TPTP Problem File: DAT328^1.p
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%------------------------------------------------------------------------------
% File : DAT328^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Analysis
% Problem : LIST_OF_SET_SING
% Version : Especial.
% English :
% Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source : [Kal16]
% Names : LIST_OF_SET_SING_.p [Kal16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 34 ( 13 unt; 17 typ; 0 def)
% Number of atoms : 45 ( 24 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 139 ( 3 ~; 1 |; 3 &; 129 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 33 ( 33 >; 0 *; 0 +; 0 <<)
% Number of symbols : 17 ( 16 usr; 1 con; 0-4 aty)
% Number of variables : 58 ( 1 ^; 45 !; 0 ?; 58 :)
% ( 12 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : Exported from core HOL Light.
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thf('thf_type_type/nums/num',type,
'type/nums/num': $tType ).
thf('thf_type_type/ind_types/list',type,
'type/ind_types/list': $tType > $tType ).
thf('thf_const_const/sets/set_of_list',type,
'const/sets/set_of_list':
!>[A: $tType] : ( ( 'type/ind_types/list' @ A ) > A > $o ) ).
thf('thf_const_const/sets/list_of_set',type,
'const/sets/list_of_set':
!>[A: $tType] : ( ( A > $o ) > ( 'type/ind_types/list' @ A ) ) ).
thf('thf_const_const/sets/INSERT',type,
'const/sets/INSERT':
!>[A: $tType] : ( A > ( A > $o ) > A > $o ) ).
thf('thf_const_const/sets/IN',type,
'const/sets/IN':
!>[A: $tType] : ( A > ( A > $o ) > $o ) ).
thf('thf_const_const/sets/FINITE',type,
'const/sets/FINITE':
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf('thf_const_const/sets/EMPTY',type,
'const/sets/EMPTY':
!>[A: $tType] : ( A > $o ) ).
thf('thf_const_const/sets/CARD',type,
'const/sets/CARD':
!>[A: $tType] : ( ( A > $o ) > 'type/nums/num' ) ).
thf('thf_const_const/nums/SUC',type,
'const/nums/SUC': 'type/nums/num' > 'type/nums/num' ).
thf('thf_const_const/nums/NUMERAL',type,
'const/nums/NUMERAL': 'type/nums/num' > 'type/nums/num' ).
thf('thf_const_const/nums/_0',type,
'const/nums/_0': 'type/nums/num' ).
thf('thf_const_const/lists/LENGTH',type,
'const/lists/LENGTH':
!>[A: $tType] : ( ( 'type/ind_types/list' @ A ) > 'type/nums/num' ) ).
thf('thf_const_const/ind_types/NIL',type,
'const/ind_types/NIL':
!>[A: $tType] : ( 'type/ind_types/list' @ A ) ).
thf('thf_const_const/ind_types/CONS',type,
'const/ind_types/CONS':
!>[A: $tType] : ( A > ( 'type/ind_types/list' @ A ) > ( 'type/ind_types/list' @ A ) ) ).
thf('thf_const_const/class/COND',type,
'const/class/COND':
!>[A: $tType] : ( $o > A > A > A ) ).
thf('thf_const_const/class/@',type,
'const/class/@':
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf('thm/ind_types/list_INDUCT_',axiom,
! [A: $tType,P: ( 'type/ind_types/list' @ A ) > $o] :
( ( ( P @ ( 'const/ind_types/NIL' @ A ) )
& ! [A0: A,A1: 'type/ind_types/list' @ A] :
( ( P @ A1 )
=> ( P @ ( 'const/ind_types/CONS' @ A @ A0 @ A1 ) ) ) )
=> ! [A0: 'type/ind_types/list' @ A] : ( P @ A0 ) ) ).
thf('thm/lists/NOT_CONS_NIL_',axiom,
! [A: $tType,A0: A,A1: 'type/ind_types/list' @ A] :
( ( 'const/ind_types/CONS' @ A @ A0 @ A1 )
!= ( 'const/ind_types/NIL' @ A ) ) ).
thf('thm/lists/CONS_11_',axiom,
! [A: $tType,A0: A,A1: A,A2: 'type/ind_types/list' @ A,A3: 'type/ind_types/list' @ A] :
( ( ( 'const/ind_types/CONS' @ A @ A0 @ A2 )
= ( 'const/ind_types/CONS' @ A @ A1 @ A3 ) )
= ( ( A0 = A1 )
& ( A2 = A3 ) ) ) ).
thf('thm/nums/NOT_SUC_',axiom,
! [A: 'type/nums/num'] :
( ( 'const/nums/SUC' @ A )
!= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ).
thf('thm/lists/LENGTH_1',axiom,
! [A: $tType,A0: A,A1: 'type/ind_types/list' @ A] :
( ( 'const/lists/LENGTH' @ A @ ( 'const/ind_types/CONS' @ A @ A0 @ A1 ) )
= ( 'const/nums/SUC' @ ( 'const/lists/LENGTH' @ A @ A1 ) ) ) ).
thf('thm/nums/SUC_INJ_',axiom,
! [A: 'type/nums/num',A0: 'type/nums/num'] :
( ( ( 'const/nums/SUC' @ A )
= ( 'const/nums/SUC' @ A0 ) )
= ( A = A0 ) ) ).
thf('thm/sets/set_of_list_1',axiom,
! [A: $tType,A0: A,A1: 'type/ind_types/list' @ A] :
( ( 'const/sets/set_of_list' @ A @ ( 'const/ind_types/CONS' @ A @ A0 @ A1 ) )
= ( 'const/sets/INSERT' @ A @ A0 @ ( 'const/sets/set_of_list' @ A @ A1 ) ) ) ).
thf('thm/sets/NOT_IN_EMPTY_',axiom,
! [A: $tType,A0: A] :
~ ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/EMPTY' @ A ) ) ).
thf('thm/sets/IN_INSERT_',axiom,
! [A: $tType,A0: A,A1: A,A2: A > $o] :
( ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/INSERT' @ A @ A1 @ A2 ) )
= ( ( A0 = A1 )
| ( 'const/sets/IN' @ A @ A0 @ A2 ) ) ) ).
thf('thm/sets/EXTENSION_',axiom,
! [A: $tType,A0: A > $o,A1: A > $o] :
( ( A0 = A1 )
= ( ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
= ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) ).
thf('thm/lists/LENGTH_0',axiom,
! [A: $tType] :
( ( 'const/lists/LENGTH' @ A @ ( 'const/ind_types/NIL' @ A ) )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ).
thf('thm/sets/set_of_list_0',axiom,
! [A: $tType] :
( ( 'const/sets/set_of_list' @ A @ ( 'const/ind_types/NIL' @ A ) )
= ( 'const/sets/EMPTY' @ A ) ) ).
thf('thm/sets/CARD_CLAUSES_0',axiom,
! [A: $tType] :
( ( 'const/sets/CARD' @ A @ ( 'const/sets/EMPTY' @ A ) )
= ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) ) ).
thf('thm/sets/CARD_CLAUSES_1',axiom,
! [A: $tType,A0: A,A1: A > $o] :
( ( 'const/sets/FINITE' @ A @ A1 )
=> ( ( 'const/sets/CARD' @ A @ ( 'const/sets/INSERT' @ A @ A0 @ A1 ) )
= ( 'const/class/COND' @ 'type/nums/num' @ ( 'const/sets/IN' @ A @ A0 @ A1 ) @ ( 'const/sets/CARD' @ A @ A1 ) @ ( 'const/nums/SUC' @ ( 'const/sets/CARD' @ A @ A1 ) ) ) ) ) ).
thf('thm/sets/FINITE_EMPTY_',axiom,
! [A: $tType] : ( 'const/sets/FINITE' @ A @ ( 'const/sets/EMPTY' @ A ) ) ).
thf('thm/sets/list_of_set_',axiom,
! [A: $tType,A0: A > $o] :
( ( 'const/sets/list_of_set' @ A @ A0 )
= ( 'const/class/@' @ ( 'type/ind_types/list' @ A )
@ ^ [A1: 'type/ind_types/list' @ A] :
( ( ( 'const/sets/set_of_list' @ A @ A1 )
= A0 )
& ( ( 'const/lists/LENGTH' @ A @ A1 )
= ( 'const/sets/CARD' @ A @ A0 ) ) ) ) ) ).
thf('thm/sets/LIST_OF_SET_SING_',conjecture,
! [A: $tType,A0: $tType,A1: A0,A2: A] :
( ( 'const/sets/list_of_set' @ A0 @ ( 'const/sets/INSERT' @ A0 @ A1 @ ( 'const/sets/EMPTY' @ A0 ) ) )
= ( 'const/ind_types/CONS' @ A0 @ A1 @ ( 'const/ind_types/NIL' @ A0 ) ) ) ).
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