SET007 Axioms: SET007+913.ax
%------------------------------------------------------------------------------
% File : SET007+913 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Theory of Sequential Files
% 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 : filerec1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 46 ( 0 unt; 0 def)
% Number of atoms : 254 ( 45 equ)
% Maximal formula atoms : 11 ( 5 avg)
% Number of connectives : 253 ( 45 ~; 7 |; 26 &)
% ( 1 <=>; 174 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-4 aty)
% Number of functors : 26 ( 26 usr; 6 con; 0-4 aty)
% Number of variables : 168 ( 167 !; 1 ?)
% 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(t1_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_finseq_1(D,A)
=> ! [E] :
( m2_finseq_1(E,A)
=> ( k1_finseq_8(A,k1_finseq_8(A,k1_finseq_8(A,B,C),D),E) = k1_finseq_8(A,k1_finseq_8(A,B,k1_finseq_8(A,C,D)),E)
& k1_finseq_8(A,k1_finseq_8(A,k1_finseq_8(A,B,C),D),E) = k1_finseq_8(A,k1_finseq_8(A,B,C),k1_finseq_8(A,D,E))
& k1_finseq_8(A,k1_finseq_8(A,B,k1_finseq_8(A,C,D)),E) = k1_finseq_8(A,k1_finseq_8(A,B,C),k1_finseq_8(A,D,E)) ) ) ) ) ) ) ).
fof(t2_filerec1,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> k16_finseq_1(A,B,k3_finseq_1(B)) = B ) ).
fof(t3_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( k3_finseq_1(B) = np__0
=> C = k1_finseq_8(A,B,C) ) ) ) ) ).
fof(t4_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [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(k3_finseq_1(k1_rfinseq(A,B,D)),k3_finseq_1(k1_rfinseq(A,B,C))) ) ) ) ) ) ).
fof(t5_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_xreal_0(np__1,k3_finseq_1(C))
=> k1_jordan3(A,k1_finseq_8(A,B,C),k1_nat_1(k3_finseq_1(B),np__1),k1_nat_1(k3_finseq_1(B),k3_finseq_1(C))) = C ) ) ) ) ).
fof(t6_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,E)
& r1_xreal_0(E,k3_finseq_1(B)) )
=> k1_jordan3(A,k1_finseq_8(A,B,C),D,E) = k1_jordan3(A,B,D,E) ) ) ) ) ) ) ).
fof(t7_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,D)
& r1_xreal_0(C,k3_finseq_1(k16_finseq_1(A,B,E)))
& r1_xreal_0(D,k3_finseq_1(k16_finseq_1(A,B,E))) )
=> k1_jordan3(A,B,C,D) = k1_jordan3(A,k16_finseq_1(A,B,E),C,D) ) ) ) ) ) ) ).
fof(t8_filerec1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( C = k9_finseq_1(A)
=> r2_hidden(A,B) ) ) ) ).
fof(t9_filerec1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_1(D,C)
=> ( D = k10_finseq_1(A,B)
=> ( r2_hidden(A,C)
& r2_hidden(B,C) ) ) ) ) ).
fof(t10_filerec1,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m2_finseq_1(E,D)
=> ( E = k11_finseq_1(A,B,C)
=> ( r2_hidden(A,D)
& r2_hidden(B,D)
& r2_hidden(C,D) ) ) ) ) ).
fof(t11_filerec1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( C = k9_finseq_1(A)
=> k16_finseq_1(B,C,np__1) = k9_finseq_1(A) ) ) ) ).
fof(t12_filerec1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m2_finseq_1(D,C)
=> ( D = k10_finseq_1(A,B)
=> k1_rfinseq(C,D,np__1) = k9_finseq_1(B) ) ) ) ).
fof(t13_filerec1,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m2_finseq_1(E,D)
=> ( E = k11_finseq_1(A,B,C)
=> k16_finseq_1(D,E,np__1) = k9_finseq_1(A) ) ) ) ).
fof(t14_filerec1,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m2_finseq_1(E,D)
=> ( E = k11_finseq_1(A,B,C)
=> k16_finseq_1(D,E,np__2) = k10_finseq_1(A,B) ) ) ) ).
fof(t15_filerec1,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m2_finseq_1(E,D)
=> ( E = k11_finseq_1(A,B,C)
=> k1_rfinseq(D,E,np__1) = k10_finseq_1(B,C) ) ) ) ).
fof(t16_filerec1,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m2_finseq_1(E,D)
=> ( E = k11_finseq_1(A,B,C)
=> k1_rfinseq(D,E,np__2) = k9_finseq_1(C) ) ) ) ).
fof(t17_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( k3_finseq_1(B) = np__0
=> k4_finseq_5(A,B) = B ) ) ) ).
fof(t18_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,k3_finseq_1(B))
=> k4_finseq_5(A,k1_rfinseq(A,B,C)) = k16_finseq_1(A,k4_finseq_5(A,B),k5_binarith(k3_finseq_1(B),C)) ) ) ) ) ).
fof(t19_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_finseq_8(A,C)
=> ( r2_finseq_8(A,B,C,np__1)
| k7_finseq_8(A,k1_finseq_8(A,B,C),C,np__1) = k1_nat_1(k3_finseq_1(B),np__1) ) ) ) ) ) ).
fof(t20_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> r3_finseq_8(A,k1_rfinseq(A,k1_finseq_8(A,B,C),k3_finseq_1(B)),C) ) ) ) ).
fof(t21_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r1_finseq_8(A,C)
=> ( r2_finseq_8(A,B,C,np__1)
| r5_finseq_8(A,k1_finseq_8(A,B,C),C) ) ) ) ) ) ).
fof(d1_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m1_finseq_1(D,A)
=> ( r1_filerec1(A,B,C,D)
<=> ( ( r2_finseq_8(A,k8_finseq_8(A,C,D),k1_finseq_8(A,D,B),np__1)
| r3_finseq_8(A,k8_finseq_8(A,C,D),B) )
& r5_finseq_8(A,B,D) ) ) ) ) ) ) ).
fof(t22_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( k3_finseq_8(A,k6_finseq_1(A),B) = k6_finseq_1(A)
& k3_finseq_8(A,B,k6_finseq_1(A)) = k6_finseq_1(A) ) ) ) ).
fof(t23_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> r1_filerec1(A,B,k6_finseq_1(A),B) ) ) ).
fof(t24_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D] :
( m1_finseq_1(D,A)
=> ! [E] :
( m1_finseq_1(E,A)
=> ! [F] :
( m1_finseq_1(F,A)
=> ( ( A = k2_tarski(B,C)
& F = k9_finseq_1(C)
& D = k11_finseq_1(C,B,C)
& E = k10_finseq_1(B,C) )
=> ( B = C
| ( r1_filerec1(A,F,D,F)
& r1_filerec1(A,E,D,F) ) ) ) ) ) ) ) ).
fof(t25_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> r3_finseq_8(A,k1_finseq_8(A,B,C),B) ) ) ) ).
fof(t26_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> r3_finseq_8(A,k8_finseq_8(A,B,C),B) ) ) ) ).
fof(t27_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ( r4_finseq_8(A,B,C)
=> r1_xreal_0(np__0,k5_real_1(k3_finseq_1(B),k3_finseq_1(C))) ) ) ) ) ).
fof(t28_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ( r4_finseq_8(A,C,B)
=> C = k8_finseq_8(A,C,B) ) ) ) ) ).
fof(t29_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ( r5_finseq_8(A,C,B)
=> C = k8_finseq_8(A,C,B) ) ) ) ) ).
fof(t30_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ( r5_finseq_8(A,B,C)
=> r1_xreal_0(k3_finseq_1(C),k3_finseq_1(B)) ) ) ) ) ).
fof(t31_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ( r1_xreal_0(k3_finseq_1(B),k3_finseq_1(k8_finseq_8(A,B,C)))
& r1_xreal_0(k3_finseq_1(C),k3_finseq_1(k8_finseq_8(A,B,C))) ) ) ) ) ).
fof(t32_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> C = k1_finseq_8(A,k3_finseq_8(A,B,C),k6_finseq_8(A,B,C)) ) ) ) ).
fof(t33_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k4_finseq_8(A,B,C) = k1_finseq_8(A,k5_finseq_8(A,B,C),C) ) ) ) ).
fof(t34_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> k8_finseq_8(A,C,B) = k1_finseq_8(A,k5_finseq_8(A,C,B),B) ) ) ) ).
fof(t35_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m1_finseq_1(D,A)
=> ( D = k1_finseq_8(A,B,C)
=> ( r2_finseq_8(A,D,B,np__1)
& r2_finseq_8(A,D,C,np__1) ) ) ) ) ) ) ).
fof(t36_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m1_finseq_1(D,A)
=> ! [E] :
( m1_finseq_1(E,A)
=> ( E = k1_finseq_8(A,k1_finseq_8(A,B,C),D)
=> ( r2_finseq_8(A,E,B,np__1)
& r2_finseq_8(A,E,C,np__1)
& r2_finseq_8(A,E,D,np__1) ) ) ) ) ) ) ) ).
fof(t37_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m1_finseq_1(D,A)
=> ( ( r5_finseq_8(A,C,B)
& r5_finseq_8(A,D,B) )
=> r2_finseq_8(A,k8_finseq_8(A,k1_finseq_8(A,C,D),B),k1_finseq_8(A,B,D),np__1) ) ) ) ) ) ).
fof(t38_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( C = k1_xboole_0
=> ( r1_xreal_0(D,np__0)
| k7_finseq_8(A,B,C,D) = D ) ) ) ) ) ) ).
fof(t39_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,k3_finseq_1(B))
=> ( r1_xreal_0(D,np__0)
| r1_xreal_0(k7_finseq_8(A,B,C,D),k3_finseq_1(B)) ) ) ) ) ) ) ).
fof(t40_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> r2_finseq_8(A,k4_finseq_8(A,B,C),C,np__1) ) ) ) ).
fof(t41_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> r2_finseq_8(A,k8_finseq_8(A,B,C),C,np__1) ) ) ) ).
fof(t42_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_finseq_8(A,k16_finseq_1(A,B,D),C,np__1)
& r1_xreal_0(k3_finseq_1(C),D) )
=> ( r1_xreal_0(k3_finseq_1(C),np__0)
| r2_finseq_8(A,B,C,np__1) ) ) ) ) ) ) ).
fof(t43_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ? [D] :
( m1_finseq_1(D,A)
& r1_filerec1(A,D,B,C) ) ) ) ) ).
fof(t44_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m1_finseq_1(D,A)
=> ( r1_filerec1(A,D,B,C)
=> r1_filerec1(A,D,D,C) ) ) ) ) ) ).
fof(t45_filerec1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_finseq_1(B,A)
=> ! [C] :
( m1_finseq_1(C,A)
=> ! [D] :
( m1_finseq_1(D,A)
=> ! [E] :
( m1_finseq_1(E,A)
=> ( ( r5_finseq_8(A,C,B)
& r5_finseq_8(A,D,B)
& E = k1_finseq_8(A,C,D) )
=> ( r1_filerec1(A,C,E,B)
& r1_filerec1(A,D,E,B) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------