SET007 Axioms: SET007+172.ax
%------------------------------------------------------------------------------
% File : SET007+172 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Two Programs for SCM. Part I - Preliminaries
% 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 : pre_ff [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 32 ( 2 unt; 0 def)
% Number of atoms : 169 ( 45 equ)
% Maximal formula atoms : 21 ( 5 avg)
% Number of connectives : 168 ( 31 ~; 5 |; 59 &)
% ( 3 <=>; 70 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-3 aty)
% Number of functors : 31 ( 31 usr; 5 con; 0-5 aty)
% Number of variables : 77 ( 74 !; 3 ?)
% 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(d1_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k3_pre_ff(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
& m2_relset_1(C,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
& B = k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,A))
& k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,np__0) = k1_domain_1(k5_numbers,k5_numbers,np__0,np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,k1_nat_1(D,np__1)) = k1_domain_1(k5_numbers,k5_numbers,k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)),k1_nat_1(k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)),k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)))) ) ) ) ) ) ).
fof(t1_pre_ff,axiom,
( k3_pre_ff(np__0) = np__0
& k3_pre_ff(np__1) = np__1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_pre_ff(k1_nat_1(k1_nat_1(A,np__1),np__1)) = k1_nat_1(k3_pre_ff(A),k3_pre_ff(k1_nat_1(A,np__1))) ) ) ).
fof(t2_pre_ff,axiom,
! [A] :
( v1_int_1(A)
=> k5_int_1(A,np__1) = A ) ).
fof(t3_pre_ff,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( ~ r1_xreal_0(B,np__0)
& k5_int_1(A,B) = np__0
& r1_xreal_0(B,A) ) ) ) ).
fof(t4_pre_ff,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_xreal_0(np__0,A)
=> ( r1_xreal_0(B,A)
| k5_int_1(A,B) = np__0 ) ) ) ) ).
fof(t5_pre_ff,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(C,np__0)
& k5_int_1(k5_int_1(A,B),C) != k5_int_1(A,k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(t6_pre_ff,axiom,
! [A] :
( v1_int_1(A)
=> ( k6_int_1(A,np__2) = np__0
| k6_int_1(A,np__2) = np__1 ) ) ).
fof(t7_pre_ff,axiom,
! [A] :
( v1_int_1(A)
=> ( m2_subset_1(A,k1_numbers,k5_numbers)
=> m2_subset_1(k5_int_1(A,np__2),k1_numbers,k5_numbers) ) ) ).
fof(t8_pre_ff,axiom,
$true ).
fof(t9_pre_ff,axiom,
$true ).
fof(t10_pre_ff,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(A,B)
=> ( r1_xreal_0(C,np__1)
| r1_xreal_0(k3_power(C,A),k3_power(C,B)) ) ) ) ) ) ).
fof(t11_pre_ff,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(B,A)
=> r1_xreal_0(k1_int_1(B),k1_int_1(A)) ) ) ) ).
fof(t12_pre_ff,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r1_xreal_0(B,C)
=> ( r1_xreal_0(A,np__1)
| r1_xreal_0(B,np__0)
| r1_xreal_0(k5_power(A,B),k5_power(A,C)) ) ) ) ) ) ).
fof(t13_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& k2_xcmplx_0(k1_int_1(k6_power(np__2,k2_nat_1(np__2,A))),np__1) = k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))) ) ) ).
fof(t14_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> r1_xreal_0(k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))),k2_xcmplx_0(k1_int_1(k6_power(np__2,k2_nat_1(np__2,A))),np__1)) ) ) ).
fof(t15_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k1_int_1(k6_power(np__2,k2_nat_1(np__2,A))) = k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))) ) ) ).
fof(t16_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k2_xcmplx_0(k1_int_1(k6_power(np__2,A)),np__1) = k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))) ) ) ).
fof(d2_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( A = np__0
=> ( B = k5_pre_ff(A)
<=> B = np__0 ) )
& ( A != np__0
=> ( B = k5_pre_ff(A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k3_finseq_2(k5_numbers))
& m2_relset_1(D,k5_numbers,k3_finseq_2(k5_numbers))
& k1_nat_1(C,np__1) = A
& B = k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,C),A)
& k4_pre_ff(D,np__0) = k12_finseq_1(k5_numbers,np__1)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k1_nat_1(E,np__2) = k2_nat_1(np__2,F)
=> k4_pre_ff(D,k1_nat_1(E,np__1)) = k8_finseq_1(k5_numbers,k4_pre_ff(D,E),k12_finseq_1(k5_numbers,k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,E),F))) ) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( k1_nat_1(E,np__2) = k1_nat_1(k2_nat_1(np__2,F),np__1)
=> k4_pre_ff(D,k1_nat_1(E,np__1)) = k8_finseq_1(k5_numbers,k4_pre_ff(D,E),k12_finseq_1(k5_numbers,k1_nat_1(k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,E),F),k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,E),k1_nat_1(F,np__1))))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_pre_ff,axiom,
( k5_pre_ff(np__0) = np__0
& k5_pre_ff(np__1) = np__1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k5_pre_ff(k2_nat_1(np__2,A)) = k5_pre_ff(A)
& k5_pre_ff(k1_nat_1(k2_nat_1(np__2,A),np__1)) = k1_nat_1(k5_pre_ff(A),k5_pre_ff(k1_nat_1(A,np__1))) ) ) ) ).
fof(t18_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& A = k2_nat_1(np__2,B)
& r1_xreal_0(A,B) ) ) ) ).
fof(t19_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A = k1_nat_1(k2_nat_1(np__2,B),np__1)
& r1_xreal_0(A,B) ) ) ) ).
fof(t20_pre_ff,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> B = k1_nat_1(k2_nat_1(A,k5_pre_ff(np__0)),k2_nat_1(B,k5_pre_ff(k1_nat_1(np__0,np__1)))) ) ) ).
fof(t21_pre_ff,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)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( A = k1_nat_1(k2_nat_1(np__2,B),np__1)
& k5_pre_ff(E) = k1_nat_1(k2_nat_1(C,k5_pre_ff(A)),k2_nat_1(D,k5_pre_ff(k1_nat_1(A,np__1)))) )
=> k5_pre_ff(E) = k1_nat_1(k2_nat_1(C,k5_pre_ff(B)),k2_nat_1(k1_nat_1(D,C),k5_pre_ff(k1_nat_1(B,np__1)))) ) ) ) ) ) ) ).
fof(t22_pre_ff,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)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( A = k2_nat_1(np__2,B)
& k5_pre_ff(E) = k1_nat_1(k2_nat_1(C,k5_pre_ff(A)),k2_nat_1(D,k5_pre_ff(k1_nat_1(A,np__1)))) )
=> k5_pre_ff(E) = k1_nat_1(k2_nat_1(k1_nat_1(C,D),k5_pre_ff(B)),k2_nat_1(D,k5_pre_ff(k1_nat_1(B,np__1)))) ) ) ) ) ) ) ).
fof(dt_k1_pre_ff,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
=> m2_subset_1(k1_pre_ff(A,B,C,D,E),A,C) ) ).
fof(redefinition_k1_pre_ff,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
=> k1_pre_ff(A,B,C,D,E) = k1_mcart_1(E) ) ).
fof(dt_k2_pre_ff,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
=> m2_subset_1(k2_pre_ff(A,B,C,D,E),B,D) ) ).
fof(redefinition_k2_pre_ff,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
=> k2_pre_ff(A,B,C,D,E) = k2_mcart_1(E) ) ).
fof(dt_k3_pre_ff,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k3_pre_ff(A),k1_numbers,k5_numbers) ) ).
fof(dt_k4_pre_ff,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_finseq_2(k5_numbers))
& m1_relset_1(A,k5_numbers,k3_finseq_2(k5_numbers))
& m1_subset_1(B,k5_numbers) )
=> m2_finseq_1(k4_pre_ff(A,B),k5_numbers) ) ).
fof(redefinition_k4_pre_ff,axiom,
! [A,B] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k3_finseq_2(k5_numbers))
& m1_relset_1(A,k5_numbers,k3_finseq_2(k5_numbers))
& m1_subset_1(B,k5_numbers) )
=> k4_pre_ff(A,B) = k1_funct_1(A,B) ) ).
fof(dt_k5_pre_ff,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k5_pre_ff(A),k1_numbers,k5_numbers) ) ).
%------------------------------------------------------------------------------