SET007 Axioms: SET007+795.ax
%------------------------------------------------------------------------------
% File : SET007+795 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Sorting Operators for 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 : rfinseq2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 44 ( 0 unt; 0 def)
% Number of atoms : 210 ( 59 equ)
% Maximal formula atoms : 18 ( 4 avg)
% Number of connectives : 178 ( 12 ~; 5 |; 49 &)
% ( 5 <=>; 107 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-3 aty)
% Number of functors : 28 ( 28 usr; 5 con; 0-5 aty)
% Number of variables : 78 ( 78 !; 0 ?)
% 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_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k1_rfinseq2(A)
<=> ( ( k3_finseq_1(A) = np__0
=> B = np__0 )
& ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> ( r2_hidden(B,k4_finseq_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(A))
& D = k2_seq_1(k5_numbers,k1_numbers,A,C)
& E = k2_seq_1(k5_numbers,k1_numbers,A,B) )
=> r1_xreal_0(D,E) ) ) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(A))
& k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,A,B) )
=> r1_xreal_0(B,C) ) ) ) ) ) ) ) ) ).
fof(d2_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k2_rfinseq2(A)
<=> ( ( k3_finseq_1(A) = np__0
=> B = np__0 )
& ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> ( r2_hidden(B,k4_finseq_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(A))
& D = k2_seq_1(k5_numbers,k1_numbers,A,C)
& E = k2_seq_1(k5_numbers,k1_numbers,A,B) )
=> r1_xreal_0(E,D) ) ) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_finseq_1(A))
& k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,A,B) )
=> r1_xreal_0(B,C) ) ) ) ) ) ) ) ) ).
fof(d3_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k3_rfinseq2(A) = k2_seq_1(k5_numbers,k1_numbers,A,k1_rfinseq2(A)) ) ).
fof(d4_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k4_rfinseq2(A) = k2_seq_1(k5_numbers,k1_numbers,A,k2_rfinseq2(A)) ) ).
fof(t1_rfinseq2,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(B,k3_finseq_1(A)) )
=> ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,A,k1_rfinseq2(A)))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k3_rfinseq2(A)) ) ) ) ) ).
fof(t2_rfinseq2,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(B,k3_finseq_1(A)) )
=> ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,k2_rfinseq2(A)),k2_seq_1(k5_numbers,k1_numbers,A,B))
& r1_xreal_0(k4_rfinseq2(A),k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) ).
fof(t3_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = k12_finseq_1(k1_numbers,B)
=> ( k1_rfinseq2(A) = np__1
& k3_rfinseq2(A) = B ) ) ) ) ).
fof(t4_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( A = k12_finseq_1(k1_numbers,B)
=> ( k2_rfinseq2(A) = np__1
& k4_rfinseq2(A) = B ) ) ) ) ).
fof(t5_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( A = k10_finseq_1(B,C)
=> ( k3_rfinseq2(A) = k4_square_1(B,C)
& k1_rfinseq2(A) = k2_cqc_lang(k1_numbers,B,k4_square_1(B,C),np__1,np__2) ) ) ) ) ) ).
fof(t6_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( A = k10_finseq_1(B,C)
=> ( k4_rfinseq2(A) = k3_square_1(B,C)
& k2_rfinseq2(A) = k2_cqc_lang(k1_numbers,B,k3_square_1(B,C),np__1,np__2) ) ) ) ) ) ).
fof(t7_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| r1_xreal_0(k3_rfinseq2(k3_rvsum_1(A,B)),k3_real_1(k3_rfinseq2(A),k3_rfinseq2(B))) ) ) ) ) ).
fof(t8_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| r1_xreal_0(k3_real_1(k4_rfinseq2(A),k4_rfinseq2(B)),k4_rfinseq2(k3_rvsum_1(A,B))) ) ) ) ) ).
fof(t9_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
& ~ r1_xreal_0(B,np__0)
& ~ ( k3_rfinseq2(k9_rvsum_1(B,A)) = k4_real_1(B,k3_rfinseq2(A))
& k1_rfinseq2(k9_rvsum_1(B,A)) = k1_rfinseq2(A) ) ) ) ) ).
fof(t10_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
& ~ r1_xreal_0(B,np__0)
& ~ ( k4_rfinseq2(k9_rvsum_1(B,A)) = k4_real_1(B,k4_rfinseq2(A))
& k2_rfinseq2(k9_rvsum_1(B,A)) = k2_rfinseq2(A) ) ) ) ) ).
fof(t11_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> ( k3_rfinseq2(k5_rvsum_1(A)) = k1_real_1(k4_rfinseq2(A))
& k1_rfinseq2(k5_rvsum_1(A)) = k2_rfinseq2(A) ) ) ) ).
fof(t12_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> ( k4_rfinseq2(k5_rvsum_1(A)) = k1_real_1(k3_rfinseq2(A))
& k2_rfinseq2(k5_rvsum_1(A)) = k1_rfinseq2(A) ) ) ) ).
fof(t13_rfinseq2,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)
| ( r1_xreal_0(k3_rfinseq2(k1_rfinseq(k1_numbers,A,B)),k3_rfinseq2(A))
& r1_xreal_0(k4_rfinseq2(A),k4_rfinseq2(k1_rfinseq(k1_numbers,A,B))) ) ) ) ) ) ).
fof(t14_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( r1_rfinseq(A,B)
=> k3_rfinseq2(A) = k3_rfinseq2(B) ) ) ) ).
fof(t15_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( r1_rfinseq(A,B)
=> k4_rfinseq2(A) = k4_rfinseq2(B) ) ) ) ).
fof(d5_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( ( v1_rfinseq(B)
& m2_finseq_1(B,k1_numbers) )
=> ( B = k5_rfinseq2(A)
<=> r1_rfinseq(A,B) ) ) ) ).
fof(t16_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( ( k3_finseq_1(A) = np__0
| k3_finseq_1(A) = np__1 )
=> v1_integra2(A) ) ) ).
fof(t17_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v1_integra2(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,B),k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ) ) ).
fof(t18_rfinseq2,axiom,
! [A] :
( ( v1_integra2(A)
& m2_finseq_1(A,k1_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( v1_integra2(k16_finseq_1(k1_numbers,A,B))
& m2_finseq_1(k16_finseq_1(k1_numbers,A,B),k1_numbers) ) ) ) ).
fof(t19_rfinseq2,axiom,
! [A] :
( ( v1_integra2(A)
& m2_finseq_1(A,k1_numbers) )
=> ! [B] :
( ( v1_integra2(B)
& m2_finseq_1(B,k1_numbers) )
=> ( r1_rfinseq(A,B)
=> A = B ) ) ) ).
fof(d6_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra2(B)
& m2_finseq_1(B,k1_numbers) )
=> ( B = k6_rfinseq2(A)
<=> r1_rfinseq(A,B) ) ) ) ).
fof(t20_rfinseq2,axiom,
! [A] :
( ( v1_rfinseq(A)
& m2_finseq_1(A,k1_numbers) )
=> k5_rfinseq2(A) = A ) ).
fof(t21_rfinseq2,axiom,
! [A] :
( ( v1_integra2(A)
& m2_finseq_1(A,k1_numbers) )
=> k6_rfinseq2(A) = A ) ).
fof(t22_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k5_rfinseq2(k5_rfinseq2(A)) = k5_rfinseq2(A) ) ).
fof(t23_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k6_rfinseq2(k6_rfinseq2(A)) = k6_rfinseq2(A) ) ).
fof(t24_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v1_rfinseq(A)
=> v1_integra2(k5_rvsum_1(A)) ) ) ).
fof(t25_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v1_integra2(A)
=> v1_rfinseq(k5_rvsum_1(A)) ) ) ).
fof(t26_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ! [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)
& r1_xreal_0(np__1,k3_finseq_1(B)) )
=> k5_rvsum_1(A) = k5_relat_1(C,k5_rvsum_1(B)) ) ) ) ) ).
fof(t27_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( r1_rfinseq(A,B)
=> r1_rfinseq(k5_rvsum_1(A),k5_rvsum_1(B)) ) ) ) ).
fof(t28_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k5_rfinseq2(k5_rvsum_1(A)) = k5_rvsum_1(k6_rfinseq2(A)) ) ).
fof(t29_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k6_rfinseq2(k5_rvsum_1(A)) = k5_rvsum_1(k5_rfinseq2(A)) ) ).
fof(t30_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( k4_finseq_1(k5_rfinseq2(A)) = k4_finseq_1(A)
& k3_finseq_1(k5_rfinseq2(A)) = k3_finseq_1(A) ) ) ).
fof(t31_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( k4_finseq_1(k6_rfinseq2(A)) = k4_finseq_1(A)
& k3_finseq_1(k6_rfinseq2(A)) = k3_finseq_1(A) ) ) ).
fof(t32_rfinseq2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( r1_xreal_0(np__1,k3_finseq_1(A))
=> ( k1_rfinseq2(k5_rfinseq2(A)) = np__1
& k2_rfinseq2(k6_rfinseq2(A)) = np__1
& k2_seq_1(k5_numbers,k1_numbers,k5_rfinseq2(A),np__1) = k3_rfinseq2(A)
& k2_seq_1(k5_numbers,k1_numbers,k6_rfinseq2(A),np__1) = k4_rfinseq2(A) ) ) ) ).
fof(dt_k1_rfinseq2,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m2_subset_1(k1_rfinseq2(A),k1_numbers,k5_numbers) ) ).
fof(dt_k2_rfinseq2,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m2_subset_1(k2_rfinseq2(A),k1_numbers,k5_numbers) ) ).
fof(dt_k3_rfinseq2,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m1_subset_1(k3_rfinseq2(A),k1_numbers) ) ).
fof(dt_k4_rfinseq2,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m1_subset_1(k4_rfinseq2(A),k1_numbers) ) ).
fof(dt_k5_rfinseq2,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> ( v1_rfinseq(k5_rfinseq2(A))
& m2_finseq_1(k5_rfinseq2(A),k1_numbers) ) ) ).
fof(dt_k6_rfinseq2,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> ( v1_integra2(k6_rfinseq2(A))
& m2_finseq_1(k6_rfinseq2(A),k1_numbers) ) ) ).
%------------------------------------------------------------------------------