SET007 Axioms: SET007+137.ax
%------------------------------------------------------------------------------
% File : SET007+137 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The de l'Hospital Theorem
% 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 : l_hospit [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 10 ( 0 unt; 0 def)
% Number of atoms : 205 ( 28 equ)
% Maximal formula atoms : 36 ( 20 avg)
% Number of connectives : 235 ( 40 ~; 0 |; 131 &)
% ( 0 <=>; 64 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 19 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 17 ( 16 usr; 0 prp; 1-3 aty)
% Number of functors : 24 ( 24 usr; 3 con; 0-4 aty)
% Number of variables : 58 ( 48 !; 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(t1_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( r1_fcont_1(A,B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,B)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ) ) ) )
=> r1_limfunc3(A,B) ) ) ) ).
fof(t2_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( ( r4_limfunc2(A,B)
& k2_limfunc2(A,B) = C )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,B)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) )
& ( ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,E)
& ~ r1_xreal_0(E,B)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) )
=> ( r4_limfunc2(A,B)
& k2_limfunc2(A,B) = C ) ) ) ) ) ) ).
fof(t3_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( ( r1_limfunc2(A,B)
& k1_limfunc2(A,B) = C )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) )
& ( ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,D)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(D)
& k2_seq_2(D) = B
& r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
=> ( v4_seq_2(k2_rfunct_2(A,D))
& k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) )
=> ( r1_limfunc2(A,B)
& k1_limfunc2(A,B) = C ) ) ) ) ) ) ).
fof(t4_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ? [C] :
( m1_rcomp_1(C,B)
& r1_tarski(k4_xboole_0(C,k1_tarski(B)),k4_relset_1(k1_numbers,k1_numbers,A)) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,C)
& ~ r1_xreal_0(B,E)
& r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
& ~ r1_xreal_0(D,F)
& ~ r1_xreal_0(F,B)
& r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ) ) ) ) ) ) ).
fof(t5_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ? [D] :
( m1_rcomp_1(D,C)
& r2_fdiff_1(A,D)
& r2_fdiff_1(B,D)
& r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
& r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
& k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
& k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
& r2_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
=> r2_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) ) ) ) ) ).
fof(t6_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ? [D] :
( m1_rcomp_1(D,C)
& r2_fdiff_1(A,D)
& r2_fdiff_1(B,D)
& r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
& r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
& k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
& k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
& r3_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
=> r3_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) ) ) ) ) ).
fof(t7_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& r2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D)))
& r2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D)))
& r1_tarski(k2_rcomp_1(C,k3_real_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
& r1_tarski(k1_rcomp_1(C,k3_real_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D))),k2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D))))))
& k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
& k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
& r1_fcont_1(A,C)
& r1_fcont_1(B,C)
& r4_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D))),k2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D)))),C) )
=> ( r4_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& k2_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k2_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D))),k2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D)))),C) ) ) ) ) ) ) ).
fof(t8_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& r2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C))
& r2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C))
& r1_tarski(k2_rcomp_1(k5_real_1(C,D),C),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
& r1_tarski(k1_rcomp_1(k5_real_1(C,D),C),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C)),k2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C)))))
& k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
& k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
& r1_fcont_1(A,C)
& r1_fcont_1(B,C)
& r1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C)),k2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C))),C) )
=> ( r1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& ~ r1_xreal_0(D,np__0)
& k1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C)),k2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C))),C) ) ) ) ) ) ) ).
fof(t9_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ? [D] :
( m1_rcomp_1(D,C)
& r2_fdiff_1(A,D)
& r2_fdiff_1(B,D)
& r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
& r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
& k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
& k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
& r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
=> ( r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
& ? [D] :
( m1_rcomp_1(D,C)
& k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) ) ) ) ) ) ) ).
fof(t10_l_hospit,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ? [D] :
( m1_rcomp_1(D,C)
& r2_fdiff_1(A,D)
& r2_fdiff_1(B,D)
& r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
& r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
& k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
& k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
& r1_fcont_1(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
=> ( r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
& k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k6_real_1(k1_fdiff_1(A,C),k1_fdiff_1(B,C)) ) ) ) ) ) ).
%------------------------------------------------------------------------------