TPTP Problem File: COM156^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : COM156^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Abstract completeness 215
% 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__215.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 0.00 v7.3.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 366 ( 143 unt; 66 typ; 0 def)
% Number of atoms : 764 ( 199 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4458 ( 62 ~; 4 |; 47 &;3974 @)
% ( 0 <=>; 371 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 344 ( 344 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 63 usr; 11 con; 0-5 aty)
% Number of variables : 1219 ( 92 ^;1023 !; 36 ?;1219 :)
% ( 63 !>; 0 ?*; 0 @-; 5 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:54:09.500
%------------------------------------------------------------------------------
%----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 (57)
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__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_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_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_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_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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( 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_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_S,type,
s: set @ state ).
thf(sy_v_eff,type,
eff: rule > state > ( fset @ state ) > $o ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_rs,type,
rs: stream @ rule ).
thf(sy_v_rsa____,type,
rsa: stream @ rule ).
thf(sy_v_rules,type,
rules: stream @ rule ).
thf(sy_v_s,type,
s2: state ).
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_s,axiom,
member @ state @ s2 @ s ).
% s
thf(fact_1_rs,axiom,
abstra928354080m_fair @ rule @ rules @ rs ).
% rs
thf(fact_2_Suc_Oprems_I2_J,axiom,
abstra928354080m_fair @ rule @ rules @ rsa ).
% Suc.prems(2)
thf(fact_3_i,axiom,
abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rs @ s2 ) @ steps ).
% i
thf(fact_4_RuleSystem__axioms,axiom,
abstra1326562878System @ rule @ state @ eff @ rules @ s ).
% RuleSystem_axioms
thf(fact_5_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_6_Suc_Oprems_I3_J,axiom,
abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rsa @ s2 ) @ stepsa ).
% Suc.prems(3)
thf(fact_7_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_8_Suc_Ohyps,axiom,
! [Rs2: stream @ rule,Steps: stream @ ( product_prod @ state @ rule )] :
( ( member @ state @ s2 @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs2 )
=> ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs2 @ s2 ) @ Steps )
=> ? [N: nat,S2: state] :
( ( member @ state @ S2 @ s )
& ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( sdrop @ rule @ N @ Rs2 ) @ S2 ) @ ( sdrop @ ( product_prod @ state @ rule ) @ ma @ Steps ) ) ) ) ) ) ).
% Suc.hyps
thf(fact_9_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_10_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_11_enabled__R,axiom,
! [S: state] :
( ( member @ state @ S @ s )
=> ? [X2: rule] :
( ( member @ rule @ X2 @ ( sset @ rule @ rules ) )
& ? [X1: fset @ state] : ( eff @ X2 @ S @ X1 ) ) ) ).
% enabled_R
thf(fact_12_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_13_sdrop__szip,axiom,
! [A: $tType,B: $tType,N2: nat,S1: stream @ A,S22: stream @ B] :
( ( sdrop @ ( product_prod @ A @ B ) @ N2 @ ( szip @ A @ B @ S1 @ S22 ) )
= ( szip @ A @ B @ ( sdrop @ A @ N2 @ S1 ) @ ( sdrop @ B @ N2 @ S22 ) ) ) ).
% sdrop_szip
thf(fact_14_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_15_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_16_pickEff,axiom,
! [R: rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
=> ( eff @ R @ S @ ( abstra1276541928ickEff @ rule @ state @ eff @ R @ S ) ) ) ).
% pickEff
thf(fact_17_ftree__no__ipath,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A,Steps2: stream @ A] :
( ( abstra668420080finite @ A @ T2 )
=> ~ ( abstra313004635_ipath @ A @ T2 @ Steps2 ) ) ).
% ftree_no_ipath
thf(fact_18_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_19_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_20_sset__fenum,axiom,
( ( sset @ rule @ ( abstra1774373515_fenum @ rule @ rules ) )
= ( sset @ rule @ rules ) ) ).
% sset_fenum
thf(fact_21_fair__fenum,axiom,
abstra928354080m_fair @ rule @ rules @ ( abstra1774373515_fenum @ rule @ rules ) ).
% fair_fenum
thf(fact_22_shd__sset,axiom,
! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% shd_sset
thf(fact_23_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 )
=> ? [X2: Rule] :
( ( member @ Rule @ X2 @ ( sset @ Rule @ Rules ) )
& ? [X1: fset @ State] : ( Eff @ X2 @ S @ X1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_24_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_25_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_26_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_27_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_28_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_29_RuleSystem__Defs_Oenabled__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1874422341nabled @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,R2: Rule,S4: State] :
( ^ [P: ( fset @ State ) > $o] :
? [X: fset @ State] : ( P @ X )
@ ( Eff2 @ R2 @ S4 ) ) ) ) ).
% RuleSystem_Defs.enabled_def
thf(fact_30_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_31_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_32_trim__def,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S )
= ( sdrop_while @ rule
@ ^ [R2: rule] :
~ ( abstra1874422341nabled @ rule @ state @ eff @ R2 @ S )
@ Rs ) ) ).
% trim_def
thf(fact_33_Saturated__def,axiom,
! [Steps2: stream @ ( product_prod @ state @ rule )] :
( ( abstra1209608345urated @ rule @ state @ eff @ rules @ Steps2 )
= ( ! [X3: rule] :
( ( member @ rule @ X3 @ ( sset @ rule @ rules ) )
=> ( abstra726722745urated @ rule @ state @ eff @ X3 @ Steps2 ) ) ) ) ).
% Saturated_def
thf(fact_34_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_35_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_36_eff__S,axiom,
! [S: state,R: rule,Sl: fset @ state,S5: state] :
( ( member @ state @ S @ s )
=> ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
=> ( ( eff @ R @ S @ Sl )
=> ( ( fmember @ state @ S5 @ Sl )
=> ( member @ state @ S5 @ s ) ) ) ) ) ).
% eff_S
thf(fact_37_NE__R,axiom,
( ( sset @ rule @ rules )
!= ( bot_bot @ ( set @ rule ) ) ) ).
% NE_R
thf(fact_38_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_39_countable__R,axiom,
countable_countable @ rule @ ( sset @ rule @ rules ) ).
% countable_R
thf(fact_40_fair__stl,axiom,
! [Rs: stream @ rule] :
( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra928354080m_fair @ rule @ rules @ ( stl @ rule @ Rs ) ) ) ).
% fair_stl
thf(fact_41_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_42_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_43_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( sdrop_while @ A )
= ( ^ [P2: A > $o,S4: stream @ A] : ( if @ ( stream @ A ) @ ( P2 @ ( shd @ A @ S4 ) ) @ ( sdrop_while @ A @ P2 @ ( stl @ A @ S4 ) ) @ S4 ) ) ) ).
% sdrop_while.simps
thf(fact_44_stream_Ocoinduct__strong,axiom,
! [A: $tType,R3: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R3 @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R3 @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( ( R3 @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) )
| ( ( stl @ A @ Stream3 )
= ( stl @ A @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
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,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P3 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_stream_Ocoinduct,axiom,
! [A: $tType,R3: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R3 @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R3 @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( R3 @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_50_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_51_stl__sset,axiom,
! [A: $tType,X4: A,A2: stream @ A] :
( ( member @ A @ X4 @ ( sset @ A @ ( stl @ A @ A2 ) ) )
=> ( member @ A @ X4 @ ( sset @ A @ A2 ) ) ) ).
% stl_sset
thf(fact_52_RuleSystem__Defs_Ocountable__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( countable_countable @ Rule @ ( sset @ Rule @ Rules ) ) ).
% RuleSystem_Defs.countable_R
thf(fact_53_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_54_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_55_RuleSystem__Defs_ONE__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] :
( ( sset @ Rule @ Rules )
!= ( bot_bot @ ( set @ Rule ) ) ) ).
% RuleSystem_Defs.NE_R
thf(fact_56_sset__induct,axiom,
! [A: $tType,Y: A,S: stream @ A,P3: A > ( stream @ A ) > $o] :
( ( member @ A @ Y @ ( sset @ A @ S ) )
=> ( ! [S6: stream @ A] : ( P3 @ ( shd @ A @ S6 ) @ S6 )
=> ( ! [S6: stream @ A,Y2: A] :
( ( member @ A @ Y2 @ ( sset @ A @ ( stl @ A @ S6 ) ) )
=> ( ( P3 @ Y2 @ ( stl @ A @ S6 ) )
=> ( P3 @ Y2 @ S6 ) ) )
=> ( P3 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_57_RuleSystem_Ointro,axiom,
! [Rule: $tType,State: $tType,S3: set @ State,Rules: stream @ Rule,Eff: Rule > State > ( fset @ State ) > $o] :
( ! [S6: State] :
( ( member @ State @ S6 @ S3 )
=> ! [R4: Rule] :
( ( member @ Rule @ R4 @ ( sset @ Rule @ Rules ) )
=> ! [Sl2: fset @ State] :
( ( Eff @ R4 @ S6 @ Sl2 )
=> ! [S2: State] :
( ( fmember @ State @ S2 @ Sl2 )
=> ( member @ State @ S2 @ S3 ) ) ) ) )
=> ( ! [S6: State] :
( ( member @ State @ S6 @ S3 )
=> ? [X5: Rule] :
( ( member @ Rule @ X5 @ ( sset @ Rule @ Rules ) )
& ? [X12: fset @ State] : ( Eff @ X5 @ S6 @ X12 ) ) )
=> ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 ) ) ) ).
% RuleSystem.intro
thf(fact_58_RuleSystem_Oeff__S,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S3: set @ State,S: State,R: Rule,Sl: fset @ State,S5: State] :
( ( abstra1326562878System @ Rule @ State @ Eff @ Rules @ S3 )
=> ( ( member @ State @ S @ S3 )
=> ( ( member @ Rule @ R @ ( sset @ Rule @ Rules ) )
=> ( ( Eff @ R @ S @ Sl )
=> ( ( fmember @ State @ S5 @ Sl )
=> ( member @ State @ S5 @ S3 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_59_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 )
=> ! [R2: Rule] :
( ( member @ Rule @ R2 @ ( sset @ Rule @ Rules2 ) )
=> ! [Sl3: fset @ State] :
( ( Eff2 @ R2 @ S4 @ Sl3 )
=> ! [S8: State] :
( ( fmember @ State @ S8 @ Sl3 )
=> ( member @ State @ S8 @ S7 ) ) ) ) )
& ! [S4: State] :
( ( member @ State @ S4 @ S7 )
=> ? [X3: Rule] :
( ( member @ Rule @ X3 @ ( sset @ Rule @ Rules2 ) )
& ( ^ [P: ( fset @ State ) > $o] :
? [X: fset @ State] : ( P @ X )
@ ( Eff2 @ X3 @ S4 ) ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_60_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 )] :
! [X3: Rule] :
( ( member @ Rule @ X3 @ ( sset @ Rule @ Rules2 ) )
=> ( abstra726722745urated @ Rule @ State @ Eff2 @ X3 @ Steps3 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_61_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_62_RuleSystem__Defs_Ofair__fenum,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( abstra928354080m_fair @ Rule @ Rules @ ( abstra1774373515_fenum @ Rule @ Rules ) ) ).
% RuleSystem_Defs.fair_fenum
thf(fact_63_RuleSystem__Defs_Otrim__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1259602206m_trim @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rs3: stream @ Rule,S4: State] :
( sdrop_while @ Rule
@ ^ [R2: Rule] :
~ ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R2 @ S4 )
@ Rs3 ) ) ) ).
% RuleSystem_Defs.trim_def
thf(fact_64_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_65_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 ) ) )
=> ? [Sl4: fset @ state,S2: state] :
( ( member @ state @ S2 @ s )
& ( eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S @ Sl4 )
& ( fmember @ state @ S2 @ Sl4 )
& ( T3
= ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S2 ) ) ) ) ) ) ).
% in_cont_mkTree
thf(fact_66_countableI__type,axiom,
! [A: $tType] :
( ( countable @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] : ( countable_countable @ A @ A3 ) ) ).
% countableI_type
thf(fact_67_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 ) ) )
=> ? [Sl4: fset @ State,S2: State] :
( ( member @ State @ S2 @ S3 )
& ( Eff @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) @ S @ Sl4 )
& ( fmember @ State @ S2 @ Sl4 )
& ( T3
= ( abstra1225283448mkTree @ Rule @ State @ Eff @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs @ S ) ) @ S2 ) ) ) ) ) ) ) ).
% RuleSystem.in_cont_mkTree
thf(fact_68_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_69_empty__Collect__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P3 ) )
= ( ! [X3: A] :
~ ( P3 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_70_Collect__empty__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( collect @ A @ P3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P3 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_71_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_72_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_73_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_74_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_75_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_76_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_77_countable__subset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( countable_countable @ A @ B2 )
=> ( countable_countable @ A @ A3 ) ) ) ).
% countable_subset
thf(fact_78_tfinite_Ocases,axiom,
! [A: $tType,A2: abstra2103299360e_tree @ A] :
( ( abstra668420080finite @ A @ A2 )
=> ! [T4: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T4 @ ( abstra1749095923e_cont @ A @ A2 ) )
=> ( abstra668420080finite @ A @ T4 ) ) ) ).
% tfinite.cases
thf(fact_79_tfinite_Osimps,axiom,
! [A: $tType] :
( ( abstra668420080finite @ A )
= ( ^ [A4: abstra2103299360e_tree @ A] :
? [T5: abstra2103299360e_tree @ A] :
( ( A4 = T5 )
& ! [X3: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ X3 @ ( abstra1749095923e_cont @ A @ T5 ) )
=> ( abstra668420080finite @ A @ X3 ) ) ) ) ) ).
% tfinite.simps
thf(fact_80_tfinite_Oinducts,axiom,
! [A: $tType,X4: abstra2103299360e_tree @ A,P3: ( abstra2103299360e_tree @ A ) > $o] :
( ( abstra668420080finite @ A @ X4 )
=> ( ! [T6: abstra2103299360e_tree @ A] :
( ! [T4: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T4 @ ( abstra1749095923e_cont @ A @ T6 ) )
=> ( abstra668420080finite @ A @ T4 ) )
=> ( ! [T4: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T4 @ ( abstra1749095923e_cont @ A @ T6 ) )
=> ( P3 @ T4 ) )
=> ( P3 @ T6 ) ) )
=> ( P3 @ X4 ) ) ) ).
% tfinite.inducts
thf(fact_81_tfinite,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A] :
( ! [T7: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ T2 ) )
=> ( abstra668420080finite @ A @ T7 ) )
=> ( abstra668420080finite @ A @ T2 ) ) ).
% tfinite
thf(fact_82_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funD
thf(fact_83_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funE
thf(fact_84_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_85_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_86_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_87_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_88_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_91_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X4 )
=> ( X4 = Y ) ) ) ) ).
% antisym
thf(fact_92_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
| ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% linear
thf(fact_93_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y: A] :
( ( X4 = Y )
=> ( ord_less_eq @ A @ X4 @ Y ) ) ) ).
% eq_refl
thf(fact_94_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ Y @ X4 ) ) ) ).
% le_cases
thf(fact_95_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_96_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X4 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_97_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X4: A] :
( ( ord_less_eq @ A @ Y @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y )
= ( X4 = Y ) ) ) ) ).
% antisym_conv
thf(fact_98_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_99_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_100_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_101_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_102_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_103_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,A2: A,B3: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P3 @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P3 @ B4 @ A5 )
=> ( P3 @ A5 @ B4 ) )
=> ( P3 @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_104_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_105_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_106_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_107_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_108_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_109_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_110_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_111_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,A2: A,Q: A > $o] :
( ( P3 @ A2 )
=> ( ! [X2: A] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2
thf(fact_112_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_113_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,Q: A > $o] :
( ? [X12: A] : ( P3 @ X12 )
=> ( ! [X2: A] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2_ex
thf(fact_114_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $false ) ) ).
% empty_def
thf(fact_115_countable__Collect,axiom,
! [A: $tType,A3: set @ A,Phi: A > $o] :
( ( countable_countable @ A @ A3 )
=> ( countable_countable @ A
@ ( collect @ A
@ ^ [A4: A] :
( ( member @ A @ A4 @ A3 )
& ( Phi @ A4 ) ) ) ) ) ).
% countable_Collect
thf(fact_116_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,K: A] :
( ( P3 @ K )
=> ( P3 @ ( ord_Least @ A @ P3 ) ) ) ) ).
% LeastI
thf(fact_117_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_118_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_119_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_120_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,X4: A,Q: A > $o] :
( ( P3 @ X4 )
=> ( ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
=> ( ! [X2: A] :
( ( P3 @ X2 )
=> ( ! [Y5: A] :
( ( P3 @ Y5 )
=> ( ord_less_eq @ A @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least @ A @ P3 ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_121_Least__equality,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,X4: A] :
( ( P3 @ X4 )
=> ( ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
=> ( ( ord_Least @ A @ P3 )
= X4 ) ) ) ) ).
% Least_equality
thf(fact_122_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,A2: A,Q: A > $o] :
( ( P3 @ A2 )
=> ( ! [A5: A] :
( ( P3 @ A5 )
=> ( ! [B5: A] :
( ( P3 @ B5 )
=> ( ord_less_eq @ A @ A5 @ B5 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_123_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P3: A > $o,Q: A > $o] :
( ? [X12: A] : ( P3 @ X12 )
=> ( ! [A5: A] :
( ( P3 @ A5 )
=> ( ! [B5: A] :
( ( P3 @ B5 )
=> ( ord_less_eq @ A @ A5 @ B5 ) )
=> ( Q @ A5 ) ) )
=> ( Q @ ( ord_Least @ A @ P3 ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_124_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_125_fsubsetI,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
=> ( fmember @ A @ X2 @ B2 ) )
=> ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 ) ) ).
% fsubsetI
thf(fact_126_ipath_Ocases,axiom,
! [A: $tType,A1: abstra2103299360e_tree @ A,A22: stream @ A] :
( ( abstra313004635_ipath @ A @ A1 @ A22 )
=> ~ ( ( ( abstra573067619e_root @ A @ A1 )
= ( shd @ A @ A22 ) )
=> ! [T7: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ A1 ) )
=> ~ ( abstra313004635_ipath @ A @ T7 @ ( stl @ A @ A22 ) ) ) ) ) ).
% ipath.cases
thf(fact_127_ipath_Osimps,axiom,
! [A: $tType] :
( ( abstra313004635_ipath @ A )
= ( ^ [A12: abstra2103299360e_tree @ A,A23: stream @ A] :
? [T5: abstra2103299360e_tree @ A,Steps3: stream @ A,T8: abstra2103299360e_tree @ A] :
( ( A12 = T5 )
& ( A23 = Steps3 )
& ( ( abstra573067619e_root @ A @ T5 )
= ( shd @ A @ Steps3 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T8 @ ( abstra1749095923e_cont @ A @ T5 ) )
& ( abstra313004635_ipath @ A @ T8 @ ( stl @ A @ Steps3 ) ) ) ) ) ).
% ipath.simps
thf(fact_128_ipath_Ointros,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A,Steps2: stream @ A,T3: abstra2103299360e_tree @ A] :
( ( ( abstra573067619e_root @ A @ T2 )
= ( shd @ A @ Steps2 ) )
=> ( ( fmember @ ( abstra2103299360e_tree @ A ) @ T3 @ ( abstra1749095923e_cont @ A @ T2 ) )
=> ( ( abstra313004635_ipath @ A @ T3 @ ( stl @ A @ Steps2 ) )
=> ( abstra313004635_ipath @ A @ T2 @ Steps2 ) ) ) ) ).
% ipath.intros
thf(fact_129_ipath_Ocoinduct,axiom,
! [A: $tType,X6: ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o,X4: abstra2103299360e_tree @ A,Xa: stream @ A] :
( ( X6 @ X4 @ Xa )
=> ( ! [X2: abstra2103299360e_tree @ A,Xa2: stream @ A] :
( ( X6 @ X2 @ Xa2 )
=> ? [T9: abstra2103299360e_tree @ A,Steps4: stream @ A,T4: abstra2103299360e_tree @ A] :
( ( X2 = T9 )
& ( Xa2 = Steps4 )
& ( ( abstra573067619e_root @ A @ T9 )
= ( shd @ A @ Steps4 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T4 @ ( abstra1749095923e_cont @ A @ T9 ) )
& ( ( X6 @ T4 @ ( stl @ A @ Steps4 ) )
| ( abstra313004635_ipath @ A @ T4 @ ( stl @ A @ Steps4 ) ) ) ) )
=> ( abstra313004635_ipath @ A @ X4 @ Xa ) ) ) ).
% ipath.coinduct
thf(fact_130_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_131_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_132_subsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% subsetI
thf(fact_133_fempty__iff,axiom,
! [A: $tType,C2: A] :
~ ( fmember @ A @ C2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% fempty_iff
thf(fact_134_all__not__fin__conv,axiom,
! [A: $tType,A3: fset @ A] :
( ( ! [X3: A] :
~ ( fmember @ A @ X3 @ A3 ) )
= ( A3
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% all_not_fin_conv
thf(fact_135_fimage__is__fempty,axiom,
! [A: $tType,B: $tType,F: B > A,A3: fset @ B] :
( ( ( fimage @ B @ A @ F @ A3 )
= ( bot_bot @ ( fset @ A ) ) )
= ( A3
= ( bot_bot @ ( fset @ B ) ) ) ) ).
% fimage_is_fempty
thf(fact_136_fempty__is__fimage,axiom,
! [A: $tType,B: $tType,F: B > A,A3: fset @ B] :
( ( ( bot_bot @ ( fset @ A ) )
= ( fimage @ B @ A @ F @ A3 ) )
= ( A3
= ( bot_bot @ ( fset @ B ) ) ) ) ).
% fempty_is_fimage
thf(fact_137_fsubset__antisym,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% fsubset_antisym
thf(fact_138_fempty__fsubsetI,axiom,
! [A: $tType,X4: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ ( bot_bot @ ( fset @ A ) ) @ X4 ) ).
% fempty_fsubsetI
thf(fact_139_fsubset__fempty,axiom,
! [A: $tType,A3: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ ( bot_bot @ ( fset @ A ) ) )
= ( A3
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% fsubset_fempty
thf(fact_140_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_141_fimage__ident,axiom,
! [A: $tType,Y6: fset @ A] :
( ( fimage @ A @ A
@ ^ [X3: A] : X3
@ Y6 )
= Y6 ) ).
% fimage_ident
thf(fact_142_fset_Omap__ident,axiom,
! [A: $tType,T2: fset @ A] :
( ( fimage @ A @ A
@ ^ [X3: A] : X3
@ T2 )
= T2 ) ).
% fset.map_ident
thf(fact_143_fimage__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X4: B,A3: fset @ B] :
( ( B3
= ( F @ X4 ) )
=> ( ( fmember @ B @ X4 @ A3 )
=> ( fmember @ A @ B3 @ ( fimage @ B @ A @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_144_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_145_subset__fimage__iff,axiom,
! [A: $tType,B: $tType,B2: fset @ A,F: B > A,A3: fset @ B] :
( ( ord_less_eq @ ( fset @ A ) @ B2 @ ( fimage @ B @ A @ F @ A3 ) )
= ( ? [AA: fset @ B] :
( ( ord_less_eq @ ( fset @ B ) @ AA @ A3 )
& ( B2
= ( fimage @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_fimage_iff
thf(fact_146_Collect__mono__iff,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P3 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_147_fset__eq__fsubset,axiom,
! [A: $tType] :
( ( ^ [Y3: fset @ A,Z: fset @ A] : Y3 = Z )
= ( ^ [A6: fset @ A,B6: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( fset @ A ) @ B6 @ A6 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_148_fimage__fsubsetI,axiom,
! [A: $tType,B: $tType,A3: fset @ A,F: A > B,B2: fset @ B] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
=> ( fmember @ B @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F @ A3 ) @ B2 ) ) ).
% fimage_fsubsetI
thf(fact_149_rev__fimage__eqI,axiom,
! [B: $tType,A: $tType,X4: A,A3: fset @ A,B3: B,F: A > B] :
( ( fmember @ A @ X4 @ A3 )
=> ( ( B3
= ( F @ X4 ) )
=> ( fmember @ B @ B3 @ ( fimage @ A @ B @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_150_equalsffemptyI,axiom,
! [A: $tType,A3: fset @ A] :
( ! [Y2: A] :
~ ( fmember @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% equalsffemptyI
thf(fact_151_equalsffemptyD,axiom,
! [A: $tType,A3: fset @ A,A2: A] :
( ( A3
= ( bot_bot @ ( fset @ A ) ) )
=> ~ ( fmember @ A @ A2 @ A3 ) ) ).
% equalsffemptyD
thf(fact_152_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ~ ( member @ A @ C2 @ A3 ) ) ) ).
% contra_subsetD
thf(fact_153_fsubset__trans,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A,C3: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( fset @ A ) @ A3 @ C3 ) ) ) ).
% fsubset_trans
thf(fact_154_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : Y3 = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_155_fsubset__refl,axiom,
! [A: $tType,A3: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ A3 @ A3 ) ).
% fsubset_refl
thf(fact_156_subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_157_Collect__mono,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_158_fimage__mono,axiom,
! [B: $tType,A: $tType,A3: fset @ A,B2: fset @ A,F: A > B] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F @ A3 ) @ ( fimage @ A @ B @ F @ B2 ) ) ) ).
% fimage_mono
thf(fact_159_fimage__cong,axiom,
! [B: $tType,A: $tType,M2: fset @ A,N4: fset @ A,F: A > B,G: A > B] :
( ( M2 = N4 )
=> ( ! [X2: A] :
( ( fmember @ A @ X2 @ N4 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( fimage @ A @ B @ F @ M2 )
= ( fimage @ A @ B @ G @ N4 ) ) ) ) ).
% fimage_cong
thf(fact_160_fequalityD2,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( fset @ A ) @ B2 @ A3 ) ) ).
% fequalityD2
thf(fact_161_fequalityD1,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 ) ) ).
% fequalityD1
thf(fact_162_ex__fin__conv,axiom,
! [A: $tType,A3: fset @ A] :
( ( ? [X3: A] : ( fmember @ A @ X3 @ A3 ) )
= ( A3
!= ( bot_bot @ ( fset @ A ) ) ) ) ).
% ex_fin_conv
thf(fact_163_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_164_rev__subsetD,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_165_fequalityE,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ~ ( ord_less_eq @ ( fset @ A ) @ B2 @ A3 ) ) ) ).
% fequalityE
thf(fact_166_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T5: A] :
( ( member @ A @ T5 @ A6 )
=> ( member @ A @ T5 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_167_set__rev__mp,axiom,
! [A: $tType,X4: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ X4 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ A @ X4 @ B2 ) ) ) ).
% set_rev_mp
thf(fact_168_equalityD2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_169_equalityD1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_170_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_171_equalityE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_172_subsetCE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetCE
thf(fact_173_fimageI,axiom,
! [B: $tType,A: $tType,X4: A,A3: fset @ A,F: A > B] :
( ( fmember @ A @ X4 @ A3 )
=> ( fmember @ B @ ( F @ X4 ) @ ( fimage @ A @ B @ F @ A3 ) ) ) ).
% fimageI
thf(fact_174_fimageE,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,A3: fset @ B] :
( ( fmember @ A @ B3 @ ( fimage @ B @ A @ F @ A3 ) )
=> ~ ! [X2: B] :
( ( B3
= ( F @ X2 ) )
=> ~ ( fmember @ B @ X2 @ A3 ) ) ) ).
% fimageE
thf(fact_175_femptyE,axiom,
! [A: $tType,A2: A] :
~ ( fmember @ A @ A2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% femptyE
thf(fact_176_subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_177_in__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_178_set__mp,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B2 ) ) ) ).
% set_mp
thf(fact_179_fimage__fimage,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A3: fset @ C] :
( ( fimage @ B @ A @ F @ ( fimage @ C @ B @ G @ A3 ) )
= ( fimage @ C @ A
@ ^ [X3: C] : ( F @ ( G @ X3 ) )
@ A3 ) ) ).
% fimage_fimage
thf(fact_180_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_181_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_182_eqfelem__imp__iff,axiom,
! [A: $tType,X4: A,Y: A,A3: fset @ A] :
( ( X4 = Y )
=> ( ( fmember @ A @ X4 @ A3 )
= ( fmember @ A @ Y @ A3 ) ) ) ).
% eqfelem_imp_iff
thf(fact_183_if__split__fmem2,axiom,
! [A: $tType,A2: A,Q: $o,X4: fset @ A,Y: fset @ A] :
( ( fmember @ A @ A2 @ ( if @ ( fset @ A ) @ Q @ X4 @ Y ) )
= ( ( Q
=> ( fmember @ A @ A2 @ X4 ) )
& ( ~ Q
=> ( fmember @ A @ A2 @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_184_if__split__fmem1,axiom,
! [A: $tType,Q: $o,X4: A,Y: A,B3: fset @ A] :
( ( fmember @ A @ ( if @ A @ Q @ X4 @ Y ) @ B3 )
= ( ( Q
=> ( fmember @ A @ X4 @ B3 ) )
& ( ~ Q
=> ( fmember @ A @ Y @ B3 ) ) ) ) ).
% if_split_fmem1
thf(fact_185_eqfset__imp__iff,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A,X4: A] :
( ( A3 = B2 )
=> ( ( fmember @ A @ X4 @ A3 )
= ( fmember @ A @ X4 @ B2 ) ) ) ).
% eqfset_imp_iff
thf(fact_186_eq__fmem__trans,axiom,
! [A: $tType,A2: A,B3: A,A3: fset @ A] :
( ( A2 = B3 )
=> ( ( fmember @ A @ B3 @ A3 )
=> ( fmember @ A @ A2 @ A3 ) ) ) ).
% eq_fmem_trans
thf(fact_187_fset__choice,axiom,
! [B: $tType,A: $tType,A3: fset @ A,P3: A > B > $o] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
=> ? [X12: B] : ( P3 @ X2 @ X12 ) )
=> ? [F3: A > B] :
! [X5: A] :
( ( fmember @ A @ X5 @ A3 )
=> ( P3 @ X5 @ ( F3 @ X5 ) ) ) ) ).
% fset_choice
thf(fact_188_fequalityCE,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A,C2: A] :
( ( A3 = B2 )
=> ( ( ( fmember @ A @ C2 @ A3 )
=> ~ ( fmember @ A @ C2 @ B2 ) )
=> ~ ( ~ ( fmember @ A @ C2 @ A3 )
=> ( fmember @ A @ C2 @ B2 ) ) ) ) ).
% fequalityCE
thf(fact_189_fset__eqI,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
= ( fmember @ A @ X2 @ B2 ) )
=> ( A3 = B2 ) ) ).
% fset_eqI
thf(fact_190_fset__mp,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A,X4: A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ( ( fmember @ A @ X4 @ A3 )
=> ( fmember @ A @ X4 @ B2 ) ) ) ).
% fset_mp
thf(fact_191_fin__mono,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A,X4: A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ( ( fmember @ A @ X4 @ A3 )
=> ( fmember @ A @ X4 @ B2 ) ) ) ).
% fin_mono
thf(fact_192_fsubsetD,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A,C2: A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ( ( fmember @ A @ C2 @ A3 )
=> ( fmember @ A @ C2 @ B2 ) ) ) ).
% fsubsetD
thf(fact_193_fset__rev__mp,axiom,
! [A: $tType,X4: A,A3: fset @ A,B2: fset @ A] :
( ( fmember @ A @ X4 @ A3 )
=> ( ( ord_less_eq @ ( fset @ A ) @ A3 @ B2 )
=> ( fmember @ A @ X4 @ B2 ) ) ) ).
% fset_rev_mp
thf(fact_194_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_195_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_196_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X2: A] :
~ ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_197_Konig,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A] :
( ~ ( abstra668420080finite @ A @ T2 )
=> ( abstra313004635_ipath @ A @ T2 @ ( abstra1918223989_konig @ A @ T2 ) ) ) ).
% Konig
thf(fact_198_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_199_predicate1I,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( A > $o ) @ P3 @ Q ) ) ).
% predicate1I
thf(fact_200_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_201_rev__predicate1D,axiom,
! [A: $tType,P3: A > $o,X4: A,Q: A > $o] :
( ( P3 @ X4 )
=> ( ( ord_less_eq @ ( A > $o ) @ P3 @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_202_predicate1D,axiom,
! [A: $tType,P3: A > $o,Q: A > $o,X4: A] :
( ( ord_less_eq @ ( A > $o ) @ P3 @ Q )
=> ( ( P3 @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_203_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
=> ( ( S5 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S5 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_204_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_205_tree_Osel_I1_J,axiom,
! [A: $tType,X13: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra573067619e_root @ A @ ( abstra388494275e_Node @ A @ X13 @ X22 ) )
= X13 ) ).
% tree.sel(1)
thf(fact_206_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_207_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_208_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_209_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
! [Rule: $tType,State: $tType,Eff: Rule > State > ( fset @ State ) > $o,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 ) ) ) ) ).
% RuleSystem_Defs.mkTree.simps(1)
thf(fact_210_RuleSystem__Defs_OmkTree_Ocode,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1225283448mkTree @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rs3: stream @ Rule,S4: State] : ( abstra388494275e_Node @ ( product_prod @ State @ Rule ) @ ( product_Pair @ State @ Rule @ S4 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs3 @ S4 ) ) ) @ ( fimage @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs3 @ S4 ) ) ) @ ( abstra1276541928ickEff @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs3 @ S4 ) ) @ S4 ) ) ) ) ) ).
% RuleSystem_Defs.mkTree.code
thf(fact_211_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P3: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( P3 @ X3 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_212_prop__restrict,axiom,
! [A: $tType,X4: A,Z3: set @ A,X6: set @ A,P3: A > $o] :
( ( member @ A @ X4 @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( P3 @ X3 ) ) ) )
=> ( P3 @ X4 ) ) ) ).
% prop_restrict
thf(fact_213_konig_Osimps_I1_J,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A] :
( ( shd @ A @ ( abstra1918223989_konig @ A @ T2 ) )
= ( abstra573067619e_root @ A @ T2 ) ) ).
% konig.simps(1)
thf(fact_214_mkTree__unfold,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S )
= ( case_stream @ rule @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) )
@ ^ [R2: rule,S8: stream @ rule] : ( abstra388494275e_Node @ ( product_prod @ state @ rule ) @ ( product_Pair @ state @ rule @ S @ R2 ) @ ( fimage @ state @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ S8 ) @ ( abstra1276541928ickEff @ rule @ state @ eff @ R2 @ S ) ) )
@ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) ).
% mkTree_unfold
thf(fact_215_RuleSystem__Defs_OmkTree__unfold,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1225283448mkTree @ Rule @ State )
= ( ^ [Eff2: Rule > State > ( fset @ State ) > $o,Rs3: stream @ Rule,S4: State] :
( case_stream @ Rule @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) )
@ ^ [R2: Rule,S8: stream @ Rule] : ( abstra388494275e_Node @ ( product_prod @ State @ Rule ) @ ( product_Pair @ State @ Rule @ S4 @ R2 ) @ ( fimage @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ S8 ) @ ( abstra1276541928ickEff @ Rule @ State @ Eff2 @ R2 @ S4 ) ) )
@ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs3 @ S4 ) ) ) ) ).
% RuleSystem_Defs.mkTree_unfold
thf(fact_216_stream_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F: A > ( stream @ A ) > B,Stream: stream @ A] :
( ( H @ ( case_stream @ A @ B @ F @ Stream ) )
= ( case_stream @ A @ C
@ ^ [X14: A,X24: stream @ A] : ( H @ ( F @ X14 @ X24 ) )
@ Stream ) ) ).
% stream.case_distrib
thf(fact_217_shd__def,axiom,
! [A: $tType] :
( ( shd @ A )
= ( case_stream @ A @ A
@ ^ [X14: A,X24: stream @ A] : X14 ) ) ).
% shd_def
thf(fact_218_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( case_stream @ A @ B )
= ( ^ [F2: A > ( stream @ A ) > B,Stream5: stream @ A] : ( F2 @ ( shd @ A @ Stream5 ) @ ( stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_219_pred__subset__eq,axiom,
! [A: $tType,R3: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R3 )
@ ^ [X3: A] : ( member @ A @ X3 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_220_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B3: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
= ( ( A2 = A7 )
& ( B3 = B7 ) ) ) ).
% old.prod.inject
thf(fact_221_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_222_stl__def,axiom,
! [A: $tType] :
( ( stl @ A )
= ( case_stream @ A @ ( stream @ A )
@ ^ [X14: A,X24: stream @ A] : X24 ) ) ).
% stl_def
thf(fact_223_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_224_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X2: A,Y2: B] :
( P4
= ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).
% surj_pair
thf(fact_225_prod__cases,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P3 @ P4 ) ) ).
% prod_cases
thf(fact_226_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B3: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B3 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ~ ( ( A2 = A7 )
=> ( B3 != B7 ) ) ) ).
% Pair_inject
thf(fact_227_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B4: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% prod_cases3
thf(fact_228_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B4: B,C4: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_229_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 ) ) )] :
~ ! [A5: A,B4: B,C4: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_230_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A5: A,B4: B,C4: C,D2: D,E2: E,F3: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_231_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
~ ! [A5: A,B4: B,C4: C,D2: D,E2: E,F3: F4,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_232_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B4: B,C4: C] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct3
thf(fact_233_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C4: C,D2: D] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct4
thf(fact_234_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,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: B,C4: C,D2: D,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct5
thf(fact_235_prod__induct6,axiom,
! [F4: $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 @ F4 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A5: A,B4: B,C4: C,D2: D,E2: E,F3: F4] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct6
thf(fact_236_prod__induct7,axiom,
! [G3: $tType,F4: $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 @ F4 @ G3 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
( ! [A5: A,B4: B,C4: C,D2: D,E2: E,F3: F4,G4: G3] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F3 @ G4 ) ) ) ) ) ) )
=> ( P3 @ X4 ) ) ).
% prod_induct7
thf(fact_237_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_238_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_239_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_240_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
@ ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_241_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 ) )
= ( ^ [X3: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_242_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_243_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B3 ) )
= ( F1 @ A2 @ B3 ) ) ).
% old.prod.rec
thf(fact_244_Collect__empty__eq__bot,axiom,
! [A: $tType,P3: A > $o] :
( ( ( collect @ A @ P3 )
= ( bot_bot @ ( set @ A ) ) )
= ( P3
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_245_predicate2I,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,Q: A > B > $o] :
( ! [X2: A,Y2: B] :
( ( P3 @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P3 @ Q ) ) ).
% predicate2I
thf(fact_246_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P3: A > B > $o,X4: A,Y: B,Q: A > B > $o] :
( ( P3 @ X4 @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P3 @ Q )
=> ( Q @ X4 @ Y ) ) ) ).
% rev_predicate2D
thf(fact_247_predicate2D,axiom,
! [A: $tType,B: $tType,P3: A > B > $o,Q: A > B > $o,X4: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P3 @ Q )
=> ( ( P3 @ X4 @ Y )
=> ( Q @ X4 @ Y ) ) ) ).
% predicate2D
thf(fact_248_bot2E,axiom,
! [A: $tType,B: $tType,X4: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X4 @ Y ) ).
% bot2E
thf(fact_249_konig_Osimps_I2_J,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A] :
( ( stl @ A @ ( abstra1918223989_konig @ A @ T2 ) )
= ( abstra1918223989_konig @ A
@ @+[T8: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T8 @ ( abstra1749095923e_cont @ A @ T2 ) )
& ~ ( abstra668420080finite @ A @ T8 ) ) ) ) ).
% konig.simps(2)
thf(fact_250_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) ) ).
% less_by_empty
thf(fact_251_some__sym__eq__trivial,axiom,
! [A: $tType,X4: A] :
( ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X4 ) )
= X4 ) ).
% some_sym_eq_trivial
thf(fact_252_some__eq__trivial,axiom,
! [A: $tType,X4: A] :
( ( @+[Y4: A] : Y4 = X4 )
= X4 ) ).
% some_eq_trivial
thf(fact_253_some__equality,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( P3 @ A2 )
=> ( ! [X2: A] :
( ( P3 @ X2 )
=> ( X2 = A2 ) )
=> ( ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 )
= A2 ) ) ) ).
% some_equality
thf(fact_254_ex__has__least__nat,axiom,
! [A: $tType,P3: A > $o,K: A,M: A > nat] :
( ( P3 @ K )
=> ? [X2: A] :
( ( P3 @ X2 )
& ! [Y5: A] :
( ( P3 @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_255_someI2,axiom,
! [A: $tType,P3: A > $o,A2: A,Q: A > $o] :
( ( P3 @ A2 )
=> ( ! [X2: A] :
( ( P3 @ X2 )
=> ( Q @ X2 ) )
=> ( Q
@ ( ^ [P: A > $o] :
@+[X: A] : ( P @ X )
@ P3 ) ) ) ) ).
% someI2
%----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,X4: A,Y: A] :
( ( if @ A @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X4: A,Y: A] :
( ( if @ A @ $true @ X4 @ Y )
= X4 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [S9: state,N5: nat] :
( ( member @ state @ S9 @ s )
=> ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( sdrop @ rule @ N5 @ rsa ) @ S9 ) @ ( sdrop @ ( product_prod @ state @ rule ) @ ma @ stepsa ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------