SET007 Axioms: SET007+65.ax
%------------------------------------------------------------------------------
% File : SET007+65 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Recursive Definitions
% 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 : recdef_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 0 unt; 0 def)
% Number of atoms : 435 ( 154 equ)
% Maximal formula atoms : 42 ( 17 avg)
% Number of connectives : 428 ( 18 ~; 32 |; 246 &)
% ( 0 <=>; 132 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 12 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 24 ( 23 usr; 0 prp; 1-3 aty)
% Number of functors : 77 ( 77 usr; 61 con; 0-4 aty)
% Number of variables : 171 ( 127 !; 44 ?)
% 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(s1_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
? [C] : p1_s1_recdef_1(A,B,C) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( ( p1_s1_recdef_1(A,B,C)
& p1_s1_recdef_1(A,B,D) )
=> C = D ) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k5_numbers
& k1_funct_1(A,np__0) = f1_s1_recdef_1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> p1_s1_recdef_1(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ).
fof(s2_recdef_1,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,f1_s2_recdef_1)
=> ? [C] :
( m1_subset_1(C,f1_s2_recdef_1)
& p1_s2_recdef_1(A,B,C) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s2_recdef_1)
& m2_relset_1(A,k5_numbers,f1_s2_recdef_1)
& k8_funct_2(k5_numbers,f1_s2_recdef_1,A,np__0) = f2_s2_recdef_1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> p1_s2_recdef_1(B,k8_funct_2(k5_numbers,f1_s2_recdef_1,A,B),k8_funct_2(k5_numbers,f1_s2_recdef_1,A,k1_nat_1(B,np__1))) ) ) ) ).
fof(s3_recdef_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k5_numbers
& k1_funct_1(A,np__0) = f1_s3_recdef_1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_funct_1(A,k1_nat_1(B,np__1)) = f2_s3_recdef_1(B,k1_funct_1(A,B)) ) ) ).
fof(s4_recdef_1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s4_recdef_1)
& m2_relset_1(A,k5_numbers,f1_s4_recdef_1)
& k8_funct_2(k5_numbers,f1_s4_recdef_1,A,np__0) = f2_s4_recdef_1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s4_recdef_1,A,k1_nat_1(B,np__1)) = f3_s4_recdef_1(B,k8_funct_2(k5_numbers,f1_s4_recdef_1,A,B)) ) ) ).
fof(s5_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f2_s5_recdef_1,A)
| ! [B] :
? [C] : p1_s5_recdef_1(A,B,C) ) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f2_s5_recdef_1,A)
| ! [B,C,D] :
( ( p1_s5_recdef_1(A,B,C)
& p1_s5_recdef_1(A,B,D) )
=> C = D ) ) ) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& k3_finseq_1(A) = f2_s5_recdef_1
& ( k1_funct_1(A,np__1) = f1_s5_recdef_1
| f2_s5_recdef_1 = np__0 )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(f2_s5_recdef_1,B)
| p1_s5_recdef_1(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ) ) ).
fof(s6_recdef_1,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f3_s6_recdef_1,A)
| ! [B] :
( m1_subset_1(B,f1_s6_recdef_1)
=> ? [C] :
( m1_subset_1(C,f1_s6_recdef_1)
& p1_s6_recdef_1(A,B,C) ) ) ) ) )
=> ? [A] :
( m2_finseq_1(A,f1_s6_recdef_1)
& k3_finseq_1(A) = f3_s6_recdef_1
& ( k1_funct_1(A,np__1) = f2_s6_recdef_1
| f3_s6_recdef_1 = np__0 )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(f3_s6_recdef_1,B)
| p1_s6_recdef_1(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ) ) ).
fof(s7_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
~ ( r1_xreal_0(np__1,A)
& ~ r1_xreal_0(k3_finseq_1(f1_s7_recdef_1),A)
& ! [C] : ~ p1_s7_recdef_1(k1_funct_1(f1_s7_recdef_1,k1_nat_1(A,np__1)),B,C) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D,E] :
( ( r1_xreal_0(np__1,A)
& E = k1_funct_1(f1_s7_recdef_1,k1_nat_1(A,np__1))
& p1_s7_recdef_1(E,B,C)
& p1_s7_recdef_1(E,B,D) )
=> ( r1_xreal_0(k3_finseq_1(f1_s7_recdef_1),A)
| C = D ) ) ) )
=> ? [A,B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& A = k1_funct_1(B,k3_finseq_1(B))
& k3_finseq_1(B) = k3_finseq_1(f1_s7_recdef_1)
& k1_funct_1(B,np__1) = k1_funct_1(f1_s7_recdef_1,np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(f1_s7_recdef_1),C)
| p1_s7_recdef_1(k1_funct_1(f1_s7_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) ) ) ) ) ).
fof(s8_recdef_1,axiom,
? [A,B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& A = k1_funct_1(B,k3_finseq_1(B))
& k3_finseq_1(B) = k3_finseq_1(f1_s8_recdef_1)
& k1_funct_1(B,np__1) = k1_funct_1(f1_s8_recdef_1,np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(f1_s8_recdef_1),C)
| k1_funct_1(B,k1_nat_1(C,np__1)) = f2_s8_recdef_1(k1_funct_1(f1_s8_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C)) ) ) ) ) ).
fof(s9_recdef_1,axiom,
( ( k1_relat_1(f2_s9_recdef_1) = k5_numbers
& k1_funct_1(f2_s9_recdef_1,np__0) = f1_s9_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> p1_s9_recdef_1(A,k1_funct_1(f2_s9_recdef_1,A),k1_funct_1(f2_s9_recdef_1,k1_nat_1(A,np__1))) )
& k1_relat_1(f3_s9_recdef_1) = k5_numbers
& k1_funct_1(f3_s9_recdef_1,np__0) = f1_s9_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> p1_s9_recdef_1(A,k1_funct_1(f3_s9_recdef_1,A),k1_funct_1(f3_s9_recdef_1,k1_nat_1(A,np__1))) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( ( p1_s9_recdef_1(A,B,C)
& p1_s9_recdef_1(A,B,D) )
=> C = D ) ) )
=> f2_s9_recdef_1 = f3_s9_recdef_1 ) ).
fof(s10_recdef_1,axiom,
( ( k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,np__0) = f2_s10_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> p1_s10_recdef_1(A,k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,A),k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,k1_nat_1(A,np__1))) )
& k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,np__0) = f2_s10_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> p1_s10_recdef_1(A,k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,A),k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,k1_nat_1(A,np__1))) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,f1_s10_recdef_1)
=> ! [C] :
( m1_subset_1(C,f1_s10_recdef_1)
=> ! [D] :
( m1_subset_1(D,f1_s10_recdef_1)
=> ( ( p1_s10_recdef_1(A,B,C)
& p1_s10_recdef_1(A,B,D) )
=> C = D ) ) ) ) ) )
=> f3_s10_recdef_1 = f4_s10_recdef_1 ) ).
fof(s11_recdef_1,axiom,
( ( k1_relat_1(f3_s11_recdef_1) = k5_numbers
& k1_funct_1(f3_s11_recdef_1,np__0) = f1_s11_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f3_s11_recdef_1,k1_nat_1(A,np__1)) = f2_s11_recdef_1(A,k1_funct_1(f3_s11_recdef_1,A)) )
& k1_relat_1(f4_s11_recdef_1) = k5_numbers
& k1_funct_1(f4_s11_recdef_1,np__0) = f1_s11_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f4_s11_recdef_1,k1_nat_1(A,np__1)) = f2_s11_recdef_1(A,k1_funct_1(f4_s11_recdef_1,A)) ) )
=> f3_s11_recdef_1 = f4_s11_recdef_1 ) ).
fof(s12_recdef_1,axiom,
( ( k8_funct_2(k5_numbers,f1_s12_recdef_1,f4_s12_recdef_1,np__0) = f2_s12_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s12_recdef_1,f4_s12_recdef_1,k1_nat_1(A,np__1)) = f3_s12_recdef_1(A,k8_funct_2(k5_numbers,f1_s12_recdef_1,f4_s12_recdef_1,A)) )
& k8_funct_2(k5_numbers,f1_s12_recdef_1,f5_s12_recdef_1,np__0) = f2_s12_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s12_recdef_1,f5_s12_recdef_1,k1_nat_1(A,np__1)) = f3_s12_recdef_1(A,k8_funct_2(k5_numbers,f1_s12_recdef_1,f5_s12_recdef_1,A)) ) )
=> f4_s12_recdef_1 = f5_s12_recdef_1 ) ).
fof(s13_recdef_1,axiom,
( ( k8_funct_2(k5_numbers,k1_numbers,f3_s13_recdef_1,np__0) = f1_s13_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_numbers,f3_s13_recdef_1,k1_nat_1(A,np__1)) = f2_s13_recdef_1(A,k8_funct_2(k5_numbers,k1_numbers,f3_s13_recdef_1,A)) )
& k8_funct_2(k5_numbers,k1_numbers,f4_s13_recdef_1,np__0) = f1_s13_recdef_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_numbers,f4_s13_recdef_1,k1_nat_1(A,np__1)) = f2_s13_recdef_1(A,k8_funct_2(k5_numbers,k1_numbers,f4_s13_recdef_1,A)) ) )
=> f3_s13_recdef_1 = f4_s13_recdef_1 ) ).
fof(s14_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f2_s14_recdef_1,A)
| ! [B,C,D] :
( ( p1_s14_recdef_1(A,B,C)
& p1_s14_recdef_1(A,B,D) )
=> C = D ) ) ) )
& k3_finseq_1(f3_s14_recdef_1) = f2_s14_recdef_1
& ( k1_funct_1(f3_s14_recdef_1,np__1) = f1_s14_recdef_1
| f2_s14_recdef_1 = np__0 )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f2_s14_recdef_1,A)
| p1_s14_recdef_1(A,k1_funct_1(f3_s14_recdef_1,A),k1_funct_1(f3_s14_recdef_1,k1_nat_1(A,np__1))) ) ) )
& k3_finseq_1(f4_s14_recdef_1) = f2_s14_recdef_1
& ( k1_funct_1(f4_s14_recdef_1,np__1) = f1_s14_recdef_1
| f2_s14_recdef_1 = np__0 )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f2_s14_recdef_1,A)
| p1_s14_recdef_1(A,k1_funct_1(f4_s14_recdef_1,A),k1_funct_1(f4_s14_recdef_1,k1_nat_1(A,np__1))) ) ) ) )
=> f3_s14_recdef_1 = f4_s14_recdef_1 ) ).
fof(s15_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f3_s15_recdef_1,A)
| ! [B] :
( m1_subset_1(B,f1_s15_recdef_1)
=> ! [C] :
( m1_subset_1(C,f1_s15_recdef_1)
=> ! [D] :
( m1_subset_1(D,f1_s15_recdef_1)
=> ( ( p1_s15_recdef_1(A,B,C)
& p1_s15_recdef_1(A,B,D) )
=> C = D ) ) ) ) ) ) )
& k3_finseq_1(f4_s15_recdef_1) = f3_s15_recdef_1
& ( k1_funct_1(f4_s15_recdef_1,np__1) = f2_s15_recdef_1
| f3_s15_recdef_1 = np__0 )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f3_s15_recdef_1,A)
| p1_s15_recdef_1(A,k1_funct_1(f4_s15_recdef_1,A),k1_funct_1(f4_s15_recdef_1,k1_nat_1(A,np__1))) ) ) )
& k3_finseq_1(f5_s15_recdef_1) = f3_s15_recdef_1
& ( k1_funct_1(f5_s15_recdef_1,np__1) = f2_s15_recdef_1
| f3_s15_recdef_1 = np__0 )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f3_s15_recdef_1,A)
| p1_s15_recdef_1(A,k1_funct_1(f5_s15_recdef_1,A),k1_funct_1(f5_s15_recdef_1,k1_nat_1(A,np__1))) ) ) ) )
=> f4_s15_recdef_1 = f5_s15_recdef_1 ) ).
fof(s16_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D,E] :
( ( r1_xreal_0(np__1,A)
& E = k1_funct_1(f1_s16_recdef_1,k1_nat_1(A,np__1))
& p1_s16_recdef_1(E,B,C)
& p1_s16_recdef_1(E,B,D) )
=> ( r1_xreal_0(k3_finseq_1(f1_s16_recdef_1),A)
| C = D ) ) )
& ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& f2_s16_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
& k3_finseq_1(A) = k3_finseq_1(f1_s16_recdef_1)
& k1_funct_1(A,np__1) = k1_funct_1(f1_s16_recdef_1,np__1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(k3_finseq_1(f1_s16_recdef_1),B)
| p1_s16_recdef_1(k1_funct_1(f1_s16_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) )
& ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& f3_s16_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
& k3_finseq_1(A) = k3_finseq_1(f1_s16_recdef_1)
& k1_funct_1(A,np__1) = k1_funct_1(f1_s16_recdef_1,np__1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(k3_finseq_1(f1_s16_recdef_1),B)
| p1_s16_recdef_1(k1_funct_1(f1_s16_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ) )
=> f2_s16_recdef_1 = f3_s16_recdef_1 ) ).
fof(s17_recdef_1,axiom,
( ( ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& f3_s17_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
& k3_finseq_1(A) = k3_finseq_1(f1_s17_recdef_1)
& k1_funct_1(A,np__1) = k1_funct_1(f1_s17_recdef_1,np__1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(k3_finseq_1(f1_s17_recdef_1),B)
| k1_funct_1(A,k1_nat_1(B,np__1)) = f2_s17_recdef_1(k1_funct_1(f1_s17_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B)) ) ) ) )
& ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& f4_s17_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
& k3_finseq_1(A) = k3_finseq_1(f1_s17_recdef_1)
& k1_funct_1(A,np__1) = k1_funct_1(f1_s17_recdef_1,np__1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(k3_finseq_1(f1_s17_recdef_1),B)
| k1_funct_1(A,k1_nat_1(B,np__1)) = f2_s17_recdef_1(k1_funct_1(f1_s17_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B)) ) ) ) ) )
=> f3_s17_recdef_1 = f4_s17_recdef_1 ) ).
fof(s18_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
? [C] : p1_s18_recdef_1(A,B,C) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( ( p1_s18_recdef_1(A,B,C)
& p1_s18_recdef_1(A,B,D) )
=> C = D ) ) )
=> ( ? [A,B] :
( v1_relat_1(B)
& v1_funct_1(B)
& A = k1_funct_1(B,f2_s18_recdef_1)
& k1_relat_1(B) = k5_numbers
& k1_funct_1(B,np__0) = f1_s18_recdef_1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> p1_s18_recdef_1(C,k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) )
& ! [A,B] :
~ ( ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& A = k1_funct_1(C,f2_s18_recdef_1)
& k1_relat_1(C) = k5_numbers
& k1_funct_1(C,np__0) = f1_s18_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> p1_s18_recdef_1(D,k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) )
& ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& B = k1_funct_1(C,f2_s18_recdef_1)
& k1_relat_1(C) = k5_numbers
& k1_funct_1(C,np__0) = f1_s18_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> p1_s18_recdef_1(D,k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) )
& A != B ) ) ) ).
fof(s19_recdef_1,axiom,
( ? [A,B] :
( v1_relat_1(B)
& v1_funct_1(B)
& A = k1_funct_1(B,f2_s19_recdef_1)
& k1_relat_1(B) = k5_numbers
& k1_funct_1(B,np__0) = f1_s19_recdef_1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_funct_1(B,k1_nat_1(C,np__1)) = f3_s19_recdef_1(C,k1_funct_1(B,C)) ) )
& ! [A,B] :
~ ( ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& A = k1_funct_1(C,f2_s19_recdef_1)
& k1_relat_1(C) = k5_numbers
& k1_funct_1(C,np__0) = f1_s19_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_funct_1(C,k1_nat_1(D,np__1)) = f3_s19_recdef_1(D,k1_funct_1(C,D)) ) )
& ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& B = k1_funct_1(C,f2_s19_recdef_1)
& k1_relat_1(C) = k5_numbers
& k1_funct_1(C,np__0) = f1_s19_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_funct_1(C,k1_nat_1(D,np__1)) = f3_s19_recdef_1(D,k1_funct_1(C,D)) ) )
& A != B ) ) ).
fof(s20_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,f1_s20_recdef_1)
=> ? [C] :
( m1_subset_1(C,f1_s20_recdef_1)
& p1_s20_recdef_1(A,B,C) ) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,f1_s20_recdef_1)
=> ! [C] :
( m1_subset_1(C,f1_s20_recdef_1)
=> ! [D] :
( m1_subset_1(D,f1_s20_recdef_1)
=> ( ( p1_s20_recdef_1(A,B,C)
& p1_s20_recdef_1(A,B,D) )
=> C = D ) ) ) ) ) )
=> ( ? [A] :
( m1_subset_1(A,f1_s20_recdef_1)
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,f1_s20_recdef_1)
& m2_relset_1(B,k5_numbers,f1_s20_recdef_1)
& A = k8_funct_2(k5_numbers,f1_s20_recdef_1,B,f3_s20_recdef_1)
& k8_funct_2(k5_numbers,f1_s20_recdef_1,B,np__0) = f2_s20_recdef_1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> p1_s20_recdef_1(C,k8_funct_2(k5_numbers,f1_s20_recdef_1,B,C),k8_funct_2(k5_numbers,f1_s20_recdef_1,B,k1_nat_1(C,np__1))) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s20_recdef_1)
=> ! [B] :
( m1_subset_1(B,f1_s20_recdef_1)
=> ~ ( ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,f1_s20_recdef_1)
& m2_relset_1(C,k5_numbers,f1_s20_recdef_1)
& A = k8_funct_2(k5_numbers,f1_s20_recdef_1,C,f3_s20_recdef_1)
& k8_funct_2(k5_numbers,f1_s20_recdef_1,C,np__0) = f2_s20_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> p1_s20_recdef_1(D,k8_funct_2(k5_numbers,f1_s20_recdef_1,C,D),k8_funct_2(k5_numbers,f1_s20_recdef_1,C,k1_nat_1(D,np__1))) ) )
& ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,f1_s20_recdef_1)
& m2_relset_1(C,k5_numbers,f1_s20_recdef_1)
& B = k8_funct_2(k5_numbers,f1_s20_recdef_1,C,f3_s20_recdef_1)
& k8_funct_2(k5_numbers,f1_s20_recdef_1,C,np__0) = f2_s20_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> p1_s20_recdef_1(D,k8_funct_2(k5_numbers,f1_s20_recdef_1,C,D),k8_funct_2(k5_numbers,f1_s20_recdef_1,C,k1_nat_1(D,np__1))) ) )
& A != B ) ) ) ) ) ).
fof(s21_recdef_1,axiom,
( ? [A] :
( m1_subset_1(A,f1_s21_recdef_1)
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,f1_s21_recdef_1)
& m2_relset_1(B,k5_numbers,f1_s21_recdef_1)
& A = k8_funct_2(k5_numbers,f1_s21_recdef_1,B,f3_s21_recdef_1)
& k8_funct_2(k5_numbers,f1_s21_recdef_1,B,np__0) = f2_s21_recdef_1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s21_recdef_1,B,k1_nat_1(C,np__1)) = f4_s21_recdef_1(C,k8_funct_2(k5_numbers,f1_s21_recdef_1,B,C)) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s21_recdef_1)
=> ! [B] :
( m1_subset_1(B,f1_s21_recdef_1)
=> ~ ( ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,f1_s21_recdef_1)
& m2_relset_1(C,k5_numbers,f1_s21_recdef_1)
& A = k8_funct_2(k5_numbers,f1_s21_recdef_1,C,f3_s21_recdef_1)
& k8_funct_2(k5_numbers,f1_s21_recdef_1,C,np__0) = f2_s21_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s21_recdef_1,C,k1_nat_1(D,np__1)) = f4_s21_recdef_1(D,k8_funct_2(k5_numbers,f1_s21_recdef_1,C,D)) ) )
& ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,f1_s21_recdef_1)
& m2_relset_1(C,k5_numbers,f1_s21_recdef_1)
& B = k8_funct_2(k5_numbers,f1_s21_recdef_1,C,f3_s21_recdef_1)
& k8_funct_2(k5_numbers,f1_s21_recdef_1,C,np__0) = f2_s21_recdef_1
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s21_recdef_1,C,k1_nat_1(D,np__1)) = f4_s21_recdef_1(D,k8_funct_2(k5_numbers,f1_s21_recdef_1,C,D)) ) )
& A != B ) ) ) ) ).
fof(s22_recdef_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
~ ( r1_xreal_0(np__1,A)
& ~ r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),A)
& ! [C] : ~ p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(A,np__1)),B,C) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D,E] :
( ( r1_xreal_0(np__1,A)
& E = k1_funct_1(f1_s22_recdef_1,k1_nat_1(A,np__1))
& p1_s22_recdef_1(E,B,C)
& p1_s22_recdef_1(E,B,D) )
=> ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),A)
| C = D ) ) ) )
=> ( ? [A,B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& A = k1_funct_1(B,k3_finseq_1(B))
& k3_finseq_1(B) = k3_finseq_1(f1_s22_recdef_1)
& k1_funct_1(B,np__1) = k1_funct_1(f1_s22_recdef_1,np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),C)
| p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) ) ) )
& ! [A,B] :
~ ( ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& A = k1_funct_1(C,k3_finseq_1(C))
& k3_finseq_1(C) = k3_finseq_1(f1_s22_recdef_1)
& k1_funct_1(C,np__1) = k1_funct_1(f1_s22_recdef_1,np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),D)
| p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) ) ) )
& ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& B = k1_funct_1(C,k3_finseq_1(C))
& k3_finseq_1(C) = k3_finseq_1(f1_s22_recdef_1)
& k1_funct_1(C,np__1) = k1_funct_1(f1_s22_recdef_1,np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),D)
| p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) ) ) )
& A != B ) ) ) ).
fof(s23_recdef_1,axiom,
( ? [A,B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& A = k1_funct_1(B,k3_finseq_1(B))
& k3_finseq_1(B) = k3_finseq_1(f1_s23_recdef_1)
& k1_funct_1(B,np__1) = k1_funct_1(f1_s23_recdef_1,np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> ( r1_xreal_0(k3_finseq_1(f1_s23_recdef_1),C)
| k1_funct_1(B,k1_nat_1(C,np__1)) = f2_s23_recdef_1(k1_funct_1(f1_s23_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C)) ) ) ) )
& ! [A,B] :
~ ( ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& A = k1_funct_1(C,k3_finseq_1(C))
& k3_finseq_1(C) = k3_finseq_1(f1_s23_recdef_1)
& k1_funct_1(C,np__1) = k1_funct_1(f1_s23_recdef_1,np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(f1_s23_recdef_1),D)
| k1_funct_1(C,k1_nat_1(D,np__1)) = f2_s23_recdef_1(k1_funct_1(f1_s23_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D)) ) ) ) )
& ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& B = k1_funct_1(C,k3_finseq_1(C))
& k3_finseq_1(C) = k3_finseq_1(f1_s23_recdef_1)
& k1_funct_1(C,np__1) = k1_funct_1(f1_s23_recdef_1,np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,D)
=> ( r1_xreal_0(k3_finseq_1(f1_s23_recdef_1),D)
| k1_funct_1(C,k1_nat_1(D,np__1)) = f2_s23_recdef_1(k1_funct_1(f1_s23_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D)) ) ) ) )
& A != B ) ) ).
fof(dt_k1_recdef_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& m1_relset_1(B,A,k5_numbers)
& m1_subset_1(C,A) )
=> m2_subset_1(k1_recdef_1(A,B,C),k1_numbers,k5_numbers) ) ).
fof(redefinition_k1_recdef_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& m1_relset_1(B,A,k5_numbers)
& m1_subset_1(C,A) )
=> k1_recdef_1(A,B,C) = k1_funct_1(B,C) ) ).
%------------------------------------------------------------------------------