B) = k1_funct_1(A,B) ) ).
fof(dt_k2_wsierp_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k2_wsierp_1(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k2_wsierp_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k2_wsierp_1(A,B) = k2_newton(A,B) ) ).
fof(dt_k3_wsierp_1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k3_wsierp_1(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k3_wsierp_1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k3_wsierp_1(A,B) = k1_funct_1(A,B) ) ).
fof(dt_k4_wsierp_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& m1_finseq_1(C,B)
& m1_finseq_1(D,B) )
=> m1_trees_4(k4_wsierp_1(A,B,C,D),A,B) ) ).
fof(redefinition_k4_wsierp_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& m1_finseq_1(C,B)
& m1_finseq_1(D,B) )
=> k4_wsierp_1(A,B,C,D) = k7_finseq_1(C,D) ) ).
fof(dt_k5_wsierp_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v1_xboole_0(k5_wsierp_1(A,B))
& m1_trees_4(k5_wsierp_1(A,B),A,B) ) ) ).
fof(redefinition_k5_wsierp_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> k5_wsierp_1(A,B) = k6_finseq_1(B) ) ).
fof(dt_k6_wsierp_1,axiom,
( ~ v1_xboole_0(k6_wsierp_1)
& m1_subset_1(k6_wsierp_1,k1_zfmisc_1(k1_numbers)) ) ).
fof(redefinition_k6_wsierp_1,axiom,
k6_wsierp_1 = k4_numbers ).
fof(dt_k7_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k6_wsierp_1)
=> m2_subset_1(k7_wsierp_1(A),k1_numbers,k6_wsierp_1) ) ).
fof(redefinition_k7_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k6_wsierp_1)
=> k7_wsierp_1(A) = k15_rvsum_1(A) ) ).
fof(dt_k8_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k6_wsierp_1)
=> m2_subset_1(k8_wsierp_1(A),k1_numbers,k6_wsierp_1) ) ).
fof(redefinition_k8_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k6_wsierp_1)
=> k8_wsierp_1(A) = k16_rvsum_1(A) ) ).
fof(dt_k9_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k5_numbers)
=> m2_subset_1(k9_wsierp_1(A),k1_numbers,k5_numbers) ) ).
fof(redefinition_k9_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k5_numbers)
=> k9_wsierp_1(A) = k15_rvsum_1(A) ) ).
fof(dt_k10_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k5_numbers)
=> m2_subset_1(k10_wsierp_1(A),k1_numbers,k5_numbers) ) ).
fof(redefinition_k10_wsierp_1,axiom,
! [A] :
( m1_finseq_1(A,k5_numbers)
=> k10_wsierp_1(A) = k16_rvsum_1(A) ) ).
fof(dt_k11_wsierp_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_finseq_1(C,A) )
=> m2_finseq_1(k11_wsierp_1(A,B,C),A) ) ).
fof(redefinition_k11_wsierp_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_finseq_1(C,A) )
=> k11_wsierp_1(A,B,C) = k2_finseq_3(B,C) ) ).
fof(dt_k12_wsierp_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers)
& m1_finseq_1(D,B) )
=> m1_trees_4(k12_wsierp_1(A,B,C,D),A,B) ) ).
fof(redefinition_k12_wsierp_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers)
& m1_finseq_1(D,B) )
=> k12_wsierp_1(A,B,C,D) = k2_finseq_3(C,D) ) ).
%------------------------------------------------------------------------------