TPTP Problem File: COM159^1.p
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%------------------------------------------------------------------------------
% File : COM159^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Abstract completeness 288
% 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__288.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 351 ( 123 unt; 63 typ; 0 def)
% Number of atoms : 806 ( 189 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4866 ( 68 ~; 7 |; 61 &;4326 @)
% ( 0 <=>; 404 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 344 ( 344 >; 0 *; 0 +; 0 <<)
% Number of symbols : 63 ( 60 usr; 12 con; 0-5 aty)
% Number of variables : 1190 ( 73 ^;1014 !; 46 ?;1190 :)
% ( 57 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:54:39.489
%------------------------------------------------------------------------------
%----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 (54)
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_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_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_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_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_Linear__Temporal__Logic__on__Streams_Oalw,type,
linear1386806755on_alw:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Oholds,type,
linear1707521579_holds:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
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_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_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_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_s_H____,type,
s3: state ).
thf(sy_v_sa____,type,
sa: state ).
thf(sy_v_steps,type,
steps: stream @ ( product_prod @ state @ rule ) ).
thf(sy_v_stepsa____,type,
stepsa: stream @ ( product_prod @ state @ rule ) ).
%----Relevant facts (256)
thf(fact_0_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_1_rt,axiom,
( ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rsa @ sa ) )
= ( shd @ ( product_prod @ state @ rule ) @ stepsa ) ) ).
% rt
thf(fact_2_s,axiom,
member @ state @ sa @ s ).
% s
thf(fact_3_pickEff,axiom,
! [R: rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
=> ( eff @ R @ S @ ( abstra1276541928ickEff @ rule @ state @ eff @ R @ S ) ) ) ).
% pickEff
thf(fact_4_RuleSystem__Defs_Oenabled__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1874422341nabled @ Rule @ State )
= ( ^ [Eff: Rule > State > ( fset @ State ) > $o,R2: Rule,S2: State] :
( ^ [P: ( fset @ State ) > $o] :
? [X: fset @ State] : ( P @ X )
@ ( Eff @ R2 @ S2 ) ) ) ) ).
% RuleSystem_Defs.enabled_def
thf(fact_5_less_Oprems_I3_J,axiom,
abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rsa @ sa ) @ stepsa ).
% less.prems(3)
thf(fact_6_False,axiom,
( ( abstra2097340358le_pos @ rule @ rsa @ r )
!= ( abstra1332369113inWait @ rule @ state @ eff @ rsa @ sa ) ) ).
% False
thf(fact_7_less_Oprems_I4_J,axiom,
( linear1386806755on_alw @ ( product_prod @ state @ rule )
@ ( linear1707521579_holds @ ( product_prod @ state @ rule )
@ ^ [Step: product_prod @ state @ rule] : ( abstra1874422341nabled @ rule @ state @ eff @ r @ ( product_fst @ state @ rule @ Step ) ) )
@ stepsa ) ).
% less.prems(4)
thf(fact_8_RuleSystem__Defs_OpickEff,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,R: Rule,S: State] :
( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S )
=> ( Eff2 @ R @ S @ ( abstra1276541928ickEff @ Rule @ State @ Eff2 @ R @ S ) ) ) ).
% RuleSystem_Defs.pickEff
thf(fact_9_local_Oalw,axiom,
( linear1386806755on_alw @ ( product_prod @ state @ rule )
@ ( linear1707521579_holds @ ( product_prod @ state @ rule )
@ ^ [Step: product_prod @ state @ rule] : ( abstra1874422341nabled @ rule @ state @ eff @ r @ ( product_fst @ state @ rule @ Step ) ) )
@ steps ) ).
% local.alw
thf(fact_10_assms_I3_J,axiom,
abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rs @ s2 ) @ steps ).
% assms(3)
thf(fact_11_trim__def_H,axiom,
( ( abstra1259602206m_trim @ rule @ state @ eff @ rsa @ sa )
= ( sdrop @ rule @ ( abstra1332369113inWait @ rule @ state @ eff @ rsa @ sa ) @ rsa ) ) ).
% trim_def'
thf(fact_12_r,axiom,
member @ rule @ r @ ( sset @ rule @ rules ) ).
% r
thf(fact_13_assms_I1_J,axiom,
member @ state @ s2 @ s ).
% assms(1)
thf(fact_14_s_H,axiom,
member @ state @ s3 @ s ).
% s'
thf(fact_15_i,axiom,
abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ rsa @ sa ) ) @ s3 ) @ ( stl @ ( product_prod @ state @ rule ) @ stepsa ) ).
% i
thf(fact_16_minWait__least,axiom,
! [N: nat,Rs: stream @ rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N @ Rs ) ) @ S )
=> ( ord_less_eq @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ N ) ) ).
% minWait_least
thf(fact_17_pos__least,axiom,
! [A: $tType,N: nat,Rs: stream @ A,R: A] :
( ( ( shd @ A @ ( sdrop @ A @ N @ Rs ) )
= R )
=> ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N ) ) ).
% pos_least
thf(fact_18_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_19_RuleSystem__axioms,axiom,
abstra1326562878System @ rule @ state @ eff @ rules @ s ).
% RuleSystem_axioms
thf(fact_20_less_Oprems_I2_J,axiom,
abstra928354080m_fair @ rule @ rules @ rsa ).
% less.prems(2)
thf(fact_21_rs,axiom,
abstra928354080m_fair @ rule @ rules @ rs ).
% rs
thf(fact_22_sset__fenum,axiom,
( ( sset @ rule @ ( abstra1774373515_fenum @ rule @ rules ) )
= ( sset @ rule @ rules ) ) ).
% sset_fenum
thf(fact_23__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_As_H_O_A_092_060lbrakk_062tree_Oroot_A_ImkTree_Ars_As_J_A_061_Ashd_Asteps_059_Aipath_A_ImkTree_A_Istl_A_Itrim_Ars_As_J_J_As_H_J_A_Istl_Asteps_J_059_As_H_A_092_060in_062_AS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ( ( abstra573067619e_root @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rsa @ sa ) )
= ( shd @ ( product_prod @ state @ rule ) @ stepsa ) )
=> ! [S3: state] :
( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ rsa @ sa ) ) @ S3 ) @ ( stl @ ( product_prod @ state @ rule ) @ stepsa ) )
=> ~ ( member @ state @ S3 @ s ) ) ) ).
% \<open>\<And>thesis. (\<And>t' s'. \<lbrakk>tree.root (mkTree rs s) = shd steps; ipath (mkTree (stl (trim rs s)) s') (stl steps); s' \<in> S\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_24_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_25_Saturated__def,axiom,
! [Steps: stream @ ( product_prod @ state @ rule )] :
( ( abstra1209608345urated @ rule @ state @ eff @ rules @ Steps )
= ( ! [X3: rule] :
( ( member @ rule @ X3 @ ( sset @ rule @ rules ) )
=> ( abstra726722745urated @ rule @ state @ eff @ X3 @ Steps ) ) ) ) ).
% Saturated_def
thf(fact_26_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_27_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_28_minWait__ex,axiom,
! [S: state,Rs: stream @ rule] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ? [N2: nat] : ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N2 @ Rs ) ) @ S ) ) ) ).
% minWait_ex
thf(fact_29_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_30_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_31_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_32_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_33_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_34_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_35_fair__stl,axiom,
! [Rs: stream @ rule] :
( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra928354080m_fair @ rule @ rules @ ( stl @ rule @ Rs ) ) ) ).
% fair_stl
thf(fact_36_fair__fenum,axiom,
abstra928354080m_fair @ rule @ rules @ ( abstra1774373515_fenum @ rule @ rules ) ).
% fair_fenum
thf(fact_37_pos__def,axiom,
! [A: $tType] :
( ( abstra2097340358le_pos @ A )
= ( ^ [Rs2: stream @ A,R2: A] :
( ord_Least @ nat
@ ^ [N3: nat] :
( ( shd @ A @ ( sdrop @ A @ N3 @ Rs2 ) )
= R2 ) ) ) ) ).
% pos_def
thf(fact_38_ipath__mkTree__sdrop,axiom,
! [S: state,Rs: stream @ rule,Steps: 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 ) @ Steps )
=> ? [N2: nat,S3: state] :
( ( member @ state @ S3 @ s )
& ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( sdrop @ rule @ N2 @ Rs ) @ S3 ) @ ( sdrop @ ( product_prod @ state @ rule ) @ M @ Steps ) ) ) ) ) ) ).
% ipath_mkTree_sdrop
thf(fact_39_RuleSystem__Defs_Ofair__fenum,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( abstra928354080m_fair @ Rule @ Rules @ ( abstra1774373515_fenum @ Rule @ Rules ) ) ).
% RuleSystem_Defs.fair_fenum
thf(fact_40_RuleSystem_Otrim__fair,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra928354080m_fair @ Rule @ Rules @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) ) ) ) ).
% RuleSystem.trim_fair
thf(fact_41_RuleSystem_Oenabled__R,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ? [X2: Rule] :
( ( member @ Rule @ X2 @ ( sset @ Rule @ Rules ) )
& ? [X1: fset @ State] : ( Eff2 @ X2 @ S @ X1 ) ) ) ) ).
% RuleSystem.enabled_R
thf(fact_42_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_43_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_44_RuleSystem_Otrim__in__R,axiom,
! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( member @ Rule @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ ( sset @ Rule @ Rules ) ) ) ) ) ).
% RuleSystem.trim_in_R
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P2 )
= ( 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_RuleSystem_Opos__def,axiom,
! [State: $tType,Rule: $tType,A: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,Rs: stream @ A,R: A] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( abstra2097340358le_pos @ A @ Rs @ R )
= ( ord_Least @ nat
@ ^ [N3: nat] :
( ( shd @ A @ ( sdrop @ A @ N3 @ Rs ) )
= R ) ) ) ) ).
% RuleSystem.pos_def
thf(fact_50_RuleSystem_OminWait__ex,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ? [N2: nat] : ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_ex
thf(fact_51_RuleSystem_Otrim__enabled,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).
% RuleSystem.trim_enabled
thf(fact_52_RuleSystem_Otrim__alt,axiom,
! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S )
= ( sdrop @ Rule @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).
% RuleSystem.trim_alt
thf(fact_53_RuleSystem_Oipath__mkTree__sdrop,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule,Steps: stream @ ( product_prod @ State @ Rule ),M: nat] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) @ Steps )
=> ? [N2: nat,S3: State] :
( ( member @ State @ S3 @ S4 )
& ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( sdrop @ Rule @ N2 @ Rs ) @ S3 ) @ ( sdrop @ ( product_prod @ State @ Rule ) @ M @ Steps ) ) ) ) ) ) ) ).
% RuleSystem.ipath_mkTree_sdrop
thf(fact_54_RuleSystem__Defs_OSaturated__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1209608345urated @ Rule @ State )
= ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,Steps2: stream @ ( product_prod @ State @ Rule )] :
! [X3: Rule] :
( ( member @ Rule @ X3 @ ( sset @ Rule @ Rules2 ) )
=> ( abstra726722745urated @ Rule @ State @ Eff @ X3 @ Steps2 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_55_RuleSystem_OminWait__def,axiom,
! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,Rs: stream @ Rule,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S )
= ( ord_Least @ nat
@ ^ [N3: nat] : ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N3 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_def
thf(fact_56_RuleSystem_Opos,axiom,
! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,Rs: stream @ Rule,R: Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( 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_57_RuleSystem_OminWait__le__pos,axiom,
! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,Rs: stream @ Rule,R: Rule,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( member @ Rule @ R @ ( sset @ Rule @ Rules ) )
=> ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S )
=> ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ ( abstra2097340358le_pos @ Rule @ Rs @ R ) ) ) ) ) ) ).
% RuleSystem.minWait_le_pos
thf(fact_58_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_59_RuleSystem__Defs_Otrim__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1259602206m_trim @ Rule @ State )
= ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rs2: stream @ Rule,S2: State] :
( sdrop_while @ Rule
@ ^ [R2: Rule] :
~ ( abstra1874422341nabled @ Rule @ State @ Eff @ R2 @ S2 )
@ Rs2 ) ) ) ).
% RuleSystem_Defs.trim_def
thf(fact_60_RuleSystem_Opos__least,axiom,
! [Rule: $tType,State: $tType,A: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,N: nat,Rs: stream @ A,R: A] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( ( shd @ A @ ( sdrop @ A @ N @ Rs ) )
= R )
=> ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N ) ) ) ).
% RuleSystem.pos_least
thf(fact_61_RuleSystem_OminWait__least,axiom,
! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,N: nat,Rs: stream @ Rule,S: State] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N @ Rs ) ) @ S )
=> ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ N ) ) ) ).
% RuleSystem.minWait_least
thf(fact_62_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_63_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_64_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_65_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( sdrop_while @ A )
= ( ^ [P3: A > $o,S2: stream @ A] : ( if @ ( stream @ A ) @ ( P3 @ ( shd @ A @ S2 ) ) @ ( sdrop_while @ A @ P3 @ ( stl @ A @ S2 ) ) @ S2 ) ) ) ).
% sdrop_while.simps
thf(fact_66_sset__induct,axiom,
! [A: $tType,Y: A,S: stream @ A,P2: A > ( stream @ A ) > $o] :
( ( member @ A @ Y @ ( sset @ A @ S ) )
=> ( ! [S6: stream @ A] : ( P2 @ ( shd @ A @ S6 ) @ S6 )
=> ( ! [S6: stream @ A,Y2: A] :
( ( member @ A @ Y2 @ ( sset @ A @ ( stl @ A @ S6 ) ) )
=> ( ( P2 @ Y2 @ ( stl @ A @ S6 ) )
=> ( P2 @ Y2 @ S6 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_67_alw__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ Phi ) )
= ( linear1386806755on_alw @ A @ Phi ) ) ).
% alw_alw
thf(fact_68_NE__R,axiom,
( ( sset @ rule @ rules )
!= ( bot_bot @ ( set @ rule ) ) ) ).
% NE_R
thf(fact_69_alw__inv,axiom,
! [B: $tType,A: $tType,F: ( stream @ A ) > ( stream @ B ),P2: ( stream @ B ) > $o,S: stream @ A] :
( ! [S6: stream @ A] :
( ( F @ ( stl @ A @ S6 ) )
= ( stl @ B @ ( F @ S6 ) ) )
=> ( ( linear1386806755on_alw @ B @ P2 @ ( F @ S ) )
= ( linear1386806755on_alw @ A
@ ^ [X3: stream @ A] : ( P2 @ ( F @ X3 ) )
@ S ) ) ) ).
% alw_inv
thf(fact_70_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,K: A] :
( ( P2 @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P2 ) @ K ) ) ) ).
% Least_le
thf(fact_71_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_72_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_73_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_74_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_75_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_76_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_77_RuleSystem__Defs_ONE__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] :
( ( sset @ Rule @ Rules )
!= ( bot_bot @ ( set @ Rule ) ) ) ).
% RuleSystem_Defs.NE_R
thf(fact_78_RuleSystem__def,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1326562878System @ Rule @ State )
= ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S7: set @ State] :
( ! [S2: State] :
( ( member @ State @ S2 @ S7 )
=> ! [R2: Rule] :
( ( member @ Rule @ R2 @ ( sset @ Rule @ Rules2 ) )
=> ! [Sl2: fset @ State] :
( ( Eff @ R2 @ S2 @ Sl2 )
=> ! [S8: State] :
( ( fmember @ State @ S8 @ Sl2 )
=> ( member @ State @ S8 @ S7 ) ) ) ) )
& ! [S2: State] :
( ( member @ State @ S2 @ S7 )
=> ? [X3: Rule] :
( ( member @ Rule @ X3 @ ( sset @ Rule @ Rules2 ) )
& ( ^ [P: ( fset @ State ) > $o] :
? [X: fset @ State] : ( P @ X )
@ ( Eff @ X3 @ S2 ) ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_79_RuleSystem_Oeff__S,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,R: Rule,Sl: fset @ State,S5: State] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( member @ Rule @ R @ ( sset @ Rule @ Rules ) )
=> ( ( Eff2 @ R @ S @ Sl )
=> ( ( fmember @ State @ S5 @ Sl )
=> ( member @ State @ S5 @ S4 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_80_RuleSystem_Ointro,axiom,
! [Rule: $tType,State: $tType,S4: set @ State,Rules: stream @ Rule,Eff2: Rule > State > ( fset @ State ) > $o] :
( ! [S6: State] :
( ( member @ State @ S6 @ S4 )
=> ! [R3: Rule] :
( ( member @ Rule @ R3 @ ( sset @ Rule @ Rules ) )
=> ! [Sl3: fset @ State] :
( ( Eff2 @ R3 @ S6 @ Sl3 )
=> ! [S3: State] :
( ( fmember @ State @ S3 @ Sl3 )
=> ( member @ State @ S3 @ S4 ) ) ) ) )
=> ( ! [S6: State] :
( ( member @ State @ S6 @ S4 )
=> ? [X5: Rule] :
( ( member @ Rule @ X5 @ ( sset @ Rule @ Rules ) )
& ? [X12: fset @ State] : ( Eff2 @ X5 @ S6 @ X12 ) ) )
=> ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 ) ) ) ).
% RuleSystem.intro
thf(fact_81_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_82_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_83_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_84_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_85_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 )
=> ( ! [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_86_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 )
=> ( ! [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_87_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 )
=> ( ! [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_88_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 )
=> ( ! [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_89_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_90_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_91_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_92_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_93_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_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_102_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P2 @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( P2 @ B3 @ A4 )
=> ( P2 @ A4 @ B3 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_103_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_104_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_105_alwD,axiom,
! [A: $tType,P2: ( stream @ A ) > $o,X4: stream @ A] :
( ( linear1386806755on_alw @ A @ P2 @ X4 )
=> ( P2 @ X4 ) ) ).
% alwD
thf(fact_106_alw__alwD,axiom,
! [A: $tType,P2: ( stream @ A ) > $o,Omega: stream @ A] :
( ( linear1386806755on_alw @ A @ P2 @ Omega )
=> ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ P2 ) @ Omega ) ) ).
% alw_alwD
thf(fact_107_alw__cong,axiom,
! [A: $tType,P2: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ P2 @ Omega )
=> ( ! [Omega2: stream @ A] :
( ( P2 @ Omega2 )
=> ( ( Q1 @ Omega2 )
= ( Q2 @ Omega2 ) ) )
=> ( ( linear1386806755on_alw @ A @ Q1 @ Omega )
= ( linear1386806755on_alw @ A @ Q2 @ Omega ) ) ) ) ).
% alw_cong
thf(fact_108_alw__mono,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
=> ( ! [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( Psi @ Xs2 ) )
=> ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ).
% alw_mono
thf(fact_109_all__imp__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A] :
( ! [X1: stream @ A] : ( Phi @ X1 )
=> ( linear1386806755on_alw @ A @ Phi @ Xs ) ) ).
% all_imp_alw
thf(fact_110_holds__mono,axiom,
! [A: $tType,P2: A > $o,Xs: stream @ A,Q: A > $o] :
( ( linear1707521579_holds @ A @ P2 @ Xs )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( linear1707521579_holds @ A @ Q @ Xs ) ) ) ).
% holds_mono
thf(fact_111_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,A2: A,Q: A > $o] :
( ( P2 @ A2 )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2
thf(fact_112_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o] :
( ? [X12: A] : ( P2 @ X12 )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% LeastI_ex
thf(fact_113_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,Q: A > $o] :
( ? [X12: A] : ( P2 @ X12 )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_ex
thf(fact_114_RuleSystem__Defs_OmkTree_Osimps_I1_J,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rs: stream @ Rule,S: State] :
( ( abstra573067619e_root @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) )
= ( product_Pair @ State @ Rule @ S @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) ) ) ).
% RuleSystem_Defs.mkTree.simps(1)
thf(fact_115_alw__mp,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs3: stream @ A] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
@ Xs )
=> ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ).
% alw_mp
thf(fact_116_alw__aand,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A
@ ^ [Xs3: stream @ A] :
( ( Phi @ Xs3 )
& ( Psi @ Xs3 ) ) )
= ( ^ [Xs3: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ Xs3 )
& ( linear1386806755on_alw @ A @ Psi @ Xs3 ) ) ) ) ).
% alw_aand
thf(fact_117_alw__False,axiom,
! [A: $tType,Omega: stream @ A] :
~ ( linear1386806755on_alw @ A
@ ^ [X3: stream @ A] : $false
@ Omega ) ).
% alw_False
thf(fact_118_holds__aand,axiom,
! [A: $tType,P2: A > $o,Steps: stream @ A,Q: A > $o] :
( ( ( linear1707521579_holds @ A @ P2 @ Steps )
& ( linear1707521579_holds @ A @ Q @ Steps ) )
= ( linear1707521579_holds @ A
@ ^ [Step: A] :
( ( P2 @ Step )
& ( Q @ Step ) )
@ Steps ) ) ).
% holds_aand
thf(fact_119_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,K: A] :
( ( P2 @ K )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% LeastI
thf(fact_120_RuleSystem_Owf__mkTree,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( abstra1874736267tem_wf @ Rule @ State @ Eff2 @ Rules @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) ) ) ) ) ).
% RuleSystem.wf_mkTree
thf(fact_121_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,X4: A,Q: A > $o] :
( ( P2 @ X4 )
=> ( ! [Y2: A] :
( ( P2 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( ! [Y5: A] :
( ( P2 @ Y5 )
=> ( ord_less_eq @ A @ X2 @ Y5 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_122_Least__equality,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,X4: A] :
( ( P2 @ X4 )
=> ( ! [Y2: A] :
( ( P2 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
=> ( ( ord_Least @ A @ P2 )
= X4 ) ) ) ) ).
% Least_equality
thf(fact_123_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,A2: A,Q: A > $o] :
( ( P2 @ A2 )
=> ( ! [A4: A] :
( ( P2 @ A4 )
=> ( ! [B4: A] :
( ( P2 @ B4 )
=> ( ord_less_eq @ A @ A4 @ B4 ) )
=> ( Q @ A4 ) ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_124_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,Q: A > $o] :
( ? [X12: A] : ( P2 @ X12 )
=> ( ! [A4: A] :
( ( P2 @ A4 )
=> ( ! [B4: A] :
( ( P2 @ B4 )
=> ( ord_less_eq @ A @ A4 @ B4 ) )
=> ( Q @ A4 ) ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_125_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_126_stream_Ocoinduct,axiom,
! [A: $tType,R4: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R4 @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R4 @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( R4 @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_127_stream_Ocoinduct__strong,axiom,
! [A: $tType,R4: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R4 @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R4 @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( ( R4 @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) )
| ( ( stl @ A @ Stream3 )
= ( stl @ A @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_128_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_129_shd__sset,axiom,
! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% shd_sset
thf(fact_130_alw_Ocases,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,A2: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ A2 )
=> ~ ( ( Phi @ A2 )
=> ~ ( linear1386806755on_alw @ A @ Phi @ ( stl @ A @ A2 ) ) ) ) ).
% alw.cases
thf(fact_131_alw_Osimps,axiom,
! [A: $tType] :
( ( linear1386806755on_alw @ A )
= ( ^ [Phi2: ( stream @ A ) > $o,A5: stream @ A] :
? [Xs3: stream @ A] :
( ( A5 = Xs3 )
& ( Phi2 @ Xs3 )
& ( linear1386806755on_alw @ A @ Phi2 @ ( stl @ A @ Xs3 ) ) ) ) ) ).
% alw.simps
thf(fact_132_alw_Ointros,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A] :
( ( Phi @ Xs )
=> ( ( linear1386806755on_alw @ A @ Phi @ ( stl @ A @ Xs ) )
=> ( linear1386806755on_alw @ A @ Phi @ Xs ) ) ) ).
% alw.intros
thf(fact_133_alw_Ocoinduct,axiom,
! [A: $tType,X6: ( stream @ A ) > $o,X4: stream @ A,Phi: ( stream @ A ) > $o] :
( ( X6 @ X4 )
=> ( ! [X2: stream @ A] :
( ( X6 @ X2 )
=> ? [Xs4: stream @ A] :
( ( X2 = Xs4 )
& ( Phi @ Xs4 )
& ( ( X6 @ ( stl @ A @ Xs4 ) )
| ( linear1386806755on_alw @ A @ Phi @ ( stl @ A @ Xs4 ) ) ) ) )
=> ( linear1386806755on_alw @ A @ Phi @ X4 ) ) ) ).
% alw.coinduct
thf(fact_134_alw__coinduct,axiom,
! [A: $tType,X6: ( stream @ A ) > $o,X4: stream @ A,Phi: ( stream @ A ) > $o] :
( ( X6 @ X4 )
=> ( ! [X2: stream @ A] :
( ( X6 @ X2 )
=> ( Phi @ X2 ) )
=> ( ! [X2: stream @ A] :
( ( X6 @ X2 )
=> ( ~ ( linear1386806755on_alw @ A @ Phi @ ( stl @ A @ X2 ) )
=> ( X6 @ ( stl @ A @ X2 ) ) ) )
=> ( linear1386806755on_alw @ A @ Phi @ X4 ) ) ) ) ).
% alw_coinduct
thf(fact_135_sdrop__stl,axiom,
! [A: $tType,N: nat,S: stream @ A] :
( ( sdrop @ A @ N @ ( stl @ A @ S ) )
= ( stl @ A @ ( sdrop @ A @ N @ S ) ) ) ).
% sdrop_stl
thf(fact_136_alw__sdrop,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,N: nat] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
=> ( linear1386806755on_alw @ A @ Phi @ ( sdrop @ A @ N @ Xs ) ) ) ).
% alw_sdrop
thf(fact_137_alw__iff__sdrop,axiom,
! [A: $tType] :
( ( linear1386806755on_alw @ A )
= ( ^ [P3: ( stream @ A ) > $o,Omega3: stream @ A] :
! [M2: nat] : ( P3 @ ( sdrop @ A @ M2 @ Omega3 ) ) ) ) ).
% alw_iff_sdrop
thf(fact_138_holds_Oelims_I3_J,axiom,
! [A: $tType,X4: A > $o,Xa: stream @ A] :
( ~ ( linear1707521579_holds @ A @ X4 @ Xa )
=> ~ ( X4 @ ( shd @ A @ Xa ) ) ) ).
% holds.elims(3)
thf(fact_139_holds_Oelims_I2_J,axiom,
! [A: $tType,X4: A > $o,Xa: stream @ A] :
( ( linear1707521579_holds @ A @ X4 @ Xa )
=> ( X4 @ ( shd @ A @ Xa ) ) ) ).
% holds.elims(2)
thf(fact_140_holds_Oelims_I1_J,axiom,
! [A: $tType,X4: A > $o,Xa: stream @ A,Y: $o] :
( ( ( linear1707521579_holds @ A @ X4 @ Xa )
= Y )
=> ( Y
= ( X4 @ ( shd @ A @ Xa ) ) ) ) ).
% holds.elims(1)
thf(fact_141_holds_Osimps,axiom,
! [A: $tType] :
( ( linear1707521579_holds @ A )
= ( ^ [P3: A > $o,Xs3: stream @ A] : ( P3 @ ( shd @ A @ Xs3 ) ) ) ) ).
% holds.simps
thf(fact_142_wf__ipath__epath,axiom,
! [T: abstra2103299360e_tree @ ( product_prod @ state @ rule ),Steps: stream @ ( product_prod @ state @ rule )] :
( ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T )
=> ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ T @ Steps )
=> ( abstra523868654_epath @ rule @ state @ eff @ rules @ Steps ) ) ) ).
% wf_ipath_epath
thf(fact_143_in__cont__mkTree,axiom,
! [S: state,Rs: stream @ rule,T2: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
( ( member @ state @ S @ s )
=> ( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ T2 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) ) )
=> ? [Sl4: fset @ state,S3: state] :
( ( member @ state @ S3 @ s )
& ( eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S @ Sl4 )
& ( fmember @ state @ S3 @ Sl4 )
& ( T2
= ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S3 ) ) ) ) ) ) ).
% in_cont_mkTree
thf(fact_144_fsubsetI,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
=> ( fmember @ A @ X2 @ B5 ) )
=> ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 ) ) ).
% fsubsetI
thf(fact_145_RuleSystem_Oin__cont__mkTree,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S4: set @ State,S: State,Rs: stream @ Rule,T2: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( abstra928354080m_fair @ Rule @ Rules @ Rs )
=> ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ T2 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) ) )
=> ? [Sl4: fset @ State,S3: State] :
( ( member @ State @ S3 @ S4 )
& ( Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S @ Sl4 )
& ( fmember @ State @ S3 @ Sl4 )
& ( T2
= ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S3 ) ) ) ) ) ) ) ).
% RuleSystem.in_cont_mkTree
thf(fact_146_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_147_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_148_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_149_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_150_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_151_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X3: A] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_152_fempty__iff,axiom,
! [A: $tType,C2: A] :
~ ( fmember @ A @ C2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% fempty_iff
thf(fact_153_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_154_fsubset__antisym,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% fsubset_antisym
thf(fact_155_fempty__fsubsetI,axiom,
! [A: $tType,X4: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ ( bot_bot @ ( fset @ A ) ) @ X4 ) ).
% fempty_fsubsetI
thf(fact_156_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_157_femptyE,axiom,
! [A: $tType,A2: A] :
~ ( fmember @ A @ A2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% femptyE
thf(fact_158_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_159_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_160_equalsffemptyD,axiom,
! [A: $tType,A3: fset @ A,A2: A] :
( ( A3
= ( bot_bot @ ( fset @ A ) ) )
=> ~ ( fmember @ A @ A2 @ A3 ) ) ).
% equalsffemptyD
thf(fact_161_equalsffemptyI,axiom,
! [A: $tType,A3: fset @ A] :
( ! [Y2: A] :
~ ( fmember @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% equalsffemptyI
thf(fact_162_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_163_fsubset__trans,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A,C3: fset @ A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( fset @ A ) @ B5 @ C3 )
=> ( ord_less_eq @ ( fset @ A ) @ A3 @ C3 ) ) ) ).
% fsubset_trans
thf(fact_164_fsubset__refl,axiom,
! [A: $tType,A3: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ A3 @ A3 ) ).
% fsubset_refl
thf(fact_165_fequalityD2,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( fset @ A ) @ B5 @ A3 ) ) ).
% fequalityD2
thf(fact_166_fequalityD1,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 ) ) ).
% fequalityD1
thf(fact_167_fequalityE,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ~ ( ord_less_eq @ ( fset @ A ) @ B5 @ A3 ) ) ) ).
% fequalityE
thf(fact_168_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_169_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 )
=> ? [T3: abstra2103299360e_tree @ A,Steps3: stream @ A,T4: abstra2103299360e_tree @ A] :
( ( X2 = T3 )
& ( Xa2 = Steps3 )
& ( ( abstra573067619e_root @ A @ T3 )
= ( shd @ A @ Steps3 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T4 @ ( abstra1749095923e_cont @ A @ T3 ) )
& ( ( X6 @ T4 @ ( stl @ A @ Steps3 ) )
| ( abstra313004635_ipath @ A @ T4 @ ( stl @ A @ Steps3 ) ) ) ) )
=> ( abstra313004635_ipath @ A @ X4 @ Xa ) ) ) ).
% ipath.coinduct
thf(fact_170_ipath_Ointros,axiom,
! [A: $tType,T: abstra2103299360e_tree @ A,Steps: stream @ A,T2: abstra2103299360e_tree @ A] :
( ( ( abstra573067619e_root @ A @ T )
= ( shd @ A @ Steps ) )
=> ( ( fmember @ ( abstra2103299360e_tree @ A ) @ T2 @ ( abstra1749095923e_cont @ A @ T ) )
=> ( ( abstra313004635_ipath @ A @ T2 @ ( stl @ A @ Steps ) )
=> ( abstra313004635_ipath @ A @ T @ Steps ) ) ) ) ).
% ipath.intros
thf(fact_171_ipath_Osimps,axiom,
! [A: $tType] :
( ( abstra313004635_ipath @ A )
= ( ^ [A1: abstra2103299360e_tree @ A,A22: stream @ A] :
? [T5: abstra2103299360e_tree @ A,Steps2: stream @ A,T6: abstra2103299360e_tree @ A] :
( ( A1 = T5 )
& ( A22 = Steps2 )
& ( ( abstra573067619e_root @ A @ T5 )
= ( shd @ A @ Steps2 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T6 @ ( abstra1749095923e_cont @ A @ T5 ) )
& ( abstra313004635_ipath @ A @ T6 @ ( stl @ A @ Steps2 ) ) ) ) ) ).
% ipath.simps
thf(fact_172_ipath_Ocases,axiom,
! [A: $tType,A12: abstra2103299360e_tree @ A,A23: stream @ A] :
( ( abstra313004635_ipath @ A @ A12 @ A23 )
=> ~ ( ( ( abstra573067619e_root @ A @ A12 )
= ( shd @ A @ A23 ) )
=> ! [T7: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ A12 ) )
=> ~ ( abstra313004635_ipath @ A @ T7 @ ( stl @ A @ A23 ) ) ) ) ) ).
% ipath.cases
thf(fact_173_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_174_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_175_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_176_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_177_fset__mp,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A,X4: A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ( ( fmember @ A @ X4 @ A3 )
=> ( fmember @ A @ X4 @ B5 ) ) ) ).
% fset_mp
thf(fact_178_fin__mono,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A,X4: A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ( ( fmember @ A @ X4 @ A3 )
=> ( fmember @ A @ X4 @ B5 ) ) ) ).
% fin_mono
thf(fact_179_fsubsetD,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A,C2: A] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ( ( fmember @ A @ C2 @ A3 )
=> ( fmember @ A @ C2 @ B5 ) ) ) ).
% fsubsetD
thf(fact_180_fset__rev__mp,axiom,
! [A: $tType,X4: A,A3: fset @ A,B5: fset @ A] :
( ( fmember @ A @ X4 @ A3 )
=> ( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ( fmember @ A @ X4 @ B5 ) ) ) ).
% fset_rev_mp
thf(fact_181_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_182_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_183_if__split__fmem1,axiom,
! [A: $tType,Q: $o,X4: A,Y: A,B2: fset @ A] :
( ( fmember @ A @ ( if @ A @ Q @ X4 @ Y ) @ B2 )
= ( ( Q
=> ( fmember @ A @ X4 @ B2 ) )
& ( ~ Q
=> ( fmember @ A @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_184_eqfset__imp__iff,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A,X4: A] :
( ( A3 = B5 )
=> ( ( fmember @ A @ X4 @ A3 )
= ( fmember @ A @ X4 @ B5 ) ) ) ).
% eqfset_imp_iff
thf(fact_185_eq__fmem__trans,axiom,
! [A: $tType,A2: A,B2: A,A3: fset @ A] :
( ( A2 = B2 )
=> ( ( fmember @ A @ B2 @ A3 )
=> ( fmember @ A @ A2 @ A3 ) ) ) ).
% eq_fmem_trans
thf(fact_186_fset__choice,axiom,
! [B: $tType,A: $tType,A3: fset @ A,P2: A > B > $o] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
=> ? [X12: B] : ( P2 @ X2 @ X12 ) )
=> ? [F3: A > B] :
! [X5: A] :
( ( fmember @ A @ X5 @ A3 )
=> ( P2 @ X5 @ ( F3 @ X5 ) ) ) ) ).
% fset_choice
thf(fact_187_fequalityCE,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A,C2: A] :
( ( A3 = B5 )
=> ( ( ( fmember @ A @ C2 @ A3 )
=> ~ ( fmember @ A @ C2 @ B5 ) )
=> ~ ( ~ ( fmember @ A @ C2 @ A3 )
=> ( fmember @ A @ C2 @ B5 ) ) ) ) ).
% fequalityCE
thf(fact_188_fset__eqI,axiom,
! [A: $tType,A3: fset @ A,B5: fset @ A] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
= ( fmember @ A @ X2 @ B5 ) )
=> ( A3 = B5 ) ) ).
% fset_eqI
thf(fact_189_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $false ) ) ).
% empty_def
thf(fact_190_RuleSystem__Defs_Owf__ipath__epath,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,T: abstra2103299360e_tree @ ( product_prod @ State @ Rule ),Steps: stream @ ( product_prod @ State @ Rule )] :
( ( abstra1874736267tem_wf @ Rule @ State @ Eff2 @ Rules @ T )
=> ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ T @ Steps )
=> ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules @ Steps ) ) ) ).
% RuleSystem_Defs.wf_ipath_epath
thf(fact_191_epath_Osimps,axiom,
! [A2: stream @ ( product_prod @ state @ rule )] :
( ( abstra523868654_epath @ rule @ state @ eff @ rules @ A2 )
= ( ? [Steps2: stream @ ( product_prod @ state @ rule ),Sl2: fset @ state] :
( ( A2 = Steps2 )
& ( 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 ) ) ) @ Sl2 )
& ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps2 ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps2 ) ) @ Sl2 )
& ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps2 ) ) ) ) ) ).
% epath.simps
thf(fact_192_epath_Ointros,axiom,
! [Steps: stream @ ( product_prod @ state @ rule ),Sl: fset @ state] :
( ( 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 ) ) ) @ Sl )
=> ( ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps ) ) @ Sl )
=> ( ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps ) )
=> ( abstra523868654_epath @ rule @ state @ eff @ rules @ Steps ) ) ) ) ) ).
% epath.intros
thf(fact_193_epath_Ocoinduct,axiom,
! [X6: ( stream @ ( product_prod @ state @ rule ) ) > $o,X4: stream @ ( product_prod @ state @ rule )] :
( ( X6 @ X4 )
=> ( ! [X2: stream @ ( product_prod @ state @ rule )] :
( ( X6 @ X2 )
=> ? [Steps3: stream @ ( product_prod @ state @ rule ),Sl5: fset @ state] :
( ( X2 = 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 ) ) ) @ Sl5 )
& ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps3 ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ Steps3 ) ) @ Sl5 )
& ( ( X6 @ ( stl @ ( product_prod @ state @ rule ) @ Steps3 ) )
| ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ Steps3 ) ) ) ) )
=> ( abstra523868654_epath @ rule @ state @ eff @ rules @ X4 ) ) ) ).
% epath.coinduct
thf(fact_194_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_195_subsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% subsetI
thf(fact_196_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 ) )
=> ! [Sl3: fset @ state] :
( ( fmember @ state @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ ( stl @ ( product_prod @ state @ rule ) @ A2 ) ) ) @ Sl3 )
=> ( ( eff @ ( product_snd @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ A2 ) ) @ ( product_fst @ state @ rule @ ( shd @ ( product_prod @ state @ rule ) @ A2 ) ) @ Sl3 )
=> ~ ( abstra523868654_epath @ rule @ state @ eff @ rules @ ( stl @ ( product_prod @ state @ rule ) @ A2 ) ) ) ) ) ) ).
% epath.cases
thf(fact_197_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_198_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_199_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ~ ( member @ A @ C2 @ B5 )
=> ~ ( member @ A @ C2 @ A3 ) ) ) ).
% contra_subsetD
thf(fact_200_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_201_subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_202_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_203_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_204_rev__subsetD,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% rev_subsetD
thf(fact_205_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_206_set__rev__mp,axiom,
! [A: $tType,X4: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ X4 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( member @ A @ X4 @ B5 ) ) ) ).
% set_rev_mp
thf(fact_207_equalityD2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_208_equalityD1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_209_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_210_equalityE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_211_subsetCE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetCE
thf(fact_212_subsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_213_in__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_214_set__mp,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B5 ) ) ) ).
% set_mp
thf(fact_215_RuleSystem__Defs_Oepath_Ocases,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,A2: stream @ ( product_prod @ State @ Rule )] :
( ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules @ A2 )
=> ~ ( ( member @ Rule @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ A2 ) ) @ ( sset @ Rule @ Rules ) )
=> ! [Sl3: fset @ State] :
( ( fmember @ State @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ ( stl @ ( product_prod @ State @ Rule ) @ A2 ) ) ) @ Sl3 )
=> ( ( Eff2 @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ A2 ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ A2 ) ) @ Sl3 )
=> ~ ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules @ ( stl @ ( product_prod @ State @ Rule ) @ A2 ) ) ) ) ) ) ).
% RuleSystem_Defs.epath.cases
thf(fact_216_RuleSystem__Defs_Oepath_Osimps,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra523868654_epath @ Rule @ State )
= ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,A5: stream @ ( product_prod @ State @ Rule )] :
? [Steps2: stream @ ( product_prod @ State @ Rule ),Sl2: fset @ State] :
( ( A5 = Steps2 )
& ( member @ Rule @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps2 ) ) @ ( sset @ Rule @ Rules2 ) )
& ( fmember @ State @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ ( stl @ ( product_prod @ State @ Rule ) @ Steps2 ) ) ) @ Sl2 )
& ( Eff @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps2 ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps2 ) ) @ Sl2 )
& ( abstra523868654_epath @ Rule @ State @ Eff @ Rules2 @ ( stl @ ( product_prod @ State @ Rule ) @ Steps2 ) ) ) ) ) ).
% RuleSystem_Defs.epath.simps
thf(fact_217_RuleSystem__Defs_Oepath_Ointros,axiom,
! [Rule: $tType,State: $tType,Steps: stream @ ( product_prod @ State @ Rule ),Rules: stream @ Rule,Sl: fset @ State,Eff2: Rule > State > ( fset @ State ) > $o] :
( ( 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 ) ) ) @ Sl )
=> ( ( Eff2 @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps ) ) @ Sl )
=> ( ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules @ ( stl @ ( product_prod @ State @ Rule ) @ Steps ) )
=> ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules @ Steps ) ) ) ) ) ).
% RuleSystem_Defs.epath.intros
thf(fact_218_RuleSystem__Defs_Oepath_Ocoinduct,axiom,
! [Rule: $tType,State: $tType,X6: ( stream @ ( product_prod @ State @ Rule ) ) > $o,X4: stream @ ( product_prod @ State @ Rule ),Rules: stream @ Rule,Eff2: Rule > State > ( fset @ State ) > $o] :
( ( X6 @ X4 )
=> ( ! [X2: stream @ ( product_prod @ State @ Rule )] :
( ( X6 @ X2 )
=> ? [Steps3: stream @ ( product_prod @ State @ Rule ),Sl5: fset @ State] :
( ( X2 = 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 ) ) ) @ Sl5 )
& ( Eff2 @ ( product_snd @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps3 ) ) @ ( product_fst @ State @ Rule @ ( shd @ ( product_prod @ State @ Rule ) @ Steps3 ) ) @ Sl5 )
& ( ( X6 @ ( stl @ ( product_prod @ State @ Rule ) @ Steps3 ) )
| ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules @ ( stl @ ( product_prod @ State @ Rule ) @ Steps3 ) ) ) ) )
=> ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules @ X4 ) ) ) ).
% RuleSystem_Defs.epath.coinduct
thf(fact_219_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_220_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_221_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
= ( ( A2 = A7 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_222_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_223_predicate1I,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% predicate1I
thf(fact_224_fset_Omap__ident,axiom,
! [A: $tType,T: fset @ A] :
( ( fimage @ A @ A
@ ^ [X3: A] : X3
@ T )
= T ) ).
% fset.map_ident
thf(fact_225_fimage__ident,axiom,
! [A: $tType,Y6: fset @ A] :
( ( fimage @ A @ A
@ ^ [X3: A] : X3
@ Y6 )
= Y6 ) ).
% fimage_ident
thf(fact_226_fimage__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X4: B,A3: fset @ B] :
( ( B2
= ( F @ X4 ) )
=> ( ( fmember @ B @ X4 @ A3 )
=> ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_227_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_228_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_229_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_230_fimageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A3: fset @ B] :
( ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A3 ) )
=> ~ ! [X2: B] :
( ( B2
= ( F @ X2 ) )
=> ~ ( fmember @ B @ X2 @ A3 ) ) ) ).
% fimageE
thf(fact_231_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_232_fimage__cong,axiom,
! [B: $tType,A: $tType,M3: fset @ A,N4: fset @ A,F: A > B,G: A > B] :
( ( M3 = N4 )
=> ( ! [X2: A] :
( ( fmember @ A @ X2 @ N4 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( fimage @ A @ B @ F @ M3 )
= ( fimage @ A @ B @ G @ N4 ) ) ) ) ).
% fimage_cong
thf(fact_233_rev__fimage__eqI,axiom,
! [B: $tType,A: $tType,X4: A,A3: fset @ A,B2: B,F: A > B] :
( ( fmember @ A @ X4 @ A3 )
=> ( ( B2
= ( F @ X4 ) )
=> ( fmember @ B @ B2 @ ( fimage @ A @ B @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_234_rev__predicate1D,axiom,
! [A: $tType,P2: A > $o,X4: A,Q: A > $o] :
( ( P2 @ X4 )
=> ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_235_subset__fimage__iff,axiom,
! [A: $tType,B: $tType,B5: fset @ A,F: B > A,A3: fset @ B] :
( ( ord_less_eq @ ( fset @ A ) @ B5 @ ( fimage @ B @ A @ F @ A3 ) )
= ( ? [AA: fset @ B] :
( ( ord_less_eq @ ( fset @ B ) @ AA @ A3 )
& ( B5
= ( fimage @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_fimage_iff
thf(fact_236_predicate1D,axiom,
! [A: $tType,P2: A > $o,Q: A > $o,X4: A] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_237_fimage__mono,axiom,
! [B: $tType,A: $tType,A3: fset @ A,B5: fset @ A,F: A > B] :
( ( ord_less_eq @ ( fset @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F @ A3 ) @ ( fimage @ A @ B @ F @ B5 ) ) ) ).
% fimage_mono
thf(fact_238_fimage__fsubsetI,axiom,
! [A: $tType,B: $tType,A3: fset @ A,F: A > B,B5: fset @ B] :
( ! [X2: A] :
( ( fmember @ A @ X2 @ A3 )
=> ( fmember @ B @ ( F @ X2 ) @ B5 ) )
=> ( ord_less_eq @ ( fset @ B ) @ ( fimage @ A @ B @ F @ A3 ) @ B5 ) ) ).
% fimage_fsubsetI
thf(fact_239_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_240_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_241_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B3: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_242_prod__induct7,axiom,
! [G3: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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 ) ) ) ) )] :
( ! [A4: A,B3: B,C4: C,D2: D,E2: E,F3: F4,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B3 @ ( 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 ) ) ) ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct7
thf(fact_243_prod__induct6,axiom,
! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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 ) ) ) )] :
( ! [A4: A,B3: B,C4: C,D2: D,E2: E,F3: F4] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B3 @ ( 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 ) ) ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct6
thf(fact_244_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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 ) ) )] :
( ! [A4: A,B3: B,C4: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct5
thf(fact_245_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B3: B,C4: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct4
thf(fact_246_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B3: B,C4: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) )
=> ( P2 @ X4 ) ) ).
% prod_induct3
thf(fact_247_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 ) ) ) ) )] :
~ ! [A4: A,B3: 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 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B3 @ ( 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_248_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 ) ) ) )] :
~ ! [A4: A,B3: 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 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B3 @ ( 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_249_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 ) ) )] :
~ ! [A4: A,B3: B,C4: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_250_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B3: B,C4: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_251_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B3: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) ) ).
% prod_cases3
thf(fact_252_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ~ ( ( A2 = A7 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_253_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P2 @ P4 ) ) ).
% prod_cases
thf(fact_254_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_255_RuleSystem__Defs_OmkTree_Osimps_I2_J,axiom,
! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rs: stream @ Rule,S: State] :
( ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) )
= ( fimage @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) ) @ ( abstra1276541928ickEff @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S ) ) ) ).
% RuleSystem_Defs.mkTree.simps(2)
%----Type constructors (28)
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___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_1,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_FSet_Ofset___Orderings_Oorder__bot_17,axiom,
! [A8: $tType] : ( order_bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Opreorder_18,axiom,
! [A8: $tType] : ( preorder @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Oorder_19,axiom,
! [A8: $tType] : ( order @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Oord_20,axiom,
! [A8: $tType] : ( ord @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Obot_21,axiom,
! [A8: $tType] : ( bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,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 (1)
thf(conj_0,conjecture,
abstra1874422341nabled @ rule @ state @ eff @ r @ sa ).
%------------------------------------------------------------------------------