SET007 Axioms: SET007+182.ax
%------------------------------------------------------------------------------
% File : SET007+182 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Preliminaries to Circuits, I
% 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_circ [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 50 ( 7 unt; 0 def)
% Number of atoms : 331 ( 45 equ)
% Maximal formula atoms : 26 ( 6 avg)
% Number of connectives : 313 ( 32 ~; 2 |; 167 &)
% ( 6 <=>; 106 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 37 ( 35 usr; 1 prp; 0-4 aty)
% Number of functors : 61 ( 61 usr; 14 con; 0-4 aty)
% Number of variables : 131 ( 121 !; 10 ?)
% 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(rc1_pre_circ,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A) ) ).
fof(fc1_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v5_membered(A) )
=> ( v4_ordinal2(k2_seq_4(A))
& v1_xcmplx_0(k2_seq_4(A))
& v1_xreal_0(k2_seq_4(A)) ) ) ).
fof(rc2_pre_circ,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v1_pre_circ(B,A) ) ).
fof(fc2_pre_circ,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v2_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B)) ) ) ).
fof(fc3_pre_circ,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k5_relat_1(B,A))
& v2_relat_1(k5_relat_1(B,A))
& v1_funct_1(k5_relat_1(B,A)) ) ) ).
fof(cc1_pre_circ,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k5_trees_3(A))
=> v1_finset_1(B) ) ) ).
fof(fc4_pre_circ,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& m1_subset_1(B,k1_relat_1(A)) )
=> ( v1_relat_1(k5_trees_2(A,B))
& v1_funct_1(k5_trees_2(A,B))
& v1_finset_1(k5_trees_2(A,B))
& v3_trees_2(k5_trees_2(A,B)) ) ) ).
fof(fc5_pre_circ,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A)
& m1_subset_1(B,k1_relat_1(A))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C)
& v3_trees_2(C) )
=> ( v1_relat_1(k8_trees_2(A,B,C))
& v1_funct_1(k8_trees_2(A,B,C))
& v1_finset_1(k8_trees_2(A,B,C))
& v3_trees_2(k8_trees_2(A,B,C)) ) ) ).
fof(fc6_pre_circ,axiom,
! [A] :
( v1_relat_1(k1_trees_4(A))
& v1_funct_1(k1_trees_4(A))
& v1_finset_1(k1_trees_4(A))
& v3_trees_2(k1_trees_4(A)) ) ).
fof(t1_pre_circ,axiom,
$true ).
fof(t2_pre_circ,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( k1_relat_1(A) = k1_tarski(B)
& k2_relat_1(A) = k1_tarski(C) )
=> A = k1_tarski(k4_tarski(B,C)) ) ) ).
fof(t3_pre_circ,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_tarski(A,B)
=> r1_tarski(k1_funct_4(A,C),k1_funct_4(B,C)) ) ) ) ) ).
fof(t4_pre_circ,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_tarski(A,B)
& r1_xboole_0(k1_relat_1(A),k1_relat_1(C)) )
=> r1_tarski(A,k1_funct_4(B,C)) ) ) ) ) ).
fof(d1_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v2_membered(A) )
=> ! [B] :
( v1_xreal_0(B)
=> ( B = k1_pre_circ(A)
<=> ( r2_hidden(B,A)
& ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r1_xreal_0(C,B) ) ) ) ) ) ) ).
fof(t5_pre_circ,axiom,
! [A,B] :
( m1_pboole(B,A)
=> k1_funct_1(k6_pboole(A,B),k6_finseq_1(A)) = k1_tarski(k1_xboole_0) ) ).
fof(d2_pre_circ,axiom,
$true ).
fof(d3_pre_circ,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_pre_circ(B,A)
<=> ! [C] :
( r2_hidden(C,A)
=> v1_finset_1(k1_funct_1(B,C)) ) ) ) ).
fof(t6_pre_circ,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ~ v1_xboole_0(k3_tarski(k2_relat_1(B))) ) ) ).
fof(t7_pre_circ,axiom,
! [A] : k4_funct_5(k2_pre_circ(A,k1_xboole_0)) = k1_xboole_0 ).
fof(t8_pre_circ,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,k2_pre_circ(A,B),C)
=> k1_relat_1(k10_funct_6(D)) = B ) ) ) ).
fof(t9_pre_circ,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( m1_subset_1(D,k4_card_3(B))
=> ( ( r1_tarski(k1_relat_1(C),k1_relat_1(B))
& ! [E] :
( r2_hidden(E,k1_relat_1(C))
=> r2_hidden(k1_funct_1(C,E),k1_funct_1(B,E)) ) )
=> m1_subset_1(k1_funct_4(D,C),k4_card_3(B)) ) ) ) ) ).
fof(t10_pre_circ,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,k2_pre_circ(A,B),C)
=> ! [E] :
( m1_subset_1(E,B)
=> ? [F] :
( m1_pboole(F,A)
& F = k1_funct_1(k10_funct_6(D),E)
& r3_pboole(A,F,C) ) ) ) ) ) ) ).
fof(t11_pre_circ,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(C,k4_card_3(B))
=> r2_hidden(k5_relat_1(D,C),k4_card_3(k5_relat_1(D,B))) ) ) ) ) ).
fof(t12_pre_circ,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] : k4_card_3(k2_finseq_2(A,k1_tarski(B))) = k1_tarski(k2_finseq_2(A,B)) ) ).
fof(t15_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B) )
=> ! [C] :
( m1_trees_1(C,A)
=> ! [D] :
( m2_finseq_1(D,k5_numbers)
=> ~ ( r2_hidden(D,k5_trees_1(A,C,B))
& r1_tarski(C,D)
& ! [E] :
( m1_trees_1(E,B)
=> D != k8_finseq_1(k5_numbers,C,E) ) ) ) ) ) ) ).
fof(t16_pre_circ,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v4_trees_3(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A))
=> k4_trees_1(k13_trees_3(A),k12_finseq_1(k5_numbers,B)) = k1_funct_1(A,k1_nat_1(B,np__1)) ) ) ) ).
fof(t17_pre_circ,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v6_trees_3(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A))
=> r2_hidden(k12_finseq_1(k5_numbers,B),k13_trees_3(k2_funct_6(A))) ) ) ) ).
fof(t18_pre_circ,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v4_trees_3(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v4_trees_3(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(A))
=> ( D = k1_nat_1(C,np__1)
| k1_funct_1(A,D) = k1_funct_1(B,D) ) ) ) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_trees_1(D) )
=> ( k1_funct_1(B,k1_nat_1(C,np__1)) = D
=> k13_trees_3(B) = k5_trees_1(k13_trees_3(A),k12_finseq_1(k5_numbers,C),D) ) ) ) ) ) ) ).
fof(t19_pre_circ,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B) )
=> ! [C,D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E)
& v6_trees_3(E) )
=> ~ ( r2_hidden(k12_finseq_1(k5_numbers,D),k1_relat_1(A))
& A = k4_trees_4(C,E)
& ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F)
& v1_finseq_1(F)
& v6_trees_3(F) )
=> ~ ( k8_trees_2(A,k12_finseq_1(k5_numbers,D),B) = k4_trees_4(C,F)
& k3_finseq_1(F) = k3_finseq_1(E)
& k1_funct_1(F,k1_nat_1(D,np__1)) = B
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k4_finseq_1(E))
=> ( G = k1_nat_1(D,np__1)
| k1_funct_1(F,G) = k1_funct_1(E,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> ~ ( B != k1_xboole_0
& r1_xreal_0(k4_card_1(A),k4_card_1(k4_trees_1(A,B))) ) ) ) ).
fof(t21_pre_circ,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k1_card_1(A) = k1_card_1(k1_relat_1(A)) ) ).
fof(t22_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B) )
=> ! [C] :
( m1_trees_1(C,A)
=> k1_nat_1(k4_card_1(k5_trees_1(A,C,B)),k4_card_1(k4_trees_1(A,C))) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ) ).
fof(t23_pre_circ,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v3_trees_2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v3_trees_2(B) )
=> ! [C] :
( m1_trees_1(C,k1_relat_1(A))
=> k1_nat_1(k4_card_1(k8_trees_2(A,C,B)),k4_card_1(k5_trees_2(A,C))) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ) ).
fof(t24_pre_circ,axiom,
! [A] : k4_card_1(k1_trees_4(A)) = np__1 ).
fof(s2_pre_circ,axiom,
? [A] :
( m1_pboole(A,f1_s2_pre_circ)
& ! [B] :
( m1_subset_1(B,f1_s2_pre_circ)
=> ( r2_hidden(B,f1_s2_pre_circ)
=> ( ( p1_s2_pre_circ(B)
=> k1_funct_1(A,B) = f2_s2_pre_circ(B) )
& ( ~ p1_s2_pre_circ(B)
=> k1_funct_1(A,B) = f3_s2_pre_circ(B) ) ) ) ) ) ).
fof(s3_pre_circ,axiom,
( ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s3_pre_circ)
& m2_relset_1(A,k5_numbers,f1_s3_pre_circ)
& k8_funct_2(k5_numbers,f1_s3_pre_circ,A,np__0) = f2_s3_pre_circ
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s3_pre_circ,A,k1_nat_1(B,np__1)) = f3_s3_pre_circ(B,k8_funct_2(k5_numbers,f1_s3_pre_circ,A,B)) ) )
& ! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s3_pre_circ)
& m2_relset_1(A,k5_numbers,f1_s3_pre_circ) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,f1_s3_pre_circ)
& m2_relset_1(B,k5_numbers,f1_s3_pre_circ) )
=> ( ( k8_funct_2(k5_numbers,f1_s3_pre_circ,A,np__0) = f2_s3_pre_circ
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s3_pre_circ,A,k1_nat_1(C,np__1)) = f3_s3_pre_circ(C,k8_funct_2(k5_numbers,f1_s3_pre_circ,A,C)) )
& k8_funct_2(k5_numbers,f1_s3_pre_circ,B,np__0) = f2_s3_pre_circ
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s3_pre_circ,B,k1_nat_1(C,np__1)) = f3_s3_pre_circ(C,k8_funct_2(k5_numbers,f1_s3_pre_circ,B,C)) ) )
=> A = B ) ) ) ) ).
fof(s4_pre_circ,axiom,
( ! [A] :
( m1_subset_1(A,f1_s4_pre_circ)
=> ( r2_hidden(A,f2_s4_pre_circ)
=> ( v1_funct_1(f5_s4_pre_circ(A))
& v1_funct_2(f5_s4_pre_circ(A),k1_funct_1(f3_s4_pre_circ,A),k1_funct_1(f4_s4_pre_circ,A))
& m2_relset_1(f5_s4_pre_circ(A),k1_funct_1(f3_s4_pre_circ,A),k1_funct_1(f4_s4_pre_circ,A)) ) ) )
=> ? [A] :
( m3_pboole(A,f2_s4_pre_circ,f3_s4_pre_circ,f4_s4_pre_circ)
& ! [B] :
( m1_subset_1(B,f1_s4_pre_circ)
=> ( r2_hidden(B,f2_s4_pre_circ)
=> k1_funct_1(A,B) = f5_s4_pre_circ(B) ) ) ) ) ).
fof(dt_k1_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v2_membered(A) )
=> v1_xreal_0(k1_pre_circ(A)) ) ).
fof(redefinition_k1_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v2_membered(A) )
=> k1_pre_circ(A) = k2_seq_4(A) ) ).
fof(dt_k2_pre_circ,axiom,
! [A,B] : m1_pboole(k2_pre_circ(A,B),A) ).
fof(redefinition_k2_pre_circ,axiom,
! [A,B] : k2_pre_circ(A,B) = k2_funcop_1(A,B) ).
fof(dt_k3_pre_circ,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_pboole(k3_pre_circ(A,B,C),C) ) ).
fof(redefinition_k3_pre_circ,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k3_pre_circ(A,B,C) = k7_relat_1(B,C) ) ).
fof(t13_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> r2_tarski(k4_trees_1(A,B),a_2_0_pre_circ(A,B)) ) ) ).
fof(t14_pre_circ,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_trees_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B) )
=> ! [C] :
( m1_trees_1(C,A)
=> k5_trees_1(A,C,B) = k2_xboole_0(a_2_1_pre_circ(A,C),a_3_0_pre_circ(A,B,C)) ) ) ) ).
fof(s1_pre_circ,axiom,
v1_finset_1(a_0_0_pre_circ) ).
fof(fraenkel_a_2_0_pre_circ,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B)
& m1_trees_1(C,B) )
=> ( r2_hidden(A,a_2_0_pre_circ(B,C))
<=> ? [D] :
( m1_trees_1(D,B)
& A = D
& r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_1_pre_circ,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B)
& m1_trees_1(C,B) )
=> ( r2_hidden(A,a_2_1_pre_circ(B,C))
<=> ? [D] :
( m1_trees_1(D,B)
& A = D
& ~ r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_3_0_pre_circ,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& v1_trees_1(B)
& ~ v1_xboole_0(C)
& v1_finset_1(C)
& v1_trees_1(C)
& m1_trees_1(D,B) )
=> ( r2_hidden(A,a_3_0_pre_circ(B,C,D))
<=> ? [E] :
( m1_trees_1(E,C)
& A = k8_finseq_1(k5_numbers,D,E) ) ) ) ).
fof(fraenkel_a_0_0_pre_circ,axiom,
! [A] :
( r2_hidden(A,a_0_0_pre_circ)
<=> ? [B] :
( m1_subset_1(B,f1_s1_pre_circ)
& A = f2_s1_pre_circ(B)
& p1_s1_pre_circ(B) ) ) ).
%------------------------------------------------------------------------------