SET007 Axioms: SET007+540.ax
%------------------------------------------------------------------------------
% File : SET007+540 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Full Trees
% 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 : bintree2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 39 ( 0 unt; 0 def)
% Number of atoms : 256 ( 34 equ)
% Maximal formula atoms : 15 ( 6 avg)
% Number of connectives : 272 ( 55 ~; 0 |; 124 &)
% ( 4 <=>; 89 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-3 aty)
% Number of functors : 44 ( 44 usr; 11 con; 0-3 aty)
% Number of variables : 87 ( 79 !; 8 ?)
% 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_bintree2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k1_bintree2(A,B))
& v1_funct_1(k1_bintree2(A,B))
& v2_funct_1(k1_bintree2(A,B))
& v1_funct_2(k1_bintree2(A,B),k2_trees_2(A,B),k5_numbers) ) ) ).
fof(fc2_bintree2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A)
& m1_subset_1(B,k5_numbers) )
=> v1_finset_1(k2_trees_2(A,B)) ) ).
fof(cc1_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_trees_2(A)
& v2_trees_9(A)
& v1_bintree1(A) ) ) ).
fof(rc1_bintree2,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_trees_2(A)
& v2_trees_9(A)
& v1_bintree1(A)
& v1_bintree2(A) ) ).
fof(fc3_bintree2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k2_bintree2(A,B))
& v1_funct_1(k2_bintree2(A,B))
& v2_funct_1(k2_bintree2(A,B))
& v1_finset_1(k2_bintree2(A,B))
& v1_finseq_1(k2_bintree2(A,B)) ) ) ).
fof(t1_bintree2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k3_finseq_2(A))
=> r2_hidden(k7_relat_1(B,k2_finseq_1(C)),k3_finseq_2(A)) ) ) ) ).
fof(t2_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A) )
=> ! [B] :
( m1_trees_1(B,A)
=> m2_finseq_1(B,k6_margrel1) ) ) ).
fof(t3_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
=> v1_bintree1(A) ) ) ).
fof(t4_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
=> k3_trees_1(A) = k1_xboole_0 ) ) ).
fof(t5_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_bintree2(C,A)
=> ( r2_hidden(C,k2_trees_2(A,B))
=> m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1)) ) ) ) ) ).
fof(t6_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( ! [B] :
( m1_trees_1(B,A)
=> k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1))) )
=> k3_trees_1(A) = k1_xboole_0 ) ) ).
fof(t7_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( ! [B] :
( m1_trees_1(B,A)
=> k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1))) )
=> v1_bintree1(A) ) ) ).
fof(t8_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
<=> ! [B] :
( m1_trees_1(B,A)
=> k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1))) ) ) ) ).
fof(d1_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_trees_2(A,B),k5_numbers)
& m2_relset_1(C,k2_trees_2(A,B),k5_numbers) )
=> ( C = k1_bintree2(A,B)
<=> ! [D] :
( m1_bintree2(D,A)
=> ( r2_hidden(D,k2_trees_2(A,B))
=> ! [E] :
( m2_finseq_2(E,k6_margrel1,k4_finseq_2(B,k6_margrel1))
=> ( E = k4_finseq_5(k6_margrel1,D)
=> k1_funct_1(C,D) = k1_nat_1(k9_binarith(B,E),np__1) ) ) ) ) ) ) ) ) ).
fof(d2_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( v1_bintree2(A)
<=> A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1)) ) ) ).
fof(t9_bintree2,axiom,
( ~ v1_xboole_0(k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1)))
& v1_trees_1(k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))) ) ).
fof(t10_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_hidden(k5_euclid(B),k2_trees_2(A,B)) ) ) ) ).
fof(t11_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A) )
=> ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
=> r2_hidden(C,k2_trees_2(A,B)) ) ) ) ) ).
fof(t12_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> r1_tarski(k2_finseq_1(k3_series_1(np__2,B)),k2_relat_1(k1_bintree2(A,B))) ) ) ).
fof(d3_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k2_bintree2(A,B) = k2_funct_1(k1_bintree2(A,B)) ) ) ).
fof(t13_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k1_funct_1(k1_bintree2(A,B),k5_euclid(B)) = np__1 ) ) ).
fof(t14_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
=> ( C = k5_euclid(B)
=> k1_funct_1(k1_bintree2(A,B),k6_binarith(B,C)) = k3_series_1(np__2,B) ) ) ) ) ).
fof(t15_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k1_funct_1(k2_bintree2(A,B),np__1) = k5_euclid(B) ) ) ).
fof(t16_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
=> ( C = k5_euclid(B)
=> k1_funct_1(k2_bintree2(A,B),k3_series_1(np__2,B)) = k6_binarith(B,C) ) ) ) ) ).
fof(t17_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(k3_series_1(np__2,B)))
=> k1_funct_1(k2_bintree2(A,B),C) = k4_finseq_5(k6_margrel1,k1_binari_3(B,k5_binarith(C,np__1))) ) ) ) ) ).
fof(t18_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_card_1(k2_trees_2(A,B)) = k3_series_1(np__2,B) ) ) ).
fof(t19_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k3_finseq_1(k2_bintree2(A,B)) = k3_series_1(np__2,B) ) ) ).
fof(t20_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k4_finseq_1(k2_bintree2(A,B)) = k2_finseq_1(k3_series_1(np__2,B)) ) ) ).
fof(t21_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k2_relat_1(k2_bintree2(A,B)) = k2_trees_2(A,B) ) ) ).
fof(t22_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> k1_funct_1(k2_bintree2(A,np__1),np__1) = k13_binarith(k5_numbers,np__0) ) ).
fof(t23_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> k1_funct_1(k2_bintree2(A,np__1),np__2) = k13_binarith(k5_numbers,np__1) ) ).
fof(t24_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r1_xreal_0(C,k3_series_1(np__2,k1_nat_1(B,np__1)))
=> ! [D] :
( m2_finseq_2(D,k6_margrel1,k4_finseq_2(B,k6_margrel1))
=> ( D = k1_funct_1(k2_bintree2(A,B),k3_nat_1(k1_nat_1(C,np__1),np__2))
=> k1_funct_1(k2_bintree2(A,k1_nat_1(B,np__1)),C) = k7_finseq_1(D,k13_binarith(k5_numbers,k4_nat_1(k1_nat_1(C,np__1),np__2))) ) ) ) ) ) ) ).
fof(s1_bintree2,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_bintree2)
=> ? [B] :
( m1_subset_1(B,f1_s1_bintree2)
& ? [C] :
( m1_subset_1(C,f1_s1_bintree2)
& p1_s1_bintree2(A,B,C) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v3_trees_2(A)
& v2_bintree1(A)
& m3_trees_2(A,f1_s1_bintree2)
& k1_relat_1(A) = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
& k1_funct_1(A,k1_xboole_0) = f2_s1_bintree2
& ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> p1_s1_bintree2(k3_trees_2(f1_s1_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0))),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1)))) ) ) ) ).
fof(s2_bintree2,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s2_bintree2)
=> ? [B] :
( m1_subset_1(B,f1_s2_bintree2)
& p1_s2_bintree2(A,B) ) )
& ! [A] :
( m1_subset_1(A,f1_s2_bintree2)
=> ? [B] :
( m1_subset_1(B,f1_s2_bintree2)
& p2_s2_bintree2(A,B) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v3_trees_2(A)
& v2_bintree1(A)
& m3_trees_2(A,f1_s2_bintree2)
& k1_relat_1(A) = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
& k1_funct_1(A,k1_xboole_0) = f2_s2_bintree2
& ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ( p1_s2_bintree2(k3_trees_2(f1_s2_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0))))
& p2_s2_bintree2(k3_trees_2(f1_s2_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1)))) ) ) ) ) ).
fof(dt_m1_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A) )
=> ! [B] :
( m1_bintree2(B,A)
=> m2_finseq_1(B,k6_margrel1) ) ) ).
fof(existence_m1_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A) )
=> ? [B] : m1_bintree2(B,A) ) ).
fof(redefinition_m1_bintree2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A) )
=> ! [B] :
( m1_bintree2(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_k1_bintree2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> ( v1_funct_1(k1_bintree2(A,B))
& v1_funct_2(k1_bintree2(A,B),k2_trees_2(A,B),k5_numbers)
& m2_relset_1(k1_bintree2(A,B),k2_trees_2(A,B),k5_numbers) ) ) ).
fof(dt_k2_bintree2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_trees_1(A)
& v1_bintree2(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> m1_trees_4(k2_bintree2(A,B),A,k2_trees_2(A,B)) ) ).
%------------------------------------------------------------------------------