TPTP Problem File: COM153^1.p
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%------------------------------------------------------------------------------
% File : COM153^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Abstract completeness 90
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BPT14] Blanchette et al. (2014), Abstract Completeness
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : abstract_completeness__90.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.67 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 350 ( 160 unt; 74 typ; 0 def)
% Number of atoms : 647 ( 317 equ; 4 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 5256 ( 89 ~; 8 |; 74 &;4840 @)
% ( 0 <=>; 245 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 480 ( 480 >; 0 *; 0 +; 0 <<)
% Number of symbols : 75 ( 71 usr; 8 con; 0-9 aty)
% Number of variables : 1336 ( 74 ^;1121 !; 42 ?;1336 :)
% ( 99 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:53:23.018
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
thf(ty_t_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree,type,
abstra2103299360e_tree: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_FSet_Ofset,type,
fset: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_state,type,
state: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_rule,type,
rule: $tType ).
%----Explicit typings (65)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OSaturated,type,
abstra1209608345urated:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oenabled,type,
abstra1874422341nabled:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > State > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oepath,type,
abstra523868654_epath:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OpickEff,type,
abstra1276541928ickEff:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > State > ( fset @ State ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Osaturated,type,
abstra726722745urated:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Owf,type,
abstra1874736267tem_wf:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Oipath,type,
abstra313004635_ipath:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Okonig,type,
abstra1918223989_konig:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( stream @ A ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otfinite,type,
abstra668420080finite:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_ONode,type,
abstra388494275e_Node:
!>[A: $tType] : ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > ( abstra2103299360e_tree @ A ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocase__tree,type,
abstra457966479e_tree:
!>[A: $tType,B: $tType] : ( ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B ) > ( abstra2103299360e_tree @ A ) > B ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocont,type,
abstra1749095923e_cont:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( fset @ ( abstra2103299360e_tree @ A ) ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Oroot,type,
abstra573067619e_root:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > A ) ).
thf(sy_c_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable__Set_Ofrom__nat__into,type,
counta609264050t_into:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Countable__Set_Oto__nat__on,type,
countable_to_nat_on:
!>[A: $tType] : ( ( set @ A ) > A > nat ) ).
thf(sy_c_FSet_Ofimage,type,
fimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( fset @ A ) > ( fset @ B ) ) ).
thf(sy_c_FSet_Ofmember,type,
fmember:
!>[A: $tType] : ( A > ( fset @ A ) > $o ) ).
thf(sy_c_FSet_Ofset_OFSet_Opred__fset,type,
pred_fset:
!>[A: $tType] : ( ( A > $o ) > ( fset @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Onxt,type,
linear1494993505on_nxt:
!>[A: $tType,B: $tType] : ( ( ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Stream_Osdrop__while,type,
sdrop_while:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osfilter,type,
sfilter:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osinterleave,type,
sinterleave:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osmap2,type,
smap2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_OSCons,type,
sCons:
!>[A: $tType] : ( A > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Ocase__stream,type,
case_stream:
!>[A: $tType,B: $tType] : ( ( A > ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Stream_Ostream_Oshd,type,
shd:
!>[A: $tType] : ( ( stream @ A ) > A ) ).
thf(sy_c_Stream_Ostream_Osset,type,
sset:
!>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
thf(sy_c_Stream_Ostream_Ostl,type,
stl:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Oszip,type,
szip:
!>[A: $tType,B: $tType] : ( ( stream @ A ) > ( stream @ B ) > ( stream @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_eff,type,
eff: rule > state > ( fset @ state ) > $o ).
thf(sy_v_rules,type,
rules: stream @ rule ).
thf(sy_v_steps,type,
steps: stream @ ( product_prod @ state @ rule ) ).
thf(sy_v_stepsa____,type,
stepsa: stream @ ( product_prod @ state @ rule ) ).
thf(sy_v_t,type,
t: abstra2103299360e_tree @ ( product_prod @ state @ rule ) ).
thf(sy_v_ta____,type,
ta: abstra2103299360e_tree @ ( product_prod @ state @ rule ) ).
%----Relevant facts (256)
thf(fact_0_local_Oepath_I1_J,axiom,
abstra1874736267tem_wf @ rule @ state @ eff @ rules @ ta ).
% local.epath(1)
thf(fact_1_local_Oepath_I2_J,axiom,
abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ta @ stepsa ).
% local.epath(2)
thf(fact_2_assms_I2_J,axiom,
abstra313004635_ipath @ ( product_prod @ state @ rule ) @ t @ steps ).
% assms(2)
thf(fact_3_assms_I1_J,axiom,
abstra1874736267tem_wf @ rule @ state @ eff @ rules @ t ).
% assms(1)
thf(fact_4_epath_Ocases,axiom,
! [A2: stream @ ( product_prod @ state @ rule )] :
( ( abstra523868654_epath @ rule @ state @ eff @ rules @ A2 )
=> ~ ( ( member @ rule @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ A2 ) ) @ ( sset @ rule @ rules ) )
=> ! [Sl: fset @ state] :
( ( fmember @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ ( stl @ ( product_prod @ state @ rule ) @ A2 ) ) ) @ Sl )
=> ( ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ A2 ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ A2 ) ) @ Sl )
=> ~ ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ A2 ) ) ) ) ) ) ).
% epath.cases
thf(fact_5_epath_Ocoinduct,axiom,
! [X: ( stream @ ( product_prod @ state @ rule ) ) > $o,X2: stream @ ( product_prod @ state @ rule )] :
( ( X @ X2 )
=> ( ! [X3: stream @ ( product_prod @ state @ rule )] :
( ( X @ X3 )
=> ? [Steps: stream @ ( product_prod @ state @ rule ),Sl2: fset @ state] :
( ( X3 = Steps )
& ( member @ rule @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ ( sset @ rule @ rules ) )
& ( fmember @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) ) ) @ Sl2 )
& ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ Sl2 )
& ( ( X @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) )
| ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) ) ) ) )
=> ( abstra523868654_epath @ rule @ state @ eff @ rules @ X2 ) ) ) ).
% epath.coinduct
thf(fact_6_epath_Ointros,axiom,
! [Steps2: stream @ ( product_prod @ state @ rule ),Sl3: fset @ state] :
( ( member @ rule @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps2 ) ) @ ( sset @ rule @ rules ) )
=> ( ( fmember @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ ( stl @ ( product_prod @ state @ rule ) @ Steps2 ) ) ) @ Sl3 )
=> ( ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps2 ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps2 ) ) @ Sl3 )
=> ( ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps2 ) )
=> ( abstra523868654_epath @ rule @ state @ eff @ rules @ Steps2 ) ) ) ) ) ).
% epath.intros
thf(fact_7_epath_Osimps,axiom,
! [A2: stream @ ( product_prod @ state @ rule )] :
( ( abstra523868654_epath @ rule @ state @ eff @ rules @ A2 )
= ( ? [Steps3: stream @ ( product_prod @ state @ rule ),Sl4: fset @ state] :
( ( A2 = Steps3 )
& ( member @ rule @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps3 ) ) @ ( sset @ rule @ rules ) )
& ( fmember @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ ( stl @ ( product_prod @ state @ rule ) @ Steps3 ) ) ) @ Sl4 )
& ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps3 ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps3 ) ) @ Sl4 )
& ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps3 ) ) ) ) ) ).
% epath.simps
thf(fact_8_enabled__def,axiom,
! [R: rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
= ( ^ [P: ( fset @ state ) > $o] :
? [X4: fset @ state] : ( P @ X4 )
@ ( eff @ R @ S ) ) ) ).
% enabled_def
thf(fact_9_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,A2: stream @ ( product_prod @ State @ Rule )] :
( ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ A2 )
=> ~ ( ( member @ Rule @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ A2 ) ) @ ( sset @ Rule @ Rules ) )
=> ! [Sl: fset @ State] :
( ( fmember @ State @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ ( stl @ ( product_prod @ State @ Rule ) @ A2 ) ) ) @ Sl )
=> ( ( Eff @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ A2 ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ A2 ) ) @ Sl )
=> ~ ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ ( stl @ ( product_prod @ State @ Rule ) @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_10_RuleSystem__Defs_Oepath_Osimps,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra523868654_epath @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,A3: stream @ ( product_prod @ State @ Rule )] :
? [Steps3: stream @ ( product_prod @ State @ Rule ),Sl4: fset @ State] :
( ( A3 = Steps3 )
& ( member @ Rule @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps3 ) ) @ ( sset @ Rule @ Rules2 ) )
& ( fmember @ State @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ ( stl @ ( product_prod @ State @ Rule ) @ Steps3 ) ) ) @ Sl4 )
& ( Eff2 @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps3 ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps3 ) ) @ Sl4 )
& ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules2 @ ( stl @ ( product_prod @ State @ Rule ) @ Steps3 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_11_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Rule: $tType,State: $tType,Steps2: stream @ ( product_prod @ State @ Rule ),Rules: stream @ Rule,Sl3: fset @ State,Eff: Rule > State > ( fset @ State ) > $o] :
( ( member @ Rule @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps2 ) ) @ ( sset @ Rule @ Rules ) )
=> ( ( fmember @ State @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ ( stl @ ( product_prod @ State @ Rule ) @ Steps2 ) ) ) @ Sl3 )
=> ( ( Eff @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps2 ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps2 ) ) @ Sl3 )
=> ( ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ ( stl @ ( product_prod @ State @ Rule ) @ Steps2 ) )
=> ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_12_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [Rule: $tType,State: $tType,X: ( stream @ ( product_prod @ State @ Rule ) ) > $o,X2: stream @ ( product_prod @ State @ Rule ),Rules: stream @ Rule,Eff: Rule > State > ( fset @ State ) > $o] :
( ( X @ X2 )
=> ( ! [X3: stream @ ( product_prod @ State @ Rule )] :
( ( X @ X3 )
=> ? [Steps: stream @ ( product_prod @ State @ Rule ),Sl2: fset @ State] :
( ( X3 = Steps )
& ( member @ Rule @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps ) ) @ ( sset @ Rule @ Rules ) )
& ( fmember @ State @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ ( stl @ ( product_prod @ State @ Rule ) @ Steps ) ) ) @ Sl2 )
& ( Eff @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps ) ) @ Sl2 )
& ( ( X @ ( stl @ ( product_prod @ State @ Rule ) @ Steps ) )
| ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ ( stl @ ( product_prod @ State @ Rule ) @ Steps ) ) ) ) )
=> ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ X2 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_13_Saturated__def,axiom,
! [Steps2: stream @ ( product_prod @ state @ rule )] :
( ( abstra1209608345urated @ rule @ state @ eff @ rules @ Steps2 )
= ( ! [X5: rule] :
( ( member @ rule @ X5 @ ( sset @ rule @ rules ) )
=> ( abstra726722745urated @ rule @ state @ eff @ X5 @ Steps2 ) ) ) ) ).
% Saturated_def
thf(fact_14__C_K_C,axiom,
! [A4: $tType,T2: abstra2103299360e_tree @ A4,St: stream @ A4] :
( ( abstra313004635_ipath @ A4 @ T2 @ St )
=> ( ( abstra573067619e_root @ A4 @ T2 )
= ( shd @ A4 @ St ) ) ) ).
% "*"
thf(fact_15_sset__induct,axiom,
! [A: $tType,Y: A,S: stream @ A,P2: A > ( stream @ A ) > $o] :
( ( member @ A @ Y @ ( sset @ A @ S ) )
=> ( ! [S2: stream @ A] : ( P2 @ ( shd @ A @ S2 ) @ S2 )
=> ( ! [S2: stream @ A,Y2: A] :
( ( member @ A @ Y2 @ ( sset @ A @ ( stl @ A @ S2 ) ) )
=> ( ( P2 @ Y2 @ ( stl @ A @ S2 ) )
=> ( P2 @ Y2 @ S2 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_16_NE__R,axiom,
( ( sset @ rule @ rules )
!= ( bot_bot @ ( set @ rule ) ) ) ).
% NE_R
thf(fact_17_shd__sset,axiom,
! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% shd_sset
thf(fact_18_stl__sset,axiom,
! [A: $tType,X2: A,A2: stream @ A] :
( ( member @ A @ X2 @ ( sset @ A @ ( stl @ A @ A2 ) ) )
=> ( member @ A @ X2 @ ( sset @ A @ A2 ) ) ) ).
% stl_sset
thf(fact_19_stream_Oexpand,axiom,
! [A: $tType,Stream: stream @ A,Stream2: stream @ A] :
( ( ( ( shd @ A @ Stream )
= ( shd @ A @ Stream2 ) )
& ( ( stl @ A @ Stream )
= ( stl @ A @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_20_stream_Ocoinduct,axiom,
! [A: $tType,R2: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R2 @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R2 @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( R2 @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_21_stream_Ocoinduct__strong,axiom,
! [A: $tType,R2: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R2 @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R2 @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( ( R2 @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) )
| ( ( stl @ A @ Stream3 )
= ( stl @ A @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_22_countable__R,axiom,
countable_countable @ rule @ ( sset @ rule @ rules ) ).
% countable_R
thf(fact_23_prod__eqI,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P3 )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P3 )
= ( product_snd @ A @ B @ Q ) )
=> ( P3 = Q ) ) ) ).
% prod_eqI
thf(fact_24_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,P3: product_prod @ B @ A] :
( ( P2 @ ( product_snd @ B @ A @ P3 ) @ ( product_fst @ B @ A @ P3 ) )
=> ~ ! [X3: B,Y2: A] :
~ ( P2 @ Y2 @ X3 ) ) ).
% exE_realizer'
thf(fact_25_pickEff,axiom,
! [R: rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
=> ( eff @ R @ S @ ( abstra1276541928ickEff @ rule @ state @ eff @ R @ S ) ) ) ).
% pickEff
thf(fact_26_RuleSystem__Defs_Oenabled__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1874422341nabled @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,R3: Rule,S3: State] :
( ^ [P: ( fset @ State ) > $o] :
? [X4: fset @ State] : ( P @ X4 )
@ ( Eff2 @ R3 @ S3 ) ) ) ) ).
% RuleSystem_Defs.enabled_def
thf(fact_27_RuleSystem__Defs_Ocountable__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( countable_countable @ Rule @ ( sset @ Rule @ Rules ) ) ).
% RuleSystem_Defs.countable_R
thf(fact_28_RuleSystem__Defs_ONE__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] :
( ( sset @ Rule @ Rules )
!= ( bot_bot @ ( set @ Rule ) ) ) ).
% RuleSystem_Defs.NE_R
thf(fact_29_RuleSystem__Defs_OSaturated__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1209608345urated @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,Steps3: stream @ ( product_prod @ State @ Rule )] :
! [X5: Rule] :
( ( member @ Rule @ X5 @ ( sset @ Rule @ Rules2 ) )
=> ( abstra726722745urated @ Rule @ State @ Eff2 @ X5 @ Steps3 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_30_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : Y3 = Z )
= ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S3 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S3 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_31_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_32_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_33_countableI__type,axiom,
! [A: $tType] :
( ( countable @ A @ ( type2 @ A ) )
=> ! [A5: set @ A] : ( countable_countable @ A @ A5 ) ) ).
% countableI_type
thf(fact_34_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_35_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X5: A] :
~ ( member @ A @ X5 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_36_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_37_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_38_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X5: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_39_RuleSystem__Defs_OpickEff,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,R: Rule,S: State] :
( ( abstra1874422341nabled @ Rule @ State @ Eff @ R @ S )
=> ( Eff @ R @ S @ ( abstra1276541928ickEff @ Rule @ State @ Eff @ R @ S ) ) ) ).
% RuleSystem_Defs.pickEff
thf(fact_40_ipath_Ocoinduct,axiom,
! [A: $tType,X: ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o,X2: abstra2103299360e_tree @ A,Xa: stream @ A] :
( ( X @ X2 @ Xa )
=> ( ! [X3: abstra2103299360e_tree @ A,Xa2: stream @ A] :
( ( X @ X3 @ Xa2 )
=> ? [T4: abstra2103299360e_tree @ A,Steps: stream @ A,T5: abstra2103299360e_tree @ A] :
( ( X3 = T4 )
& ( Xa2 = Steps )
& ( ( abstra573067619e_root @ A @ T4 )
= ( shd @ A @ Steps ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ T4 ) )
& ( ( X @ T5 @ ( stl @ A @ Steps ) )
| ( abstra313004635_ipath @ A @ T5 @ ( stl @ A @ Steps ) ) ) ) )
=> ( abstra313004635_ipath @ A @ X2 @ Xa ) ) ) ).
% ipath.coinduct
thf(fact_41_ipath_Ointros,axiom,
! [A: $tType,T6: abstra2103299360e_tree @ A,Steps2: stream @ A,T7: abstra2103299360e_tree @ A] :
( ( ( abstra573067619e_root @ A @ T6 )
= ( shd @ A @ Steps2 ) )
=> ( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ T6 ) )
=> ( ( abstra313004635_ipath @ A @ T7 @ ( stl @ A @ Steps2 ) )
=> ( abstra313004635_ipath @ A @ T6 @ Steps2 ) ) ) ) ).
% ipath.intros
thf(fact_42_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_43_tree_Oexpand,axiom,
! [A: $tType,Tree: abstra2103299360e_tree @ A,Tree2: abstra2103299360e_tree @ A] :
( ( ( ( abstra573067619e_root @ A @ Tree )
= ( abstra573067619e_root @ A @ Tree2 ) )
& ( ( abstra1749095923e_cont @ A @ Tree )
= ( abstra1749095923e_cont @ A @ Tree2 ) ) )
=> ( Tree = Tree2 ) ) ).
% tree.expand
thf(fact_44_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X5: A] : ( member @ A @ X5 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_50_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_51_equals0D,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A5 ) ) ).
% equals0D
thf(fact_52_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_53_ipath_Ocases,axiom,
! [A: $tType,A1: abstra2103299360e_tree @ A,A22: stream @ A] :
( ( abstra313004635_ipath @ A @ A1 @ A22 )
=> ~ ( ( ( abstra573067619e_root @ A @ A1 )
= ( shd @ A @ A22 ) )
=> ! [T8: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T8 @ ( abstra1749095923e_cont @ A @ A1 ) )
=> ~ ( abstra313004635_ipath @ A @ T8 @ ( stl @ A @ A22 ) ) ) ) ) ).
% ipath.cases
thf(fact_54_ipath_Osimps,axiom,
! [A: $tType] :
( ( abstra313004635_ipath @ A )
= ( ^ [A12: abstra2103299360e_tree @ A,A23: stream @ A] :
? [T3: abstra2103299360e_tree @ A,Steps3: stream @ A,T9: abstra2103299360e_tree @ A] :
( ( A12 = T3 )
& ( A23 = Steps3 )
& ( ( abstra573067619e_root @ A @ T3 )
= ( shd @ A @ Steps3 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T9 @ ( abstra1749095923e_cont @ A @ T3 ) )
& ( abstra313004635_ipath @ A @ T9 @ ( stl @ A @ Steps3 ) ) ) ) ) ).
% ipath.simps
thf(fact_55_fempty__iff,axiom,
! [A: $tType,C2: A] :
~ ( fmember @ A @ C2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% fempty_iff
thf(fact_56_all__not__fin__conv,axiom,
! [A: $tType,A5: fset @ A] :
( ( ! [X5: A] :
~ ( fmember @ A @ X5 @ A5 ) )
= ( A5
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% all_not_fin_conv
thf(fact_57_wf_Ocases,axiom,
! [A2: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ A2 )
=> ~ ( ( member @ rule @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ A2 ) ) @ ( sset @ rule @ rules ) )
=> ( ( eff @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ A2 ) ) @ ( product_fst @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ A2 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ state @ ( comp @ ( product_prod @ state @ rule ) @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( product_fst @ state @ rule ) @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ A2 ) ) )
=> ~ ! [T5: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ T5 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ A2 ) )
=> ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T5 ) ) ) ) ) ).
% wf.cases
thf(fact_58_wf_Ocoinduct,axiom,
! [X: ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) > $o,X2: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( X @ X2 )
=> ( ! [X3: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( X @ X3 )
=> ? [T4: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( X3 = T4 )
& ( member @ rule @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T4 ) ) @ ( sset @ rule @ rules ) )
& ( eff @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T4 ) ) @ ( product_fst @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T4 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ state @ ( comp @ ( product_prod @ state @ rule ) @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( product_fst @ state @ rule ) @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ T4 ) ) )
& ! [Xa2: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ Xa2 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ T4 ) )
=> ( ( X @ Xa2 )
| ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ Xa2 ) ) ) ) )
=> ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ X2 ) ) ) ).
% wf.coinduct
thf(fact_59_wf_Osimps,axiom,
! [A2: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ A2 )
= ( ? [T3: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( A2 = T3 )
& ( member @ rule @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T3 ) ) @ ( sset @ rule @ rules ) )
& ( eff @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T3 ) ) @ ( product_fst @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T3 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ state @ ( comp @ ( product_prod @ state @ rule ) @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( product_fst @ state @ rule ) @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ T3 ) ) )
& ! [X5: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ X5 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ T3 ) )
=> ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ X5 ) ) ) ) ) ).
% wf.simps
thf(fact_60_wf_Owf,axiom,
! [T6: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( member @ rule @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T6 ) ) @ ( sset @ rule @ rules ) )
=> ( ( eff @ ( product_snd @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T6 ) ) @ ( product_fst @ state @ rule @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ T6 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ state @ ( comp @ ( product_prod @ state @ rule ) @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( product_fst @ state @ rule ) @ ( abstra573067619e_root @ ( product_prod @ state @ rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ T6 ) ) )
=> ( ! [T8: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ T8 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ T6 ) )
=> ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T8 ) )
=> ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T6 ) ) ) ) ).
% wf.wf
thf(fact_61_konig_Osimps_I1_J,axiom,
! [A: $tType,T6: abstra2103299360e_tree @ A] :
( ( shd @ A @ ( abstra1918223989_konig @ A @ T6 ) )
= ( abstra573067619e_root @ A @ T6 ) ) ).
% konig.simps(1)
thf(fact_62_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_63_fset_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: fset @ A] :
( ( fimage @ B @ C @ G @ ( fimage @ A @ B @ F @ V ) )
= ( fimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fset.map_comp
thf(fact_64_fimage__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A5: fset @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( fmember @ B @ X2 @ A5 )
=> ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A5 ) ) ) ) ).
% fimage_eqI
thf(fact_65_fimage__fempty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( fimage @ B @ A @ F @ ( bot_bot @ ( fset @ B ) ) )
= ( bot_bot @ ( fset @ A ) ) ) ).
% fimage_fempty
thf(fact_66_fempty__is__fimage,axiom,
! [A: $tType,B: $tType,F: B > A,A5: fset @ B] :
( ( ( bot_bot @ ( fset @ A ) )
= ( fimage @ B @ A @ F @ A5 ) )
= ( A5
= ( bot_bot @ ( fset @ B ) ) ) ) ).
% fempty_is_fimage
thf(fact_67_fimage__is__fempty,axiom,
! [A: $tType,B: $tType,F: B > A,A5: fset @ B] :
( ( ( fimage @ B @ A @ F @ A5 )
= ( bot_bot @ ( fset @ A ) ) )
= ( A5
= ( bot_bot @ ( fset @ B ) ) ) ) ).
% fimage_is_fempty
thf(fact_68_fimageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A5: fset @ B] :
( ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A5 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F @ X3 ) )
=> ~ ( fmember @ B @ X3 @ A5 ) ) ) ).
% fimageE
thf(fact_69_fimageI,axiom,
! [B: $tType,A: $tType,X2: A,A5: fset @ A,F: A > B] :
( ( fmember @ A @ X2 @ A5 )
=> ( fmember @ B @ ( F @ X2 ) @ ( fimage @ A @ B @ F @ A5 ) ) ) ).
% fimageI
thf(fact_70_fimage__cong,axiom,
! [B: $tType,A: $tType,M: fset @ A,N: fset @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( fmember @ A @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( fimage @ A @ B @ F @ M )
= ( fimage @ A @ B @ G @ N ) ) ) ) ).
% fimage_cong
thf(fact_71_rev__fimage__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A5: fset @ A,B2: B,F: A > B] :
( ( fmember @ A @ X2 @ A5 )
=> ( ( B2
= ( F @ X2 ) )
=> ( fmember @ B @ B2 @ ( fimage @ A @ B @ F @ A5 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_72_RuleSystem__Defs_Owf_Ocoinduct,axiom,
! [Rule: $tType,State: $tType,X: ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) > $o,X2: abstra2103299360e_tree @ ( product_prod @ State @ Rule ),Rules: stream @ Rule,Eff: Rule > State > ( fset @ State ) > $o] :
( ( X @ X2 )
=> ( ! [X3: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( X @ X3 )
=> ? [T4: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( X3 = T4 )
& ( member @ Rule @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T4 ) ) @ ( sset @ Rule @ Rules ) )
& ( Eff @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T4 ) ) @ ( product_fst @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T4 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ State @ ( comp @ ( product_prod @ State @ Rule ) @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( product_fst @ State @ Rule ) @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ T4 ) ) )
& ! [Xa2: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ Xa2 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ T4 ) )
=> ( ( X @ Xa2 )
| ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ Xa2 ) ) ) ) )
=> ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ X2 ) ) ) ).
% RuleSystem_Defs.wf.coinduct
thf(fact_73_RuleSystem__Defs_Owf_Ointros,axiom,
! [Rule: $tType,State: $tType,T6: abstra2103299360e_tree @ ( product_prod @ State @ Rule ),Rules: stream @ Rule,Eff: Rule > State > ( fset @ State ) > $o] :
( ( member @ Rule @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T6 ) ) @ ( sset @ Rule @ Rules ) )
=> ( ( Eff @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T6 ) ) @ ( product_fst @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T6 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ State @ ( comp @ ( product_prod @ State @ Rule ) @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( product_fst @ State @ Rule ) @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ T6 ) ) )
=> ( ! [T8: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ T8 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ T6 ) )
=> ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ T8 ) )
=> ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ T6 ) ) ) ) ).
% RuleSystem_Defs.wf.intros
thf(fact_74_RuleSystem__Defs_Owf_Osimps,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1874736267tem_wf @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,A3: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
? [T3: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( A3 = T3 )
& ( member @ Rule @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T3 ) ) @ ( sset @ Rule @ Rules2 ) )
& ( Eff2 @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T3 ) ) @ ( product_fst @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ T3 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ State @ ( comp @ ( product_prod @ State @ Rule ) @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( product_fst @ State @ Rule ) @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ T3 ) ) )
& ! [X5: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ X5 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ T3 ) )
=> ( abstra1874736267tem_wf @ Rule @ State @ Eff2 @ Rules2 @ X5 ) ) ) ) ) ).
% RuleSystem_Defs.wf.simps
thf(fact_75_RuleSystem__Defs_Owf_Ocases,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,A2: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ A2 )
=> ~ ( ( member @ Rule @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ A2 ) ) @ ( sset @ Rule @ Rules ) )
=> ( ( Eff @ ( product_snd @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ A2 ) ) @ ( product_fst @ State @ Rule @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ A2 ) ) @ ( fimage @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ State @ ( comp @ ( product_prod @ State @ Rule ) @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( product_fst @ State @ Rule ) @ ( abstra573067619e_root @ ( product_prod @ State @ Rule ) ) ) @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ A2 ) ) )
=> ~ ! [T5: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ T5 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ A2 ) )
=> ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ T5 ) ) ) ) ) ).
% RuleSystem_Defs.wf.cases
thf(fact_76_eqfelem__imp__iff,axiom,
! [A: $tType,X2: A,Y: A,A5: fset @ A] :
( ( X2 = Y )
=> ( ( fmember @ A @ X2 @ A5 )
= ( fmember @ A @ Y @ A5 ) ) ) ).
% eqfelem_imp_iff
thf(fact_77_if__split__fmem2,axiom,
! [A: $tType,A2: A,Q2: $o,X2: fset @ A,Y: fset @ A] :
( ( fmember @ A @ A2 @ ( if @ ( fset @ A ) @ Q2 @ X2 @ Y ) )
= ( ( Q2
=> ( fmember @ A @ A2 @ X2 ) )
& ( ~ Q2
=> ( fmember @ A @ A2 @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_78_if__split__fmem1,axiom,
! [A: $tType,Q2: $o,X2: A,Y: A,B2: fset @ A] :
( ( fmember @ A @ ( if @ A @ Q2 @ X2 @ Y ) @ B2 )
= ( ( Q2
=> ( fmember @ A @ X2 @ B2 ) )
& ( ~ Q2
=> ( fmember @ A @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_79_eqfset__imp__iff,axiom,
! [A: $tType,A5: fset @ A,B3: fset @ A,X2: A] :
( ( A5 = B3 )
=> ( ( fmember @ A @ X2 @ A5 )
= ( fmember @ A @ X2 @ B3 ) ) ) ).
% eqfset_imp_iff
thf(fact_80_eq__fmem__trans,axiom,
! [A: $tType,A2: A,B2: A,A5: fset @ A] :
( ( A2 = B2 )
=> ( ( fmember @ A @ B2 @ A5 )
=> ( fmember @ A @ A2 @ A5 ) ) ) ).
% eq_fmem_trans
thf(fact_81_fset__choice,axiom,
! [B: $tType,A: $tType,A5: fset @ A,P2: A > B > $o] :
( ! [X3: A] :
( ( fmember @ A @ X3 @ A5 )
=> ? [X1: B] : ( P2 @ X3 @ X1 ) )
=> ? [F2: A > B] :
! [X6: A] :
( ( fmember @ A @ X6 @ A5 )
=> ( P2 @ X6 @ ( F2 @ X6 ) ) ) ) ).
% fset_choice
thf(fact_82_fequalityCE,axiom,
! [A: $tType,A5: fset @ A,B3: fset @ A,C2: A] :
( ( A5 = B3 )
=> ( ( ( fmember @ A @ C2 @ A5 )
=> ~ ( fmember @ A @ C2 @ B3 ) )
=> ~ ( ~ ( fmember @ A @ C2 @ A5 )
=> ( fmember @ A @ C2 @ B3 ) ) ) ) ).
% fequalityCE
thf(fact_83_fset__eqI,axiom,
! [A: $tType,A5: fset @ A,B3: fset @ A] :
( ! [X3: A] :
( ( fmember @ A @ X3 @ A5 )
= ( fmember @ A @ X3 @ B3 ) )
=> ( A5 = B3 ) ) ).
% fset_eqI
thf(fact_84_equalsffemptyI,axiom,
! [A: $tType,A5: fset @ A] :
( ! [Y2: A] :
~ ( fmember @ A @ Y2 @ A5 )
=> ( A5
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% equalsffemptyI
thf(fact_85_equalsffemptyD,axiom,
! [A: $tType,A5: fset @ A,A2: A] :
( ( A5
= ( bot_bot @ ( fset @ A ) ) )
=> ~ ( fmember @ A @ A2 @ A5 ) ) ).
% equalsffemptyD
thf(fact_86_ex__fin__conv,axiom,
! [A: $tType,A5: fset @ A] :
( ( ? [X5: A] : ( fmember @ A @ X5 @ A5 ) )
= ( A5
!= ( bot_bot @ ( fset @ A ) ) ) ) ).
% ex_fin_conv
thf(fact_87_femptyE,axiom,
! [A: $tType,A2: A] :
~ ( fmember @ A @ A2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% femptyE
thf(fact_88_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F3: B > A,G2: C > B,X5: C] : ( F3 @ ( G2 @ X5 ) ) ) ) ).
% comp_apply
thf(fact_89_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_90_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X5: A] : ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_91_Konig,axiom,
! [A: $tType,T6: abstra2103299360e_tree @ A] :
( ~ ( abstra668420080finite @ A @ T6 )
=> ( abstra313004635_ipath @ A @ T6 @ ( abstra1918223989_konig @ A @ T6 ) ) ) ).
% Konig
thf(fact_92_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
= ( product_fst @ A @ B @ X2 ) ) ).
% snd_swap
thf(fact_93_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
= ( product_snd @ B @ A @ X2 ) ) ).
% fst_swap
thf(fact_94_tree_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( abstra457966479e_tree @ A @ B )
= ( ^ [F3: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree3: abstra2103299360e_tree @ A] : ( F3 @ ( abstra573067619e_root @ A @ Tree3 ) @ ( abstra1749095923e_cont @ A @ Tree3 ) ) ) ) ).
% tree.case_eq_if
thf(fact_95_swap__swap,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P3 ) )
= P3 ) ).
% swap_swap
thf(fact_96_ftree__no__ipath,axiom,
! [A: $tType,T6: abstra2103299360e_tree @ A,Steps2: stream @ A] :
( ( abstra668420080finite @ A @ T6 )
=> ~ ( abstra313004635_ipath @ A @ T6 @ Steps2 ) ) ).
% ftree_no_ipath
thf(fact_97_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_98_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_99_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_100_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_101_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F3: B > C,G2: A > B,X5: A] : ( F3 @ ( G2 @ X5 ) ) ) ) ).
% comp_def
thf(fact_102_tfinite_Ocases,axiom,
! [A: $tType,A2: abstra2103299360e_tree @ A] :
( ( abstra668420080finite @ A @ A2 )
=> ! [T5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ A2 ) )
=> ( abstra668420080finite @ A @ T5 ) ) ) ).
% tfinite.cases
thf(fact_103_tfinite_Osimps,axiom,
! [A: $tType] :
( ( abstra668420080finite @ A )
= ( ^ [A3: abstra2103299360e_tree @ A] :
? [T3: abstra2103299360e_tree @ A] :
( ( A3 = T3 )
& ! [X5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ X5 @ ( abstra1749095923e_cont @ A @ T3 ) )
=> ( abstra668420080finite @ A @ X5 ) ) ) ) ) ).
% tfinite.simps
thf(fact_104_tfinite_Oinducts,axiom,
! [A: $tType,X2: abstra2103299360e_tree @ A,P2: ( abstra2103299360e_tree @ A ) > $o] :
( ( abstra668420080finite @ A @ X2 )
=> ( ! [T10: abstra2103299360e_tree @ A] :
( ! [T5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ T10 ) )
=> ( abstra668420080finite @ A @ T5 ) )
=> ( ! [T5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ T10 ) )
=> ( P2 @ T5 ) )
=> ( P2 @ T10 ) ) )
=> ( P2 @ X2 ) ) ) ).
% tfinite.inducts
thf(fact_105_tfinite,axiom,
! [A: $tType,T6: abstra2103299360e_tree @ A] :
( ! [T8: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T8 @ ( abstra1749095923e_cont @ A @ T6 ) )
=> ( abstra668420080finite @ A @ T8 ) )
=> ( abstra668420080finite @ A @ T6 ) ) ).
% tfinite
thf(fact_106_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_107_tree_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree: abstra2103299360e_tree @ A] :
( ( P2 @ ( abstra457966479e_tree @ A @ B @ F @ Tree ) )
= ( ( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) )
=> ( P2 @ ( F @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ) ) ).
% tree.split_sel
thf(fact_108_tree_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree: abstra2103299360e_tree @ A] :
( ( P2 @ ( abstra457966479e_tree @ A @ B @ F @ Tree ) )
= ( ~ ( ( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) )
& ~ ( P2 @ ( F @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ) ) ) ).
% tree.split_sel_asm
thf(fact_109_Stream_Osmember__def,axiom,
! [A: $tType] :
( ( smember @ A )
= ( ^ [X5: A,S3: stream @ A] : ( member @ A @ X5 @ ( sset @ A @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_110_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( case_stream @ A @ B )
= ( ^ [F3: A > ( stream @ A ) > B,Stream5: stream @ A] : ( F3 @ ( shd @ A @ Stream5 ) @ ( stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_111_tree_Oinject,axiom,
! [A: $tType,X12: A,X22: fset @ ( abstra2103299360e_tree @ A ),Y1: A,Y22: fset @ ( abstra2103299360e_tree @ A )] :
( ( ( abstra388494275e_Node @ A @ X12 @ X22 )
= ( abstra388494275e_Node @ A @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% tree.inject
thf(fact_112_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,F: A > B,G: A > B] :
( ( member @ A @ A2 @ A5 )
=> ( ( override_on @ A @ B @ F @ G @ A5 @ A2 )
= ( G @ A2 ) ) ) ).
% override_on_apply_in
thf(fact_113_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,F: A > B,G: A > B] :
( ~ ( member @ A @ A2 @ A5 )
=> ( ( override_on @ A @ B @ F @ G @ A5 @ A2 )
= ( F @ A2 ) ) ) ).
% override_on_apply_notin
thf(fact_114_tree_Ocollapse,axiom,
! [A: $tType,Tree: abstra2103299360e_tree @ A] :
( ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) )
= Tree ) ).
% tree.collapse
thf(fact_115_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F3: A > B,G2: A > B,A6: set @ A,A3: A] : ( if @ B @ ( member @ A @ A3 @ A6 ) @ ( G2 @ A3 ) @ ( F3 @ A3 ) ) ) ) ).
% override_on_def
thf(fact_116_tree_Oexhaust,axiom,
! [A: $tType,Y: abstra2103299360e_tree @ A] :
~ ! [X13: A,X23: fset @ ( abstra2103299360e_tree @ A )] :
( Y
!= ( abstra388494275e_Node @ A @ X13 @ X23 ) ) ).
% tree.exhaust
thf(fact_117_tree_Osel_I1_J,axiom,
! [A: $tType,X12: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra573067619e_root @ A @ ( abstra388494275e_Node @ A @ X12 @ X22 ) )
= X12 ) ).
% tree.sel(1)
thf(fact_118_tree_Osel_I2_J,axiom,
! [A: $tType,X12: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra1749095923e_cont @ A @ ( abstra388494275e_Node @ A @ X12 @ X22 ) )
= X22 ) ).
% tree.sel(2)
thf(fact_119_tree_Ocase,axiom,
! [B: $tType,A: $tType,F: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,X12: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra457966479e_tree @ A @ B @ F @ ( abstra388494275e_Node @ A @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% tree.case
thf(fact_120_tree_Oexhaust__sel,axiom,
! [A: $tType,Tree: abstra2103299360e_tree @ A] :
( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ).
% tree.exhaust_sel
thf(fact_121_from__nat__into__inject,axiom,
! [A: $tType,A5: set @ A,B3: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A5 )
=> ( ( B3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ B3 )
=> ( ( ( counta609264050t_into @ A @ A5 )
= ( counta609264050t_into @ A @ B3 ) )
= ( A5 = B3 ) ) ) ) ) ) ).
% from_nat_into_inject
thf(fact_122_szip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,S1: stream @ A,S22: stream @ B] :
( ( stl @ ( product_prod @ A @ B ) @ ( szip @ A @ B @ S1 @ S22 ) )
= ( szip @ A @ B @ ( stl @ A @ S1 ) @ ( stl @ B @ S22 ) ) ) ).
% szip.simps(2)
thf(fact_123_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( sdrop_while @ A )
= ( ^ [P4: A > $o,S3: stream @ A] : ( if @ ( stream @ A ) @ ( P4 @ ( shd @ A @ S3 ) ) @ ( sdrop_while @ A @ P4 @ ( stl @ A @ S3 ) ) @ S3 ) ) ) ).
% sdrop_while.simps
thf(fact_124_stream_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F: A > ( stream @ A ) > B,Stream: stream @ A] :
( ( P2 @ ( case_stream @ A @ B @ F @ Stream ) )
= ( ~ ( ( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) )
& ~ ( P2 @ ( F @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ) ) ) ).
% stream.split_sel_asm
thf(fact_125_stream_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F: A > ( stream @ A ) > B,Stream: stream @ A] :
( ( P2 @ ( case_stream @ A @ B @ F @ Stream ) )
= ( ( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) )
=> ( P2 @ ( F @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ) ) ).
% stream.split_sel
thf(fact_126_stream_Oinject,axiom,
! [A: $tType,X12: A,X22: stream @ A,Y1: A,Y22: stream @ A] :
( ( ( sCons @ A @ X12 @ X22 )
= ( sCons @ A @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_127_smember__code,axiom,
! [A: $tType,X2: A,Y: A,S: stream @ A] :
( ( smember @ A @ X2 @ ( sCons @ A @ Y @ S ) )
= ( ( X2 != Y )
=> ( smember @ A @ X2 @ S ) ) ) ).
% smember_code
thf(fact_128_stream_Ocollapse,axiom,
! [A: $tType,Stream: stream @ A] :
( ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_129_stream_Osel_I2_J,axiom,
! [A: $tType,X12: A,X22: stream @ A] :
( ( stl @ A @ ( sCons @ A @ X12 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_130_stream_Osel_I1_J,axiom,
! [A: $tType,X12: A,X22: stream @ A] :
( ( shd @ A @ ( sCons @ A @ X12 @ X22 ) )
= X12 ) ).
% stream.sel(1)
thf(fact_131_stream_Oset__induct,axiom,
! [A: $tType,X2: A,A2: stream @ A,P2: A > ( stream @ A ) > $o] :
( ( member @ A @ X2 @ ( sset @ A @ A2 ) )
=> ( ! [Z1: A,Z2: stream @ A] : ( P2 @ Z1 @ ( sCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: stream @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( sset @ A @ Z2 ) )
=> ( ( P2 @ Xa2 @ Z2 )
=> ( P2 @ Xa2 @ ( sCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P2 @ X2 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_132_stream_Oset__cases,axiom,
! [A: $tType,E: A,A2: stream @ A] :
( ( member @ A @ E @ ( sset @ A @ A2 ) )
=> ( ! [Z2: stream @ A] :
( A2
!= ( sCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: stream @ A] :
( ( A2
= ( sCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( sset @ A @ Z2 ) ) ) ) ) ).
% stream.set_cases
thf(fact_133_stream_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: stream @ A] : ( member @ A @ A1 @ ( sset @ A @ ( sCons @ A @ A1 @ A22 ) ) ) ).
% stream.set_intros(1)
thf(fact_134_stream_Oset__intros_I2_J,axiom,
! [A: $tType,X2: A,A22: stream @ A,A1: A] :
( ( member @ A @ X2 @ ( sset @ A @ A22 ) )
=> ( member @ A @ X2 @ ( sset @ A @ ( sCons @ A @ A1 @ A22 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_135_stream_Oexhaust,axiom,
! [A: $tType,Y: stream @ A] :
~ ! [X13: A,X23: stream @ A] :
( Y
!= ( sCons @ A @ X13 @ X23 ) ) ).
% stream.exhaust
thf(fact_136_sdrop__while__SCons,axiom,
! [A: $tType,P2: A > $o,A2: A,S: stream @ A] :
( ( ( P2 @ A2 )
=> ( ( sdrop_while @ A @ P2 @ ( sCons @ A @ A2 @ S ) )
= ( sdrop_while @ A @ P2 @ S ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( sdrop_while @ A @ P2 @ ( sCons @ A @ A2 @ S ) )
= ( sCons @ A @ A2 @ S ) ) ) ) ).
% sdrop_while_SCons
thf(fact_137_stream_Ocase,axiom,
! [B: $tType,A: $tType,F: A > ( stream @ A ) > B,X12: A,X22: stream @ A] :
( ( case_stream @ A @ B @ F @ ( sCons @ A @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% stream.case
thf(fact_138_from__nat__into,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( counta609264050t_into @ A @ A5 @ N2 ) @ A5 ) ) ).
% from_nat_into
thf(fact_139_from__nat__into__surj,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [N3: nat] :
( ( counta609264050t_into @ A @ A5 @ N3 )
= A2 ) ) ) ).
% from_nat_into_surj
thf(fact_140_countable__all,axiom,
! [A: $tType,S4: set @ A,P2: A > $o] :
( ( countable_countable @ A @ S4 )
=> ( ( ! [X5: A] :
( ( member @ A @ X5 @ S4 )
=> ( P2 @ X5 ) ) )
= ( ! [N4: nat] :
( ( member @ A @ ( counta609264050t_into @ A @ S4 @ N4 ) @ S4 )
=> ( P2 @ ( counta609264050t_into @ A @ S4 @ N4 ) ) ) ) ) ) ).
% countable_all
thf(fact_141_stream_Oexhaust__sel,axiom,
! [A: $tType,Stream: stream @ A] :
( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_142_sfilter_Ocode,axiom,
! [A: $tType] :
( ( sfilter @ A )
= ( ^ [P4: A > $o,S3: stream @ A] : ( sCons @ A @ ( shd @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ S3 ) ) @ ( sfilter @ A @ P4 @ ( stl @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ S3 ) ) ) ) ) ) ).
% sfilter.code
thf(fact_143_sinterleave_Ocode,axiom,
! [A: $tType] :
( ( sinterleave @ A )
= ( ^ [S12: stream @ A,S23: stream @ A] : ( sCons @ A @ ( shd @ A @ S12 ) @ ( sinterleave @ A @ S23 @ ( stl @ A @ S12 ) ) ) ) ) ).
% sinterleave.code
thf(fact_144_smap2_Ocode,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( smap2 @ A @ B @ C )
= ( ^ [F3: A > B > C,S12: stream @ A,S23: stream @ B] : ( sCons @ C @ ( F3 @ ( shd @ A @ S12 ) @ ( shd @ B @ S23 ) ) @ ( smap2 @ A @ B @ C @ F3 @ ( stl @ A @ S12 ) @ ( stl @ B @ S23 ) ) ) ) ) ).
% smap2.code
thf(fact_145_from__nat__into__to__nat__on,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( counta609264050t_into @ A @ A5 @ ( countable_to_nat_on @ A @ A5 @ A2 ) )
= A2 ) ) ) ).
% from_nat_into_to_nat_on
thf(fact_146_to__nat__on__inj,axiom,
! [A: $tType,A5: set @ A,A2: A,B2: A] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( member @ A @ B2 @ A5 )
=> ( ( ( countable_to_nat_on @ A @ A5 @ A2 )
= ( countable_to_nat_on @ A @ A5 @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% to_nat_on_inj
thf(fact_147_sfilter__Stream,axiom,
! [A: $tType,P2: A > $o,X2: A,S: stream @ A] :
( ( ( P2 @ X2 )
=> ( ( sfilter @ A @ P2 @ ( sCons @ A @ X2 @ S ) )
= ( sCons @ A @ X2 @ ( sfilter @ A @ P2 @ S ) ) ) )
& ( ~ ( P2 @ X2 )
=> ( ( sfilter @ A @ P2 @ ( sCons @ A @ X2 @ S ) )
= ( sfilter @ A @ P2 @ S ) ) ) ) ).
% sfilter_Stream
thf(fact_148_smap2_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,S1: stream @ A,S22: stream @ B] :
( ( stl @ C @ ( smap2 @ A @ B @ C @ F @ S1 @ S22 ) )
= ( smap2 @ A @ B @ C @ F @ ( stl @ A @ S1 ) @ ( stl @ B @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_149_smap2_Osimps_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,S1: stream @ A,S22: stream @ B] :
( ( shd @ C @ ( smap2 @ A @ B @ C @ F @ S1 @ S22 ) )
= ( F @ ( shd @ A @ S1 ) @ ( shd @ B @ S22 ) ) ) ).
% smap2.simps(1)
thf(fact_150_smap2__unfold,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,S1: stream @ B,B2: C,S22: stream @ C] :
( ( smap2 @ B @ C @ A @ F @ ( sCons @ B @ A2 @ S1 ) @ ( sCons @ C @ B2 @ S22 ) )
= ( sCons @ A @ ( F @ A2 @ B2 ) @ ( smap2 @ B @ C @ A @ F @ S1 @ S22 ) ) ) ).
% smap2_unfold
thf(fact_151_sinterleave_Osimps_I2_J,axiom,
! [A: $tType,S1: stream @ A,S22: stream @ A] :
( ( stl @ A @ ( sinterleave @ A @ S1 @ S22 ) )
= ( sinterleave @ A @ S22 @ ( stl @ A @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_152_sinterleave_Osimps_I1_J,axiom,
! [A: $tType,S1: stream @ A,S22: stream @ A] :
( ( shd @ A @ ( sinterleave @ A @ S1 @ S22 ) )
= ( shd @ A @ S1 ) ) ).
% sinterleave.simps(1)
thf(fact_153_sinterleave__code,axiom,
! [A: $tType,X2: A,S1: stream @ A,S22: stream @ A] :
( ( sinterleave @ A @ ( sCons @ A @ X2 @ S1 ) @ S22 )
= ( sCons @ A @ X2 @ ( sinterleave @ A @ S22 @ S1 ) ) ) ).
% sinterleave_code
thf(fact_154_sfilter_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ( stl @ A @ ( sfilter @ A @ P2 @ S ) )
= ( sfilter @ A @ P2 @ ( stl @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) ) ) ).
% sfilter.simps(2)
thf(fact_155_sfilter_Osimps_I1_J,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ( shd @ A @ ( sfilter @ A @ P2 @ S ) )
= ( shd @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) ) ).
% sfilter.simps(1)
thf(fact_156_sfilter__P,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ( P2 @ ( shd @ A @ S ) )
=> ( ( sfilter @ A @ P2 @ S )
= ( sCons @ A @ ( shd @ A @ S ) @ ( sfilter @ A @ P2 @ ( stl @ A @ S ) ) ) ) ) ).
% sfilter_P
thf(fact_157_sfilter__not__P,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ~ ( P2 @ ( shd @ A @ S ) )
=> ( ( sfilter @ A @ P2 @ S )
= ( sfilter @ A @ P2 @ ( stl @ A @ S ) ) ) ) ).
% sfilter_not_P
thf(fact_158_to__nat__on__from__nat__into__infinite,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( countable_to_nat_on @ A @ A5 @ ( counta609264050t_into @ A @ A5 @ N2 ) )
= N2 ) ) ) ).
% to_nat_on_from_nat_into_infinite
thf(fact_159_from__nat__into__inj__infinite,axiom,
! [A: $tType,A5: set @ A,M2: nat,N2: nat] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( ( counta609264050t_into @ A @ A5 @ M2 )
= ( counta609264050t_into @ A @ A5 @ N2 ) )
= ( M2 = N2 ) ) ) ) ).
% from_nat_into_inj_infinite
thf(fact_160_countable__finite,axiom,
! [A: $tType,S4: set @ A] :
( ( finite_finite2 @ A @ S4 )
=> ( countable_countable @ A @ S4 ) ) ).
% countable_finite
thf(fact_161_uncountable__infinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( countable_countable @ A @ A5 )
=> ~ ( finite_finite2 @ A @ A5 ) ) ).
% uncountable_infinite
thf(fact_162_countable__Collect__finite,axiom,
! [A: $tType] :
( ( countable @ A @ ( type2 @ A ) )
=> ( countable_countable @ ( set @ A ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ).
% countable_Collect_finite
thf(fact_163_to__nat__on__surj,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ( countable_to_nat_on @ A @ A5 @ X3 )
= N2 ) ) ) ) ).
% to_nat_on_surj
thf(fact_164_infinite__imp__nonempty,axiom,
! [A: $tType,S4: set @ A] :
( ~ ( finite_finite2 @ A @ S4 )
=> ( S4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_165_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_166_nxt_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( linear1494993505on_nxt @ A @ B )
= ( ^ [Phi: ( stream @ A ) > B,Xs: stream @ A] : ( Phi @ ( stl @ A @ Xs ) ) ) ) ).
% nxt.simps
thf(fact_167_nxt_Oelims,axiom,
! [B: $tType,A: $tType,X2: ( stream @ A ) > B,Xa: stream @ A,Y: B] :
( ( ( linear1494993505on_nxt @ A @ B @ X2 @ Xa )
= Y )
=> ( Y
= ( X2 @ ( stl @ A @ Xa ) ) ) ) ).
% nxt.elims
thf(fact_168_szip_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,S1: stream @ A,S22: stream @ B] :
( ( shd @ ( product_prod @ A @ B ) @ ( szip @ A @ B @ S1 @ S22 ) )
= ( product_Pair @ A @ B @ ( shd @ A @ S1 ) @ ( shd @ B @ S22 ) ) ) ).
% szip.simps(1)
thf(fact_169_fset_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: B > $o,F: A > B,X2: fset @ A] :
( ( pred_fset @ B @ Q2 @ ( fimage @ A @ B @ F @ X2 ) )
= ( pred_fset @ A @ ( comp @ B @ $o @ A @ Q2 @ F ) @ X2 ) ) ).
% fset.pred_map
thf(fact_170_szip_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( szip @ A @ B )
= ( ^ [S12: stream @ A,S23: stream @ B] : ( sCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( shd @ A @ S12 ) @ ( shd @ B @ S23 ) ) @ ( szip @ A @ B @ ( stl @ A @ S12 ) @ ( stl @ B @ S23 ) ) ) ) ) ).
% szip.code
thf(fact_171_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X12 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_172_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B4 ) )
= ( ( A2 = A7 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_173_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% swap_simp
thf(fact_174_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_175_snd__conv,axiom,
! [Aa: $tType,A: $tType,X12: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X12 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_176_fst__conv,axiom,
! [B: $tType,A: $tType,X12: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X12 @ X22 ) )
= X12 ) ).
% fst_conv
thf(fact_177_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_178_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_179_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X3: A,Y2: B] :
( P3
= ( product_Pair @ A @ B @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_180_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A8: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A8 @ B5 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_181_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B4 ) )
=> ~ ( ( A2 = A7 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_182_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A8: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_183_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A8: A,B5: B,C3: C,D3: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_184_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
~ ! [A8: A,B5: B,C3: C,D3: D,E3: E2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) ) ).
% prod_cases5
thf(fact_185_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) ) )] :
~ ! [A8: A,B5: B,C3: C,D3: D,E3: E2,F2: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F4 ) @ D3 @ ( product_Pair @ E2 @ F4 @ E3 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_186_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
~ ! [A8: A,B5: B,C3: C,D3: D,E3: E2,F2: F4,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F4 @ G3 ) @ E3 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_187_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A8: A,B5: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct3
thf(fact_188_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A8: A,B5: B,C3: C,D3: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D3 ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct4
thf(fact_189_prod__induct5,axiom,
! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
( ! [A8: A,B5: B,C3: C,D3: D,E3: E2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C3 @ ( product_Pair @ D @ E2 @ D3 @ E3 ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct5
thf(fact_190_prod__induct6,axiom,
! [F4: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) ) )] :
( ! [A8: A,B5: B,C3: C,D3: D,E3: E2,F2: F4] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ F4 ) @ D3 @ ( product_Pair @ E2 @ F4 @ E3 @ F2 ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct6
thf(fact_191_prod__induct7,axiom,
! [G3: $tType,F4: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
( ! [A8: A,B5: B,C3: C,D3: D,E3: E2,F2: F4,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F4 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F4 @ G3 ) @ E3 @ ( product_Pair @ F4 @ G3 @ F2 @ G4 ) ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct7
thf(fact_192_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A8: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A8 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_193_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A8: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A8 @ B5 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_194_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X2: B] :
( ( P2 @ Y @ X2 )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_195_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P3: A,Q2: B > $o,Q: B] :
( ( P2 @ P3 )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_196_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_197_surjective__pairing,axiom,
! [B: $tType,A: $tType,T6: product_prod @ A @ B] :
( T6
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T6 ) @ ( product_snd @ A @ B @ T6 ) ) ) ).
% surjective_pairing
thf(fact_198_szip__unfold,axiom,
! [A: $tType,B: $tType,A2: A,S1: stream @ A,B2: B,S22: stream @ B] :
( ( szip @ A @ B @ ( sCons @ A @ A2 @ S1 ) @ ( sCons @ B @ B2 @ S22 ) )
= ( sCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( szip @ A @ B @ S1 @ S22 ) ) ) ).
% szip_unfold
thf(fact_199_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P5 ) @ ( product_fst @ A @ B @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_200_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_201_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
( ( P2 @ X2 @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_202_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P3: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P3 ) )
= ( ? [A3: B] :
( P3
= ( product_Pair @ B @ A @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_203_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P3: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P3 ) )
= ( ? [B6: B] :
( P3
= ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_204_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z3: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z3 ) )
=> ( ( product_fst @ A @ B @ X2 )
= Y ) ) ).
% fstI
thf(fact_205_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z3: B] :
( ( X2
= ( product_Pair @ A @ B @ Y @ Z3 ) )
=> ( ( product_snd @ A @ B @ X2 )
= Z3 ) ) ).
% sndI
thf(fact_206_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X2: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X2 )
= ( comp @ D @ A @ C @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_207_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_208_sdrop__while_Oraw__induct,axiom,
! [A: $tType,Pa: ( product_prod @ ( A > $o ) @ ( stream @ A ) ) > ( stream @ A ) > $o,P2: A > $o,S: stream @ A,Y: stream @ A] :
( ! [Sdrop_while: ( A > $o ) > ( stream @ A ) > ( stream @ A )] :
( ! [S5: A > $o,B7: stream @ A] :
( ( ( Sdrop_while @ S5 @ B7 )
!= ( undefined @ ( stream @ A ) ) )
=> ( Pa @ ( product_Pair @ ( A > $o ) @ ( stream @ A ) @ S5 @ B7 ) @ ( Sdrop_while @ S5 @ B7 ) ) )
=> ! [P6: A > $o,S2: stream @ A,Pa2: stream @ A] :
( ( ( ( P6 @ ( shd @ A @ S2 ) )
=> ( ( Sdrop_while @ P6 @ ( stl @ A @ S2 ) )
= Pa2 ) )
& ( ~ ( P6 @ ( shd @ A @ S2 ) )
=> ( S2 = Pa2 ) ) )
=> ( ( Pa2
!= ( undefined @ ( stream @ A ) ) )
=> ( Pa @ ( product_Pair @ ( A > $o ) @ ( stream @ A ) @ P6 @ S2 ) @ Pa2 ) ) ) )
=> ( ( ( sdrop_while @ A @ P2 @ S )
= Y )
=> ( ( Y
!= ( undefined @ ( stream @ A ) ) )
=> ( Pa @ ( product_Pair @ ( A > $o ) @ ( stream @ A ) @ P2 @ S ) @ Y ) ) ) ) ).
% sdrop_while.raw_induct
thf(fact_209_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_210_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_211_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P2: B > C > $o,Q2: C > A > $o] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P2 @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_212_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P2: A > C > $o,Q2: C > B > $o] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_213_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q2: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q2 ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_214_from__nat__into__inj,axiom,
! [A: $tType,A5: set @ A,M2: nat,N2: nat] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ nat @ M2 @ ( image @ A @ nat @ ( countable_to_nat_on @ A @ A5 ) @ A5 ) )
=> ( ( member @ nat @ N2 @ ( image @ A @ nat @ ( countable_to_nat_on @ A @ A5 ) @ A5 ) )
=> ( ( ( counta609264050t_into @ A @ A5 @ M2 )
= ( counta609264050t_into @ A @ A5 @ N2 ) )
= ( M2 = N2 ) ) ) ) ) ).
% from_nat_into_inj
thf(fact_215_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A5: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_216_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( bot_bot @ ( A > C > $o ) ) @ R2 )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot1
thf(fact_217_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R2: A > C > $o] :
( ( relcompp @ A @ C @ B @ R2 @ ( bot_bot @ ( C > B > $o ) ) )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot2
thf(fact_218_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( image @ B @ A @ F @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_219_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_220_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_221_countable__image,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( countable_countable @ A @ A5 )
=> ( countable_countable @ B @ ( image @ A @ B @ F @ A5 ) ) ) ).
% countable_image
thf(fact_222_to__nat__on__from__nat__into,axiom,
! [A: $tType,N2: nat,A5: set @ A] :
( ( member @ nat @ N2 @ ( image @ A @ nat @ ( countable_to_nat_on @ A @ A5 ) @ A5 ) )
=> ( ( countable_to_nat_on @ A @ A5 @ ( counta609264050t_into @ A @ A5 @ N2 ) )
= N2 ) ) ).
% to_nat_on_from_nat_into
thf(fact_223_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_224_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A5: set @ B,G: C > A,B3: set @ C,H: A > D] :
( ( ( image @ B @ A @ F @ A5 )
= ( image @ C @ A @ G @ B3 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A5 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B3 ) ) ) ).
% image_eq_imp_comp
thf(fact_225_relcompp_OrelcompI,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > B > $o,A2: A,B2: B,S: B > C > $o,C2: C] :
( ( R @ A2 @ B2 )
=> ( ( S @ B2 @ C2 )
=> ( relcompp @ A @ B @ C @ R @ S @ A2 @ C2 ) ) ) ).
% relcompp.relcompI
thf(fact_226_relcompp_Oinducts,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > $o,S: B > C > $o,X12: A,X22: C,P2: A > C > $o] :
( ( relcompp @ A @ B @ C @ R @ S @ X12 @ X22 )
=> ( ! [A8: A,B5: B,C3: C] :
( ( R @ A8 @ B5 )
=> ( ( S @ B5 @ C3 )
=> ( P2 @ A8 @ C3 ) ) )
=> ( P2 @ X12 @ X22 ) ) ) ).
% relcompp.inducts
thf(fact_227_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R: A > D > $o,S: D > C > $o,T6: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R @ S ) @ T6 )
= ( relcompp @ A @ D @ B @ R @ ( relcompp @ D @ C @ B @ S @ T6 ) ) ) ).
% relcompp_assoc
thf(fact_228_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R4: A > B > $o,S6: B > C > $o,A3: A,C4: C] :
? [B6: B] :
( ( R4 @ A3 @ B6 )
& ( S6 @ B6 @ C4 ) ) ) ) ).
% relcompp_apply
thf(fact_229_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R3: A > B > $o,S3: B > C > $o,A12: A,A23: C] :
? [A3: A,B6: B,C4: C] :
( ( A12 = A3 )
& ( A23 = C4 )
& ( R3 @ A3 @ B6 )
& ( S3 @ B6 @ C4 ) ) ) ) ).
% relcompp.simps
thf(fact_230_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > B > $o,S: B > C > $o,A1: A,A22: C] :
( ( relcompp @ A @ B @ C @ R @ S @ A1 @ A22 )
=> ~ ! [B5: B] :
( ( R @ A1 @ B5 )
=> ~ ( S @ B5 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_231_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > B > $o,S: B > C > $o,A2: A,C2: C] :
( ( relcompp @ A @ B @ C @ R @ S @ A2 @ C2 )
=> ~ ! [B5: B] :
( ( R @ A2 @ B5 )
=> ~ ( S @ B5 @ C2 ) ) ) ).
% relcomppE
thf(fact_232_eq__OO,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R2 )
= R2 ) ).
% eq_OO
thf(fact_233_OO__eq,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( relcompp @ A @ B @ B @ R2
@ ^ [Y3: B,Z: B] : Y3 = Z )
= R2 ) ).
% OO_eq
thf(fact_234_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,B2: B,F: A > B] :
( ( member @ A @ X2 @ A5 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_235_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A5 ) )
=> ( P2 @ X3 ) )
=> ! [X6: B] :
( ( member @ B @ X6 @ A5 )
=> ( P2 @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_236_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_237_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P2: A > $o] :
( ? [X6: A] :
( ( member @ A @ X6 @ ( image @ B @ A @ F @ A5 ) )
& ( P2 @ X6 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_238_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F: B > A,A5: set @ B] :
( ( member @ A @ Z3 @ ( image @ B @ A @ F @ A5 ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( Z3
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_239_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,F: A > B] :
( ( member @ A @ X2 @ A5 )
=> ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A5 ) ) ) ).
% imageI
thf(fact_240_eq__from__nat__into__iff,axiom,
! [A: $tType,A5: set @ A,X2: A,I: nat] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( member @ nat @ I @ ( image @ A @ nat @ ( countable_to_nat_on @ A @ A5 ) @ A5 ) )
=> ( ( X2
= ( counta609264050t_into @ A @ A5 @ I ) )
= ( I
= ( countable_to_nat_on @ A @ A5 @ X2 ) ) ) ) ) ) ).
% eq_from_nat_into_iff
thf(fact_241_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% Inf.INF_image
thf(fact_242_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A5 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% Sup.SUP_image
thf(fact_243_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X2: B,A5: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X2 ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A5 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ A5 ) ) ).
% pair_in_swap_image
thf(fact_244_Stream__image,axiom,
! [A: $tType,X2: A,Y: stream @ A,X7: A,Y4: set @ ( stream @ A )] :
( ( member @ ( stream @ A ) @ ( sCons @ A @ X2 @ Y ) @ ( image @ ( stream @ A ) @ ( stream @ A ) @ ( sCons @ A @ X7 ) @ Y4 ) )
= ( ( X2 = X7 )
& ( member @ ( stream @ A ) @ Y @ Y4 ) ) ) ).
% Stream_image
thf(fact_245_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_246_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_247_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X2: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X2 @ Y ) )
= ( product_Pair @ A @ B @ ( F @ X2 ) @ Y ) ) ).
% apfst_conv
thf(fact_248_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X2: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X2 @ Y ) )
= ( product_Pair @ A @ B @ X2 @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_249_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X2: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X2 ) )
= ( F @ ( product_fst @ C @ B @ X2 ) ) ) ).
% fst_apfst
thf(fact_250_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X2: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X2 ) )
= ( product_snd @ C @ A @ X2 ) ) ).
% snd_apfst
thf(fact_251_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X2: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X2 )
= ( product_apfst @ C @ A @ B @ G @ X2 ) )
= ( ( F @ ( product_fst @ C @ B @ X2 ) )
= ( G @ ( product_fst @ C @ B @ X2 ) ) ) ) ).
% apfst_eq_conv
thf(fact_252_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X2 ) )
= ( product_fst @ A @ C @ X2 ) ) ).
% fst_apsnd
thf(fact_253_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X2: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X2 ) )
= ( F @ ( product_snd @ B @ C @ X2 ) ) ) ).
% snd_apsnd
thf(fact_254_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X2 )
= ( product_apsnd @ C @ B @ A @ G @ X2 ) )
= ( ( F @ ( product_snd @ A @ C @ X2 ) )
= ( G @ ( product_snd @ A @ C @ X2 ) ) ) ) ).
% apsnd_eq_conv
thf(fact_255_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
%----Type constructors (16)
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite,axiom,
! [A9: $tType,A4: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A4 @ ( type2 @ A4 ) ) )
=> ( finite_finite @ ( product_prod @ A9 @ A4 ) @ ( type2 @ ( product_prod @ A9 @ A4 ) ) ) ) ).
thf(tcon_FSet_Ofset___Finite__Set_Ofinite_1,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( finite_finite @ ( fset @ A9 ) @ ( type2 @ ( fset @ A9 ) ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_2,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( finite_finite @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite_4,axiom,
! [A9: $tType,A4: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( finite_finite @ A4 @ ( type2 @ A4 ) ) )
=> ( finite_finite @ ( A9 > A4 ) @ ( type2 @ ( A9 > A4 ) ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A9: $tType,A4: $tType] :
( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
& ( countable @ A4 @ ( type2 @ A4 ) ) )
=> ( countable @ ( A9 > A4 ) @ ( type2 @ ( A9 > A4 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A4: $tType] :
( ( bot @ A4 @ ( type2 @ A4 ) )
=> ( bot @ ( A9 > A4 ) @ ( type2 @ ( A9 > A4 ) ) ) ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_5,axiom,
countable @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_6,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Countable_Ocountable_7,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( countable @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Countable_Ocountable_9,axiom,
countable @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_FSet_Ofset___Countable_Ocountable_11,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 @ ( type2 @ A9 ) )
=> ( countable @ ( fset @ A9 ) @ ( type2 @ ( fset @ A9 ) ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Obot_12,axiom,
! [A9: $tType] : ( bot @ ( fset @ A9 ) @ ( type2 @ ( fset @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_13,axiom,
! [A9: $tType,A4: $tType] :
( ( ( countable @ A9 @ ( type2 @ A9 ) )
& ( countable @ A4 @ ( type2 @ A4 ) ) )
=> ( countable @ ( product_prod @ A9 @ A4 ) @ ( type2 @ ( product_prod @ A9 @ A4 ) ) ) ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
? [Steps: stream @ ( product_prod @ state @ rule ),Sl2: fset @ state] :
( ( stepsa = Steps )
& ( member @ rule @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ ( sset @ rule @ rules ) )
& ( fmember @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) ) ) @ Sl2 )
& ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ Sl2 )
& ( ? [T4: abstra2103299360e_tree @ ( product_prod @ state @ rule ),Stepsa: stream @ ( product_prod @ state @ rule )] :
( ( ( stl @ ( product_prod @ state @ rule ) @ Steps )
= Stepsa )
& ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T4 )
& ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ T4 @ Stepsa ) )
| ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) ) ) ) ).
%------------------------------------------------------------------------------