SET007 Axioms: SET007+753.ax
%------------------------------------------------------------------------------
% File : SET007+753 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Processes in Petri nets
% 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 : pnproc_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 147 ( 7 unt; 0 def)
% Number of atoms : 763 ( 106 equ)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 645 ( 29 ~; 0 |; 255 &)
% ( 18 <=>; 343 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 67 ( 67 usr; 7 con; 0-5 aty)
% Number of variables : 475 ( 459 !; 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_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v2_finseq_1(A) ) ) ).
fof(cc2_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v2_finseq_1(A) ) ) ).
fof(fc1_pnproc_1,axiom,
! [A,B,C,D] :
( ( m3_pnproc_1(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B))) )
=> ~ v1_xboole_0(k12_pnproc_1(A,B,C,D)) ) ).
fof(fc2_pnproc_1,axiom,
! [A,B,C,D] :
( ( m3_pnproc_1(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B))) )
=> ~ v1_xboole_0(k13_pnproc_1(A,B,C,D)) ) ).
fof(t1_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ r1_xreal_0(A,np__0)
=> ( v1_relat_1(k1_tarski(k4_tarski(A,B)))
& v1_funct_1(k1_tarski(k4_tarski(A,B)))
& v2_finseq_1(k1_tarski(k4_tarski(A,B))) ) ) ) ).
fof(t2_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A) )
=> ( A = k1_xboole_0
<=> k15_finseq_1(A) = k1_xboole_0 ) ) ).
fof(t3_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ( C = k1_tarski(k4_tarski(A,B))
=> k15_finseq_1(C) = k9_finseq_1(B) ) ) ) ).
fof(t4_pnproc_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> ~ ( k15_finseq_1(B) = k9_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> B != k1_tarski(k4_tarski(C,A)) ) ) ) ).
fof(t5_pnproc_1,axiom,
! [A,B,C,D] :
~ ( v1_relat_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D)))
& v1_funct_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D)))
& v1_finseq_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D)))
& ~ ( A = np__1
& C = np__1
& B = D )
& ~ ( A = np__1
& C = np__2 )
& ~ ( A = np__2
& C = np__1 ) ) ).
fof(t6_pnproc_1,axiom,
! [A,B] : k10_finseq_1(A,B) = k2_tarski(k4_tarski(np__1,A),k4_tarski(np__2,B)) ).
fof(t7_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A) )
=> k1_card_1(A) = k3_finseq_1(k15_finseq_1(A)) ) ).
fof(t8_pnproc_1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> r1_xboole_0(A,B) ) ) ) ).
fof(t9_pnproc_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ! [D] :
( v1_relat_1(D)
=> ( r1_xboole_0(A,B)
=> r1_xboole_0(k7_relat_1(C,A),k7_relat_1(D,B)) ) ) ) ).
fof(t10_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(A,C)
& r1_tarski(B,C)
& r1_xboole_0(A,B) )
=> r1_xboole_0(k1_relat_1(A),k1_relat_1(B)) ) ) ) ) ).
fof(t11_pnproc_1,axiom,
! [A,B] :
( v1_relat_1(B)
=> r1_tarski(k8_relat_1(A,B),k7_relat_1(B,k10_relat_1(B,A))) ) ).
fof(t12_pnproc_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k8_relat_1(A,B) = k7_relat_1(B,k10_relat_1(B,A)) ) ).
fof(d1_pnproc_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( m1_pnproc_1(B,A)
<=> ( k1_relat_1(B) = A
& r1_tarski(k2_relat_1(B),k5_numbers) ) ) ) ).
fof(d2_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ( r1_pnproc_1(A,B,C)
<=> ! [D] :
( r2_hidden(D,A)
=> k1_pnproc_1(A,B,D) = k1_pnproc_1(A,C,D) ) ) ) ) ).
fof(d3_pnproc_1,axiom,
! [A] : k3_pnproc_1(A) = k2_funcop_1(A,np__0) ).
fof(d4_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ( r2_pnproc_1(A,B,C)
<=> ! [D] :
( r2_hidden(D,A)
=> r1_xreal_0(k1_pnproc_1(A,B,D),k1_pnproc_1(A,C,D)) ) ) ) ) ).
fof(t13_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> r2_pnproc_1(A,k3_pnproc_1(A),B) ) ).
fof(t14_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( ( r2_pnproc_1(A,B,C)
& r2_pnproc_1(A,C,D) )
=> r2_pnproc_1(A,B,D) ) ) ) ) ).
fof(d5_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( D = k4_pnproc_1(A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> k1_pnproc_1(A,D,E) = k1_nat_1(k1_pnproc_1(A,B,E),k1_pnproc_1(A,C,E)) ) ) ) ) ) ).
fof(t15_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> r1_pnproc_1(A,k4_pnproc_1(A,B,k3_pnproc_1(A)),B) ) ).
fof(d6_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ( r2_pnproc_1(A,C,B)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( D = k5_pnproc_1(A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> k1_pnproc_1(A,D,E) = k6_xcmplx_0(k1_pnproc_1(A,B,E),k1_pnproc_1(A,C,E)) ) ) ) ) ) ) ).
fof(t16_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> r2_pnproc_1(A,B,k4_pnproc_1(A,B,C)) ) ) ).
fof(t17_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> r1_pnproc_1(A,k5_pnproc_1(A,B,k3_pnproc_1(A)),B) ) ).
fof(t18_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( ( r2_pnproc_1(A,B,C)
& r2_pnproc_1(A,C,D) )
=> r2_pnproc_1(A,k5_pnproc_1(A,D,C),k5_pnproc_1(A,D,B)) ) ) ) ) ).
fof(t19_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> r1_pnproc_1(A,k5_pnproc_1(A,k4_pnproc_1(A,B,C),C),B) ) ) ).
fof(t20_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( ( r2_pnproc_1(A,B,C)
& r2_pnproc_1(A,C,D) )
=> r2_pnproc_1(A,k5_pnproc_1(A,C,B),k5_pnproc_1(A,D,B)) ) ) ) ) ).
fof(t21_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( r2_pnproc_1(A,B,C)
=> r1_pnproc_1(A,k5_pnproc_1(A,k4_pnproc_1(A,C,D),B),k4_pnproc_1(A,k5_pnproc_1(A,C,B),D)) ) ) ) ) ).
fof(t22_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ( ( r2_pnproc_1(A,B,C)
& r2_pnproc_1(A,C,B) )
=> r1_pnproc_1(A,B,C) ) ) ) ).
fof(t23_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> r1_pnproc_1(A,k4_pnproc_1(A,k4_pnproc_1(A,B,C),D),k4_pnproc_1(A,B,k4_pnproc_1(A,C,D))) ) ) ) ).
fof(t24_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ! [E] :
( m1_pnproc_1(E,A)
=> ( ( r2_pnproc_1(A,B,C)
& r2_pnproc_1(A,D,E) )
=> r2_pnproc_1(A,k4_pnproc_1(A,B,D),k4_pnproc_1(A,C,E)) ) ) ) ) ) ).
fof(t25_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ( r2_pnproc_1(A,B,C)
=> r2_pnproc_1(A,k5_pnproc_1(A,C,B),C) ) ) ) ).
fof(t26_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ! [E] :
( m1_pnproc_1(E,A)
=> ( ( r2_pnproc_1(A,B,C)
& r2_pnproc_1(A,D,E)
& r2_pnproc_1(A,E,B) )
=> r2_pnproc_1(A,k5_pnproc_1(A,B,E),k5_pnproc_1(A,C,D)) ) ) ) ) ) ).
fof(t27_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ( r2_pnproc_1(A,B,C)
=> r1_pnproc_1(A,C,k4_pnproc_1(A,k5_pnproc_1(A,C,B),B)) ) ) ) ).
fof(t28_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> r1_pnproc_1(A,k5_pnproc_1(A,k4_pnproc_1(A,B,C),B),C) ) ) ).
fof(t29_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( r2_pnproc_1(A,k4_pnproc_1(A,B,C),D)
=> r1_pnproc_1(A,k5_pnproc_1(A,k5_pnproc_1(A,D,B),C),k5_pnproc_1(A,D,k4_pnproc_1(A,B,C))) ) ) ) ) ).
fof(t30_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m1_pnproc_1(C,A)
=> ! [D] :
( m1_pnproc_1(D,A)
=> ( ( r2_pnproc_1(A,B,C)
& r2_pnproc_1(A,C,D) )
=> r1_pnproc_1(A,k5_pnproc_1(A,D,k5_pnproc_1(A,C,B)),k4_pnproc_1(A,k5_pnproc_1(A,D,C),B)) ) ) ) ) ).
fof(t31_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> r2_hidden(B,k1_funct_2(A,k5_numbers)) ) ).
fof(t32_pnproc_1,axiom,
! [A,B] :
( r2_hidden(A,k1_funct_2(B,k5_numbers))
=> m1_pnproc_1(A,B) ) ).
fof(d7_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
<=> ? [C] :
( m1_pnproc_1(C,A)
& ? [D] :
( m1_pnproc_1(D,A)
& B = k4_tarski(C,D) ) ) ) ).
fof(d8_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m2_pnproc_1(C,A)
=> ( ( r2_pnproc_1(A,k6_pnproc_1(A,C),B)
=> k8_pnproc_1(A,B,C) = k4_pnproc_1(A,k5_pnproc_1(A,B,k6_pnproc_1(A,C)),k7_pnproc_1(A,C)) )
& ( ~ r2_pnproc_1(A,k6_pnproc_1(A,C),B)
=> k8_pnproc_1(A,B,C) = B ) ) ) ) ).
fof(t33_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m2_pnproc_1(C,A)
=> ! [D] :
( m2_pnproc_1(D,A)
=> ( r2_pnproc_1(A,k4_pnproc_1(A,k6_pnproc_1(A,C),k6_pnproc_1(A,D)),B)
=> r1_pnproc_1(A,k8_pnproc_1(A,k8_pnproc_1(A,B,C),D),k4_pnproc_1(A,k4_pnproc_1(A,k5_pnproc_1(A,k5_pnproc_1(A,B,k6_pnproc_1(A,C)),k6_pnproc_1(A,D)),k7_pnproc_1(A,C)),k7_pnproc_1(A,D))) ) ) ) ) ).
fof(d9_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k9_pnproc_1(A,B)
<=> ( k1_relat_1(C) = k1_funct_2(A,k5_numbers)
& ! [D] :
( m1_pnproc_1(D,A)
=> k1_funct_1(C,D) = k8_pnproc_1(A,D,B) ) ) ) ) ) ).
fof(t34_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
=> r1_tarski(k2_relat_1(k9_pnproc_1(A,B)),k1_funct_2(A,k5_numbers)) ) ).
fof(t35_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ! [C] :
( m2_pnproc_1(C,A)
=> ! [D] :
( m2_pnproc_1(D,A)
=> k8_pnproc_1(A,k8_pnproc_1(A,B,D),C) = k1_funct_1(k5_relat_1(k9_pnproc_1(A,D),k9_pnproc_1(A,C)),B) ) ) ) ).
fof(d10_pnproc_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( m3_pnproc_1(B,A)
<=> ! [C] :
( r2_hidden(C,B)
=> m2_pnproc_1(C,A) ) ) ) ).
fof(d11_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m2_finseq_2(C,B,k3_finseq_2(B))
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( D = k10_pnproc_1(A,B,C)
<=> ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E)
& v1_funcop_1(E)
& D = k4_funct_7(k1_funct_2(A,k5_numbers),E)
& k3_finseq_1(E) = k3_finseq_1(C)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k4_finseq_1(C))
=> k1_funct_1(E,F) = k9_pnproc_1(A,k4_finseq_4(k5_numbers,B,C,F)) ) ) ) ) ) ) ) ).
fof(t36_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> k10_pnproc_1(A,B,k2_lang1(B)) = k6_relat_1(k1_funct_2(A,k5_numbers)) ) ).
fof(t37_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> k10_pnproc_1(A,B,k3_lang1(B,C)) = k9_pnproc_1(A,C) ) ) ).
fof(t38_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> k5_relat_1(k6_relat_1(k1_funct_2(A,k5_numbers)),k9_pnproc_1(A,C)) = k9_pnproc_1(A,C) ) ) ).
fof(t39_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> ! [D] :
( m4_pnproc_1(D,A,B)
=> k10_pnproc_1(A,B,k4_lang1(B,C,D)) = k5_relat_1(k9_pnproc_1(A,C),k9_pnproc_1(A,D)) ) ) ) ).
fof(t40_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m2_finseq_2(C,B,k3_finseq_2(B))
=> ( k1_relat_1(k10_pnproc_1(A,B,C)) = k1_funct_2(A,k5_numbers)
& r1_tarski(k2_relat_1(k10_pnproc_1(A,B,C)),k1_funct_2(A,k5_numbers)) ) ) ) ).
fof(t41_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m2_finseq_2(C,B,k3_finseq_2(B))
=> ! [D] :
( m2_finseq_2(D,B,k3_finseq_2(B))
=> k10_pnproc_1(A,B,k1_lang1(B,C,D)) = k5_relat_1(k10_pnproc_1(A,B,C),k10_pnproc_1(A,B,D)) ) ) ) ).
fof(t42_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> ! [D] :
( m2_finseq_2(D,B,k3_finseq_2(B))
=> k10_pnproc_1(A,B,k1_lang1(B,D,k3_lang1(B,C))) = k5_relat_1(k10_pnproc_1(A,B,D),k9_pnproc_1(A,C)) ) ) ) ).
fof(d12_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m2_finseq_2(C,B,k3_finseq_2(B))
=> ! [D] :
( m1_pnproc_1(D,A)
=> k11_pnproc_1(A,B,C,D) = k1_funct_1(k10_pnproc_1(A,B,C),D) ) ) ) ).
fof(t43_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> k12_pnproc_1(A,B,k4_subset_1(k3_finseq_2(B),C,D),E) = k4_subset_1(k3_finseq_2(B),k12_pnproc_1(A,B,C,E),k12_pnproc_1(A,B,D,E)) ) ) ) ) ).
fof(t44_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> k12_pnproc_1(A,B,C,k4_subset_1(k3_finseq_2(B),D,E)) = k4_subset_1(k3_finseq_2(B),k12_pnproc_1(A,B,C,D),k12_pnproc_1(A,B,C,E)) ) ) ) ) ).
fof(t45_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m2_finseq_2(C,B,k3_finseq_2(B))
=> ! [D] :
( m2_finseq_2(D,B,k3_finseq_2(B))
=> k12_pnproc_1(A,B,k6_domain_1(k3_finseq_2(B),C),k6_domain_1(k3_finseq_2(B),D)) = k6_domain_1(k3_finseq_2(B),k1_lang1(B,C,D)) ) ) ) ).
fof(t46_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m2_finseq_2(C,B,k3_finseq_2(B))
=> ! [D] :
( m2_finseq_2(D,B,k3_finseq_2(B))
=> ! [E] :
( m2_finseq_2(E,B,k3_finseq_2(B))
=> k12_pnproc_1(A,B,k7_domain_1(k3_finseq_2(B),C,D),k6_domain_1(k3_finseq_2(B),E)) = k7_domain_1(k3_finseq_2(B),k1_lang1(B,C,E),k1_lang1(B,D,E)) ) ) ) ) ).
fof(t47_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m2_finseq_2(C,B,k3_finseq_2(B))
=> ! [D] :
( m2_finseq_2(D,B,k3_finseq_2(B))
=> ! [E] :
( m2_finseq_2(E,B,k3_finseq_2(B))
=> k12_pnproc_1(A,B,k6_domain_1(k3_finseq_2(B),C),k7_domain_1(k3_finseq_2(B),D,E)) = k7_domain_1(k3_finseq_2(B),k1_lang1(B,C,D),k1_lang1(B,C,E)) ) ) ) ) ).
fof(t48_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> k13_pnproc_1(A,B,k4_subset_1(k3_finseq_2(B),C,D),E) = k4_subset_1(k3_finseq_2(B),k13_pnproc_1(A,B,C,E),k13_pnproc_1(A,B,D,E)) ) ) ) ) ).
fof(t49_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> ! [D] :
( m4_pnproc_1(D,A,B)
=> k13_pnproc_1(A,B,k6_domain_1(k3_finseq_2(B),k3_lang1(B,C)),k6_domain_1(k3_finseq_2(B),k3_lang1(B,D))) = k7_domain_1(k3_finseq_2(B),k4_lang1(B,C,D),k4_lang1(B,D,C)) ) ) ) ).
fof(t50_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> ! [D] :
( m4_pnproc_1(D,A,B)
=> ! [E] :
( m4_pnproc_1(E,A,B)
=> k13_pnproc_1(A,B,k7_domain_1(k3_finseq_2(B),k3_lang1(B,C),k3_lang1(B,D)),k6_domain_1(k3_finseq_2(B),k3_lang1(B,E))) = k9_domain_1(k3_finseq_2(B),k4_lang1(B,C,E),k4_lang1(B,D,E),k4_lang1(B,E,C),k4_lang1(B,E,D)) ) ) ) ) ).
fof(t51_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> k12_pnproc_1(A,B,k12_pnproc_1(A,B,C,D),E) = k12_pnproc_1(A,B,C,k12_pnproc_1(A,B,D,E)) ) ) ) ) ).
fof(t52_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A) )
=> k14_pnproc_1(A,np__0) = A ) ).
fof(t53_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> k14_pnproc_1(C,k1_nat_1(A,B)) = k14_pnproc_1(k14_pnproc_1(C,B),A) ) ) ) ).
fof(t55_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> ( B = k1_xboole_0
<=> k14_pnproc_1(B,A) = k1_xboole_0 ) ) ) ).
fof(t56_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C)
& k1_relat_1(C) = k1_relat_1(B)
& k2_relat_1(C) = k1_relat_1(k14_pnproc_1(B,A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k1_nat_1(A,D) ) )
& v2_funct_1(C) ) ) ) ).
fof(t57_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> k1_card_1(B) = k1_card_1(k14_pnproc_1(B,A)) ) ) ).
fof(t58_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k4_finseq_1(B) = k4_finseq_1(k15_finseq_1(k14_pnproc_1(B,A))) ) ) ).
fof(t59_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r2_hidden(A,k4_finseq_1(C))
=> k1_funct_1(k14_finseq_1(k1_relat_1(k14_pnproc_1(C,B))),A) = k1_nat_1(B,A) ) ) ) ) ).
fof(t60_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r2_hidden(A,k4_finseq_1(C))
=> k1_funct_1(k15_finseq_1(k14_pnproc_1(C,B)),A) = k1_funct_1(C,A) ) ) ) ) ).
fof(t61_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k15_finseq_1(k14_pnproc_1(B,A)) = B ) ) ).
fof(t62_pnproc_1,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) )
=> k1_relat_1(k2_xboole_0(A,k14_pnproc_1(B,k3_finseq_1(A)))) = k2_finseq_1(k1_nat_1(k3_finseq_1(A),k3_finseq_1(B))) ) ) ).
fof(t63_pnproc_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)
& v2_finseq_1(C) )
=> ( r1_xreal_0(k3_finseq_1(B),A)
=> r1_xboole_0(k4_finseq_1(B),k1_relat_1(k14_pnproc_1(C,A))) ) ) ) ) ).
fof(t64_pnproc_1,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) )
=> k7_finseq_1(A,B) = k2_xboole_0(A,k14_pnproc_1(B,k3_finseq_1(A))) ) ) ).
fof(t65_pnproc_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)
& v2_finseq_1(C) )
=> ( r1_xreal_0(k3_finseq_1(B),A)
=> r1_xboole_0(B,k14_pnproc_1(C,A)) ) ) ) ) ).
fof(t66_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> k13_pnproc_1(A,B,k13_pnproc_1(A,B,C,D),E) = k13_pnproc_1(A,B,C,k13_pnproc_1(A,B,D,E)) ) ) ) ) ).
fof(t67_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> r1_tarski(k12_pnproc_1(A,B,C,D),k13_pnproc_1(A,B,C,D)) ) ) ) ).
fof(t68_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k3_finseq_2(B)))
=> ( ( r1_tarski(C,D)
& r1_tarski(E,F) )
=> r1_tarski(k12_pnproc_1(A,B,C,E),k12_pnproc_1(A,B,D,F)) ) ) ) ) ) ) ).
fof(t69_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k3_finseq_2(B)))
=> ( ( r1_tarski(C,D)
& r1_tarski(E,F) )
=> r1_tarski(k13_pnproc_1(A,B,C,E),k13_pnproc_1(A,B,D,F)) ) ) ) ) ) ) ).
fof(t70_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ( r1_tarski(C,B)
=> r1_tarski(k14_pnproc_1(C,A),k14_pnproc_1(B,A)) ) ) ) ) ).
fof(t71_pnproc_1,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) )
=> r1_tarski(k14_pnproc_1(B,k3_finseq_1(A)),k7_finseq_1(A,B)) ) ) ).
fof(t72_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ( r1_xboole_0(k1_relat_1(B),k1_relat_1(C))
=> r1_xboole_0(k1_relat_1(k14_pnproc_1(B,A)),k1_relat_1(k14_pnproc_1(C,A))) ) ) ) ) ).
fof(t73_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v2_finseq_1(D) )
=> ( ( B = k2_xboole_0(C,D)
& r1_xboole_0(C,D) )
=> k2_xboole_0(k14_pnproc_1(C,A),k14_pnproc_1(D,A)) = k14_pnproc_1(B,A) ) ) ) ) ) ).
fof(t74_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> k4_finseq_1(k15_finseq_1(B)) = k4_finseq_1(k15_finseq_1(k14_pnproc_1(B,A))) ) ) ).
fof(t75_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ~ ( r2_hidden(A,k4_finseq_1(k15_finseq_1(C)))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( D = k1_funct_1(k14_finseq_1(k1_relat_1(C)),A)
& k1_funct_1(k14_finseq_1(k1_relat_1(k14_pnproc_1(C,B))),A) = k1_nat_1(B,D) ) ) ) ) ) ) ).
fof(t76_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ( r2_hidden(A,k4_finseq_1(k15_finseq_1(C)))
=> k1_funct_1(k15_finseq_1(k14_pnproc_1(C,B)),A) = k1_funct_1(k15_finseq_1(C),A) ) ) ) ) ).
fof(t77_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> k15_finseq_1(B) = k15_finseq_1(k14_pnproc_1(B,A)) ) ) ).
fof(t78_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ( r1_tarski(k1_relat_1(C),k2_finseq_1(A))
=> r1_tarski(k1_relat_1(k14_pnproc_1(C,B)),k2_finseq_1(k1_nat_1(B,A))) ) ) ) ) ).
fof(t79_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ~ ( r1_tarski(B,A)
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v2_finseq_1(D) )
=> D != k2_xboole_0(B,k14_pnproc_1(C,k3_finseq_1(A))) ) ) ) ) ) ).
fof(t80_pnproc_1,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)
& v2_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v2_finseq_1(D) )
=> ~ ( r1_tarski(C,A)
& r1_tarski(D,B)
& ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v2_finseq_1(E) )
=> ~ ( E = k2_xboole_0(C,k14_pnproc_1(D,k3_finseq_1(A)))
& k4_finseq_1(k15_finseq_1(E)) = k2_finseq_1(k1_nat_1(k3_finseq_1(k15_finseq_1(C)),k3_finseq_1(k15_finseq_1(D)))) ) ) ) ) ) ) ) ).
fof(t81_pnproc_1,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)
& v2_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v2_finseq_1(D) )
=> ~ ( r1_tarski(C,A)
& r1_tarski(D,B)
& ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v2_finseq_1(E) )
=> ~ ( E = k2_xboole_0(C,k14_pnproc_1(D,k3_finseq_1(A)))
& k4_finseq_1(k15_finseq_1(E)) = k2_finseq_1(k1_nat_1(k3_finseq_1(k15_finseq_1(C)),k3_finseq_1(k15_finseq_1(D))))
& k15_finseq_1(E) = k2_xboole_0(k15_finseq_1(C),k14_pnproc_1(k15_finseq_1(D),k3_finseq_1(k15_finseq_1(C)))) ) ) ) ) ) ) ) ).
fof(t82_pnproc_1,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)
& v2_finseq_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v2_finseq_1(D) )
=> ~ ( r1_tarski(C,A)
& r1_tarski(D,B)
& ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v2_finseq_1(E) )
=> ~ ( E = k2_xboole_0(C,k14_pnproc_1(D,k3_finseq_1(A)))
& k7_finseq_1(k15_finseq_1(C),k15_finseq_1(D)) = k15_finseq_1(E) ) ) ) ) ) ) ) ).
fof(t83_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k3_finseq_2(B)))
=> r1_tarski(k12_pnproc_1(A,B,k13_pnproc_1(A,B,C,D),k13_pnproc_1(A,B,E,F)),k13_pnproc_1(A,B,k12_pnproc_1(A,B,C,E),k12_pnproc_1(A,B,D,F))) ) ) ) ) ) ).
fof(d16_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> k15_pnproc_1(A,B) = k6_domain_1(k3_finseq_2(B),k2_lang1(B)) ) ).
fof(d17_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> k16_pnproc_1(A,B,C) = k6_domain_1(k3_finseq_2(B),k3_lang1(B,C)) ) ) ).
fof(t84_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> k12_pnproc_1(A,B,k15_pnproc_1(A,B),C) = C ) ) ).
fof(t85_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> k12_pnproc_1(A,B,C,k15_pnproc_1(A,B)) = C ) ) ).
fof(t86_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> k13_pnproc_1(A,B,k15_pnproc_1(A,B),C) = C ) ) ).
fof(s1_pnproc_1,axiom,
? [A] :
( m1_pnproc_1(A,f1_s1_pnproc_1)
& ! [B] :
( r2_hidden(B,f1_s1_pnproc_1)
=> k1_pnproc_1(f1_s1_pnproc_1,A,B) = f2_s1_pnproc_1(B) ) ) ).
fof(dt_m1_pnproc_1,axiom,
! [A,B] :
( m1_pnproc_1(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ).
fof(existence_m1_pnproc_1,axiom,
! [A] :
? [B] : m1_pnproc_1(B,A) ).
fof(dt_m2_pnproc_1,axiom,
$true ).
fof(existence_m2_pnproc_1,axiom,
! [A] :
? [B] : m2_pnproc_1(B,A) ).
fof(dt_m3_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ~ v1_xboole_0(B) ) ).
fof(existence_m3_pnproc_1,axiom,
! [A] :
? [B] : m3_pnproc_1(B,A) ).
fof(dt_m4_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
=> m2_pnproc_1(C,A) ) ) ).
fof(existence_m4_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ? [C] : m4_pnproc_1(C,A,B) ) ).
fof(redefinition_m4_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m4_pnproc_1(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(symmetry_r1_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m1_pnproc_1(C,A) )
=> ( r1_pnproc_1(A,B,C)
=> r1_pnproc_1(A,C,B) ) ) ).
fof(reflexivity_r1_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m1_pnproc_1(C,A) )
=> r1_pnproc_1(A,B,B) ) ).
fof(redefinition_r1_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m1_pnproc_1(C,A) )
=> ( r1_pnproc_1(A,B,C)
<=> B = C ) ) ).
fof(reflexivity_r2_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m1_pnproc_1(C,A) )
=> r2_pnproc_1(A,B,B) ) ).
fof(dt_k1_pnproc_1,axiom,
! [A,B,C] :
( m1_pnproc_1(B,A)
=> m2_subset_1(k1_pnproc_1(A,B,C),k1_numbers,k5_numbers) ) ).
fof(redefinition_k1_pnproc_1,axiom,
! [A,B,C] :
( m1_pnproc_1(B,A)
=> k1_pnproc_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k2_pnproc_1,axiom,
! [A,B,C] :
( m1_pnproc_1(B,A)
=> m2_subset_1(k2_pnproc_1(A,B,C),k1_numbers,k5_numbers) ) ).
fof(redefinition_k2_pnproc_1,axiom,
! [A,B,C] :
( m1_pnproc_1(B,A)
=> k2_pnproc_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k3_pnproc_1,axiom,
! [A] : m1_pnproc_1(k3_pnproc_1(A),A) ).
fof(dt_k4_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m1_pnproc_1(C,A) )
=> m1_pnproc_1(k4_pnproc_1(A,B,C),A) ) ).
fof(commutativity_k4_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m1_pnproc_1(C,A) )
=> k4_pnproc_1(A,B,C) = k4_pnproc_1(A,C,B) ) ).
fof(dt_k5_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m1_pnproc_1(C,A) )
=> m1_pnproc_1(k5_pnproc_1(A,B,C),A) ) ).
fof(dt_k6_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
=> m1_pnproc_1(k6_pnproc_1(A,B),A) ) ).
fof(redefinition_k6_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
=> k6_pnproc_1(A,B) = k1_mcart_1(B) ) ).
fof(dt_k7_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
=> m1_pnproc_1(k7_pnproc_1(A,B),A) ) ).
fof(redefinition_k7_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
=> k7_pnproc_1(A,B) = k2_mcart_1(B) ) ).
fof(dt_k8_pnproc_1,axiom,
! [A,B,C] :
( ( m1_pnproc_1(B,A)
& m2_pnproc_1(C,A) )
=> m1_pnproc_1(k8_pnproc_1(A,B,C),A) ) ).
fof(dt_k9_pnproc_1,axiom,
! [A,B] :
( m2_pnproc_1(B,A)
=> ( v1_relat_1(k9_pnproc_1(A,B))
& v1_funct_1(k9_pnproc_1(A,B)) ) ) ).
fof(dt_k10_pnproc_1,axiom,
! [A,B,C] :
( ( m3_pnproc_1(B,A)
& m1_subset_1(C,k3_finseq_2(B)) )
=> ( v1_relat_1(k10_pnproc_1(A,B,C))
& v1_funct_1(k10_pnproc_1(A,B,C)) ) ) ).
fof(dt_k11_pnproc_1,axiom,
! [A,B,C,D] :
( ( m3_pnproc_1(B,A)
& m1_subset_1(C,k3_finseq_2(B))
& m1_pnproc_1(D,A) )
=> m1_pnproc_1(k11_pnproc_1(A,B,C,D),A) ) ).
fof(dt_k12_pnproc_1,axiom,
! [A,B,C,D] :
( ( m3_pnproc_1(B,A)
& m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
& m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B))) )
=> m1_subset_1(k12_pnproc_1(A,B,C,D),k1_zfmisc_1(k3_finseq_2(B))) ) ).
fof(dt_k13_pnproc_1,axiom,
! [A,B,C,D] :
( ( m3_pnproc_1(B,A)
& m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
& m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B))) )
=> m1_subset_1(k13_pnproc_1(A,B,C,D),k1_zfmisc_1(k3_finseq_2(B))) ) ).
fof(commutativity_k13_pnproc_1,axiom,
! [A,B,C,D] :
( ( m3_pnproc_1(B,A)
& m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
& m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B))) )
=> k13_pnproc_1(A,B,C,D) = k13_pnproc_1(A,B,D,C) ) ).
fof(dt_k14_pnproc_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k14_pnproc_1(A,B))
& v1_funct_1(k14_pnproc_1(A,B))
& v2_finseq_1(k14_pnproc_1(A,B)) ) ) ).
fof(dt_k15_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ( ~ v1_xboole_0(k15_pnproc_1(A,B))
& m1_subset_1(k15_pnproc_1(A,B),k1_zfmisc_1(k3_finseq_2(B))) ) ) ).
fof(dt_k16_pnproc_1,axiom,
! [A,B,C] :
( ( m3_pnproc_1(B,A)
& m1_subset_1(C,B) )
=> ( ~ v1_xboole_0(k16_pnproc_1(A,B,C))
& m1_subset_1(k16_pnproc_1(A,B,C),k1_zfmisc_1(k3_finseq_2(B))) ) ) ).
fof(d13_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> k12_pnproc_1(A,B,C,D) = a_4_0_pnproc_1(A,B,C,D) ) ) ) ).
fof(d14_pnproc_1,axiom,
! [A,B] :
( m3_pnproc_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_finseq_2(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(B)))
=> k13_pnproc_1(A,B,C,D) = a_4_1_pnproc_1(A,B,C,D) ) ) ) ).
fof(d15_pnproc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_finseq_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v2_finseq_1(C) )
=> ( C = k14_pnproc_1(A,B)
<=> ( k1_relat_1(C) = a_2_0_pnproc_1(A,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k1_relat_1(A))
=> k1_funct_1(C,k1_nat_1(B,D)) = k1_funct_1(A,D) ) ) ) ) ) ) ) ).
fof(t54_pnproc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( B != k1_xboole_0
=> k1_relat_1(k14_pnproc_1(B,A)) = a_2_1_pnproc_1(A,B) ) ) ) ).
fof(fraenkel_a_4_0_pnproc_1,axiom,
! [A,B,C,D,E] :
( ( m3_pnproc_1(C,B)
& m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(C)))
& m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(C))) )
=> ( r2_hidden(A,a_4_0_pnproc_1(B,C,D,E))
<=> ? [F,G] :
( m2_finseq_2(F,C,k3_finseq_2(C))
& m2_finseq_2(G,C,k3_finseq_2(C))
& A = k1_lang1(C,F,G)
& r2_hidden(F,D)
& r2_hidden(G,E) ) ) ) ).
fof(fraenkel_a_4_1_pnproc_1,axiom,
! [A,B,C,D,E] :
( ( m3_pnproc_1(C,B)
& m1_subset_1(D,k1_zfmisc_1(k3_finseq_2(C)))
& m1_subset_1(E,k1_zfmisc_1(k3_finseq_2(C))) )
=> ( r2_hidden(A,a_4_1_pnproc_1(B,C,D,E))
<=> ? [F] :
( m2_finseq_2(F,C,k3_finseq_2(C))
& A = F
& ? [G] :
( v1_relat_1(G)
& v1_funct_1(G)
& v2_finseq_1(G)
& ? [H] :
( v1_relat_1(H)
& v1_funct_1(H)
& v2_finseq_1(H)
& F = k2_xboole_0(G,H)
& r1_xboole_0(G,H)
& r2_hidden(k15_finseq_1(G),D)
& r2_hidden(k15_finseq_1(H),E) ) ) ) ) ) ).
fof(fraenkel_a_2_0_pnproc_1,axiom,
! [A,B,C] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_finseq_1(B)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_0_pnproc_1(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k1_nat_1(C,D)
& r2_hidden(D,k1_relat_1(B)) ) ) ) ).
fof(fraenkel_a_2_1_pnproc_1,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r2_hidden(A,a_2_1_pnproc_1(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = D
& r1_xreal_0(k1_nat_1(B,np__1),D)
& r1_xreal_0(D,k1_nat_1(B,k3_finseq_1(C))) ) ) ) ).
%------------------------------------------------------------------------------