SET007 Axioms: SET007+189.ax
%------------------------------------------------------------------------------
% File : SET007+189 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Reduction Relations
% 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 : rewrite1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 124 ( 2 unt; 0 def)
% Number of atoms : 622 ( 50 equ)
% Maximal formula atoms : 23 ( 5 avg)
% Number of connectives : 529 ( 31 ~; 15 |; 219 &)
% ( 43 <=>; 221 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 40 ( 38 usr; 1 prp; 0-3 aty)
% Number of functors : 26 ( 26 usr; 8 con; 0-2 aty)
% Number of variables : 326 ( 310 !; 16 ?)
% 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(cc1_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m1_rewrite1(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(cc2_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v3_rewrite1(A) )
=> ( v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A) ) ) ).
fof(cc3_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A) )
=> ( v1_relat_1(A)
& v3_rewrite1(A) ) ) ).
fof(cc4_rewrite1,axiom,
! [A] :
( ( v1_xboole_0(A)
& v1_relat_1(A) )
=> ( v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A)
& v2_rewrite1(A)
& v3_rewrite1(A) ) ) ).
fof(rc1_rewrite1,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A)
& v2_rewrite1(A)
& v3_rewrite1(A) ) ).
fof(cc5_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v3_rewrite1(A) )
=> ( v1_relat_1(A)
& v2_rewrite1(A) ) ) ).
fof(cc6_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v8_rewrite1(A) )
=> ( v1_relat_1(A)
& v7_rewrite1(A) ) ) ).
fof(cc7_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v7_rewrite1(A) )
=> ( v1_relat_1(A)
& v8_rewrite1(A)
& v9_rewrite1(A) ) ) ).
fof(cc8_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v6_rewrite1(A) )
=> ( v1_relat_1(A)
& v7_rewrite1(A) ) ) ).
fof(cc9_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v8_rewrite1(A) )
=> ( v1_relat_1(A)
& v5_rewrite1(A) ) ) ).
fof(cc10_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v5_rewrite1(A) )
=> ( v1_relat_1(A)
& v4_rewrite1(A) ) ) ).
fof(cc11_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_rewrite1(A)
& v4_rewrite1(A) )
=> ( v1_relat_1(A)
& v8_rewrite1(A) ) ) ).
fof(cc12_rewrite1,axiom,
! [A] :
( ( v1_xboole_0(A)
& v1_relat_1(A) )
=> ( v1_relat_1(A)
& v4_rewrite1(A)
& v5_rewrite1(A)
& v6_rewrite1(A)
& v7_rewrite1(A)
& v8_rewrite1(A)
& v9_rewrite1(A) ) ) ).
fof(rc2_rewrite1,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A)
& v2_rewrite1(A)
& v3_rewrite1(A)
& v4_rewrite1(A)
& v5_rewrite1(A)
& v6_rewrite1(A)
& v7_rewrite1(A)
& v8_rewrite1(A)
& v9_rewrite1(A) ) ).
fof(cc13_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v3_rewrite1(A)
& v9_rewrite1(A) )
=> ( v1_relat_1(A)
& v4_rewrite1(A)
& v5_rewrite1(A)
& v7_rewrite1(A)
& v8_rewrite1(A)
& v9_rewrite1(A) ) ) ).
fof(cc14_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v10_rewrite1(A) )
=> ( v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A)
& v2_rewrite1(A)
& v3_rewrite1(A)
& v4_rewrite1(A)
& v5_rewrite1(A)
& v7_rewrite1(A)
& v8_rewrite1(A)
& v9_rewrite1(A) ) ) ).
fof(cc15_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v3_rewrite1(A)
& v7_rewrite1(A) )
=> ( v1_relat_1(A)
& v10_rewrite1(A) ) ) ).
fof(rc3_rewrite1,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A)
& v2_rewrite1(A)
& v3_rewrite1(A)
& v4_rewrite1(A)
& v5_rewrite1(A)
& v6_rewrite1(A)
& v7_rewrite1(A)
& v8_rewrite1(A)
& v9_rewrite1(A)
& v10_rewrite1(A) ) ).
fof(rc4_rewrite1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_2(A)
& v1_rewrite1(A)
& v2_rewrite1(A)
& v3_rewrite1(A)
& v4_rewrite1(A)
& v5_rewrite1(A)
& v7_rewrite1(A)
& v8_rewrite1(A)
& v9_rewrite1(A)
& v10_rewrite1(A) ) ).
fof(d1_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [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) )
=> ( ( ( A = k1_xboole_0
| B = k1_xboole_0 )
=> ( C = k1_rewrite1(A,B)
<=> C = k7_finseq_1(A,B) ) )
& ~ ( A != k1_xboole_0
& B != k1_xboole_0
& ~ ( C = k1_rewrite1(A,B)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E)
& k3_finseq_1(A) = k1_nat_1(D,np__1)
& E = k7_relat_1(A,k2_finseq_1(D))
& C = k7_finseq_1(E,B) ) ) ) ) ) ) ) ) ).
fof(t1_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( k1_rewrite1(k1_xboole_0,A) = A
& k1_rewrite1(A,k1_xboole_0) = A ) ) ).
fof(t2_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( A != k1_xboole_0
=> k1_rewrite1(k7_finseq_1(B,k9_finseq_1(C)),A) = k7_finseq_1(B,A) ) ) ) ).
fof(t3_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C,D] : k1_rewrite1(k7_finseq_1(A,k9_finseq_1(C)),k7_finseq_1(k9_finseq_1(D),B)) = k7_finseq_1(k7_finseq_1(A,k9_finseq_1(D)),B) ) ) ).
fof(t4_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( A != k1_xboole_0
=> k1_rewrite1(k9_finseq_1(B),A) = A ) ) ).
fof(t5_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ~ ( A != k1_xboole_0
& ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ~ ( A = k7_finseq_1(k9_finseq_1(B),C)
& k3_finseq_1(A) = k1_nat_1(k3_finseq_1(C),np__1) ) ) ) ) ).
fof(d2_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( m1_rewrite1(B,A)
<=> ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(B)) )
=> r2_hidden(k4_tarski(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))),A) ) ) ) ) ) ) ).
fof(t6_rewrite1,axiom,
$true ).
fof(t7_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] : m1_rewrite1(k9_finseq_1(B),A) ) ).
fof(t8_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r2_hidden(k4_tarski(B,C),A)
=> m1_rewrite1(k10_finseq_1(B,C),A) ) ) ).
fof(t9_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m1_rewrite1(B,A)
=> ! [C] :
( m1_rewrite1(C,A)
=> ( k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1)
=> m1_rewrite1(k1_rewrite1(B,C),A) ) ) ) ) ).
fof(t10_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m1_rewrite1(B,A)
=> m1_rewrite1(k3_finseq_5(B),k4_relat_1(A)) ) ) ).
fof(t11_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r1_tarski(A,B)
=> ! [C] :
( m1_rewrite1(C,A)
=> m1_rewrite1(C,B) ) ) ) ) ).
fof(d3_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
<=> ? [D] :
( m1_rewrite1(D,A)
& k1_funct_1(D,np__1) = B
& k1_funct_1(D,k3_finseq_1(D)) = C ) ) ) ).
fof(d4_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r2_rewrite1(A,B,C)
<=> r1_rewrite1(k2_xboole_0(A,k4_relat_1(A)),B,C) ) ) ).
fof(t12_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& ~ r1_xreal_0(k3_finseq_1(D),np__0)
& k1_funct_1(D,np__1) = B
& k1_funct_1(D,k3_finseq_1(D)) = C
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r2_hidden(E,k4_finseq_1(D))
& r2_hidden(k1_nat_1(E,np__1),k4_finseq_1(D)) )
=> r2_hidden(k4_tarski(k1_funct_1(D,E),k1_funct_1(D,k1_nat_1(E,np__1))),A) ) ) ) ) ) ).
fof(t13_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] : r1_rewrite1(A,B,B) ) ).
fof(t14_rewrite1,axiom,
! [A,B] :
( r1_rewrite1(k1_xboole_0,A,B)
=> A = B ) ).
fof(t15_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
=> ( r2_hidden(B,k3_relat_1(A))
| B = C ) ) ) ).
fof(t16_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r2_hidden(k4_tarski(B,C),A)
=> r1_rewrite1(A,B,C) ) ) ).
fof(t17_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C,D] :
( ( r1_rewrite1(A,B,C)
& r1_rewrite1(A,C,D) )
=> r1_rewrite1(A,B,D) ) ) ).
fof(t18_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m1_rewrite1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k4_finseq_1(B))
& r1_xreal_0(C,D) )
=> r1_rewrite1(A,k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ).
fof(t19_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
=> ( B = C
| ( r2_hidden(B,k3_relat_1(A))
& r2_hidden(C,k3_relat_1(A)) ) ) ) ) ).
fof(t20_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
=> ( r2_hidden(B,k3_relat_1(A))
<=> r2_hidden(C,k3_relat_1(A)) ) ) ) ).
fof(t21_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
<=> ( B = C
| r2_hidden(k4_tarski(B,C),k17_finseq_1(A)) ) ) ) ).
fof(t22_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
<=> r1_rewrite1(k17_finseq_1(A),B,C) ) ) ).
fof(t23_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r1_tarski(A,B)
=> ! [C,D] :
( r1_rewrite1(A,C,D)
=> r1_rewrite1(B,C,D) ) ) ) ) ).
fof(t24_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C,D] :
( r1_rewrite1(A,C,D)
<=> r1_rewrite1(k2_xboole_0(A,k6_relat_1(B)),C,D) ) ) ).
fof(t25_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
=> r1_rewrite1(k4_relat_1(A),C,B) ) ) ).
fof(t26_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
=> ( r2_rewrite1(A,B,C)
& r2_rewrite1(A,C,B) ) ) ) ).
fof(t27_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] : r2_rewrite1(A,B,B) ) ).
fof(t28_rewrite1,axiom,
! [A,B] :
( r2_rewrite1(k1_xboole_0,A,B)
=> A = B ) ).
fof(t29_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r2_rewrite1(A,B,C)
=> ( r2_hidden(B,k3_relat_1(A))
| B = C ) ) ) ).
fof(t30_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r2_hidden(k4_tarski(B,C),A)
=> r2_rewrite1(A,B,C) ) ) ).
fof(t31_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C,D] :
( ( r2_rewrite1(A,B,C)
& r2_rewrite1(A,C,D) )
=> r2_rewrite1(A,B,D) ) ) ).
fof(t32_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r2_rewrite1(A,B,C)
=> r2_rewrite1(A,C,B) ) ) ).
fof(t33_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r2_rewrite1(A,B,C)
=> ( B = C
| ( r2_hidden(B,k3_relat_1(A))
& r2_hidden(C,k3_relat_1(A)) ) ) ) ) ).
fof(d5_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r3_rewrite1(A,B)
<=> ! [C] : ~ r2_hidden(k4_tarski(B,C),A) ) ) ).
fof(t34_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r3_rewrite1(A,B)
& r1_rewrite1(A,B,C) )
=> B = C ) ) ).
fof(t35_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ~ r2_hidden(B,k3_relat_1(A))
=> r3_rewrite1(A,B) ) ) ).
fof(d6_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r4_rewrite1(A,B,C)
<=> ( r3_rewrite1(A,C)
& r1_rewrite1(A,B,C) ) ) ) ).
fof(d7_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r5_rewrite1(A,B,C)
<=> ? [D] :
( r1_rewrite1(A,B,D)
& r1_rewrite1(A,C,D) ) ) ) ).
fof(d8_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r6_rewrite1(A,B,C)
<=> ? [D] :
( r1_rewrite1(A,D,B)
& r1_rewrite1(A,D,C) ) ) ) ).
fof(d9_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r7_rewrite1(A,B,C)
<=> ? [D] :
( ( r2_hidden(k4_tarski(B,D),A)
| B = D )
& ( r2_hidden(k4_tarski(C,D),A)
| C = D ) ) ) ) ).
fof(d10_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r8_rewrite1(A,B,C)
<=> ? [D] :
( ( r2_hidden(k4_tarski(D,B),A)
| B = D )
& ( r2_hidden(k4_tarski(D,C),A)
| C = D ) ) ) ) ).
fof(t36_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ~ r2_hidden(B,k3_relat_1(A))
=> r4_rewrite1(A,B,B) ) ) ).
fof(t37_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r1_rewrite1(A,B,C)
=> ( r5_rewrite1(A,B,C)
& r6_rewrite1(A,B,C)
& r5_rewrite1(A,C,B)
& r6_rewrite1(A,C,B) ) ) ) ).
fof(t38_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( ( r5_rewrite1(A,B,C)
| r6_rewrite1(A,B,C) )
=> r2_rewrite1(A,B,C) ) ) ).
fof(t39_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r5_rewrite1(A,B,B)
& r6_rewrite1(A,B,B) ) ) ).
fof(t40_rewrite1,axiom,
! [A,B] :
( ( r5_rewrite1(k1_xboole_0,A,B)
| r6_rewrite1(k1_xboole_0,A,B) )
=> A = B ) ).
fof(t41_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r5_rewrite1(A,B,C)
=> r5_rewrite1(A,C,B) ) ) ).
fof(t42_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r6_rewrite1(A,B,C)
=> r6_rewrite1(A,C,B) ) ) ).
fof(t43_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C,D] :
( ( ( r1_rewrite1(A,B,C)
& r5_rewrite1(A,C,D) )
| ( r5_rewrite1(A,B,C)
& r1_rewrite1(A,D,C) ) )
=> r5_rewrite1(A,B,D) ) ) ).
fof(t44_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C,D] :
( ( ( r1_rewrite1(A,C,B)
& r6_rewrite1(A,C,D) )
| ( r6_rewrite1(A,B,C)
& r1_rewrite1(A,C,D) ) )
=> r6_rewrite1(A,B,D) ) ) ).
fof(t45_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r7_rewrite1(A,B,C)
=> r5_rewrite1(A,B,C) ) ) ).
fof(t46_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r8_rewrite1(A,B,C)
=> r6_rewrite1(A,B,C) ) ) ).
fof(d11_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r9_rewrite1(A,B)
<=> ? [C] : r4_rewrite1(A,B,C) ) ) ).
fof(t47_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ~ r2_hidden(B,k3_relat_1(A))
=> r9_rewrite1(A,B) ) ) ).
fof(d12_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( r9_rewrite1(A,B)
& ! [C,D] :
( ( r4_rewrite1(A,B,C)
& r4_rewrite1(A,B,D) )
=> C = D ) )
=> ! [C] :
( C = k2_rewrite1(A,B)
<=> r4_rewrite1(A,B,C) ) ) ) ).
fof(d13_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_rewrite1(A)
<=> v1_wellord1(k4_relat_1(A)) ) ) ).
fof(d14_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_rewrite1(A)
<=> ! [B] :
( r2_hidden(B,k3_relat_1(A))
=> r9_rewrite1(A,B) ) ) ) ).
fof(d15_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v3_rewrite1(A)
<=> ! [B] :
( m1_pboole(B,k5_numbers)
=> ~ ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r2_hidden(k4_tarski(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))),A) ) ) ) ) ).
fof(d16_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v1_rewrite1(A)
<=> ! [B] :
~ ( r1_tarski(B,k3_relat_1(A))
& B != k1_xboole_0
& ! [C] :
~ ( r2_hidden(C,B)
& ! [D] :
~ ( r2_hidden(D,B)
& C != D
& r2_hidden(k4_tarski(C,D),A) ) ) ) ) ) ).
fof(t48_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_rewrite1(A) )
=> ! [B] :
( v1_relat_1(B)
=> ( r1_tarski(B,A)
=> v1_rewrite1(B) ) ) ) ).
fof(d17_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r10_rewrite1(A,B)
<=> ! [C,D,E] :
~ ( r2_hidden(k4_tarski(C,D),A)
& r2_hidden(k4_tarski(C,E),B)
& ! [F] :
~ ( r1_rewrite1(B,D,F)
& r1_rewrite1(A,E,F) ) ) ) ) ) ).
fof(d18_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r11_rewrite1(A,B)
<=> ! [C,D,E] :
~ ( r1_rewrite1(A,C,D)
& r1_rewrite1(B,C,E)
& ! [F] :
~ ( r1_rewrite1(B,D,F)
& r1_rewrite1(A,E,F) ) ) ) ) ) ).
fof(t49_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r11_rewrite1(A,B)
=> r10_rewrite1(A,B) ) ) ) ).
fof(d19_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v4_rewrite1(A)
<=> ! [B,C] :
( ( r3_rewrite1(A,B)
& r3_rewrite1(A,C)
& r2_rewrite1(A,B,C) )
=> B = C ) ) ) ).
fof(d20_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v5_rewrite1(A)
<=> ! [B,C] :
( ( r3_rewrite1(A,B)
& r2_rewrite1(A,B,C) )
=> r1_rewrite1(A,C,B) ) ) ) ).
fof(d21_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v6_rewrite1(A)
<=> ! [B,C,D] :
( ( r2_hidden(k4_tarski(B,C),A)
& r2_hidden(k4_tarski(B,D),A) )
=> r7_rewrite1(A,C,D) ) ) ) ).
fof(d22_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_rewrite1(A)
<=> ! [B,C] :
( r6_rewrite1(A,B,C)
=> r5_rewrite1(A,B,C) ) ) ) ).
fof(d23_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v8_rewrite1(A)
<=> ! [B,C] :
( r2_rewrite1(A,B,C)
=> r5_rewrite1(A,B,C) ) ) ) ).
fof(d24_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v9_rewrite1(A)
<=> ! [B,C,D] :
( ( r2_hidden(k4_tarski(B,C),A)
& r2_hidden(k4_tarski(B,D),A) )
=> r5_rewrite1(A,C,D) ) ) ) ).
fof(t50_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v6_rewrite1(A)
=> ! [B,C,D] :
( ( r1_rewrite1(A,B,C)
& r2_hidden(k4_tarski(B,D),A) )
=> r5_rewrite1(A,C,D) ) ) ) ).
fof(t51_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_rewrite1(A)
<=> r11_rewrite1(A,A) ) ) ).
fof(t52_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_rewrite1(A)
<=> ! [B,C,D] :
( ( r1_rewrite1(A,B,C)
& r2_hidden(k4_tarski(B,D),A) )
=> r5_rewrite1(A,C,D) ) ) ) ).
fof(t53_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v9_rewrite1(A)
<=> r10_rewrite1(A,A) ) ) ).
fof(t54_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v4_rewrite1(A) )
=> ! [B,C,D] :
( ( r4_rewrite1(A,B,C)
& r4_rewrite1(A,B,D) )
=> C = D ) ) ).
fof(t55_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_rewrite1(A)
& v4_rewrite1(A) )
=> ! [B] : r4_rewrite1(A,B,k2_rewrite1(A,B)) ) ).
fof(t56_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_rewrite1(A)
& v4_rewrite1(A) )
=> ! [B,C] :
( r2_rewrite1(A,B,C)
=> k2_rewrite1(A,B) = k2_rewrite1(A,C) ) ) ).
fof(d25_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v10_rewrite1(A)
<=> ( v7_rewrite1(A)
& v3_rewrite1(A) ) ) ) ).
fof(t57_rewrite1,axiom,
! [A] :
( ( v1_relat_1(A)
& v8_rewrite1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v8_rewrite1(B) )
=> ( r11_rewrite1(A,B)
=> v8_rewrite1(k2_xboole_0(A,B)) ) ) ) ).
fof(t58_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_rewrite1(A)
<=> v9_rewrite1(k17_finseq_1(A)) ) ) ).
fof(t59_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v7_rewrite1(A)
<=> v6_rewrite1(k17_finseq_1(A)) ) ) ).
fof(d26_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r12_rewrite1(A,B)
<=> ! [C,D] :
( r2_rewrite1(A,C,D)
<=> r2_rewrite1(B,C,D) ) ) ) ) ).
fof(d27_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r13_rewrite1(A,B,C)
<=> ( r8_rewrite1(A,B,C)
& ~ r5_rewrite1(A,B,C) ) ) ) ).
fof(t60_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B,C] :
( r13_rewrite1(A,B,C)
=> r2_rewrite1(A,B,C) ) ) ).
fof(t61_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ( ! [B,C] : ~ r13_rewrite1(A,B,C)
=> v9_rewrite1(A) ) ) ).
fof(t62_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
=> r13_rewrite1(A,C,D) )
=> r12_rewrite1(A,k2_xboole_0(A,B)) ) ) ) ).
fof(t63_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ? [B] :
( v1_relat_1(B)
& v10_rewrite1(B)
& r1_tarski(k3_relat_1(B),k3_relat_1(A))
& ! [C,D] :
( r2_rewrite1(A,C,D)
<=> r5_rewrite1(B,C,D) ) ) ) ).
fof(d28_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v10_rewrite1(B) )
=> ( m2_rewrite1(B,A)
<=> ! [C,D] :
( r2_rewrite1(A,C,D)
<=> r5_rewrite1(B,C,D) ) ) ) ) ).
fof(t64_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m2_rewrite1(B,A)
=> r12_rewrite1(A,B) ) ) ).
fof(t65_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v10_rewrite1(B) )
=> ( r12_rewrite1(A,B)
=> m2_rewrite1(B,A) ) ) ) ).
fof(t66_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m2_rewrite1(B,A)
=> ! [C,D] :
( r2_rewrite1(A,C,D)
<=> k2_rewrite1(B,C) = k2_rewrite1(B,D) ) ) ) ).
fof(s1_rewrite1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r2_hidden(A,k4_finseq_1(f1_s1_rewrite1))
& r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(f1_s1_rewrite1)) )
=> p1_s1_rewrite1(k1_funct_1(f1_s1_rewrite1,A),k1_funct_1(f1_s1_rewrite1,k1_nat_1(A,np__1))) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r2_hidden(A,k4_finseq_1(f2_s1_rewrite1))
& r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(f2_s1_rewrite1)) )
=> p1_s1_rewrite1(k1_funct_1(f2_s1_rewrite1,A),k1_funct_1(f2_s1_rewrite1,k1_nat_1(A,np__1))) ) )
& ~ r1_xreal_0(k3_finseq_1(f1_s1_rewrite1),np__0)
& ~ r1_xreal_0(k3_finseq_1(f2_s1_rewrite1),np__0)
& k1_funct_1(f1_s1_rewrite1,k3_finseq_1(f1_s1_rewrite1)) = k1_funct_1(f2_s1_rewrite1,np__1) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r2_hidden(A,k4_finseq_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1)))
& r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1))) )
=> ! [B,C] :
( ( B = k1_funct_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1),A)
& C = k1_funct_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1),k1_nat_1(A,np__1)) )
=> p1_s1_rewrite1(B,C) ) ) ) ) ).
fof(s2_rewrite1,axiom,
( ( v1_rewrite1(f1_s2_rewrite1)
& ! [A] :
( ! [B] :
( r2_hidden(k4_tarski(A,B),f1_s2_rewrite1)
=> ( A = B
| p1_s2_rewrite1(B) ) )
=> p1_s2_rewrite1(A) ) )
=> ! [A] :
( r2_hidden(A,k3_relat_1(f1_s2_rewrite1))
=> p1_s2_rewrite1(A) ) ) ).
fof(dt_m1_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m1_rewrite1(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) ) ) ) ).
fof(existence_m1_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ? [B] : m1_rewrite1(B,A) ) ).
fof(dt_m2_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( m2_rewrite1(B,A)
=> ( v1_relat_1(B)
& v10_rewrite1(B) ) ) ) ).
fof(existence_m2_rewrite1,axiom,
! [A] :
( v1_relat_1(A)
=> ? [B] : m2_rewrite1(B,A) ) ).
fof(symmetry_r10_rewrite1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> ( r10_rewrite1(A,B)
=> r10_rewrite1(B,A) ) ) ).
fof(symmetry_r11_rewrite1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> ( r11_rewrite1(A,B)
=> r11_rewrite1(B,A) ) ) ).
fof(symmetry_r12_rewrite1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> ( r12_rewrite1(A,B)
=> r12_rewrite1(B,A) ) ) ).
fof(dt_k1_rewrite1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_relat_1(k1_rewrite1(A,B))
& v1_funct_1(k1_rewrite1(A,B))
& v1_finseq_1(k1_rewrite1(A,B)) ) ) ).
fof(dt_k2_rewrite1,axiom,
$true ).
%------------------------------------------------------------------------------