TPTP Problem File: SEV500^1.p
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% File : SEV500^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Analysis
% Problem : ITERATE_OP_GEN
% Version : Especial.
% English :
% Refs : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source : [Kal16]
% Names : ITERATE_OP_GEN_.p [Kal16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 22 ( 8 unt; 11 typ; 0 def)
% Number of atoms : 47 ( 16 equ; 0 cnn)
% Maximal formula atoms : 6 ( 4 avg)
% Number of connectives : 197 ( 2 ~; 1 |; 7 &; 179 @)
% ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 1 ( 0 usr)
% Number of type conns : 74 ( 74 >; 0 *; 0 +; 0 <<)
% Number of symbols : 12 ( 11 usr; 0 con; 2-6 aty)
% Number of variables : 72 ( 4 ^; 53 !; 2 ?; 72 :)
% ( 13 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : Exported from core HOL Light.
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thf('thf_const_const/trivia/I',type,
'const/trivia/I':
!>[A: $tType] : ( A > A ) ).
thf('thf_const_const/sets/UNION',type,
'const/sets/UNION':
!>[A: $tType] : ( ( A > $o ) > ( A > $o ) > A > $o ) ).
thf('thf_const_const/sets/SUBSET',type,
'const/sets/SUBSET':
!>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
thf('thf_const_const/sets/SETSPEC',type,
'const/sets/SETSPEC':
!>[A: $tType] : ( 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/GSPEC',type,
'const/sets/GSPEC':
!>[A: $tType] : ( ( A > $o ) > A > $o ) ).
thf('thf_const_const/sets/FINITE',type,
'const/sets/FINITE':
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf('thf_const_const/iterate/support',type,
'const/iterate/support':
!>[B: $tType,A: $tType] : ( ( B > B > B ) > ( A > B ) > ( A > $o ) > A > $o ) ).
thf('thf_const_const/iterate/neutral',type,
'const/iterate/neutral':
!>[A: $tType] : ( ( A > A > A ) > A ) ).
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('thm/iterate/ITERATE_SUPERSET_',axiom,
! [A: $tType,B: $tType,A0: B > B > B] :
( ( 'const/iterate/monoidal' @ B @ A0 )
=> ! [A1: A > B,A2: A > $o,A3: A > $o] :
( ( ( 'const/sets/SUBSET' @ A @ A2 @ A3 )
& ! [A4: A] :
( ( ( 'const/sets/IN' @ A @ A4 @ A3 )
& ~ ( 'const/sets/IN' @ A @ A4 @ A2 ) )
=> ( ( A1 @ A4 )
= ( 'const/iterate/neutral' @ B @ A0 ) ) ) )
=> ( ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A1 )
= ( 'const/iterate/iterate' @ A @ B @ A0 @ A2 @ A1 ) ) ) ) ).
thf('thm/trivia/I_THM_',axiom,
! [A: $tType,A0: A] :
( ( 'const/trivia/I' @ A @ A0 )
= A0 ) ).
thf('thm/iterate/monoidal_',axiom,
! [A: $tType,A0: A > A > A] :
( ( 'const/iterate/monoidal' @ A @ A0 )
= ( ! [A1: A,A2: A] :
( ( A0 @ A1 @ A2 )
= ( A0 @ A2 @ A1 ) )
& ! [A1: A,A2: A,A3: A] :
( ( A0 @ A1 @ ( A0 @ A2 @ A3 ) )
= ( A0 @ ( A0 @ A1 @ A2 ) @ A3 ) )
& ! [A1: A] :
( ( A0 @ ( 'const/iterate/neutral' @ A @ A0 ) @ A1 )
= A1 ) ) ) ).
thf('thm/sets/IN_ELIM_THM_1',axiom,
! [A: $tType,A0: A > $o,A1: A] :
( ( 'const/sets/IN' @ A @ A1
@ ( 'const/sets/GSPEC' @ A
@ ^ [A2: A] :
? [A3: A] : ( 'const/sets/SETSPEC' @ A @ A2 @ ( A0 @ A3 ) @ A3 ) ) )
= ( A0 @ A1 ) ) ).
thf('thm/iterate/support_',axiom,
! [B: $tType,A: $tType,A0: A > $o,A1: A > B,A2: B > B > B] :
( ( 'const/iterate/support' @ B @ A @ A2 @ A1 @ A0 )
= ( 'const/sets/GSPEC' @ A
@ ^ [A3: A] :
? [A4: A] :
( 'const/sets/SETSPEC' @ A @ A3
@ ( ( 'const/sets/IN' @ A @ A4 @ A0 )
& ( ( A1 @ A4 )
!= ( 'const/iterate/neutral' @ B @ A2 ) ) )
@ A4 ) ) ) ).
thf('thm/sets/IN_UNION_',axiom,
! [A: $tType,A0: A > $o,A1: A > $o,A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ ( 'const/sets/UNION' @ A @ A0 @ A1 ) )
= ( ( 'const/sets/IN' @ A @ A2 @ A0 )
| ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ).
thf('thm/sets/SUBSET_',axiom,
! [A: $tType,A0: A > $o,A1: A > $o] :
( ( 'const/sets/SUBSET' @ A @ A0 @ A1 )
= ( ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
=> ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) ).
thf('thm/iterate/ITERATE_OP_',axiom,
! [A: $tType,A0: $tType,A1: A0 > A0 > A0] :
( ( 'const/iterate/monoidal' @ A0 @ A1 )
=> ! [A2: A > A0,A3: A > A0,A4: A > $o] :
( ( 'const/sets/FINITE' @ A @ A4 )
=> ( ( 'const/iterate/iterate' @ A @ A0 @ A1 @ A4
@ ^ [A5: A] : ( A1 @ ( A2 @ A5 ) @ ( A3 @ A5 ) ) )
= ( A1 @ ( 'const/iterate/iterate' @ A @ A0 @ A1 @ A4 @ A2 ) @ ( 'const/iterate/iterate' @ A @ A0 @ A1 @ A4 @ A3 ) ) ) ) ) ).
thf('thm/sets/FINITE_UNION_',axiom,
! [A: $tType,A0: A > $o,A1: A > $o] :
( ( 'const/sets/FINITE' @ A @ ( 'const/sets/UNION' @ A @ A0 @ A1 ) )
= ( ( 'const/sets/FINITE' @ A @ A0 )
& ( 'const/sets/FINITE' @ A @ A1 ) ) ) ).
thf('thm/iterate/ITERATE_SUPPORT_',axiom,
! [A: $tType,A0: $tType,A1: A0 > A0 > A0,A2: A > A0,A3: A > $o] :
( ( 'const/iterate/iterate' @ A @ A0 @ A1 @ ( 'const/iterate/support' @ A0 @ A @ A1 @ A2 @ A3 ) @ A2 )
= ( 'const/iterate/iterate' @ A @ A0 @ A1 @ A3 @ A2 ) ) ).
thf('thm/iterate/ITERATE_OP_GEN_',conjecture,
! [A: $tType,B: $tType,A0: B > B > B] :
( ( 'const/iterate/monoidal' @ B @ A0 )
=> ! [A1: A > B,A2: A > B,A3: A > $o] :
( ( ( 'const/sets/FINITE' @ A @ ( 'const/iterate/support' @ B @ A @ A0 @ A1 @ A3 ) )
& ( 'const/sets/FINITE' @ A @ ( 'const/iterate/support' @ B @ A @ A0 @ A2 @ A3 ) ) )
=> ( ( 'const/iterate/iterate' @ A @ B @ A0 @ A3
@ ^ [A4: A] : ( A0 @ ( A1 @ A4 ) @ ( A2 @ A4 ) ) )
= ( A0 @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A1 ) @ ( 'const/iterate/iterate' @ A @ B @ A0 @ A3 @ A2 ) ) ) ) ) ).
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