TPTP Problem File: COM160^1.p
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%------------------------------------------------------------------------------
% File : COM160^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Abstract completeness 317
% 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__317.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 365 ( 132 unt; 65 typ; 0 def)
% Number of atoms : 824 ( 205 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 5149 ( 60 ~; 6 |; 68 &;4631 @)
% ( 0 <=>; 384 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 327 ( 327 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 62 usr; 8 con; 0-6 aty)
% Number of variables : 1224 ( 65 ^;1058 !; 36 ?;1224 :)
% ( 65 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:54:49.610
%------------------------------------------------------------------------------
%----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 (56)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem,type,
abstra1326562878System:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_OminWait,type,
abstra1332369113inWait:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > nat ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Oper,type,
abstra2096684367le_per:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > Rule > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Opos,type,
abstra2097340358le_pos:
!>[A: $tType] : ( ( stream @ A ) > A > nat ) ).
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_Ofair,type,
abstra928354080m_fair:
!>[Rule: $tType] : ( ( stream @ Rule ) > ( stream @ Rule ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofenum,type,
abstra1774373515_fenum:
!>[Rule: $tType] : ( ( stream @ Rule ) > ( stream @ Rule ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OmkTree,type,
abstra1225283448mkTree:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) ) ).
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_Otrim,type,
abstra1259602206m_trim:
!>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > ( stream @ Rule ) ) ).
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_Otree_ONode,type,
abstra388494275e_Node:
!>[A: $tType] : ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > ( abstra2103299360e_tree @ A ) ) ).
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_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Stream_Osdrop,type,
sdrop:
!>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osdrop__while,type,
sdrop_while:
!>[A: $tType] : ( ( A > $o ) > ( 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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_S,type,
s: set @ state ).
thf(sy_v_eff,type,
eff: rule > state > ( fset @ state ) > $o ).
thf(sy_v_r,type,
r: rule ).
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_thesis____,type,
thesis: $o ).
%----Relevant facts (256)
thf(fact_0_calculation_I1_J,axiom,
member @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ stepsa ) ) @ s ).
% calculation(1)
thf(fact_1_calculation_I3_J,axiom,
( ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ stepsa ) )
!= r ) ).
% calculation(3)
thf(fact_2_e,axiom,
( ( abstra523868654_epath @ rule @ state @ eff @ rules @ steps )
& ( member @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ steps ) ) @ s ) ) ).
% e
thf(fact_3_calculation_I2_J,axiom,
abstra1874422341nabled @ rule @ state @ eff @ r @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ stepsa ) ) ).
% calculation(2)
thf(fact_4_enabled__def,axiom,
! [R: rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
= ( ^ [P: ( fset @ state ) > $o] :
? [X: fset @ state] : ( P @ X )
@ ( eff @ R @ S ) ) ) ).
% enabled_def
thf(fact_5_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_6_epath_Ocoinduct,axiom,
! [X2: ( stream @ ( product_prod @ state @ rule ) ) > $o,X3: stream @ ( product_prod @ state @ rule )] :
( ( X2 @ X3 )
=> ( ! [X4: stream @ ( product_prod @ state @ rule )] :
( ( X2 @ X4 )
=> ? [Steps: stream @ ( product_prod @ state @ rule ),Sl2: fset @ state] :
( ( X4 = 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 )
& ( ( X2 @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) )
| ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) ) ) ) )
=> ( abstra523868654_epath @ rule @ state @ eff @ rules @ X3 ) ) ) ).
% epath.coinduct
thf(fact_7_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_8_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_9_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_10_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_11_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_12_RuleSystem__axioms,axiom,
abstra1326562878System @ rule @ state @ eff @ rules @ s ).
% RuleSystem_axioms
thf(fact_13_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_14_p,axiom,
abstra2096684367le_per @ rule @ state @ eff @ rules @ s @ r ).
% p
thf(fact_15_enabled__R,axiom,
! [S: state] :
( ( member @ state @ S @ s )
=> ? [X4: rule] :
( ( member @ rule @ X4 @ ( sset @ rule @ rules ) )
& ? [X1: fset @ state] : ( eff @ X4 @ S @ X1 ) ) ) ).
% enabled_R
thf(fact_16_calculation_I4_J,axiom,
member @ rule @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ stepsa ) ) @ ( sset @ rule @ rules ) ).
% calculation(4)
thf(fact_17_eff__S,axiom,
! [S: state,R: rule,Sl3: fset @ state,S2: state] :
( ( member @ state @ S @ s )
=> ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
=> ( ( eff @ R @ S @ Sl3 )
=> ( ( fmember @ state @ S2 @ Sl3 )
=> ( member @ state @ S2 @ s ) ) ) ) ) ).
% eff_S
thf(fact_18_sset__fenum,axiom,
( ( sset @ rule @ ( abstra1774373515_fenum @ rule @ rules ) )
= ( sset @ rule @ rules ) ) ).
% sset_fenum
thf(fact_19_pickEff,axiom,
! [R: rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
=> ( eff @ R @ S @ ( abstra1276541928ickEff @ rule @ state @ eff @ R @ S ) ) ) ).
% pickEff
thf(fact_20_RuleSystem_Oenabled__R,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ? [X4: Rule] :
( ( member @ Rule @ X4 @ ( sset @ Rule @ Rules ) )
& ? [X1: fset @ State] : ( Eff @ X4 @ S @ X1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_21_RuleSystem__Defs_Oenabled__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1874422341nabled @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,R3: Rule,S4: State] :
( ^ [P: ( fset @ State ) > $o] :
? [X: fset @ State] : ( P @ X )
@ ( Eff2 @ R3 @ S4 ) ) ) ) ).
% RuleSystem_Defs.enabled_def
thf(fact_22_RuleSystem_Ointro,axiom,
! [Rule: $tType,State: $tType,S3: set @ State,Rules: stream @ Rule,Eff: Rule > State > ( fset @ State ) > $o] :
( ! [S5: State] :
( ( member @ State @ S5 @ S3 )
=> ! [R4: Rule] :
( ( member @ Rule @ R4 @ ( sset @ Rule @ Rules ) )
=> ! [Sl: fset @ State] :
( ( Eff @ R4 @ S5 @ Sl )
=> ! [S6: State] :
( ( fmember @ State @ S6 @ Sl )
=> ( member @ State @ S6 @ S3 ) ) ) ) )
=> ( ! [S5: State] :
( ( member @ State @ S5 @ S3 )
=> ? [X5: Rule] :
( ( member @ Rule @ X5 @ ( sset @ Rule @ Rules ) )
& ? [X12: fset @ State] : ( Eff @ X5 @ S5 @ X12 ) ) )
=> ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 ) ) ) ).
% RuleSystem.intro
thf(fact_23_RuleSystem_Oeff__S,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,R: Rule,Sl3: fset @ State,S2: State] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( member @ Rule @ R @ ( sset @ Rule @ Rules ) )
=> ( ( Eff @ R @ S @ Sl3 )
=> ( ( fmember @ State @ S2 @ Sl3 )
=> ( member @ State @ S2 @ S3 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_24_RuleSystem__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1326562878System @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S7: set @ State] :
( ! [S4: State] :
( ( member @ State @ S4 @ S7 )
=> ! [R3: Rule] :
( ( member @ Rule @ R3 @ ( sset @ Rule @ Rules2 ) )
=> ! [Sl4: fset @ State] :
( ( Eff2 @ R3 @ S4 @ Sl4 )
=> ! [S8: State] :
( ( fmember @ State @ S8 @ Sl4 )
=> ( member @ State @ S8 @ S7 ) ) ) ) )
& ! [S4: State] :
( ( member @ State @ S4 @ S7 )
=> ? [X6: Rule] :
( ( member @ Rule @ X6 @ ( sset @ Rule @ Rules2 ) )
& ( ^ [P: ( fset @ State ) > $o] :
? [X: fset @ State] : ( P @ X )
@ ( Eff2 @ X6 @ S4 ) ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_25_stl__sset,axiom,
! [A: $tType,X3: A,A2: stream @ A] :
( ( member @ A @ X3 @ ( sset @ A @ ( stl @ A @ A2 ) ) )
=> ( member @ A @ X3 @ ( sset @ A @ A2 ) ) ) ).
% stl_sset
thf(fact_26_shd__sset,axiom,
! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% shd_sset
thf(fact_27_sset__induct,axiom,
! [A: $tType,Y: A,S: stream @ A,P3: A > ( stream @ A ) > $o] :
( ( member @ A @ Y @ ( sset @ A @ S ) )
=> ( ! [S5: stream @ A] : ( P3 @ ( shd @ A @ S5 ) @ S5 )
=> ( ! [S5: stream @ A,Y2: A] :
( ( member @ A @ Y2 @ ( sset @ A @ ( stl @ A @ S5 ) ) )
=> ( ( P3 @ Y2 @ ( stl @ A @ S5 ) )
=> ( P3 @ Y2 @ S5 ) ) )
=> ( P3 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_28_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [Rule: $tType,State: $tType,X2: ( stream @ ( product_prod @ State @ Rule ) ) > $o,X3: stream @ ( product_prod @ State @ Rule ),Rules: stream @ Rule,Eff: Rule > State > ( fset @ State ) > $o] :
( ( X2 @ X3 )
=> ( ! [X4: stream @ ( product_prod @ State @ Rule )] :
( ( X2 @ X4 )
=> ? [Steps: stream @ ( product_prod @ State @ Rule ),Sl2: fset @ State] :
( ( X4 = 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 )
& ( ( X2 @ ( stl @ ( product_prod @ State @ Rule ) @ Steps ) )
| ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ ( stl @ ( product_prod @ State @ Rule ) @ Steps ) ) ) ) )
=> ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ X3 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_29_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_30_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_31_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_32_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y3: product_prod @ A @ B,Z: product_prod @ A @ B] : ( Y3 = Z ) )
= ( ^ [S4: product_prod @ A @ B,T: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S4 )
= ( product_fst @ A @ B @ T ) )
& ( ( product_snd @ A @ B @ S4 )
= ( product_snd @ A @ B @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_33_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_34_Saturated__def,axiom,
! [Steps2: stream @ ( product_prod @ state @ rule )] :
( ( abstra1209608345urated @ rule @ state @ eff @ rules @ Steps2 )
= ( ! [X6: rule] :
( ( member @ rule @ X6 @ ( sset @ rule @ rules ) )
=> ( abstra726722745urated @ rule @ state @ eff @ X6 @ Steps2 ) ) ) ) ).
% Saturated_def
thf(fact_35_local_Oalw,axiom,
( ( abstra523868654_epath @ rule @ state @ eff @ rules @ stepsa )
& ( member @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ stepsa ) ) @ s ) ) ).
% local.alw
thf(fact_36_NE__R,axiom,
( ( sset @ rule @ rules )
!= ( bot_bot @ ( set @ rule ) ) ) ).
% NE_R
thf(fact_37_fair__stl,axiom,
! [Rs: stream @ rule] :
( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra928354080m_fair @ rule @ rules @ ( stl @ rule @ Rs ) ) ) ).
% fair_stl
thf(fact_38_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P3: A > B > $o,P2: product_prod @ B @ A] :
( ( P3 @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ~ ! [X4: B,Y2: A] :
~ ( P3 @ Y2 @ X4 ) ) ).
% exE_realizer'
thf(fact_39_countable__R,axiom,
countable_countable @ rule @ ( sset @ rule @ rules ) ).
% countable_R
thf(fact_40_trim__enabled,axiom,
! [S: state,Rs: stream @ rule] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S ) ) ) ).
% trim_enabled
thf(fact_41_trim__in__R,axiom,
! [S: state,Rs: stream @ rule] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( member @ rule @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ ( sset @ rule @ rules ) ) ) ) ).
% trim_in_R
thf(fact_42_wf__ipath__epath,axiom,
! [T2: abstra2103299360e_tree @ ( product_prod @ state @ rule ),Steps2: stream @ ( product_prod @ state @ rule )] :
( ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T2 )
=> ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ T2 @ Steps2 )
=> ( abstra523868654_epath @ rule @ state @ eff @ rules @ Steps2 ) ) ) ).
% wf_ipath_epath
thf(fact_43_minWait__ex,axiom,
! [S: state,Rs: stream @ rule] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ? [N: nat] : ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N @ Rs ) ) @ S ) ) ) ).
% minWait_ex
thf(fact_44_trim__fair,axiom,
! [S: state,Rs: stream @ rule] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra928354080m_fair @ rule @ rules @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) ) ).
% trim_fair
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P3: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X6: A] : ( member @ A @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ! [X4: A] :
( ( P3 @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_sdrop__fair,axiom,
! [Rs: stream @ rule,M: nat] :
( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra928354080m_fair @ rule @ rules @ ( sdrop @ rule @ M @ Rs ) ) ) ).
% sdrop_fair
thf(fact_50_fair__fenum,axiom,
abstra928354080m_fair @ rule @ rules @ ( abstra1774373515_fenum @ rule @ rules ) ).
% fair_fenum
thf(fact_51_RuleSystem__Defs_Osdrop__fair,axiom,
! [Rule: $tType,Rules: stream @ Rule,Rs: stream @ Rule,M: nat] :
( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra928354080m_fair @ Rule @ Rules @ ( sdrop @ Rule @ M @ Rs ) ) ) ).
% RuleSystem_Defs.sdrop_fair
thf(fact_52_RuleSystem__Defs_Ofair__fenum,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( abstra928354080m_fair @ Rule @ Rules @ ( abstra1774373515_fenum @ Rule @ Rules ) ) ).
% RuleSystem_Defs.fair_fenum
thf(fact_53_RuleSystem_Otrim__fair,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra928354080m_fair @ Rule @ Rules @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) ) ) ) ).
% RuleSystem.trim_fair
thf(fact_54_RuleSystem__Defs_Owf__ipath__epath,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,T2: abstra2103299360e_tree @ ( product_prod @ State @ Rule ),Steps2: stream @ ( product_prod @ State @ Rule )] :
( ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ T2 )
=> ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ T2 @ Steps2 )
=> ( abstra523868654_epath @ Rule @ State @ Eff @ Rules @ Steps2 ) ) ) ).
% RuleSystem_Defs.wf_ipath_epath
thf(fact_55_RuleSystem__Defs_Ocountable__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( countable_countable @ Rule @ ( sset @ Rule @ Rules ) ) ).
% RuleSystem_Defs.countable_R
thf(fact_56_sdrop__stl,axiom,
! [A: $tType,N2: nat,S: stream @ A] :
( ( sdrop @ A @ N2 @ ( stl @ A @ S ) )
= ( stl @ A @ ( sdrop @ A @ N2 @ S ) ) ) ).
% sdrop_stl
thf(fact_57_RuleSystem__Defs_ONE__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] :
( ( sset @ Rule @ Rules )
!= ( bot_bot @ ( set @ Rule ) ) ) ).
% RuleSystem_Defs.NE_R
thf(fact_58_RuleSystem__Defs_Ofair__stl,axiom,
! [Rule: $tType,Rules: stream @ Rule,Rs: stream @ Rule] :
( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra928354080m_fair @ Rule @ Rules @ ( stl @ Rule @ Rs ) ) ) ).
% RuleSystem_Defs.fair_stl
thf(fact_59_RuleSystem_OminWait__ex,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ? [N: nat] : ( abstra1874422341nabled @ Rule @ State @ Eff @ ( shd @ Rule @ ( sdrop @ Rule @ N @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_60_RuleSystem_Otrim__in__R,axiom,
! [State: $tType,Rule: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( member @ Rule @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) @ ( sset @ Rule @ Rules ) ) ) ) ) ).
% RuleSystem.trim_in_R
thf(fact_61_RuleSystem_Otrim__enabled,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra1874422341nabled @ Rule @ State @ Eff @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) @ S ) ) ) ) ).
% RuleSystem.trim_enabled
thf(fact_62_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 )] :
! [X6: Rule] :
( ( member @ Rule @ X6 @ ( sset @ Rule @ Rules2 ) )
=> ( abstra726722745urated @ Rule @ State @ Eff2 @ X6 @ Steps3 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_63_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_64_RuleSystem__Defs_Osset__fenum,axiom,
! [Rule: $tType,Rules: stream @ Rule] :
( ( sset @ Rule @ ( abstra1774373515_fenum @ Rule @ Rules ) )
= ( sset @ Rule @ Rules ) ) ).
% RuleSystem_Defs.sset_fenum
thf(fact_65_trim__alt,axiom,
! [S: state,Rs: stream @ rule] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S )
= ( sdrop @ rule @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ Rs ) ) ) ) ).
% trim_alt
thf(fact_66_pos,axiom,
! [Rs: stream @ rule,R: rule] :
( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
=> ( ( shd @ rule @ ( sdrop @ rule @ ( abstra2097340358le_pos @ rule @ Rs @ R ) @ Rs ) )
= R ) ) ) ).
% pos
thf(fact_67_trim__def,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S )
= ( sdrop_while @ rule
@ ^ [R3: rule] :
~ ( abstra1874422341nabled @ rule @ state @ eff @ R3 @ S )
@ Rs ) ) ).
% trim_def
thf(fact_68_wf__mkTree,axiom,
! [S: state,Rs: stream @ rule] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) ) ) ) ).
% wf_mkTree
thf(fact_69_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_70_RuleSystem_Opos,axiom,
! [State: $tType,Rule: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,Rs: stream @ Rule,R: Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( member @ Rule @ R @ ( sset @ Rule @ Rules ) )
=> ( ( shd @ Rule @ ( sdrop @ Rule @ ( abstra2097340358le_pos @ Rule @ Rs @ R ) @ Rs ) )
= R ) ) ) ) ).
% RuleSystem.pos
thf(fact_71_in__cont__mkTree,axiom,
! [S: state,Rs: stream @ rule,T3: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ T3 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) ) )
=> ? [Sl5: fset @ state,S6: state] :
( ( member @ state @ S6 @ s )
& ( eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S @ Sl5 )
& ( fmember @ state @ S6 @ Sl5 )
& ( T3
= ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S6 ) ) ) ) ) ) ).
% in_cont_mkTree
thf(fact_72_RuleSystem_Otrim__alt,axiom,
! [State: $tType,Rule: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S )
= ( sdrop @ Rule @ ( abstra1332369113inWait @ Rule @ State @ Eff @ Rs @ S ) @ Rs ) ) ) ) ) ).
% RuleSystem.trim_alt
thf(fact_73_countableI__type,axiom,
! [A: $tType] :
( ( countable @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] : ( countable_countable @ A @ A4 ) ) ).
% countableI_type
thf(fact_74_ipath__mkTree__sdrop,axiom,
! [S: state,Rs: stream @ rule,Steps2: stream @ ( product_prod @ state @ rule ),M: nat] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) @ Steps2 )
=> ? [N: nat,S6: state] :
( ( member @ state @ S6 @ s )
& ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( sdrop @ rule @ N @ Rs ) @ S6 ) @ ( sdrop @ ( product_prod @ state @ rule ) @ M @ Steps2 ) ) ) ) ) ) ).
% ipath_mkTree_sdrop
thf(fact_75_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( sdrop_while @ A )
= ( ^ [P4: A > $o,S4: stream @ A] : ( if @ ( stream @ A ) @ ( P4 @ ( shd @ A @ S4 ) ) @ ( sdrop_while @ A @ P4 @ ( stl @ A @ S4 ) ) @ S4 ) ) ) ).
% sdrop_while.simps
thf(fact_76_RuleSystem_Oin__cont__mkTree,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,T3: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ T3 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ Rs @ S ) ) )
=> ? [Sl5: fset @ State,S6: State] :
( ( member @ State @ S6 @ S3 )
& ( Eff @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) @ S @ Sl5 )
& ( fmember @ State @ S6 @ Sl5 )
& ( T3
= ( abstra1225283448mkTree @ Rule @ State @ Eff @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) @ S6 ) ) ) ) ) ) ) ).
% RuleSystem.in_cont_mkTree
thf(fact_77_RuleSystem__Defs_Otrim__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1259602206m_trim @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rs2: stream @ Rule,S4: State] :
( sdrop_while @ Rule
@ ^ [R3: Rule] :
~ ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R3 @ S4 )
@ Rs2 ) ) ) ).
% RuleSystem_Defs.trim_def
thf(fact_78_countable__Collect,axiom,
! [A: $tType,A4: set @ A,Phi: A > $o] :
( ( countable_countable @ A @ A4 )
=> ( countable_countable @ A
@ ( collect @ A
@ ^ [A3: A] :
( ( member @ A @ A3 @ A4 )
& ( Phi @ A3 ) ) ) ) ) ).
% countable_Collect
thf(fact_79_RuleSystem_Owf__mkTree,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra1874736267tem_wf @ Rule @ State @ Eff @ Rules @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ Rs @ S ) ) ) ) ) ).
% RuleSystem.wf_mkTree
thf(fact_80_RuleSystem_Oipath__mkTree__sdrop,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,Steps2: stream @ ( product_prod @ State @ Rule ),M: nat] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ Rs @ S ) @ Steps2 )
=> ? [N: nat,S6: State] :
( ( member @ State @ S6 @ S3 )
& ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ ( sdrop @ Rule @ N @ Rs ) @ S6 ) @ ( sdrop @ ( product_prod @ State @ Rule ) @ M @ Steps2 ) ) ) ) ) ) ) ).
% RuleSystem.ipath_mkTree_sdrop
thf(fact_81_minWait__le__pos,axiom,
! [Rs: stream @ rule,R: rule,S: state] :
( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
=> ( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
=> ( ord_less_eq @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ ( abstra2097340358le_pos @ rule @ Rs @ R ) ) ) ) ) ).
% minWait_le_pos
thf(fact_82_minWait__def,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S )
= ( ord_Least @ nat
@ ^ [N3: nat] : ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N3 @ Rs ) ) @ S ) ) ) ).
% minWait_def
thf(fact_83_mkTree_Osimps_I2_J,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) )
= ( fimage @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) @ ( abstra1276541928ickEff @ rule @ state @ eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S ) ) ) ).
% mkTree.simps(2)
thf(fact_84_minWait__least,axiom,
! [N2: nat,Rs: stream @ rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N2 @ Rs ) ) @ S )
=> ( ord_less_eq @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ N2 ) ) ).
% minWait_least
thf(fact_85_pos__def,axiom,
! [A: $tType] :
( ( abstra2097340358le_pos @ A )
= ( ^ [Rs2: stream @ A,R3: A] :
( ord_Least @ nat
@ ^ [N3: nat] :
( ( shd @ A @ ( sdrop @ A @ N3 @ Rs2 ) )
= R3 ) ) ) ) ).
% pos_def
thf(fact_86_pos__least,axiom,
! [A: $tType,N2: nat,Rs: stream @ A,R: A] :
( ( ( shd @ A @ ( sdrop @ A @ N2 @ Rs ) )
= R )
=> ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N2 ) ) ).
% pos_least
thf(fact_87_RuleSystem_Opos__least,axiom,
! [Rule: $tType,State: $tType,A: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,N2: nat,Rs: stream @ A,R: A] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( ( shd @ A @ ( sdrop @ A @ N2 @ Rs ) )
= R )
=> ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N2 ) ) ) ).
% RuleSystem.pos_least
thf(fact_88_RuleSystem_Opos__def,axiom,
! [State: $tType,Rule: $tType,A: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,Rs: stream @ A,R: A] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( abstra2097340358le_pos @ A @ Rs @ R )
= ( ord_Least @ nat
@ ^ [N3: nat] :
( ( shd @ A @ ( sdrop @ A @ N3 @ Rs ) )
= R ) ) ) ) ).
% RuleSystem.pos_def
thf(fact_89_RuleSystem_OminWait__least,axiom,
! [State: $tType,Rule: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,N2: nat,Rs: stream @ Rule,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( abstra1874422341nabled @ Rule @ State @ Eff @ ( shd @ Rule @ ( sdrop @ Rule @ N2 @ Rs ) ) @ S )
=> ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff @ Rs @ S ) @ N2 ) ) ) ).
% RuleSystem.minWait_least
thf(fact_90_RuleSystem__Defs_OmkTree_Osimps_I2_J,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rs: stream @ Rule,S: State] :
( ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ Rs @ S ) )
= ( fimage @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) ) @ ( abstra1276541928ickEff @ Rule @ State @ Eff @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) @ S ) ) ) ).
% RuleSystem_Defs.mkTree.simps(2)
thf(fact_91_RuleSystem_OminWait__def,axiom,
! [State: $tType,Rule: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,Rs: stream @ Rule,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( abstra1332369113inWait @ Rule @ State @ Eff @ Rs @ S )
= ( ord_Least @ nat
@ ^ [N3: nat] : ( abstra1874422341nabled @ Rule @ State @ Eff @ ( shd @ Rule @ ( sdrop @ Rule @ N3 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_def
thf(fact_92_RuleSystem_OminWait__le__pos,axiom,
! [State: $tType,Rule: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,Rs: stream @ Rule,R: Rule,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( member @ Rule @ R @ ( sset @ Rule @ Rules ) )
=> ( ( abstra1874422341nabled @ Rule @ State @ Eff @ R @ S )
=> ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff @ Rs @ S ) @ ( abstra2097340358le_pos @ Rule @ Rs @ R ) ) ) ) ) ) ).
% RuleSystem.minWait_le_pos
thf(fact_93_fimage__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X3: B,A4: fset @ B] :
( ( B2
= ( F @ X3 ) )
=> ( ( fmember @ B @ X3 @ A4 )
=> ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A4 ) ) ) ) ).
% fimage_eqI
thf(fact_94_fimage__ident,axiom,
! [A: $tType,Y4: fset @ A] :
( ( fimage @ A @ A
@ ^ [X6: A] : X6
@ Y4 )
= Y4 ) ).
% fimage_ident
thf(fact_95_fset_Omap__ident,axiom,
! [A: $tType,T2: fset @ A] :
( ( fimage @ A @ A
@ ^ [X6: A] : X6
@ T2 )
= T2 ) ).
% fset.map_ident
thf(fact_96_per__def,axiom,
! [R: rule] :
( ( abstra2096684367le_per @ rule @ state @ eff @ rules @ s @ R )
= ( ! [S4: state,R1: rule,Sl6: fset @ state,S8: state] :
( ( ( member @ state @ S4 @ s )
& ( abstra1874422341nabled @ rule @ state @ eff @ R @ S4 )
& ( member @ rule @ R1 @ ( minus_minus @ ( set @ rule ) @ ( sset @ rule @ rules ) @ ( insert @ rule @ R @ ( bot_bot @ ( set @ rule ) ) ) ) )
& ( eff @ R1 @ S4 @ Sl6 )
& ( fmember @ state @ S8 @ Sl6 ) )
=> ( abstra1874422341nabled @ rule @ state @ eff @ R @ S8 ) ) ) ) ).
% per_def
thf(fact_97_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,K: A] :
( ( P3 @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P3 ) @ K ) ) ) ).
% Least_le
thf(fact_98_mkTree_Octr,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S )
= ( abstra388494275e_Node @ ( product_prod @ state @ rule ) @ ( product_Pair @ state @ rule @ S @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) @ ( fimage @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) @ ( abstra1276541928ickEff @ rule @ state @ eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S ) ) ) ) ).
% mkTree.ctr
thf(fact_99_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_100_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_101_prod_Oinject,axiom,
! [A: $tType,B: $tType,X13: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X13 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_102_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X6: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_103_fsubsetI,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A] :
( ! [X4: A] :
( ( fmember @ A @ X4 @ A4 )
=> ( fmember @ A @ X4 @ B4 ) )
=> ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 ) ) ).
% fsubsetI
thf(fact_104_tree_Oinject,axiom,
! [A: $tType,X13: A,X22: fset @ ( abstra2103299360e_tree @ A ),Y1: A,Y22: fset @ ( abstra2103299360e_tree @ A )] :
( ( ( abstra388494275e_Node @ A @ X13 @ X22 )
= ( abstra388494275e_Node @ A @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X22 = Y22 ) ) ) ).
% tree.inject
thf(fact_105_countable__insert,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( countable_countable @ A @ A4 )
=> ( countable_countable @ A @ ( insert @ A @ A2 @ A4 ) ) ) ).
% countable_insert
thf(fact_106_countable__insert__eq,axiom,
! [A: $tType,X3: A,A4: set @ A] :
( ( countable_countable @ A @ ( insert @ A @ X3 @ A4 ) )
= ( countable_countable @ A @ A4 ) ) ).
% countable_insert_eq
thf(fact_107_countable__Diff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% countable_Diff
thf(fact_108_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_109_countable__Diff__eq,axiom,
! [A: $tType,A4: set @ A,X3: A] :
( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( countable_countable @ A @ A4 ) ) ).
% countable_Diff_eq
thf(fact_110_fset__mp,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A,X3: A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ( ( fmember @ A @ X3 @ A4 )
=> ( fmember @ A @ X3 @ B4 ) ) ) ).
% fset_mp
thf(fact_111_fin__mono,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A,X3: A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ( ( fmember @ A @ X3 @ A4 )
=> ( fmember @ A @ X3 @ B4 ) ) ) ).
% fin_mono
thf(fact_112_fsubsetD,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A,C2: A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ( ( fmember @ A @ C2 @ A4 )
=> ( fmember @ A @ C2 @ B4 ) ) ) ).
% fsubsetD
thf(fact_113_fset__rev__mp,axiom,
! [A: $tType,X3: A,A4: fset @ A,B4: fset @ A] :
( ( fmember @ A @ X3 @ A4 )
=> ( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ( fmember @ A @ X3 @ B4 ) ) ) ).
% fset_rev_mp
thf(fact_114_subset__fimage__iff,axiom,
! [A: $tType,B: $tType,B4: fset @ A,F: B > A,A4: fset @ B] :
( ( ord_less_eq @ ( fset @ A ) @ B4 @ ( fimage @ B @ A @ F @ A4 ) )
= ( ? [AA: fset @ B] :
( ( ord_less_eq @ ( fset @ B ) @ AA @ A4 )
& ( B4
= ( fimage @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_fimage_iff
thf(fact_115_fimage__mono,axiom,
! [B: $tType,A: $tType,A4: fset @ A,B4: fset @ A,F: A > B] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F @ A4 ) @ ( fimage @ A @ B @ F @ B4 ) ) ) ).
% fimage_mono
thf(fact_116_tree_Oexhaust,axiom,
! [A: $tType,Y: abstra2103299360e_tree @ A] :
~ ! [X1: A,X23: fset @ ( abstra2103299360e_tree @ A )] :
( Y
!= ( abstra388494275e_Node @ A @ X1 @ X23 ) ) ).
% tree.exhaust
thf(fact_117_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_118_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_119_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P3 @ X3 ) ) ).
% prod_induct7
thf(fact_120_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P3 @ X3 ) ) ).
% prod_induct6
thf(fact_121_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P3 @ X3 ) ) ).
% prod_induct5
thf(fact_122_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B5: B,C3: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P3 @ X3 ) ) ).
% prod_induct4
thf(fact_123_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B5: B,C3: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P3 @ X3 ) ) ).
% prod_induct3
thf(fact_124_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_125_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F3: F2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_126_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_127_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_128_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_129_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_130_prod__cases,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_131_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X4: A,Y2: B] :
( P2
= ( product_Pair @ A @ B @ X4 @ Y2 ) ) ).
% surj_pair
thf(fact_132_fst__conv,axiom,
! [B: $tType,A: $tType,X13: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X13 @ X22 ) )
= X13 ) ).
% fst_conv
thf(fact_133_fst__eqD,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= A2 )
=> ( X3 = A2 ) ) ).
% fst_eqD
thf(fact_134_snd__conv,axiom,
! [Aa: $tType,A: $tType,X13: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X13 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_135_snd__eqD,axiom,
! [B: $tType,A: $tType,X3: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_136_uncountable__minus__countable,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ~ ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B4 )
=> ~ ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% uncountable_minus_countable
thf(fact_137_countable__subset,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( countable_countable @ A @ B4 )
=> ( countable_countable @ A @ A4 ) ) ) ).
% countable_subset
thf(fact_138_fimage__fsubsetI,axiom,
! [A: $tType,B: $tType,A4: fset @ A,F: A > B,B4: fset @ B] :
( ! [X4: A] :
( ( fmember @ A @ X4 @ A4 )
=> ( fmember @ B @ ( F @ X4 ) @ B4 ) )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F @ A4 ) @ B4 ) ) ).
% fimage_fsubsetI
thf(fact_139_tree_Osel_I2_J,axiom,
! [A: $tType,X13: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra1749095923e_cont @ A @ ( abstra388494275e_Node @ A @ X13 @ X22 ) )
= X22 ) ).
% tree.sel(2)
thf(fact_140_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_141_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_142_conjI__realizer,axiom,
! [A: $tType,B: $tType,P3: A > $o,P2: A,Q2: B > $o,Q: B] :
( ( P3 @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P3 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_143_exI__realizer,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,Y: A,X3: B] :
( ( P3 @ Y @ X3 )
=> ( P3 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_144_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_145_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_146_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_147_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X6: A] : ( ord_less_eq @ B @ ( F4 @ X6 ) @ ( G4 @ X6 ) ) ) ) ) ).
% le_fun_def
thf(fact_148_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y2: B] :
( ( ord_less_eq @ B @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_150_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y2: B] :
( ( ord_less_eq @ B @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_152_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
= ( ^ [X6: A,Y5: A] :
( ( ord_less_eq @ A @ X6 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X6 ) ) ) ) ) ).
% eq_iff
thf(fact_153_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ).
% antisym
thf(fact_154_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linear
thf(fact_155_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( X3 = Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% eq_refl
thf(fact_156_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% le_cases
thf(fact_157_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_158_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_159_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% antisym_conv
thf(fact_160_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_161_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_162_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_163_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_164_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_165_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P3 @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P3 @ B5 @ A6 )
=> ( P3 @ A6 @ B5 ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_166_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_167_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_168_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X6: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_169_eqfelem__imp__iff,axiom,
! [A: $tType,X3: A,Y: A,A4: fset @ A] :
( ( X3 = Y )
=> ( ( fmember @ A @ X3 @ A4 )
= ( fmember @ A @ Y @ A4 ) ) ) ).
% eqfelem_imp_iff
thf(fact_170_if__split__fmem2,axiom,
! [A: $tType,A2: A,Q2: $o,X3: fset @ A,Y: fset @ A] :
( ( fmember @ A @ A2 @ ( if @ ( fset @ A ) @ Q2 @ X3 @ Y ) )
= ( ( Q2
=> ( fmember @ A @ A2 @ X3 ) )
& ( ~ Q2
=> ( fmember @ A @ A2 @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_171_if__split__fmem1,axiom,
! [A: $tType,Q2: $o,X3: A,Y: A,B2: fset @ A] :
( ( fmember @ A @ ( if @ A @ Q2 @ X3 @ Y ) @ B2 )
= ( ( Q2
=> ( fmember @ A @ X3 @ B2 ) )
& ( ~ Q2
=> ( fmember @ A @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_172_eqfset__imp__iff,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A,X3: A] :
( ( A4 = B4 )
=> ( ( fmember @ A @ X3 @ A4 )
= ( fmember @ A @ X3 @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_173_eq__fmem__trans,axiom,
! [A: $tType,A2: A,B2: A,A4: fset @ A] :
( ( A2 = B2 )
=> ( ( fmember @ A @ B2 @ A4 )
=> ( fmember @ A @ A2 @ A4 ) ) ) ).
% eq_fmem_trans
thf(fact_174_fset__choice,axiom,
! [B: $tType,A: $tType,A4: fset @ A,P3: A > B > $o] :
( ! [X4: A] :
( ( fmember @ A @ X4 @ A4 )
=> ? [X12: B] : ( P3 @ X4 @ X12 ) )
=> ? [F3: A > B] :
! [X5: A] :
( ( fmember @ A @ X5 @ A4 )
=> ( P3 @ X5 @ ( F3 @ X5 ) ) ) ) ).
% fset_choice
thf(fact_175_fequalityCE,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A,C2: A] :
( ( A4 = B4 )
=> ( ( ( fmember @ A @ C2 @ A4 )
=> ~ ( fmember @ A @ C2 @ B4 ) )
=> ~ ( ~ ( fmember @ A @ C2 @ A4 )
=> ( fmember @ A @ C2 @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_176_fset__eqI,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A] :
( ! [X4: A] :
( ( fmember @ A @ X4 @ A4 )
= ( fmember @ A @ X4 @ B4 ) )
=> ( A4 = B4 ) ) ).
% fset_eqI
thf(fact_177_RuleSystem_Oper__def,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,R: Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( abstra2096684367le_per @ Rule @ State @ Eff @ Rules @ S3 @ R )
= ( ! [S4: State,R1: Rule,Sl6: fset @ State,S8: State] :
( ( ( member @ State @ S4 @ S3 )
& ( abstra1874422341nabled @ Rule @ State @ Eff @ R @ S4 )
& ( member @ Rule @ R1 @ ( minus_minus @ ( set @ Rule ) @ ( sset @ Rule @ Rules ) @ ( insert @ Rule @ R @ ( bot_bot @ ( set @ Rule ) ) ) ) )
& ( Eff @ R1 @ S4 @ Sl6 )
& ( fmember @ State @ S8 @ Sl6 ) )
=> ( abstra1874422341nabled @ Rule @ State @ Eff @ R @ S8 ) ) ) ) ) ).
% RuleSystem.per_def
thf(fact_178_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,A2: A,Q2: A > $o] :
( ( P3 @ A2 )
=> ( ! [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) )
=> ( Q2 @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2
thf(fact_179_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o] :
( ? [X12: A] : ( P3 @ X12 )
=> ( P3 @ ( ord_Least @ A @ P3 ) ) ) ) ).
% LeastI_ex
thf(fact_180_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,Q2: A > $o] :
( ? [X12: A] : ( P3 @ X12 )
=> ( ! [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) )
=> ( Q2 @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2_ex
thf(fact_181_RuleSystem__Defs_OmkTree_Ocode,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1225283448mkTree @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rs2: stream @ Rule,S4: State] : ( abstra388494275e_Node @ ( product_prod @ State @ Rule ) @ ( product_Pair @ State @ Rule @ S4 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs2 @ S4 ) ) ) @ ( fimage @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs2 @ S4 ) ) ) @ ( abstra1276541928ickEff @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs2 @ S4 ) ) @ S4 ) ) ) ) ) ).
% RuleSystem_Defs.mkTree.code
thf(fact_182_fimage__fimage,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A4: fset @ C] :
( ( fimage @ B @ A @ F @ ( fimage @ C @ B @ G @ A4 ) )
= ( fimage @ C @ A
@ ^ [X6: C] : ( F @ ( G @ X6 ) )
@ A4 ) ) ).
% fimage_fimage
thf(fact_183_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,K: A] :
( ( P3 @ K )
=> ( P3 @ ( ord_Least @ A @ P3 ) ) ) ) ).
% LeastI
thf(fact_184_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_185_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_186_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_187_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,X3: A,Q2: A > $o] :
( ( P3 @ X3 )
=> ( ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) )
=> ( ! [X4: A] :
( ( P3 @ X4 )
=> ( ! [Y6: A] :
( ( P3 @ Y6 )
=> ( ord_less_eq @ A @ X4 @ Y6 ) )
=> ( Q2 @ X4 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P3 ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_188_Least__equality,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,X3: A] :
( ( P3 @ X3 )
=> ( ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) )
=> ( ( ord_Least @ A @ P3 )
= X3 ) ) ) ) ).
% Least_equality
thf(fact_189_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,A2: A,Q2: A > $o] :
( ( P3 @ A2 )
=> ( ! [A6: A] :
( ( P3 @ A6 )
=> ( ! [B6: A] :
( ( P3 @ B6 )
=> ( ord_less_eq @ A @ A6 @ B6 ) )
=> ( Q2 @ A6 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_190_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,Q2: A > $o] :
( ? [X12: A] : ( P3 @ X12 )
=> ( ! [A6: A] :
( ( P3 @ A6 )
=> ( ! [B6: A] :
( ( P3 @ B6 )
=> ( ord_less_eq @ A @ A6 @ B6 ) )
=> ( Q2 @ A6 ) ) )
=> ( Q2 @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_191_rev__fimage__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A4: fset @ A,B2: B,F: A > B] :
( ( fmember @ A @ X3 @ A4 )
=> ( ( B2
= ( F @ X3 ) )
=> ( fmember @ B @ B2 @ ( fimage @ A @ B @ F @ A4 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_192_fimage__cong,axiom,
! [B: $tType,A: $tType,M2: fset @ A,N4: fset @ A,F: A > B,G: A > B] :
( ( M2 = N4 )
=> ( ! [X4: A] :
( ( fmember @ A @ X4 @ N4 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( fimage @ A @ B @ F @ M2 )
= ( fimage @ A @ B @ G @ N4 ) ) ) ) ).
% fimage_cong
thf(fact_193_fimageI,axiom,
! [B: $tType,A: $tType,X3: A,A4: fset @ A,F: A > B] :
( ( fmember @ A @ X3 @ A4 )
=> ( fmember @ B @ ( F @ X3 ) @ ( fimage @ A @ B @ F @ A4 ) ) ) ).
% fimageI
thf(fact_194_fimageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A4: fset @ B] :
( ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A4 ) )
=> ~ ! [X4: B] :
( ( B2
= ( F @ X4 ) )
=> ~ ( fmember @ B @ X4 @ A4 ) ) ) ).
% fimageE
thf(fact_195_mkTree__unfold,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S )
= ( case_stream @ rule @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) )
@ ^ [R3: rule,S8: stream @ rule] : ( abstra388494275e_Node @ ( product_prod @ state @ rule ) @ ( product_Pair @ state @ rule @ S @ R3 ) @ ( fimage @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ S8 ) @ ( abstra1276541928ickEff @ rule @ state @ eff @ R3 @ S ) ) )
@ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) ).
% mkTree_unfold
thf(fact_196_insert__Diff__single,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A4 ) ) ).
% insert_Diff_single
thf(fact_197_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y3: A,Z: A] : ( Y3 = Z )
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_198_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X6: A] : ( X6 = A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_199_mkTree_Osimps_I1_J,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) )
= ( product_Pair @ state @ rule @ S @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) ) ).
% mkTree.simps(1)
thf(fact_200_empty__Collect__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P3 ) )
= ( ! [X6: A] :
~ ( P3 @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_201_Collect__empty__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( collect @ A @ P3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X6: A] :
~ ( P3 @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_202_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X6: A] :
~ ( member @ A @ X6 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_203_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_204_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_205_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ A @ X4 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_206_insert__absorb2,axiom,
! [A: $tType,X3: A,A4: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A4 ) )
= ( insert @ A @ X3 @ A4 ) ) ).
% insert_absorb2
thf(fact_207_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_208_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_209_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ).
% Diff_idemp
thf(fact_210_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_211_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_212_fempty__iff,axiom,
! [A: $tType,C2: A] :
~ ( fmember @ A @ C2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% fempty_iff
thf(fact_213_all__not__fin__conv,axiom,
! [A: $tType,A4: fset @ A] :
( ( ! [X6: A] :
~ ( fmember @ A @ X6 @ A4 ) )
= ( A4
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% all_not_fin_conv
thf(fact_214_fempty__fsubsetI,axiom,
! [A: $tType,X3: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ ( bot_bot @ ( fset @ A ) ) @ X3 ) ).
% fempty_fsubsetI
thf(fact_215_fsubset__fempty,axiom,
! [A: $tType,A4: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ ( bot_bot @ ( fset @ A ) ) )
= ( A4
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% fsubset_fempty
thf(fact_216_fsubset__antisym,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% fsubset_antisym
thf(fact_217_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_218_fempty__is__fimage,axiom,
! [A: $tType,B: $tType,F: B > A,A4: fset @ B] :
( ( ( bot_bot @ ( fset @ A ) )
= ( fimage @ B @ A @ F @ A4 ) )
= ( A4
= ( bot_bot @ ( fset @ B ) ) ) ) ).
% fempty_is_fimage
thf(fact_219_fimage__is__fempty,axiom,
! [A: $tType,B: $tType,F: B > A,A4: fset @ B] :
( ( ( fimage @ B @ A @ F @ A4 )
= ( bot_bot @ ( fset @ A ) ) )
= ( A4
= ( bot_bot @ ( fset @ B ) ) ) ) ).
% fimage_is_fempty
thf(fact_220_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_221_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_222_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_223_insert__subset,axiom,
! [A: $tType,X3: A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B4 )
= ( ( member @ A @ X3 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_224_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_225_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_226_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_227_insert__Diff1,axiom,
! [A: $tType,X3: A,B4: set @ A,A4: set @ A] :
( ( member @ A @ X3 @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_228_Diff__insert0,axiom,
! [A: $tType,X3: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X3 @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X3 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_229_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A4: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A4 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_230_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_231_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_232_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_233_Collect__mono__iff,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) )
= ( ! [X6: A] :
( ( P3 @ X6 )
=> ( Q2 @ X6 ) ) ) ) ).
% Collect_mono_iff
thf(fact_234_fset__eq__fsubset,axiom,
! [A: $tType] :
( ( ^ [Y3: fset @ A,Z: fset @ A] : ( Y3 = Z ) )
= ( ^ [A7: fset @ A,B7: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( fset @ A ) @ B7 @ A7 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_235_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ~ ( member @ A @ C2 @ A4 ) ) ) ).
% contra_subsetD
thf(fact_236_fsubset__trans,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A,C4: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( fset @ A ) @ A4 @ C4 ) ) ) ).
% fsubset_trans
thf(fact_237_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : ( Y3 = Z ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_238_fsubset__refl,axiom,
! [A: $tType,A4: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ A4 @ A4 ) ).
% fsubset_refl
thf(fact_239_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_240_Collect__mono,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ! [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_mono
thf(fact_241_fequalityD2,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( fset @ A ) @ B4 @ A4 ) ) ).
% fequalityD2
thf(fact_242_fequalityD1,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 ) ) ).
% fequalityD1
thf(fact_243_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_244_rev__subsetD,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% rev_subsetD
thf(fact_245_fequalityE,axiom,
! [A: $tType,A4: fset @ A,B4: fset @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( fset @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( fset @ A ) @ B4 @ A4 ) ) ) ).
% fequalityE
thf(fact_246_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T: A] :
( ( member @ A @ T @ A7 )
=> ( member @ A @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_247_set__rev__mp,axiom,
! [A: $tType,X3: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ X3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% set_rev_mp
thf(fact_248_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_249_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_250_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X6: A] :
( ( member @ A @ X6 @ A7 )
=> ( member @ A @ X6 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_251_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_252_subsetCE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetCE
thf(fact_253_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_254_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_255_set__mp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% set_mp
%----Type constructors (39)
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( product_prod @ A8 @ A9 ) @ ( type2 @ ( product_prod @ A8 @ A9 ) ) ) ) ).
thf(tcon_FSet_Ofset___Finite__Set_Ofinite_1,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_2,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite_4,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 @ ( type2 @ A9 ) )
=> ( order_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( countable @ A9 @ ( type2 @ A9 ) ) )
=> ( countable @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_5,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_6,axiom,
countable @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_7,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_9,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_10,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_11,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_12,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( countable @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_13,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_14,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_15,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_16,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_17,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Countable_Ocountable_18,axiom,
countable @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_19,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_20,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_21,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_22,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_23,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_FSet_Ofset___Orderings_Oorder__bot_24,axiom,
! [A8: $tType] : ( order_bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Countable_Ocountable_25,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( countable @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Opreorder_26,axiom,
! [A8: $tType] : ( preorder @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Oorder_27,axiom,
! [A8: $tType] : ( order @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Oord_28,axiom,
! [A8: $tType] : ( ord @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Obot_29,axiom,
! [A8: $tType] : ( bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_30,axiom,
! [A8: $tType,A9: $tType] :
( ( ( countable @ A8 @ ( type2 @ A8 ) )
& ( countable @ A9 @ ( type2 @ A9 ) ) )
=> ( countable @ ( product_prod @ A8 @ A9 ) @ ( type2 @ ( product_prod @ A8 @ A9 ) ) ) ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [Sl7: fset @ state] :
( ( ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ stepsa ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ stepsa ) ) @ Sl7 )
& ( fmember @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ ( stl @ ( product_prod @ state @ rule ) @ stepsa ) ) ) @ Sl7 ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------