## TPTP Problem File: SEV499^1.p

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```%------------------------------------------------------------------------------
% File     : SEV499^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : ITERATE_EQ_GENERAL
% Version  : Especial.
% English  :

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

% Status   : Theorem
% Rating   : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax   : Number of formulae    :   14 (   0 unit;   7 type;   0 defn)
%            Number of atoms       :  157 (  15 equality; 116 variable)
%            Maximal formula depth :   21 (  10 average)
%            Number of connectives :  120 (   0   ~;   0   |;   6   &; 104   @)
%                                         (   0 <=>;  10  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   49 (  49   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   7   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   57 (   0 sgn;  44   !;   1   ?;   1   ^)
%                                         (  57   :;  11  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_const_const/trivia/o',type,(
'const/trivia/o':
!>[B: \$tType,A: \$tType,C: \$tType] :
( ( B > C ) > ( A > B ) > A > C ) )).

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/IMAGE',type,(
'const/sets/IMAGE':
!>[A: \$tType,B: \$tType] :
( ( A > B ) > ( A > \$o ) > B > \$o ) )).

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

thf('thf_const_const/iterate/iterate',type,(
'const/iterate/iterate':
!>[A: \$tType,A0: \$tType] :
( ( A0 > A0 > A0 ) > ( A > \$o ) > ( A > A0 ) > A0 ) )).

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

thf('thm/sets/IN_IMAGE_',axiom,(
! [B: \$tType,A: \$tType,A0: B,A1: A > \$o,A2: A > B] :
( ( 'const/sets/IN' @ B @ A0 @ ( 'const/sets/IMAGE' @ A @ B @ A2 @ A1 ) )
= ( ? [A3: A] :
( ( A0
= ( A2 @ A3 ) )
& ( 'const/sets/IN' @ A @ A3 @ A1 ) ) ) ) )).

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/iterate/ITERATE_IMAGE_',axiom,(
! [C: \$tType,A: \$tType,B: \$tType,A0: C > C > C] :
( ( 'const/iterate/monoidal' @ C @ A0 )
=> ! [A1: A > B,A2: B > C,A3: A > \$o] :
( ! [A4: A,A5: A] :
( ( ( 'const/sets/IN' @ A @ A4 @ A3 )
& ( 'const/sets/IN' @ A @ A5 @ A3 )
& ( ( A1 @ A4 )
= ( A1 @ A5 ) ) )
=> ( A4 = A5 ) )
=> ( ( 'const/iterate/iterate' @ B @ C @ A0 @ ( 'const/sets/IMAGE' @ A @ B @ A1 @ A3 ) @ A2 )
= ( 'const/iterate/iterate' @ A @ C @ A0 @ A3 @ ( 'const/trivia/o' @ B @ A @ C @ A2 @ A1 ) ) ) ) ) )).

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

thf('thm/trivia/o_THM_',axiom,(
! [C: \$tType,B: \$tType,A: \$tType,A0: B > C,A1: A > B,A2: A] :
( ( 'const/trivia/o' @ B @ A @ C @ A0 @ A1 @ A2 )
= ( A0 @ ( A1 @ A2 ) ) ) )).

thf('thm/iterate/ITERATE_EQ_',axiom,(
! [A: \$tType,B: \$tType,A0: B > B > B] :
( ( 'const/iterate/monoidal' @ B @ A0 )
=> ! [A1: A > B,A2: A > B,A3: A > \$o] :
( ! [A4: A] :
( ( 'const/sets/IN' @ A @ A4 @ A3 )
=> ( ( A1 @ A4 )
= ( A2 @ A4 ) ) )
=> ( ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A1 )
= ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A2 ) ) ) ) )).

thf('thm/iterate/ITERATE_EQ_GENERAL_',conjecture,(
! [A: \$tType,B: \$tType,C: \$tType,A0: C > C > C] :
( ( 'const/iterate/monoidal' @ C @ A0 )
=> ! [A1: A > \$o,A2: B > \$o,A3: A > C,A4: B > C,A5: A > B] :
( ( ! [A6: B] :
( ( 'const/sets/IN' @ B @ A6 @ A2 )
=> ( 'const/bool/?!' @ A
@ ^ [A7: A] :
( ( 'const/sets/IN' @ A @ A7 @ A1 )
& ( ( A5 @ A7 )
= A6 ) ) ) )
& ! [A6: A] :
( ( 'const/sets/IN' @ A @ A6 @ A1 )
=> ( ( 'const/sets/IN' @ B @ ( A5 @ A6 ) @ A2 )
& ( ( A4 @ ( A5 @ A6 ) )
= ( A3 @ A6 ) ) ) ) )
=> ( ( 'const/iterate/iterate' @ A @ C @ A0 @ A1 @ A3 )
= ( 'const/iterate/iterate' @ B @ C @ A0 @ A2 @ A4 ) ) ) ) )).

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