## TPTP Problem File: SEV477^1.p

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```%------------------------------------------------------------------------------
% File     : SEV477^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : FINREC_UNIQUE_LEMMA
% Version  : Especial.
% English  :

% Refs     : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source   : [Kal16]
% Names    : FINREC_UNIQUE_LEMMA_.p [Kal16]

% Status   : Theorem
% Rating   : 1.00 v7.1.0
% Syntax   : Number of formulae    :   16 (   0 unit;   9 type;   0 defn)
%            Number of atoms       :  146 (  13 equality; 111 variable)
%            Maximal formula depth :   21 (   8 average)
%            Number of connectives :  116 (   3   ~;   0   |;   7   &;  97   @)
%                                         (   0 <=>;   9  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   32 (  32   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   11 (   9   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   52 (   0 sgn;  43   !;   3   ?;   0   ^)
%                                         (  52   :;   6  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_type_type/nums/num',type,(
'type/nums/num': \$tType )).

thf('thf_const_const/trivia/I',type,(
'const/trivia/I':
!>[A: \$tType] :
( A > A ) )).

thf('thf_const_const/sets/IN',type,(
'const/sets/IN':
!>[A: \$tType] :
( A > ( A > \$o ) > \$o ) )).

thf('thf_const_const/sets/FINREC',type,(
'const/sets/FINREC':
!>[A: \$tType,B: \$tType] :
( ( A > B > B ) > B > ( A > \$o ) > B > 'type/nums/num' > \$o ) )).

thf('thf_const_const/sets/EMPTY',type,(
'const/sets/EMPTY':
!>[A: \$tType] :
( A > \$o ) )).

thf('thf_const_const/sets/DELETE',type,(
'const/sets/DELETE':
!>[A: \$tType] :
( ( A > \$o ) > A > A > \$o ) )).

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('thm/nums/num_INDUCTION_',axiom,(
! [P: 'type/nums/num' > \$o] :
( ( ( P @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
& ! [A: 'type/nums/num'] :
( ( P @ A )
=> ( P @ ( 'const/nums/SUC' @ A ) ) ) )
=> ! [A: 'type/nums/num'] :
( P @ A ) ) )).

thf('thm/trivia/I_THM_',axiom,(
! [A: \$tType,A0: A] :
( ( 'const/trivia/I' @ A @ A0 )
= A0 ) )).

thf('thm/sets/FINREC_SUC_LEMMA_',axiom,(
! [A: \$tType,B: \$tType,A0: A > B > B,A1: B] :
( ! [A2: A,A3: A,A4: B] :
( ( A2 != A3 )
=> ( ( A0 @ A2 @ ( A0 @ A3 @ A4 ) )
= ( A0 @ A3 @ ( A0 @ A2 @ A4 ) ) ) )
=> ! [A2: 'type/nums/num',A3: A > \$o,A4: B] :
( ( 'const/sets/FINREC' @ A @ B @ A0 @ A1 @ A3 @ A4 @ ( 'const/nums/SUC' @ A2 ) )
=> ! [A5: A] :
( ( 'const/sets/IN' @ A @ A5 @ A3 )
=> ? [A6: B] :
( ( 'const/sets/FINREC' @ A @ B @ A0 @ A1 @ ( 'const/sets/DELETE' @ A @ A3 @ A5 ) @ A6 @ A2 )
& ( A4
= ( A0 @ A5 @ A6 ) ) ) ) ) ) )).

thf('thm/sets/FINREC_1',axiom,(
! [A: \$tType,B: \$tType,A0: B,A1: A > \$o,A2: 'type/nums/num',A3: B,A4: A > B > B] :
( ( 'const/sets/FINREC' @ A @ B @ A4 @ A0 @ A1 @ A3 @ ( 'const/nums/SUC' @ A2 ) )
= ( ? [A5: A,A6: B] :
( ( 'const/sets/IN' @ A @ A5 @ A1 )
& ( 'const/sets/FINREC' @ A @ B @ A4 @ A0 @ ( 'const/sets/DELETE' @ A @ A1 @ A5 ) @ A6 @ A2 )
& ( A3
= ( A4 @ A5 @ A6 ) ) ) ) ) )).

thf('thm/sets/NOT_IN_EMPTY_',axiom,(
! [A: \$tType,A0: A] :
~ ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/EMPTY' @ A ) ) )).

thf('thm/sets/FINREC_0',axiom,(
! [A: \$tType,B: \$tType,A0: A > B > B,A1: A > \$o,A2: B,A3: B] :
( ( 'const/sets/FINREC' @ A @ B @ A0 @ A3 @ A1 @ A2 @ ( 'const/nums/NUMERAL' @ 'const/nums/_0' ) )
= ( ( A1
= ( 'const/sets/EMPTY' @ A ) )
& ( A2 = A3 ) ) ) )).

thf('thm/sets/FINREC_UNIQUE_LEMMA_',conjecture,(
! [A: \$tType,B: \$tType,A0: A > B > B,A1: B] :
( ! [A2: A,A3: A,A4: B] :
( ( A2 != A3 )
=> ( ( A0 @ A2 @ ( A0 @ A3 @ A4 ) )
= ( A0 @ A3 @ ( A0 @ A2 @ A4 ) ) ) )
=> ! [A2: 'type/nums/num',A3: 'type/nums/num',A4: A > \$o,A5: B,A6: B] :
( ( ( 'const/sets/FINREC' @ A @ B @ A0 @ A1 @ A4 @ A5 @ A2 )
& ( 'const/sets/FINREC' @ A @ B @ A0 @ A1 @ A4 @ A6 @ A3 ) )
=> ( ( A5 = A6 )
& ( A2 = A3 ) ) ) ) )).

%------------------------------------------------------------------------------
```