SET007 Axioms: SET007+800.ax
%------------------------------------------------------------------------------
% File : SET007+800 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Correctness of Non Overwriting Programs. Part I
% 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 : prgcor_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 17 ( 0 unt; 0 def)
% Number of atoms : 248 ( 90 equ)
% Maximal formula atoms : 80 ( 14 avg)
% Number of connectives : 270 ( 39 ~; 17 |; 119 &)
% ( 2 <=>; 93 =>; 0 <=; 0 <~>)
% Maximal formula depth : 48 ( 13 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 23 ( 23 usr; 6 con; 0-3 aty)
% Number of variables : 57 ( 52 !; 5 ?)
% 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_prgcor_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& m1_relset_1(B,A,k4_numbers) )
=> ( v1_xcmplx_0(k1_funct_1(B,C))
& v1_xreal_0(k1_funct_1(B,C))
& v1_int_1(k1_funct_1(B,C)) ) ) ).
fof(t1_prgcor_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)
=> k5_binarith(k1_nat_1(A,C),k1_nat_1(B,C)) = k5_binarith(A,B) ) ) ) ).
fof(t2_prgcor_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
=> ( r1_xreal_0(B,np__0)
| ( k6_xcmplx_0(k4_nat_1(A,k2_nat_1(np__2,B)),B) = k4_nat_1(A,B)
& k1_nat_1(k4_nat_1(A,B),B) = k4_nat_1(A,k2_nat_1(np__2,B)) ) ) ) ) ) ).
fof(t3_prgcor_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
=> ( r1_xreal_0(B,np__0)
| k3_nat_1(A,B) = k1_nat_1(k2_nat_1(k3_nat_1(A,k2_nat_1(np__2,B)),np__2),np__1) ) ) ) ) ).
fof(t4_prgcor_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
& k4_nat_1(A,k2_nat_1(np__2,B)) != k4_nat_1(A,B) ) ) ) ).
fof(t5_prgcor_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
& k3_nat_1(A,B) != k2_nat_1(k3_nat_1(A,k2_nat_1(np__2,B)),np__2) ) ) ) ).
fof(t6_prgcor_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,D)
=> r1_xreal_0(k2_nat_1(A,k3_euler_2(np__2,D)),B) ) )
& ~ r1_xreal_0(k2_nat_1(A,k3_euler_2(np__2,C)),B) ) ) ) ) ) ).
fof(t7_prgcor_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(B)) )
=> r2_hidden(A,k4_finseq_1(B)) ) ) ) ).
fof(d1_prgcor_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,np__0)
| ! [C] :
( v1_int_1(C)
=> ( C = k1_prgcor_1(A,B)
<=> ? [D] :
( m2_finseq_1(D,k4_numbers)
& ? [E] :
( m2_finseq_1(E,k4_numbers)
& ? [F] :
( m2_finseq_1(F,k4_numbers)
& k3_finseq_1(D) = k2_xcmplx_0(A,np__1)
& k3_finseq_1(E) = k2_xcmplx_0(A,np__1)
& k3_finseq_1(F) = k2_xcmplx_0(A,np__1)
& ( ~ r1_xreal_0(B,A)
=> C = np__0 )
& ( r1_xreal_0(B,A)
=> ( k1_funct_1(D,np__1) = B
& ? [G] :
( v1_int_1(G)
& r1_xreal_0(np__1,G)
& r1_xreal_0(G,A)
& ! [H] :
( v1_int_1(H)
=> ( r1_xreal_0(np__1,H)
=> ( r1_xreal_0(G,H)
| ( k1_funct_1(D,k2_xcmplx_0(H,np__1)) = k3_xcmplx_0(k1_funct_1(D,H),np__2)
& r1_xreal_0(k1_funct_1(D,k2_xcmplx_0(H,np__1)),A) ) ) ) )
& k1_funct_1(D,k2_xcmplx_0(G,np__1)) = k3_xcmplx_0(k1_funct_1(D,G),np__2)
& ~ r1_xreal_0(k1_funct_1(D,k2_xcmplx_0(G,np__1)),A)
& k1_funct_1(F,k2_xcmplx_0(G,np__1)) = np__0
& k1_funct_1(E,k2_xcmplx_0(G,np__1)) = A
& ! [H] :
( v1_int_1(H)
=> ( ( r1_xreal_0(np__1,H)
& r1_xreal_0(H,G) )
=> ( ( r1_xreal_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)),k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))))
=> ( k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k6_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)))
& k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k2_xcmplx_0(k3_xcmplx_0(k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))),np__2),np__1) ) )
& ( ~ r1_xreal_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)),k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))))
=> ( k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1)))
& k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k3_xcmplx_0(k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))),np__2) ) ) ) ) )
& C = k1_funct_1(F,np__1) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_prgcor_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,np__0)
| ! [C] :
( m2_finseq_1(C,k4_numbers)
=> ! [D] :
( m2_finseq_1(D,k4_numbers)
=> ! [E] :
( m2_finseq_1(E,k4_numbers)
=> ! [F] :
( v1_int_1(F)
=> ( ( k3_finseq_1(C) = k2_xcmplx_0(A,np__1)
& k3_finseq_1(D) = k2_xcmplx_0(A,np__1)
& k3_finseq_1(E) = k2_xcmplx_0(A,np__1) )
=> ( ( r1_xreal_0(B,A)
& ~ ( k1_funct_1(C,np__1) = B
& r1_xreal_0(np__1,F)
& r1_xreal_0(F,A)
& ! [G] :
( v1_int_1(G)
=> ( r1_xreal_0(np__1,G)
=> ( r1_xreal_0(F,G)
| ( k1_funct_1(C,k2_xcmplx_0(G,np__1)) = k3_xcmplx_0(k1_funct_1(C,G),np__2)
& r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(G,np__1)),A) ) ) ) )
& k1_funct_1(C,k2_xcmplx_0(F,np__1)) = k3_xcmplx_0(k1_funct_1(C,F),np__2)
& ~ r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(F,np__1)),A)
& k1_funct_1(E,k2_xcmplx_0(F,np__1)) = np__0
& k1_funct_1(D,k2_xcmplx_0(F,np__1)) = A
& ! [G] :
( v1_int_1(G)
=> ( ( r1_xreal_0(np__1,G)
& r1_xreal_0(G,F) )
=> ( ( r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
=> ( k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k6_xcmplx_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)))
& k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k2_xcmplx_0(k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2),np__1) ) )
& ( ~ r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
=> ( k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)))
& k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2) ) ) ) ) )
& k1_prgcor_1(A,B) = k1_funct_1(E,np__1) ) )
| ( k3_finseq_1(C) = k2_xcmplx_0(A,np__1)
& k3_finseq_1(D) = k2_xcmplx_0(A,np__1)
& k3_finseq_1(E) = k2_xcmplx_0(A,np__1)
& ( ~ r1_xreal_0(B,A)
=> k1_prgcor_1(A,B) = np__0 )
& ( r1_xreal_0(B,A)
=> ( r2_hidden(np__1,k4_relset_1(k5_numbers,k4_numbers,C))
& k1_funct_1(C,np__1) = B
& r1_xreal_0(np__1,F)
& r1_xreal_0(F,A)
& ! [G] :
( v1_int_1(G)
=> ( r1_xreal_0(np__1,G)
=> ( r1_xreal_0(F,G)
| ( r2_hidden(k2_xcmplx_0(G,np__1),k4_relset_1(k5_numbers,k4_numbers,C))
& r2_hidden(G,k4_relset_1(k5_numbers,k4_numbers,C))
& k1_funct_1(C,k2_xcmplx_0(G,np__1)) = k3_xcmplx_0(k1_funct_1(C,G),np__2)
& r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(G,np__1)),A) ) ) ) )
& r2_hidden(k2_xcmplx_0(F,np__1),k4_relset_1(k5_numbers,k4_numbers,C))
& r2_hidden(F,k4_relset_1(k5_numbers,k4_numbers,C))
& k1_funct_1(C,k2_xcmplx_0(F,np__1)) = k3_xcmplx_0(k1_funct_1(C,F),np__2)
& ~ r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(F,np__1)),A)
& r2_hidden(k2_xcmplx_0(F,np__1),k4_relset_1(k5_numbers,k4_numbers,E))
& k1_funct_1(E,k2_xcmplx_0(F,np__1)) = np__0
& r2_hidden(k2_xcmplx_0(F,np__1),k4_relset_1(k5_numbers,k4_numbers,D))
& k1_funct_1(D,k2_xcmplx_0(F,np__1)) = A
& ! [G] :
( v1_int_1(G)
=> ( ( r1_xreal_0(np__1,G)
& r1_xreal_0(G,F) )
=> ( r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,D))
& r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,C))
& ( r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
=> ( r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,D))
& r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,C))
& k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k6_xcmplx_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)))
& r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,E))
& r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,E))
& k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k2_xcmplx_0(k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2),np__1) ) )
& ( ~ r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
=> ( r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,D))
& r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,D))
& k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)))
& r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,E))
& r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,E))
& k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2) ) ) ) ) )
& r2_hidden(np__1,k4_relset_1(k5_numbers,k4_numbers,E))
& k1_prgcor_1(A,B) = k1_funct_1(E,np__1) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t9_prgcor_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k1_prgcor_1(A,B) = k3_nat_1(A,B) ) ) ) ).
fof(t10_prgcor_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,np__0)
| k1_prgcor_1(A,B) = k5_int_1(A,B) ) ) ) ) ).
fof(t11_prgcor_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( ( B = np__0
& C = A
& D = B )
=> ( k5_int_1(A,B) = np__0
& k3_nat_1(C,D) = np__0 ) )
& ( ( r1_xreal_0(np__0,A)
& C = A
& D = B )
=> ( r1_xreal_0(B,np__0)
| k5_int_1(A,B) = k3_nat_1(C,D) ) )
& ( ( r1_xreal_0(np__0,A)
& C = A
& D = k4_xcmplx_0(B) )
=> ( r1_xreal_0(np__0,B)
| ( ( k2_nat_1(D,k3_nat_1(C,D)) = C
=> k5_int_1(A,B) = k4_xcmplx_0(k3_nat_1(C,D)) )
& ( k2_nat_1(D,k3_nat_1(C,D)) != C
=> k5_int_1(A,B) = k6_xcmplx_0(k4_xcmplx_0(k3_nat_1(C,D)),np__1) ) ) ) )
& ( ( C = k4_xcmplx_0(A)
& D = B )
=> ( r1_xreal_0(np__0,A)
| r1_xreal_0(B,np__0)
| ( ( k2_nat_1(D,k3_nat_1(C,D)) = C
=> k5_int_1(A,B) = k4_xcmplx_0(k3_nat_1(C,D)) )
& ( k2_nat_1(D,k3_nat_1(C,D)) != C
=> k5_int_1(A,B) = k6_xcmplx_0(k4_xcmplx_0(k3_nat_1(C,D)),np__1) ) ) ) )
& ( ( C = k4_xcmplx_0(A)
& D = k4_xcmplx_0(B) )
=> ( r1_xreal_0(np__0,A)
| r1_xreal_0(np__0,B)
| k5_int_1(A,B) = k3_nat_1(C,D) ) ) ) ) ) ) ) ).
fof(d2_prgcor_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( C = k2_prgcor_1(A,B)
<=> ? [D] :
( v1_int_1(D)
& ( B = np__0
=> C = np__0 )
& ( B != np__0
=> ( ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,np__0)
| C = k1_prgcor_1(A,B) ) )
& ( ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,np__0) )
=> ( ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(np__0,B)
| ( D = k1_prgcor_1(A,k4_xcmplx_0(B))
& ( k3_xcmplx_0(k4_xcmplx_0(B),D) = A
=> C = k4_xcmplx_0(D) )
& ( k3_xcmplx_0(k4_xcmplx_0(B),D) != A
=> C = k6_xcmplx_0(k4_xcmplx_0(D),np__1) ) ) ) )
& ( ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(np__0,B) )
=> ( ~ ( ~ r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,np__0)
& ~ ( D = k1_prgcor_1(k4_xcmplx_0(A),B)
& ( k3_xcmplx_0(B,D) = k4_xcmplx_0(A)
=> C = k4_xcmplx_0(D) )
& ( k3_xcmplx_0(B,D) != k4_xcmplx_0(A)
=> C = k6_xcmplx_0(k4_xcmplx_0(D),np__1) ) ) )
& ( ~ ( ~ r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,np__0) )
=> C = k1_prgcor_1(k4_xcmplx_0(A),k4_xcmplx_0(B)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_prgcor_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> k2_prgcor_1(A,B) = k5_int_1(A,B) ) ) ).
fof(dt_k1_prgcor_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> v1_int_1(k1_prgcor_1(A,B)) ) ).
fof(dt_k2_prgcor_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> v1_int_1(k2_prgcor_1(A,B)) ) ).
%------------------------------------------------------------------------------