SET007 Axioms: SET007+165.ax
%------------------------------------------------------------------------------
% File : SET007+165 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Functions and Finite Sequences of Real Numbers
% 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 : rfinseq [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 48 ( 3 unt; 0 def)
% Number of atoms : 290 ( 52 equ)
% Maximal formula atoms : 12 ( 6 avg)
% Number of connectives : 250 ( 8 ~; 3 |; 103 &)
% ( 15 <=>; 121 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-3 aty)
% Number of functors : 33 ( 33 usr; 5 con; 0-4 aty)
% Number of variables : 113 ( 107 !; 6 ?)
% 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(fc1_rfinseq,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(B) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B))
& v1_finset_1(k7_relat_1(A,B)) ) ) ).
fof(rc1_rfinseq,axiom,
? [A] :
( m1_finseq_1(A,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_rfinseq(A) ) ).
fof(d1_rfinseq,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r1_rfinseq(A,B)
<=> ! [C] : k1_card_1(k10_relat_1(A,k1_tarski(C))) = k1_card_1(k10_relat_1(B,k1_tarski(C))) ) ) ) ).
fof(t1_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_rfinseq(A,B)
=> k2_relat_1(A) = k2_relat_1(B) ) ) ) ).
fof(t2_rfinseq,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_rfinseq(A,B)
& r1_rfinseq(A,C) )
=> r1_rfinseq(B,C) ) ) ) ) ).
fof(t3_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_rfinseq(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& k1_relat_1(C) = k1_relat_1(A)
& k2_relat_1(C) = k1_relat_1(B)
& v2_funct_1(C)
& A = k5_relat_1(C,B) ) ) ) ) ).
fof(t4_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_rfinseq(A,B)
<=> ! [C] : k1_card_1(k10_relat_1(A,C)) = k1_card_1(k10_relat_1(B,C)) ) ) ) ).
fof(t5_rfinseq,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k2_relat_1(B),A)
& r1_tarski(k2_relat_1(C),A) )
=> ( r1_rfinseq(B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> k1_card_1(k10_relat_1(B,k1_tarski(D))) = k1_card_1(k10_relat_1(C,k1_tarski(D))) ) ) ) ) ) ) ).
fof(t6_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(A) = k1_relat_1(B)
=> ( r1_rfinseq(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k1_relat_1(A),k1_relat_1(A))
& v3_funct_2(C,k1_relat_1(A),k1_relat_1(A))
& m2_relset_1(C,k1_relat_1(A),k1_relat_1(A))
& A = k5_relat_1(C,B) ) ) ) ) ) ).
fof(t7_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_rfinseq(A,B)
=> k1_card_1(k1_relat_1(A)) = k1_card_1(k1_relat_1(B)) ) ) ) ).
fof(t8_rfinseq,axiom,
$true ).
fof(t9_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B) )
=> ( r1_rfinseq(A,B)
<=> ! [C] : k4_card_1(k10_relat_1(A,C)) = k4_card_1(k10_relat_1(B,C)) ) ) ) ).
fof(t10_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B) )
=> ( r1_rfinseq(A,B)
=> k4_card_1(k1_relat_1(A)) = k4_card_1(k1_relat_1(B)) ) ) ) ).
fof(t11_rfinseq,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C) )
=> ( ( r1_tarski(k2_relat_1(B),A)
& r1_tarski(k2_relat_1(C),A) )
=> ( r1_rfinseq(B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> k4_card_1(k10_relat_1(B,k1_tarski(D))) = k4_card_1(k10_relat_1(C,k1_tarski(D))) ) ) ) ) ) ) ).
fof(t12_rfinseq,axiom,
$true ).
fof(t13_rfinseq,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_rfinseq(A,B)
<=> ! [C] : k4_card_1(k10_relat_1(A,C)) = k4_card_1(k10_relat_1(B,C)) ) ) ) ).
fof(t14_rfinseq,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) )
=> ( r1_rfinseq(A,B)
<=> r1_rfinseq(k7_finseq_1(A,C),k7_finseq_1(B,C)) ) ) ) ) ).
fof(t15_rfinseq,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_rfinseq(k7_finseq_1(A,B),k7_finseq_1(B,A)) ) ) ).
fof(t16_rfinseq,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_rfinseq(A,B)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
& k4_finseq_1(A) = k4_finseq_1(B) ) ) ) ) ).
fof(t17_rfinseq,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_rfinseq(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k4_finseq_1(B),k4_finseq_1(B))
& v3_funct_2(C,k4_finseq_1(B),k4_finseq_1(B))
& m2_relset_1(C,k4_finseq_1(B),k4_finseq_1(B))
& A = k5_relat_1(C,B) ) ) ) ) ).
fof(t18_rfinseq,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( v1_finset_1(B)
=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& r1_rfinseq(k7_relat_1(A,B),C) ) ) ) ).
fof(d2_rfinseq,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_finseq_1(D,A)
=> ( ( r1_xreal_0(C,k3_finseq_1(B))
=> ( D = k1_rfinseq(A,B,C)
<=> ( k3_finseq_1(D) = k5_real_1(k3_finseq_1(B),C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> k1_funct_1(D,E) = k1_funct_1(B,k1_nat_1(E,C)) ) ) ) ) )
& ( ~ r1_xreal_0(C,k3_finseq_1(B))
=> ( D = k1_rfinseq(A,B,C)
<=> D = k6_finseq_1(A) ) ) ) ) ) ) ).
fof(t19_rfinseq,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)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k2_finseq_1(C)) )
=> ( k1_funct_1(k16_finseq_1(A,B,C),D) = k1_funct_1(B,D)
& r2_hidden(D,k4_finseq_1(B)) ) ) ) ) ) ) ).
fof(t20_rfinseq,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( k3_finseq_1(B) = k1_nat_1(C,np__1)
& D = k1_funct_1(B,k1_nat_1(C,np__1)) )
=> B = k7_finseq_1(k16_finseq_1(A,B,C),k9_finseq_1(D)) ) ) ) ) ).
fof(t21_rfinseq,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_finseq_1(A,k16_finseq_1(A,B,C),k1_rfinseq(A,B,C)) = B ) ) ) ).
fof(t22_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( r1_rfinseq(A,B)
=> k15_rvsum_1(A) = k15_rvsum_1(B) ) ) ) ).
fof(d3_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( B = k2_rfinseq(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(A)
& k2_seq_1(k5_numbers,k1_numbers,B,k3_finseq_1(B)) = k2_seq_1(k5_numbers,k1_numbers,A,k3_finseq_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k5_real_1(k3_finseq_1(B),np__1)) )
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k5_real_1(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) ) ) ) ) ).
fof(t23_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k3_finseq_1(A) = k1_nat_1(C,np__2)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1)) = B )
=> k2_rfinseq(k16_finseq_1(k1_numbers,A,k1_nat_1(C,np__1))) = k8_finseq_1(k1_numbers,k16_finseq_1(k1_numbers,k2_rfinseq(A),C),k12_finseq_1(k1_numbers,B)) ) ) ) ) ).
fof(t24_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( k3_finseq_1(A) = k1_nat_1(D,np__2)
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(D,np__1)) = B
& k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(D,np__2)) = C )
=> k2_rfinseq(A) = k8_finseq_1(k1_numbers,k16_finseq_1(k1_numbers,k2_rfinseq(A),D),k2_finseq_4(k1_numbers,k5_real_1(B,C),C)) ) ) ) ) ) ).
fof(t25_rfinseq,axiom,
k2_rfinseq(k6_finseq_1(k1_numbers)) = k6_finseq_1(k1_numbers) ).
fof(t26_rfinseq,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k2_rfinseq(k12_finseq_1(k1_numbers,A)) = k12_finseq_1(k1_numbers,A) ) ).
fof(t27_rfinseq,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_rfinseq(k2_finseq_4(k1_numbers,A,B)) = k2_finseq_4(k1_numbers,k5_real_1(A,B),B) ) ) ).
fof(t28_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_rfinseq(k1_numbers,k2_rfinseq(A),B) = k2_rfinseq(k1_rfinseq(k1_numbers,A,B)) ) ) ).
fof(t29_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( k3_finseq_1(A) != np__0
=> k15_rvsum_1(k2_rfinseq(A)) = k2_seq_1(k5_numbers,k1_numbers,A,np__1) ) ) ).
fof(t30_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(k3_finseq_1(A),B)
| k15_rvsum_1(k2_rfinseq(k1_rfinseq(k1_numbers,A,B))) = k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1)) ) ) ) ) ).
fof(d4_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v1_rfinseq(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A)) )
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1)),k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t31_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( ( k3_finseq_1(A) = np__0
| k3_finseq_1(A) = np__1 )
=> v1_rfinseq(A) ) ) ).
fof(t32_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v1_rfinseq(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& r2_hidden(C,k4_finseq_1(A)) )
=> ( r1_xreal_0(C,B)
| r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) ) ) ).
fof(t33_rfinseq,axiom,
! [A] :
( ( v1_rfinseq(A)
& m2_finseq_1(A,k1_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( v1_rfinseq(k16_finseq_1(k1_numbers,A,B))
& m2_finseq_1(k16_finseq_1(k1_numbers,A,B),k1_numbers) ) ) ) ).
fof(t34_rfinseq,axiom,
! [A] :
( ( v1_rfinseq(A)
& m2_finseq_1(A,k1_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( v1_rfinseq(k1_rfinseq(k1_numbers,A,B))
& m2_finseq_1(k1_rfinseq(k1_numbers,A,B),k1_numbers) ) ) ) ).
fof(t35_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ? [B] :
( v1_rfinseq(B)
& m2_finseq_1(B,k1_numbers)
& r1_rfinseq(A,B) ) ) ).
fof(t36_rfinseq,axiom,
! [A] :
( ( v1_rfinseq(A)
& m2_finseq_1(A,k1_numbers) )
=> ! [B] :
( ( v1_rfinseq(B)
& m2_finseq_1(B,k1_numbers) )
=> ( r1_rfinseq(A,B)
=> A = B ) ) ) ).
fof(t37_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( B != np__0
=> k3_funct_2(k5_numbers,k1_numbers,A,k1_tarski(k6_real_1(C,B))) = k3_funct_2(k5_numbers,k1_numbers,k9_rvsum_1(B,A),k1_tarski(C)) ) ) ) ) ).
fof(t38_rfinseq,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k3_funct_2(k5_numbers,k1_numbers,k9_rvsum_1(np__0,A),k1_tarski(np__0)) = k4_finseq_1(A) ) ).
fof(symmetry_r1_rfinseq,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> ( r1_rfinseq(A,B)
=> r1_rfinseq(B,A) ) ) ).
fof(reflexivity_r1_rfinseq,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> r1_rfinseq(A,A) ) ).
fof(dt_k1_rfinseq,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_1(k1_rfinseq(A,B,C),A) ) ).
fof(dt_k2_rfinseq,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m2_finseq_1(k2_rfinseq(A),k1_numbers) ) ).
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