%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM161^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:48 AM UTC 2026
% Result : Theorem 3.29s 1.88s
% Output : Refutation 3.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : COM161^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n006.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 04:15:30 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.28/0.49 ---- Original DTF file ---
% 0.28/0.49 thf(spec,logic,$$dhol).
% 0.28/0.49 %------------------------------------------------------------------------------
% 0.28/0.49 % File : COM161^1 : TPTP v9.2.1. Released v7.0.0.
% 0.28/0.49 % Domain : Computing Theory
% 0.28/0.49 % Problem : Abstract completeness 355
% 0.28/0.49 % Version : [Bla16] axioms : Especial.
% 0.28/0.49 % English :
% 0.28/0.49
% 0.28/0.49 % Refs : [BPT14] Blanchette et al. (2014), Abstract Completeness
% 0.28/0.49 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.28/0.49 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.28/0.49 % Source : [Bla16]
% 0.28/0.49 % Names : abstract_completeness__355.p [Bla16]
% 0.28/0.49
% 0.28/0.49 % Status : Theorem
% 0.28/0.49 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.28/0.49 % Syntax : Number of formulae : 364 ( 112 unt; 61 typ; 0 def)
% 0.28/0.49 % Number of atoms : 925 ( 197 equ; 0 cnn)
% 0.28/0.49 % Maximal formula atoms : 13 ( 3 avg)
% 0.28/0.49 % Number of connectives : 4443 ( 73 ~; 11 |; 56 &;3888 @)
% 0.28/0.49 % ( 0 <=>; 415 =>; 0 <=; 0 <~>)
% 0.28/0.49 % Maximal formula depth : 26 ( 8 avg)
% 0.28/0.49 % Number of types : 4 ( 3 usr)
% 0.28/0.49 % Number of type conns : 273 ( 273 >; 0 *; 0 +; 0 <<)
% 0.28/0.49 % Number of symbols : 61 ( 58 usr; 7 con; 0-6 aty)
% 0.28/0.49 % Number of variables : 1085 ( 43 ^; 956 !; 28 ?;1085 :)
% 0.28/0.49 % ( 58 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.49 % SPC : TH1_THM_EQU_NAR
% 0.28/0.49
% 0.28/0.49 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.28/0.49 % 2016-07-13 14:54:58.683
% 0.28/0.49 %------------------------------------------------------------------------------
% 0.28/0.49 %----Could-be-implicit typings (9)
% 0.28/0.49 thf(ty_t_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree,type,
% 0.28/0.49 abstra2103299360e_tree: $tType > $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_t_Product__Type_Oprod,type,
% 0.28/0.49 product_prod: $tType > $tType > $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_t_Stream_Ostream,type,
% 0.28/0.49 stream: $tType > $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_t_FSet_Ofset,type,
% 0.28/0.49 fset: $tType > $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_t_Set_Oset,type,
% 0.28/0.49 set: $tType > $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_t_Nat_Onat,type,
% 0.28/0.49 nat: $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_tf_state,type,
% 0.28/0.49 state: $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_t_itself,type,
% 0.28/0.49 itself: $tType > $tType ).
% 0.28/0.49
% 0.28/0.49 thf(ty_tf_rule,type,
% 0.28/0.49 rule: $tType ).
% 0.28/0.49
% 0.28/0.49 %----Explicit typings (52)
% 0.28/0.49 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.28/0.49 finite_finite:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_HOL_Otype,type,
% 0.28/0.49 type:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Groups_Ominus,type,
% 0.28/0.49 minus:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Orderings_Obot,type,
% 0.28/0.49 bot:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Orderings_Oord,type,
% 0.28/0.49 ord:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Orderings_Oorder,type,
% 0.28/0.49 order:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Orderings_Olinorder,type,
% 0.28/0.49 linorder:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Orderings_Opreorder,type,
% 0.28/0.49 preorder:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Countable_Ocountable,type,
% 0.28/0.49 countable:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.28/0.49 order_bot:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.28/0.49 dense_linorder:
% 0.28/0.49 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_OPersistentRuleSystem,type,
% 0.28/0.49 abstra612932245System:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem,type,
% 0.28/0.49 abstra1326562878System:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_OminWait,type,
% 0.28/0.49 abstra1332369113inWait:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Oper,type,
% 0.28/0.49 abstra2096684367le_per:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > Rule > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Opos,type,
% 0.28/0.49 abstra2097340358le_pos:
% 0.28/0.49 !>[A: $tType] : ( ( stream @ A ) > A > nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OSaturated,type,
% 0.28/0.49 abstra1209608345urated:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oenabled,type,
% 0.28/0.49 abstra1874422341nabled:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > State > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oepath,type,
% 0.28/0.49 abstra523868654_epath:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofair,type,
% 0.28/0.49 abstra928354080m_fair:
% 0.28/0.49 !>[Rule: $tType] : ( ( stream @ Rule ) > ( stream @ Rule ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofenum,type,
% 0.28/0.49 abstra1774373515_fenum:
% 0.28/0.49 !>[Rule: $tType] : ( ( stream @ Rule ) > ( stream @ Rule ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OmkTree,type,
% 0.28/0.49 abstra1225283448mkTree:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OpickEff,type,
% 0.28/0.49 abstra1276541928ickEff:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > State > ( fset @ State ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Osaturated,type,
% 0.28/0.49 abstra726722745urated:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Otrim,type,
% 0.28/0.49 abstra1259602206m_trim:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > ( stream @ Rule ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Owf,type,
% 0.28/0.49 abstra1874736267tem_wf:
% 0.28/0.49 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Oipath,type,
% 0.28/0.49 abstra313004635_ipath:
% 0.28/0.49 !>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocont,type,
% 0.28/0.49 abstra1749095923e_cont:
% 0.28/0.49 !>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( fset @ ( abstra2103299360e_tree @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Countable__Set_Ocountable,type,
% 0.28/0.49 countable_countable:
% 0.28/0.49 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_FSet_Ofimage,type,
% 0.28/0.49 fimage:
% 0.28/0.49 !>[A: $tType,B: $tType] : ( ( A > B ) > ( fset @ A ) > ( fset @ B ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_FSet_Ofmember,type,
% 0.28/0.49 fmember:
% 0.28/0.49 !>[A: $tType] : ( A > ( fset @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.28/0.49 minus_minus:
% 0.28/0.49 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_If,type,
% 0.28/0.49 if:
% 0.28/0.49 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Nat_OSuc,type,
% 0.28/0.49 suc: nat > nat ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.28/0.49 bot_bot:
% 0.28/0.49 !>[A: $tType] : A ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/0.49 ord_less:
% 0.28/0.49 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.49 ord_less_eq:
% 0.28/0.49 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Pure_Otype,type,
% 0.28/0.49 type2:
% 0.28/0.49 !>[A: $tType] : ( itself @ A ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Set_OCollect,type,
% 0.28/0.49 collect:
% 0.28/0.49 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Set_Oinsert,type,
% 0.28/0.49 insert:
% 0.28/0.49 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Set_Ois__singleton,type,
% 0.28/0.49 is_singleton:
% 0.28/0.49 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Stream_Osdrop,type,
% 0.28/0.49 sdrop:
% 0.28/0.49 !>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Stream_Ostream_Oshd,type,
% 0.28/0.49 shd:
% 0.28/0.49 !>[A: $tType] : ( ( stream @ A ) > A ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Stream_Ostream_Osset,type,
% 0.28/0.49 sset:
% 0.28/0.49 !>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_Stream_Ostream_Ostl,type,
% 0.28/0.49 stl:
% 0.28/0.49 !>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_c_member,type,
% 0.28/0.49 member:
% 0.28/0.49 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(sy_v_S,type,
% 0.28/0.49 s: set @ state ).
% 0.28/0.49
% 0.28/0.49 thf(sy_v_eff,type,
% 0.28/0.49 eff: rule > state > ( fset @ state ) > $o ).
% 0.28/0.49
% 0.28/0.49 thf(sy_v_rs,type,
% 0.28/0.49 rs: stream @ rule ).
% 0.28/0.49
% 0.28/0.49 thf(sy_v_rules,type,
% 0.28/0.49 rules: stream @ rule ).
% 0.28/0.49
% 0.28/0.49 thf(sy_v_s,type,
% 0.28/0.49 s2: state ).
% 0.28/0.49
% 0.28/0.49 thf(sy_v_steps,type,
% 0.28/0.49 steps: stream @ ( product_prod @ state @ rule ) ).
% 0.28/0.49
% 0.28/0.49 %----Relevant facts (256)
% 0.28/0.49 thf(fact_0__092_060open_062_092_060And_062ra_O_Ara_A_092_060in_062_AR_A_092_060Longrightarrow_062_Asaturated_Ara_Asteps_092_060close_062,axiom,
% 0.28/0.49 ! [R: rule] :
% 0.28/0.49 ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( abstra726722745urated @ rule @ state @ eff @ R @ steps ) ) ).
% 0.28/0.49
% 0.28/0.49 % \<open>\<And>ra. ra \<in> R \<Longrightarrow> saturated ra steps\<close>
% 0.28/0.49 thf(fact_1_Saturated__def,axiom,
% 0.28/0.49 ! [Steps: stream @ ( product_prod @ state @ rule )] :
% 0.28/0.49 ( ( abstra1209608345urated @ rule @ state @ eff @ rules @ Steps )
% 0.28/0.49 = ( ! [X: rule] :
% 0.28/0.49 ( ( member @ rule @ X @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( abstra726722745urated @ rule @ state @ eff @ X @ Steps ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Saturated_def
% 0.28/0.49 thf(fact_2_sset__fenum,axiom,
% 0.28/0.49 ( ( sset @ rule @ ( abstra1774373515_fenum @ rule @ rules ) )
% 0.28/0.49 = ( sset @ rule @ rules ) ) ).
% 0.28/0.49
% 0.28/0.49 % sset_fenum
% 0.28/0.49 thf(fact_3_assms_I2_J,axiom,
% 0.28/0.49 abstra928354080m_fair @ rule @ rules @ rs ).
% 0.28/0.49
% 0.28/0.49 % assms(2)
% 0.28/0.49 thf(fact_4_RuleSystem__Defs_OSaturated__def,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType] :
% 0.28/0.49 ( ( abstra1209608345urated @ Rule @ State )
% 0.28/0.49 = ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,Steps2: stream @ ( product_prod @ State @ Rule )] :
% 0.28/0.49 ! [X: Rule] :
% 0.28/0.49 ( ( member @ Rule @ X @ ( sset @ Rule @ Rules ) )
% 0.28/0.49 => ( abstra726722745urated @ Rule @ State @ Eff @ X @ Steps2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.Saturated_def
% 0.28/0.49 thf(fact_5_enabled__R,axiom,
% 0.28/0.49 ! [S: state] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ? [X2: rule] :
% 0.28/0.49 ( ( member @ rule @ X2 @ ( sset @ rule @ rules ) )
% 0.28/0.49 & ? [X1: fset @ state] : ( eff @ X2 @ S @ X1 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % enabled_R
% 0.28/0.49 thf(fact_6_NE__R,axiom,
% 0.28/0.49 ( ( sset @ rule @ rules )
% 0.28/0.49 != ( bot_bot @ ( set @ rule ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % NE_R
% 0.28/0.49 thf(fact_7_PersistentRuleSystem__axioms,axiom,
% 0.28/0.49 abstra612932245System @ rule @ state @ eff @ rules @ s ).
% 0.28/0.49
% 0.28/0.49 % PersistentRuleSystem_axioms
% 0.28/0.49 thf(fact_8_enabled__def,axiom,
% 0.28/0.49 ! [R: rule,S: state] :
% 0.28/0.49 ( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
% 0.28/0.49 = ( ^ [P: ( fset @ state ) > $o] :
% 0.28/0.49 ? [X3: fset @ state] : ( P @ X3 )
% 0.28/0.49 @ ( eff @ R @ S ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % enabled_def
% 0.28/0.49 thf(fact_9_countable__R,axiom,
% 0.28/0.49 countable_countable @ rule @ ( sset @ rule @ rules ) ).
% 0.28/0.49
% 0.28/0.49 % countable_R
% 0.28/0.49 thf(fact_10_per,axiom,
% 0.28/0.49 ! [R: rule] :
% 0.28/0.49 ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( abstra2096684367le_per @ rule @ state @ eff @ rules @ s @ R ) ) ).
% 0.28/0.49
% 0.28/0.49 % per
% 0.28/0.49 thf(fact_11_assms_I3_J,axiom,
% 0.28/0.49 abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rs @ s2 ) @ steps ).
% 0.28/0.49
% 0.28/0.49 % assms(3)
% 0.28/0.49 thf(fact_12_RuleSystem__Defs_Osset__fenum,axiom,
% 0.28/0.49 ! [Rule: $tType,Rules2: stream @ Rule] :
% 0.28/0.49 ( ( sset @ Rule @ ( abstra1774373515_fenum @ Rule @ Rules2 ) )
% 0.28/0.49 = ( sset @ Rule @ Rules2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.sset_fenum
% 0.28/0.49 thf(fact_13_eff__S,axiom,
% 0.28/0.49 ! [S: state,R: rule,Sl: fset @ state,S2: state] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( ( eff @ R @ S @ Sl )
% 0.28/0.49 => ( ( fmember @ state @ S2 @ Sl )
% 0.28/0.49 => ( member @ state @ S2 @ s ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % eff_S
% 0.28/0.49 thf(fact_14_RuleSystem__axioms,axiom,
% 0.28/0.49 abstra1326562878System @ rule @ state @ eff @ rules @ s ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_axioms
% 0.28/0.49 thf(fact_15_assms_I1_J,axiom,
% 0.28/0.49 member @ state @ s2 @ s ).
% 0.28/0.49
% 0.28/0.49 % assms(1)
% 0.28/0.49 thf(fact_16_fair__fenum,axiom,
% 0.28/0.49 abstra928354080m_fair @ rule @ rules @ ( abstra1774373515_fenum @ rule @ rules ) ).
% 0.28/0.49
% 0.28/0.49 % fair_fenum
% 0.28/0.49 thf(fact_17_ipath__mkTree__saturated,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule,Steps: stream @ ( product_prod @ state @ rule ),R: rule] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) @ Steps )
% 0.28/0.49 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( abstra726722745urated @ rule @ state @ eff @ R @ Steps ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ipath_mkTree_saturated
% 0.28/0.49 thf(fact_18_pickEff,axiom,
% 0.28/0.49 ! [R: rule,S: state] :
% 0.28/0.49 ( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
% 0.28/0.49 => ( eff @ R @ S @ ( abstra1276541928ickEff @ rule @ state @ eff @ R @ S ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % pickEff
% 0.28/0.49 thf(fact_19_PersistentRuleSystem_Oaxioms_I1_J,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State] :
% 0.28/0.49 ( ( abstra612932245System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % PersistentRuleSystem.axioms(1)
% 0.28/0.49 thf(fact_20_RuleSystem__Defs_Ofair__fenum,axiom,
% 0.28/0.49 ! [Rule: $tType,Rules2: stream @ Rule] : ( abstra928354080m_fair @ Rule @ Rules2 @ ( abstra1774373515_fenum @ Rule @ Rules2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.fair_fenum
% 0.28/0.49 thf(fact_21_RuleSystem__Defs_Oenabled__def,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType] :
% 0.28/0.49 ( ( abstra1874422341nabled @ Rule @ State )
% 0.28/0.49 = ( ^ [Eff: Rule > State > ( fset @ State ) > $o,R2: Rule,S4: State] :
% 0.28/0.49 ( ^ [P: ( fset @ State ) > $o] :
% 0.28/0.49 ? [X3: fset @ State] : ( P @ X3 )
% 0.28/0.49 @ ( Eff @ R2 @ S4 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.enabled_def
% 0.28/0.49 thf(fact_22_RuleSystem_Ointro,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,S3: set @ State,Rules2: stream @ Rule,Eff2: Rule > State > ( fset @ State ) > $o] :
% 0.28/0.49 ( ! [S5: State] :
% 0.28/0.49 ( ( member @ State @ S5 @ S3 )
% 0.28/0.49 => ! [R3: Rule] :
% 0.28/0.49 ( ( member @ Rule @ R3 @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 => ! [Sl2: fset @ State] :
% 0.28/0.49 ( ( Eff2 @ R3 @ S5 @ Sl2 )
% 0.28/0.49 => ! [S6: State] :
% 0.28/0.49 ( ( fmember @ State @ S6 @ Sl2 )
% 0.28/0.49 => ( member @ State @ S6 @ S3 ) ) ) ) )
% 0.28/0.49 => ( ! [S5: State] :
% 0.28/0.49 ( ( member @ State @ S5 @ S3 )
% 0.28/0.49 => ? [X4: Rule] :
% 0.28/0.49 ( ( member @ Rule @ X4 @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 & ? [X12: fset @ State] : ( Eff2 @ X4 @ S5 @ X12 ) ) )
% 0.28/0.49 => ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.intro
% 0.28/0.49 thf(fact_23_RuleSystem_Oeff__S,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,R: Rule,Sl: fset @ State,S2: State] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 => ( ( Eff2 @ R @ S @ Sl )
% 0.28/0.49 => ( ( fmember @ State @ S2 @ Sl )
% 0.28/0.49 => ( member @ State @ S2 @ S3 ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.eff_S
% 0.28/0.49 thf(fact_24_RuleSystem__def,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State )
% 0.28/0.49 = ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S7: set @ State] :
% 0.28/0.49 ( ! [S4: State] :
% 0.28/0.49 ( ( member @ State @ S4 @ S7 )
% 0.28/0.49 => ! [R2: Rule] :
% 0.28/0.49 ( ( member @ Rule @ R2 @ ( sset @ Rule @ Rules ) )
% 0.28/0.49 => ! [Sl3: fset @ State] :
% 0.28/0.49 ( ( Eff @ R2 @ S4 @ Sl3 )
% 0.28/0.49 => ! [S8: State] :
% 0.28/0.49 ( ( fmember @ State @ S8 @ Sl3 )
% 0.28/0.49 => ( member @ State @ S8 @ S7 ) ) ) ) )
% 0.28/0.49 & ! [S4: State] :
% 0.28/0.49 ( ( member @ State @ S4 @ S7 )
% 0.28/0.49 => ? [X: Rule] :
% 0.28/0.49 ( ( member @ Rule @ X @ ( sset @ Rule @ Rules ) )
% 0.28/0.49 & ( ^ [P: ( fset @ State ) > $o] :
% 0.28/0.49 ? [X3: fset @ State] : ( P @ X3 )
% 0.28/0.49 @ ( Eff @ X @ S4 ) ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_def
% 0.28/0.49 thf(fact_25_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,Steps: stream @ ( product_prod @ State @ Rule ),R: Rule] :
% 0.28/0.49 ( ( abstra612932245System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) @ Steps )
% 0.28/0.49 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 => ( abstra726722745urated @ Rule @ State @ Eff2 @ R @ Steps ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % PersistentRuleSystem.ipath_mkTree_saturated
% 0.28/0.49 thf(fact_26_RuleSystem__Defs_Ocountable__R,axiom,
% 0.28/0.49 ! [Rule: $tType,Rules2: stream @ Rule] : ( countable_countable @ Rule @ ( sset @ Rule @ Rules2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.countable_R
% 0.28/0.49 thf(fact_27_RuleSystem__Defs_ONE__R,axiom,
% 0.28/0.49 ! [Rule: $tType,Rules2: stream @ Rule] :
% 0.28/0.49 ( ( sset @ Rule @ Rules2 )
% 0.28/0.49 != ( bot_bot @ ( set @ Rule ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.NE_R
% 0.28/0.49 thf(fact_28_PersistentRuleSystem_Oper,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,R: Rule] :
% 0.28/0.49 ( ( abstra612932245System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 => ( abstra2096684367le_per @ Rule @ State @ Eff2 @ Rules2 @ S3 @ R ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % PersistentRuleSystem.per
% 0.28/0.49 thf(fact_29_RuleSystem_Oenabled__R,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ? [X2: Rule] :
% 0.28/0.49 ( ( member @ Rule @ X2 @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 & ? [X1: fset @ State] : ( Eff2 @ X2 @ S @ X1 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.enabled_R
% 0.28/0.49 thf(fact_30_wf__mkTree,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % wf_mkTree
% 0.28/0.49 thf(fact_31_trim__fair,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( abstra928354080m_fair @ rule @ rules @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % trim_fair
% 0.28/0.49 thf(fact_32_countable__empty,axiom,
% 0.28/0.49 ! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % countable_empty
% 0.28/0.49 thf(fact_33_per__def,axiom,
% 0.28/0.49 ! [R: rule] :
% 0.28/0.49 ( ( abstra2096684367le_per @ rule @ state @ eff @ rules @ s @ R )
% 0.28/0.49 = ( ! [S4: state,R1: rule,Sl4: fset @ state,S8: state] :
% 0.28/0.49 ( ( ( member @ state @ S4 @ s )
% 0.28/0.49 & ( abstra1874422341nabled @ rule @ state @ eff @ R @ S4 )
% 0.28/0.49 & ( member @ rule @ R1 @ ( minus_minus @ ( set @ rule ) @ ( sset @ rule @ rules ) @ ( insert @ rule @ R @ ( bot_bot @ ( set @ rule ) ) ) ) )
% 0.28/0.49 & ( eff @ R1 @ S4 @ Sl4 )
% 0.28/0.49 & ( fmember @ state @ S8 @ Sl4 ) )
% 0.28/0.49 => ( abstra1874422341nabled @ rule @ state @ eff @ R @ S8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % per_def
% 0.28/0.49 thf(fact_34_countableI__type,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( countable @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A2: set @ A] : ( countable_countable @ A @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % countableI_type
% 0.28/0.49 thf(fact_35_sdrop__fair,axiom,
% 0.28/0.49 ! [Rs: stream @ rule,M: nat] :
% 0.28/0.49 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( abstra928354080m_fair @ rule @ rules @ ( sdrop @ rule @ M @ Rs ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % sdrop_fair
% 0.28/0.49 thf(fact_36_empty__iff,axiom,
% 0.28/0.49 ! [A: $tType,C: A] :
% 0.28/0.49 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % empty_iff
% 0.28/0.49 thf(fact_37_all__not__in__conv,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( ! [X: A] :
% 0.28/0.49 ~ ( member @ A @ X @ A2 ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % all_not_in_conv
% 0.28/0.49 thf(fact_38_Collect__empty__eq,axiom,
% 0.28/0.49 ! [A: $tType,P2: A > $o] :
% 0.28/0.49 ( ( ( collect @ A @ P2 )
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 = ( ! [X: A] :
% 0.28/0.49 ~ ( P2 @ X ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Collect_empty_eq
% 0.28/0.49 thf(fact_39_empty__Collect__eq,axiom,
% 0.28/0.49 ! [A: $tType,P2: A > $o] :
% 0.28/0.49 ( ( ( bot_bot @ ( set @ A ) )
% 0.28/0.49 = ( collect @ A @ P2 ) )
% 0.28/0.49 = ( ! [X: A] :
% 0.28/0.49 ~ ( P2 @ X ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % empty_Collect_eq
% 0.28/0.49 thf(fact_40_insert__absorb2,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A] :
% 0.28/0.49 ( ( insert @ A @ X5 @ ( insert @ A @ X5 @ A2 ) )
% 0.28/0.49 = ( insert @ A @ X5 @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_absorb2
% 0.28/0.49 thf(fact_41_insert__iff,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 0.28/0.49 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 0.28/0.49 = ( ( A3 = B2 )
% 0.28/0.49 | ( member @ A @ A3 @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_iff
% 0.28/0.49 thf(fact_42_insertCI,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B3: set @ A,B2: A] :
% 0.28/0.49 ( ( ~ ( member @ A @ A3 @ B3 )
% 0.28/0.49 => ( A3 = B2 ) )
% 0.28/0.49 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insertCI
% 0.28/0.49 thf(fact_43_Diff__idemp,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ B3 )
% 0.28/0.49 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_idemp
% 0.28/0.49 thf(fact_44_Diff__iff,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 0.28/0.49 = ( ( member @ A @ C @ A2 )
% 0.28/0.49 & ~ ( member @ A @ C @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_iff
% 0.28/0.49 thf(fact_45_mem__Collect__eq,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,P2: A > $o] :
% 0.28/0.49 ( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
% 0.28/0.49 = ( P2 @ A3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % mem_Collect_eq
% 0.28/0.49 thf(fact_46_Collect__mem__eq,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( collect @ A
% 0.28/0.49 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 0.28/0.49 = A2 ) ).
% 0.28/0.49
% 0.28/0.49 % Collect_mem_eq
% 0.28/0.49 thf(fact_47_Collect__cong,axiom,
% 0.28/0.49 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( P2 @ X2 )
% 0.28/0.49 = ( Q @ X2 ) )
% 0.28/0.49 => ( ( collect @ A @ P2 )
% 0.28/0.49 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Collect_cong
% 0.28/0.49 thf(fact_48_ext,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( F @ X2 )
% 0.28/0.49 = ( G @ X2 ) )
% 0.28/0.49 => ( F = G ) ) ).
% 0.28/0.49
% 0.28/0.49 % ext
% 0.28/0.49 thf(fact_49_DiffI,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( member @ A @ C @ A2 )
% 0.28/0.49 => ( ~ ( member @ A @ C @ B3 )
% 0.28/0.49 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % DiffI
% 0.28/0.49 thf(fact_50_singletonI,axiom,
% 0.28/0.49 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % singletonI
% 0.28/0.49 thf(fact_51_Diff__cancel,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_cancel
% 0.28/0.49 thf(fact_52_empty__Diff,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % empty_Diff
% 0.28/0.49 thf(fact_53_Diff__empty,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 = A2 ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_empty
% 0.28/0.49 thf(fact_54_insert__Diff1,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,B3: set @ A,A2: set @ A] :
% 0.28/0.49 ( ( member @ A @ X5 @ B3 )
% 0.28/0.49 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 0.28/0.49 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_Diff1
% 0.28/0.49 thf(fact_55_Diff__insert0,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ~ ( member @ A @ X5 @ A2 )
% 0.28/0.49 => ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) )
% 0.28/0.49 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_insert0
% 0.28/0.49 thf(fact_56_countable__insert__eq,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A] :
% 0.28/0.49 ( ( countable_countable @ A @ ( insert @ A @ X5 @ A2 ) )
% 0.28/0.49 = ( countable_countable @ A @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % countable_insert_eq
% 0.28/0.49 thf(fact_57_countable__insert,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,A3: A] :
% 0.28/0.49 ( ( countable_countable @ A @ A2 )
% 0.28/0.49 => ( countable_countable @ A @ ( insert @ A @ A3 @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % countable_insert
% 0.28/0.49 thf(fact_58_countable__Diff,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( countable_countable @ A @ A2 )
% 0.28/0.49 => ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % countable_Diff
% 0.28/0.49 thf(fact_59_insert__Diff__single,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,A2: set @ A] :
% 0.28/0.49 ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.49 = ( insert @ A @ A3 @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_Diff_single
% 0.28/0.49 thf(fact_60_countable__Diff__eq,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,X5: A] :
% 0.28/0.49 ( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.49 = ( countable_countable @ A @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % countable_Diff_eq
% 0.28/0.49 thf(fact_61_Diff__insert,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,A3: A,B3: set @ A] :
% 0.28/0.49 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B3 ) )
% 0.28/0.49 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_insert
% 0.28/0.49 thf(fact_62_insert__Diff,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,A2: set @ A] :
% 0.28/0.49 ( ( member @ A @ A3 @ A2 )
% 0.28/0.49 => ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.49 = A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_Diff
% 0.28/0.49 thf(fact_63_Diff__insert2,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,A3: A,B3: set @ A] :
% 0.28/0.49 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B3 ) )
% 0.28/0.49 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_insert2
% 0.28/0.49 thf(fact_64_Diff__insert__absorb,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A] :
% 0.28/0.49 ( ~ ( member @ A @ X5 @ A2 )
% 0.28/0.49 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 = A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_insert_absorb
% 0.28/0.49 thf(fact_65_mk__disjoint__insert,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,A2: set @ A] :
% 0.28/0.49 ( ( member @ A @ A3 @ A2 )
% 0.28/0.49 => ? [B4: set @ A] :
% 0.28/0.49 ( ( A2
% 0.28/0.49 = ( insert @ A @ A3 @ B4 ) )
% 0.28/0.49 & ~ ( member @ A @ A3 @ B4 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % mk_disjoint_insert
% 0.28/0.49 thf(fact_66_insert__commute,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,Y: A,A2: set @ A] :
% 0.28/0.49 ( ( insert @ A @ X5 @ ( insert @ A @ Y @ A2 ) )
% 0.28/0.49 = ( insert @ A @ Y @ ( insert @ A @ X5 @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_commute
% 0.28/0.49 thf(fact_67_insert__Diff__if,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,B3: set @ A,A2: set @ A] :
% 0.28/0.49 ( ( ( member @ A @ X5 @ B3 )
% 0.28/0.49 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 0.28/0.49 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) )
% 0.28/0.49 & ( ~ ( member @ A @ X5 @ B3 )
% 0.28/0.49 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 0.28/0.49 = ( insert @ A @ X5 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_Diff_if
% 0.28/0.49 thf(fact_68_insert__eq__iff,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,A2: set @ A,B2: A,B3: set @ A] :
% 0.28/0.49 ( ~ ( member @ A @ A3 @ A2 )
% 0.28/0.49 => ( ~ ( member @ A @ B2 @ B3 )
% 0.28/0.49 => ( ( ( insert @ A @ A3 @ A2 )
% 0.28/0.49 = ( insert @ A @ B2 @ B3 ) )
% 0.28/0.49 = ( ( ( A3 = B2 )
% 0.28/0.49 => ( A2 = B3 ) )
% 0.28/0.49 & ( ( A3 != B2 )
% 0.28/0.49 => ? [C2: set @ A] :
% 0.28/0.49 ( ( A2
% 0.28/0.49 = ( insert @ A @ B2 @ C2 ) )
% 0.28/0.49 & ~ ( member @ A @ B2 @ C2 )
% 0.28/0.49 & ( B3
% 0.28/0.49 = ( insert @ A @ A3 @ C2 ) )
% 0.28/0.49 & ~ ( member @ A @ A3 @ C2 ) ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_eq_iff
% 0.28/0.49 thf(fact_69_insert__absorb,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,A2: set @ A] :
% 0.28/0.49 ( ( member @ A @ A3 @ A2 )
% 0.28/0.49 => ( ( insert @ A @ A3 @ A2 )
% 0.28/0.49 = A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_absorb
% 0.28/0.49 thf(fact_70_insert__ident,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ~ ( member @ A @ X5 @ A2 )
% 0.28/0.49 => ( ~ ( member @ A @ X5 @ B3 )
% 0.28/0.49 => ( ( ( insert @ A @ X5 @ A2 )
% 0.28/0.49 = ( insert @ A @ X5 @ B3 ) )
% 0.28/0.49 = ( A2 = B3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_ident
% 0.28/0.49 thf(fact_71_Set_Oset__insert,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A] :
% 0.28/0.49 ( ( member @ A @ X5 @ A2 )
% 0.28/0.49 => ~ ! [B4: set @ A] :
% 0.28/0.49 ( ( A2
% 0.28/0.49 = ( insert @ A @ X5 @ B4 ) )
% 0.28/0.49 => ( member @ A @ X5 @ B4 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Set.set_insert
% 0.28/0.49 thf(fact_72_insertI2,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B3: set @ A,B2: A] :
% 0.28/0.49 ( ( member @ A @ A3 @ B3 )
% 0.28/0.49 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insertI2
% 0.28/0.49 thf(fact_73_insertI1,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B3: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % insertI1
% 0.28/0.49 thf(fact_74_insertE,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 0.28/0.49 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 0.28/0.49 => ( ( A3 != B2 )
% 0.28/0.49 => ( member @ A @ A3 @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insertE
% 0.28/0.49 thf(fact_75_DiffD2,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 0.28/0.49 => ~ ( member @ A @ C @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % DiffD2
% 0.28/0.49 thf(fact_76_DiffD1,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 0.28/0.49 => ( member @ A @ C @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % DiffD1
% 0.28/0.49 thf(fact_77_DiffE,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 0.28/0.49 => ~ ( ( member @ A @ C @ A2 )
% 0.28/0.49 => ( member @ A @ C @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % DiffE
% 0.28/0.49 thf(fact_78_singleton__inject,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B2: A] :
% 0.28/0.49 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 => ( A3 = B2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % singleton_inject
% 0.28/0.49 thf(fact_79_insert__not__empty,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,A2: set @ A] :
% 0.28/0.49 ( ( insert @ A @ A3 @ A2 )
% 0.28/0.49 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_not_empty
% 0.28/0.49 thf(fact_80_doubleton__eq__iff,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B2: A,C: A,D: A] :
% 0.28/0.49 ( ( ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 = ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.28/0.49 = ( ( ( A3 = C )
% 0.28/0.49 & ( B2 = D ) )
% 0.28/0.49 | ( ( A3 = D )
% 0.28/0.49 & ( B2 = C ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % doubleton_eq_iff
% 0.28/0.49 thf(fact_81_singleton__iff,axiom,
% 0.28/0.49 ! [A: $tType,B2: A,A3: A] :
% 0.28/0.49 ( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 = ( B2 = A3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % singleton_iff
% 0.28/0.49 thf(fact_82_singletonD,axiom,
% 0.28/0.49 ! [A: $tType,B2: A,A3: A] :
% 0.28/0.49 ( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 => ( B2 = A3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % singletonD
% 0.28/0.49 thf(fact_83_uncountable__minus__countable,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ~ ( countable_countable @ A @ A2 )
% 0.28/0.49 => ( ( countable_countable @ A @ B3 )
% 0.28/0.49 => ~ ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % uncountable_minus_countable
% 0.28/0.49 thf(fact_84_RuleSystem__Defs_Osdrop__fair,axiom,
% 0.28/0.49 ! [Rule: $tType,Rules2: stream @ Rule,Rs: stream @ Rule,M: nat] :
% 0.28/0.49 ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( abstra928354080m_fair @ Rule @ Rules2 @ ( sdrop @ Rule @ M @ Rs ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.sdrop_fair
% 0.28/0.49 thf(fact_85_RuleSystem_Oipath__mkTree__sdrop,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,Steps: stream @ ( product_prod @ State @ Rule ),M: nat] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) @ Steps )
% 0.28/0.49 => ? [N: nat,S6: State] :
% 0.28/0.49 ( ( member @ State @ S6 @ S3 )
% 0.28/0.49 & ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( sdrop @ Rule @ N @ Rs ) @ S6 ) @ ( sdrop @ ( product_prod @ State @ Rule ) @ M @ Steps ) ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.ipath_mkTree_sdrop
% 0.28/0.49 thf(fact_86_RuleSystem__Defs_OpickEff,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,R: Rule,S: State] :
% 0.28/0.49 ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S )
% 0.28/0.49 => ( Eff2 @ R @ S @ ( abstra1276541928ickEff @ Rule @ State @ Eff2 @ R @ S ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.pickEff
% 0.28/0.49 thf(fact_87_ex__in__conv,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( ? [X: A] : ( member @ A @ X @ A2 ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ex_in_conv
% 0.28/0.49 thf(fact_88_equals0I,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ! [Y2: A] :
% 0.28/0.49 ~ ( member @ A @ Y2 @ A2 )
% 0.28/0.49 => ( A2
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % equals0I
% 0.28/0.49 thf(fact_89_equals0D,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,A3: A] :
% 0.28/0.49 ( ( A2
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % equals0D
% 0.28/0.49 thf(fact_90_emptyE,axiom,
% 0.28/0.49 ! [A: $tType,A3: A] :
% 0.28/0.49 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % emptyE
% 0.28/0.49 thf(fact_91_RuleSystem_Oper__def,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,R: Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( abstra2096684367le_per @ Rule @ State @ Eff2 @ Rules2 @ S3 @ R )
% 0.28/0.49 = ( ! [S4: State,R1: Rule,Sl4: fset @ State,S8: State] :
% 0.28/0.49 ( ( ( member @ State @ S4 @ S3 )
% 0.28/0.49 & ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S4 )
% 0.28/0.49 & ( member @ Rule @ R1 @ ( minus_minus @ ( set @ Rule ) @ ( sset @ Rule @ Rules2 ) @ ( insert @ Rule @ R @ ( bot_bot @ ( set @ Rule ) ) ) ) )
% 0.28/0.49 & ( Eff2 @ R1 @ S4 @ Sl4 )
% 0.28/0.49 & ( fmember @ State @ S8 @ Sl4 ) )
% 0.28/0.49 => ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S8 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.per_def
% 0.28/0.49 thf(fact_92_RuleSystem_Otrim__fair,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( abstra928354080m_fair @ Rule @ Rules2 @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.trim_fair
% 0.28/0.49 thf(fact_93_RuleSystem_Owf__mkTree,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( abstra1874736267tem_wf @ Rule @ State @ Eff2 @ Rules2 @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.wf_mkTree
% 0.28/0.49 thf(fact_94_trim__alt,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S )
% 0.28/0.49 = ( sdrop @ rule @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ Rs ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % trim_alt
% 0.28/0.49 thf(fact_95_wf__ipath__epath,axiom,
% 0.28/0.49 ! [T: abstra2103299360e_tree @ ( product_prod @ state @ rule ),Steps: stream @ ( product_prod @ state @ rule )] :
% 0.28/0.49 ( ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T )
% 0.28/0.49 => ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ T @ Steps )
% 0.28/0.49 => ( abstra523868654_epath @ rule @ state @ eff @ rules @ Steps ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % wf_ipath_epath
% 0.28/0.49 thf(fact_96_ipath__mkTree__sdrop,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule,Steps: stream @ ( product_prod @ state @ rule ),M: nat] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) @ Steps )
% 0.28/0.49 => ? [N: nat,S6: state] :
% 0.28/0.49 ( ( member @ state @ S6 @ s )
% 0.28/0.49 & ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( sdrop @ rule @ N @ Rs ) @ S6 ) @ ( sdrop @ ( product_prod @ state @ rule ) @ M @ Steps ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ipath_mkTree_sdrop
% 0.28/0.49 thf(fact_97_minWait__ex,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ? [N: nat] : ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N @ Rs ) ) @ S ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % minWait_ex
% 0.28/0.49 thf(fact_98_trim__enabled,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % trim_enabled
% 0.28/0.49 thf(fact_99_trim__in__R,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( member @ rule @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ ( sset @ rule @ rules ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % trim_in_R
% 0.28/0.49 thf(fact_100_RuleSystem_Otrim__alt,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S )
% 0.28/0.49 = ( sdrop @ Rule @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.trim_alt
% 0.28/0.49 thf(fact_101_bot__apply,axiom,
% 0.28/0.49 ! [C3: $tType,D2: $tType] :
% 0.28/0.49 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 0.28/0.49 => ( ( bot_bot @ ( D2 > C3 ) )
% 0.28/0.49 = ( ^ [X: D2] : ( bot_bot @ C3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % bot_apply
% 0.28/0.49 thf(fact_102_bot__set__def,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( bot_bot @ ( set @ A ) )
% 0.28/0.49 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % bot_set_def
% 0.28/0.49 thf(fact_103_bot__fun__def,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType] :
% 0.28/0.49 ( ( bot @ B @ ( type2 @ B ) )
% 0.28/0.49 => ( ( bot_bot @ ( A > B ) )
% 0.28/0.49 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % bot_fun_def
% 0.28/0.49 thf(fact_104_RuleSystem_Otrim__in__R,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( member @ Rule @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ ( sset @ Rule @ Rules2 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.trim_in_R
% 0.28/0.49 thf(fact_105_RuleSystem_OminWait__ex,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ? [N: nat] : ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N @ Rs ) ) @ S ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.minWait_ex
% 0.28/0.49 thf(fact_106_RuleSystem_Otrim__enabled,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.trim_enabled
% 0.28/0.49 thf(fact_107_RuleSystem__Defs_Owf__ipath__epath,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,T: abstra2103299360e_tree @ ( product_prod @ State @ Rule ),Steps: stream @ ( product_prod @ State @ Rule )] :
% 0.28/0.49 ( ( abstra1874736267tem_wf @ Rule @ State @ Eff2 @ Rules2 @ T )
% 0.28/0.49 => ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ T @ Steps )
% 0.28/0.49 => ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules2 @ Steps ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.wf_ipath_epath
% 0.28/0.49 thf(fact_108_pos,axiom,
% 0.28/0.49 ! [Rs: stream @ rule,R: rule] :
% 0.28/0.49 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( ( shd @ rule @ ( sdrop @ rule @ ( abstra2097340358le_pos @ rule @ Rs @ R ) @ Rs ) )
% 0.28/0.49 = R ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % pos
% 0.28/0.49 thf(fact_109_minWait__least,axiom,
% 0.28/0.49 ! [N2: nat,Rs: stream @ rule,S: state] :
% 0.28/0.49 ( ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N2 @ Rs ) ) @ S )
% 0.28/0.49 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ N2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % minWait_least
% 0.28/0.49 thf(fact_110_RuleSystem_Opos,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,Rs: stream @ Rule,R: Rule] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 => ( ( shd @ Rule @ ( sdrop @ Rule @ ( abstra2097340358le_pos @ Rule @ Rs @ R ) @ Rs ) )
% 0.28/0.49 = R ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.pos
% 0.28/0.49 thf(fact_111_fempty__iff,axiom,
% 0.28/0.49 ! [A: $tType,C: A] :
% 0.28/0.49 ~ ( fmember @ A @ C @ ( bot_bot @ ( fset @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fempty_iff
% 0.28/0.49 thf(fact_112_all__not__fin__conv,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A] :
% 0.28/0.49 ( ( ! [X: A] :
% 0.28/0.49 ~ ( fmember @ A @ X @ A2 ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % all_not_fin_conv
% 0.28/0.49 thf(fact_113_order__refl,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A] : ( ord_less_eq @ A @ X5 @ X5 ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_refl
% 0.28/0.49 thf(fact_114_pos__least,axiom,
% 0.28/0.49 ! [A: $tType,N2: nat,Rs: stream @ A,R: A] :
% 0.28/0.49 ( ( ( shd @ A @ ( sdrop @ A @ N2 @ Rs ) )
% 0.28/0.49 = R )
% 0.28/0.49 => ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % pos_least
% 0.28/0.49 thf(fact_115_minWait__le__pos,axiom,
% 0.28/0.49 ! [Rs: stream @ rule,R: rule,S: state] :
% 0.28/0.49 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
% 0.28/0.49 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ ( abstra2097340358le_pos @ rule @ Rs @ R ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % minWait_le_pos
% 0.28/0.49 thf(fact_116_le__funD,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType] :
% 0.28/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.49 => ! [F: A > B,G: A > B,X5: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.28/0.49 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_funD
% 0.28/0.49 thf(fact_117_le__funE,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType] :
% 0.28/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.49 => ! [F: A > B,G: A > B,X5: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.28/0.49 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_funE
% 0.28/0.49 thf(fact_118_le__funI,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType] :
% 0.28/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.49 => ! [F: A > B,G: A > B] :
% 0.28/0.49 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.28/0.49 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_funI
% 0.28/0.49 thf(fact_119_le__fun__def,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType] :
% 0.28/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.49 => ( ( ord_less_eq @ ( A > B ) )
% 0.28/0.49 = ( ^ [F2: A > B,G2: A > B] :
% 0.28/0.49 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_fun_def
% 0.28/0.49 thf(fact_120_order__subst1,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType] :
% 0.28/0.49 ( ( ( order @ B @ ( type2 @ B ) )
% 0.28/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.28/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.28/0.49 => ( ! [X2: B,Y2: B] :
% 0.28/0.49 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_subst1
% 0.28/0.49 thf(fact_121_order__subst2,axiom,
% 0.28/0.49 ! [A: $tType,C3: $tType] :
% 0.28/0.49 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.28/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,B2: A,F: A > C3,C: C3] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
% 0.28/0.49 => ( ! [X2: A,Y2: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less_eq @ C3 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_subst2
% 0.28/0.49 thf(fact_122_ord__eq__le__subst,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType] :
% 0.28/0.49 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.49 & ( ord @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.28/0.49 ( ( A3
% 0.28/0.49 = ( F @ B2 ) )
% 0.28/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.28/0.49 => ( ! [X2: B,Y2: B] :
% 0.28/0.49 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ord_eq_le_subst
% 0.28/0.49 thf(fact_123_ord__le__eq__subst,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType] :
% 0.28/0.49 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.28/0.49 & ( ord @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,B2: A,F: A > B,C: B] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ( F @ B2 )
% 0.28/0.49 = C )
% 0.28/0.49 => ( ! [X2: A,Y2: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ord_le_eq_subst
% 0.28/0.49 thf(fact_124_eq__iff,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 0.28/0.49 = ( ^ [X: A,Y4: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/0.49 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % eq_iff
% 0.28/0.49 thf(fact_125_antisym,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 => ( ( ord_less_eq @ A @ Y @ X5 )
% 0.28/0.49 => ( X5 = Y ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % antisym
% 0.28/0.49 thf(fact_126_linear,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 | ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % linear
% 0.28/0.49 thf(fact_127_eq__refl,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( X5 = Y )
% 0.28/0.49 => ( ord_less_eq @ A @ X5 @ Y ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % eq_refl
% 0.28/0.49 thf(fact_128_le__cases,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ~ ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_cases
% 0.28/0.49 thf(fact_129_order_Otrans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A,C: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.28/0.49 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order.trans
% 0.28/0.49 thf(fact_130_le__cases3,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A,Z2: A] :
% 0.28/0.49 ( ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.28/0.49 => ( ( ( ord_less_eq @ A @ Y @ X5 )
% 0.28/0.49 => ~ ( ord_less_eq @ A @ X5 @ Z2 ) )
% 0.28/0.49 => ( ( ( ord_less_eq @ A @ X5 @ Z2 )
% 0.28/0.49 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.28/0.49 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.28/0.49 => ~ ( ord_less_eq @ A @ Y @ X5 ) )
% 0.28/0.49 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.28/0.49 => ~ ( ord_less_eq @ A @ Z2 @ X5 ) )
% 0.28/0.49 => ~ ( ( ord_less_eq @ A @ Z2 @ X5 )
% 0.28/0.49 => ~ ( ord_less_eq @ A @ X5 @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_cases3
% 0.28/0.49 thf(fact_131_antisym__conv,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [Y: A,X5: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ Y @ X5 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 = ( X5 = Y ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % antisym_conv
% 0.28/0.49 thf(fact_132_ord__eq__le__trans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A,C: A] :
% 0.28/0.49 ( ( A3 = B2 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.28/0.49 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ord_eq_le_trans
% 0.28/0.49 thf(fact_133_ord__le__eq__trans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( ord @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A,C: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( B2 = C )
% 0.28/0.49 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ord_le_eq_trans
% 0.28/0.49 thf(fact_134_order__class_Oorder_Oantisym,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.28/0.49 => ( A3 = B2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_class.order.antisym
% 0.28/0.49 thf(fact_135_order__trans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A,Z2: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.28/0.49 => ( ord_less_eq @ A @ X5 @ Z2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_trans
% 0.28/0.49 thf(fact_136_dual__order_Orefl,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.refl
% 0.28/0.49 thf(fact_137_linorder__wlog,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [P2: A > A > $o,A3: A,B2: A] :
% 0.28/0.49 ( ! [A4: A,B5: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A4 @ B5 )
% 0.28/0.49 => ( P2 @ A4 @ B5 ) )
% 0.28/0.49 => ( ! [A4: A,B5: A] :
% 0.28/0.49 ( ( P2 @ B5 @ A4 )
% 0.28/0.49 => ( P2 @ A4 @ B5 ) )
% 0.28/0.49 => ( P2 @ A3 @ B2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % linorder_wlog
% 0.28/0.49 thf(fact_138_dual__order_Otrans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [B2: A,A3: A,C: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.28/0.49 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.trans
% 0.28/0.49 thf(fact_139_dual__order_Oantisym,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [B2: A,A3: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( A3 = B2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.antisym
% 0.28/0.49 thf(fact_140_bot_Oextremum,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % bot.extremum
% 0.28/0.49 thf(fact_141_bot_Oextremum__unique,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.28/0.49 = ( A3
% 0.28/0.49 = ( bot_bot @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % bot.extremum_unique
% 0.28/0.49 thf(fact_142_bot_Oextremum__uniqueI,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.28/0.49 => ( A3
% 0.28/0.49 = ( bot_bot @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % bot.extremum_uniqueI
% 0.28/0.49 thf(fact_143_RuleSystem_Opos__least,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,A: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,N2: nat,Rs: stream @ A,R: A] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( ( shd @ A @ ( sdrop @ A @ N2 @ Rs ) )
% 0.28/0.49 = R )
% 0.28/0.49 => ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.pos_least
% 0.28/0.49 thf(fact_144_eqfelem__imp__iff,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,Y: A,A2: fset @ A] :
% 0.28/0.49 ( ( X5 = Y )
% 0.28/0.49 => ( ( fmember @ A @ X5 @ A2 )
% 0.28/0.49 = ( fmember @ A @ Y @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % eqfelem_imp_iff
% 0.28/0.49 thf(fact_145_if__split__fmem2,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,Q: $o,X5: fset @ A,Y: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ A3 @ ( if @ ( fset @ A ) @ Q @ X5 @ Y ) )
% 0.28/0.49 = ( ( Q
% 0.28/0.49 => ( fmember @ A @ A3 @ X5 ) )
% 0.28/0.49 & ( ~ Q
% 0.28/0.49 => ( fmember @ A @ A3 @ Y ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % if_split_fmem2
% 0.28/0.49 thf(fact_146_if__split__fmem1,axiom,
% 0.28/0.49 ! [A: $tType,Q: $o,X5: A,Y: A,B2: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ ( if @ A @ Q @ X5 @ Y ) @ B2 )
% 0.28/0.49 = ( ( Q
% 0.28/0.49 => ( fmember @ A @ X5 @ B2 ) )
% 0.28/0.49 & ( ~ Q
% 0.28/0.49 => ( fmember @ A @ Y @ B2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % if_split_fmem1
% 0.28/0.49 thf(fact_147_eqfset__imp__iff,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A,X5: A] :
% 0.28/0.49 ( ( A2 = B3 )
% 0.28/0.49 => ( ( fmember @ A @ X5 @ A2 )
% 0.28/0.49 = ( fmember @ A @ X5 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % eqfset_imp_iff
% 0.28/0.49 thf(fact_148_eq__fmem__trans,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,B2: A,A2: fset @ A] :
% 0.28/0.49 ( ( A3 = B2 )
% 0.28/0.49 => ( ( fmember @ A @ B2 @ A2 )
% 0.28/0.49 => ( fmember @ A @ A3 @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % eq_fmem_trans
% 0.28/0.49 thf(fact_149_fset__choice,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType,A2: fset @ A,P2: A > B > $o] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( fmember @ A @ X2 @ A2 )
% 0.28/0.49 => ? [X12: B] : ( P2 @ X2 @ X12 ) )
% 0.28/0.49 => ? [F3: A > B] :
% 0.28/0.49 ! [X4: A] :
% 0.28/0.49 ( ( fmember @ A @ X4 @ A2 )
% 0.28/0.49 => ( P2 @ X4 @ ( F3 @ X4 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fset_choice
% 0.28/0.49 thf(fact_150_fequalityCE,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A,C: A] :
% 0.28/0.49 ( ( A2 = B3 )
% 0.28/0.49 => ( ( ( fmember @ A @ C @ A2 )
% 0.28/0.49 => ~ ( fmember @ A @ C @ B3 ) )
% 0.28/0.49 => ~ ( ~ ( fmember @ A @ C @ A2 )
% 0.28/0.49 => ( fmember @ A @ C @ B3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fequalityCE
% 0.28/0.49 thf(fact_151_fset__eqI,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( fmember @ A @ X2 @ A2 )
% 0.28/0.49 = ( fmember @ A @ X2 @ B3 ) )
% 0.28/0.49 => ( A2 = B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % fset_eqI
% 0.28/0.49 thf(fact_152_femptyE,axiom,
% 0.28/0.49 ! [A: $tType,A3: A] :
% 0.28/0.49 ~ ( fmember @ A @ A3 @ ( bot_bot @ ( fset @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % femptyE
% 0.28/0.49 thf(fact_153_ex__fin__conv,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A] :
% 0.28/0.49 ( ( ? [X: A] : ( fmember @ A @ X @ A2 ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 != ( bot_bot @ ( fset @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % ex_fin_conv
% 0.28/0.49 thf(fact_154_equalsffemptyD,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,A3: A] :
% 0.28/0.49 ( ( A2
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) )
% 0.28/0.49 => ~ ( fmember @ A @ A3 @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % equalsffemptyD
% 0.28/0.49 thf(fact_155_equalsffemptyI,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A] :
% 0.28/0.49 ( ! [Y2: A] :
% 0.28/0.49 ~ ( fmember @ A @ Y2 @ A2 )
% 0.28/0.49 => ( A2
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % equalsffemptyI
% 0.28/0.49 thf(fact_156_RuleSystem_OminWait__le__pos,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,Rs: stream @ Rule,R: Rule,S: State] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 0.28/0.49 => ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S )
% 0.28/0.49 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ ( abstra2097340358le_pos @ Rule @ Rs @ R ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.minWait_le_pos
% 0.28/0.49 thf(fact_157_RuleSystem_OminWait__least,axiom,
% 0.28/0.49 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,N2: nat,Rs: stream @ Rule,S: State] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N2 @ Rs ) ) @ S )
% 0.28/0.49 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ N2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.minWait_least
% 0.28/0.49 thf(fact_158_minus__apply,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType] :
% 0.28/0.49 ( ( minus @ B @ ( type2 @ B ) )
% 0.28/0.49 => ( ( minus_minus @ ( A > B ) )
% 0.28/0.49 = ( ^ [A5: A > B,B6: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B6 @ X ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % minus_apply
% 0.28/0.49 thf(fact_159_bot__empty__eq,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( bot_bot @ ( A > $o ) )
% 0.28/0.49 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % bot_empty_eq
% 0.28/0.49 thf(fact_160_Collect__empty__eq__bot,axiom,
% 0.28/0.49 ! [A: $tType,P2: A > $o] :
% 0.28/0.49 ( ( ( collect @ A @ P2 )
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 = ( P2
% 0.28/0.49 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Collect_empty_eq_bot
% 0.28/0.49 thf(fact_161_in__cont__mkTree,axiom,
% 0.28/0.49 ! [S: state,Rs: stream @ rule,T2: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
% 0.28/0.49 ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ T2 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) ) )
% 0.28/0.49 => ? [Sl5: fset @ state,S6: state] :
% 0.28/0.49 ( ( member @ state @ S6 @ s )
% 0.28/0.49 & ( eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S @ Sl5 )
% 0.28/0.49 & ( fmember @ state @ S6 @ Sl5 )
% 0.28/0.49 & ( T2
% 0.28/0.49 = ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S6 ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % in_cont_mkTree
% 0.28/0.49 thf(fact_162_is__singletonI,axiom,
% 0.28/0.49 ! [A: $tType,X5: A] : ( is_singleton @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % is_singletonI
% 0.28/0.49 thf(fact_163_fair__stl,axiom,
% 0.28/0.49 ! [Rs: stream @ rule] :
% 0.28/0.49 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( abstra928354080m_fair @ rule @ rules @ ( stl @ rule @ Rs ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fair_stl
% 0.28/0.49 thf(fact_164_subset__empty,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_empty
% 0.28/0.49 thf(fact_165_empty__subsetI,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.28/0.49
% 0.28/0.49 % empty_subsetI
% 0.28/0.49 thf(fact_166_insert__subset,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 0.28/0.49 = ( ( member @ A @ X5 @ B3 )
% 0.28/0.49 & ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_subset
% 0.28/0.49 thf(fact_167_fminus__iff,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 0.28/0.49 = ( ( fmember @ A @ C @ A2 )
% 0.28/0.49 & ~ ( fmember @ A @ C @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminus_iff
% 0.28/0.49 thf(fact_168_fminusI,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ C @ A2 )
% 0.28/0.49 => ( ~ ( fmember @ A @ C @ B3 )
% 0.28/0.49 => ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminusI
% 0.28/0.49 thf(fact_169_fsubsetI,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( fmember @ A @ X2 @ A2 )
% 0.28/0.49 => ( fmember @ A @ X2 @ B3 ) )
% 0.28/0.49 => ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % fsubsetI
% 0.28/0.49 thf(fact_170_fminus__fempty,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A] :
% 0.28/0.49 ( ( minus_minus @ ( fset @ A ) @ A2 @ ( bot_bot @ ( fset @ A ) ) )
% 0.28/0.49 = A2 ) ).
% 0.28/0.49
% 0.28/0.49 % fminus_fempty
% 0.28/0.49 thf(fact_171_fminus__cancel,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A] :
% 0.28/0.49 ( ( minus_minus @ ( fset @ A ) @ A2 @ A2 )
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminus_cancel
% 0.28/0.49 thf(fact_172_fempty__fminus,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A] :
% 0.28/0.49 ( ( minus_minus @ ( fset @ A ) @ ( bot_bot @ ( fset @ A ) ) @ A2 )
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fempty_fminus
% 0.28/0.49 thf(fact_173_fempty__fsubsetI,axiom,
% 0.28/0.49 ! [A: $tType,X5: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ ( bot_bot @ ( fset @ A ) ) @ X5 ) ).
% 0.28/0.49
% 0.28/0.49 % fempty_fsubsetI
% 0.28/0.49 thf(fact_174_fsubset__fempty,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ ( bot_bot @ ( fset @ A ) ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fsubset_fempty
% 0.28/0.49 thf(fact_175_singleton__insert__inj__eq_H,axiom,
% 0.28/0.49 ! [A: $tType,A3: A,A2: set @ A,B2: A] :
% 0.28/0.49 ( ( ( insert @ A @ A3 @ A2 )
% 0.28/0.49 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 = ( ( A3 = B2 )
% 0.28/0.49 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % singleton_insert_inj_eq'
% 0.28/0.49 thf(fact_176_singleton__insert__inj__eq,axiom,
% 0.28/0.49 ! [A: $tType,B2: A,A3: A,A2: set @ A] :
% 0.28/0.49 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 = ( insert @ A @ A3 @ A2 ) )
% 0.28/0.49 = ( ( A3 = B2 )
% 0.28/0.49 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % singleton_insert_inj_eq
% 0.28/0.49 thf(fact_177_Diff__eq__empty__iff,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( ( minus_minus @ ( set @ A ) @ A2 @ B3 )
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 = ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_eq_empty_iff
% 0.28/0.49 thf(fact_178_fminusD2,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 0.28/0.49 => ~ ( fmember @ A @ C @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminusD2
% 0.28/0.49 thf(fact_179_fminusD1,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 0.28/0.49 => ( fmember @ A @ C @ A2 ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminusD1
% 0.28/0.49 thf(fact_180_fminusE,axiom,
% 0.28/0.49 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 0.28/0.49 => ~ ( ( fmember @ A @ C @ A2 )
% 0.28/0.49 => ( fmember @ A @ C @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminusE
% 0.28/0.49 thf(fact_181_fset__mp,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A,X5: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( fmember @ A @ X5 @ A2 )
% 0.28/0.49 => ( fmember @ A @ X5 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fset_mp
% 0.28/0.49 thf(fact_182_fin__mono,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A,X5: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( fmember @ A @ X5 @ A2 )
% 0.28/0.49 => ( fmember @ A @ X5 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fin_mono
% 0.28/0.49 thf(fact_183_fsubsetD,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A,C: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( fmember @ A @ C @ A2 )
% 0.28/0.49 => ( fmember @ A @ C @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fsubsetD
% 0.28/0.49 thf(fact_184_fset__rev__mp,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( fmember @ A @ X5 @ A2 )
% 0.28/0.49 => ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( fmember @ A @ X5 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fset_rev_mp
% 0.28/0.49 thf(fact_185_subset__insertI2,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,B2: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_insertI2
% 0.28/0.49 thf(fact_186_subset__insertI,axiom,
% 0.28/0.49 ! [A: $tType,B3: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A3 @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_insertI
% 0.28/0.49 thf(fact_187_subset__insert,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ~ ( member @ A @ X5 @ A2 )
% 0.28/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) )
% 0.28/0.49 = ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_insert
% 0.28/0.49 thf(fact_188_insert__mono,axiom,
% 0.28/0.49 ! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 0.28/0.49 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % insert_mono
% 0.28/0.49 thf(fact_189_RuleSystem__Defs_Ofair__stl,axiom,
% 0.28/0.49 ! [Rule: $tType,Rules2: stream @ Rule,Rs: stream @ Rule] :
% 0.28/0.49 ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( abstra928354080m_fair @ Rule @ Rules2 @ ( stl @ Rule @ Rs ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem_Defs.fair_stl
% 0.28/0.49 thf(fact_190_double__diff,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,C4: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 0.28/0.49 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C4 @ A2 ) )
% 0.28/0.49 = A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % double_diff
% 0.28/0.49 thf(fact_191_Diff__subset,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ A2 ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_subset
% 0.28/0.49 thf(fact_192_Diff__mono,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,C4: set @ A,D3: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 0.28/0.49 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B3 )
% 0.28/0.49 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_mono
% 0.28/0.49 thf(fact_193_countable__subset,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( countable_countable @ A @ B3 )
% 0.28/0.49 => ( countable_countable @ A @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % countable_subset
% 0.28/0.49 thf(fact_194_subset__singleton__iff,axiom,
% 0.28/0.49 ! [A: $tType,X6: set @ A,A3: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 = ( ( X6
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 | ( X6
% 0.28/0.49 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_singleton_iff
% 0.28/0.49 thf(fact_195_subset__singletonD,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,X5: A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.28/0.49 => ( ( A2
% 0.28/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 | ( A2
% 0.28/0.49 = ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_singletonD
% 0.28/0.49 thf(fact_196_subset__Diff__insert,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A,X5: A,C4: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X5 @ C4 ) ) )
% 0.28/0.49 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B3 @ C4 ) )
% 0.28/0.49 & ~ ( member @ A @ X5 @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_Diff_insert
% 0.28/0.49 thf(fact_197_is__singletonI_H,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( A2
% 0.28/0.49 != ( bot_bot @ ( set @ A ) ) )
% 0.28/0.49 => ( ! [X2: A,Y2: A] :
% 0.28/0.49 ( ( member @ A @ X2 @ A2 )
% 0.28/0.49 => ( ( member @ A @ Y2 @ A2 )
% 0.28/0.49 => ( X2 = Y2 ) ) )
% 0.28/0.49 => ( is_singleton @ A @ A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % is_singletonI'
% 0.28/0.49 thf(fact_198_Diff__single__insert,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,X5: A,B3: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 )
% 0.28/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % Diff_single_insert
% 0.28/0.49 thf(fact_199_subset__insert__iff,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,X5: A,B3: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) )
% 0.28/0.49 = ( ( ( member @ A @ X5 @ A2 )
% 0.28/0.49 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
% 0.28/0.49 & ( ~ ( member @ A @ X5 @ A2 )
% 0.28/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_insert_iff
% 0.28/0.49 thf(fact_200_fun__diff__def,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType] :
% 0.28/0.49 ( ( minus @ B @ ( type2 @ B ) )
% 0.28/0.49 => ( ( minus_minus @ ( A > B ) )
% 0.28/0.49 = ( ^ [A5: A > B,B6: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B6 @ X ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fun_diff_def
% 0.28/0.49 thf(fact_201_RuleSystem_Oin__cont__mkTree,axiom,
% 0.28/0.49 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,T2: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
% 0.28/0.49 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 0.28/0.49 => ( ( member @ State @ S @ S3 )
% 0.28/0.49 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 0.28/0.49 => ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ T2 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) ) )
% 0.28/0.49 => ? [Sl5: fset @ State,S6: State] :
% 0.28/0.49 ( ( member @ State @ S6 @ S3 )
% 0.28/0.49 & ( Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
% 0.28/0.49 & ( fmember @ State @ S6 @ Sl5 )
% 0.28/0.49 & ( T2
% 0.28/0.49 = ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S6 ) ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % RuleSystem.in_cont_mkTree
% 0.28/0.49 thf(fact_202_is__singleton__def,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( is_singleton @ A )
% 0.28/0.49 = ( ^ [A5: set @ A] :
% 0.28/0.49 ? [X: A] :
% 0.28/0.49 ( A5
% 0.28/0.49 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % is_singleton_def
% 0.28/0.49 thf(fact_203_is__singletonE,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A] :
% 0.28/0.49 ( ( is_singleton @ A @ A2 )
% 0.28/0.49 => ~ ! [X2: A] :
% 0.28/0.49 ( A2
% 0.28/0.49 != ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % is_singletonE
% 0.28/0.49 thf(fact_204_mkTree_Osimps_I2_J,axiom,
% 0.28/0.49 ! [Rs: stream @ rule,S: state] :
% 0.28/0.49 ( ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) )
% 0.28/0.49 = ( 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 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % mkTree.simps(2)
% 0.28/0.49 thf(fact_205_sset__induct,axiom,
% 0.28/0.49 ! [A: $tType,Y: A,S: stream @ A,P2: A > ( stream @ A ) > $o] :
% 0.28/0.49 ( ( member @ A @ Y @ ( sset @ A @ S ) )
% 0.28/0.49 => ( ! [S5: stream @ A] : ( P2 @ ( shd @ A @ S5 ) @ S5 )
% 0.28/0.49 => ( ! [S5: stream @ A,Y2: A] :
% 0.28/0.49 ( ( member @ A @ Y2 @ ( sset @ A @ ( stl @ A @ S5 ) ) )
% 0.28/0.49 => ( ( P2 @ Y2 @ ( stl @ A @ S5 ) )
% 0.28/0.49 => ( P2 @ Y2 @ S5 ) ) )
% 0.28/0.49 => ( P2 @ Y @ S ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % sset_induct
% 0.28/0.49 thf(fact_206_stake__pos__minWait,axiom,
% 0.28/0.49 ! [Rs: stream @ rule,S: state,R: rule] :
% 0.28/0.49 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 0.28/0.49 => ( ( ord_less @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ ( abstra2097340358le_pos @ rule @ Rs @ R ) )
% 0.28/0.49 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( ( member @ state @ S @ s )
% 0.28/0.49 => ( ( abstra2097340358le_pos @ rule @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ R )
% 0.28/0.49 = ( minus_minus @ nat @ ( abstra2097340358le_pos @ rule @ Rs @ R ) @ ( suc @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) ) ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % stake_pos_minWait
% 0.28/0.49 thf(fact_207_subsetI,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( member @ A @ X2 @ A2 )
% 0.28/0.49 => ( member @ A @ X2 @ B3 ) )
% 0.28/0.49 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % subsetI
% 0.28/0.49 thf(fact_208_subset__antisym,axiom,
% 0.28/0.49 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 0.28/0.49 => ( A2 = B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_antisym
% 0.28/0.49 thf(fact_209_fsubset__antisym,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( ord_less_eq @ ( fset @ A ) @ B3 @ A2 )
% 0.28/0.49 => ( A2 = B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fsubset_antisym
% 0.28/0.49 thf(fact_210_fminus__idemp,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( minus_minus @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) @ B3 )
% 0.28/0.49 = ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminus_idemp
% 0.28/0.49 thf(fact_211_fimage__eqI,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType,B2: A,F: B > A,X5: B,A2: fset @ B] :
% 0.28/0.49 ( ( B2
% 0.28/0.49 = ( F @ X5 ) )
% 0.28/0.49 => ( ( fmember @ B @ X5 @ A2 )
% 0.28/0.49 => ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fimage_eqI
% 0.28/0.49 thf(fact_212_fimage__is__fempty,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType,F: B > A,A2: fset @ B] :
% 0.28/0.49 ( ( ( fimage @ B @ A @ F @ A2 )
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 = ( bot_bot @ ( fset @ B ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fimage_is_fempty
% 0.28/0.49 thf(fact_213_fempty__is__fimage,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType,F: B > A,A2: fset @ B] :
% 0.28/0.49 ( ( ( bot_bot @ ( fset @ A ) )
% 0.28/0.49 = ( fimage @ B @ A @ F @ A2 ) )
% 0.28/0.49 = ( A2
% 0.28/0.49 = ( bot_bot @ ( fset @ B ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fempty_is_fimage
% 0.28/0.49 thf(fact_214_fimage__fempty,axiom,
% 0.28/0.49 ! [B: $tType,A: $tType,F: B > A] :
% 0.28/0.49 ( ( fimage @ B @ A @ F @ ( bot_bot @ ( fset @ B ) ) )
% 0.28/0.49 = ( bot_bot @ ( fset @ A ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fimage_fempty
% 0.28/0.49 thf(fact_215_sdrop__simps_I2_J,axiom,
% 0.28/0.49 ! [A: $tType,N2: nat,S: stream @ A] :
% 0.28/0.49 ( ( stl @ A @ ( sdrop @ A @ N2 @ S ) )
% 0.28/0.49 = ( sdrop @ A @ ( suc @ N2 ) @ S ) ) ).
% 0.28/0.49
% 0.28/0.49 % sdrop_simps(2)
% 0.28/0.49 thf(fact_216_fimage__fminus__fsubset,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType,F: B > A,A2: fset @ B,B3: fset @ B] : ( ord_less_eq @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ ( fimage @ B @ A @ F @ A2 ) @ ( fimage @ B @ A @ F @ B3 ) ) @ ( fimage @ B @ A @ F @ ( minus_minus @ ( fset @ B ) @ A2 @ B3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fimage_fminus_fsubset
% 0.28/0.49 thf(fact_217_fminus__fsubset,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) @ A2 ) ).
% 0.28/0.49
% 0.28/0.49 % fminus_fsubset
% 0.28/0.49 thf(fact_218_double__fminus,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,B3: fset @ A,C4: fset @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 0.28/0.49 => ( ( ord_less_eq @ ( fset @ A ) @ B3 @ C4 )
% 0.28/0.49 => ( ( minus_minus @ ( fset @ A ) @ B3 @ ( minus_minus @ ( fset @ A ) @ C4 @ A2 ) )
% 0.28/0.49 = A2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % double_fminus
% 0.28/0.49 thf(fact_219_fminus__mono,axiom,
% 0.28/0.49 ! [A: $tType,A2: fset @ A,C4: fset @ A,D3: fset @ A,B3: fset @ A] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ C4 )
% 0.28/0.49 => ( ( ord_less_eq @ ( fset @ A ) @ D3 @ B3 )
% 0.28/0.49 => ( ord_less_eq @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( fset @ A ) @ C4 @ D3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % fminus_mono
% 0.28/0.49 thf(fact_220_leD,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [Y: A,X5: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ Y @ X5 )
% 0.28/0.49 => ~ ( ord_less @ A @ X5 @ Y ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % leD
% 0.28/0.49 thf(fact_221_leI,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ~ ( ord_less @ A @ X5 @ Y )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % leI
% 0.28/0.49 thf(fact_222_le__less,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ord_less_eq @ A )
% 0.28/0.49 = ( ^ [X: A,Y4: A] :
% 0.28/0.49 ( ( ord_less @ A @ X @ Y4 )
% 0.28/0.49 | ( X = Y4 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_less
% 0.28/0.49 thf(fact_223_less__le,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ord_less @ A )
% 0.28/0.49 = ( ^ [X: A,Y4: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/0.49 & ( X != Y4 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % less_le
% 0.28/0.49 thf(fact_224_order__le__less__subst1,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType] :
% 0.28/0.49 ( ( ( order @ B @ ( type2 @ B ) )
% 0.28/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 0.28/0.49 => ( ( ord_less @ B @ B2 @ C )
% 0.28/0.49 => ( ! [X2: B,Y2: B] :
% 0.28/0.49 ( ( ord_less @ B @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_le_less_subst1
% 0.28/0.49 thf(fact_225_order__le__less__subst2,axiom,
% 0.28/0.49 ! [A: $tType,C3: $tType] :
% 0.28/0.49 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.28/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,B2: A,F: A > C3,C: C3] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ord_less @ C3 @ ( F @ B2 ) @ C )
% 0.28/0.49 => ( ! [X2: A,Y2: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less_eq @ C3 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_le_less_subst2
% 0.28/0.49 thf(fact_226_order__less__le__subst1,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType] :
% 0.28/0.49 ( ( ( order @ B @ ( type2 @ B ) )
% 0.28/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,F: B > A,B2: B,C: B] :
% 0.28/0.49 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 0.28/0.49 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.28/0.49 => ( ! [X2: B,Y2: B] :
% 0.28/0.49 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_less_le_subst1
% 0.28/0.49 thf(fact_227_order__less__le__subst2,axiom,
% 0.28/0.49 ! [A: $tType,C3: $tType] :
% 0.28/0.49 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.28/0.49 & ( order @ A @ ( type2 @ A ) ) )
% 0.28/0.49 => ! [A3: A,B2: A,F: A > C3,C: C3] :
% 0.28/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
% 0.28/0.49 => ( ! [X2: A,Y2: A] :
% 0.28/0.49 ( ( ord_less @ A @ X2 @ Y2 )
% 0.28/0.49 => ( ord_less @ C3 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.28/0.49 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order_less_le_subst2
% 0.28/0.49 thf(fact_228_not__le,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ~ ( ord_less_eq @ A @ X5 @ Y ) )
% 0.28/0.49 = ( ord_less @ A @ Y @ X5 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % not_le
% 0.28/0.49 thf(fact_229_not__less,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ~ ( ord_less @ A @ X5 @ Y ) )
% 0.28/0.49 = ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % not_less
% 0.28/0.49 thf(fact_230_le__neq__trans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( A3 != B2 )
% 0.28/0.49 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_neq_trans
% 0.28/0.49 thf(fact_231_less__imp__le,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ord_less @ A @ X5 @ Y )
% 0.28/0.49 => ( ord_less_eq @ A @ X5 @ Y ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % less_imp_le
% 0.28/0.49 thf(fact_232_antisym__conv1,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ~ ( ord_less @ A @ X5 @ Y )
% 0.28/0.49 => ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 = ( X5 = Y ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % antisym_conv1
% 0.28/0.49 thf(fact_233_antisym__conv2,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 => ( ( ~ ( ord_less @ A @ X5 @ Y ) )
% 0.28/0.49 = ( X5 = Y ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % antisym_conv2
% 0.28/0.49 thf(fact_234_le__less__trans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A,Z2: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 => ( ( ord_less @ A @ Y @ Z2 )
% 0.28/0.49 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_less_trans
% 0.28/0.49 thf(fact_235_less__le__trans,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A,Z2: A] :
% 0.28/0.49 ( ( ord_less @ A @ X5 @ Y )
% 0.28/0.49 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.28/0.49 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % less_le_trans
% 0.28/0.49 thf(fact_236_dense__ge,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [Z2: A,Y: A] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( ord_less @ A @ Z2 @ X2 )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ X2 ) )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dense_ge
% 0.28/0.49 thf(fact_237_dense__le,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [Y: A,Z2: A] :
% 0.28/0.49 ( ! [X2: A] :
% 0.28/0.49 ( ( ord_less @ A @ X2 @ Y )
% 0.28/0.49 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dense_le
% 0.28/0.49 thf(fact_238_le__less__linear,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 | ( ord_less @ A @ Y @ X5 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_less_linear
% 0.28/0.49 thf(fact_239_le__imp__less__or__eq,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X5 @ Y )
% 0.28/0.49 => ( ( ord_less @ A @ X5 @ Y )
% 0.28/0.49 | ( X5 = Y ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % le_imp_less_or_eq
% 0.28/0.49 thf(fact_240_less__le__not__le,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( preorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ord_less @ A )
% 0.28/0.49 = ( ^ [X: A,Y4: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.28/0.49 & ~ ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % less_le_not_le
% 0.28/0.49 thf(fact_241_not__le__imp__less,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [Y: A,X5: A] :
% 0.28/0.49 ( ~ ( ord_less_eq @ A @ Y @ X5 )
% 0.28/0.49 => ( ord_less @ A @ X5 @ Y ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % not_le_imp_less
% 0.28/0.49 thf(fact_242_order_Ostrict__trans1,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A,C: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ord_less @ A @ B2 @ C )
% 0.28/0.49 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order.strict_trans1
% 0.28/0.49 thf(fact_243_order_Ostrict__trans2,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A,C: A] :
% 0.28/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.28/0.49 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order.strict_trans2
% 0.28/0.49 thf(fact_244_order_Oorder__iff__strict,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ord_less_eq @ A )
% 0.28/0.49 = ( ^ [A6: A,B7: A] :
% 0.28/0.49 ( ( ord_less @ A @ A6 @ B7 )
% 0.28/0.49 | ( A6 = B7 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order.order_iff_strict
% 0.28/0.49 thf(fact_245_order_Ostrict__iff__order,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ord_less @ A )
% 0.28/0.49 = ( ^ [A6: A,B7: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ A6 @ B7 )
% 0.28/0.49 & ( A6 != B7 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order.strict_iff_order
% 0.28/0.49 thf(fact_246_dual__order_Ostrict__trans1,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [B2: A,A3: A,C: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.28/0.49 => ( ( ord_less @ A @ C @ B2 )
% 0.28/0.49 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.strict_trans1
% 0.28/0.49 thf(fact_247_dual__order_Ostrict__trans2,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [B2: A,A3: A,C: A] :
% 0.28/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.28/0.49 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.strict_trans2
% 0.28/0.49 thf(fact_248_dense__ge__bounded,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [Z2: A,X5: A,Y: A] :
% 0.28/0.49 ( ( ord_less @ A @ Z2 @ X5 )
% 0.28/0.49 => ( ! [W: A] :
% 0.28/0.49 ( ( ord_less @ A @ Z2 @ W )
% 0.28/0.49 => ( ( ord_less @ A @ W @ X5 )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dense_ge_bounded
% 0.28/0.49 thf(fact_249_dense__le__bounded,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [X5: A,Y: A,Z2: A] :
% 0.28/0.49 ( ( ord_less @ A @ X5 @ Y )
% 0.28/0.49 => ( ! [W: A] :
% 0.28/0.49 ( ( ord_less @ A @ X5 @ W )
% 0.28/0.49 => ( ( ord_less @ A @ W @ Y )
% 0.28/0.49 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.28/0.49 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dense_le_bounded
% 0.28/0.49 thf(fact_250_order_Ostrict__implies__order,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A] :
% 0.28/0.49 ( ( ord_less @ A @ A3 @ B2 )
% 0.28/0.49 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order.strict_implies_order
% 0.28/0.49 thf(fact_251_dual__order_Oorder__iff__strict,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ord_less_eq @ A )
% 0.28/0.49 = ( ^ [B7: A,A6: A] :
% 0.28/0.49 ( ( ord_less @ A @ B7 @ A6 )
% 0.28/0.49 | ( A6 = B7 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.order_iff_strict
% 0.28/0.49 thf(fact_252_dual__order_Ostrict__iff__order,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ( ( ord_less @ A )
% 0.28/0.49 = ( ^ [B7: A,A6: A] :
% 0.28/0.49 ( ( ord_less_eq @ A @ B7 @ A6 )
% 0.28/0.49 & ( A6 != B7 ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.strict_iff_order
% 0.28/0.49 thf(fact_253_dual__order_Ostrict__implies__order,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [B2: A,A3: A] :
% 0.28/0.49 ( ( ord_less @ A @ B2 @ A3 )
% 0.28/0.49 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % dual_order.strict_implies_order
% 0.28/0.49 thf(fact_254_order_Onot__eq__order__implies__strict,axiom,
% 0.28/0.49 ! [A: $tType] :
% 0.28/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.28/0.49 => ! [A3: A,B2: A] :
% 0.28/0.49 ( ( A3 != B2 )
% 0.28/0.49 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.28/0.49 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % order.not_eq_order_implies_strict
% 0.28/0.49 thf(fact_255_subset__fimage__iff,axiom,
% 0.28/0.49 ! [A: $tType,B: $tType,B3: fset @ A,F: B > A,A2: fset @ B] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ A ) @ B3 @ ( fimage @ B @ A @ F @ A2 ) )
% 0.28/0.49 = ( ? [AA: fset @ B] :
% 0.28/0.49 ( ( ord_less_eq @ ( fset @ B ) @ AA @ A2 )
% 0.28/0.49 & ( B3
% 0.28/0.49 = ( fimage @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 % subset_fimage_iff
% 0.28/0.49
% 0.28/0.49 %----Type constructors (43)
% 0.28/0.49 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 & ( finite_finite @ A8 @ ( type2 @ A8 ) ) )
% 0.28/0.49 => ( finite_finite @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_1,axiom,
% 0.28/0.49 ! [A7: $tType] :
% 0.28/0.49 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 => ( finite_finite @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Finite__Set_Ofinite_2,axiom,
% 0.28/0.49 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 0.28/0.49 ! [A7: $tType] :
% 0.28/0.49 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 => ( finite_finite @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Finite__Set_Ofinite_4,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 & ( finite_finite @ A8 @ ( type2 @ A8 ) ) )
% 0.28/0.49 => ( finite_finite @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 0.28/0.49 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Countable_Ocountable,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 & ( countable @ A8 @ ( type2 @ A8 ) ) )
% 0.28/0.49 => ( countable @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.28/0.49 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.28/0.49 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Orderings_Oord,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.28/0.49 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Orderings_Obot,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 0.28/0.49 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_fun___Groups_Ominus,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( minus @ A8 @ ( type2 @ A8 ) )
% 0.28/0.49 => ( minus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Orderings_Oorder__bot_5,axiom,
% 0.28/0.49 order_bot @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Countable_Ocountable_6,axiom,
% 0.28/0.49 countable @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Orderings_Opreorder_7,axiom,
% 0.28/0.49 preorder @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.28/0.49 linorder @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
% 0.28/0.49 order @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Orderings_Oord_9,axiom,
% 0.28/0.49 ord @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Orderings_Obot_10,axiom,
% 0.28/0.49 bot @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Nat_Onat___Groups_Ominus_11,axiom,
% 0.28/0.49 minus @ nat @ ( type2 @ nat ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Orderings_Oorder__bot_12,axiom,
% 0.28/0.49 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Countable_Ocountable_13,axiom,
% 0.28/0.49 ! [A7: $tType] :
% 0.28/0.49 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 => ( countable @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Orderings_Opreorder_14,axiom,
% 0.28/0.49 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Orderings_Oorder_15,axiom,
% 0.28/0.49 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
% 0.28/0.49 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Orderings_Obot_17,axiom,
% 0.28/0.49 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Set_Oset___Groups_Ominus_18,axiom,
% 0.28/0.49 ! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Orderings_Oorder__bot_19,axiom,
% 0.28/0.49 order_bot @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Countable_Ocountable_20,axiom,
% 0.28/0.49 countable @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
% 0.28/0.49 preorder @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
% 0.28/0.49 linorder @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Orderings_Oorder_23,axiom,
% 0.28/0.49 order @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Orderings_Oord_24,axiom,
% 0.28/0.49 ord @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Orderings_Obot_25,axiom,
% 0.28/0.49 bot @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_HOL_Obool___Groups_Ominus_26,axiom,
% 0.28/0.49 minus @ $o @ ( type2 @ $o ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Orderings_Oorder__bot_27,axiom,
% 0.28/0.49 ! [A7: $tType] : ( order_bot @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Countable_Ocountable_28,axiom,
% 0.28/0.49 ! [A7: $tType] :
% 0.28/0.49 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 => ( countable @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Orderings_Opreorder_29,axiom,
% 0.28/0.49 ! [A7: $tType] : ( preorder @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Orderings_Oorder_30,axiom,
% 0.28/0.49 ! [A7: $tType] : ( order @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Orderings_Oord_31,axiom,
% 0.28/0.49 ! [A7: $tType] : ( ord @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Orderings_Obot_32,axiom,
% 0.28/0.49 ! [A7: $tType] : ( bot @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_FSet_Ofset___Groups_Ominus_33,axiom,
% 0.28/0.49 ! [A7: $tType] : ( minus @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(tcon_Product__Type_Oprod___Countable_Ocountable_34,axiom,
% 0.28/0.49 ! [A7: $tType,A8: $tType] :
% 0.28/0.49 ( ( ( countable @ A7 @ ( type2 @ A7 ) )
% 0.28/0.49 & ( countable @ A8 @ ( type2 @ A8 ) ) )
% 0.28/0.49 => ( countable @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
% 0.28/0.49
% 0.28/0.49 %----Helper facts (3)
% 0.28/0.49 thf(help_If_3_1_T,axiom,
% 0.28/0.49 ! [P2: $o] :
% 0.28/0.49 ( ( P2 = $true )
% 0.28/0.49 | ( P2 = $false ) ) ).
% 0.28/0.49
% 0.28/0.49 thf(help_If_2_1_T,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,Y: A] :
% 0.28/0.49 ( ( if @ A @ $false @ X5 @ Y )
% 0.28/0.49 = Y ) ).
% 0.28/0.49
% 0.28/0.49 thf(help_If_1_1_T,axiom,
% 0.28/0.49 ! [A: $tType,X5: A,Y: A] :
% 0.28/0.49 ( ( if @ A @ $true @ X5 @ Y )
% 0.28/0.49 = X5 ) ).
% 0.28/0.49
% 0.28/0.49 %----Conjectures (1)
% 0.28/0.49 thf(conj_0,conjecture,
% 0.28/0.49 ! [X2: rule] :
% 0.28/0.49 ( ( member @ rule @ X2 @ ( sset @ rule @ rules ) )
% 0.28/0.49 => ( abstra726722745urated @ rule @ state @ eff @ X2 @ steps ) ) ).
% 0.28/0.49
% 0.28/0.49 %------------------------------------------------------------------------------
% 0.28/0.49 ------------------------
% 3.16/1.75 ---- Embedded in THF ---
% 3.16/1.75 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.16/1.75 %%% Generated on Tue Feb 24 04:15:31 EST 2026
% 3.16/1.75 %%% using '$$dhol' embedding, version 1.3.0.
% 3.16/1.75 %%% Logic specification used:
% 3.16/1.75 %%% thf(spec, logic, $$dhol).
% 3.16/1.75
% 3.16/1.75 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.JVLNTEOK4C/DTF2DTF19524.p
% 3.16/1.75 thf(ty_t_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree, type, abstra2103299360e_tree: ($tType > $tType)).
% 3.16/1.75 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.16/1.75 thf(ty_t_Stream_Ostream, type, stream: ($tType > $tType)).
% 3.16/1.75 thf(ty_t_FSet_Ofset, type, fset: ($tType > $tType)).
% 3.16/1.75 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.16/1.75 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.16/1.75 thf(ty_tf_state, type, state: $tType).
% 3.16/1.75 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.16/1.75 thf(ty_tf_rule, type, rule: $tType).
% 3.16/1.75 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_OPersistentRuleSystem, type, abstra612932245System: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > ((set @ State) > $o)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem, type, abstra1326562878System: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > ((set @ State) > $o)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_OminWait, type, abstra1332369113inWait: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > (State > nat)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Oper, type, abstra2096684367le_per: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > ((set @ State) > (Rule > $o))))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Opos, type, abstra2097340358le_pos: (!> [A:$tType]: (((stream @ A) > (A > nat))))).
% 3.16/1.75 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)) > $o)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oenabled, type, abstra1874422341nabled: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > (Rule > (State > $o)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oepath, type, abstra523868654_epath: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > ((stream @ ((product_prod @ State) @ Rule)) > $o)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofair, type, abstra928354080m_fair: (!> [Rule:$tType]: (((stream @ Rule) > ((stream @ Rule) > $o))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofenum, type, abstra1774373515_fenum: (!> [Rule:$tType]: (((stream @ Rule) > (stream @ Rule))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OmkTree, type, abstra1225283448mkTree: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > (State > (abstra2103299360e_tree @ ((product_prod @ State) @ Rule)))))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OpickEff, type, abstra1276541928ickEff: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > (Rule > (State > (fset @ State))))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Osaturated, type, abstra726722745urated: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > (Rule > ((stream @ ((product_prod @ State) @ Rule)) > $o)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Otrim, type, abstra1259602206m_trim: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > (State > (stream @ Rule))))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Owf, type, abstra1874736267tem_wf: (!> [Rule:$tType,State:$tType]: (((Rule > (State > ((fset @ State) > $o))) > ((stream @ Rule) > ((abstra2103299360e_tree @ ((product_prod @ State) @ Rule)) > $o)))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Oipath, type, abstra313004635_ipath: (!> [A:$tType]: (((abstra2103299360e_tree @ A) > ((stream @ A) > $o))))).
% 3.16/1.75 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocont, type, abstra1749095923e_cont: (!> [A:$tType]: (((abstra2103299360e_tree @ A) > (fset @ (abstra2103299360e_tree @ A)))))).
% 3.16/1.75 thf(sy_c_Countable__Set_Ocountable, type, countable_countable: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.16/1.75 thf(sy_c_FSet_Ofimage, type, fimage: (!> [A:$tType,B:$tType]: (((A > B) > ((fset @ A) > (fset @ B)))))).
% 3.16/1.75 thf(sy_c_FSet_Ofmember, type, fmember: (!> [A:$tType]: ((A > ((fset @ A) > $o))))).
% 3.16/1.75 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 3.16/1.75 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 3.16/1.75 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 3.16/1.75 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 3.16/1.75 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.16/1.75 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.16/1.75 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 3.16/1.75 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.16/1.75 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 3.16/1.75 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.16/1.75 thf(sy_c_Stream_Osdrop, type, sdrop: (!> [A:$tType]: ((nat > ((stream @ A) > (stream @ A)))))).
% 3.16/1.75 thf(sy_c_Stream_Ostream_Oshd, type, shd: (!> [A:$tType]: (((stream @ A) > A)))).
% 3.16/1.75 thf(sy_c_Stream_Ostream_Osset, type, sset: (!> [A:$tType]: (((stream @ A) > (set @ A))))).
% 3.16/1.75 thf(sy_c_Stream_Ostream_Ostl, type, stl: (!> [A:$tType]: (((stream @ A) > (stream @ A))))).
% 3.16/1.75 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.16/1.75 thf(sy_v_S, type, s: (set @ state)).
% 3.16/1.75 thf(sy_v_eff, type, eff: (rule > (state > ((fset @ state) > $o)))).
% 3.16/1.75 thf(sy_v_rs, type, rs: (stream @ rule)).
% 3.16/1.75 thf(sy_v_rules, type, rules: (stream @ rule)).
% 3.16/1.75 thf(sy_v_s, type, s2: state).
% 3.16/1.75 thf(sy_v_steps, type, steps: (stream @ ((product_prod @ state) @ rule))).
% 3.16/1.75 thf(fact_0__092_060open_062_092_060And_062ra_O_Ara_A_092_060in_062_AR_A_092_060Longrightarrow_062_Asaturated_Ara_Asteps_092_060close_062, axiom, (! [R:rule]: (((((member @ rule) @ R) @ ((sset @ rule) @ rules)) => (((((abstra726722745urated @ rule) @ state) @ eff) @ R) @ steps))))).
% 3.16/1.75 thf(fact_1_Saturated__def, axiom, (! [Steps:(stream @ ((product_prod @ state) @ rule))]: (((((((abstra1209608345urated @ rule) @ state) @ eff) @ rules) @ Steps) = (! [X:rule]: (((((member @ rule) @ X) @ ((sset @ rule) @ rules)) => (((((abstra726722745urated @ rule) @ state) @ eff) @ X) @ Steps)))))))).
% 3.16/1.75 thf(fact_2_sset__fenum, axiom, (((sset @ rule) @ ((abstra1774373515_fenum @ rule) @ rules)) = ((sset @ rule) @ rules))).
% 3.16/1.75 thf(fact_3_assms_I2_J, axiom, (((abstra928354080m_fair @ rule) @ rules) @ rs)).
% 3.16/1.75 thf(fact_4_RuleSystem__Defs_OSaturated__def, axiom, (! [State:$tType,Rule:$tType]: ((((abstra1209608345urated @ Rule) @ State) = (^ [Eff:(Rule > (State > ((fset @ State) > $o))),Rules:(stream @ Rule),Steps2:(stream @ ((product_prod @ State) @ Rule))]: ((! [X:Rule]: (((((member @ Rule) @ X) @ ((sset @ Rule) @ Rules)) => (((((abstra726722745urated @ Rule) @ State) @ Eff) @ X) @ Steps2)))))))))).
% 3.16/1.75 thf(fact_5_enabled__R, axiom, (! [S:state]: (((((member @ state) @ S) @ s) => (? [X2:rule]: (((((member @ rule) @ X2) @ ((sset @ rule) @ rules)) & (? [X1:(fset @ state)]: ((((eff @ X2) @ S) @ X1)))))))))).
% 3.16/1.75 thf(fact_6_NE__R, axiom, (((sset @ rule) @ rules) != (bot_bot @ (set @ rule)))).
% 3.16/1.75 thf(fact_7_PersistentRuleSystem__axioms, axiom, (((((abstra612932245System @ rule) @ state) @ eff) @ rules) @ s)).
% 3.16/1.75 thf(fact_8_enabled__def, axiom, (! [R:rule,S:state]: (((((((abstra1874422341nabled @ rule) @ state) @ eff) @ R) @ S) = ((^ [P:((fset @ state) > $o)]: ((? [X3:(fset @ state)]: ((P @ X3))))) @ ((eff @ R) @ S)))))).
% 3.16/1.75 thf(fact_9_countable__R, axiom, ((countable_countable @ rule) @ ((sset @ rule) @ rules))).
% 3.16/1.75 thf(fact_10_per, axiom, (! [R:rule]: (((((member @ rule) @ R) @ ((sset @ rule) @ rules)) => ((((((abstra2096684367le_per @ rule) @ state) @ eff) @ rules) @ s) @ R))))).
% 3.16/1.75 thf(fact_11_assms_I3_J, axiom, (((abstra313004635_ipath @ ((product_prod @ state) @ rule)) @ (((((abstra1225283448mkTree @ rule) @ state) @ eff) @ rs) @ s2)) @ steps)).
% 3.16/1.75 thf(fact_12_RuleSystem__Defs_Osset__fenum, axiom, (! [Rule:$tType,Rules2:(stream @ Rule)]: ((((sset @ Rule) @ ((abstra1774373515_fenum @ Rule) @ Rules2)) = ((sset @ Rule) @ Rules2))))).
% 3.16/1.75 thf(fact_13_eff__S, axiom, (! [S:state,R:rule,Sl:(fset @ state),S2:state]: (((((member @ state) @ S) @ s) => ((((member @ rule) @ R) @ ((sset @ rule) @ rules)) => ((((eff @ R) @ S) @ Sl) => ((((fmember @ state) @ S2) @ Sl) => (((member @ state) @ S2) @ s)))))))).
% 3.16/1.75 thf(fact_14_RuleSystem__axioms, axiom, (((((abstra1326562878System @ rule) @ state) @ eff) @ rules) @ s)).
% 3.16/1.75 thf(fact_15_assms_I1_J, axiom, (((member @ state) @ s2) @ s)).
% 3.16/1.75 thf(fact_16_fair__fenum, axiom, (((abstra928354080m_fair @ rule) @ rules) @ ((abstra1774373515_fenum @ rule) @ rules))).
% 3.16/1.75 thf(fact_17_ipath__mkTree__saturated, axiom, (! [S:state,Rs:(stream @ rule),Steps:(stream @ ((product_prod @ state) @ rule)),R:rule]: (((((member @ state) @ S) @ s) => ((((abstra928354080m_fair @ rule) @ rules) @ Rs) => ((((abstra313004635_ipath @ ((product_prod @ state) @ rule)) @ (((((abstra1225283448mkTree @ rule) @ state) @ eff) @ Rs) @ S)) @ Steps) => ((((member @ rule) @ R) @ ((sset @ rule) @ rules)) => (((((abstra726722745urated @ rule) @ state) @ eff) @ R) @ Steps)))))))).
% 3.16/1.75 thf(fact_18_pickEff, axiom, (! [R:rule,S:state]: (((((((abstra1874422341nabled @ rule) @ state) @ eff) @ R) @ S) => (((eff @ R) @ S) @ (((((abstra1276541928ickEff @ rule) @ state) @ eff) @ R) @ S)))))).
% 3.16/1.75 thf(fact_19_PersistentRuleSystem_Oaxioms_I1_J, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State)]: (((((((abstra612932245System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => (((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3))))).
% 3.16/1.75 thf(fact_20_RuleSystem__Defs_Ofair__fenum, axiom, (! [Rule:$tType,Rules2:(stream @ Rule)]: ((((abstra928354080m_fair @ Rule) @ Rules2) @ ((abstra1774373515_fenum @ Rule) @ Rules2))))).
% 3.16/1.75 thf(fact_21_RuleSystem__Defs_Oenabled__def, axiom, (! [State:$tType,Rule:$tType]: ((((abstra1874422341nabled @ Rule) @ State) = (^ [Eff:(Rule > (State > ((fset @ State) > $o))),R2:Rule,S4:State]: (((^ [P:((fset @ State) > $o)]: ((? [X3:(fset @ State)]: ((P @ X3))))) @ ((Eff @ R2) @ S4)))))))).
% 3.16/1.75 thf(fact_22_RuleSystem_Ointro, axiom, (! [Rule:$tType,State:$tType,S3:(set @ State),Rules2:(stream @ Rule),Eff2:(Rule > (State > ((fset @ State) > $o)))]: (((! [S5:State]: (((((member @ State) @ S5) @ S3) => (! [R3:Rule]: (((((member @ Rule) @ R3) @ ((sset @ Rule) @ Rules2)) => (! [Sl2:(fset @ State)]: (((((Eff2 @ R3) @ S5) @ Sl2) => (! [S6:State]: (((((fmember @ State) @ S6) @ Sl2) => (((member @ State) @ S6) @ S3))))))))))))) => ((! [S5:State]: (((((member @ State) @ S5) @ S3) => (? [X4:Rule]: (((((member @ Rule) @ X4) @ ((sset @ Rule) @ Rules2)) & (? [X12:(fset @ State)]: ((((Eff2 @ X4) @ S5) @ X12))))))))) => (((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3)))))).
% 3.16/1.75 thf(fact_23_RuleSystem_Oeff__S, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,R:Rule,Sl:(fset @ State),S2:State]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((member @ Rule) @ R) @ ((sset @ Rule) @ Rules2)) => ((((Eff2 @ R) @ S) @ Sl) => ((((fmember @ State) @ S2) @ Sl) => (((member @ State) @ S2) @ S3))))))))).
% 3.16/1.75 thf(fact_24_RuleSystem__def, axiom, (! [State:$tType,Rule:$tType]: ((((abstra1326562878System @ Rule) @ State) = (^ [Eff:(Rule > (State > ((fset @ State) > $o))),Rules:(stream @ Rule),S7:(set @ State)]: (((! [S4:State]: (((((member @ State) @ S4) @ S7) => (! [R2:Rule]: (((((member @ Rule) @ R2) @ ((sset @ Rule) @ Rules)) => (! [Sl3:(fset @ State)]: (((((Eff @ R2) @ S4) @ Sl3) => (! [S8:State]: (((((fmember @ State) @ S8) @ Sl3) => (((member @ State) @ S8) @ S7))))))))))))) & (! [S4:State]: (((((member @ State) @ S4) @ S7) => (? [X:Rule]: (((((member @ Rule) @ X) @ ((sset @ Rule) @ Rules)) & ((^ [P:((fset @ State) > $o)]: ((? [X3:(fset @ State)]: ((P @ X3))))) @ ((Eff @ X) @ S4))))))))))))))).
% 3.16/1.75 thf(fact_25_PersistentRuleSystem_Oipath__mkTree__saturated, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule),Steps:(stream @ ((product_prod @ State) @ Rule)),R:Rule]: (((((((abstra612932245System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => ((((abstra313004635_ipath @ ((product_prod @ State) @ Rule)) @ (((((abstra1225283448mkTree @ Rule) @ State) @ Eff2) @ Rs) @ S)) @ Steps) => ((((member @ Rule) @ R) @ ((sset @ Rule) @ Rules2)) => (((((abstra726722745urated @ Rule) @ State) @ Eff2) @ R) @ Steps))))))))).
% 3.16/1.75 thf(fact_26_RuleSystem__Defs_Ocountable__R, axiom, (! [Rule:$tType,Rules2:(stream @ Rule)]: (((countable_countable @ Rule) @ ((sset @ Rule) @ Rules2))))).
% 3.16/1.75 thf(fact_27_RuleSystem__Defs_ONE__R, axiom, (! [Rule:$tType,Rules2:(stream @ Rule)]: ((((sset @ Rule) @ Rules2) != (bot_bot @ (set @ Rule)))))).
% 3.16/1.75 thf(fact_28_PersistentRuleSystem_Oper, axiom, (! [State:$tType,Rule:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),R:Rule]: (((((((abstra612932245System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ Rule) @ R) @ ((sset @ Rule) @ Rules2)) => ((((((abstra2096684367le_per @ Rule) @ State) @ Eff2) @ Rules2) @ S3) @ R)))))).
% 3.16/1.75 thf(fact_29_RuleSystem_Oenabled__R, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => (? [X2:Rule]: (((((member @ Rule) @ X2) @ ((sset @ Rule) @ Rules2)) & (? [X1:(fset @ State)]: ((((Eff2 @ X2) @ S) @ X1))))))))))).
% 3.16/1.75 thf(fact_30_wf__mkTree, axiom, (! [S:state,Rs:(stream @ rule)]: (((((member @ state) @ S) @ s) => ((((abstra928354080m_fair @ rule) @ rules) @ Rs) => (((((abstra1874736267tem_wf @ rule) @ state) @ eff) @ rules) @ (((((abstra1225283448mkTree @ rule) @ state) @ eff) @ Rs) @ S))))))).
% 3.16/1.75 thf(fact_31_trim__fair, axiom, (! [S:state,Rs:(stream @ rule)]: (((((member @ state) @ S) @ s) => ((((abstra928354080m_fair @ rule) @ rules) @ Rs) => (((abstra928354080m_fair @ rule) @ rules) @ (((((abstra1259602206m_trim @ rule) @ state) @ eff) @ Rs) @ S))))))).
% 3.16/1.75 thf(fact_32_countable__empty, axiom, (! [A:$tType]: (((countable_countable @ A) @ (bot_bot @ (set @ A)))))).
% 3.16/1.75 thf(fact_33_per__def, axiom, (! [R:rule]: ((((((((abstra2096684367le_per @ rule) @ state) @ eff) @ rules) @ s) @ R) = (! [S4:state,R1:rule,Sl4:(fset @ state),S8:state]: ((((((member @ state) @ S4) @ s) & ((((((abstra1874422341nabled @ rule) @ state) @ eff) @ R) @ S4) & ((((member @ rule) @ R1) @ (((minus_minus @ (set @ rule)) @ ((sset @ rule) @ rules)) @ (((insert @ rule) @ R) @ (bot_bot @ (set @ rule))))) & ((((eff @ R1) @ S4) @ Sl4) & (((fmember @ state) @ S8) @ Sl4))))) => (((((abstra1874422341nabled @ rule) @ state) @ eff) @ R) @ S8)))))))).
% 3.16/1.75 thf(fact_34_countableI__type, axiom, (! [A:$tType]: ((((countable @ A) @ (type2 @ A)) => (! [A2:(set @ A)]: (((countable_countable @ A) @ A2))))))).
% 3.16/1.75 thf(fact_35_sdrop__fair, axiom, (! [Rs:(stream @ rule),M:nat]: (((((abstra928354080m_fair @ rule) @ rules) @ Rs) => (((abstra928354080m_fair @ rule) @ rules) @ (((sdrop @ rule) @ M) @ Rs)))))).
% 3.16/1.75 thf(fact_36_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 3.16/1.75 thf(fact_37_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 3.16/1.75 thf(fact_38_Collect__empty__eq, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (! [X:A]: ((~ ((P2 @ X))))))))).
% 3.16/1.75 thf(fact_39_empty__Collect__eq, axiom, (! [A:$tType,P2:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P2)) = (! [X:A]: ((~ ((P2 @ X))))))))).
% 3.16/1.75 thf(fact_40_insert__absorb2, axiom, (! [A:$tType,X5:A,A2:(set @ A)]: (((((insert @ A) @ X5) @ (((insert @ A) @ X5) @ A2)) = (((insert @ A) @ X5) @ A2))))).
% 3.16/1.75 thf(fact_41_insert__iff, axiom, (! [A:$tType,A3:A,B2:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B2) @ A2)) = ((A3 = B2) | (((member @ A) @ A3) @ A2)))))).
% 3.16/1.75 thf(fact_42_insertCI, axiom, (! [A:$tType,A3:A,B3:(set @ A),B2:A]: ((((~ ((((member @ A) @ A3) @ B3))) => (A3 = B2)) => (((member @ A) @ A3) @ (((insert @ A) @ B2) @ B3)))))).
% 3.16/1.75 thf(fact_43_Diff__idemp, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) @ B3) = (((minus_minus @ (set @ A)) @ A2) @ B3))))).
% 3.16/1.75 thf(fact_44_Diff__iff, axiom, (! [A:$tType,C:A,A2:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) = ((((member @ A) @ C) @ A2) & (~ ((((member @ A) @ C) @ B3)))))))).
% 3.16/1.75 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P2:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P2)) = (P2 @ A3))))).
% 3.16/1.75 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) = A2)))).
% 3.16/1.75 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) = (Q @ X2)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 3.16/1.75 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 3.16/1.75 thf(fact_49_DiffI, axiom, (! [A:$tType,C:A,A2:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ A2) => ((~ ((((member @ A) @ C) @ B3))) => (((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B3))))))).
% 3.16/1.75 thf(fact_50_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 3.16/1.75 thf(fact_51_Diff__cancel, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ A2) = (bot_bot @ (set @ A)))))).
% 3.16/1.75 thf(fact_52_empty__Diff, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2) = (bot_bot @ (set @ A)))))).
% 3.16/1.75 thf(fact_53_Diff__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = A2)))).
% 3.16/1.75 thf(fact_54_insert__Diff1, axiom, (! [A:$tType,X5:A,B3:(set @ A),A2:(set @ A)]: (((((member @ A) @ X5) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X5) @ A2)) @ B3) = (((minus_minus @ (set @ A)) @ A2) @ B3)))))).
% 3.16/1.75 thf(fact_55_Diff__insert0, axiom, (! [A:$tType,X5:A,A2:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X5) @ A2))) => ((((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ B3)) = (((minus_minus @ (set @ A)) @ A2) @ B3)))))).
% 3.16/1.75 thf(fact_56_countable__insert__eq, axiom, (! [A:$tType,X5:A,A2:(set @ A)]: ((((countable_countable @ A) @ (((insert @ A) @ X5) @ A2)) = ((countable_countable @ A) @ A2))))).
% 3.16/1.75 thf(fact_57_countable__insert, axiom, (! [A:$tType,A2:(set @ A),A3:A]: ((((countable_countable @ A) @ A2) => ((countable_countable @ A) @ (((insert @ A) @ A3) @ A2)))))).
% 3.16/1.75 thf(fact_58_countable__Diff, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: ((((countable_countable @ A) @ A2) => ((countable_countable @ A) @ (((minus_minus @ (set @ A)) @ A2) @ B3)))))).
% 3.16/1.75 thf(fact_59_insert__Diff__single, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((insert @ A) @ A3) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A3) @ A2))))).
% 3.16/1.75 thf(fact_60_countable__Diff__eq, axiom, (! [A:$tType,A2:(set @ A),X5:A]: ((((countable_countable @ A) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))) = ((countable_countable @ A) @ A2))))).
% 3.16/1.75 thf(fact_61_Diff__insert, axiom, (! [A:$tType,A2:(set @ A),A3:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 3.16/1.75 thf(fact_62_insert__Diff, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((insert @ A) @ A3) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) = A2))))).
% 3.16/1.75 thf(fact_63_Diff__insert2, axiom, (! [A:$tType,A2:(set @ A),A3:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))) @ B3))))).
% 3.16/1.75 thf(fact_64_Diff__insert__absorb, axiom, (! [A:$tType,X5:A,A2:(set @ A)]: (((~ ((((member @ A) @ X5) @ A2))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X5) @ A2)) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))) = A2))))).
% 3.16/1.75 thf(fact_65_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => (? [B4:(set @ A)]: (((A2 = (((insert @ A) @ A3) @ B4)) & (~ ((((member @ A) @ A3) @ B4)))))))))).
% 3.16/1.75 thf(fact_66_insert__commute, axiom, (! [A:$tType,X5:A,Y:A,A2:(set @ A)]: (((((insert @ A) @ X5) @ (((insert @ A) @ Y) @ A2)) = (((insert @ A) @ Y) @ (((insert @ A) @ X5) @ A2)))))).
% 3.16/1.75 thf(fact_67_insert__Diff__if, axiom, (! [A:$tType,X5:A,B3:(set @ A),A2:(set @ A)]: ((((((member @ A) @ X5) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X5) @ A2)) @ B3) = (((minus_minus @ (set @ A)) @ A2) @ B3))) & ((~ ((((member @ A) @ X5) @ B3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X5) @ A2)) @ B3) = (((insert @ A) @ X5) @ (((minus_minus @ (set @ A)) @ A2) @ B3)))))))).
% 3.16/1.75 thf(fact_68_insert__eq__iff, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:A,B3:(set @ A)]: (((~ ((((member @ A) @ A3) @ A2))) => ((~ ((((member @ A) @ B2) @ B3))) => (((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B2) @ B3)) = (((A3 = B2) => (A2 = B3)) & ((A3 != B2) => (? [C2:(set @ A)]: (((A2 = (((insert @ A) @ B2) @ C2)) & ((~ ((((member @ A) @ B2) @ C2))) & ((B3 = (((insert @ A) @ A3) @ C2)) & (~ ((((member @ A) @ A3) @ C2)))))))))))))))).
% 3.16/1.75 thf(fact_69_insert__absorb, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((insert @ A) @ A3) @ A2) = A2))))).
% 3.16/1.75 thf(fact_70_insert__ident, axiom, (! [A:$tType,X5:A,A2:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X5) @ A2))) => ((~ ((((member @ A) @ X5) @ B3))) => (((((insert @ A) @ X5) @ A2) = (((insert @ A) @ X5) @ B3)) = (A2 = B3))))))).
% 3.16/1.75 thf(fact_71_Set_Oset__insert, axiom, (! [A:$tType,X5:A,A2:(set @ A)]: (((((member @ A) @ X5) @ A2) => (~ ((! [B4:(set @ A)]: (((A2 = (((insert @ A) @ X5) @ B4)) => (((member @ A) @ X5) @ B4)))))))))).
% 3.16/1.75 thf(fact_72_insertI2, axiom, (! [A:$tType,A3:A,B3:(set @ A),B2:A]: (((((member @ A) @ A3) @ B3) => (((member @ A) @ A3) @ (((insert @ A) @ B2) @ B3)))))).
% 3.16/1.75 thf(fact_73_insertI1, axiom, (! [A:$tType,A3:A,B3:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B3))))).
% 3.16/1.75 thf(fact_74_insertE, axiom, (! [A:$tType,A3:A,B2:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B2) @ A2)) => ((A3 != B2) => (((member @ A) @ A3) @ A2)))))).
% 3.16/1.75 thf(fact_75_DiffD2, axiom, (! [A:$tType,C:A,A2:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) => (~ ((((member @ A) @ C) @ B3))))))).
% 3.16/1.75 thf(fact_76_DiffD1, axiom, (! [A:$tType,C:A,A2:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) => (((member @ A) @ C) @ A2))))).
% 3.16/1.75 thf(fact_77_DiffE, axiom, (! [A:$tType,C:A,A2:(set @ A),B3:(set @ A)]: (((((member @ A) @ C) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) => (~ (((((member @ A) @ C) @ A2) => (((member @ A) @ C) @ B3)))))))).
% 3.16/1.75 thf(fact_78_singleton__inject, axiom, (! [A:$tType,A3:A,B2:A]: ((((((insert @ A) @ A3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A3 = B2))))).
% 3.16/1.75 thf(fact_79_insert__not__empty, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((insert @ A) @ A3) @ A2) != (bot_bot @ (set @ A)))))).
% 3.16/1.75 thf(fact_80_doubleton__eq__iff, axiom, (! [A:$tType,A3:A,B2:A,C:A,D:A]: ((((((insert @ A) @ A3) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C) @ (((insert @ A) @ D) @ (bot_bot @ (set @ A))))) = (((A3 = C) & (B2 = D)) | ((A3 = D) & (B2 = C))))))).
% 3.16/1.75 thf(fact_81_singleton__iff, axiom, (! [A:$tType,B2:A,A3:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = (B2 = A3))))).
% 3.16/1.75 thf(fact_82_singletonD, axiom, (! [A:$tType,B2:A,A3:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => (B2 = A3))))).
% 3.16/1.75 thf(fact_83_uncountable__minus__countable, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((~ (((countable_countable @ A) @ A2))) => (((countable_countable @ A) @ B3) => (~ (((countable_countable @ A) @ (((minus_minus @ (set @ A)) @ A2) @ B3))))))))).
% 3.16/1.75 thf(fact_84_RuleSystem__Defs_Osdrop__fair, axiom, (! [Rule:$tType,Rules2:(stream @ Rule),Rs:(stream @ Rule),M:nat]: (((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => (((abstra928354080m_fair @ Rule) @ Rules2) @ (((sdrop @ Rule) @ M) @ Rs)))))).
% 3.16/1.75 thf(fact_85_RuleSystem_Oipath__mkTree__sdrop, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule),Steps:(stream @ ((product_prod @ State) @ Rule)),M:nat]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => ((((abstra313004635_ipath @ ((product_prod @ State) @ Rule)) @ (((((abstra1225283448mkTree @ Rule) @ State) @ Eff2) @ Rs) @ S)) @ Steps) => (? [N:nat,S6:State]: (((((member @ State) @ S6) @ S3) & (((abstra313004635_ipath @ ((product_prod @ State) @ Rule)) @ (((((abstra1225283448mkTree @ Rule) @ State) @ Eff2) @ (((sdrop @ Rule) @ N) @ Rs)) @ S6)) @ (((sdrop @ ((product_prod @ State) @ Rule)) @ M) @ Steps)))))))))))).
% 3.16/1.75 thf(fact_86_RuleSystem__Defs_OpickEff, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),R:Rule,S:State]: (((((((abstra1874422341nabled @ Rule) @ State) @ Eff2) @ R) @ S) => (((Eff2 @ R) @ S) @ (((((abstra1276541928ickEff @ Rule) @ State) @ Eff2) @ R) @ S)))))).
% 3.16/1.75 thf(fact_87_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 3.16/1.75 thf(fact_88_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 3.16/1.75 thf(fact_89_equals0D, axiom, (! [A:$tType,A2:(set @ A),A3:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A2))))))).
% 3.16/1.75 thf(fact_90_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 3.16/1.75 thf(fact_91_RuleSystem_Oper__def, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),R:Rule]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => (((((((abstra2096684367le_per @ Rule) @ State) @ Eff2) @ Rules2) @ S3) @ R) = (! [S4:State,R1:Rule,Sl4:(fset @ State),S8:State]: ((((((member @ State) @ S4) @ S3) & ((((((abstra1874422341nabled @ Rule) @ State) @ Eff2) @ R) @ S4) & ((((member @ Rule) @ R1) @ (((minus_minus @ (set @ Rule)) @ ((sset @ Rule) @ Rules2)) @ (((insert @ Rule) @ R) @ (bot_bot @ (set @ Rule))))) & ((((Eff2 @ R1) @ S4) @ Sl4) & (((fmember @ State) @ S8) @ Sl4))))) => (((((abstra1874422341nabled @ Rule) @ State) @ Eff2) @ R) @ S8))))))))).
% 3.16/1.75 thf(fact_92_RuleSystem_Otrim__fair, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule)]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => (((abstra928354080m_fair @ Rule) @ Rules2) @ (((((abstra1259602206m_trim @ Rule) @ State) @ Eff2) @ Rs) @ S)))))))).
% 3.16/1.75 thf(fact_93_RuleSystem_Owf__mkTree, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule)]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => (((((abstra1874736267tem_wf @ Rule) @ State) @ Eff2) @ Rules2) @ (((((abstra1225283448mkTree @ Rule) @ State) @ Eff2) @ Rs) @ S)))))))).
% 3.16/1.75 thf(fact_94_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))))))).
% 3.16/1.75 thf(fact_95_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)))))).
% 3.16/1.75 thf(fact_96_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) => (? [N:nat,S6:state]: (((((member @ state) @ S6) @ s) & (((abstra313004635_ipath @ ((product_prod @ state) @ rule)) @ (((((abstra1225283448mkTree @ rule) @ state) @ eff) @ (((sdrop @ rule) @ N) @ Rs)) @ S6)) @ (((sdrop @ ((product_prod @ state) @ rule)) @ M) @ Steps))))))))))).
% 3.16/1.75 thf(fact_97_minWait__ex, axiom, (! [S:state,Rs:(stream @ rule)]: (((((member @ state) @ S) @ s) => ((((abstra928354080m_fair @ rule) @ rules) @ Rs) => (? [N:nat]: ((((((abstra1874422341nabled @ rule) @ state) @ eff) @ ((shd @ rule) @ (((sdrop @ rule) @ N) @ Rs))) @ S)))))))).
% 3.16/1.75 thf(fact_98_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)))))).
% 3.16/1.75 thf(fact_99_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))))))).
% 3.16/1.75 thf(fact_100_RuleSystem_Otrim__alt, axiom, (! [State:$tType,Rule:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule)]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => ((((((abstra1259602206m_trim @ Rule) @ State) @ Eff2) @ Rs) @ S) = (((sdrop @ Rule) @ (((((abstra1332369113inWait @ Rule) @ State) @ Eff2) @ Rs) @ S)) @ Rs)))))))).
% 3.16/1.75 thf(fact_101_bot__apply, axiom, (! [C3:$tType,D2:$tType]: ((((bot @ C3) @ (type2 @ C3)) => ((bot_bot @ (D2 > C3)) = (^ [X:D2]: ((bot_bot @ C3)))))))).
% 3.16/1.75 thf(fact_102_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 3.16/1.75 thf(fact_103_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X:A]: ((bot_bot @ B)))))))).
% 3.16/1.75 thf(fact_104_RuleSystem_Otrim__in__R, axiom, (! [State:$tType,Rule:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule)]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => (((member @ Rule) @ ((shd @ Rule) @ (((((abstra1259602206m_trim @ Rule) @ State) @ Eff2) @ Rs) @ S))) @ ((sset @ Rule) @ Rules2)))))))).
% 3.16/1.75 thf(fact_105_RuleSystem_OminWait__ex, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule)]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => (? [N:nat]: ((((((abstra1874422341nabled @ Rule) @ State) @ Eff2) @ ((shd @ Rule) @ (((sdrop @ Rule) @ N) @ Rs))) @ S))))))))).
% 3.16/1.75 thf(fact_106_RuleSystem_Otrim__enabled, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule)]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => (((((abstra1874422341nabled @ Rule) @ State) @ Eff2) @ ((shd @ Rule) @ (((((abstra1259602206m_trim @ Rule) @ State) @ Eff2) @ Rs) @ S))) @ S))))))).
% 3.16/1.75 thf(fact_107_RuleSystem__Defs_Owf__ipath__epath, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),T:(abstra2103299360e_tree @ ((product_prod @ State) @ Rule)),Steps:(stream @ ((product_prod @ State) @ Rule))]: (((((((abstra1874736267tem_wf @ Rule) @ State) @ Eff2) @ Rules2) @ T) => ((((abstra313004635_ipath @ ((product_prod @ State) @ Rule)) @ T) @ Steps) => (((((abstra523868654_epath @ Rule) @ State) @ Eff2) @ Rules2) @ Steps)))))).
% 3.16/1.75 thf(fact_108_pos, axiom, (! [Rs:(stream @ rule),R:rule]: (((((abstra928354080m_fair @ rule) @ rules) @ Rs) => ((((member @ rule) @ R) @ ((sset @ rule) @ rules)) => (((shd @ rule) @ (((sdrop @ rule) @ (((abstra2097340358le_pos @ rule) @ Rs) @ R)) @ Rs)) = R)))))).
% 3.16/1.75 thf(fact_109_minWait__least, axiom, (! [N2:nat,Rs:(stream @ rule),S:state]: (((((((abstra1874422341nabled @ rule) @ state) @ eff) @ ((shd @ rule) @ (((sdrop @ rule) @ N2) @ Rs))) @ S) => (((ord_less_eq @ nat) @ (((((abstra1332369113inWait @ rule) @ state) @ eff) @ Rs) @ S)) @ N2))))).
% 3.16/1.75 thf(fact_110_RuleSystem_Opos, axiom, (! [State:$tType,Rule:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),Rs:(stream @ Rule),R:Rule]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => ((((member @ Rule) @ R) @ ((sset @ Rule) @ Rules2)) => (((shd @ Rule) @ (((sdrop @ Rule) @ (((abstra2097340358le_pos @ Rule) @ Rs) @ R)) @ Rs)) = R))))))).
% 3.16/1.75 thf(fact_111_fempty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((fmember @ A) @ C) @ (bot_bot @ (fset @ A)))))))).
% 3.16/1.75 thf(fact_112_all__not__fin__conv, axiom, (! [A:$tType,A2:(fset @ A)]: (((! [X:A]: ((~ ((((fmember @ A) @ X) @ A2))))) = (A2 = (bot_bot @ (fset @ A))))))).
% 3.16/1.75 thf(fact_113_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A]: ((((ord_less_eq @ A) @ X5) @ X5))))))).
% 3.16/1.75 thf(fact_114_pos__least, axiom, (! [A:$tType,N2:nat,Rs:(stream @ A),R:A]: (((((shd @ A) @ (((sdrop @ A) @ N2) @ Rs)) = R) => (((ord_less_eq @ nat) @ (((abstra2097340358le_pos @ A) @ Rs) @ R)) @ N2))))).
% 3.16/1.75 thf(fact_115_minWait__le__pos, axiom, (! [Rs:(stream @ rule),R:rule,S:state]: (((((abstra928354080m_fair @ rule) @ rules) @ Rs) => ((((member @ rule) @ R) @ ((sset @ rule) @ rules)) => ((((((abstra1874422341nabled @ rule) @ state) @ eff) @ R) @ S) => (((ord_less_eq @ nat) @ (((((abstra1332369113inWait @ rule) @ state) @ eff) @ Rs) @ S)) @ (((abstra2097340358le_pos @ rule) @ Rs) @ R)))))))).
% 3.16/1.75 thf(fact_116_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X5:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X5)) @ (G @ X5))))))))).
% 3.16/1.75 thf(fact_117_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X5:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X5)) @ (G @ X5))))))))).
% 3.16/1.75 thf(fact_118_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 3.16/1.75 thf(fact_119_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))))).
% 3.16/1.75 thf(fact_120_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 3.16/1.75 thf(fact_121_order__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B2:A,F:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ C3) @ (F @ B2)) @ C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C3) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ C3) @ (F @ A3)) @ C)))))))))).
% 3.16/1.75 thf(fact_122_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B2:B,C:B]: (((A3 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A3) @ (F @ C))))))))))).
% 3.16/1.75 thf(fact_123_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A3:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A3) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C)))))))))).
% 3.16/1.75 thf(fact_124_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y3:A,Z:A]: ((Y3 = Z))) = (^ [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X))))))))).
% 3.16/1.75 thf(fact_125_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((((ord_less_eq @ A) @ X5) @ Y) => ((((ord_less_eq @ A) @ Y) @ X5) => (X5 = Y))))))))).
% 3.16/1.75 thf(fact_126_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((((ord_less_eq @ A) @ X5) @ Y) | (((ord_less_eq @ A) @ Y) @ X5)))))))).
% 3.16/1.75 thf(fact_127_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((X5 = Y) => (((ord_less_eq @ A) @ X5) @ Y)))))))).
% 3.16/1.75 thf(fact_128_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X5) @ Y))) => (((ord_less_eq @ A) @ Y) @ X5)))))))).
% 3.16/1.75 thf(fact_129_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 3.16/1.75 thf(fact_130_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X5) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ Z2)))) => (((((ord_less_eq @ A) @ X5) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X5)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X5)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ Y))))))))))))))))).
% 3.16/1.75 thf(fact_131_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X5:A]: (((((ord_less_eq @ A) @ Y) @ X5) => ((((ord_less_eq @ A) @ X5) @ Y) = (X5 = Y))))))))).
% 3.16/1.75 thf(fact_132_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B2:A,C:A]: (((A3 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 3.16/1.75 thf(fact_133_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A3:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A3) @ C))))))))).
% 3.16/1.75 thf(fact_134_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ A3) => (A3 = B2))))))))).
% 3.16/1.75 thf(fact_135_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X5) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X5) @ Z2))))))))).
% 3.16/1.75 thf(fact_136_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 3.16/1.75 thf(fact_137_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P2:(A > (A > $o)),A3:A,B2:A]: (((! [A4:A,B5:A]: (((((ord_less_eq @ A) @ A4) @ B5) => ((P2 @ A4) @ B5)))) => ((! [A4:A,B5:A]: ((((P2 @ B5) @ A4) => ((P2 @ A4) @ B5)))) => ((P2 @ A3) @ B2))))))))).
% 3.16/1.75 thf(fact_138_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A3))))))))).
% 3.16/1.75 thf(fact_139_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ A3) @ B2) => (A3 = B2))))))))).
% 3.16/1.75 thf(fact_140_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 3.16/1.75 thf(fact_141_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 3.16/1.75 thf(fact_142_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 3.16/1.75 thf(fact_143_RuleSystem_Opos__least, axiom, (! [Rule:$tType,State:$tType,A:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),N2:nat,Rs:(stream @ A),R:A]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((shd @ A) @ (((sdrop @ A) @ N2) @ Rs)) = R) => (((ord_less_eq @ nat) @ (((abstra2097340358le_pos @ A) @ Rs) @ R)) @ N2)))))).
% 3.16/1.75 thf(fact_144_eqfelem__imp__iff, axiom, (! [A:$tType,X5:A,Y:A,A2:(fset @ A)]: (((X5 = Y) => ((((fmember @ A) @ X5) @ A2) = (((fmember @ A) @ Y) @ A2)))))).
% 3.16/1.75 thf(fact_145_if__split__fmem2, axiom, (! [A:$tType,A3:A,Q:$o,X5:(fset @ A),Y:(fset @ A)]: (((((fmember @ A) @ A3) @ ((((if @ (fset @ A)) @ Q) @ X5) @ Y)) = ((Q => (((fmember @ A) @ A3) @ X5)) & ((~ (Q)) => (((fmember @ A) @ A3) @ Y))))))).
% 3.16/1.75 thf(fact_146_if__split__fmem1, axiom, (! [A:$tType,Q:$o,X5:A,Y:A,B2:(fset @ A)]: (((((fmember @ A) @ ((((if @ A) @ Q) @ X5) @ Y)) @ B2) = ((Q => (((fmember @ A) @ X5) @ B2)) & ((~ (Q)) => (((fmember @ A) @ Y) @ B2))))))).
% 3.16/1.75 thf(fact_147_eqfset__imp__iff, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A),X5:A]: (((A2 = B3) => ((((fmember @ A) @ X5) @ A2) = (((fmember @ A) @ X5) @ B3)))))).
% 3.16/1.75 thf(fact_148_eq__fmem__trans, axiom, (! [A:$tType,A3:A,B2:A,A2:(fset @ A)]: (((A3 = B2) => ((((fmember @ A) @ B2) @ A2) => (((fmember @ A) @ A3) @ A2)))))).
% 3.16/1.75 thf(fact_149_fset__choice, axiom, (! [B:$tType,A:$tType,A2:(fset @ A),P2:(A > (B > $o))]: (((! [X2:A]: (((((fmember @ A) @ X2) @ A2) => (? [X12:B]: (((P2 @ X2) @ X12)))))) => (? [F3:(A > B)]: ((! [X4:A]: (((((fmember @ A) @ X4) @ A2) => ((P2 @ X4) @ (F3 @ X4))))))))))).
% 3.16/1.75 thf(fact_150_fequalityCE, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A),C:A]: (((A2 = B3) => (((((fmember @ A) @ C) @ A2) => (~ ((((fmember @ A) @ C) @ B3)))) => (~ (((~ ((((fmember @ A) @ C) @ A2))) => (((fmember @ A) @ C) @ B3))))))))).
% 3.16/1.75 thf(fact_151_fset__eqI, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A)]: (((! [X2:A]: (((((fmember @ A) @ X2) @ A2) = (((fmember @ A) @ X2) @ B3)))) => (A2 = B3))))).
% 3.16/1.75 thf(fact_152_femptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((fmember @ A) @ A3) @ (bot_bot @ (fset @ A)))))))).
% 3.16/1.75 thf(fact_153_ex__fin__conv, axiom, (! [A:$tType,A2:(fset @ A)]: (((? [X:A]: ((((fmember @ A) @ X) @ A2))) = (A2 != (bot_bot @ (fset @ A))))))).
% 3.16/1.75 thf(fact_154_equalsffemptyD, axiom, (! [A:$tType,A2:(fset @ A),A3:A]: (((A2 = (bot_bot @ (fset @ A))) => (~ ((((fmember @ A) @ A3) @ A2))))))).
% 3.16/1.75 thf(fact_155_equalsffemptyI, axiom, (! [A:$tType,A2:(fset @ A)]: (((! [Y2:A]: ((~ ((((fmember @ A) @ Y2) @ A2))))) => (A2 = (bot_bot @ (fset @ A))))))).
% 3.16/1.75 thf(fact_156_RuleSystem_OminWait__le__pos, axiom, (! [State:$tType,Rule:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),Rs:(stream @ Rule),R:Rule,S:State]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => ((((member @ Rule) @ R) @ ((sset @ Rule) @ Rules2)) => ((((((abstra1874422341nabled @ Rule) @ State) @ Eff2) @ R) @ S) => (((ord_less_eq @ nat) @ (((((abstra1332369113inWait @ Rule) @ State) @ Eff2) @ Rs) @ S)) @ (((abstra2097340358le_pos @ Rule) @ Rs) @ R))))))))).
% 3.16/1.75 thf(fact_157_RuleSystem_OminWait__least, axiom, (! [State:$tType,Rule:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),N2:nat,Rs:(stream @ Rule),S:State]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((((abstra1874422341nabled @ Rule) @ State) @ Eff2) @ ((shd @ Rule) @ (((sdrop @ Rule) @ N2) @ Rs))) @ S) => (((ord_less_eq @ nat) @ (((((abstra1332369113inWait @ Rule) @ State) @ Eff2) @ Rs) @ S)) @ N2)))))).
% 3.16/1.75 thf(fact_158_minus__apply, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B6:(A > B),X:A]: ((((minus_minus @ B) @ (A5 @ X)) @ (B6 @ X))))))))).
% 3.16/1.75 thf(fact_159_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 3.16/1.75 thf(fact_160_Collect__empty__eq__bot, axiom, (! [A:$tType,P2:(A > $o)]: (((((collect @ A) @ P2) = (bot_bot @ (set @ A))) = (P2 = (bot_bot @ (A > $o))))))).
% 3.16/1.75 thf(fact_161_in__cont__mkTree, axiom, (! [S:state,Rs:(stream @ rule),T2:(abstra2103299360e_tree @ ((product_prod @ state) @ rule))]: (((((member @ state) @ S) @ s) => ((((abstra928354080m_fair @ rule) @ rules) @ Rs) => ((((fmember @ (abstra2103299360e_tree @ ((product_prod @ state) @ rule))) @ T2) @ ((abstra1749095923e_cont @ ((product_prod @ state) @ rule)) @ (((((abstra1225283448mkTree @ rule) @ state) @ eff) @ Rs) @ S))) => (? [Sl5:(fset @ state),S6:state]: (((((member @ state) @ S6) @ s) & ((((eff @ ((shd @ rule) @ (((((abstra1259602206m_trim @ rule) @ state) @ eff) @ Rs) @ S))) @ S) @ Sl5) & ((((fmember @ state) @ S6) @ Sl5) & (T2 = (((((abstra1225283448mkTree @ rule) @ state) @ eff) @ ((stl @ rule) @ (((((abstra1259602206m_trim @ rule) @ state) @ eff) @ Rs) @ S))) @ S6))))))))))))).
% 3.16/1.75 thf(fact_162_is__singletonI, axiom, (! [A:$tType,X5:A]: (((is_singleton @ A) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))))).
% 3.16/1.75 thf(fact_163_fair__stl, axiom, (! [Rs:(stream @ rule)]: (((((abstra928354080m_fair @ rule) @ rules) @ Rs) => (((abstra928354080m_fair @ rule) @ rules) @ ((stl @ rule) @ Rs)))))).
% 3.16/1.75 thf(fact_164_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 3.16/1.75 thf(fact_165_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 3.16/1.75 thf(fact_166_insert__subset, axiom, (! [A:$tType,X5:A,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X5) @ A2)) @ B3) = ((((member @ A) @ X5) @ B3) & (((ord_less_eq @ (set @ A)) @ A2) @ B3)))))).
% 3.16/1.75 thf(fact_167_fminus__iff, axiom, (! [A:$tType,C:A,A2:(fset @ A),B3:(fset @ A)]: (((((fmember @ A) @ C) @ (((minus_minus @ (fset @ A)) @ A2) @ B3)) = ((((fmember @ A) @ C) @ A2) & (~ ((((fmember @ A) @ C) @ B3)))))))).
% 3.16/1.75 thf(fact_168_fminusI, axiom, (! [A:$tType,C:A,A2:(fset @ A),B3:(fset @ A)]: (((((fmember @ A) @ C) @ A2) => ((~ ((((fmember @ A) @ C) @ B3))) => (((fmember @ A) @ C) @ (((minus_minus @ (fset @ A)) @ A2) @ B3))))))).
% 3.16/1.75 thf(fact_169_fsubsetI, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A)]: (((! [X2:A]: (((((fmember @ A) @ X2) @ A2) => (((fmember @ A) @ X2) @ B3)))) => (((ord_less_eq @ (fset @ A)) @ A2) @ B3))))).
% 3.16/1.75 thf(fact_170_fminus__fempty, axiom, (! [A:$tType,A2:(fset @ A)]: (((((minus_minus @ (fset @ A)) @ A2) @ (bot_bot @ (fset @ A))) = A2)))).
% 3.16/1.75 thf(fact_171_fminus__cancel, axiom, (! [A:$tType,A2:(fset @ A)]: (((((minus_minus @ (fset @ A)) @ A2) @ A2) = (bot_bot @ (fset @ A)))))).
% 3.16/1.75 thf(fact_172_fempty__fminus, axiom, (! [A:$tType,A2:(fset @ A)]: (((((minus_minus @ (fset @ A)) @ (bot_bot @ (fset @ A))) @ A2) = (bot_bot @ (fset @ A)))))).
% 3.16/1.75 thf(fact_173_fempty__fsubsetI, axiom, (! [A:$tType,X5:(fset @ A)]: ((((ord_less_eq @ (fset @ A)) @ (bot_bot @ (fset @ A))) @ X5)))).
% 3.16/1.75 thf(fact_174_fsubset__fempty, axiom, (! [A:$tType,A2:(fset @ A)]: (((((ord_less_eq @ (fset @ A)) @ A2) @ (bot_bot @ (fset @ A))) = (A2 = (bot_bot @ (fset @ A))))))).
% 3.16/1.75 thf(fact_175_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A2:(set @ A),B2:A]: ((((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A3 = B2) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 3.16/1.75 thf(fact_176_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A3:A,A2:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A3) @ A2)) = ((A3 = B2) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 3.16/1.75 thf(fact_177_Diff__eq__empty__iff, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A2) @ B3) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A2) @ B3))))).
% 3.16/1.75 thf(fact_178_fminusD2, axiom, (! [A:$tType,C:A,A2:(fset @ A),B3:(fset @ A)]: (((((fmember @ A) @ C) @ (((minus_minus @ (fset @ A)) @ A2) @ B3)) => (~ ((((fmember @ A) @ C) @ B3))))))).
% 3.16/1.75 thf(fact_179_fminusD1, axiom, (! [A:$tType,C:A,A2:(fset @ A),B3:(fset @ A)]: (((((fmember @ A) @ C) @ (((minus_minus @ (fset @ A)) @ A2) @ B3)) => (((fmember @ A) @ C) @ A2))))).
% 3.16/1.75 thf(fact_180_fminusE, axiom, (! [A:$tType,C:A,A2:(fset @ A),B3:(fset @ A)]: (((((fmember @ A) @ C) @ (((minus_minus @ (fset @ A)) @ A2) @ B3)) => (~ (((((fmember @ A) @ C) @ A2) => (((fmember @ A) @ C) @ B3)))))))).
% 3.16/1.75 thf(fact_181_fset__mp, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A),X5:A]: (((((ord_less_eq @ (fset @ A)) @ A2) @ B3) => ((((fmember @ A) @ X5) @ A2) => (((fmember @ A) @ X5) @ B3)))))).
% 3.16/1.75 thf(fact_182_fin__mono, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A),X5:A]: (((((ord_less_eq @ (fset @ A)) @ A2) @ B3) => ((((fmember @ A) @ X5) @ A2) => (((fmember @ A) @ X5) @ B3)))))).
% 3.16/1.75 thf(fact_183_fsubsetD, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A),C:A]: (((((ord_less_eq @ (fset @ A)) @ A2) @ B3) => ((((fmember @ A) @ C) @ A2) => (((fmember @ A) @ C) @ B3)))))).
% 3.16/1.75 thf(fact_184_fset__rev__mp, axiom, (! [A:$tType,X5:A,A2:(fset @ A),B3:(fset @ A)]: (((((fmember @ A) @ X5) @ A2) => ((((ord_less_eq @ (fset @ A)) @ A2) @ B3) => (((fmember @ A) @ X5) @ B3)))))).
% 3.16/1.75 thf(fact_185_subset__insertI2, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B2) @ B3)))))).
% 3.16/1.75 thf(fact_186_subset__insertI, axiom, (! [A:$tType,B3:(set @ A),A3:A]: ((((ord_less_eq @ (set @ A)) @ B3) @ (((insert @ A) @ A3) @ B3))))).
% 3.16/1.75 thf(fact_187_subset__insert, axiom, (! [A:$tType,X5:A,A2:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X5) @ A2))) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ B3)) = (((ord_less_eq @ (set @ A)) @ A2) @ B3)))))).
% 3.16/1.75 thf(fact_188_insert__mono, axiom, (! [A:$tType,C4:(set @ A),D3:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ C4) @ D3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A3) @ C4)) @ (((insert @ A) @ A3) @ D3)))))).
% 3.16/1.75 thf(fact_189_RuleSystem__Defs_Ofair__stl, axiom, (! [Rule:$tType,Rules2:(stream @ Rule),Rs:(stream @ Rule)]: (((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => (((abstra928354080m_fair @ Rule) @ Rules2) @ ((stl @ Rule) @ Rs)))))).
% 3.16/1.75 thf(fact_190_double__diff, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C4) => ((((minus_minus @ (set @ A)) @ B3) @ (((minus_minus @ (set @ A)) @ C4) @ A2)) = A2)))))).
% 3.16/1.75 thf(fact_191_Diff__subset, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) @ A2)))).
% 3.16/1.75 thf(fact_192_Diff__mono, axiom, (! [A:$tType,A2:(set @ A),C4:(set @ A),D3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C4) => ((((ord_less_eq @ (set @ A)) @ D3) @ B3) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ B3)) @ (((minus_minus @ (set @ A)) @ C4) @ D3))))))).
% 3.16/1.75 thf(fact_193_countable__subset, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => (((countable_countable @ A) @ B3) => ((countable_countable @ A) @ A2)))))).
% 3.16/1.75 thf(fact_194_subset__singleton__iff, axiom, (! [A:$tType,X6:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ X6) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((X6 = (bot_bot @ (set @ A))) | (X6 = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))).
% 3.16/1.75 thf(fact_195_subset__singletonD, axiom, (! [A:$tType,A2:(set @ A),X5:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))) => ((A2 = (bot_bot @ (set @ A))) | (A2 = (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))))))).
% 3.16/1.75 thf(fact_196_subset__Diff__insert, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A),X5:A,C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B3) @ (((insert @ A) @ X5) @ C4))) = ((((ord_less_eq @ (set @ A)) @ A2) @ (((minus_minus @ (set @ A)) @ B3) @ C4)) & (~ ((((member @ A) @ X5) @ A2)))))))).
% 3.16/1.75 thf(fact_197_is__singletonI_H, axiom, (! [A:$tType,A2:(set @ A)]: (((A2 != (bot_bot @ (set @ A))) => ((! [X2:A,Y2:A]: (((((member @ A) @ X2) @ A2) => ((((member @ A) @ Y2) @ A2) => (X2 = Y2))))) => ((is_singleton @ A) @ A2)))))).
% 3.16/1.75 thf(fact_198_Diff__single__insert, axiom, (! [A:$tType,A2:(set @ A),X5:A,B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))) @ B3) => (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ B3)))))).
% 3.16/1.75 thf(fact_199_subset__insert__iff, axiom, (! [A:$tType,A2:(set @ A),X5:A,B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ B3)) = (((((member @ A) @ X5) @ A2) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A2) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A))))) @ B3)) & ((~ ((((member @ A) @ X5) @ A2))) => (((ord_less_eq @ (set @ A)) @ A2) @ B3))))))).
% 3.16/1.75 thf(fact_200_fun__diff__def, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A5:(A > B),B6:(A > B),X:A]: ((((minus_minus @ B) @ (A5 @ X)) @ (B6 @ X))))))))).
% 3.16/1.75 thf(fact_201_RuleSystem_Oin__cont__mkTree, axiom, (! [Rule:$tType,State:$tType,Eff2:(Rule > (State > ((fset @ State) > $o))),Rules2:(stream @ Rule),S3:(set @ State),S:State,Rs:(stream @ Rule),T2:(abstra2103299360e_tree @ ((product_prod @ State) @ Rule))]: (((((((abstra1326562878System @ Rule) @ State) @ Eff2) @ Rules2) @ S3) => ((((member @ State) @ S) @ S3) => ((((abstra928354080m_fair @ Rule) @ Rules2) @ Rs) => ((((fmember @ (abstra2103299360e_tree @ ((product_prod @ State) @ Rule))) @ T2) @ ((abstra1749095923e_cont @ ((product_prod @ State) @ Rule)) @ (((((abstra1225283448mkTree @ Rule) @ State) @ Eff2) @ Rs) @ S))) => (? [Sl5:(fset @ State),S6:State]: (((((member @ State) @ S6) @ S3) & ((((Eff2 @ ((shd @ Rule) @ (((((abstra1259602206m_trim @ Rule) @ State) @ Eff2) @ Rs) @ S))) @ S) @ Sl5) & ((((fmember @ State) @ S6) @ Sl5) & (T2 = (((((abstra1225283448mkTree @ Rule) @ State) @ Eff2) @ ((stl @ Rule) @ (((((abstra1259602206m_trim @ Rule) @ State) @ Eff2) @ Rs) @ S))) @ S6)))))))))))))).
% 3.16/1.75 thf(fact_202_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A5:(set @ A)]: ((? [X:A]: ((A5 = (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))))).
% 3.16/1.75 thf(fact_203_is__singletonE, axiom, (! [A:$tType,A2:(set @ A)]: ((((is_singleton @ A) @ A2) => (~ ((! [X2:A]: ((A2 != (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 3.16/1.75 thf(fact_204_mkTree_Osimps_I2_J, axiom, (! [Rs:(stream @ rule),S:state]: ((((abstra1749095923e_cont @ ((product_prod @ state) @ rule)) @ (((((abstra1225283448mkTree @ rule) @ state) @ eff) @ Rs) @ S)) = ((((fimage @ state) @ (abstra2103299360e_tree @ ((product_prod @ state) @ rule))) @ ((((abstra1225283448mkTree @ rule) @ state) @ eff) @ ((stl @ rule) @ (((((abstra1259602206m_trim @ rule) @ state) @ eff) @ Rs) @ S)))) @ (((((abstra1276541928ickEff @ rule) @ state) @ eff) @ ((shd @ rule) @ (((((abstra1259602206m_trim @ rule) @ state) @ eff) @ Rs) @ S))) @ S)))))).
% 3.16/1.75 thf(fact_205_sset__induct, axiom, (! [A:$tType,Y:A,S:(stream @ A),P2:(A > ((stream @ A) > $o))]: (((((member @ A) @ Y) @ ((sset @ A) @ S)) => ((! [S5:(stream @ A)]: (((P2 @ ((shd @ A) @ S5)) @ S5))) => ((! [S5:(stream @ A),Y2:A]: (((((member @ A) @ Y2) @ ((sset @ A) @ ((stl @ A) @ S5))) => (((P2 @ Y2) @ ((stl @ A) @ S5)) => ((P2 @ Y2) @ S5))))) => ((P2 @ Y) @ S))))))).
% 3.16/1.75 thf(fact_206_stake__pos__minWait, axiom, (! [Rs:(stream @ rule),S:state,R:rule]: (((((abstra928354080m_fair @ rule) @ rules) @ Rs) => ((((ord_less @ nat) @ (((((abstra1332369113inWait @ rule) @ state) @ eff) @ Rs) @ S)) @ (((abstra2097340358le_pos @ rule) @ Rs) @ R)) => ((((member @ rule) @ R) @ ((sset @ rule) @ rules)) => ((((member @ state) @ S) @ s) => ((((abstra2097340358le_pos @ rule) @ ((stl @ rule) @ (((((abstra1259602206m_trim @ rule) @ state) @ eff) @ Rs) @ S))) @ R) = (((minus_minus @ nat) @ (((abstra2097340358le_pos @ rule) @ Rs) @ R)) @ (suc @ (((((abstra1332369113inWait @ rule) @ state) @ eff) @ Rs) @ S))))))))))).
% 3.16/1.75 thf(fact_207_subsetI, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B3))))).
% 3.16/1.75 thf(fact_208_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A2) => (A2 = B3)))))).
% 3.16/1.75 thf(fact_209_fsubset__antisym, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A)]: (((((ord_less_eq @ (fset @ A)) @ A2) @ B3) => ((((ord_less_eq @ (fset @ A)) @ B3) @ A2) => (A2 = B3)))))).
% 3.16/1.75 thf(fact_210_fminus__idemp, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A)]: (((((minus_minus @ (fset @ A)) @ (((minus_minus @ (fset @ A)) @ A2) @ B3)) @ B3) = (((minus_minus @ (fset @ A)) @ A2) @ B3))))).
% 3.16/1.75 thf(fact_211_fimage__eqI, axiom, (! [A:$tType,B:$tType,B2:A,F:(B > A),X5:B,A2:(fset @ B)]: (((B2 = (F @ X5)) => ((((fmember @ B) @ X5) @ A2) => (((fmember @ A) @ B2) @ ((((fimage @ B) @ A) @ F) @ A2))))))).
% 3.16/1.75 thf(fact_212_fimage__is__fempty, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(fset @ B)]: (((((((fimage @ B) @ A) @ F) @ A2) = (bot_bot @ (fset @ A))) = (A2 = (bot_bot @ (fset @ B))))))).
% 3.16/1.75 thf(fact_213_fempty__is__fimage, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(fset @ B)]: ((((bot_bot @ (fset @ A)) = ((((fimage @ B) @ A) @ F) @ A2)) = (A2 = (bot_bot @ (fset @ B))))))).
% 3.16/1.75 thf(fact_214_fimage__fempty, axiom, (! [B:$tType,A:$tType,F:(B > A)]: ((((((fimage @ B) @ A) @ F) @ (bot_bot @ (fset @ B))) = (bot_bot @ (fset @ A)))))).
% 3.16/1.75 thf(fact_215_sdrop__simps_I2_J, axiom, (! [A:$tType,N2:nat,S:(stream @ A)]: ((((stl @ A) @ (((sdrop @ A) @ N2) @ S)) = (((sdrop @ A) @ (suc @ N2)) @ S))))).
% 3.16/1.75 thf(fact_216_fimage__fminus__fsubset, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(fset @ B),B3:(fset @ B)]: ((((ord_less_eq @ (fset @ A)) @ (((minus_minus @ (fset @ A)) @ ((((fimage @ B) @ A) @ F) @ A2)) @ ((((fimage @ B) @ A) @ F) @ B3))) @ ((((fimage @ B) @ A) @ F) @ (((minus_minus @ (fset @ B)) @ A2) @ B3)))))).
% 3.16/1.75 thf(fact_217_fminus__fsubset, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A)]: ((((ord_less_eq @ (fset @ A)) @ (((minus_minus @ (fset @ A)) @ A2) @ B3)) @ A2)))).
% 3.16/1.75 thf(fact_218_double__fminus, axiom, (! [A:$tType,A2:(fset @ A),B3:(fset @ A),C4:(fset @ A)]: (((((ord_less_eq @ (fset @ A)) @ A2) @ B3) => ((((ord_less_eq @ (fset @ A)) @ B3) @ C4) => ((((minus_minus @ (fset @ A)) @ B3) @ (((minus_minus @ (fset @ A)) @ C4) @ A2)) = A2)))))).
% 3.16/1.75 thf(fact_219_fminus__mono, axiom, (! [A:$tType,A2:(fset @ A),C4:(fset @ A),D3:(fset @ A),B3:(fset @ A)]: (((((ord_less_eq @ (fset @ A)) @ A2) @ C4) => ((((ord_less_eq @ (fset @ A)) @ D3) @ B3) => (((ord_less_eq @ (fset @ A)) @ (((minus_minus @ (fset @ A)) @ A2) @ B3)) @ (((minus_minus @ (fset @ A)) @ C4) @ D3))))))).
% 3.16/1.75 thf(fact_220_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X5:A]: (((((ord_less_eq @ A) @ Y) @ X5) => (~ ((((ord_less @ A) @ X5) @ Y)))))))))).
% 3.16/1.75 thf(fact_221_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((~ ((((ord_less @ A) @ X5) @ Y))) => (((ord_less_eq @ A) @ Y) @ X5)))))))).
% 3.16/1.75 thf(fact_222_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) | (X = Y4))))))))).
% 3.16/1.75 thf(fact_223_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) & (X != Y4))))))))).
% 3.16/1.75 thf(fact_224_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y2:B]: (((((ord_less @ B) @ X2) @ Y2) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 3.16/1.75 thf(fact_225_order__le__less__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B2:A,F:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ C3) @ (F @ B2)) @ C) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C3) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ C3) @ (F @ A3)) @ C)))))))))).
% 3.16/1.75 thf(fact_226_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A3) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ A) @ A3) @ (F @ C))))))))))).
% 3.16/1.75 thf(fact_227_order__less__le__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A3:A,B2:A,F:(A > C3),C:C3]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ C3) @ (F @ B2)) @ C) => ((! [X2:A,Y2:A]: (((((ord_less @ A) @ X2) @ Y2) => (((ord_less @ C3) @ (F @ X2)) @ (F @ Y2))))) => (((ord_less @ C3) @ (F @ A3)) @ C)))))))))).
% 3.16/1.75 thf(fact_228_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X5) @ Y))) = (((ord_less @ A) @ Y) @ X5)))))))).
% 3.16/1.75 thf(fact_229_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((~ ((((ord_less @ A) @ X5) @ Y))) = (((ord_less_eq @ A) @ Y) @ X5)))))))).
% 3.16/1.75 thf(fact_230_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((A3 != B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 3.16/1.75 thf(fact_231_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((((ord_less @ A) @ X5) @ Y) => (((ord_less_eq @ A) @ X5) @ Y)))))))).
% 3.16/1.75 thf(fact_232_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((~ ((((ord_less @ A) @ X5) @ Y))) => ((((ord_less_eq @ A) @ X5) @ Y) = (X5 = Y))))))))).
% 3.16/1.75 thf(fact_233_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((((ord_less_eq @ A) @ X5) @ Y) => ((~ ((((ord_less @ A) @ X5) @ Y))) = (X5 = Y))))))))).
% 3.16/1.75 thf(fact_234_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X5) @ Y) => ((((ord_less @ A) @ Y) @ Z2) => (((ord_less @ A) @ X5) @ Z2))))))))).
% 3.16/1.75 thf(fact_235_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A,Z2:A]: (((((ord_less @ A) @ X5) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less @ A) @ X5) @ Z2))))))))).
% 3.16/1.75 thf(fact_236_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y:A]: (((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => (((ord_less_eq @ A) @ Y) @ X2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 3.16/1.75 thf(fact_237_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y:A,Z2:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Z2)))) => (((ord_less_eq @ A) @ Y) @ Z2)))))))).
% 3.16/1.75 thf(fact_238_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((((ord_less_eq @ A) @ X5) @ Y) | (((ord_less @ A) @ Y) @ X5)))))))).
% 3.16/1.75 thf(fact_239_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X5:A,Y:A]: (((((ord_less_eq @ A) @ X5) @ Y) => ((((ord_less @ A) @ X5) @ Y) | (X5 = Y))))))))).
% 3.16/1.75 thf(fact_240_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) & (~ ((((ord_less_eq @ A) @ Y4) @ X))))))))))).
% 3.16/1.75 thf(fact_241_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X5:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X5))) => (((ord_less @ A) @ X5) @ Y)))))))).
% 3.16/1.75 thf(fact_242_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 3.16/1.75 thf(fact_243_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B2:A,C:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ A3) @ C))))))))).
% 3.16/1.75 thf(fact_244_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A6:A,B7:A]: (((((ord_less @ A) @ A6) @ B7) | (A6 = B7))))))))).
% 3.16/1.75 thf(fact_245_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A6:A,B7:A]: (((((ord_less_eq @ A) @ A6) @ B7) & (A6 != B7))))))))).
% 3.16/1.75 thf(fact_246_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A3:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A3))))))))).
% 3.16/1.75 thf(fact_247_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A3:A,C:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A3))))))))).
% 3.16/1.75 thf(fact_248_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X5:A,Y:A]: (((((ord_less @ A) @ Z2) @ X5) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X5) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 3.16/1.75 thf(fact_249_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X5:A,Y:A,Z2:A]: (((((ord_less @ A) @ X5) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X5) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y) @ Z2))))))))).
% 3.16/1.75 thf(fact_250_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 3.16/1.75 thf(fact_251_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B7:A,A6:A]: (((((ord_less @ A) @ B7) @ A6) | (A6 = B7))))))))).
% 3.16/1.75 thf(fact_252_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B7:A,A6:A]: (((((ord_less_eq @ A) @ B7) @ A6) & (A6 != B7))))))))).
% 3.16/1.75 thf(fact_253_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 3.16/1.75 thf(fact_254_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A3:A,B2:A]: (((A3 != B2) => ((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less @ A) @ A3) @ B2))))))))).
% 3.16/1.75 thf(fact_255_subset__fimage__iff, axiom, (! [A:$tType,B:$tType,B3:(fset @ A),F:(B > A),A2:(fset @ B)]: (((((ord_less_eq @ (fset @ A)) @ B3) @ ((((fimage @ B) @ A) @ F) @ A2)) = (? [AA:(fset @ B)]: (((((ord_less_eq @ (fset @ B)) @ AA) @ A2) & (B3 = ((((fimage @ B) @ A) @ F) @ AA))))))))).
% 3.16/1.75 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite, axiom, (! [A7:$tType,A8:$tType]: (((((finite_finite @ A7) @ (type2 @ A7)) & ((finite_finite @ A8) @ (type2 @ A8))) => ((finite_finite @ ((product_prod @ A7) @ A8)) @ (type2 @ ((product_prod @ A7) @ A8))))))).
% 3.16/1.75 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_1, axiom, (! [A7:$tType]: ((((finite_finite @ A7) @ (type2 @ A7)) => ((finite_finite @ (fset @ A7)) @ (type2 @ (fset @ A7))))))).
% 3.16/1.75 thf(tcon_HOL_Obool___Finite__Set_Ofinite_2, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_Set_Oset___Finite__Set_Ofinite_3, axiom, (! [A7:$tType]: ((((finite_finite @ A7) @ (type2 @ A7)) => ((finite_finite @ (set @ A7)) @ (type2 @ (set @ A7))))))).
% 3.16/1.75 thf(tcon_fun___Finite__Set_Ofinite_4, axiom, (! [A7:$tType,A8:$tType]: (((((finite_finite @ A7) @ (type2 @ A7)) & ((finite_finite @ A8) @ (type2 @ A8))) => ((finite_finite @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: ((((order_bot @ A8) @ (type2 @ A8)) => ((order_bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_fun___Countable_Ocountable, axiom, (! [A7:$tType,A8:$tType]: (((((finite_finite @ A7) @ (type2 @ A7)) & ((countable @ A8) @ (type2 @ A8))) => ((countable @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: ((((bot @ A8) @ (type2 @ A8)) => ((bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_fun___Groups_Ominus, axiom, (! [A7:$tType,A8:$tType]: ((((minus @ A8) @ (type2 @ A8)) => ((minus @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.16/1.75 thf(tcon_Nat_Onat___Orderings_Oorder__bot_5, axiom, ((order_bot @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Nat_Onat___Countable_Ocountable_6, axiom, ((countable @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Nat_Onat___Orderings_Opreorder_7, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Nat_Onat___Orderings_Oorder_8, axiom, ((order @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Nat_Onat___Orderings_Oord_9, axiom, ((ord @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Nat_Onat___Orderings_Obot_10, axiom, ((bot @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Nat_Onat___Groups_Ominus_11, axiom, ((minus @ nat) @ (type2 @ nat))).
% 3.16/1.75 thf(tcon_Set_Oset___Orderings_Oorder__bot_12, axiom, (! [A7:$tType]: (((order_bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.16/1.75 thf(tcon_Set_Oset___Countable_Ocountable_13, axiom, (! [A7:$tType]: ((((finite_finite @ A7) @ (type2 @ A7)) => ((countable @ (set @ A7)) @ (type2 @ (set @ A7))))))).
% 3.16/1.75 thf(tcon_Set_Oset___Orderings_Opreorder_14, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.16/1.75 thf(tcon_Set_Oset___Orderings_Oorder_15, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.16/1.75 thf(tcon_Set_Oset___Orderings_Oord_16, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.16/1.75 thf(tcon_Set_Oset___Orderings_Obot_17, axiom, (! [A7:$tType]: (((bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.16/1.75 thf(tcon_Set_Oset___Groups_Ominus_18, axiom, (! [A7:$tType]: (((minus @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.16/1.75 thf(tcon_HOL_Obool___Orderings_Oorder__bot_19, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_HOL_Obool___Countable_Ocountable_20, axiom, ((countable @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_HOL_Obool___Orderings_Opreorder_21, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_HOL_Obool___Orderings_Olinorder_22, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_HOL_Obool___Orderings_Oorder_23, axiom, ((order @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_HOL_Obool___Orderings_Oord_24, axiom, ((ord @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_HOL_Obool___Orderings_Obot_25, axiom, ((bot @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_HOL_Obool___Groups_Ominus_26, axiom, ((minus @ $o) @ (type2 @ $o))).
% 3.16/1.75 thf(tcon_FSet_Ofset___Orderings_Oorder__bot_27, axiom, (! [A7:$tType]: (((order_bot @ (fset @ A7)) @ (type2 @ (fset @ A7)))))).
% 3.16/1.75 thf(tcon_FSet_Ofset___Countable_Ocountable_28, axiom, (! [A7:$tType]: ((((finite_finite @ A7) @ (type2 @ A7)) => ((countable @ (fset @ A7)) @ (type2 @ (fset @ A7))))))).
% 3.16/1.75 thf(tcon_FSet_Ofset___Orderings_Opreorder_29, axiom, (! [A7:$tType]: (((preorder @ (fset @ A7)) @ (type2 @ (fset @ A7)))))).
% 3.16/1.75 thf(tcon_FSet_Ofset___Orderings_Oorder_30, axiom, (! [A7:$tType]: (((order @ (fset @ A7)) @ (type2 @ (fset @ A7)))))).
% 3.16/1.76 thf(tcon_FSet_Ofset___Orderings_Oord_31, axiom, (! [A7:$tType]: (((ord @ (fset @ A7)) @ (type2 @ (fset @ A7)))))).
% 3.16/1.76 thf(tcon_FSet_Ofset___Orderings_Obot_32, axiom, (! [A7:$tType]: (((bot @ (fset @ A7)) @ (type2 @ (fset @ A7)))))).
% 3.16/1.76 thf(tcon_FSet_Ofset___Groups_Ominus_33, axiom, (! [A7:$tType]: (((minus @ (fset @ A7)) @ (type2 @ (fset @ A7)))))).
% 3.16/1.76 thf(tcon_Product__Type_Oprod___Countable_Ocountable_34, axiom, (! [A7:$tType,A8:$tType]: (((((countable @ A7) @ (type2 @ A7)) & ((countable @ A8) @ (type2 @ A8))) => ((countable @ ((product_prod @ A7) @ A8)) @ (type2 @ ((product_prod @ A7) @ A8))))))).
% 3.16/1.76 thf(help_If_3_1_T, axiom, (! [P2:$o]: (((P2 = $true) | (P2 = $false))))).
% 3.16/1.76 thf(help_If_2_1_T, axiom, (! [A:$tType,X5:A,Y:A]: ((((((if @ A) @ $false) @ X5) @ Y) = Y)))).
% 3.16/1.76 thf(help_If_1_1_T, axiom, (! [A:$tType,X5:A,Y:A]: ((((((if @ A) @ $true) @ X5) @ Y) = X5)))).
% 3.16/1.76 thf(conj_0, conjecture, (! [X2:rule]: (((((member @ rule) @ X2) @ ((sset @ rule) @ rules)) => (((((abstra726722745urated @ rule) @ state) @ eff) @ X2) @ steps))))).
% 3.16/1.76 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.JVLNTEOK4C/DTF2DTF19524.p
% 3.16/1.76 ------------------------
% 3.29/1.81 ---- Cleaned THF ---
% 3.29/1.81 thf(ty_t_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree,type,
% 3.29/1.81 abstra2103299360e_tree: $tType > $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_t_Product__Type_Oprod,type,
% 3.29/1.81 product_prod: $tType > $tType > $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_t_Stream_Ostream,type,
% 3.29/1.81 stream: $tType > $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_t_FSet_Ofset,type,
% 3.29/1.81 fset: $tType > $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_t_Set_Oset,type,
% 3.29/1.81 set: $tType > $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_t_Nat_Onat,type,
% 3.29/1.81 nat: $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_tf_state,type,
% 3.29/1.81 state: $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_t_itself,type,
% 3.29/1.81 itself: $tType > $tType ).
% 3.29/1.81
% 3.29/1.81 thf(ty_tf_rule,type,
% 3.29/1.81 rule: $tType ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Finite__Set_Ofinite,type,
% 3.29/1.81 finite_finite:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_HOL_Otype,type,
% 3.29/1.81 type:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Groups_Ominus,type,
% 3.29/1.81 minus:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Orderings_Obot,type,
% 3.29/1.81 bot:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Orderings_Oord,type,
% 3.29/1.81 ord:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Orderings_Oorder,type,
% 3.29/1.81 order:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Orderings_Olinorder,type,
% 3.29/1.81 linorder:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Orderings_Opreorder,type,
% 3.29/1.81 preorder:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Countable_Ocountable,type,
% 3.29/1.81 countable:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Orderings_Oorder__bot,type,
% 3.29/1.81 order_bot:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_cl_Orderings_Odense__linorder,type,
% 3.29/1.81 dense_linorder:
% 3.29/1.81 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_OPersistentRuleSystem,type,
% 3.29/1.81 abstra612932245System:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem,type,
% 3.29/1.81 abstra1326562878System:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_OminWait,type,
% 3.29/1.81 abstra1332369113inWait:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Oper,type,
% 3.29/1.81 abstra2096684367le_per:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( set @ State ) > Rule > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem_Opos,type,
% 3.29/1.81 abstra2097340358le_pos:
% 3.29/1.81 !>[A: $tType] : ( ( stream @ A ) > A > nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OSaturated,type,
% 3.29/1.81 abstra1209608345urated:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oenabled,type,
% 3.29/1.81 abstra1874422341nabled:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > State > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Oepath,type,
% 3.29/1.81 abstra523868654_epath:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofair,type,
% 3.29/1.81 abstra928354080m_fair:
% 3.29/1.81 !>[Rule: $tType] : ( ( stream @ Rule ) > ( stream @ Rule ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofenum,type,
% 3.29/1.81 abstra1774373515_fenum:
% 3.29/1.81 !>[Rule: $tType] : ( ( stream @ Rule ) > ( stream @ Rule ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OmkTree,type,
% 3.29/1.81 abstra1225283448mkTree:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_OpickEff,type,
% 3.29/1.81 abstra1276541928ickEff:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > State > ( fset @ State ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Osaturated,type,
% 3.29/1.81 abstra726722745urated:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > Rule > ( stream @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Otrim,type,
% 3.29/1.81 abstra1259602206m_trim:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > State > ( stream @ Rule ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Owf,type,
% 3.29/1.81 abstra1874736267tem_wf:
% 3.29/1.81 !>[Rule: $tType,State: $tType] : ( ( Rule > State > ( fset @ State ) > $o ) > ( stream @ Rule ) > ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Oipath,type,
% 3.29/1.81 abstra313004635_ipath:
% 3.29/1.81 !>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( stream @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_Otree_Ocont,type,
% 3.29/1.81 abstra1749095923e_cont:
% 3.29/1.81 !>[A: $tType] : ( ( abstra2103299360e_tree @ A ) > ( fset @ ( abstra2103299360e_tree @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Countable__Set_Ocountable,type,
% 3.29/1.81 countable_countable:
% 3.29/1.81 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_FSet_Ofimage,type,
% 3.29/1.81 fimage:
% 3.29/1.81 !>[A: $tType,B: $tType] : ( ( A > B ) > ( fset @ A ) > ( fset @ B ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_FSet_Ofmember,type,
% 3.29/1.81 fmember:
% 3.29/1.81 !>[A: $tType] : ( A > ( fset @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 3.29/1.81 minus_minus:
% 3.29/1.81 !>[A: $tType] : ( A > A > A ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_If,type,
% 3.29/1.81 if:
% 3.29/1.81 !>[A: $tType] : ( $o > A > A > A ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Nat_OSuc,type,
% 3.29/1.81 suc: nat > nat ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Orderings_Obot__class_Obot,type,
% 3.29/1.81 bot_bot:
% 3.29/1.81 !>[A: $tType] : A ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.29/1.81 ord_less:
% 3.29/1.81 !>[A: $tType] : ( A > A > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.29/1.81 ord_less_eq:
% 3.29/1.81 !>[A: $tType] : ( A > A > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Pure_Otype,type,
% 3.29/1.81 type2:
% 3.29/1.81 !>[A: $tType] : ( itself @ A ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Set_OCollect,type,
% 3.29/1.81 collect:
% 3.29/1.81 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Set_Oinsert,type,
% 3.29/1.81 insert:
% 3.29/1.81 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Set_Ois__singleton,type,
% 3.29/1.81 is_singleton:
% 3.29/1.81 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Stream_Osdrop,type,
% 3.29/1.81 sdrop:
% 3.29/1.81 !>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Stream_Ostream_Oshd,type,
% 3.29/1.81 shd:
% 3.29/1.81 !>[A: $tType] : ( ( stream @ A ) > A ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Stream_Ostream_Osset,type,
% 3.29/1.81 sset:
% 3.29/1.81 !>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_Stream_Ostream_Ostl,type,
% 3.29/1.81 stl:
% 3.29/1.81 !>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_c_member,type,
% 3.29/1.81 member:
% 3.29/1.81 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(sy_v_S,type,
% 3.29/1.81 s: set @ state ).
% 3.29/1.81
% 3.29/1.81 thf(sy_v_eff,type,
% 3.29/1.81 eff: rule > state > ( fset @ state ) > $o ).
% 3.29/1.81
% 3.29/1.81 thf(sy_v_rs,type,
% 3.29/1.81 rs: stream @ rule ).
% 3.29/1.81
% 3.29/1.81 thf(sy_v_rules,type,
% 3.29/1.81 rules: stream @ rule ).
% 3.29/1.81
% 3.29/1.81 thf(sy_v_s,type,
% 3.29/1.81 s2: state ).
% 3.29/1.81
% 3.29/1.81 thf(sy_v_steps,type,
% 3.29/1.81 steps: stream @ ( product_prod @ state @ rule ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_0__092_060open_062_092_060And_062ra_O_Ara_A_092_060in_062_AR_A_092_060Longrightarrow_062_Asaturated_Ara_Asteps_092_060close_062,axiom,
% 3.29/1.81 ! [R: rule] :
% 3.29/1.81 ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( abstra726722745urated @ rule @ state @ eff @ R @ steps ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_1_Saturated__def,axiom,
% 3.29/1.81 ! [Steps: stream @ ( product_prod @ state @ rule )] :
% 3.29/1.81 ( ( abstra1209608345urated @ rule @ state @ eff @ rules @ Steps )
% 3.29/1.81 = ( ! [X: rule] :
% 3.29/1.81 ( ( member @ rule @ X @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( abstra726722745urated @ rule @ state @ eff @ X @ Steps ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_2_sset__fenum,axiom,
% 3.29/1.81 ( ( sset @ rule @ ( abstra1774373515_fenum @ rule @ rules ) )
% 3.29/1.81 = ( sset @ rule @ rules ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_3_assms_I2_J,axiom,
% 3.29/1.81 abstra928354080m_fair @ rule @ rules @ rs ).
% 3.29/1.81
% 3.29/1.81 thf(fact_4_RuleSystem__Defs_OSaturated__def,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType] :
% 3.29/1.81 ( ( abstra1209608345urated @ Rule @ State )
% 3.29/1.81 = ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,Steps2: stream @ ( product_prod @ State @ Rule )] :
% 3.29/1.81 ! [X: Rule] :
% 3.29/1.81 ( ( member @ Rule @ X @ ( sset @ Rule @ Rules ) )
% 3.29/1.81 => ( abstra726722745urated @ Rule @ State @ Eff @ X @ Steps2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_5_enabled__R,axiom,
% 3.29/1.81 ! [S: state] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ? [X2: rule] :
% 3.29/1.81 ( ( member @ rule @ X2 @ ( sset @ rule @ rules ) )
% 3.29/1.81 & ? [X1: fset @ state] : ( eff @ X2 @ S @ X1 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_6_NE__R,axiom,
% 3.29/1.81 ( ( sset @ rule @ rules )
% 3.29/1.81 != ( bot_bot @ ( set @ rule ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_7_PersistentRuleSystem__axioms,axiom,
% 3.29/1.81 abstra612932245System @ rule @ state @ eff @ rules @ s ).
% 3.29/1.81
% 3.29/1.81 thf(fact_8_enabled__def,axiom,
% 3.29/1.81 ! [R: rule,S: state] :
% 3.29/1.81 ( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
% 3.29/1.81 = ( ^ [P: ( fset @ state ) > $o] :
% 3.29/1.81 ? [X3: fset @ state] : ( P @ X3 )
% 3.29/1.81 @ ( eff @ R @ S ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_9_countable__R,axiom,
% 3.29/1.81 countable_countable @ rule @ ( sset @ rule @ rules ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_10_per,axiom,
% 3.29/1.81 ! [R: rule] :
% 3.29/1.81 ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( abstra2096684367le_per @ rule @ state @ eff @ rules @ s @ R ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_11_assms_I3_J,axiom,
% 3.29/1.81 abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ rs @ s2 ) @ steps ).
% 3.29/1.81
% 3.29/1.81 thf(fact_12_RuleSystem__Defs_Osset__fenum,axiom,
% 3.29/1.81 ! [Rule: $tType,Rules2: stream @ Rule] :
% 3.29/1.81 ( ( sset @ Rule @ ( abstra1774373515_fenum @ Rule @ Rules2 ) )
% 3.29/1.81 = ( sset @ Rule @ Rules2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_13_eff__S,axiom,
% 3.29/1.81 ! [S: state,R: rule,Sl: fset @ state,S2: state] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( ( eff @ R @ S @ Sl )
% 3.29/1.81 => ( ( fmember @ state @ S2 @ Sl )
% 3.29/1.81 => ( member @ state @ S2 @ s ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_14_RuleSystem__axioms,axiom,
% 3.29/1.81 abstra1326562878System @ rule @ state @ eff @ rules @ s ).
% 3.29/1.81
% 3.29/1.81 thf(fact_15_assms_I1_J,axiom,
% 3.29/1.81 member @ state @ s2 @ s ).
% 3.29/1.81
% 3.29/1.81 thf(fact_16_fair__fenum,axiom,
% 3.29/1.81 abstra928354080m_fair @ rule @ rules @ ( abstra1774373515_fenum @ rule @ rules ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_17_ipath__mkTree__saturated,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule,Steps: stream @ ( product_prod @ state @ rule ),R: rule] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) @ Steps )
% 3.29/1.81 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( abstra726722745urated @ rule @ state @ eff @ R @ Steps ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_18_pickEff,axiom,
% 3.29/1.81 ! [R: rule,S: state] :
% 3.29/1.81 ( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
% 3.29/1.81 => ( eff @ R @ S @ ( abstra1276541928ickEff @ rule @ state @ eff @ R @ S ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_19_PersistentRuleSystem_Oaxioms_I1_J,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State] :
% 3.29/1.81 ( ( abstra612932245System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_20_RuleSystem__Defs_Ofair__fenum,axiom,
% 3.29/1.81 ! [Rule: $tType,Rules2: stream @ Rule] : ( abstra928354080m_fair @ Rule @ Rules2 @ ( abstra1774373515_fenum @ Rule @ Rules2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_21_RuleSystem__Defs_Oenabled__def,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType] :
% 3.29/1.81 ( ( abstra1874422341nabled @ Rule @ State )
% 3.29/1.81 = ( ^ [Eff: Rule > State > ( fset @ State ) > $o,R2: Rule,S4: State] :
% 3.29/1.81 ( ^ [P: ( fset @ State ) > $o] :
% 3.29/1.81 ? [X3: fset @ State] : ( P @ X3 )
% 3.29/1.81 @ ( Eff @ R2 @ S4 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_22_RuleSystem_Ointro,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,S3: set @ State,Rules2: stream @ Rule,Eff2: Rule > State > ( fset @ State ) > $o] :
% 3.29/1.81 ( ! [S5: State] :
% 3.29/1.81 ( ( member @ State @ S5 @ S3 )
% 3.29/1.81 => ! [R3: Rule] :
% 3.29/1.81 ( ( member @ Rule @ R3 @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 => ! [Sl2: fset @ State] :
% 3.29/1.81 ( ( Eff2 @ R3 @ S5 @ Sl2 )
% 3.29/1.81 => ! [S6: State] :
% 3.29/1.81 ( ( fmember @ State @ S6 @ Sl2 )
% 3.29/1.81 => ( member @ State @ S6 @ S3 ) ) ) ) )
% 3.29/1.81 => ( ! [S5: State] :
% 3.29/1.81 ( ( member @ State @ S5 @ S3 )
% 3.29/1.81 => ? [X4: Rule] :
% 3.29/1.81 ( ( member @ Rule @ X4 @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 & ? [X12: fset @ State] : ( Eff2 @ X4 @ S5 @ X12 ) ) )
% 3.29/1.81 => ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_23_RuleSystem_Oeff__S,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,R: Rule,Sl: fset @ State,S2: State] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 => ( ( Eff2 @ R @ S @ Sl )
% 3.29/1.81 => ( ( fmember @ State @ S2 @ Sl )
% 3.29/1.81 => ( member @ State @ S2 @ S3 ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_24_RuleSystem__def,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State )
% 3.29/1.81 = ( ^ [Eff: Rule > State > ( fset @ State ) > $o,Rules: stream @ Rule,S7: set @ State] :
% 3.29/1.81 ( ! [S4: State] :
% 3.29/1.81 ( ( member @ State @ S4 @ S7 )
% 3.29/1.81 => ! [R2: Rule] :
% 3.29/1.81 ( ( member @ Rule @ R2 @ ( sset @ Rule @ Rules ) )
% 3.29/1.81 => ! [Sl3: fset @ State] :
% 3.29/1.81 ( ( Eff @ R2 @ S4 @ Sl3 )
% 3.29/1.81 => ! [S8: State] :
% 3.29/1.81 ( ( fmember @ State @ S8 @ Sl3 )
% 3.29/1.81 => ( member @ State @ S8 @ S7 ) ) ) ) )
% 3.29/1.81 & ! [S4: State] :
% 3.29/1.81 ( ( member @ State @ S4 @ S7 )
% 3.29/1.81 => ? [X: Rule] :
% 3.29/1.81 ( ( member @ Rule @ X @ ( sset @ Rule @ Rules ) )
% 3.29/1.81 & ( ^ [P: ( fset @ State ) > $o] :
% 3.29/1.81 ? [X3: fset @ State] : ( P @ X3 )
% 3.29/1.81 @ ( Eff @ X @ S4 ) ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_25_PersistentRuleSystem_Oipath__mkTree__saturated,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,Steps: stream @ ( product_prod @ State @ Rule ),R: Rule] :
% 3.29/1.81 ( ( abstra612932245System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) @ Steps )
% 3.29/1.81 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 => ( abstra726722745urated @ Rule @ State @ Eff2 @ R @ Steps ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_26_RuleSystem__Defs_Ocountable__R,axiom,
% 3.29/1.81 ! [Rule: $tType,Rules2: stream @ Rule] : ( countable_countable @ Rule @ ( sset @ Rule @ Rules2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_27_RuleSystem__Defs_ONE__R,axiom,
% 3.29/1.81 ! [Rule: $tType,Rules2: stream @ Rule] :
% 3.29/1.81 ( ( sset @ Rule @ Rules2 )
% 3.29/1.81 != ( bot_bot @ ( set @ Rule ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_28_PersistentRuleSystem_Oper,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,R: Rule] :
% 3.29/1.81 ( ( abstra612932245System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 => ( abstra2096684367le_per @ Rule @ State @ Eff2 @ Rules2 @ S3 @ R ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_29_RuleSystem_Oenabled__R,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ? [X2: Rule] :
% 3.29/1.81 ( ( member @ Rule @ X2 @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 & ? [X1: fset @ State] : ( Eff2 @ X2 @ S @ X1 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_30_wf__mkTree,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_31_trim__fair,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( abstra928354080m_fair @ rule @ rules @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_32_countable__empty,axiom,
% 3.29/1.81 ! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_33_per__def,axiom,
% 3.29/1.81 ! [R: rule] :
% 3.29/1.81 ( ( abstra2096684367le_per @ rule @ state @ eff @ rules @ s @ R )
% 3.29/1.81 = ( ! [S4: state,R1: rule,Sl4: fset @ state,S8: state] :
% 3.29/1.81 ( ( ( member @ state @ S4 @ s )
% 3.29/1.81 & ( abstra1874422341nabled @ rule @ state @ eff @ R @ S4 )
% 3.29/1.81 & ( member @ rule @ R1 @ ( minus_minus @ ( set @ rule ) @ ( sset @ rule @ rules ) @ ( insert @ rule @ R @ ( bot_bot @ ( set @ rule ) ) ) ) )
% 3.29/1.81 & ( eff @ R1 @ S4 @ Sl4 )
% 3.29/1.81 & ( fmember @ state @ S8 @ Sl4 ) )
% 3.29/1.81 => ( abstra1874422341nabled @ rule @ state @ eff @ R @ S8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_34_countableI__type,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( countable @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A2: set @ A] : ( countable_countable @ A @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_35_sdrop__fair,axiom,
% 3.29/1.81 ! [Rs: stream @ rule,M: nat] :
% 3.29/1.81 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( abstra928354080m_fair @ rule @ rules @ ( sdrop @ rule @ M @ Rs ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_36_empty__iff,axiom,
% 3.29/1.81 ! [A: $tType,C: A] :
% 3.29/1.81 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_37_all__not__in__conv,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( ! [X: A] :
% 3.29/1.81 ~ ( member @ A @ X @ A2 ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_38_Collect__empty__eq,axiom,
% 3.29/1.81 ! [A: $tType,P2: A > $o] :
% 3.29/1.81 ( ( ( collect @ A @ P2 )
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 = ( ! [X: A] :
% 3.29/1.81 ~ ( P2 @ X ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_39_empty__Collect__eq,axiom,
% 3.29/1.81 ! [A: $tType,P2: A > $o] :
% 3.29/1.81 ( ( ( bot_bot @ ( set @ A ) )
% 3.29/1.81 = ( collect @ A @ P2 ) )
% 3.29/1.81 = ( ! [X: A] :
% 3.29/1.81 ~ ( P2 @ X ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_40_insert__absorb2,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A] :
% 3.29/1.81 ( ( insert @ A @ X5 @ ( insert @ A @ X5 @ A2 ) )
% 3.29/1.81 = ( insert @ A @ X5 @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_41_insert__iff,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 3.29/1.81 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 3.29/1.81 = ( ( A3 = B2 )
% 3.29/1.81 | ( member @ A @ A3 @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_42_insertCI,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B3: set @ A,B2: A] :
% 3.29/1.81 ( ( ~ ( member @ A @ A3 @ B3 )
% 3.29/1.81 => ( A3 = B2 ) )
% 3.29/1.81 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_43_Diff__idemp,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ B3 )
% 3.29/1.81 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_44_Diff__iff,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 3.29/1.81 = ( ( member @ A @ C @ A2 )
% 3.29/1.81 & ~ ( member @ A @ C @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_45_mem__Collect__eq,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,P2: A > $o] :
% 3.29/1.81 ( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
% 3.29/1.81 = ( P2 @ A3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_46_Collect__mem__eq,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( collect @ A
% 3.29/1.81 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 3.29/1.81 = A2 ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_47_Collect__cong,axiom,
% 3.29/1.81 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( P2 @ X2 )
% 3.29/1.81 = ( Q @ X2 ) )
% 3.29/1.81 => ( ( collect @ A @ P2 )
% 3.29/1.81 = ( collect @ A @ Q ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_48_ext,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( F @ X2 )
% 3.29/1.81 = ( G @ X2 ) )
% 3.29/1.81 => ( F = G ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_49_DiffI,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( member @ A @ C @ A2 )
% 3.29/1.81 => ( ~ ( member @ A @ C @ B3 )
% 3.29/1.81 => ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_50_singletonI,axiom,
% 3.29/1.81 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_51_Diff__cancel,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( minus_minus @ ( set @ A ) @ A2 @ A2 )
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_52_empty__Diff,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 )
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_53_Diff__empty,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( minus_minus @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 = A2 ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_54_insert__Diff1,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,B3: set @ A,A2: set @ A] :
% 3.29/1.81 ( ( member @ A @ X5 @ B3 )
% 3.29/1.81 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 3.29/1.81 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_55_Diff__insert0,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ~ ( member @ A @ X5 @ A2 )
% 3.29/1.81 => ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) )
% 3.29/1.81 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_56_countable__insert__eq,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A] :
% 3.29/1.81 ( ( countable_countable @ A @ ( insert @ A @ X5 @ A2 ) )
% 3.29/1.81 = ( countable_countable @ A @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_57_countable__insert,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,A3: A] :
% 3.29/1.81 ( ( countable_countable @ A @ A2 )
% 3.29/1.81 => ( countable_countable @ A @ ( insert @ A @ A3 @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_58_countable__Diff,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( countable_countable @ A @ A2 )
% 3.29/1.81 => ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_59_insert__Diff__single,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,A2: set @ A] :
% 3.29/1.81 ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.29/1.81 = ( insert @ A @ A3 @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_60_countable__Diff__eq,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,X5: A] :
% 3.29/1.81 ( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.29/1.81 = ( countable_countable @ A @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_61_Diff__insert,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,A3: A,B3: set @ A] :
% 3.29/1.81 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B3 ) )
% 3.29/1.81 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_62_insert__Diff,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,A2: set @ A] :
% 3.29/1.81 ( ( member @ A @ A3 @ A2 )
% 3.29/1.81 => ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.29/1.81 = A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_63_Diff__insert2,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,A3: A,B3: set @ A] :
% 3.29/1.81 ( ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ B3 ) )
% 3.29/1.81 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_64_Diff__insert__absorb,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A] :
% 3.29/1.81 ( ~ ( member @ A @ X5 @ A2 )
% 3.29/1.81 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 = A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_65_mk__disjoint__insert,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,A2: set @ A] :
% 3.29/1.81 ( ( member @ A @ A3 @ A2 )
% 3.29/1.81 => ? [B4: set @ A] :
% 3.29/1.81 ( ( A2
% 3.29/1.81 = ( insert @ A @ A3 @ B4 ) )
% 3.29/1.81 & ~ ( member @ A @ A3 @ B4 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_66_insert__commute,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,Y: A,A2: set @ A] :
% 3.29/1.81 ( ( insert @ A @ X5 @ ( insert @ A @ Y @ A2 ) )
% 3.29/1.81 = ( insert @ A @ Y @ ( insert @ A @ X5 @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_67_insert__Diff__if,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,B3: set @ A,A2: set @ A] :
% 3.29/1.81 ( ( ( member @ A @ X5 @ B3 )
% 3.29/1.81 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 3.29/1.81 = ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) )
% 3.29/1.81 & ( ~ ( member @ A @ X5 @ B3 )
% 3.29/1.81 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 3.29/1.81 = ( insert @ A @ X5 @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_68_insert__eq__iff,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,A2: set @ A,B2: A,B3: set @ A] :
% 3.29/1.81 ( ~ ( member @ A @ A3 @ A2 )
% 3.29/1.81 => ( ~ ( member @ A @ B2 @ B3 )
% 3.29/1.81 => ( ( ( insert @ A @ A3 @ A2 )
% 3.29/1.81 = ( insert @ A @ B2 @ B3 ) )
% 3.29/1.81 = ( ( ( A3 = B2 )
% 3.29/1.81 => ( A2 = B3 ) )
% 3.29/1.81 & ( ( A3 != B2 )
% 3.29/1.81 => ? [C2: set @ A] :
% 3.29/1.81 ( ( A2
% 3.29/1.81 = ( insert @ A @ B2 @ C2 ) )
% 3.29/1.81 & ~ ( member @ A @ B2 @ C2 )
% 3.29/1.81 & ( B3
% 3.29/1.81 = ( insert @ A @ A3 @ C2 ) )
% 3.29/1.81 & ~ ( member @ A @ A3 @ C2 ) ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_69_insert__absorb,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,A2: set @ A] :
% 3.29/1.81 ( ( member @ A @ A3 @ A2 )
% 3.29/1.81 => ( ( insert @ A @ A3 @ A2 )
% 3.29/1.81 = A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_70_insert__ident,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ~ ( member @ A @ X5 @ A2 )
% 3.29/1.81 => ( ~ ( member @ A @ X5 @ B3 )
% 3.29/1.81 => ( ( ( insert @ A @ X5 @ A2 )
% 3.29/1.81 = ( insert @ A @ X5 @ B3 ) )
% 3.29/1.81 = ( A2 = B3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_71_Set_Oset__insert,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A] :
% 3.29/1.81 ( ( member @ A @ X5 @ A2 )
% 3.29/1.81 => ~ ! [B4: set @ A] :
% 3.29/1.81 ( ( A2
% 3.29/1.81 = ( insert @ A @ X5 @ B4 ) )
% 3.29/1.81 => ( member @ A @ X5 @ B4 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_72_insertI2,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B3: set @ A,B2: A] :
% 3.29/1.81 ( ( member @ A @ A3 @ B3 )
% 3.29/1.81 => ( member @ A @ A3 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_73_insertI1,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B3: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_74_insertE,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B2: A,A2: set @ A] :
% 3.29/1.81 ( ( member @ A @ A3 @ ( insert @ A @ B2 @ A2 ) )
% 3.29/1.81 => ( ( A3 != B2 )
% 3.29/1.81 => ( member @ A @ A3 @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_75_DiffD2,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 3.29/1.81 => ~ ( member @ A @ C @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_76_DiffD1,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 3.29/1.81 => ( member @ A @ C @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_77_DiffE,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
% 3.29/1.81 => ~ ( ( member @ A @ C @ A2 )
% 3.29/1.81 => ( member @ A @ C @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_78_singleton__inject,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B2: A] :
% 3.29/1.81 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 => ( A3 = B2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_79_insert__not__empty,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,A2: set @ A] :
% 3.29/1.81 ( ( insert @ A @ A3 @ A2 )
% 3.29/1.81 != ( bot_bot @ ( set @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_80_doubleton__eq__iff,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B2: A,C: A,D: A] :
% 3.29/1.81 ( ( ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 = ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
% 3.29/1.81 = ( ( ( A3 = C )
% 3.29/1.81 & ( B2 = D ) )
% 3.29/1.81 | ( ( A3 = D )
% 3.29/1.81 & ( B2 = C ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_81_singleton__iff,axiom,
% 3.29/1.81 ! [A: $tType,B2: A,A3: A] :
% 3.29/1.81 ( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 = ( B2 = A3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_82_singletonD,axiom,
% 3.29/1.81 ! [A: $tType,B2: A,A3: A] :
% 3.29/1.81 ( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 => ( B2 = A3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_83_uncountable__minus__countable,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ~ ( countable_countable @ A @ A2 )
% 3.29/1.81 => ( ( countable_countable @ A @ B3 )
% 3.29/1.81 => ~ ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_84_RuleSystem__Defs_Osdrop__fair,axiom,
% 3.29/1.81 ! [Rule: $tType,Rules2: stream @ Rule,Rs: stream @ Rule,M: nat] :
% 3.29/1.81 ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( abstra928354080m_fair @ Rule @ Rules2 @ ( sdrop @ Rule @ M @ Rs ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_85_RuleSystem_Oipath__mkTree__sdrop,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,Steps: stream @ ( product_prod @ State @ Rule ),M: nat] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) @ Steps )
% 3.29/1.81 => ? [N: nat,S6: State] :
% 3.29/1.81 ( ( member @ State @ S6 @ S3 )
% 3.29/1.81 & ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( sdrop @ Rule @ N @ Rs ) @ S6 ) @ ( sdrop @ ( product_prod @ State @ Rule ) @ M @ Steps ) ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_86_RuleSystem__Defs_OpickEff,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,R: Rule,S: State] :
% 3.29/1.81 ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S )
% 3.29/1.81 => ( Eff2 @ R @ S @ ( abstra1276541928ickEff @ Rule @ State @ Eff2 @ R @ S ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_87_ex__in__conv,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( ? [X: A] : ( member @ A @ X @ A2 ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_88_equals0I,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ! [Y2: A] :
% 3.29/1.81 ~ ( member @ A @ Y2 @ A2 )
% 3.29/1.81 => ( A2
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_89_equals0D,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,A3: A] :
% 3.29/1.81 ( ( A2
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_90_emptyE,axiom,
% 3.29/1.81 ! [A: $tType,A3: A] :
% 3.29/1.81 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_91_RuleSystem_Oper__def,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,R: Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( abstra2096684367le_per @ Rule @ State @ Eff2 @ Rules2 @ S3 @ R )
% 3.29/1.81 = ( ! [S4: State,R1: Rule,Sl4: fset @ State,S8: State] :
% 3.29/1.81 ( ( ( member @ State @ S4 @ S3 )
% 3.29/1.81 & ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S4 )
% 3.29/1.81 & ( member @ Rule @ R1 @ ( minus_minus @ ( set @ Rule ) @ ( sset @ Rule @ Rules2 ) @ ( insert @ Rule @ R @ ( bot_bot @ ( set @ Rule ) ) ) ) )
% 3.29/1.81 & ( Eff2 @ R1 @ S4 @ Sl4 )
% 3.29/1.81 & ( fmember @ State @ S8 @ Sl4 ) )
% 3.29/1.81 => ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S8 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_92_RuleSystem_Otrim__fair,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( abstra928354080m_fair @ Rule @ Rules2 @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_93_RuleSystem_Owf__mkTree,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( abstra1874736267tem_wf @ Rule @ State @ Eff2 @ Rules2 @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_94_trim__alt,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S )
% 3.29/1.81 = ( sdrop @ rule @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ Rs ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_95_wf__ipath__epath,axiom,
% 3.29/1.81 ! [T: abstra2103299360e_tree @ ( product_prod @ state @ rule ),Steps: stream @ ( product_prod @ state @ rule )] :
% 3.29/1.81 ( ( abstra1874736267tem_wf @ rule @ state @ eff @ rules @ T )
% 3.29/1.81 => ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ T @ Steps )
% 3.29/1.81 => ( abstra523868654_epath @ rule @ state @ eff @ rules @ Steps ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_96_ipath__mkTree__sdrop,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule,Steps: stream @ ( product_prod @ state @ rule ),M: nat] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) @ Steps )
% 3.29/1.81 => ? [N: nat,S6: state] :
% 3.29/1.81 ( ( member @ state @ S6 @ s )
% 3.29/1.81 & ( abstra313004635_ipath @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ ( sdrop @ rule @ N @ Rs ) @ S6 ) @ ( sdrop @ ( product_prod @ state @ rule ) @ M @ Steps ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_97_minWait__ex,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ? [N: nat] : ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N @ Rs ) ) @ S ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_98_trim__enabled,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_99_trim__in__R,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( member @ rule @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ ( sset @ rule @ rules ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_100_RuleSystem_Otrim__alt,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S )
% 3.29/1.81 = ( sdrop @ Rule @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ Rs ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_101_bot__apply,axiom,
% 3.29/1.81 ! [C3: $tType,D2: $tType] :
% 3.29/1.81 ( ( bot @ C3 @ ( type2 @ C3 ) )
% 3.29/1.81 => ( ( bot_bot @ ( D2 > C3 ) )
% 3.29/1.81 = ( ^ [X: D2] : ( bot_bot @ C3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_102_bot__set__def,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( bot_bot @ ( set @ A ) )
% 3.29/1.81 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_103_bot__fun__def,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType] :
% 3.29/1.81 ( ( bot @ B @ ( type2 @ B ) )
% 3.29/1.81 => ( ( bot_bot @ ( A > B ) )
% 3.29/1.81 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_104_RuleSystem_Otrim__in__R,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( member @ Rule @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ ( sset @ Rule @ Rules2 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_105_RuleSystem_OminWait__ex,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ? [N: nat] : ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N @ Rs ) ) @ S ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_106_RuleSystem_Otrim__enabled,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_107_RuleSystem__Defs_Owf__ipath__epath,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,T: abstra2103299360e_tree @ ( product_prod @ State @ Rule ),Steps: stream @ ( product_prod @ State @ Rule )] :
% 3.29/1.81 ( ( abstra1874736267tem_wf @ Rule @ State @ Eff2 @ Rules2 @ T )
% 3.29/1.81 => ( ( abstra313004635_ipath @ ( product_prod @ State @ Rule ) @ T @ Steps )
% 3.29/1.81 => ( abstra523868654_epath @ Rule @ State @ Eff2 @ Rules2 @ Steps ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_108_pos,axiom,
% 3.29/1.81 ! [Rs: stream @ rule,R: rule] :
% 3.29/1.81 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( ( shd @ rule @ ( sdrop @ rule @ ( abstra2097340358le_pos @ rule @ Rs @ R ) @ Rs ) )
% 3.29/1.81 = R ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_109_minWait__least,axiom,
% 3.29/1.81 ! [N2: nat,Rs: stream @ rule,S: state] :
% 3.29/1.81 ( ( abstra1874422341nabled @ rule @ state @ eff @ ( shd @ rule @ ( sdrop @ rule @ N2 @ Rs ) ) @ S )
% 3.29/1.81 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ N2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_110_RuleSystem_Opos,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,Rs: stream @ Rule,R: Rule] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 => ( ( shd @ Rule @ ( sdrop @ Rule @ ( abstra2097340358le_pos @ Rule @ Rs @ R ) @ Rs ) )
% 3.29/1.81 = R ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_111_fempty__iff,axiom,
% 3.29/1.81 ! [A: $tType,C: A] :
% 3.29/1.81 ~ ( fmember @ A @ C @ ( bot_bot @ ( fset @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_112_all__not__fin__conv,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A] :
% 3.29/1.81 ( ( ! [X: A] :
% 3.29/1.81 ~ ( fmember @ A @ X @ A2 ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_113_order__refl,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A] : ( ord_less_eq @ A @ X5 @ X5 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_114_pos__least,axiom,
% 3.29/1.81 ! [A: $tType,N2: nat,Rs: stream @ A,R: A] :
% 3.29/1.81 ( ( ( shd @ A @ ( sdrop @ A @ N2 @ Rs ) )
% 3.29/1.81 = R )
% 3.29/1.81 => ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_115_minWait__le__pos,axiom,
% 3.29/1.81 ! [Rs: stream @ rule,R: rule,S: state] :
% 3.29/1.81 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( ( abstra1874422341nabled @ rule @ state @ eff @ R @ S )
% 3.29/1.81 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ ( abstra2097340358le_pos @ rule @ Rs @ R ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_116_le__funD,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType] :
% 3.29/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 3.29/1.81 => ! [F: A > B,G: A > B,X5: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.29/1.81 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_117_le__funE,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType] :
% 3.29/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 3.29/1.81 => ! [F: A > B,G: A > B,X5: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.29/1.81 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_118_le__funI,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType] :
% 3.29/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 3.29/1.81 => ! [F: A > B,G: A > B] :
% 3.29/1.81 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 3.29/1.81 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_119_le__fun__def,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType] :
% 3.29/1.81 ( ( ord @ B @ ( type2 @ B ) )
% 3.29/1.81 => ( ( ord_less_eq @ ( A > B ) )
% 3.29/1.81 = ( ^ [F2: A > B,G2: A > B] :
% 3.29/1.81 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_120_order__subst1,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType] :
% 3.29/1.81 ( ( ( order @ B @ ( type2 @ B ) )
% 3.29/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 3.29/1.81 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.29/1.81 => ( ! [X2: B,Y2: B] :
% 3.29/1.81 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_121_order__subst2,axiom,
% 3.29/1.81 ! [A: $tType,C3: $tType] :
% 3.29/1.81 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 3.29/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,B2: A,F: A > C3,C: C3] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
% 3.29/1.81 => ( ! [X2: A,Y2: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less_eq @ C3 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_122_ord__eq__le__subst,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType] :
% 3.29/1.81 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.29/1.81 & ( ord @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.29/1.81 ( ( A3
% 3.29/1.81 = ( F @ B2 ) )
% 3.29/1.81 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.29/1.81 => ( ! [X2: B,Y2: B] :
% 3.29/1.81 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_123_ord__le__eq__subst,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType] :
% 3.29/1.81 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.29/1.81 & ( ord @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,B2: A,F: A > B,C: B] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ( F @ B2 )
% 3.29/1.81 = C )
% 3.29/1.81 => ( ! [X2: A,Y2: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_124_eq__iff,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ^ [Y3: A,Z: A] : ( Y3 = Z ) )
% 3.29/1.81 = ( ^ [X: A,Y4: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.29/1.81 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_125_antisym,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 => ( ( ord_less_eq @ A @ Y @ X5 )
% 3.29/1.81 => ( X5 = Y ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_126_linear,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 | ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_127_eq__refl,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( X5 = Y )
% 3.29/1.81 => ( ord_less_eq @ A @ X5 @ Y ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_128_le__cases,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ~ ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_129_order_Otrans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A,C: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.29/1.81 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_130_le__cases3,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A,Z2: A] :
% 3.29/1.81 ( ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 3.29/1.81 => ( ( ( ord_less_eq @ A @ Y @ X5 )
% 3.29/1.81 => ~ ( ord_less_eq @ A @ X5 @ Z2 ) )
% 3.29/1.81 => ( ( ( ord_less_eq @ A @ X5 @ Z2 )
% 3.29/1.81 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 3.29/1.81 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 3.29/1.81 => ~ ( ord_less_eq @ A @ Y @ X5 ) )
% 3.29/1.81 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.29/1.81 => ~ ( ord_less_eq @ A @ Z2 @ X5 ) )
% 3.29/1.81 => ~ ( ( ord_less_eq @ A @ Z2 @ X5 )
% 3.29/1.81 => ~ ( ord_less_eq @ A @ X5 @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_131_antisym__conv,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [Y: A,X5: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ Y @ X5 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 = ( X5 = Y ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_132_ord__eq__le__trans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( ord @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A,C: A] :
% 3.29/1.81 ( ( A3 = B2 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.29/1.81 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_133_ord__le__eq__trans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( ord @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A,C: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( B2 = C )
% 3.29/1.81 => ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_134_order__class_Oorder_Oantisym,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.29/1.81 => ( A3 = B2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_135_order__trans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A,Z2: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.29/1.81 => ( ord_less_eq @ A @ X5 @ Z2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_136_dual__order_Orefl,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_137_linorder__wlog,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [P2: A > A > $o,A3: A,B2: A] :
% 3.29/1.81 ( ! [A4: A,B5: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A4 @ B5 )
% 3.29/1.81 => ( P2 @ A4 @ B5 ) )
% 3.29/1.81 => ( ! [A4: A,B5: A] :
% 3.29/1.81 ( ( P2 @ B5 @ A4 )
% 3.29/1.81 => ( P2 @ A4 @ B5 ) )
% 3.29/1.81 => ( P2 @ A3 @ B2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_138_dual__order_Otrans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [B2: A,A3: A,C: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.29/1.81 => ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_139_dual__order_Oantisym,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [B2: A,A3: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( A3 = B2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_140_bot_Oextremum,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_141_bot_Oextremum__unique,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 3.29/1.81 = ( A3
% 3.29/1.81 = ( bot_bot @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_142_bot_Oextremum__uniqueI,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order_bot @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 3.29/1.81 => ( A3
% 3.29/1.81 = ( bot_bot @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_143_RuleSystem_Opos__least,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,A: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,N2: nat,Rs: stream @ A,R: A] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( ( shd @ A @ ( sdrop @ A @ N2 @ Rs ) )
% 3.29/1.81 = R )
% 3.29/1.81 => ( ord_less_eq @ nat @ ( abstra2097340358le_pos @ A @ Rs @ R ) @ N2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_144_eqfelem__imp__iff,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,Y: A,A2: fset @ A] :
% 3.29/1.81 ( ( X5 = Y )
% 3.29/1.81 => ( ( fmember @ A @ X5 @ A2 )
% 3.29/1.81 = ( fmember @ A @ Y @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_145_if__split__fmem2,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,Q: $o,X5: fset @ A,Y: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ A3 @ ( if @ ( fset @ A ) @ Q @ X5 @ Y ) )
% 3.29/1.81 = ( ( Q
% 3.29/1.81 => ( fmember @ A @ A3 @ X5 ) )
% 3.29/1.81 & ( ~ Q
% 3.29/1.81 => ( fmember @ A @ A3 @ Y ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_146_if__split__fmem1,axiom,
% 3.29/1.81 ! [A: $tType,Q: $o,X5: A,Y: A,B2: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ ( if @ A @ Q @ X5 @ Y ) @ B2 )
% 3.29/1.81 = ( ( Q
% 3.29/1.81 => ( fmember @ A @ X5 @ B2 ) )
% 3.29/1.81 & ( ~ Q
% 3.29/1.81 => ( fmember @ A @ Y @ B2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_147_eqfset__imp__iff,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A,X5: A] :
% 3.29/1.81 ( ( A2 = B3 )
% 3.29/1.81 => ( ( fmember @ A @ X5 @ A2 )
% 3.29/1.81 = ( fmember @ A @ X5 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_148_eq__fmem__trans,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,B2: A,A2: fset @ A] :
% 3.29/1.81 ( ( A3 = B2 )
% 3.29/1.81 => ( ( fmember @ A @ B2 @ A2 )
% 3.29/1.81 => ( fmember @ A @ A3 @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_149_fset__choice,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType,A2: fset @ A,P2: A > B > $o] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( fmember @ A @ X2 @ A2 )
% 3.29/1.81 => ? [X12: B] : ( P2 @ X2 @ X12 ) )
% 3.29/1.81 => ? [F3: A > B] :
% 3.29/1.81 ! [X4: A] :
% 3.29/1.81 ( ( fmember @ A @ X4 @ A2 )
% 3.29/1.81 => ( P2 @ X4 @ ( F3 @ X4 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_150_fequalityCE,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A,C: A] :
% 3.29/1.81 ( ( A2 = B3 )
% 3.29/1.81 => ( ( ( fmember @ A @ C @ A2 )
% 3.29/1.81 => ~ ( fmember @ A @ C @ B3 ) )
% 3.29/1.81 => ~ ( ~ ( fmember @ A @ C @ A2 )
% 3.29/1.81 => ( fmember @ A @ C @ B3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_151_fset__eqI,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( fmember @ A @ X2 @ A2 )
% 3.29/1.81 = ( fmember @ A @ X2 @ B3 ) )
% 3.29/1.81 => ( A2 = B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_152_femptyE,axiom,
% 3.29/1.81 ! [A: $tType,A3: A] :
% 3.29/1.81 ~ ( fmember @ A @ A3 @ ( bot_bot @ ( fset @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_153_ex__fin__conv,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A] :
% 3.29/1.81 ( ( ? [X: A] : ( fmember @ A @ X @ A2 ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 != ( bot_bot @ ( fset @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_154_equalsffemptyD,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,A3: A] :
% 3.29/1.81 ( ( A2
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) )
% 3.29/1.81 => ~ ( fmember @ A @ A3 @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_155_equalsffemptyI,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A] :
% 3.29/1.81 ( ! [Y2: A] :
% 3.29/1.81 ~ ( fmember @ A @ Y2 @ A2 )
% 3.29/1.81 => ( A2
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_156_RuleSystem_OminWait__le__pos,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,Rs: stream @ Rule,R: Rule,S: State] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( ( member @ Rule @ R @ ( sset @ Rule @ Rules2 ) )
% 3.29/1.81 => ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ R @ S )
% 3.29/1.81 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ ( abstra2097340358le_pos @ Rule @ Rs @ R ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_157_RuleSystem_OminWait__least,axiom,
% 3.29/1.81 ! [State: $tType,Rule: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,N2: nat,Rs: stream @ Rule,S: State] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( abstra1874422341nabled @ Rule @ State @ Eff2 @ ( shd @ Rule @ ( sdrop @ Rule @ N2 @ Rs ) ) @ S )
% 3.29/1.81 => ( ord_less_eq @ nat @ ( abstra1332369113inWait @ Rule @ State @ Eff2 @ Rs @ S ) @ N2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_158_minus__apply,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType] :
% 3.29/1.81 ( ( minus @ B @ ( type2 @ B ) )
% 3.29/1.81 => ( ( minus_minus @ ( A > B ) )
% 3.29/1.81 = ( ^ [A5: A > B,B6: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B6 @ X ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_159_bot__empty__eq,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( bot_bot @ ( A > $o ) )
% 3.29/1.81 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_160_Collect__empty__eq__bot,axiom,
% 3.29/1.81 ! [A: $tType,P2: A > $o] :
% 3.29/1.81 ( ( ( collect @ A @ P2 )
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 = ( P2
% 3.29/1.81 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_161_in__cont__mkTree,axiom,
% 3.29/1.81 ! [S: state,Rs: stream @ rule,T2: abstra2103299360e_tree @ ( product_prod @ state @ rule )] :
% 3.29/1.81 ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ state @ rule ) ) @ T2 @ ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) ) )
% 3.29/1.81 => ? [Sl5: fset @ state,S6: state] :
% 3.29/1.81 ( ( member @ state @ S6 @ s )
% 3.29/1.81 & ( eff @ ( shd @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S @ Sl5 )
% 3.29/1.81 & ( fmember @ state @ S6 @ Sl5 )
% 3.29/1.81 & ( T2
% 3.29/1.81 = ( abstra1225283448mkTree @ rule @ state @ eff @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ S6 ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_162_is__singletonI,axiom,
% 3.29/1.81 ! [A: $tType,X5: A] : ( is_singleton @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_163_fair__stl,axiom,
% 3.29/1.81 ! [Rs: stream @ rule] :
% 3.29/1.81 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( abstra928354080m_fair @ rule @ rules @ ( stl @ rule @ Rs ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_164_subset__empty,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_165_empty__subsetI,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_166_insert__subset,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X5 @ A2 ) @ B3 )
% 3.29/1.81 = ( ( member @ A @ X5 @ B3 )
% 3.29/1.81 & ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_167_fminus__iff,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 3.29/1.81 = ( ( fmember @ A @ C @ A2 )
% 3.29/1.81 & ~ ( fmember @ A @ C @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_168_fminusI,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ C @ A2 )
% 3.29/1.81 => ( ~ ( fmember @ A @ C @ B3 )
% 3.29/1.81 => ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_169_fsubsetI,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( fmember @ A @ X2 @ A2 )
% 3.29/1.81 => ( fmember @ A @ X2 @ B3 ) )
% 3.29/1.81 => ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_170_fminus__fempty,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A] :
% 3.29/1.81 ( ( minus_minus @ ( fset @ A ) @ A2 @ ( bot_bot @ ( fset @ A ) ) )
% 3.29/1.81 = A2 ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_171_fminus__cancel,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A] :
% 3.29/1.81 ( ( minus_minus @ ( fset @ A ) @ A2 @ A2 )
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_172_fempty__fminus,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A] :
% 3.29/1.81 ( ( minus_minus @ ( fset @ A ) @ ( bot_bot @ ( fset @ A ) ) @ A2 )
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_173_fempty__fsubsetI,axiom,
% 3.29/1.81 ! [A: $tType,X5: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ ( bot_bot @ ( fset @ A ) ) @ X5 ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_174_fsubset__fempty,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ ( bot_bot @ ( fset @ A ) ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_175_singleton__insert__inj__eq_H,axiom,
% 3.29/1.81 ! [A: $tType,A3: A,A2: set @ A,B2: A] :
% 3.29/1.81 ( ( ( insert @ A @ A3 @ A2 )
% 3.29/1.81 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 = ( ( A3 = B2 )
% 3.29/1.81 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_176_singleton__insert__inj__eq,axiom,
% 3.29/1.81 ! [A: $tType,B2: A,A3: A,A2: set @ A] :
% 3.29/1.81 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 = ( insert @ A @ A3 @ A2 ) )
% 3.29/1.81 = ( ( A3 = B2 )
% 3.29/1.81 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_177_Diff__eq__empty__iff,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( ( minus_minus @ ( set @ A ) @ A2 @ B3 )
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 = ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_178_fminusD2,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 3.29/1.81 => ~ ( fmember @ A @ C @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_179_fminusD1,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 3.29/1.81 => ( fmember @ A @ C @ A2 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_180_fminusE,axiom,
% 3.29/1.81 ! [A: $tType,C: A,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ C @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) )
% 3.29/1.81 => ~ ( ( fmember @ A @ C @ A2 )
% 3.29/1.81 => ( fmember @ A @ C @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_181_fset__mp,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A,X5: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( fmember @ A @ X5 @ A2 )
% 3.29/1.81 => ( fmember @ A @ X5 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_182_fin__mono,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A,X5: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( fmember @ A @ X5 @ A2 )
% 3.29/1.81 => ( fmember @ A @ X5 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_183_fsubsetD,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A,C: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( fmember @ A @ C @ A2 )
% 3.29/1.81 => ( fmember @ A @ C @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_184_fset__rev__mp,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( fmember @ A @ X5 @ A2 )
% 3.29/1.81 => ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( fmember @ A @ X5 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_185_subset__insertI2,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A,B2: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_186_subset__insertI,axiom,
% 3.29/1.81 ! [A: $tType,B3: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A3 @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_187_subset__insert,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ~ ( member @ A @ X5 @ A2 )
% 3.29/1.81 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) )
% 3.29/1.81 = ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_188_insert__mono,axiom,
% 3.29/1.81 ! [A: $tType,C4: set @ A,D3: set @ A,A3: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 3.29/1.81 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_189_RuleSystem__Defs_Ofair__stl,axiom,
% 3.29/1.81 ! [Rule: $tType,Rules2: stream @ Rule,Rs: stream @ Rule] :
% 3.29/1.81 ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( abstra928354080m_fair @ Rule @ Rules2 @ ( stl @ Rule @ Rs ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_190_double__diff,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A,C4: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
% 3.29/1.81 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C4 @ A2 ) )
% 3.29/1.81 = A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_191_Diff__subset,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ A2 ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_192_Diff__mono,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,C4: set @ A,D3: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
% 3.29/1.81 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B3 )
% 3.29/1.81 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_193_countable__subset,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( countable_countable @ A @ B3 )
% 3.29/1.81 => ( countable_countable @ A @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_194_subset__singleton__iff,axiom,
% 3.29/1.81 ! [A: $tType,X6: set @ A,A3: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 = ( ( X6
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 | ( X6
% 3.29/1.81 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_195_subset__singletonD,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,X5: A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) )
% 3.29/1.81 => ( ( A2
% 3.29/1.81 = ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 | ( A2
% 3.29/1.81 = ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_196_subset__Diff__insert,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A,X5: A,C4: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X5 @ C4 ) ) )
% 3.29/1.81 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B3 @ C4 ) )
% 3.29/1.81 & ~ ( member @ A @ X5 @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_197_is__singletonI_H,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( A2
% 3.29/1.81 != ( bot_bot @ ( set @ A ) ) )
% 3.29/1.81 => ( ! [X2: A,Y2: A] :
% 3.29/1.81 ( ( member @ A @ X2 @ A2 )
% 3.29/1.81 => ( ( member @ A @ Y2 @ A2 )
% 3.29/1.81 => ( X2 = Y2 ) ) )
% 3.29/1.81 => ( is_singleton @ A @ A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_198_Diff__single__insert,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,X5: A,B3: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 )
% 3.29/1.81 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_199_subset__insert__iff,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,X5: A,B3: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ B3 ) )
% 3.29/1.81 = ( ( ( member @ A @ X5 @ A2 )
% 3.29/1.81 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) )
% 3.29/1.81 & ( ~ ( member @ A @ X5 @ A2 )
% 3.29/1.81 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_200_fun__diff__def,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType] :
% 3.29/1.81 ( ( minus @ B @ ( type2 @ B ) )
% 3.29/1.81 => ( ( minus_minus @ ( A > B ) )
% 3.29/1.81 = ( ^ [A5: A > B,B6: A > B,X: A] : ( minus_minus @ B @ ( A5 @ X ) @ ( B6 @ X ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_201_RuleSystem_Oin__cont__mkTree,axiom,
% 3.29/1.81 ! [Rule: $tType,State: $tType,Eff2: Rule > State > ( fset @ State ) > $o,Rules2: stream @ Rule,S3: set @ State,S: State,Rs: stream @ Rule,T2: abstra2103299360e_tree @ ( product_prod @ State @ Rule )] :
% 3.29/1.81 ( ( abstra1326562878System @ Rule @ State @ Eff2 @ Rules2 @ S3 )
% 3.29/1.81 => ( ( member @ State @ S @ S3 )
% 3.29/1.81 => ( ( abstra928354080m_fair @ Rule @ Rules2 @ Rs )
% 3.29/1.81 => ( ( fmember @ ( abstra2103299360e_tree @ ( product_prod @ State @ Rule ) ) @ T2 @ ( abstra1749095923e_cont @ ( product_prod @ State @ Rule ) @ ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ Rs @ S ) ) )
% 3.29/1.81 => ? [Sl5: fset @ State,S6: State] :
% 3.29/1.81 ( ( member @ State @ S6 @ S3 )
% 3.29/1.81 & ( Eff2 @ ( shd @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S @ Sl5 )
% 3.29/1.81 & ( fmember @ State @ S6 @ Sl5 )
% 3.29/1.81 & ( T2
% 3.29/1.81 = ( abstra1225283448mkTree @ Rule @ State @ Eff2 @ ( stl @ Rule @ ( abstra1259602206m_trim @ Rule @ State @ Eff2 @ Rs @ S ) ) @ S6 ) ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_202_is__singleton__def,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( is_singleton @ A )
% 3.29/1.81 = ( ^ [A5: set @ A] :
% 3.29/1.81 ? [X: A] :
% 3.29/1.81 ( A5
% 3.29/1.81 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_203_is__singletonE,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A] :
% 3.29/1.81 ( ( is_singleton @ A @ A2 )
% 3.29/1.81 => ~ ! [X2: A] :
% 3.29/1.81 ( A2
% 3.29/1.81 != ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_204_mkTree_Osimps_I2_J,axiom,
% 3.29/1.81 ! [Rs: stream @ rule,S: state] :
% 3.29/1.81 ( ( abstra1749095923e_cont @ ( product_prod @ state @ rule ) @ ( abstra1225283448mkTree @ rule @ state @ eff @ Rs @ S ) )
% 3.29/1.81 = ( 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 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_205_sset__induct,axiom,
% 3.29/1.81 ! [A: $tType,Y: A,S: stream @ A,P2: A > ( stream @ A ) > $o] :
% 3.29/1.81 ( ( member @ A @ Y @ ( sset @ A @ S ) )
% 3.29/1.81 => ( ! [S5: stream @ A] : ( P2 @ ( shd @ A @ S5 ) @ S5 )
% 3.29/1.81 => ( ! [S5: stream @ A,Y2: A] :
% 3.29/1.81 ( ( member @ A @ Y2 @ ( sset @ A @ ( stl @ A @ S5 ) ) )
% 3.29/1.81 => ( ( P2 @ Y2 @ ( stl @ A @ S5 ) )
% 3.29/1.81 => ( P2 @ Y2 @ S5 ) ) )
% 3.29/1.81 => ( P2 @ Y @ S ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_206_stake__pos__minWait,axiom,
% 3.29/1.81 ! [Rs: stream @ rule,S: state,R: rule] :
% 3.29/1.81 ( ( abstra928354080m_fair @ rule @ rules @ Rs )
% 3.29/1.81 => ( ( ord_less @ nat @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) @ ( abstra2097340358le_pos @ rule @ Rs @ R ) )
% 3.29/1.81 => ( ( member @ rule @ R @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( ( member @ state @ S @ s )
% 3.29/1.81 => ( ( abstra2097340358le_pos @ rule @ ( stl @ rule @ ( abstra1259602206m_trim @ rule @ state @ eff @ Rs @ S ) ) @ R )
% 3.29/1.81 = ( minus_minus @ nat @ ( abstra2097340358le_pos @ rule @ Rs @ R ) @ ( suc @ ( abstra1332369113inWait @ rule @ state @ eff @ Rs @ S ) ) ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_207_subsetI,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( member @ A @ X2 @ A2 )
% 3.29/1.81 => ( member @ A @ X2 @ B3 ) )
% 3.29/1.81 => ( ord_less_eq @ ( set @ A ) @ A2 @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_208_subset__antisym,axiom,
% 3.29/1.81 ! [A: $tType,A2: set @ A,B3: set @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A2 )
% 3.29/1.81 => ( A2 = B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_209_fsubset__antisym,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( ord_less_eq @ ( fset @ A ) @ B3 @ A2 )
% 3.29/1.81 => ( A2 = B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_210_fminus__idemp,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( minus_minus @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) @ B3 )
% 3.29/1.81 = ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_211_fimage__eqI,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType,B2: A,F: B > A,X5: B,A2: fset @ B] :
% 3.29/1.81 ( ( B2
% 3.29/1.81 = ( F @ X5 ) )
% 3.29/1.81 => ( ( fmember @ B @ X5 @ A2 )
% 3.29/1.81 => ( fmember @ A @ B2 @ ( fimage @ B @ A @ F @ A2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_212_fimage__is__fempty,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType,F: B > A,A2: fset @ B] :
% 3.29/1.81 ( ( ( fimage @ B @ A @ F @ A2 )
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 = ( bot_bot @ ( fset @ B ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_213_fempty__is__fimage,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType,F: B > A,A2: fset @ B] :
% 3.29/1.81 ( ( ( bot_bot @ ( fset @ A ) )
% 3.29/1.81 = ( fimage @ B @ A @ F @ A2 ) )
% 3.29/1.81 = ( A2
% 3.29/1.81 = ( bot_bot @ ( fset @ B ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_214_fimage__fempty,axiom,
% 3.29/1.81 ! [B: $tType,A: $tType,F: B > A] :
% 3.29/1.81 ( ( fimage @ B @ A @ F @ ( bot_bot @ ( fset @ B ) ) )
% 3.29/1.81 = ( bot_bot @ ( fset @ A ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_215_sdrop__simps_I2_J,axiom,
% 3.29/1.81 ! [A: $tType,N2: nat,S: stream @ A] :
% 3.29/1.81 ( ( stl @ A @ ( sdrop @ A @ N2 @ S ) )
% 3.29/1.81 = ( sdrop @ A @ ( suc @ N2 ) @ S ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_216_fimage__fminus__fsubset,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType,F: B > A,A2: fset @ B,B3: fset @ B] : ( ord_less_eq @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ ( fimage @ B @ A @ F @ A2 ) @ ( fimage @ B @ A @ F @ B3 ) ) @ ( fimage @ B @ A @ F @ ( minus_minus @ ( fset @ B ) @ A2 @ B3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_217_fminus__fsubset,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A] : ( ord_less_eq @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) @ A2 ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_218_double__fminus,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,B3: fset @ A,C4: fset @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ B3 )
% 3.29/1.81 => ( ( ord_less_eq @ ( fset @ A ) @ B3 @ C4 )
% 3.29/1.81 => ( ( minus_minus @ ( fset @ A ) @ B3 @ ( minus_minus @ ( fset @ A ) @ C4 @ A2 ) )
% 3.29/1.81 = A2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_219_fminus__mono,axiom,
% 3.29/1.81 ! [A: $tType,A2: fset @ A,C4: fset @ A,D3: fset @ A,B3: fset @ A] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ A2 @ C4 )
% 3.29/1.81 => ( ( ord_less_eq @ ( fset @ A ) @ D3 @ B3 )
% 3.29/1.81 => ( ord_less_eq @ ( fset @ A ) @ ( minus_minus @ ( fset @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( fset @ A ) @ C4 @ D3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_220_leD,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [Y: A,X5: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ Y @ X5 )
% 3.29/1.81 => ~ ( ord_less @ A @ X5 @ Y ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_221_leI,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ~ ( ord_less @ A @ X5 @ Y )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_222_le__less,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ord_less_eq @ A )
% 3.29/1.81 = ( ^ [X: A,Y4: A] :
% 3.29/1.81 ( ( ord_less @ A @ X @ Y4 )
% 3.29/1.81 | ( X = Y4 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_223_less__le,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ord_less @ A )
% 3.29/1.81 = ( ^ [X: A,Y4: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.29/1.81 & ( X != Y4 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_224_order__le__less__subst1,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType] :
% 3.29/1.81 ( ( ( order @ B @ ( type2 @ B ) )
% 3.29/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
% 3.29/1.81 => ( ( ord_less @ B @ B2 @ C )
% 3.29/1.81 => ( ! [X2: B,Y2: B] :
% 3.29/1.81 ( ( ord_less @ B @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_225_order__le__less__subst2,axiom,
% 3.29/1.81 ! [A: $tType,C3: $tType] :
% 3.29/1.81 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 3.29/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,B2: A,F: A > C3,C: C3] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ord_less @ C3 @ ( F @ B2 ) @ C )
% 3.29/1.81 => ( ! [X2: A,Y2: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less_eq @ C3 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_226_order__less__le__subst1,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType] :
% 3.29/1.81 ( ( ( order @ B @ ( type2 @ B ) )
% 3.29/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,F: B > A,B2: B,C: B] :
% 3.29/1.81 ( ( ord_less @ A @ A3 @ ( F @ B2 ) )
% 3.29/1.81 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.29/1.81 => ( ! [X2: B,Y2: B] :
% 3.29/1.81 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_227_order__less__le__subst2,axiom,
% 3.29/1.81 ! [A: $tType,C3: $tType] :
% 3.29/1.81 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 3.29/1.81 & ( order @ A @ ( type2 @ A ) ) )
% 3.29/1.81 => ! [A3: A,B2: A,F: A > C3,C: C3] :
% 3.29/1.81 ( ( ord_less @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
% 3.29/1.81 => ( ! [X2: A,Y2: A] :
% 3.29/1.81 ( ( ord_less @ A @ X2 @ Y2 )
% 3.29/1.81 => ( ord_less @ C3 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 3.29/1.81 => ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_228_not__le,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ~ ( ord_less_eq @ A @ X5 @ Y ) )
% 3.29/1.81 = ( ord_less @ A @ Y @ X5 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_229_not__less,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ~ ( ord_less @ A @ X5 @ Y ) )
% 3.29/1.81 = ( ord_less_eq @ A @ Y @ X5 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_230_le__neq__trans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( A3 != B2 )
% 3.29/1.81 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_231_less__imp__le,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ord_less @ A @ X5 @ Y )
% 3.29/1.81 => ( ord_less_eq @ A @ X5 @ Y ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_232_antisym__conv1,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ~ ( ord_less @ A @ X5 @ Y )
% 3.29/1.81 => ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 = ( X5 = Y ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_233_antisym__conv2,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 => ( ( ~ ( ord_less @ A @ X5 @ Y ) )
% 3.29/1.81 = ( X5 = Y ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_234_le__less__trans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A,Z2: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 => ( ( ord_less @ A @ Y @ Z2 )
% 3.29/1.81 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_235_less__le__trans,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A,Z2: A] :
% 3.29/1.81 ( ( ord_less @ A @ X5 @ Y )
% 3.29/1.81 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 3.29/1.81 => ( ord_less @ A @ X5 @ Z2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_236_dense__ge,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [Z2: A,Y: A] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( ord_less @ A @ Z2 @ X2 )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ X2 ) )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_237_dense__le,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [Y: A,Z2: A] :
% 3.29/1.81 ( ! [X2: A] :
% 3.29/1.81 ( ( ord_less @ A @ X2 @ Y )
% 3.29/1.81 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_238_le__less__linear,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 | ( ord_less @ A @ Y @ X5 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_239_le__imp__less__or__eq,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X5 @ Y )
% 3.29/1.81 => ( ( ord_less @ A @ X5 @ Y )
% 3.29/1.81 | ( X5 = Y ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_240_less__le__not__le,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( preorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ord_less @ A )
% 3.29/1.81 = ( ^ [X: A,Y4: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.29/1.81 & ~ ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_241_not__le__imp__less,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [Y: A,X5: A] :
% 3.29/1.81 ( ~ ( ord_less_eq @ A @ Y @ X5 )
% 3.29/1.81 => ( ord_less @ A @ X5 @ Y ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_242_order_Ostrict__trans1,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A,C: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ord_less @ A @ B2 @ C )
% 3.29/1.81 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_243_order_Ostrict__trans2,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A,C: A] :
% 3.29/1.81 ( ( ord_less @ A @ A3 @ B2 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.29/1.81 => ( ord_less @ A @ A3 @ C ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_244_order_Oorder__iff__strict,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ord_less_eq @ A )
% 3.29/1.81 = ( ^ [A6: A,B7: A] :
% 3.29/1.81 ( ( ord_less @ A @ A6 @ B7 )
% 3.29/1.81 | ( A6 = B7 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_245_order_Ostrict__iff__order,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ord_less @ A )
% 3.29/1.81 = ( ^ [A6: A,B7: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ A6 @ B7 )
% 3.29/1.81 & ( A6 != B7 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_246_dual__order_Ostrict__trans1,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [B2: A,A3: A,C: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ B2 @ A3 )
% 3.29/1.81 => ( ( ord_less @ A @ C @ B2 )
% 3.29/1.81 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_247_dual__order_Ostrict__trans2,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [B2: A,A3: A,C: A] :
% 3.29/1.81 ( ( ord_less @ A @ B2 @ A3 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.29/1.81 => ( ord_less @ A @ C @ A3 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_248_dense__ge__bounded,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [Z2: A,X5: A,Y: A] :
% 3.29/1.81 ( ( ord_less @ A @ Z2 @ X5 )
% 3.29/1.81 => ( ! [W: A] :
% 3.29/1.81 ( ( ord_less @ A @ Z2 @ W )
% 3.29/1.81 => ( ( ord_less @ A @ W @ X5 )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ W ) ) )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_249_dense__le__bounded,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [X5: A,Y: A,Z2: A] :
% 3.29/1.81 ( ( ord_less @ A @ X5 @ Y )
% 3.29/1.81 => ( ! [W: A] :
% 3.29/1.81 ( ( ord_less @ A @ X5 @ W )
% 3.29/1.81 => ( ( ord_less @ A @ W @ Y )
% 3.29/1.81 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 3.29/1.81 => ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_250_order_Ostrict__implies__order,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A] :
% 3.29/1.81 ( ( ord_less @ A @ A3 @ B2 )
% 3.29/1.81 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_251_dual__order_Oorder__iff__strict,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ord_less_eq @ A )
% 3.29/1.81 = ( ^ [B7: A,A6: A] :
% 3.29/1.81 ( ( ord_less @ A @ B7 @ A6 )
% 3.29/1.81 | ( A6 = B7 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_252_dual__order_Ostrict__iff__order,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ( ( ord_less @ A )
% 3.29/1.81 = ( ^ [B7: A,A6: A] :
% 3.29/1.81 ( ( ord_less_eq @ A @ B7 @ A6 )
% 3.29/1.81 & ( A6 != B7 ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_253_dual__order_Ostrict__implies__order,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [B2: A,A3: A] :
% 3.29/1.81 ( ( ord_less @ A @ B2 @ A3 )
% 3.29/1.81 => ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_254_order_Onot__eq__order__implies__strict,axiom,
% 3.29/1.81 ! [A: $tType] :
% 3.29/1.81 ( ( order @ A @ ( type2 @ A ) )
% 3.29/1.81 => ! [A3: A,B2: A] :
% 3.29/1.81 ( ( A3 != B2 )
% 3.29/1.81 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.29/1.81 => ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(fact_255_subset__fimage__iff,axiom,
% 3.29/1.81 ! [A: $tType,B: $tType,B3: fset @ A,F: B > A,A2: fset @ B] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ A ) @ B3 @ ( fimage @ B @ A @ F @ A2 ) )
% 3.29/1.81 = ( ? [AA: fset @ B] :
% 3.29/1.81 ( ( ord_less_eq @ ( fset @ B ) @ AA @ A2 )
% 3.29/1.81 & ( B3
% 3.29/1.81 = ( fimage @ B @ A @ F @ AA ) ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 & ( finite_finite @ A8 @ ( type2 @ A8 ) ) )
% 3.29/1.81 => ( finite_finite @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_1,axiom,
% 3.29/1.81 ! [A7: $tType] :
% 3.29/1.81 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 => ( finite_finite @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Finite__Set_Ofinite_2,axiom,
% 3.29/1.81 finite_finite @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
% 3.29/1.81 ! [A7: $tType] :
% 3.29/1.81 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 => ( finite_finite @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Finite__Set_Ofinite_4,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 & ( finite_finite @ A8 @ ( type2 @ A8 ) ) )
% 3.29/1.81 => ( finite_finite @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 3.29/1.81 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Countable_Ocountable,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 & ( countable @ A8 @ ( type2 @ A8 ) ) )
% 3.29/1.81 => ( countable @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 3.29/1.81 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( order @ A8 @ ( type2 @ A8 ) )
% 3.29/1.81 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Orderings_Oord,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 3.29/1.81 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Orderings_Obot,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 3.29/1.81 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_fun___Groups_Ominus,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( minus @ A8 @ ( type2 @ A8 ) )
% 3.29/1.81 => ( minus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Orderings_Oorder__bot_5,axiom,
% 3.29/1.81 order_bot @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Countable_Ocountable_6,axiom,
% 3.29/1.81 countable @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Orderings_Opreorder_7,axiom,
% 3.29/1.81 preorder @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.29/1.81 linorder @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
% 3.29/1.81 order @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Orderings_Oord_9,axiom,
% 3.29/1.81 ord @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Orderings_Obot_10,axiom,
% 3.29/1.81 bot @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Nat_Onat___Groups_Ominus_11,axiom,
% 3.29/1.81 minus @ nat @ ( type2 @ nat ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Orderings_Oorder__bot_12,axiom,
% 3.29/1.81 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Countable_Ocountable_13,axiom,
% 3.29/1.81 ! [A7: $tType] :
% 3.29/1.81 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 => ( countable @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Orderings_Opreorder_14,axiom,
% 3.29/1.81 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Orderings_Oorder_15,axiom,
% 3.29/1.81 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
% 3.29/1.81 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Orderings_Obot_17,axiom,
% 3.29/1.81 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Set_Oset___Groups_Ominus_18,axiom,
% 3.29/1.81 ! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Orderings_Oorder__bot_19,axiom,
% 3.29/1.81 order_bot @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Countable_Ocountable_20,axiom,
% 3.29/1.81 countable @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
% 3.29/1.81 preorder @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
% 3.29/1.81 linorder @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Orderings_Oorder_23,axiom,
% 3.29/1.81 order @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Orderings_Oord_24,axiom,
% 3.29/1.81 ord @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Orderings_Obot_25,axiom,
% 3.29/1.81 bot @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_HOL_Obool___Groups_Ominus_26,axiom,
% 3.29/1.81 minus @ $o @ ( type2 @ $o ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Orderings_Oorder__bot_27,axiom,
% 3.29/1.81 ! [A7: $tType] : ( order_bot @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Countable_Ocountable_28,axiom,
% 3.29/1.81 ! [A7: $tType] :
% 3.29/1.81 ( ( finite_finite @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 => ( countable @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Orderings_Opreorder_29,axiom,
% 3.29/1.81 ! [A7: $tType] : ( preorder @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Orderings_Oorder_30,axiom,
% 3.29/1.81 ! [A7: $tType] : ( order @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Orderings_Oord_31,axiom,
% 3.29/1.81 ! [A7: $tType] : ( ord @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Orderings_Obot_32,axiom,
% 3.29/1.81 ! [A7: $tType] : ( bot @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_FSet_Ofset___Groups_Ominus_33,axiom,
% 3.29/1.81 ! [A7: $tType] : ( minus @ ( fset @ A7 ) @ ( type2 @ ( fset @ A7 ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(tcon_Product__Type_Oprod___Countable_Ocountable_34,axiom,
% 3.29/1.81 ! [A7: $tType,A8: $tType] :
% 3.29/1.81 ( ( ( countable @ A7 @ ( type2 @ A7 ) )
% 3.29/1.81 & ( countable @ A8 @ ( type2 @ A8 ) ) )
% 3.29/1.81 => ( countable @ ( product_prod @ A7 @ A8 ) @ ( type2 @ ( product_prod @ A7 @ A8 ) ) ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(help_If_3_1_T,axiom,
% 3.29/1.81 ! [P2: $o] :
% 3.29/1.81 ( ( P2 = $true )
% 3.29/1.81 | ( P2 = $false ) ) ).
% 3.29/1.81
% 3.29/1.81 thf(help_If_2_1_T,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,Y: A] :
% 3.29/1.81 ( ( if @ A @ $false @ X5 @ Y )
% 3.29/1.81 = Y ) ).
% 3.29/1.81
% 3.29/1.81 thf(help_If_1_1_T,axiom,
% 3.29/1.81 ! [A: $tType,X5: A,Y: A] :
% 3.29/1.81 ( ( if @ A @ $true @ X5 @ Y )
% 3.29/1.81 = X5 ) ).
% 3.29/1.81
% 3.29/1.81 thf(conj_0,conjecture,
% 3.29/1.81 ! [X2: rule] :
% 3.29/1.81 ( ( member @ rule @ X2 @ ( sset @ rule @ rules ) )
% 3.29/1.81 => ( abstra726722745urated @ rule @ state @ eff @ X2 @ steps ) ) ).
% 3.29/1.81 ------------------------
% 3.29/1.85 % (19623)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/183Mi)
% 3.29/1.85 % (19627)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/2Mi)
% 3.29/1.85 % (19624)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/4Mi)
% 3.29/1.85 % (19625)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/27Mi)
% 3.29/1.85 % (19628)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/275Mi)
% 3.29/1.85 % (19629)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/18Mi)
% 3.29/1.85 % (19626)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/2Mi)
% 3.29/1.85 % (19626)Instruction limit reached!
% 3.29/1.85 % (19626)------------------------------
% 3.29/1.85 % (19626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.85 % (19626)Termination reason: Unknown
% 3.29/1.85 % (19626)Termination phase: shuffling
% 3.29/1.85 % (19627)Instruction limit reached!
% 3.29/1.85 % (19627)------------------------------
% 3.29/1.85 % (19627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.85
% 3.29/1.85 % (19626)Memory used [KB]: 1407
% 3.29/1.85 % (19626)Time elapsed: 0.003 s
% 3.29/1.85 % (19626)Instructions burned: 2 (million)
% 3.29/1.85 % (19626)------------------------------
% 3.29/1.85 % (19626)------------------------------
% 3.29/1.85 % (19627)Termination reason: Unknown
% 3.29/1.85 % (19627)Termination phase: shuffling
% 3.29/1.85
% 3.29/1.85 % (19627)Memory used [KB]: 1407
% 3.29/1.85 % (19627)Time elapsed: 0.003 s
% 3.29/1.85 % (19627)Instructions burned: 2 (million)
% 3.29/1.85 % (19627)------------------------------
% 3.29/1.85 % (19627)------------------------------
% 3.29/1.85 % (19624)Instruction limit reached!
% 3.29/1.85 % (19624)------------------------------
% 3.29/1.85 % (19624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.85 % (19624)Termination reason: Unknown
% 3.29/1.85 % (19624)Termination phase: shuffling
% 3.29/1.85
% 3.29/1.85 % (19624)Memory used [KB]: 1407
% 3.29/1.85 % (19624)Time elapsed: 0.004 s
% 3.29/1.85 % (19624)Instructions burned: 4 (million)
% 3.29/1.85 % (19624)------------------------------
% 3.29/1.85 % (19624)------------------------------
% 3.29/1.86 % (19629)Instruction limit reached!
% 3.29/1.86 % (19629)------------------------------
% 3.29/1.86 % (19629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.86 % (19629)Termination reason: Unknown
% 3.29/1.86 % (19629)Termination phase: shuffling
% 3.29/1.86
% 3.29/1.86 % (19629)Memory used [KB]: 1663
% 3.29/1.86 % (19629)Time elapsed: 0.011 s
% 3.29/1.86 % (19629)Instructions burned: 18 (million)
% 3.29/1.86 % (19629)------------------------------
% 3.29/1.86 % (19629)------------------------------
% 3.29/1.86 % (19625)Instruction limit reached!
% 3.29/1.86 % (19625)------------------------------
% 3.29/1.86 % (19625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.86 % (19625)Termination reason: Unknown
% 3.29/1.86 % (19625)Termination phase: shuffling
% 3.29/1.86
% 3.29/1.86 % (19625)Memory used [KB]: 1918
% 3.29/1.86 % (19625)Time elapsed: 0.016 s
% 3.29/1.86 % (19625)Instructions burned: 27 (million)
% 3.29/1.86 % (19625)------------------------------
% 3.29/1.86 % (19625)------------------------------
% 3.29/1.87 % (19630)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/3Mi)
% 3.29/1.87 % (19631)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/37Mi)
% 3.29/1.87 % (19632)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/15Mi)
% 3.29/1.87 % (19630)Instruction limit reached!
% 3.29/1.87 % (19630)------------------------------
% 3.29/1.87 % (19630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.87 % (19630)Termination reason: Unknown
% 3.29/1.87 % (19630)Termination phase: shuffling
% 3.29/1.87
% 3.29/1.87 % (19630)Memory used [KB]: 1407
% 3.29/1.87 % (19630)Time elapsed: 0.003 s
% 3.29/1.87 % (19630)Instructions burned: 3 (million)
% 3.29/1.87 % (19630)------------------------------
% 3.29/1.87 % (19630)------------------------------
% 3.29/1.87 % (19633)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19524 for (2999ds/3Mi)
% 3.29/1.87 % (19632)Instruction limit reached!
% 3.29/1.87 % (19632)------------------------------
% 3.29/1.87 % (19632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.87 % (19632)Termination reason: Unknown
% 3.29/1.87 % (19632)Termination phase: shuffling
% 3.29/1.87
% 3.29/1.87 % (19632)Memory used [KB]: 1663
% 3.29/1.87 % (19632)Time elapsed: 0.009 s
% 3.29/1.87 % (19632)Instructions burned: 15 (million)
% 3.29/1.87 % (19632)------------------------------
% 3.29/1.87 % (19632)------------------------------
% 3.29/1.88 % (19633)Instruction limit reached!
% 3.29/1.88 % (19633)------------------------------
% 3.29/1.88 % (19633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.88 % (19633)Termination reason: Unknown
% 3.29/1.88 % (19633)Termination phase: shuffling
% 3.29/1.88
% 3.29/1.88 % (19633)Memory used [KB]: 1407
% 3.29/1.88 % (19633)Time elapsed: 0.003 s
% 3.29/1.88 % (19633)Instructions burned: 3 (million)
% 3.29/1.88 % (19633)------------------------------
% 3.29/1.88 % (19633)------------------------------
% 3.29/1.88 % (19628)First to succeed.
% 3.29/1.88 % (19628)Refutation found. Thanks to Tanya!
% 3.29/1.88 % SZS status Theorem for DTF2THF_19524
% 3.29/1.88 % SZS output start Proof for DTF2THF_19524
% 3.29/1.88 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.29/1.88 thf(type_def_7, type, fset: ($tType) > $tType).
% 3.29/1.88 thf(type_def_8, type, stream: ($tType) > $tType).
% 3.29/1.88 thf(type_def_9, type, set: ($tType) > $tType).
% 3.29/1.88 thf(type_def_10, type, nat: $tType).
% 3.29/1.88 thf(type_def_11, type, product_prod: ($tType * $tType) > $tType).
% 3.29/1.88 thf(type_def_12, type, abstra2103299360e_tree: ($tType) > $tType).
% 3.29/1.88 thf(type_def_13, type, state: $tType).
% 3.29/1.88 thf(type_def_14, type, rule: $tType).
% 3.29/1.88 thf(func_def_0, type, abstra2103299360e_tree: ($tType) > $tType).
% 3.29/1.88 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 3.29/1.88 thf(func_def_2, type, stream: ($tType) > $tType).
% 3.29/1.88 thf(func_def_3, type, fset: ($tType) > $tType).
% 3.29/1.88 thf(func_def_4, type, set: ($tType) > $tType).
% 3.29/1.88 thf(func_def_5, type, nat: $tType).
% 3.29/1.88 thf(func_def_6, type, state: $tType).
% 3.29/1.88 thf(func_def_7, type, itself: ($tType) > $tType).
% 3.29/1.88 thf(func_def_8, type, rule: $tType).
% 3.29/1.88 thf(func_def_9, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_10, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_11, type, minus: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_12, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_13, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_14, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_15, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_16, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_17, type, countable: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_18, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_19, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.29/1.88 thf(func_def_20, type, abstra612932245System: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > set @ X1 > $o)).
% 3.29/1.88 thf(func_def_21, type, abstra1326562878System: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > set @ X1 > $o)).
% 3.29/1.88 thf(func_def_22, type, abstra1332369113inWait: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > X1 > nat)).
% 3.29/1.88 thf(func_def_23, type, abstra2096684367le_per: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > set @ X1 > X0 > $o)).
% 3.29/1.88 thf(func_def_24, type, abstra2097340358le_pos: !>[X0: $tType]:(stream @ X0 > X0 > nat)).
% 3.29/1.88 thf(func_def_25, type, abstra1209608345urated: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > stream @ product_prod @ X1 @ X0 > $o)).
% 3.29/1.88 thf(func_def_26, type, abstra1874422341nabled: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > X0 > X1 > $o)).
% 3.29/1.88 thf(func_def_27, type, abstra523868654_epath: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > stream @ product_prod @ X1 @ X0 > $o)).
% 3.29/1.88 thf(func_def_28, type, abstra928354080m_fair: !>[X0: $tType]:(stream @ X0 > stream @ X0 > $o)).
% 3.29/1.88 thf(func_def_29, type, abstra1774373515_fenum: !>[X0: $tType]:(stream @ X0 > stream @ X0)).
% 3.29/1.88 thf(func_def_30, type, abstra1225283448mkTree: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > X1 > abstra2103299360e_tree @ product_prod @ X1 @ X0)).
% 3.29/1.88 thf(func_def_31, type, abstra1276541928ickEff: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > X0 > X1 > fset @ X1)).
% 3.29/1.88 thf(func_def_32, type, abstra726722745urated: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > X0 > stream @ product_prod @ X1 @ X0 > $o)).
% 3.29/1.88 thf(func_def_33, type, abstra1259602206m_trim: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > X1 > stream @ X0)).
% 3.29/1.88 thf(func_def_34, type, abstra1874736267tem_wf: !>[X0: $tType, X1: $tType]:((X0 > X1 > fset @ X1 > $o) > stream @ X0 > abstra2103299360e_tree @ product_prod @ X1 @ X0 > $o)).
% 3.29/1.88 thf(func_def_35, type, abstra313004635_ipath: !>[X0: $tType]:(abstra2103299360e_tree @ X0 > stream @ X0 > $o)).
% 3.29/1.88 thf(func_def_36, type, abstra1749095923e_cont: !>[X0: $tType]:(abstra2103299360e_tree @ X0 > fset @ abstra2103299360e_tree @ X0)).
% 3.29/1.88 thf(func_def_37, type, countable_countable: !>[X0: $tType]:(set @ X0 > $o)).
% 3.29/1.88 thf(func_def_38, type, fimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > fset @ X0 > fset @ X1)).
% 3.29/1.88 thf(func_def_39, type, fmember: !>[X0: $tType]:(X0 > fset @ X0 > $o)).
% 3.29/1.88 thf(func_def_40, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.29/1.88 thf(func_def_41, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.29/1.88 thf(func_def_42, type, suc: nat > nat).
% 3.29/1.88 thf(func_def_43, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.29/1.88 thf(func_def_44, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.29/1.88 thf(func_def_45, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.29/1.88 thf(func_def_46, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.29/1.88 thf(func_def_47, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.29/1.88 thf(func_def_48, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.29/1.88 thf(func_def_49, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 3.29/1.88 thf(func_def_50, type, sdrop: !>[X0: $tType]:(nat > stream @ X0 > stream @ X0)).
% 3.29/1.88 thf(func_def_51, type, shd: !>[X0: $tType]:(stream @ X0 > X0)).
% 3.29/1.88 thf(func_def_52, type, sset: !>[X0: $tType]:(stream @ X0 > set @ X0)).
% 3.29/1.88 thf(func_def_53, type, stl: !>[X0: $tType]:(stream @ X0 > stream @ X0)).
% 3.29/1.88 thf(func_def_54, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.29/1.88 thf(func_def_55, type, s: set @ state).
% 3.29/1.88 thf(func_def_56, type, eff: rule > state > fset @ state > $o).
% 3.29/1.88 thf(func_def_57, type, rs: stream @ rule).
% 3.29/1.88 thf(func_def_58, type, rules: stream @ rule).
% 3.29/1.88 thf(func_def_59, type, s2: state).
% 3.29/1.88 thf(func_def_60, type, steps: stream @ product_prod @ state @ rule).
% 3.29/1.88 thf(func_def_78, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.29/1.88 thf(func_def_79, type, sK1: nat > stream @ product_prod @ state @ rule > stream @ rule > nat).
% 3.29/1.88 thf(func_def_80, type, sK2: nat > stream @ product_prod @ state @ rule > stream @ rule > state).
% 3.29/1.88 thf(func_def_81, type, sK3: !>[X0: $tType]:(set @ X0 > X0)).
% 3.29/1.88 thf(func_def_82, type, sK4: !>[X0: $tType, X1: $tType]:(nat > stream @ product_prod @ X1 @ X0 > stream @ X0 > set @ X1 > (X0 > X1 > fset @ X1 > $o) > nat)).
% 3.29/1.88 thf(func_def_83, type, sK5: !>[X0: $tType, X1: $tType]:(nat > stream @ product_prod @ X1 @ X0 > stream @ X0 > set @ X1 > (X0 > X1 > fset @ X1 > $o) > X1)).
% 3.29/1.88 thf(func_def_84, type, sK6: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.29/1.88 thf(func_def_85, type, sK7: !>[X0: $tType]:(set @ X0 > X0)).
% 3.29/1.88 thf(func_def_86, type, sK8: !>[X0: $tType]:(set @ X0 > X0)).
% 3.29/1.88 thf(func_def_87, type, sK9: rule).
% 3.29/1.88 thf(func_def_88, type, sK10: state > rule).
% 3.29/1.88 thf(func_def_89, type, sK11: state > fset @ state).
% 3.29/1.88 thf(func_def_91, type, ph13: !>[X0: $tType]:(X0)).
% 3.29/1.88 thf(f945,plain,(
% 3.29/1.88 $false),
% 3.29/1.88 inference(subsumption_resolution,[status(thm)],[f944,f932])).
% 3.29/1.88 thf(f932,plain,(
% 3.29/1.88 ((member @ rule @ sK9 @ (sset @ rule @ rules)) = $true)),
% 3.29/1.88 inference(cnf_transformation,[status(thm)],[f899])).
% 3.29/1.88 thf(f899,plain,(
% 3.29/1.88 ((member @ rule @ sK9 @ (sset @ rule @ rules)) = $true) & ((abstra726722745urated @ rule @ state @ eff @ sK9 @ steps) != $true)),
% 3.29/1.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK9])],[f877,f898])).
% 3.29/1.88 thf(f898,plain,(
% 3.29/1.88 ? [X0 : rule] : (((member @ rule @ X0 @ (sset @ rule @ rules)) = $true) & ((abstra726722745urated @ rule @ state @ eff @ X0 @ steps) != $true)) => (((member @ rule @ sK9 @ (sset @ rule @ rules)) = $true) & ((abstra726722745urated @ rule @ state @ eff @ sK9 @ steps) != $true))),
% 3.29/1.88 introduced(definition,[],[choice_axiom])).
% 3.29/1.88 thf(f877,plain,(
% 3.29/1.88 ? [X0 : rule] : (((member @ rule @ X0 @ (sset @ rule @ rules)) = $true) & ((abstra726722745urated @ rule @ state @ eff @ X0 @ steps) != $true))),
% 3.29/1.88 inference(ennf_transformation,[status(thm)],[f753])).
% 3.29/1.88 thf(f753,plain,(
% 3.29/1.88 ~! [X0 : rule] : (((member @ rule @ X0 @ (sset @ rule @ rules)) = $true) => ((abstra726722745urated @ rule @ state @ eff @ X0 @ steps) = $true))),
% 3.29/1.88 inference(fool_elimination,[status(thm)],[f752])).
% 3.29/1.88 thf(f752,plain,(
% 3.29/1.88 ~! [X0 : rule] : ((member @ rule @ X0 @ (sset @ rule @ rules)) => (abstra726722745urated @ rule @ state @ eff @ X0 @ steps))),
% 3.29/1.88 inference(rectify,[status(thm)],[f304])).
% 3.29/1.88 thf(f304,negated_conjecture,(
% 3.29/1.88 ~! [X11 : rule] : ((member @ rule @ X11 @ (sset @ rule @ rules)) => (abstra726722745urated @ rule @ state @ eff @ X11 @ steps))),
% 3.29/1.88 inference(negated_conjecture,[status(cth)],[f303])).
% 3.29/1.88 thf(f303,conjecture,(
% 3.29/1.88 ! [X11 : rule] : ((member @ rule @ X11 @ (sset @ rule @ rules)) => (abstra726722745urated @ rule @ state @ eff @ X11 @ steps))),
% 3.29/1.88 file('/export/starexec/sandbox/tmp/tmp.JVLNTEOK4C/DTF2THF_19524.p',conj_0)).
% 3.29/1.88 thf(f944,plain,(
% 3.29/1.88 ((member @ rule @ sK9 @ (sset @ rule @ rules)) != $true)),
% 3.29/1.88 inference(trivial_inequality_removal,[status(thm)],[f943])).
% 3.29/1.88 thf(f943,plain,(
% 3.29/1.88 ($true != $true) | ((member @ rule @ sK9 @ (sset @ rule @ rules)) != $true)),
% 3.29/1.88 inference(superposition,[status(thm)],[f931,f933])).
% 3.29/1.88 thf(f933,plain,(
% 3.29/1.88 ( ! [X0 : rule] : (((abstra726722745urated @ rule @ state @ eff @ X0 @ steps) = $true) | ((member @ rule @ X0 @ (sset @ rule @ rules)) != $true)) )),
% 3.29/1.88 inference(cnf_transformation,[status(thm)],[f870])).
% 3.29/1.88 thf(f870,plain,(
% 3.29/1.88 ! [X0 : rule] : (((abstra726722745urated @ rule @ state @ eff @ X0 @ steps) = $true) | ((member @ rule @ X0 @ (sset @ rule @ rules)) != $true))),
% 3.29/1.88 inference(ennf_transformation,[status(thm)],[f561])).
% 3.29/1.88 thf(f561,plain,(
% 3.29/1.88 ! [X0 : rule] : (((member @ rule @ X0 @ (sset @ rule @ rules)) = $true) => ((abstra726722745urated @ rule @ state @ eff @ X0 @ steps) = $true))),
% 3.29/1.88 inference(fool_elimination,[status(thm)],[f560])).
% 3.29/1.88 thf(f560,plain,(
% 3.29/1.88 ! [X0 : rule] : ((member @ rule @ X0 @ (sset @ rule @ rules)) => (abstra726722745urated @ rule @ state @ eff @ X0 @ steps))),
% 3.29/1.88 inference(rectify,[status(thm)],[f1])).
% 3.29/1.88 thf(f1,axiom,(
% 3.29/1.88 ! [X4 : rule] : ((member @ rule @ X4 @ (sset @ rule @ rules)) => (abstra726722745urated @ rule @ state @ eff @ X4 @ steps))),
% 3.29/1.88 file('/export/starexec/sandbox/tmp/tmp.JVLNTEOK4C/DTF2THF_19524.p',fact_0__092_060open_062_092_060And_062ra_O_Ara_A_092_060in_062_AR_A_092_060Longrightarrow_062_Asaturated_Ara_Asteps_092_060close_062)).
% 3.29/1.88 thf(f931,plain,(
% 3.29/1.88 ((abstra726722745urated @ rule @ state @ eff @ sK9 @ steps) != $true)),
% 3.29/1.88 inference(cnf_transformation,[status(thm)],[f899])).
% 3.29/1.88 % SZS output end Proof for DTF2THF_19524
% 3.29/1.88 % (19628)------------------------------
% 3.29/1.88 % (19628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.88 % (19628)Termination reason: Refutation
% 3.29/1.88
% 3.29/1.88 % (19628)Memory used [KB]: 6524
% 3.29/1.88 % (19628)Time elapsed: 0.030 s
% 3.29/1.88 % (19628)Instructions burned: 52 (million)
% 3.29/1.88 % (19628)------------------------------
% 3.29/1.88 % (19628)------------------------------
% 3.29/1.88 % (19622)Success in time 0.056 s
%------------------------------------------------------------------------------