SET007 Axioms: SET007+98.ax
%------------------------------------------------------------------------------
% File : SET007+98 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Binary Operations on Finite Sequences
% 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 : finsop_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 1 unt; 0 def)
% Number of atoms : 290 ( 53 equ)
% Maximal formula atoms : 24 ( 8 avg)
% Number of connectives : 325 ( 68 ~; 11 |; 124 &)
% ( 2 <=>; 120 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-3 aty)
% Number of functors : 33 ( 33 usr; 5 con; 0-6 aty)
% Number of variables : 118 ( 117 !; 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(d1_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( ( v1_setwiseo(C,A)
| r1_xreal_0(np__1,k3_finseq_1(B)) )
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( ( v1_setwiseo(C,A)
& k3_finseq_1(B) = np__0 )
=> ( D = k2_finsop_1(A,B,C)
<=> D = k3_binop_1(A,C) ) )
& ( ~ ( v1_setwiseo(C,A)
& k3_finseq_1(B) = np__0 )
=> ( D = k2_finsop_1(A,B,C)
<=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,A)
& m2_relset_1(E,k5_numbers,A)
& k8_funct_2(k5_numbers,A,E,np__1) = k1_funct_1(B,np__1)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( np__0 != F
& ~ r1_xreal_0(k3_finseq_1(B),F)
& k8_funct_2(k5_numbers,A,E,k1_nat_1(F,np__1)) != k1_binop_1(C,k8_funct_2(k5_numbers,A,E,F),k1_funct_1(B,k1_nat_1(F,np__1))) ) )
& D = k8_funct_2(k5_numbers,A,E,k3_finseq_1(B)) ) ) ) ) ) ) ) ) ) ).
fof(t1_finsop_1,axiom,
$true ).
fof(t2_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,A)
& m2_relset_1(D,k5_numbers,A) )
=> ~ ( k8_funct_2(k5_numbers,A,D,np__1) = k1_funct_1(B,np__1)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( np__0 != E
& ~ r1_xreal_0(k3_finseq_1(B),E)
& k8_funct_2(k5_numbers,A,D,k1_nat_1(E,np__1)) != k1_binop_1(C,k8_funct_2(k5_numbers,A,D,E),k1_funct_1(B,k1_nat_1(E,np__1))) ) )
& k2_finsop_1(A,B,C) = k8_funct_2(k5_numbers,A,D,k3_finseq_1(B)) ) ) ) ) ) ) ).
fof(t3_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( r1_xreal_0(np__1,k3_finseq_1(C))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,A)
& m2_relset_1(E,k5_numbers,A) )
=> ~ ( k8_funct_2(k5_numbers,A,E,np__1) = k1_funct_1(C,np__1)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( np__0 != F
& ~ r1_xreal_0(k3_finseq_1(C),F)
& k8_funct_2(k5_numbers,A,E,k1_nat_1(F,np__1)) != k1_binop_1(D,k8_funct_2(k5_numbers,A,E,F),k1_funct_1(C,k1_nat_1(F,np__1))) ) )
& B = k8_funct_2(k5_numbers,A,E,k3_finseq_1(C)) ) )
| B = k2_finsop_1(A,C,D) ) ) ) ) ) ) ).
fof(t4_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( ( v2_binop_1(C,A)
& v1_binop_1(C,A) )
=> ( ( ~ v1_setwiseo(C,A)
& ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
| k2_finsop_1(A,B,C) = k7_setwiseo(k5_numbers,A,C,k5_finsop_1(B),k4_finsop_1(A,k3_finsop_1(A,k5_numbers,k3_binop_1(A,C)),B)) ) ) ) ) ) ).
fof(t5_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ( v1_setwiseo(D,A)
| r1_xreal_0(np__1,k3_finseq_1(C)) )
=> k2_finsop_1(A,k8_finseq_1(A,C,k12_finseq_1(A,B)),D) = k2_binop_1(A,A,A,D,k2_finsop_1(A,C,D),B) ) ) ) ) ) ).
fof(t6_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( v2_binop_1(D,A)
=> ( ( ~ v1_setwiseo(D,A)
& ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
& r1_xreal_0(np__1,k3_finseq_1(C)) ) )
| k2_finsop_1(A,k8_finseq_1(A,B,C),D) = k2_binop_1(A,A,A,D,k2_finsop_1(A,B,D),k2_finsop_1(A,C,D)) ) ) ) ) ) ) ).
fof(t7_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( v2_binop_1(D,A)
=> ( ( ~ v1_setwiseo(D,A)
& ~ r1_xreal_0(np__1,k3_finseq_1(C)) )
| k2_finsop_1(A,k8_finseq_1(A,k12_finseq_1(A,B),C),D) = k2_binop_1(A,A,A,D,B,k2_finsop_1(A,C,D)) ) ) ) ) ) ) ).
fof(t8_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
& v3_funct_2(E,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
& m2_relset_1(E,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B)) )
=> ( ( v1_binop_1(D,A)
& v2_binop_1(D,A)
& C = k5_relat_1(E,B) )
=> ( ( ~ v1_setwiseo(D,A)
& ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
| k2_finsop_1(A,B,D) = k2_finsop_1(A,C,D) ) ) ) ) ) ) ) ).
fof(t9_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ( v2_binop_1(D,A)
& v1_binop_1(D,A)
& v2_funct_1(B)
& v2_funct_1(C)
& k2_relat_1(B) = k2_relat_1(C) )
=> ( ( ~ v1_setwiseo(D,A)
& ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
| k2_finsop_1(A,B,D) = k2_finsop_1(A,C,D) ) ) ) ) ) ) ).
fof(t10_finsop_1,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] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m2_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ( ( v2_binop_1(E,A)
& v1_binop_1(E,A)
& k3_finseq_1(B) = k3_finseq_1(C)
& k3_finseq_1(B) = k3_finseq_1(D)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k5_finsop_1(B))
=> k1_funct_1(B,F) = k1_binop_1(E,k1_funct_1(C,F),k1_funct_1(D,F)) ) ) )
=> ( ( ~ v1_setwiseo(E,A)
& ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
| k2_finsop_1(A,B,E) = k2_binop_1(A,A,A,E,k2_finsop_1(A,C,E),k2_finsop_1(A,D,E)) ) ) ) ) ) ) ) ).
fof(t11_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_setwiseo(B,A)
=> k2_finsop_1(A,k6_finseq_1(A),B) = k3_binop_1(A,B) ) ) ) ).
fof(t12_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> k2_finsop_1(A,k12_finseq_1(A,B),C) = B ) ) ) ).
fof(t13_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> k2_finsop_1(A,k2_finseq_4(A,B,C),D) = k2_binop_1(A,A,A,D,B,C) ) ) ) ) ).
fof(t14_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(D,A)
=> k2_finsop_1(A,k2_finseq_4(A,B,C),D) = k2_finsop_1(A,k2_finseq_4(A,C,B),D) ) ) ) ) ) ).
fof(t15_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m2_relset_1(E,k2_zfmisc_1(A,A),A) )
=> k2_finsop_1(A,k3_finseq_4(A,B,C,D),E) = k2_binop_1(A,A,A,E,k2_binop_1(A,A,A,E,B,C),D) ) ) ) ) ) ).
fof(t16_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m2_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(E,A)
=> k2_finsop_1(A,k3_finseq_4(A,B,C,D),E) = k2_finsop_1(A,k3_finseq_4(A,C,B,D),E) ) ) ) ) ) ) ).
fof(t17_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> k2_finsop_1(A,k1_finsop_1(A,np__1,B),C) = B ) ) ) ).
fof(t18_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> k2_finsop_1(A,k1_finsop_1(A,np__2,B),C) = k2_binop_1(A,A,A,C,B,B) ) ) ) ).
fof(t19_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( v2_binop_1(C,A)
=> ( ( ~ v1_setwiseo(C,A)
& ~ ( D != np__0
& E != np__0 ) )
| k2_finsop_1(A,k1_finsop_1(A,k1_nat_1(D,E),B),C) = k2_binop_1(A,A,A,C,k2_finsop_1(A,k1_finsop_1(A,D,B),C),k2_finsop_1(A,k1_finsop_1(A,E,B),C)) ) ) ) ) ) ) ) ).
fof(t20_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( v2_binop_1(C,A)
=> ( ( ~ v1_setwiseo(C,A)
& ~ ( D != np__0
& E != np__0 ) )
| k2_finsop_1(A,k1_finsop_1(A,k2_nat_1(D,E),B),C) = k2_finsop_1(A,k1_finsop_1(A,E,k2_finsop_1(A,k1_finsop_1(A,D,B),C)),C) ) ) ) ) ) ) ) ).
fof(t21_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( k3_finseq_1(B) = np__1
=> k2_finsop_1(A,B,C) = k1_funct_1(B,np__1) ) ) ) ) ).
fof(t22_finsop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( k3_finseq_1(B) = np__2
=> k2_finsop_1(A,B,C) = k1_binop_1(C,k1_funct_1(B,np__1),k1_funct_1(B,np__2)) ) ) ) ) ).
fof(dt_k1_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,A) )
=> m2_finseq_1(k1_finsop_1(A,B,C),A) ) ).
fof(redefinition_k1_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,A) )
=> k1_finsop_1(A,B,C) = k2_finseq_2(B,C) ) ).
fof(dt_k2_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A) )
=> m1_subset_1(k2_finsop_1(A,B,C),A) ) ).
fof(dt_k3_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,A) )
=> ( v1_funct_1(k3_finsop_1(A,B,C))
& v1_funct_2(k3_finsop_1(A,B,C),B,A)
& m2_relset_1(k3_finsop_1(A,B,C),B,A) ) ) ).
fof(redefinition_k3_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,A) )
=> k3_finsop_1(A,B,C) = k2_funcop_1(B,C) ) ).
fof(dt_k4_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,A)
& m1_relset_1(B,k5_numbers,A)
& m1_finseq_1(C,A) )
=> ( v1_funct_1(k4_finsop_1(A,B,C))
& v1_funct_2(k4_finsop_1(A,B,C),k5_numbers,A)
& m2_relset_1(k4_finsop_1(A,B,C),k5_numbers,A) ) ) ).
fof(idempotence_k4_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,A)
& m1_relset_1(B,k5_numbers,A)
& m1_finseq_1(C,A) )
=> k4_finsop_1(A,B,B) = B ) ).
fof(redefinition_k4_finsop_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,A)
& m1_relset_1(B,k5_numbers,A)
& m1_finseq_1(C,A) )
=> k4_finsop_1(A,B,C) = k1_funct_4(B,C) ) ).
fof(dt_k5_finsop_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> m1_subset_1(k5_finsop_1(A),k5_finsub_1(k5_numbers)) ) ).
fof(redefinition_k5_finsop_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> k5_finsop_1(A) = k1_relat_1(A) ) ).
%------------------------------------------------------------------------------