SET007 Axioms: SET007+321.ax
%------------------------------------------------------------------------------
% File : SET007+321 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Cyclic Groups and Some of Their Properties - Part 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 : gr_cy_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 70 ( 21 unt; 0 def)
% Number of atoms : 312 ( 38 equ)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 288 ( 46 ~; 2 |; 134 &)
% ( 9 <=>; 97 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 31 ( 29 usr; 1 prp; 0-3 aty)
% Number of functors : 45 ( 45 usr; 8 con; 0-6 aty)
% Number of variables : 86 ( 76 !; 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_gr_cy_1,axiom,
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A) ) ).
fof(cc1_gr_cy_1,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v7_group_1(A) ) ) ) ).
fof(t1_gr_cy_1,axiom,
$true ).
fof(t2_gr_cy_1,axiom,
$true ).
fof(t3_gr_cy_1,axiom,
$true ).
fof(t4_gr_cy_1,axiom,
$true ).
fof(t5_gr_cy_1,axiom,
$true ).
fof(t6_gr_cy_1,axiom,
$true ).
fof(t7_gr_cy_1,axiom,
$true ).
fof(t8_gr_cy_1,axiom,
$true ).
fof(t9_gr_cy_1,axiom,
$true ).
fof(t10_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,np__0)
=> ( r2_hidden(B,k1_gr_cy_1(A))
<=> ~ r1_xreal_0(A,B) ) ) ) ) ).
fof(t11_gr_cy_1,axiom,
$true ).
fof(t12_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> r2_hidden(np__0,k1_gr_cy_1(A)) ) ) ).
fof(t13_gr_cy_1,axiom,
k1_gr_cy_1(np__1) = k18_group_2(k5_numbers,np__0) ).
fof(d2_gr_cy_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers)
& m2_relset_1(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers) )
=> ( A = k44_binop_2
<=> ! [B] :
( m1_subset_1(B,k4_numbers)
=> ! [C] :
( m1_subset_1(C,k4_numbers)
=> k2_binop_1(k4_numbers,k4_numbers,k4_numbers,A,B,C) = k1_binop_1(k33_binop_2,B,C) ) ) ) ) ).
fof(t14_gr_cy_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> k1_binop_1(k44_binop_2,A,B) = k2_xcmplx_0(A,B) ) ) ).
fof(t15_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,k4_numbers)
=> ( A = np__0
=> r3_binop_1(k4_numbers,A,k44_binop_2) ) ) ).
fof(d3_gr_cy_1,axiom,
! [A] :
( m2_finseq_1(A,k4_numbers)
=> k2_gr_cy_1(A) = k2_finsop_1(k4_numbers,A,k44_binop_2) ) ).
fof(t16_gr_cy_1,axiom,
$true ).
fof(t17_gr_cy_1,axiom,
$true ).
fof(t18_gr_cy_1,axiom,
$true ).
fof(t19_gr_cy_1,axiom,
$true ).
fof(t20_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,k4_numbers)
=> ! [B] :
( m2_finseq_1(B,k4_numbers)
=> k2_gr_cy_1(k8_finseq_1(k4_numbers,B,k12_finseq_1(k4_numbers,A))) = k2_xcmplx_0(k2_gr_cy_1(B),k2_group_4(A)) ) ) ).
fof(t21_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,k4_numbers)
=> k2_gr_cy_1(k12_finseq_1(k4_numbers,A)) = A ) ).
fof(t22_gr_cy_1,axiom,
k2_gr_cy_1(k6_finseq_1(k4_numbers)) = np__0 ).
fof(t23_gr_cy_1,axiom,
$true ).
fof(t24_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,k4_numbers)
=> k3_group_4(A,k4_group_4(A,C,k1_finsop_1(u1_struct_0(A),k3_finseq_1(C),B))) = k6_group_1(A,k2_gr_cy_1(C),B) ) ) ) ).
fof(t25_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_rlvect_1(k5_group_4(A,k1_struct_0(A,C)),B)
<=> ? [D] :
( v1_int_1(D)
& B = k6_group_1(A,D,C) ) ) ) ) ) ).
fof(t26_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( v6_group_1(A)
& v5_group_1(B,A) ) ) ) ).
fof(t27_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v6_group_1(A)
=> k7_group_1(A,B) = k9_group_1(k5_group_4(A,k1_struct_0(A,B))) ) ) ) ).
fof(t28_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v6_group_1(A)
=> r1_nat_1(k7_group_1(A,B),k9_group_1(A)) ) ) ) ).
fof(t29_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v6_group_1(A)
=> k6_group_1(A,k9_group_1(A),B) = k2_group_1(A) ) ) ) ).
fof(t30_gr_cy_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( v6_group_1(B)
=> k3_group_1(B,k6_group_1(B,A,C)) = k6_group_1(B,k10_binop_2(k9_group_1(B),k4_nat_1(A,k9_group_1(B))),C) ) ) ) ) ).
fof(t31_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ~ ( ~ r1_xreal_0(k9_group_1(A),np__1)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> B = k2_group_1(A) ) ) ) ).
fof(t32_gr_cy_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ( ( v6_group_1(B)
& k9_group_1(B) = A
& v1_int_2(A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,B) )
=> ( r1_group_2(B,C,k5_group_2(B))
| C = B ) ) ) ) ) ).
fof(t33_gr_cy_1,axiom,
( v4_group_1(g1_group_1(k4_numbers,k44_binop_2))
& v3_group_1(g1_group_1(k4_numbers,k44_binop_2)) ) ).
fof(d4_gr_cy_1,axiom,
k3_gr_cy_1 = g1_group_1(k4_numbers,k44_binop_2) ).
fof(d5_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A))
& m2_relset_1(B,k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A)) )
=> ( B = k4_gr_cy_1(A)
<=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_gr_cy_1(A))
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_gr_cy_1(A))
=> k2_binop_1(k1_gr_cy_1(A),k1_gr_cy_1(A),k1_gr_cy_1(A),B,C,D) = k4_nat_1(k1_nat_1(C,D),A) ) ) ) ) ) ) ).
fof(t34_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ( v4_group_1(g1_group_1(k1_gr_cy_1(A),k4_gr_cy_1(A)))
& v3_group_1(g1_group_1(k1_gr_cy_1(A),k4_gr_cy_1(A))) ) ) ) ).
fof(d6_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k5_gr_cy_1(A) = g1_group_1(k1_gr_cy_1(A),k4_gr_cy_1(A)) ) ) ).
fof(t35_gr_cy_1,axiom,
k2_group_1(k3_gr_cy_1) = np__0 ).
fof(t36_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k2_group_1(k5_gr_cy_1(A)) = np__0 ) ) ).
fof(d7_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
=> k6_gr_cy_1(A) = A ) ).
fof(d8_gr_cy_1,axiom,
! [A] :
( v1_int_1(A)
=> k7_gr_cy_1(A) = A ) ).
fof(t37_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
=> k3_group_1(k3_gr_cy_1,A) = k4_xcmplx_0(k6_gr_cy_1(A)) ) ).
fof(t38_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
=> ( A = np__1
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k6_group_1(k3_gr_cy_1,B,A) = B ) ) ) ).
fof(t39_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
=> ! [B] :
( v1_int_1(B)
=> ( A = np__1
=> B = k6_group_1(k3_gr_cy_1,B,A) ) ) ) ).
fof(d9_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_gr_cy_1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& g1_group_1(u1_struct_0(A),u1_group_1(A)) = k5_group_4(A,k1_struct_0(A,B)) ) ) ) ).
fof(t40_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> v1_gr_cy_1(k5_group_2(A)) ) ).
fof(t41_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) )
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( v1_int_1(D)
& C = k6_group_1(A,D,B) ) ) ) ) ) ).
fof(t42_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v6_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) )
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& C = k6_group_1(A,D,B) ) ) ) ) ) ) ).
fof(t43_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v6_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) )
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& k7_group_1(A,B) = k9_group_1(A) ) ) ) ) ).
fof(t44_gr_cy_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ( ( v6_group_1(A)
& ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) ) ) ) ) ).
fof(t45_gr_cy_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ( ( v6_group_1(B)
& k9_group_1(B) = A
& v1_int_2(A) )
=> ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) ) ) ) ) ).
fof(t46_gr_cy_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ! [B] :
( m1_subset_1(B,u1_struct_0(k5_gr_cy_1(A)))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k5_gr_cy_1(A)))
& ! [D] :
( v1_int_1(D)
=> C != k6_group_1(k5_gr_cy_1(A),D,B) ) ) ) ) ) ).
fof(t47_gr_cy_1,axiom,
$true ).
fof(t48_gr_cy_1,axiom,
v1_gr_cy_1(k3_gr_cy_1) ).
fof(t49_gr_cy_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ( ~ v3_struct_0(k5_gr_cy_1(A))
& v3_group_1(k5_gr_cy_1(A))
& v4_group_1(k5_gr_cy_1(A))
& v1_gr_cy_1(k5_gr_cy_1(A))
& l1_group_1(k5_gr_cy_1(A)) ) ) ) ).
fof(t50_gr_cy_1,axiom,
( ~ v3_struct_0(k3_gr_cy_1)
& v3_group_1(k3_gr_cy_1)
& v4_group_1(k3_gr_cy_1)
& v7_group_1(k3_gr_cy_1)
& l1_group_1(k3_gr_cy_1) ) ).
fof(t51_gr_cy_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> ( r1_xreal_0(A,np__0)
| r1_tarski(k1_gr_cy_1(A),k1_gr_cy_1(B)) ) ) ) ) ).
fof(dt_k1_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ v1_xboole_0(k1_gr_cy_1(A))
& m1_subset_1(k1_gr_cy_1(A),k1_zfmisc_1(k5_numbers)) ) ) ).
fof(dt_k2_gr_cy_1,axiom,
! [A] :
( m1_finseq_1(A,k4_numbers)
=> v1_int_1(k2_gr_cy_1(A)) ) ).
fof(dt_k3_gr_cy_1,axiom,
( ~ v3_struct_0(k3_gr_cy_1)
& v1_group_1(k3_gr_cy_1)
& v3_group_1(k3_gr_cy_1)
& v4_group_1(k3_gr_cy_1)
& l1_group_1(k3_gr_cy_1) ) ).
fof(dt_k4_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_funct_1(k4_gr_cy_1(A))
& v1_funct_2(k4_gr_cy_1(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A))
& m2_relset_1(k4_gr_cy_1(A),k2_zfmisc_1(k1_gr_cy_1(A),k1_gr_cy_1(A)),k1_gr_cy_1(A)) ) ) ).
fof(dt_k5_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ v3_struct_0(k5_gr_cy_1(A))
& v1_group_1(k5_gr_cy_1(A))
& v3_group_1(k5_gr_cy_1(A))
& v4_group_1(k5_gr_cy_1(A))
& l1_group_1(k5_gr_cy_1(A)) ) ) ).
fof(dt_k6_gr_cy_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_gr_cy_1))
=> v1_int_1(k6_gr_cy_1(A)) ) ).
fof(dt_k7_gr_cy_1,axiom,
! [A] :
( v1_int_1(A)
=> m1_subset_1(k7_gr_cy_1(A),u1_struct_0(k3_gr_cy_1)) ) ).
fof(d1_gr_cy_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k1_gr_cy_1(A) = a_1_0_gr_cy_1(A) ) ) ).
fof(fraenkel_a_1_0_gr_cy_1,axiom,
! [A,B] :
( v4_ordinal2(B)
=> ( r2_hidden(A,a_1_0_gr_cy_1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& ~ r1_xreal_0(B,C) ) ) ) ).
%------------------------------------------------------------------------------