TPTP Problem File: COM155^1.p
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%------------------------------------------------------------------------------
% File : COM155^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Abstract completeness 181
% 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__181.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 362 ( 180 unt; 76 typ; 0 def)
% Number of atoms : 641 ( 297 equ; 3 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3997 ( 88 ~; 6 |; 49 &;3650 @)
% ( 0 <=>; 204 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 338 ( 338 >; 0 *; 0 +; 0 <<)
% Number of symbols : 77 ( 73 usr; 9 con; 0-7 aty)
% Number of variables : 1163 ( 106 ^; 945 !; 29 ?;1163 :)
% ( 80 !>; 0 ?*; 0 @-; 3 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:53:50.994
%------------------------------------------------------------------------------
%----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 (67)
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_Groups_Ouminus,type,
uminus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[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_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_Oipath,type,
abstra313004635_ipath:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Okonig,type,
abstra1918223989_konig:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( stream @ A ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otfinite,type,
abstra668420080finite:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_ONode,type,
abstra388494275e_Node:
!>[A: $tType] : ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > ( abstra2103299360e_tree @ A ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocase__tree,type,
abstra457966479e_tree:
!>[A: $tType,B: $tType] : ( ( A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B ) > ( abstra2103299360e_tree @ A ) > B ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocont,type,
abstra1749095923e_cont:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( fset @ ( abstra2103299360e_tree @ A ) ) ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Oroot,type,
abstra573067619e_root:
!>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > A ) ).
thf(sy_c_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_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_OHLD,type,
linear1178169911on_HLD:
!>[A: $tType] : ( ( set @ A ) > ( stream @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev__at,type,
linear2074238637_ev_at:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > nat > ( 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_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_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > 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_Osfilter,type,
sfilter:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osinterleave,type,
sinterleave:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osmap2,type,
smap2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Osmerge,type,
smerge:
!>[A: $tType] : ( ( stream @ ( stream @ A ) ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osnth,type,
snth:
!>[A: $tType] : ( ( stream @ A ) > nat > A ) ).
thf(sy_c_Stream_Ostream_OSCons,type,
sCons:
!>[A: $tType] : ( A > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Ocase__stream,type,
case_stream:
!>[A: $tType,B: $tType] : ( ( A > ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Stream_Ostream_Ocorec__stream,type,
corec_stream:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( C > $o ) > ( C > ( stream @ A ) ) > ( C > C ) > C > ( 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_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_r____,type,
r: rule ).
thf(sy_v_rs,type,
rs: stream @ rule ).
thf(sy_v_rules,type,
rules: stream @ rule ).
thf(sy_v_s,type,
s2: state ).
thf(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (256)
thf(fact_0_s,axiom,
member @ state @ s2 @ s ).
% s
thf(fact_1_r,axiom,
member @ rule @ r @ ( sset @ rule @ rules ) ).
% r
thf(fact_2_e,axiom,
abstra1874422341nabled @ rule @ state @ eff @ r @ s2 ).
% e
thf(fact_3__092_060open_062_092_060And_062m_O_Afair_A_Isdrop_Am_Ars_J_092_060close_062,axiom,
! [M: nat] : ( abstra928354080m_fair @ rule @ rules @ ( sdrop @ rule @ M @ rs ) ) ).
% \<open>\<And>m. fair (sdrop m rs)\<close>
thf(fact_4_rs,axiom,
abstra928354080m_fair @ rule @ rules @ rs ).
% rs
thf(fact_5_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_6_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_7_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_8_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_9_sdrop__simps_I1_J,axiom,
! [A: $tType,N: nat,S: stream @ A] :
( ( shd @ A @ ( sdrop @ A @ N @ S ) )
= ( snth @ A @ S @ N ) ) ).
% sdrop_simps(1)
thf(fact_10_shd__sset,axiom,
! [A: $tType,A2: stream @ A] : ( member @ A @ ( shd @ A @ A2 ) @ ( sset @ A @ A2 ) ) ).
% shd_sset
thf(fact_11_sdrop__szip,axiom,
! [A: $tType,B: $tType,N: nat,S1: stream @ A,S22: stream @ B] :
( ( sdrop @ ( product_prod @ A @ B ) @ N @ ( szip @ A @ B @ S1 @ S22 ) )
= ( szip @ A @ B @ ( sdrop @ A @ N @ S1 ) @ ( sdrop @ B @ N @ S22 ) ) ) ).
% sdrop_szip
thf(fact_12_sdrop__smap2,axiom,
! [B: $tType,A: $tType,C: $tType,N: nat,F: B > C > A,S1: stream @ B,S22: stream @ C] :
( ( sdrop @ A @ N @ ( smap2 @ B @ C @ A @ F @ S1 @ S22 ) )
= ( smap2 @ B @ C @ A @ F @ ( sdrop @ B @ N @ S1 ) @ ( sdrop @ C @ N @ S22 ) ) ) ).
% sdrop_smap2
thf(fact_13_sinterleave_Osimps_I1_J,axiom,
! [A: $tType,S1: stream @ A,S22: stream @ A] :
( ( shd @ A @ ( sinterleave @ A @ S1 @ S22 ) )
= ( shd @ A @ S1 ) ) ).
% sinterleave.simps(1)
thf(fact_14_stream_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,G1: C > A,Q2: C > $o,G21: C > ( stream @ A ),G22: C > C,A2: C] :
( ( shd @ A @ ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ A2 ) )
= ( G1 @ A2 ) ) ).
% stream.corec_sel(1)
thf(fact_15_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_16_HLD__iff,axiom,
! [A: $tType] :
( ( linear1178169911on_HLD @ A )
= ( ^ [S2: set @ A,Omega: stream @ A] : ( member @ A @ ( shd @ A @ Omega ) @ S2 ) ) ) ).
% HLD_iff
thf(fact_17_smap2_Osimps_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,S1: stream @ A,S22: stream @ B] :
( ( shd @ C @ ( smap2 @ A @ B @ C @ F @ S1 @ S22 ) )
= ( F @ ( shd @ A @ S1 ) @ ( shd @ B @ S22 ) ) ) ).
% smap2.simps(1)
thf(fact_18_ev__at__imp__snth,axiom,
! [A: $tType,P2: ( stream @ A ) > $o,N: nat,Omega2: stream @ A] :
( ( linear2074238637_ev_at @ A @ P2 @ N @ Omega2 )
=> ( P2 @ ( sdrop @ A @ N @ Omega2 ) ) ) ).
% ev_at_imp_snth
thf(fact_19_RuleSystem__axioms,axiom,
abstra1326562878System @ rule @ state @ eff @ rules @ s ).
% RuleSystem_axioms
thf(fact_20_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_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_O_A_092_060lbrakk_062r_A_092_060in_062_AR_059_Aenabled_Ar_As_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [R3: rule] :
( ( member @ rule @ R3 @ ( sset @ rule @ rules ) )
=> ~ ( abstra1874422341nabled @ rule @ state @ eff @ R3 @ s2 ) ) ).
% \<open>\<And>thesis. (\<And>r. \<lbrakk>r \<in> R; enabled r s\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_22_snth__smap2,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,S1: stream @ B,S22: stream @ C,N: nat] :
( ( snth @ A @ ( smap2 @ B @ C @ A @ F @ S1 @ S22 ) @ N )
= ( F @ ( snth @ B @ S1 @ N ) @ ( snth @ C @ S22 @ N ) ) ) ).
% snth_smap2
thf(fact_23_sset__fenum,axiom,
( ( sset @ rule @ ( abstra1774373515_fenum @ rule @ rules ) )
= ( sset @ rule @ rules ) ) ).
% sset_fenum
thf(fact_24_fair__fenum,axiom,
abstra928354080m_fair @ rule @ rules @ ( abstra1774373515_fenum @ rule @ rules ) ).
% fair_fenum
thf(fact_25_pickEff,axiom,
! [R: rule,S: state] :
( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
=> ( eff @ R @ S @ ( abstra1276541928ickEff @ rule @ state @ eff @ R @ S ) ) ) ).
% pickEff
thf(fact_26_smap2__alt,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > C > A,S1: stream @ B,S22: stream @ C,S: stream @ A] :
( ( ( smap2 @ B @ C @ A @ F @ S1 @ S22 )
= S )
= ( ! [N2: nat] :
( ( F @ ( snth @ B @ S1 @ N2 ) @ ( snth @ C @ S22 @ N2 ) )
= ( snth @ A @ S @ N2 ) ) ) ) ).
% smap2_alt
thf(fact_27_snth__sset,axiom,
! [A: $tType,S: stream @ A,N: nat] : ( member @ A @ ( snth @ A @ S @ N ) @ ( sset @ A @ S ) ) ).
% snth_sset
thf(fact_28_stream_Ocorec__disc,axiom,
! [A: $tType,C: $tType] :
( ( corec_stream @ C @ A )
= ( corec_stream @ C @ A ) ) ).
% stream.corec_disc
thf(fact_29_ev__at__unique,axiom,
! [A: $tType,P2: ( stream @ A ) > $o,N: nat,Omega2: stream @ A,M: nat] :
( ( linear2074238637_ev_at @ A @ P2 @ N @ Omega2 )
=> ( ( linear2074238637_ev_at @ A @ P2 @ M @ Omega2 )
=> ( N = M ) ) ) ).
% ev_at_unique
thf(fact_30_ev__at__HLD__imp__snth,axiom,
! [A: $tType,X3: set @ A,N: nat,Omega2: stream @ A] :
( ( linear2074238637_ev_at @ A @ ( linear1178169911on_HLD @ A @ X3 ) @ N @ Omega2 )
=> ( member @ A @ ( snth @ A @ Omega2 @ N ) @ X3 ) ) ).
% ev_at_HLD_imp_snth
thf(fact_31_Saturated__def,axiom,
! [Steps: stream @ ( product_prod @ state @ rule )] :
( ( abstra1209608345urated @ rule @ state @ eff @ rules @ Steps )
= ( ! [X4: rule] :
( ( member @ rule @ X4 @ ( sset @ rule @ rules ) )
=> ( abstra726722745urated @ rule @ state @ eff @ X4 @ Steps ) ) ) ) ).
% Saturated_def
thf(fact_32_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_33_eff__S,axiom,
! [S: state,R: rule,Sl: fset @ state,S3: state] :
( ( member @ state @ S @ s )
=> ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
=> ( ( eff @ R @ S @ Sl )
=> ( ( fmember @ state @ S3 @ Sl )
=> ( member @ state @ S3 @ s ) ) ) ) ) ).
% eff_S
thf(fact_34_NE__R,axiom,
( ( sset @ rule @ rules )
!= ( bot_bot @ ( set @ rule ) ) ) ).
% NE_R
thf(fact_35_snth__sset__smerge,axiom,
! [A: $tType,Ss: stream @ ( stream @ A ),N: nat,M: nat] : ( member @ A @ ( snth @ A @ ( snth @ ( stream @ A ) @ Ss @ N ) @ M ) @ ( sset @ A @ ( smerge @ A @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_36_fair__stl,axiom,
! [Rs: stream @ rule] :
( ( abstra928354080m_fair @ rule @ rules @ Rs )
=> ( abstra928354080m_fair @ rule @ rules @ ( stl @ rule @ Rs ) ) ) ).
% fair_stl
thf(fact_37_countable__R,axiom,
countable_countable @ rule @ ( sset @ rule @ rules ) ).
% countable_R
thf(fact_38_RuleSystem__Defs_Ofair__fenum,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( abstra928354080m_fair @ Rule @ Rules @ ( abstra1774373515_fenum @ Rule @ Rules ) ) ).
% RuleSystem_Defs.fair_fenum
thf(fact_39_Stream_Osmember__def,axiom,
! [A: $tType] :
( ( smember @ A )
= ( ^ [X4: A,S2: stream @ A] : ( member @ A @ X4 @ ( sset @ A @ S2 ) ) ) ) ).
% Stream.smember_def
thf(fact_40_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_41_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_42_RuleSystem_Ointro,axiom,
! [Rule: $tType,State: $tType,S4: set @ State,Rules: stream @ Rule,Eff2: Rule > State > ( fset @ State ) > $o] :
( ! [S5: State] :
( ( member @ State @ S5 @ S4 )
=> ! [R3: Rule] :
( ( member @ Rule @ R3 @ ( sset @ Rule @ Rules ) )
=> ! [Sl2: fset @ State] :
( ( Eff2 @ R3 @ S5 @ Sl2 )
=> ! [S6: State] :
( ( fmember @ State @ S6 @ Sl2 )
=> ( member @ State @ S6 @ S4 ) ) ) ) )
=> ( ! [S5: State] :
( ( member @ State @ S5 @ S4 )
=> ? [X5: Rule] :
( ( member @ Rule @ X5 @ ( sset @ Rule @ Rules ) )
& ? [X12: fset @ State] : ( Eff2 @ X5 @ S5 @ X12 ) ) )
=> ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 ) ) ) ).
% RuleSystem.intro
thf(fact_43_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,S3: State] :
( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules @ S4 )
=> ( ( member @ State @ S @ S4 )
=> ( ( member @ Rule @ R @ ( sset @ Rule @ Rules ) )
=> ( ( Eff2 @ R @ S @ Sl )
=> ( ( fmember @ State @ S3 @ Sl )
=> ( member @ State @ S3 @ S4 ) ) ) ) ) ) ).
% RuleSystem.eff_S
thf(fact_44_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 ) )
=> ! [Sl3: fset @ State] :
( ( Eff @ R2 @ S2 @ Sl3 )
=> ! [S8: State] :
( ( fmember @ State @ S8 @ Sl3 )
=> ( member @ State @ S8 @ S7 ) ) ) ) )
& ! [S2: State] :
( ( member @ State @ S2 @ S7 )
=> ? [X4: Rule] :
( ( member @ Rule @ X4 @ ( sset @ Rule @ Rules2 ) )
& ( ^ [P: ( fset @ State ) > $o] :
? [X: fset @ State] : ( P @ X )
@ ( Eff @ X4 @ S2 ) ) ) ) ) ) ) ).
% RuleSystem_def
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ 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_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_50_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_51_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_52_stl__sset,axiom,
! [A: $tType,X6: A,A2: stream @ A] :
( ( member @ A @ X6 @ ( sset @ A @ ( stl @ A @ A2 ) ) )
=> ( member @ A @ X6 @ ( sset @ A @ A2 ) ) ) ).
% stl_sset
thf(fact_53_RuleSystem__Defs_Ocountable__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] : ( countable_countable @ Rule @ ( sset @ Rule @ Rules ) ) ).
% RuleSystem_Defs.countable_R
thf(fact_54_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_55_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_56_smap2_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,S1: stream @ A,S22: stream @ B] :
( ( stl @ C @ ( smap2 @ A @ B @ C @ F @ S1 @ S22 ) )
= ( smap2 @ A @ B @ C @ F @ ( stl @ A @ S1 ) @ ( stl @ B @ S22 ) ) ) ).
% smap2.simps(2)
thf(fact_57_RuleSystem__Defs_ONE__R,axiom,
! [Rule: $tType,Rules: stream @ Rule] :
( ( sset @ Rule @ Rules )
!= ( bot_bot @ ( set @ Rule ) ) ) ).
% RuleSystem_Defs.NE_R
thf(fact_58_stream_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,Q2: C > $o,A2: C,G1: C > A,G21: C > ( stream @ A ),G22: C > C] :
( ( ( Q2 @ A2 )
=> ( ( stl @ A @ ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) )
& ( ~ ( Q2 @ A2 )
=> ( ( stl @ A @ ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ A2 ) )
= ( corec_stream @ C @ A @ G1 @ Q2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% stream.corec_sel(2)
thf(fact_59_sinterleave_Osimps_I2_J,axiom,
! [A: $tType,S1: stream @ A,S22: stream @ A] :
( ( stl @ A @ ( sinterleave @ A @ S1 @ S22 ) )
= ( sinterleave @ A @ S22 @ ( stl @ A @ S1 ) ) ) ).
% sinterleave.simps(2)
thf(fact_60_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_61_sset__induct,axiom,
! [A: $tType,Y: A,S: stream @ A,P2: A > ( stream @ A ) > $o] :
( ( member @ A @ Y @ ( sset @ A @ S ) )
=> ( ! [S5: stream @ A] : ( P2 @ ( shd @ A @ S5 ) @ S5 )
=> ( ! [S5: stream @ A,Y2: A] :
( ( member @ A @ Y2 @ ( sset @ A @ ( stl @ A @ S5 ) ) )
=> ( ( P2 @ Y2 @ ( stl @ A @ S5 ) )
=> ( P2 @ Y2 @ S5 ) ) )
=> ( P2 @ Y @ S ) ) ) ) ).
% sset_induct
thf(fact_62_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_63_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 )] :
! [X4: Rule] :
( ( member @ Rule @ X4 @ ( sset @ Rule @ Rules2 ) )
=> ( abstra726722745urated @ Rule @ State @ Eff @ X4 @ Steps2 ) ) ) ) ).
% RuleSystem_Defs.Saturated_def
thf(fact_64_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_65_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_66_countableI__type,axiom,
! [A: $tType] :
( ( countable @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] : ( countable_countable @ A @ A3 ) ) ).
% countableI_type
thf(fact_67_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_68_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_69_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_70_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_71_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_72_ev__at__HLD__single__imp__snth,axiom,
! [A: $tType,X6: A,N: nat,Omega2: stream @ A] :
( ( linear2074238637_ev_at @ A @ ( linear1178169911on_HLD @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) @ N @ Omega2 )
=> ( ( snth @ A @ Omega2 @ N )
= X6 ) ) ).
% ev_at_HLD_single_imp_snth
thf(fact_73_insertCI,axiom,
! [A: $tType,A2: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B2 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_74_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_75_insert__absorb2,axiom,
! [A: $tType,X6: A,A3: set @ A] :
( ( insert @ A @ X6 @ ( insert @ A @ X6 @ A3 ) )
= ( insert @ A @ X6 @ A3 ) ) ).
% insert_absorb2
thf(fact_76_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_77_countable__insert,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( countable_countable @ A @ A3 )
=> ( countable_countable @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% countable_insert
thf(fact_78_countable__insert__eq,axiom,
! [A: $tType,X6: A,A3: set @ A] :
( ( countable_countable @ A @ ( insert @ A @ X6 @ A3 ) )
= ( countable_countable @ A @ A3 ) ) ).
% countable_insert_eq
thf(fact_79_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_80_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_81_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C2: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_82_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_83_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_84_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_85_insertE,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_86_insertI1,axiom,
! [A: $tType,A2: A,B2: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B2 ) ) ).
% insertI1
thf(fact_87_insertI2,axiom,
! [A: $tType,A2: A,B2: set @ A,B3: A] :
( ( member @ A @ A2 @ B2 )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_88_Set_Oset__insert,axiom,
! [A: $tType,X6: A,A3: set @ A] :
( ( member @ A @ X6 @ A3 )
=> ~ ! [B4: set @ A] :
( ( A3
= ( insert @ A @ X6 @ B4 ) )
=> ( member @ A @ X6 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_89_insert__ident,axiom,
! [A: $tType,X6: A,A3: set @ A,B2: set @ A] :
( ~ ( member @ A @ X6 @ A3 )
=> ( ~ ( member @ A @ X6 @ B2 )
=> ( ( ( insert @ A @ X6 @ A3 )
= ( insert @ A @ X6 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_90_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_91_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B3 @ B2 ) )
= ( ( ( A2 = B3 )
=> ( A3 = B2 ) )
& ( ( A2 != B3 )
=> ? [C3: set @ A] :
( ( A3
= ( insert @ A @ B3 @ C3 ) )
& ~ ( member @ A @ B3 @ C3 )
& ( B2
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_92_insert__commute,axiom,
! [A: $tType,X6: A,Y: A,A3: set @ A] :
( ( insert @ A @ X6 @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X6 @ A3 ) ) ) ).
% insert_commute
thf(fact_93_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B4: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B4 ) )
& ~ ( member @ A @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_94_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_95_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_96_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_97_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_98_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_99_fempty__iff,axiom,
! [A: $tType,C2: A] :
~ ( fmember @ A @ C2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% fempty_iff
thf(fact_100_all__not__fin__conv,axiom,
! [A: $tType,A3: fset @ A] :
( ( ! [X4: A] :
~ ( fmember @ A @ X4 @ A3 ) )
= ( A3
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% all_not_fin_conv
thf(fact_101_is__singletonI,axiom,
! [A: $tType,X6: A] : ( is_singleton @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_102_the__elem__eq,axiom,
! [A: $tType,X6: A] :
( ( the_elem @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
= X6 ) ).
% the_elem_eq
thf(fact_103_holds__eq1,axiom,
! [A: $tType,X6: A] :
( ( linear1707521579_holds @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X6 ) )
= ( linear1178169911on_HLD @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% holds_eq1
thf(fact_104_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_105_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_106_holds_Oelims_I3_J,axiom,
! [A: $tType,X6: A > $o,Xa: stream @ A] :
( ~ ( linear1707521579_holds @ A @ X6 @ Xa )
=> ~ ( X6 @ ( shd @ A @ Xa ) ) ) ).
% holds.elims(3)
thf(fact_107_holds_Oelims_I2_J,axiom,
! [A: $tType,X6: A > $o,Xa: stream @ A] :
( ( linear1707521579_holds @ A @ X6 @ Xa )
=> ( X6 @ ( shd @ A @ Xa ) ) ) ).
% holds.elims(2)
thf(fact_108_holds_Oelims_I1_J,axiom,
! [A: $tType,X6: A > $o,Xa: stream @ A,Y: $o] :
( ( ( linear1707521579_holds @ A @ X6 @ Xa )
= Y )
=> ( Y
= ( X6 @ ( shd @ A @ Xa ) ) ) ) ).
% holds.elims(1)
thf(fact_109_holds_Osimps,axiom,
! [A: $tType] :
( ( linear1707521579_holds @ A )
= ( ^ [P3: A > $o,Xs2: stream @ A] : ( P3 @ ( shd @ A @ Xs2 ) ) ) ) ).
% holds.simps
thf(fact_110_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_111_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_112_eqfelem__imp__iff,axiom,
! [A: $tType,X6: A,Y: A,A3: fset @ A] :
( ( X6 = Y )
=> ( ( fmember @ A @ X6 @ A3 )
= ( fmember @ A @ Y @ A3 ) ) ) ).
% eqfelem_imp_iff
thf(fact_113_if__split__fmem2,axiom,
! [A: $tType,A2: A,Q: $o,X6: fset @ A,Y: fset @ A] :
( ( fmember @ A @ A2 @ ( if @ ( fset @ A ) @ Q @ X6 @ Y ) )
= ( ( Q
=> ( fmember @ A @ A2 @ X6 ) )
& ( ~ Q
=> ( fmember @ A @ A2 @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_114_if__split__fmem1,axiom,
! [A: $tType,Q: $o,X6: A,Y: A,B3: fset @ A] :
( ( fmember @ A @ ( if @ A @ Q @ X6 @ Y ) @ B3 )
= ( ( Q
=> ( fmember @ A @ X6 @ B3 ) )
& ( ~ Q
=> ( fmember @ A @ Y @ B3 ) ) ) ) ).
% if_split_fmem1
thf(fact_115_eqfset__imp__iff,axiom,
! [A: $tType,A3: fset @ A,B2: fset @ A,X6: A] :
( ( A3 = B2 )
=> ( ( fmember @ A @ X6 @ A3 )
= ( fmember @ A @ X6 @ B2 ) ) ) ).
% eqfset_imp_iff
thf(fact_116_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_117_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 ) )
=> ? [F2: A > B] :
! [X5: A] :
( ( fmember @ A @ X5 @ A3 )
=> ( P2 @ X5 @ ( F2 @ X5 ) ) ) ) ).
% fset_choice
thf(fact_118_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_119_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_120_femptyE,axiom,
! [A: $tType,A2: A] :
~ ( fmember @ A @ A2 @ ( bot_bot @ ( fset @ A ) ) ) ).
% femptyE
thf(fact_121_ex__fin__conv,axiom,
! [A: $tType,A3: fset @ A] :
( ( ? [X4: A] : ( fmember @ A @ X4 @ A3 ) )
= ( A3
!= ( bot_bot @ ( fset @ A ) ) ) ) ).
% ex_fin_conv
thf(fact_122_equalsffemptyD,axiom,
! [A: $tType,A3: fset @ A,A2: A] :
( ( A3
= ( bot_bot @ ( fset @ A ) ) )
=> ~ ( fmember @ A @ A2 @ A3 ) ) ).
% equalsffemptyD
thf(fact_123_equalsffemptyI,axiom,
! [A: $tType,A3: fset @ A] :
( ! [Y2: A] :
~ ( fmember @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( fset @ A ) ) ) ) ).
% equalsffemptyI
thf(fact_124_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X2: A] :
( A3
!= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_125_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X4: A] :
( A4
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_126_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_127_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_128_not__holds__eq,axiom,
! [A: $tType,X6: A] :
( ( linear1707521579_holds @ A
@ ( uminus_uminus @ ( A > $o )
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X6 ) ) )
= ( ^ [Xs2: stream @ A] :
~ ( linear1178169911on_HLD @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) @ Xs2 ) ) ) ).
% not_holds_eq
thf(fact_129_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_130_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( case_stream @ A @ B )
= ( ^ [F3: A > ( stream @ A ) > B,Stream5: stream @ A] : ( F3 @ ( shd @ A @ Stream5 ) @ ( stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_131_snth__szip,axiom,
! [A: $tType,B: $tType,S1: stream @ A,S22: stream @ B,N: nat] :
( ( snth @ ( product_prod @ A @ B ) @ ( szip @ A @ B @ S1 @ S22 ) @ N )
= ( product_Pair @ A @ B @ ( snth @ A @ S1 @ N ) @ ( snth @ B @ S22 @ N ) ) ) ).
% snth_szip
thf(fact_132_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_133_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_134_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X4: A] : ( uminus_uminus @ B @ ( A4 @ X4 ) ) ) ) ) ).
% uminus_apply
thf(fact_135_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X4: A] : ( uminus_uminus @ B @ ( A4 @ X4 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_136_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_137_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_138_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X6: A,Y: A] :
( ( ( uminus_uminus @ A @ X6 )
= ( uminus_uminus @ A @ Y ) )
= ( X6 = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_139_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X6: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X6 ) )
= X6 ) ) ).
% double_compl
thf(fact_140_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B3 ) )
= ( A2 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_141_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_142_fimage__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X6: B,A3: fset @ B] :
( ( B3
= ( F @ X6 ) )
=> ( ( fmember @ B @ X6 @ A3 )
=> ( fmember @ A @ B3 @ ( fimage @ B @ A @ F @ A3 ) ) ) ) ).
% fimage_eqI
thf(fact_143_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_144_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_145_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_146_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_147_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_148_fimageI,axiom,
! [B: $tType,A: $tType,X6: A,A3: fset @ A,F: A > B] :
( ( fmember @ A @ X6 @ A3 )
=> ( fmember @ B @ ( F @ X6 ) @ ( fimage @ A @ B @ F @ A3 ) ) ) ).
% fimageI
thf(fact_149_fimage__cong,axiom,
! [B: $tType,A: $tType,M2: fset @ A,N3: fset @ A,F: A > B,G: A > B] :
( ( M2 = N3 )
=> ( ! [X2: A] :
( ( fmember @ A @ X2 @ N3 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( fimage @ A @ B @ F @ M2 )
= ( fimage @ A @ B @ G @ N3 ) ) ) ) ).
% fimage_cong
thf(fact_150_rev__fimage__eqI,axiom,
! [B: $tType,A: $tType,X6: A,A3: fset @ A,B3: B,F: A > B] :
( ( fmember @ A @ X6 @ A3 )
=> ( ( B3
= ( F @ X6 ) )
=> ( fmember @ B @ B3 @ ( fimage @ A @ B @ F @ A3 ) ) ) ) ).
% rev_fimage_eqI
thf(fact_151_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_152_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ( uminus_uminus @ A @ A2 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_153_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_154_RuleSystem__Defs_OmkTree_Ocode,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1225283448mkTree @ Rule @ State )
= ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rs2: stream @ Rule,S2: State] : ( abstra388494275e_Node @ ( product_prod @ State @ Rule ) @ ( product_Pair @ State @ Rule @ S2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs2 @ S2 ) ) ) @ ( fimage @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs2 @ S2 ) ) ) @ ( abstra1276541928ickEff @ Rule @ State @ Eff @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs2 @ S2 ) ) @ S2 ) ) ) ) ) ).
% RuleSystem_Defs.mkTree.code
thf(fact_155_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_156_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)
thf(fact_157_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B3 = B5 ) ) ) ).
% old.prod.inject
thf(fact_158_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_159_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_160_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_161_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_162_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_163_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_164_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A6: A,B6: B] : ( P2 @ ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ( P2 @ P4 ) ) ).
% prod_cases
thf(fact_165_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B3: B,A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B3 )
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B3 != B5 ) ) ) ).
% Pair_inject
thf(fact_166_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B6: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) ) ).
% prod_cases3
thf(fact_167_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B6: B,C4: C,D3: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_168_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B6: B,C4: C,D3: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_169_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 ) ) ) )] :
~ ! [A6: A,B6: B,C4: C,D3: D,E2: E,F2: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_170_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
~ ! [A6: A,B6: B,C4: C,D3: D,E2: E,F2: F4,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_171_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B6: B,C4: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B6 @ C4 ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct3
thf(fact_172_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B6: B,C4: C,D3: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B6 @ ( product_Pair @ C @ D @ C4 @ D3 ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct4
thf(fact_173_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,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B6: B,C4: C,D3: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D3 @ E2 ) ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct5
thf(fact_174_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,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A6: A,B6: B,C4: C,D3: D,E2: E,F2: F4] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct6
thf(fact_175_prod__induct7,axiom,
! [G2: $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 @ G2 ) ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
( ! [A6: A,B6: B,C4: C,D3: D,E2: E,F2: F4,G3: G2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B6 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F2 @ G3 ) ) ) ) ) ) )
=> ( P2 @ X6 ) ) ).
% prod_induct7
thf(fact_176_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B6: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_177_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B6: B] : ( P2 @ ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_178_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_179_ipath_Ocases,axiom,
! [A: $tType,A1: abstra2103299360e_tree @ A,A22: stream @ A] :
( ( abstra313004635_ipath @ A @ A1 @ A22 )
=> ~ ( ( ( abstra573067619e_root @ A @ A1 )
= ( shd @ A @ A22 ) )
=> ! [T3: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T3 @ ( abstra1749095923e_cont @ A @ A1 ) )
=> ~ ( abstra313004635_ipath @ A @ T3 @ ( stl @ A @ A22 ) ) ) ) ) ).
% ipath.cases
thf(fact_180_ipath_Osimps,axiom,
! [A: $tType] :
( ( abstra313004635_ipath @ A )
= ( ^ [A12: abstra2103299360e_tree @ A,A23: stream @ A] :
? [T4: abstra2103299360e_tree @ A,Steps2: stream @ A,T5: abstra2103299360e_tree @ A] :
( ( A12 = T4 )
& ( A23 = Steps2 )
& ( ( abstra573067619e_root @ A @ T4 )
= ( shd @ A @ Steps2 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ T4 ) )
& ( abstra313004635_ipath @ A @ T5 @ ( stl @ A @ Steps2 ) ) ) ) ) ).
% ipath.simps
thf(fact_181_ipath_Ocoinduct,axiom,
! [A: $tType,X3: ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o,X6: abstra2103299360e_tree @ A,Xa: stream @ A] :
( ( X3 @ X6 @ Xa )
=> ( ! [X2: abstra2103299360e_tree @ A,Xa2: stream @ A] :
( ( X3 @ X2 @ Xa2 )
=> ? [T6: abstra2103299360e_tree @ A,Steps3: stream @ A,T7: abstra2103299360e_tree @ A] :
( ( X2 = T6 )
& ( Xa2 = Steps3 )
& ( ( abstra573067619e_root @ A @ T6 )
= ( shd @ A @ Steps3 ) )
& ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ T6 ) )
& ( ( X3 @ T7 @ ( stl @ A @ Steps3 ) )
| ( abstra313004635_ipath @ A @ T7 @ ( stl @ A @ Steps3 ) ) ) ) )
=> ( abstra313004635_ipath @ A @ X6 @ Xa ) ) ) ).
% ipath.coinduct
thf(fact_182_ipath_Ointros,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A,Steps: stream @ A,T8: abstra2103299360e_tree @ A] :
( ( ( abstra573067619e_root @ A @ T2 )
= ( shd @ A @ Steps ) )
=> ( ( fmember @ ( abstra2103299360e_tree @ A ) @ T8 @ ( abstra1749095923e_cont @ A @ T2 ) )
=> ( ( abstra313004635_ipath @ A @ T8 @ ( stl @ A @ Steps ) )
=> ( abstra313004635_ipath @ A @ T2 @ Steps ) ) ) ) ).
% ipath.intros
thf(fact_183_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_184_tree_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree: abstra2103299360e_tree @ A] :
( ( P2 @ ( abstra457966479e_tree @ A @ B @ F @ Tree ) )
= ( ~ ( ( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) )
& ~ ( P2 @ ( F @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ) ) ) ).
% tree.split_sel_asm
thf(fact_185_tree_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree: abstra2103299360e_tree @ A] :
( ( P2 @ ( abstra457966479e_tree @ A @ B @ F @ Tree ) )
= ( ( Tree
= ( abstra388494275e_Node @ A @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) )
=> ( P2 @ ( F @ ( abstra573067619e_root @ A @ Tree ) @ ( abstra1749095923e_cont @ A @ Tree ) ) ) ) ) ).
% tree.split_sel
thf(fact_186_fset_Omap__ident,axiom,
! [A: $tType,T2: fset @ A] :
( ( fimage @ A @ A
@ ^ [X4: A] : X4
@ T2 )
= T2 ) ).
% fset.map_ident
thf(fact_187_fimage__ident,axiom,
! [A: $tType,Y4: fset @ A] :
( ( fimage @ A @ A
@ ^ [X4: A] : X4
@ Y4 )
= Y4 ) ).
% fimage_ident
thf(fact_188_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_189_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_190_holds__eq2,axiom,
! [A: $tType,X6: A] :
( ( linear1707521579_holds @ A
@ ^ [Y5: A] : Y5 = X6 )
= ( linear1178169911on_HLD @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% holds_eq2
thf(fact_191_cont__def,axiom,
! [A: $tType] :
( ( abstra1749095923e_cont @ A )
= ( abstra457966479e_tree @ A @ ( fset @ ( abstra2103299360e_tree @ A ) )
@ ^ [X14: A,X24: fset @ ( abstra2103299360e_tree @ A )] : X24 ) ) ).
% cont_def
thf(fact_192_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R4 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S4 ) ) )
= ( R4 = S4 ) ) ).
% pred_equals_eq2
thf(fact_193_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_194_tree_Oroot__def,axiom,
! [A: $tType] :
( ( abstra573067619e_root @ A )
= ( abstra457966479e_tree @ A @ A
@ ^ [X14: A,X24: fset @ ( abstra2103299360e_tree @ A )] : X14 ) ) ).
% tree.root_def
thf(fact_195_not__HLD,axiom,
! [A: $tType,X3: set @ A] :
( ( ^ [Xs2: stream @ A] :
~ ( linear1178169911on_HLD @ A @ X3 @ Xs2 ) )
= ( linear1178169911on_HLD @ A @ ( uminus_uminus @ ( set @ A ) @ X3 ) ) ) ).
% not_HLD
thf(fact_196_shd__def,axiom,
! [A: $tType] :
( ( shd @ A )
= ( case_stream @ A @ A
@ ^ [X14: A,X24: stream @ A] : X14 ) ) ).
% shd_def
thf(fact_197_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A4: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) ) ) ) ) ).
% uminus_set_def
thf(fact_198_Collect__conv__if2,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P2 @ X4 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P2 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_199_Collect__conv__if,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P2 @ X4 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P2 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_200_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_201_HLD__def,axiom,
! [A: $tType] :
( ( linear1178169911on_HLD @ A )
= ( ^ [S2: set @ A] :
( linear1707521579_holds @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S2 ) ) ) ) ).
% HLD_def
thf(fact_202_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_203_tree_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree: abstra2103299360e_tree @ A] :
( ( H @ ( abstra457966479e_tree @ A @ B @ F @ Tree ) )
= ( abstra457966479e_tree @ A @ C
@ ^ [X14: A,X24: fset @ ( abstra2103299360e_tree @ A )] : ( H @ ( F @ X14 @ X24 ) )
@ Tree ) ) ).
% tree.case_distrib
thf(fact_204_insert__Collect,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P2 ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A2 )
=> ( P2 @ U ) ) ) ) ).
% insert_Collect
thf(fact_205_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A7: A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A7 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_206_countable__Collect,axiom,
! [A: $tType,A3: set @ A,Phi: A > $o] :
( ( countable_countable @ A @ A3 )
=> ( countable_countable @ A
@ ( collect @ A
@ ^ [A7: A] :
( ( member @ A @ A7 @ A3 )
& ( Phi @ A7 ) ) ) ) ) ).
% countable_Collect
thf(fact_207_stl__def,axiom,
! [A: $tType] :
( ( stl @ A )
= ( case_stream @ A @ ( stream @ A )
@ ^ [X14: A,X24: stream @ A] : X24 ) ) ).
% stl_def
thf(fact_208_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_209_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
@ ^ [X4: C] : ( F @ ( G @ X4 ) )
@ A3 ) ) ).
% fimage_fimage
thf(fact_210_tree_Ocase,axiom,
! [B: $tType,A: $tType,F: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,X13: A,X22: fset @ ( abstra2103299360e_tree @ A )] :
( ( abstra457966479e_tree @ A @ B @ F @ ( abstra388494275e_Node @ A @ X13 @ X22 ) )
= ( F @ X13 @ X22 ) ) ).
% tree.case
thf(fact_211_RuleSystem__Defs_OmkTree__unfold,axiom,
! [State: $tType,Rule: $tType] :
( ( abstra1225283448mkTree @ Rule @ State )
= ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rs2: stream @ Rule,S2: 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 @ S2 @ R2 ) @ ( fimage @ State @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff @ S8 ) @ ( abstra1276541928ickEff @ Rule @ State @ Eff @ R2 @ S2 ) ) )
@ ( abstra1259602206m_trim @ Rule @ State @ Eff @ Rs2 @ S2 ) ) ) ) ).
% RuleSystem_Defs.mkTree_unfold
thf(fact_212_tree_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( abstra457966479e_tree @ A @ B )
= ( ^ [F3: A > ( fset @ ( abstra2103299360e_tree @ A ) ) > B,Tree3: abstra2103299360e_tree @ A] : ( F3 @ ( abstra573067619e_root @ A @ Tree3 ) @ ( abstra1749095923e_cont @ A @ Tree3 ) ) ) ) ).
% tree.case_eq_if
thf(fact_213_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_214_minWait__def,axiom,
! [Rs: stream @ rule,S: state] :
( ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S )
= ( ord_Least @ nat
@ ^ [N2: nat] : ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N2 @ Rs ) ) @ S ) ) ) ).
% minWait_def
thf(fact_215_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
@ ^ [N2: nat] : ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N2 @ Rs ) ) @ S ) ) ) ) ).
% RuleSystem.minWait_def
thf(fact_216_ComplI,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_217_Compl__iff,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_218_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A3 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_219_Collect__neg__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P2 @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P2 ) ) ) ).
% Collect_neg_eq
thf(fact_220_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A4: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A4 ) ) ) ) ).
% Compl_eq
thf(fact_221_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,K: A] :
( ( P2 @ K )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% LeastI
thf(fact_222_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_223_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_224_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_225_ComplD,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C2 @ A3 ) ) ).
% ComplD
thf(fact_226_double__complement,axiom,
! [A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_227_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_228_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_229_Konig,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A] :
( ~ ( abstra668420080finite @ A @ T2 )
=> ( abstra313004635_ipath @ A @ T2 @ ( abstra1918223989_konig @ A @ T2 ) ) ) ).
% Konig
thf(fact_230_sdrop__while__sdrop__LEAST,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ? [N4: nat] : ( P2 @ ( snth @ A @ S @ N4 ) )
=> ( ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S )
= ( sdrop @ A
@ ( ord_Least @ nat
@ ^ [N2: nat] : ( P2 @ ( snth @ A @ S @ N2 ) ) )
@ S ) ) ) ).
% sdrop_while_sdrop_LEAST
thf(fact_231_fset_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: fset @ A] :
( ( fimage @ B @ C @ G @ ( fimage @ A @ B @ F @ V ) )
= ( fimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fset.map_comp
thf(fact_232_ftree__no__ipath,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A,Steps: stream @ A] :
( ( abstra668420080finite @ A @ T2 )
=> ~ ( abstra313004635_ipath @ A @ T2 @ Steps ) ) ).
% ftree_no_ipath
thf(fact_233_tfinite_Ocases,axiom,
! [A: $tType,A2: abstra2103299360e_tree @ A] :
( ( abstra668420080finite @ A @ A2 )
=> ! [T7: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ A2 ) )
=> ( abstra668420080finite @ A @ T7 ) ) ) ).
% tfinite.cases
thf(fact_234_tfinite_Osimps,axiom,
! [A: $tType] :
( ( abstra668420080finite @ A )
= ( ^ [A7: abstra2103299360e_tree @ A] :
? [T4: abstra2103299360e_tree @ A] :
( ( A7 = T4 )
& ! [X4: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ X4 @ ( abstra1749095923e_cont @ A @ T4 ) )
=> ( abstra668420080finite @ A @ X4 ) ) ) ) ) ).
% tfinite.simps
thf(fact_235_tfinite_Oinducts,axiom,
! [A: $tType,X6: abstra2103299360e_tree @ A,P2: ( abstra2103299360e_tree @ A ) > $o] :
( ( abstra668420080finite @ A @ X6 )
=> ( ! [T9: abstra2103299360e_tree @ A] :
( ! [T7: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ T9 ) )
=> ( abstra668420080finite @ A @ T7 ) )
=> ( ! [T7: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T7 @ ( abstra1749095923e_cont @ A @ T9 ) )
=> ( P2 @ T7 ) )
=> ( P2 @ T9 ) ) )
=> ( P2 @ X6 ) ) ) ).
% tfinite.inducts
thf(fact_236_tfinite,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A] :
( ! [T3: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T3 @ ( abstra1749095923e_cont @ A @ T2 ) )
=> ( abstra668420080finite @ A @ T3 ) )
=> ( abstra668420080finite @ A @ T2 ) ) ).
% tfinite
thf(fact_237_konig_Osimps_I2_J,axiom,
! [A: $tType,T2: abstra2103299360e_tree @ A] :
( ( stl @ A @ ( abstra1918223989_konig @ A @ T2 ) )
= ( abstra1918223989_konig @ A
@ @+[T5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ T2 ) )
& ~ ( abstra668420080finite @ A @ T5 ) ) ) ) ).
% konig.simps(2)
thf(fact_238_sfilter_Osimps_I1_J,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ( shd @ A @ ( sfilter @ A @ P2 @ S ) )
= ( shd @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) ) ).
% sfilter.simps(1)
thf(fact_239_sfilter__not__P,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ~ ( P2 @ ( shd @ A @ S ) )
=> ( ( sfilter @ A @ P2 @ S )
= ( sfilter @ A @ P2 @ ( stl @ A @ S ) ) ) ) ).
% sfilter_not_P
thf(fact_240_sfilter_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ( stl @ A @ ( sfilter @ A @ P2 @ S ) )
= ( sfilter @ A @ P2 @ ( stl @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) ) ) ).
% sfilter.simps(2)
thf(fact_241_some__in__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( member @ A
@ @+[X4: A] : ( member @ A @ X4 @ A3 )
@ A3 )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_242_konig_Ocode,axiom,
! [A: $tType] :
( ( abstra1918223989_konig @ A )
= ( ^ [T4: abstra2103299360e_tree @ A] :
( sCons @ A @ ( abstra573067619e_root @ A @ T4 )
@ ( abstra1918223989_konig @ A
@ @+[T5: abstra2103299360e_tree @ A] :
( ( fmember @ ( abstra2103299360e_tree @ A ) @ T5 @ ( abstra1749095923e_cont @ A @ T4 ) )
& ~ ( abstra668420080finite @ A @ T5 ) ) ) ) ) ) ).
% konig.code
thf(fact_243_stream_Oinject,axiom,
! [A: $tType,X13: A,X22: stream @ A,Y1: A,Y22: stream @ A] :
( ( ( sCons @ A @ X13 @ X22 )
= ( sCons @ A @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X22 = Y22 ) ) ) ).
% stream.inject
thf(fact_244_HLD__Stream,axiom,
! [A: $tType,X3: set @ A,X6: A,Omega2: stream @ A] :
( ( linear1178169911on_HLD @ A @ X3 @ ( sCons @ A @ X6 @ Omega2 ) )
= ( member @ A @ X6 @ X3 ) ) ).
% HLD_Stream
thf(fact_245_smember__code,axiom,
! [A: $tType,X6: A,Y: A,S: stream @ A] :
( ( smember @ A @ X6 @ ( sCons @ A @ Y @ S ) )
= ( ( X6 != Y )
=> ( smember @ A @ X6 @ S ) ) ) ).
% smember_code
thf(fact_246_stream_Oset,axiom,
! [A: $tType,X13: A,X22: stream @ A] :
( ( sset @ A @ ( sCons @ A @ X13 @ X22 ) )
= ( insert @ A @ X13 @ ( sset @ A @ X22 ) ) ) ).
% stream.set
thf(fact_247_stream_Ocollapse,axiom,
! [A: $tType,Stream: stream @ A] :
( ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_248_sfilter__P,axiom,
! [A: $tType,P2: A > $o,S: stream @ A] :
( ( P2 @ ( shd @ A @ S ) )
=> ( ( sfilter @ A @ P2 @ S )
= ( sCons @ A @ ( shd @ A @ S ) @ ( sfilter @ A @ P2 @ ( stl @ A @ S ) ) ) ) ) ).
% sfilter_P
thf(fact_249_sfilter__Stream,axiom,
! [A: $tType,P2: A > $o,X6: A,S: stream @ A] :
( ( ( P2 @ X6 )
=> ( ( sfilter @ A @ P2 @ ( sCons @ A @ X6 @ S ) )
= ( sCons @ A @ X6 @ ( sfilter @ A @ P2 @ S ) ) ) )
& ( ~ ( P2 @ X6 )
=> ( ( sfilter @ A @ P2 @ ( sCons @ A @ X6 @ S ) )
= ( sfilter @ A @ P2 @ S ) ) ) ) ).
% sfilter_Stream
thf(fact_250_stream_Ocase,axiom,
! [B: $tType,A: $tType,F: A > ( stream @ A ) > B,X13: A,X22: stream @ A] :
( ( case_stream @ A @ B @ F @ ( sCons @ A @ X13 @ X22 ) )
= ( F @ X13 @ X22 ) ) ).
% stream.case
thf(fact_251_szip__unfold,axiom,
! [A: $tType,B: $tType,A2: A,S1: stream @ A,B3: B,S22: stream @ B] :
( ( szip @ A @ B @ ( sCons @ A @ A2 @ S1 ) @ ( sCons @ B @ B3 @ S22 ) )
= ( sCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( szip @ A @ B @ S1 @ S22 ) ) ) ).
% szip_unfold
thf(fact_252_stream_Osel_I1_J,axiom,
! [A: $tType,X13: A,X22: stream @ A] :
( ( shd @ A @ ( sCons @ A @ X13 @ X22 ) )
= X13 ) ).
% stream.sel(1)
thf(fact_253_sinterleave__code,axiom,
! [A: $tType,X6: A,S1: stream @ A,S22: stream @ A] :
( ( sinterleave @ A @ ( sCons @ A @ X6 @ S1 ) @ S22 )
= ( sCons @ A @ X6 @ ( sinterleave @ A @ S22 @ S1 ) ) ) ).
% sinterleave_code
thf(fact_254_stream_Osel_I2_J,axiom,
! [A: $tType,X13: A,X22: stream @ A] :
( ( stl @ A @ ( sCons @ A @ X13 @ X22 ) )
= X22 ) ).
% stream.sel(2)
thf(fact_255_stream_Oexhaust,axiom,
! [A: $tType,Y: stream @ A] :
~ ! [X1: A,X23: stream @ A] :
( Y
!= ( sCons @ A @ X1 @ X23 ) ) ).
% stream.exhaust
%----Type constructors (25)
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___Lattices_Oboolean__algebra,axiom,
! [A8: $tType,A9: $tType] :
( ( boolean_algebra @ A9 @ ( type2 @ A9 ) )
=> ( boolean_algebra @ ( 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_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A8: $tType,A9: $tType] :
( ( uminus @ A9 @ ( type2 @ A9 ) )
=> ( uminus @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_5,axiom,
countable @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_6,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_7,axiom,
! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_8,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( countable @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_10,axiom,
! [A8: $tType] : ( uminus @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Countable_Ocountable_12,axiom,
countable @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_13,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_14,axiom,
uminus @ $o @ ( type2 @ $o ) ).
thf(tcon_FSet_Ofset___Lattices_Oboolean__algebra_15,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( boolean_algebra @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
thf(tcon_FSet_Ofset___Countable_Ocountable_16,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( countable @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
thf(tcon_FSet_Ofset___Orderings_Obot_17,axiom,
! [A8: $tType] : ( bot @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ).
thf(tcon_FSet_Ofset___Groups_Ouminus_18,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( uminus @ ( fset @ A8 ) @ ( type2 @ ( fset @ A8 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_19,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,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X6: A,Y: A] :
( ( if @ A @ $false @ X6 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X6: A,Y: A] :
( ( if @ A @ $true @ X6 @ Y )
= X6 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [N4: nat] :
( ( ( shd @ rule @ ( sdrop @ rule @ N4 @ rs ) )
= r )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------