TPTP Problem File: ITP183^2.p
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%------------------------------------------------------------------------------
% File : ITP183^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_329__3411594_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Strong_Late_Sim_SC/prob_329__3411594_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 324 ( 136 unt; 50 typ; 0 def)
% Number of atoms : 735 ( 233 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 4007 ( 134 ~; 5 |; 49 &;3430 @)
% ( 0 <=>; 389 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 147 ( 147 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 45 usr; 9 con; 0-5 aty)
% Number of variables : 1242 ( 28 ^;1179 !; 13 ?;1242 :)
% ( 22 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:12.846
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_t_Late__Semantics_Oresidual,type,
late_residual: $tType ).
thf(ty_t_Late__Semantics_Osubject,type,
late_subject: $tType ).
thf(ty_t_Late__Semantics_OfreeRes,type,
late_freeRes: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Nominal_Onoption,type,
noption: $tType > $tType ).
thf(ty_t_Agent_Oname,type,
name: $tType ).
thf(ty_t_Agent_Opi,type,
pi: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Agent_Ofs__name,type,
fs_name:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_c_Agent_Opi_OInput,type,
input: name > name > pi > pi ).
thf(sy_c_Agent_Opi_OMismatch,type,
mismatch: name > name > pi > pi ).
thf(sy_c_Agent_Opi_OOutput,type,
output: name > name > pi > pi ).
thf(sy_c_Agent_Opi_OPar,type,
par: pi > pi > pi ).
thf(sy_c_Agent_Opi_OPiNil,type,
piNil: pi ).
thf(sy_c_Agent_Opi_ORes,type,
res: name > pi > pi ).
thf(sy_c_Agent_Opi_OSum,type,
sum: pi > pi > pi ).
thf(sy_c_Agent_Opi_OTau,type,
tau: pi > pi ).
thf(sy_c_Agent_Osubs,type,
subs: pi > name > name > pi ).
thf(sy_c_Late__Semantics_OfreeRes_OOutputR,type,
late_OutputR: name > name > late_freeRes ).
thf(sy_c_Late__Semantics_OfreeRes_OTauR,type,
late_TauR: late_freeRes ).
thf(sy_c_Late__Semantics_OfreeRes_OfreeRes__rec,type,
late_freeRes_rec:
!>[T: $tType] : ( ( name > name > T ) > T > late_freeRes > T ) ).
thf(sy_c_Late__Semantics_Oresidual_OBoundR,type,
late_BoundR: late_subject > name > pi > late_residual ).
thf(sy_c_Late__Semantics_Oresidual_OFreeR,type,
late_FreeR: late_freeRes > pi > late_residual ).
thf(sy_c_Late__Semantics_Osubject_OBoundOutputS,type,
late_BoundOutputS: name > late_subject ).
thf(sy_c_Late__Semantics_Osubject_OInputS,type,
late_InputS: name > late_subject ).
thf(sy_c_Late__Semantics_Osubject_Osubject__rec,type,
late_subject_rec:
!>[T: $tType] : ( ( name > T ) > ( name > T ) > late_subject > T ) ).
thf(sy_c_Late__Semantics_Otransitions,type,
late_transitions: pi > late_residual > $o ).
thf(sy_c_Nominal_Oabs__fun,type,
abs_fun:
!>[X: $tType,A: $tType] : ( X > A > X > ( noption @ A ) ) ).
thf(sy_c_Nominal_Ofresh,type,
fresh:
!>[X: $tType,A: $tType] : ( X > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Rel_Oeqvt,type,
eqvt:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_OId,type,
id:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Strong__Late__Sim_Oderivative,type,
strong2129052853vative: pi > pi > late_subject > name > ( set @ ( product_prod @ pi @ pi ) ) > $o ).
thf(sy_c_Strong__Late__Sim_Osimulation,type,
strong743114133lation: pi > ( set @ ( product_prod @ pi @ pi ) ) > pi > $o ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: pi ).
thf(sy_v_PQ____,type,
pq: pi ).
thf(sy_v_Q,type,
q: pi ).
thf(sy_v_Q_H____,type,
q2: pi ).
thf(sy_v_Rel,type,
rel: set @ ( product_prod @ pi @ pi ) ).
thf(sy_v__092_060alpha_062____,type,
alpha: late_freeRes ).
thf(sy_v_x,type,
x: name ).
% Relevant facts (256)
thf(fact_0_cRes_Ohyps_I2_J,axiom,
fresh @ name @ late_freeRes @ x @ alpha ).
% cRes.hyps(2)
thf(fact_1_Eqvt,axiom,
eqvt @ pi @ rel ).
% Eqvt
thf(fact_2__092_060open_062P_A_092_060oplus_062_AQ_A_092_060longmapsto_062_A_092_060alpha_062_A_092_060prec_062_AQ_H_092_060close_062,axiom,
late_transitions @ ( sum @ p @ q ) @ ( late_FreeR @ alpha @ q2 ) ).
% \<open>P \<oplus> Q \<longmapsto> \<alpha> \<prec> Q'\<close>
thf(fact_3_cRes_Ohyps_I1_J,axiom,
late_transitions @ q @ ( late_FreeR @ alpha @ q2 ) ).
% cRes.hyps(1)
thf(fact_4_cSum2_Ohyps,axiom,
late_transitions @ ( res @ x @ q ) @ ( late_FreeR @ alpha @ pq ) ).
% cSum2.hyps
thf(fact_5_Free_Ohyps,axiom,
late_transitions @ ( sum @ ( res @ x @ p ) @ ( res @ x @ q ) ) @ ( late_FreeR @ alpha @ pq ) ).
% Free.hyps
thf(fact_6_pi_Odistinct_I81_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(81)
thf(fact_7_sumCases,axiom,
! [P: pi,Q: pi,Rs: late_residual] :
( ( late_transitions @ ( sum @ P @ Q ) @ Rs )
=> ( ~ ( late_transitions @ P @ Rs )
=> ( late_transitions @ Q @ Rs ) ) ) ).
% sumCases
thf(fact_8_sumCases_H,axiom,
! [P: pi,Q: pi,Rs: late_residual] :
( ( late_transitions @ ( sum @ P @ Q ) @ Rs )
=> ( ~ ( late_transitions @ P @ Rs )
=> ( late_transitions @ Q @ Rs ) ) ) ).
% sumCases'
thf(fact_9_Sum1,axiom,
! [P: pi,Rs: late_residual,Q: pi] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum1
thf(fact_10_Sum2,axiom,
! [Q: pi,Rs: late_residual,P: pi] :
( ( late_transitions @ Q @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum2
thf(fact_11_resCasesF,axiom,
! [X2: name,P: pi,Alpha: late_freeRes,XP: pi,F: pi > $o] :
( ( late_transitions @ ( res @ X2 @ P ) @ ( late_FreeR @ Alpha @ XP ) )
=> ( ! [P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh @ name @ late_freeRes @ X2 @ Alpha )
=> ( F @ ( res @ X2 @ P2 ) ) ) )
=> ( F @ XP ) ) ) ).
% resCasesF
thf(fact_12_resCasesF_H,axiom,
! [X2: name,P: pi,Alpha: late_freeRes,P3: pi] :
( ( late_transitions @ ( res @ X2 @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P2: pi,Y: name] :
( ( ( res @ X2 @ P )
= ( res @ Y @ P4 ) )
=> ( ( ( late_FreeR @ Alpha @ P3 )
= ( late_FreeR @ Alpha2 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P2 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y @ Alpha2 ) ) ) ) ) ).
% resCasesF'
thf(fact_13_ResF,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh @ name @ late_freeRes @ Y2 @ Alpha )
=> ( late_transitions @ ( res @ Y2 @ P ) @ ( late_FreeR @ Alpha @ ( res @ Y2 @ P3 ) ) ) ) ) ).
% ResF
thf(fact_14_pi_Oinject_I6_J,axiom,
! [X22: pi,X1: pi,Y22: pi,Y1: pi] :
( ( ( sum @ X22 @ X1 )
= ( sum @ Y22 @ Y1 ) )
= ( ( X22 = Y22 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(6)
thf(fact_15_residual_Oinject_I2_J,axiom,
! [X22: late_freeRes,X1: pi,Y22: late_freeRes,Y1: pi] :
( ( ( late_FreeR @ X22 @ X1 )
= ( late_FreeR @ Y22 @ Y1 ) )
= ( ( X22 = Y22 )
& ( X1 = Y1 ) ) ) ).
% residual.inject(2)
thf(fact_16_resInputFreeTrans,axiom,
! [X2: name,A2: name,Y2: name,P: pi,Alpha: late_freeRes,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( input @ A2 @ Y2 @ P ) ) @ ( late_FreeR @ Alpha @ P3 ) ) ).
% resInputFreeTrans
thf(fact_17_resZeroTrans,axiom,
! [X2: name,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ piNil ) @ Rs ) ).
% resZeroTrans
thf(fact_18_pi_Odistinct_I5_J,axiom,
! [Name1: name,Name2: name,Pi: pi] :
( piNil
!= ( input @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(5)
thf(fact_19_name__exists__fresh,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [X2: A] :
~ ! [C: name] :
~ ( fresh @ name @ A @ C @ X2 ) ) ).
% name_exists_fresh
thf(fact_20_pi_Odistinct_I57_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( input @ Name12 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(57)
thf(fact_21_zeroTrans,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% zeroTrans
thf(fact_22_nilCases_H,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% nilCases'
thf(fact_23_pi_Odistinct_I53_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name12 @ Name22 @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(53)
thf(fact_24_pi_Odistinct_I15_J,axiom,
! [Name: name,Pi: pi] :
( piNil
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(15)
thf(fact_25_pi_Odistinct_I11_J,axiom,
! [Pi12: pi,Pi22: pi] :
( piNil
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(11)
thf(fact_26_resTrans_I2_J,axiom,
! [X2: name,Y2: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ ( input @ X2 @ Y2 @ P ) ) @ Rs ) ).
% resTrans(2)
thf(fact_27_inputFreeTrans,axiom,
! [A2: name,X2: name,P: pi,Alpha: late_freeRes,P3: pi] :
~ ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_FreeR @ Alpha @ P3 ) ) ).
% inputFreeTrans
thf(fact_28_Id,axiom,
ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ rel ).
% Id
thf(fact_29_outputFreshTrans,axiom,
! [A2: name,B2: name,P: pi,Alpha: late_freeRes,P3: pi] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ~ ( ( fresh @ name @ late_freeRes @ A2 @ Alpha )
| ( fresh @ name @ late_freeRes @ B2 @ Alpha ) ) ) ).
% outputFreshTrans
thf(fact_30_freshFreeDerivative_I1_J,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh @ name @ pi @ Y2 @ P )
=> ( fresh @ name @ late_freeRes @ Y2 @ Alpha ) ) ) ).
% freshFreeDerivative(1)
thf(fact_31_nilSim_I2_J,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),A2: name,X2: name,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( input @ A2 @ X2 @ P ) ) ).
% nilSim(2)
thf(fact_32_resTrans_I1_J,axiom,
! [X2: name,B2: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ ( output @ X2 @ B2 @ P ) ) @ Rs ) ).
% resTrans(1)
thf(fact_33_freeRes_Ofresh_I2_J,axiom,
! [A2: name] : ( fresh @ name @ late_freeRes @ A2 @ late_TauR ) ).
% freeRes.fresh(2)
thf(fact_34_freshFreeDerivative_I2_J,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh @ name @ pi @ Y2 @ P )
=> ( fresh @ name @ pi @ Y2 @ P3 ) ) ) ).
% freshFreeDerivative(2)
thf(fact_35_residual_Ostrong__induct,axiom,
! [N: $tType] :
( ( fs_name @ N )
=> ! [P: N > late_residual > $o,Z: N,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: N] :
( ( fresh @ name @ N @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: N] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_induct
thf(fact_36_residual_Ostrong__inducts,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [P: A > late_residual > $o,Z: A,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: A] :
( ( fresh @ name @ A @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: A] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_inducts
thf(fact_37_Par2F,axiom,
! [Q: pi,Alpha: late_freeRes,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha @ ( par @ P @ Q2 ) ) ) ) ).
% Par2F
thf(fact_38_pi_Ofresh_I8_J,axiom,
! [A2: name,X22: pi,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( par @ X22 @ X1 ) )
= ( ( fresh @ name @ pi @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(8)
thf(fact_39_pi_Ofresh_I7_J,axiom,
! [A2: name,X22: pi,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( sum @ X22 @ X1 ) )
= ( ( fresh @ name @ pi @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(7)
thf(fact_40_pi_Ofresh_I2_J,axiom,
! [A2: name,X3: name,X22: name,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( output @ X3 @ X22 @ X1 ) )
= ( ( fresh @ name @ name @ A2 @ X3 )
& ( fresh @ name @ name @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(2)
thf(fact_41_pi_Ofresh_I1_J,axiom,
! [A2: name] : ( fresh @ name @ pi @ A2 @ piNil ) ).
% pi.fresh(1)
thf(fact_42_residual_Ofresh_I2_J,axiom,
! [A2: name,X22: late_freeRes,X1: pi] :
( ( fresh @ name @ late_residual @ A2 @ ( late_FreeR @ X22 @ X1 ) )
= ( ( fresh @ name @ late_freeRes @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% residual.fresh(2)
thf(fact_43_pi_Oinject_I7_J,axiom,
! [X22: pi,X1: pi,Y22: pi,Y1: pi] :
( ( ( par @ X22 @ X1 )
= ( par @ Y22 @ Y1 ) )
= ( ( X22 = Y22 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(7)
thf(fact_44_pi_Oinject_I1_J,axiom,
! [X3: name,X22: name,X1: pi,Y3: name,Y22: name,Y1: pi] :
( ( ( output @ X3 @ X22 @ X1 )
= ( output @ Y3 @ Y22 @ Y1 ) )
= ( ( X3 = Y3 )
& ( X22 = Y22 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(1)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_pi_Odistinct_I29_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(29)
thf(fact_49_freshBoundDerivative_I2_J,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ Y2 @ P )
=> ( ( Y2 != X2 )
=> ( fresh @ name @ pi @ Y2 @ P3 ) ) ) ) ).
% freshBoundDerivative(2)
thf(fact_50_freshBoundDerivative_I1_J,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ Y2 @ P )
=> ( fresh @ name @ late_subject @ Y2 @ A2 ) ) ) ).
% freshBoundDerivative(1)
thf(fact_51_Late__Semantics_OPar1B,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P3 @ Q ) ) ) ) ) ).
% Late_Semantics.Par1B
thf(fact_52_Late__Semantics_OPar2B,axiom,
! [Q: pi,A2: late_subject,X2: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P @ Q2 ) ) ) ) ) ).
% Late_Semantics.Par2B
thf(fact_53_parCasesB,axiom,
! [P: pi,Q: pi,A2: late_subject,X2: name,PQ: pi,Prop: pi > $o] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ PQ ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ! [P2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
=> ( Prop @ ( par @ P2 @ Q ) ) )
=> ( ! [Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q3 ) )
=> ( Prop @ ( par @ P @ Q3 ) ) )
=> ( Prop @ PQ ) ) ) ) ) ) ).
% parCasesB
thf(fact_54_parCasesB_H,axiom,
! [P: pi,Q: pi,B2: late_subject,Y2: name,P3: pi] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ B2 @ Y2 @ P3 ) )
=> ( ! [P4: pi,A4: late_subject,X5: name,P2: pi,Q4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B2 @ Y2 @ P3 )
= ( late_BoundR @ A4 @ X5 @ ( par @ P2 @ Q4 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X5 @ P2 ) )
=> ( ( fresh @ name @ pi @ X5 @ P4 )
=> ( ( fresh @ name @ pi @ X5 @ Q4 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) )
=> ~ ! [Q4: pi,A4: late_subject,X5: name,Q3: pi,P4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B2 @ Y2 @ P3 )
= ( late_BoundR @ A4 @ X5 @ ( par @ P4 @ Q3 ) ) )
=> ( ( late_transitions @ Q4 @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh @ name @ pi @ X5 @ P4 )
=> ( ( fresh @ name @ pi @ X5 @ Q4 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) ) ) ).
% parCasesB'
thf(fact_55_freshResidual,axiom,
! [P: pi,Rs: late_residual,X2: name] :
( ( late_transitions @ P @ Rs )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( fresh @ name @ late_residual @ X2 @ Rs ) ) ) ).
% freshResidual
thf(fact_56_transitions_OPar1B,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ late_subject @ X2 @ A2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P3 @ Q ) ) ) ) ) ) ) ).
% transitions.Par1B
thf(fact_57_transitions_OPar2B,axiom,
! [Q: pi,A2: late_subject,X2: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ late_subject @ X2 @ A2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P @ Q2 ) ) ) ) ) ) ) ).
% transitions.Par2B
thf(fact_58_sumIdempRight,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ P @ P ) @ Rel @ P ) ) ).
% sumIdempRight
thf(fact_59_sumAssocRight,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi,Q: pi,R: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ P @ ( sum @ Q @ R ) ) @ Rel @ ( sum @ ( sum @ P @ Q ) @ R ) ) ) ).
% sumAssocRight
thf(fact_60_sumIdempLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ P @ Rel @ ( sum @ P @ P ) ) ) ).
% sumIdempLeft
thf(fact_61_sumAssocLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi,Q: pi,R: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ ( sum @ P @ Q ) @ R ) @ Rel @ ( sum @ P @ ( sum @ Q @ R ) ) ) ) ).
% sumAssocLeft
thf(fact_62_sumSym,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi,Q: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ P @ Q ) @ Rel @ ( sum @ Q @ P ) ) ) ).
% sumSym
thf(fact_63_outputBoundTrans,axiom,
! [A2: name,B2: name,P: pi,C2: late_subject,X2: name,P3: pi] :
~ ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_BoundR @ C2 @ X2 @ P3 ) ) ).
% outputBoundTrans
thf(fact_64_transitions_OResB,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ late_subject @ Y2 @ A2 )
=> ( ( Y2 != X2 )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ late_subject @ X2 @ A2 )
=> ( late_transitions @ ( res @ Y2 @ P ) @ ( late_BoundR @ A2 @ X2 @ ( res @ Y2 @ P3 ) ) ) ) ) ) ) ) ).
% transitions.ResB
thf(fact_65_nilSim_I3_J,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),A2: name,B2: name,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( output @ A2 @ B2 @ P ) ) ).
% nilSim(3)
thf(fact_66_sumZeroRight,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ P @ Rel @ ( sum @ P @ piNil ) ) ) ).
% sumZeroRight
thf(fact_67_sumZeroLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ P @ piNil ) @ Rel @ P ) ) ).
% sumZeroLeft
thf(fact_68_Late__Semantics_OResB,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ late_subject @ Y2 @ A2 )
=> ( ( Y2 != X2 )
=> ( late_transitions @ ( res @ Y2 @ P ) @ ( late_BoundR @ A2 @ X2 @ ( res @ Y2 @ P3 ) ) ) ) ) ) ).
% Late_Semantics.ResB
thf(fact_69_inputIneqTrans,axiom,
! [A2: name,X2: name,P: pi,B2: late_subject,Y2: name,P3: pi] :
( ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ B2 @ Y2 @ P3 ) )
=> ~ ( fresh @ name @ late_subject @ A2 @ B2 ) ) ).
% inputIneqTrans
thf(fact_70_pi_Odistinct_I85_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( par @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(85)
thf(fact_71_freshRes,axiom,
! [A2: name,P: pi] : ( fresh @ name @ pi @ A2 @ ( res @ A2 @ P ) ) ).
% freshRes
thf(fact_72_residual_Odistinct_I1_J,axiom,
! [Subject2: late_subject,Name4: name,Pi3: pi,FreeRes2: late_freeRes,Pi: pi] :
( ( late_BoundR @ Subject2 @ Name4 @ Pi3 )
!= ( late_FreeR @ FreeRes2 @ Pi ) ) ).
% residual.distinct(1)
thf(fact_73_residual_Oinducts,axiom,
! [P: late_residual > $o,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi] : ( P @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi] : ( P @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Residual ) ) ) ).
% residual.inducts
thf(fact_74_pi_Odistinct_I79_J,axiom,
! [Pi1: pi,Pi2: pi,Pi12: pi,Pi22: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(79)
thf(fact_75_pi_Odistinct_I55_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name12 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(55)
thf(fact_76_sumResLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),X2: name,P: pi,Q: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( ( eqvt @ pi @ Rel )
=> ( strong743114133lation @ ( sum @ ( res @ X2 @ P ) @ ( res @ X2 @ Q ) ) @ Rel @ ( res @ X2 @ ( sum @ P @ Q ) ) ) ) ) ).
% sumResLeft
thf(fact_77_pi_Odistinct_I13_J,axiom,
! [Pi12: pi,Pi22: pi] :
( piNil
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(13)
thf(fact_78_outputTauTrans,axiom,
! [A2: name,B2: name,P: pi,P3: pi] :
~ ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ late_TauR @ P3 ) ) ).
% outputTauTrans
thf(fact_79_pi_Odistinct_I31_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(31)
thf(fact_80_pi_Odistinct_I27_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(27)
thf(fact_81_pi_Odistinct_I21_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name1: name,Name2: name,Pi: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( input @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(21)
thf(fact_82_pi_Odistinct_I1_J,axiom,
! [Name1: name,Name2: name,Pi: pi] :
( piNil
!= ( output @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(1)
thf(fact_83_nilSimRight,axiom,
! [P: pi,Rel: set @ ( product_prod @ pi @ pi )] : ( strong743114133lation @ P @ Rel @ piNil ) ).
% nilSimRight
thf(fact_84_resOutputTauTrans,axiom,
! [X2: name,A2: name,B2: name,P: pi,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ B2 @ P ) ) @ ( late_FreeR @ late_TauR @ P3 ) ) ).
% resOutputTauTrans
thf(fact_85_Par1F,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha @ ( par @ P3 @ Q ) ) ) ) ).
% Par1F
thf(fact_86_Strong__Late__Sim_Oreflexive,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ P @ Rel @ P ) ) ).
% Strong_Late_Sim.reflexive
thf(fact_87_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_88_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ A @ X5 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_89_monotonic,axiom,
! [P: pi,A3: set @ ( product_prod @ pi @ pi ),P3: pi,B3: set @ ( product_prod @ pi @ pi )] :
( ( strong743114133lation @ P @ A3 @ P3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ A3 @ B3 )
=> ( strong743114133lation @ P @ B3 @ P3 ) ) ) ).
% monotonic
thf(fact_90_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_91_resOutputInputTrans,axiom,
! [X2: name,A2: name,B2: name,P: pi,C2: name,Y2: name,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ B2 @ P ) ) @ ( late_BoundR @ ( late_InputS @ C2 ) @ Y2 @ P3 ) ) ).
% resOutputInputTrans
thf(fact_92_resInputBoundOutputTrans,axiom,
! [X2: name,A2: name,Y2: name,P: pi,B2: name,Z: name,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( input @ A2 @ Y2 @ P ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Z @ P3 ) ) ).
% resInputBoundOutputTrans
thf(fact_93_freeRes_Orecs_I2_J,axiom,
! [T: $tType,P: T > $o,F1: name > name > T,F2: T] :
( ! [X12: name,X23: name] : ( P @ ( F1 @ X12 @ X23 ) )
=> ( ( P @ F2 )
=> ( ( late_freeRes_rec @ T @ F1 @ F2 @ late_TauR )
= F2 ) ) ) ).
% freeRes.recs(2)
thf(fact_94_resOutputOutputTrans,axiom,
! [X2: name,A2: name,P: pi,B2: name,Y2: name,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ X2 @ P ) ) @ ( late_FreeR @ ( late_OutputR @ B2 @ Y2 ) @ P3 ) ) ).
% resOutputOutputTrans
thf(fact_95_Late__Semantics_OfreeRes_Oinject,axiom,
! [X22: name,X1: name,Y22: name,Y1: name] :
( ( ( late_OutputR @ X22 @ X1 )
= ( late_OutputR @ Y22 @ Y1 ) )
= ( ( X22 = Y22 )
& ( X1 = Y1 ) ) ) ).
% Late_Semantics.freeRes.inject
thf(fact_96_Late__Semantics_Osubject_Oinject_I2_J,axiom,
! [X1: name,Y1: name] :
( ( ( late_BoundOutputS @ X1 )
= ( late_BoundOutputS @ Y1 ) )
= ( X1 = Y1 ) ) ).
% Late_Semantics.subject.inject(2)
thf(fact_97_name__fresh,axiom,
( ( fresh @ name @ name )
= ( ^ [A5: name,B4: name] : A5 != B4 ) ) ).
% name_fresh
thf(fact_98_subject_Ofresh_I1_J,axiom,
! [A2: name,X1: name] :
( ( fresh @ name @ late_subject @ A2 @ ( late_InputS @ X1 ) )
= ( fresh @ name @ name @ A2 @ X1 ) ) ).
% subject.fresh(1)
thf(fact_99_subject_Ofresh_I2_J,axiom,
! [A2: name,X1: name] :
( ( fresh @ name @ late_subject @ A2 @ ( late_BoundOutputS @ X1 ) )
= ( fresh @ name @ name @ A2 @ X1 ) ) ).
% subject.fresh(2)
thf(fact_100_freeRes_Ofresh_I1_J,axiom,
! [A2: name,X22: name,X1: name] :
( ( fresh @ name @ late_freeRes @ A2 @ ( late_OutputR @ X22 @ X1 ) )
= ( ( fresh @ name @ name @ A2 @ X22 )
& ( fresh @ name @ name @ A2 @ X1 ) ) ) ).
% freeRes.fresh(1)
thf(fact_101_subject_Ostrong__inducts,axiom,
! [A: $tType,P: A > late_subject > $o,Z: A,Subject2: late_subject] :
( ! [Name3: name,Z2: A] : ( P @ Z2 @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name,Z2: A] : ( P @ Z2 @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Z @ Subject2 ) ) ) ).
% subject.strong_inducts
thf(fact_102_subject_Ostrong__induct_H,axiom,
! [N: $tType,P: N > late_subject > $o,Z: N,Subject2: late_subject] :
( ! [Name3: name,Z2: N] : ( P @ Z2 @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name,Z2: N] : ( P @ Z2 @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Z @ Subject2 ) ) ) ).
% subject.strong_induct'
thf(fact_103_subject_Oinducts,axiom,
! [P: late_subject > $o,Subject2: late_subject] :
( ! [Name3: name] : ( P @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name] : ( P @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Subject2 ) ) ) ).
% subject.inducts
thf(fact_104_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name4: name,Name: name] :
( ( late_InputS @ Name4 )
!= ( late_BoundOutputS @ Name ) ) ).
% Late_Semantics.subject.distinct(1)
thf(fact_105_freeRes_Orecs_I1_J,axiom,
! [T: $tType,P: T > $o,F1: name > name > T,F2: T,Name12: name,Name22: name] :
( ! [X12: name,X23: name] : ( P @ ( F1 @ X12 @ X23 ) )
=> ( ( P @ F2 )
=> ( ( late_freeRes_rec @ T @ F1 @ F2 @ ( late_OutputR @ Name12 @ Name22 ) )
= ( F1 @ Name12 @ Name22 ) ) ) ) ).
% freeRes.recs(1)
thf(fact_106_freeRes_Ostrong__inducts,axiom,
! [A: $tType,P: A > late_freeRes > $o,Z: A,FreeRes3: late_freeRes] :
( ! [Name13: name,Name23: name,Z2: A] : ( P @ Z2 @ ( late_OutputR @ Name13 @ Name23 ) )
=> ( ! [Z2: A] : ( P @ Z2 @ late_TauR )
=> ( P @ Z @ FreeRes3 ) ) ) ).
% freeRes.strong_inducts
thf(fact_107_freeRes_Ostrong__induct_H,axiom,
! [N: $tType,P: N > late_freeRes > $o,Z: N,FreeRes3: late_freeRes] :
( ! [Name13: name,Name23: name,Z2: N] : ( P @ Z2 @ ( late_OutputR @ Name13 @ Name23 ) )
=> ( ! [Z2: N] : ( P @ Z2 @ late_TauR )
=> ( P @ Z @ FreeRes3 ) ) ) ).
% freeRes.strong_induct'
thf(fact_108_freeRes_Oinducts,axiom,
! [P: late_freeRes > $o,FreeRes3: late_freeRes] :
( ! [Name13: name,Name23: name] : ( P @ ( late_OutputR @ Name13 @ Name23 ) )
=> ( ( P @ late_TauR )
=> ( P @ FreeRes3 ) ) ) ).
% freeRes.inducts
thf(fact_109_Late__Semantics_OfreeRes_Odistinct_I1_J,axiom,
! [Name12: name,Name22: name] :
( ( late_OutputR @ Name12 @ Name22 )
!= late_TauR ) ).
% Late_Semantics.freeRes.distinct(1)
thf(fact_110_Open,axiom,
! [P: pi,A2: name,B2: name,P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) )
=> ( ( A2 != B2 )
=> ( late_transitions @ ( res @ B2 @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ B2 @ P3 ) ) ) ) ).
% Open
thf(fact_111_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_112_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_113_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% le_funI
thf(fact_114_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_115_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y: B] :
( ( ord_less_eq @ B @ X5 @ Y )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_116_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C3,C2: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C2 )
=> ( ! [X5: A,Y: A] :
( ( ord_less_eq @ A @ X5 @ Y )
=> ( ord_less_eq @ C3 @ ( F3 @ X5 ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ C3 @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_117_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y: B] :
( ( ord_less_eq @ B @ X5 @ Y )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C2 )
=> ( ! [X5: A,Y: A] :
( ( ord_less_eq @ A @ X5 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_119_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_120_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_121_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_122_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_123_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_124_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_125_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_126_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_127_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_128_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_129_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_130_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_131_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_132_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_133_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B5: A] :
( ( ord_less_eq @ A @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: A,B5: A] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_134_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_135_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_136_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_137_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_138_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_139_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_140_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_141_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_142_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_143_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A6 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_144_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_145_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_146_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% subset_trans
thf(fact_147_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z3: set @ A] : Y4 = Z3 )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_148_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_149_resCasesB_H,axiom,
! [X6: name,P: pi,A2: late_subject,Y6: name,P3: pi] :
( ( late_transitions @ ( res @ X6 @ P ) @ ( late_BoundR @ A2 @ Y6 @ P3 ) )
=> ( ! [P4: pi,A4: name,B5: name] :
( ( ( res @ X6 @ P )
= ( res @ B5 @ P4 ) )
=> ! [P2: pi] :
( ( ( late_BoundR @ A2 @ Y6 @ P3 )
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B5 @ P2 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A4 @ B5 ) @ P2 ) )
=> ( A4 = B5 ) ) ) )
=> ~ ! [P4: pi,A4: late_subject,X5: name,P2: pi,Y: name] :
( ( ( res @ X6 @ P )
= ( res @ Y @ P4 ) )
=> ( ( ( late_BoundR @ A2 @ Y6 @ P3 )
= ( late_BoundR @ A4 @ X5 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X5 @ P2 ) )
=> ( ( fresh @ name @ late_subject @ Y @ A4 )
=> ( ( Y != X5 )
=> ( ( fresh @ name @ pi @ X5 @ P4 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_150_Output,axiom,
! [A2: name,B2: name,P: pi] : ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P ) ) ).
% Output
thf(fact_151_outputIneqTrans,axiom,
! [A2: name,B2: name,P: pi,C2: name,D: name,P3: pi] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ ( late_OutputR @ C2 @ D ) @ P3 ) )
=> ~ ( ( A2 != C2 )
| ( B2 != D ) ) ) ).
% outputIneqTrans
thf(fact_152_outputCases_H,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ Rs )
=> ~ ! [A4: name,B5: name,P4: pi] :
( ( ( A2 = A4 )
& ( B2 = B5 )
& ( P = P4 ) )
=> ( Rs
!= ( late_FreeR @ ( late_OutputR @ A4 @ B5 ) @ P4 ) ) ) ) ).
% outputCases'
thf(fact_153_outputCases,axiom,
! [A2: name,B2: name,P: pi,Alpha: late_freeRes,P3: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( ( Alpha
= ( late_OutputR @ A2 @ B2 ) )
=> ( ( P = P3 )
=> ( Prop @ ( late_OutputR @ A2 @ B2 ) @ P ) ) )
=> ( Prop @ Alpha @ P3 ) ) ) ).
% outputCases
thf(fact_154_Late__Semantics_OInput,axiom,
! [A2: name,X2: name,P: pi] : ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P ) ) ).
% Late_Semantics.Input
thf(fact_155_transitions_OInput,axiom,
! [X2: name,A2: name,P: pi] :
( ( X2 != A2 )
=> ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P ) ) ) ).
% transitions.Input
thf(fact_156_inputBoundOutputTrans,axiom,
! [A2: name,X2: name,P: pi,B2: name,Y2: name,P3: pi] :
~ ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Y2 @ P3 ) ) ).
% inputBoundOutputTrans
thf(fact_157_resCases_H,axiom,
! [X2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X2 @ P ) @ Rs )
=> ( ! [P4: pi,A4: name,B5: name] :
( ( ( res @ X2 @ P )
= ( res @ B5 @ P4 ) )
=> ! [P2: pi] :
( ( Rs
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B5 @ P2 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A4 @ B5 ) @ P2 ) )
=> ( A4 = B5 ) ) ) )
=> ( ! [P4: pi,A4: late_subject,X5: name,P2: pi,Y: name] :
( ( ( res @ X2 @ P )
= ( res @ Y @ P4 ) )
=> ( ( Rs
= ( late_BoundR @ A4 @ X5 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X5 @ P2 ) )
=> ( ( fresh @ name @ late_subject @ Y @ A4 )
=> ( ( Y != X5 )
=> ( ( fresh @ name @ pi @ X5 @ P4 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P2: pi,Y: name] :
( ( ( res @ X2 @ P )
= ( res @ Y @ P4 ) )
=> ( ( Rs
= ( late_FreeR @ Alpha2 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P2 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y @ Alpha2 ) ) ) ) ) ) ) ).
% resCases'
thf(fact_158_transitions_OClose2,axiom,
! [P: pi,A2: name,Y2: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ pi @ Y2 @ P )
=> ( ( fresh @ name @ pi @ Y2 @ Q )
=> ( ( X2 != A2 )
=> ( ( fresh @ name @ pi @ X2 @ P3 )
=> ( ( Y2 != A2 )
=> ( ( fresh @ name @ pi @ Y2 @ Q2 )
=> ( ( X2 != Y2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ P3 @ ( subs @ Q2 @ X2 @ Y2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close2
thf(fact_159_transitions_OClose1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,Y2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ pi @ Y2 @ P )
=> ( ( fresh @ name @ pi @ Y2 @ Q )
=> ( ( X2 != A2 )
=> ( ( fresh @ name @ pi @ X2 @ Q2 )
=> ( ( Y2 != A2 )
=> ( ( fresh @ name @ pi @ Y2 @ P3 )
=> ( ( X2 != Y2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ ( subs @ P3 @ X2 @ Y2 ) @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close1
thf(fact_160_Late__Semantics_OClose2,axiom,
! [P: pi,A2: name,Y2: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ Y2 @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ P3 @ ( subs @ Q2 @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% Late_Semantics.Close2
thf(fact_161_Late__Semantics_OClose1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,Y2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ Y2 @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ ( subs @ P3 @ X2 @ Y2 ) @ Q2 ) ) ) ) ) ) ) ).
% Late_Semantics.Close1
thf(fact_162_Late__Semantics1_Osubject_Oinject_I2_J,axiom,
! [X22: name,Y22: name] :
( ( ( late_BoundOutputS @ X22 )
= ( late_BoundOutputS @ Y22 ) )
= ( X22 = Y22 ) ) ).
% Late_Semantics1.subject.inject(2)
thf(fact_163_subst__identity,axiom,
! [P: pi,A2: name] :
( ( subs @ P @ A2 @ A2 )
= P ) ).
% subst_identity
thf(fact_164_Late__Semantics1_OfreeRes_Oinject,axiom,
! [X11: name,X122: name,Y11: name,Y12: name] :
( ( ( late_OutputR @ X11 @ X122 )
= ( late_OutputR @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 ) ) ) ).
% Late_Semantics1.freeRes.inject
thf(fact_165_Late__Semantics1_Osubject_Oinject_I1_J,axiom,
! [X1: name,Y1: name] :
( ( ( late_InputS @ X1 )
= ( late_InputS @ Y1 ) )
= ( X1 = Y1 ) ) ).
% Late_Semantics1.subject.inject(1)
thf(fact_166_simps_I9_J,axiom,
! [X2: name,C2: name,D: name,P: pi] :
( ( X2 != C2 )
=> ( ( X2 != D )
=> ( ( subs @ ( res @ X2 @ P ) @ C2 @ D )
= ( res @ X2 @ ( subs @ P @ C2 @ D ) ) ) ) ) ).
% simps(9)
thf(fact_167_simps_I8_J,axiom,
! [P: pi,Q: pi,C2: name,D: name] :
( ( subs @ ( par @ P @ Q ) @ C2 @ D )
= ( par @ ( subs @ P @ C2 @ D ) @ ( subs @ Q @ C2 @ D ) ) ) ).
% simps(8)
thf(fact_168_substTrans,axiom,
! [B2: name,P: pi,A2: name,C2: name] :
( ( fresh @ name @ pi @ B2 @ P )
=> ( ( subs @ ( subs @ P @ A2 @ B2 ) @ B2 @ C2 )
= ( subs @ P @ A2 @ C2 ) ) ) ).
% substTrans
thf(fact_169_simps_I7_J,axiom,
! [P: pi,Q: pi,C2: name,D: name] :
( ( subs @ ( sum @ P @ Q ) @ C2 @ D )
= ( sum @ ( subs @ P @ C2 @ D ) @ ( subs @ Q @ C2 @ D ) ) ) ).
% simps(7)
thf(fact_170_simps_I1_J,axiom,
! [C2: name,D: name] :
( ( subs @ piNil @ C2 @ D )
= piNil ) ).
% simps(1)
thf(fact_171_forget,axiom,
! [A2: name,P: pi,B2: name] :
( ( fresh @ name @ pi @ A2 @ P )
=> ( ( subs @ P @ A2 @ B2 )
= P ) ) ).
% forget
thf(fact_172_fresh__fact1,axiom,
! [A2: name,P: pi,C2: name,B2: name] :
( ( fresh @ name @ pi @ A2 @ P )
=> ( ( A2 != C2 )
=> ( fresh @ name @ pi @ A2 @ ( subs @ P @ B2 @ C2 ) ) ) ) ).
% fresh_fact1
thf(fact_173_fresh__fact2,axiom,
! [A2: name,B2: name,P: pi] :
( ( A2 != B2 )
=> ( fresh @ name @ pi @ A2 @ ( subs @ P @ A2 @ B2 ) ) ) ).
% fresh_fact2
thf(fact_174_substRes3,axiom,
! [B2: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ B2 @ P )
=> ( ( subs @ ( res @ A2 @ P ) @ A2 @ B2 )
= ( res @ B2 @ ( subs @ P @ A2 @ B2 ) ) ) ) ).
% substRes3
thf(fact_175_substRes2,axiom,
! [B2: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ B2 @ P )
=> ( ( res @ A2 @ P )
= ( res @ B2 @ ( subs @ P @ A2 @ B2 ) ) ) ) ).
% substRes2
thf(fact_176_freeRes_Oexhaust,axiom,
! [Y2: late_freeRes] :
( ! [X112: name,X123: name] :
( Y2
!= ( late_OutputR @ X112 @ X123 ) )
=> ( Y2 = late_TauR ) ) ).
% freeRes.exhaust
thf(fact_177_Late__Semantics1_OfreeRes_Odistinct_I1_J,axiom,
! [X11: name,X122: name] :
( ( late_OutputR @ X11 @ X122 )
!= late_TauR ) ).
% Late_Semantics1.freeRes.distinct(1)
thf(fact_178_subject_Oexhaust,axiom,
! [Y2: late_subject] :
( ! [X12: name] :
( Y2
!= ( late_InputS @ X12 ) )
=> ~ ! [X23: name] :
( Y2
!= ( late_BoundOutputS @ X23 ) ) ) ).
% subject.exhaust
thf(fact_179_Late__Semantics1_Osubject_Odistinct_I1_J,axiom,
! [X1: name,X22: name] :
( ( late_InputS @ X1 )
!= ( late_BoundOutputS @ X22 ) ) ).
% Late_Semantics1.subject.distinct(1)
thf(fact_180_Late__Semantics_OComm1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,B2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P3 @ X2 @ B2 ) @ Q2 ) ) ) ) ) ).
% Late_Semantics.Comm1
thf(fact_181_Late__Semantics_OComm2,axiom,
! [P: pi,A2: name,B2: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P3 @ ( subs @ Q2 @ X2 @ B2 ) ) ) ) ) ) ).
% Late_Semantics.Comm2
thf(fact_182_transitions_OComm1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,B2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( X2 != A2 )
=> ( ( X2 != B2 )
=> ( ( fresh @ name @ pi @ X2 @ Q2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P3 @ X2 @ B2 ) @ Q2 ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm1
thf(fact_183_transitions_OComm2,axiom,
! [P: pi,A2: name,B2: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( X2 != A2 )
=> ( ( X2 != B2 )
=> ( ( fresh @ name @ pi @ X2 @ P3 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P3 @ ( subs @ Q2 @ X2 @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm2
thf(fact_184_resSimCases,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,X2: name,P: pi,C4: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [Q3: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X2 ) @ Q3 ) )
=> ( ( A4 != X2 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X2 @ P5 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P5 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A4: late_subject,Y: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y @ Q3 ) )
=> ( ( fresh @ name @ late_subject @ X2 @ A4 )
=> ( ( X2 != Y )
=> ( ( fresh @ name @ A @ Y @ C4 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y @ P5 ) )
& ( strong2129052853vative @ P5 @ ( res @ X2 @ Q3 ) @ A4 @ Y @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh @ name @ late_freeRes @ X2 @ Alpha2 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P5 @ ( res @ X2 @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X2 @ Q ) ) ) ) ) ) ) ).
% resSimCases
thf(fact_185_simE_I1_J,axiom,
! [P: pi,Rel: set @ ( product_prod @ pi @ pi ),Q: pi,A2: late_subject,X2: name,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ? [P2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
& ( strong2129052853vative @ P2 @ Q2 @ A2 @ X2 @ Rel ) ) ) ) ) ).
% simE(1)
thf(fact_186_simCasesCont,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,P: pi,C4: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [A4: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh @ name @ pi @ X5 @ P )
=> ( ( fresh @ name @ pi @ X5 @ Q )
=> ( ( fresh @ name @ late_subject @ X5 @ A4 )
=> ( ( fresh @ name @ A @ X5 @ C4 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P5 ) )
& ( strong2129052853vative @ P5 @ Q3 @ A4 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P5 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ) ).
% simCasesCont
thf(fact_187_derivativeMonotonic,axiom,
! [P: pi,Q: pi,A2: late_subject,X2: name,A3: set @ ( product_prod @ pi @ pi ),B3: set @ ( product_prod @ pi @ pi )] :
( ( strong2129052853vative @ P @ Q @ A2 @ X2 @ A3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ A3 @ B3 )
=> ( strong2129052853vative @ P @ Q @ A2 @ X2 @ B3 ) ) ) ).
% derivativeMonotonic
thf(fact_188_simulation__def,axiom,
( strong743114133lation
= ( ^ [P6: pi,Rel2: set @ ( product_prod @ pi @ pi ),Q5: pi] :
( ! [A5: late_subject,X4: name,Q6: pi] :
( ( ( late_transitions @ Q5 @ ( late_BoundR @ A5 @ X4 @ Q6 ) )
& ( fresh @ name @ pi @ X4 @ P6 ) )
=> ? [P7: pi] :
( ( late_transitions @ P6 @ ( late_BoundR @ A5 @ X4 @ P7 ) )
& ( strong2129052853vative @ P7 @ Q6 @ A5 @ X4 @ Rel2 ) ) )
& ! [Alpha3: late_freeRes,Q6: pi] :
( ( late_transitions @ Q5 @ ( late_FreeR @ Alpha3 @ Q6 ) )
=> ? [P7: pi] :
( ( late_transitions @ P6 @ ( late_FreeR @ Alpha3 @ P7 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P7 @ Q6 ) @ Rel2 ) ) ) ) ) ) ).
% simulation_def
thf(fact_189_simCases,axiom,
! [Q: pi,P: pi,Rel: set @ ( product_prod @ pi @ pi )] :
( ! [A4: late_subject,Y: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y @ Q3 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y @ P5 ) )
& ( strong2129052853vative @ P5 @ Q3 @ A4 @ Y @ Rel ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P5 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ).
% simCases
thf(fact_190_simE_I2_J,axiom,
! [P: pi,Rel: set @ ( product_prod @ pi @ pi ),Q: pi,Alpha: late_freeRes,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q2 ) )
=> ? [P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P2 @ Q2 ) @ Rel ) ) ) ) ).
% simE(2)
thf(fact_191_derivativeReflexive,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi,A2: late_subject,X2: name] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong2129052853vative @ P @ P @ A2 @ X2 @ Rel ) ) ).
% derivativeReflexive
thf(fact_192_pair__in__Id__conv,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
= ( A2 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_193_IdI,axiom,
! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id @ A ) ) ).
% IdI
thf(fact_194_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_195_IdE,axiom,
! [A: $tType,P8: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P8 @ ( id @ A ) )
=> ~ ! [X5: A] :
( P8
!= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_196_fresh__prod,axiom,
! [A: $tType,X: $tType,B: $tType,A2: X,X2: A,Y2: B] :
( ( fresh @ X @ ( product_prod @ A @ B ) @ A2 @ ( product_Pair @ A @ B @ X2 @ Y2 ) )
= ( ( fresh @ X @ A @ A2 @ X2 )
& ( fresh @ X @ B @ A2 @ Y2 ) ) ) ).
% fresh_prod
thf(fact_197_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
= ( ( A2 = A7 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_198_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_199_surj__pair,axiom,
! [A: $tType,B: $tType,P8: product_prod @ A @ B] :
? [X5: A,Y: B] :
( P8
= ( product_Pair @ A @ B @ X5 @ Y ) ) ).
% surj_pair
thf(fact_200_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P8: product_prod @ A @ B] :
( ! [A4: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B5 ) )
=> ( P @ P8 ) ) ).
% prod_cases
thf(fact_201_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ~ ( ( A2 = A7 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_202_prod__cases3,axiom,
! [A: $tType,B: $tType,C3: $tType,Y2: product_prod @ A @ ( product_prod @ B @ C3 )] :
~ ! [A4: A,B5: B,C: C3] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ C3 ) @ A4 @ ( product_Pair @ B @ C3 @ B5 @ C ) ) ) ).
% prod_cases3
thf(fact_203_prod__cases4,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) )] :
~ ! [A4: A,B5: B,C: C3,D3: D2] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ D2 ) @ B5 @ ( product_Pair @ C3 @ D2 @ C @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_204_prod__cases5,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,E: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) )] :
~ ! [A4: A,B5: B,C: C3,D3: D2,E2: E] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) @ B5 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ E ) @ C @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_205_prod__cases6,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,E: $tType,F5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
~ ! [A4: A,B5: B,C: C3,D3: D2,E2: E,F6: F5] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B5 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F6 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_206_prod__cases7,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,E: $tType,F5: $tType,G3: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
~ ! [A4: A,B5: B,C: C3,D3: D2,E2: E,F6: F5,G4: G3] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E2 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_207_prod__induct3,axiom,
! [C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C3 ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C3 )] :
( ! [A4: A,B5: B,C: C3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C3 ) @ A4 @ ( product_Pair @ B @ C3 @ B5 @ C ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_208_prod__induct4,axiom,
! [D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) )] :
( ! [A4: A,B5: B,C: C3,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ D2 ) @ B5 @ ( product_Pair @ C3 @ D2 @ C @ D3 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_209_prod__induct5,axiom,
! [E: $tType,D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) )] :
( ! [A4: A,B5: B,C: C3,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) @ B5 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ E ) @ C @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_210_prod__induct6,axiom,
! [F5: $tType,E: $tType,D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
( ! [A4: A,B5: B,C: C3,D3: D2,E2: E,F6: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B5 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E2 @ F6 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_211_prod__induct7,axiom,
! [G3: $tType,F5: $tType,E: $tType,D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
( ! [A4: A,B5: B,C: C3,D3: D2,E2: E,F6: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E2 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_212_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod @ A @ B] :
~ ! [A4: A,B5: B] :
( Y2
!= ( product_Pair @ A @ B @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_213_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_214_fresh__prodD_I2_J,axiom,
! [B: $tType,A: $tType,C3: $tType,A2: A,X2: B,Y2: C3] :
( ( fresh @ A @ ( product_prod @ B @ C3 ) @ A2 @ ( product_Pair @ B @ C3 @ X2 @ Y2 ) )
=> ( fresh @ A @ C3 @ A2 @ Y2 ) ) ).
% fresh_prodD(2)
thf(fact_215_fresh__prodD_I1_J,axiom,
! [C3: $tType,A: $tType,B: $tType,A2: A,X2: B,Y2: C3] :
( ( fresh @ A @ ( product_prod @ B @ C3 ) @ A2 @ ( product_Pair @ B @ C3 @ X2 @ Y2 ) )
=> ( fresh @ A @ B @ A2 @ X2 ) ) ).
% fresh_prodD(1)
thf(fact_216_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_217_IdD,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
=> ( A2 = B2 ) ) ).
% IdD
thf(fact_218_subject_Orecs_I2_J,axiom,
! [T: $tType,P: T > $o,F1: name > T,F2: name > T,Name4: name] :
( ! [X12: name] : ( P @ ( F1 @ X12 ) )
=> ( ! [X12: name] : ( P @ ( F2 @ X12 ) )
=> ( ( late_subject_rec @ T @ F1 @ F2 @ ( late_BoundOutputS @ Name4 ) )
= ( F2 @ Name4 ) ) ) ) ).
% subject.recs(2)
thf(fact_219_subject_Orecs_I1_J,axiom,
! [T: $tType,P: T > $o,F1: name > T,F2: name > T,Name4: name] :
( ! [X12: name] : ( P @ ( F1 @ X12 ) )
=> ( ! [X12: name] : ( P @ ( F2 @ X12 ) )
=> ( ( late_subject_rec @ T @ F1 @ F2 @ ( late_InputS @ Name4 ) )
= ( F1 @ Name4 ) ) ) ) ).
% subject.recs(1)
thf(fact_220_resTauOutputTrans,axiom,
! [X2: name,P: pi,A2: name,B2: name,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( tau @ P ) ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) ) ).
% resTauOutputTrans
thf(fact_221_residual_Ofresh_I1_J,axiom,
! [A2: name,X3: late_subject,X1: name,X22: pi] :
( ( fresh @ name @ late_residual @ A2 @ ( late_BoundR @ X3 @ X1 @ X22 ) )
= ( ( fresh @ name @ late_subject @ A2 @ X3 )
& ( fresh @ name @ ( name > ( noption @ pi ) ) @ A2 @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ) ).
% residual.fresh(1)
thf(fact_222_pi_Ofresh_I3_J,axiom,
! [A2: name,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( tau @ X1 ) )
= ( fresh @ name @ pi @ A2 @ X1 ) ) ).
% pi.fresh(3)
thf(fact_223_simps_I2_J,axiom,
! [P: pi,C2: name,D: name] :
( ( subs @ ( tau @ P ) @ C2 @ D )
= ( tau @ ( subs @ P @ C2 @ D ) ) ) ).
% simps(2)
thf(fact_224_pi_Ofresh_I9_J,axiom,
! [A2: name,X1: name,X22: pi] :
( ( fresh @ name @ pi @ A2 @ ( res @ X1 @ X22 ) )
= ( fresh @ name @ ( name > ( noption @ pi ) ) @ A2 @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ).
% pi.fresh(9)
thf(fact_225_pi_Ofresh_I4_J,axiom,
! [A2: name,X3: name,X1: name,X22: pi] :
( ( fresh @ name @ pi @ A2 @ ( input @ X3 @ X1 @ X22 ) )
= ( ( fresh @ name @ name @ A2 @ X3 )
& ( fresh @ name @ ( name > ( noption @ pi ) ) @ A2 @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ) ).
% pi.fresh(4)
thf(fact_226_name__fresh__abs,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [B2: name,A2: name,X2: A] :
( ( fresh @ name @ ( name > ( noption @ A ) ) @ B2 @ ( abs_fun @ name @ A @ A2 @ X2 ) )
= ( ( B2 = A2 )
| ( fresh @ name @ A @ B2 @ X2 ) ) ) ) ).
% name_fresh_abs
thf(fact_227_abs__fresh_I1_J,axiom,
! [X7: $tType] :
( ( fs_name @ X7 )
=> ! [B2: name,A2: name,X2: X7] :
( ( fresh @ name @ ( name > ( noption @ X7 ) ) @ B2 @ ( abs_fun @ name @ X7 @ A2 @ X2 ) )
= ( ( B2 = A2 )
| ( fresh @ name @ X7 @ B2 @ X2 ) ) ) ) ).
% abs_fresh(1)
thf(fact_228_pi_Oinject_I2_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( tau @ X1 )
= ( tau @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(2)
thf(fact_229_pi_Odistinct_I3_J,axiom,
! [Pi: pi] :
( piNil
!= ( tau @ Pi ) ) ).
% pi.distinct(3)
thf(fact_230_pi_Odistinct_I19_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( tau @ Pi ) ) ).
% pi.distinct(19)
thf(fact_231_pi_Odistinct_I35_J,axiom,
! [Pi3: pi,Name1: name,Name2: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( input @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(35)
thf(fact_232_pi_Odistinct_I41_J,axiom,
! [Pi3: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(41)
thf(fact_233_pi_Odistinct_I43_J,axiom,
! [Pi3: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(43)
thf(fact_234_pi_Odistinct_I45_J,axiom,
! [Pi3: pi,Name: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(45)
thf(fact_235_pi_Oinject_I3_J,axiom,
! [X3: name,X1: name,X22: pi,Y3: name,Y1: name,Y22: pi] :
( ( ( input @ X3 @ X1 @ X22 )
= ( input @ Y3 @ Y1 @ Y22 ) )
= ( ( X3 = Y3 )
& ( ( abs_fun @ name @ pi @ X1 @ X22 )
= ( abs_fun @ name @ pi @ Y1 @ Y22 ) ) ) ) ).
% pi.inject(3)
thf(fact_236_residual_Oinject_I1_J,axiom,
! [X3: late_subject,X1: name,X22: pi,Y3: late_subject,Y1: name,Y22: pi] :
( ( ( late_BoundR @ X3 @ X1 @ X22 )
= ( late_BoundR @ Y3 @ Y1 @ Y22 ) )
= ( ( X3 = Y3 )
& ( ( abs_fun @ name @ pi @ X1 @ X22 )
= ( abs_fun @ name @ pi @ Y1 @ Y22 ) ) ) ) ).
% residual.inject(1)
thf(fact_237_pi_Oinject_I8_J,axiom,
! [X1: name,X22: pi,Y1: name,Y22: pi] :
( ( ( res @ X1 @ X22 )
= ( res @ Y1 @ Y22 ) )
= ( ( abs_fun @ name @ pi @ X1 @ X22 )
= ( abs_fun @ name @ pi @ Y1 @ Y22 ) ) ) ).
% pi.inject(8)
thf(fact_238_tauBoundTrans,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_BoundR @ A2 @ X2 @ P3 ) ) ).
% tauBoundTrans
thf(fact_239_nilSim_I1_J,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( tau @ P ) ) ).
% nilSim(1)
thf(fact_240_resTauBoundTrans,axiom,
! [X2: name,P: pi,A2: late_subject,Y2: name,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( tau @ P ) ) @ ( late_BoundR @ A2 @ Y2 @ P3 ) ) ).
% resTauBoundTrans
thf(fact_241_tauOutputTrans,axiom,
! [P: pi,A2: name,B2: name,P3: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) ) ).
% tauOutputTrans
thf(fact_242_tauCases,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( ( Alpha = late_TauR )
=> ( ( P = P3 )
=> ( Prop @ late_TauR @ P ) ) )
=> ( Prop @ Alpha @ P3 ) ) ) ).
% tauCases
thf(fact_243_tauCases_H,axiom,
! [P: pi,Rs: late_residual] :
( ( late_transitions @ ( tau @ P ) @ Rs )
=> ~ ! [P4: pi] :
( ( ( tau @ P )
= ( tau @ P4 ) )
=> ( Rs
!= ( late_FreeR @ late_TauR @ P4 ) ) ) ) ).
% tauCases'
thf(fact_244_Tau,axiom,
! [P: pi] : ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ late_TauR @ P ) ) ).
% Tau
thf(fact_245_inputCases_H,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( input @ A2 @ B2 @ P ) @ Rs )
=> ~ ! [X5: name,A4: name,P4: pi] :
( ( ( A2 = A4 )
& ( ( abs_fun @ name @ pi @ B2 @ P )
= ( abs_fun @ name @ pi @ X5 @ P4 ) ) )
=> ( ( Rs
= ( late_BoundR @ ( late_InputS @ A4 ) @ X5 @ P4 ) )
=> ( X5 = A4 ) ) ) ) ).
% inputCases'
thf(fact_246_mismatchIdRight,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),A2: name,B2: name,P: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( ( A2 != B2 )
=> ( strong743114133lation @ P @ Rel @ ( mismatch @ A2 @ B2 @ P ) ) ) ) ).
% mismatchIdRight
thf(fact_247_mismatchIdLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),A2: name,B2: name,P: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( ( A2 != B2 )
=> ( strong743114133lation @ ( mismatch @ A2 @ B2 @ P ) @ Rel @ P ) ) ) ).
% mismatchIdLeft
thf(fact_248_pi_Ofresh_I6_J,axiom,
! [A2: name,X3: name,X22: name,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( mismatch @ X3 @ X22 @ X1 ) )
= ( ( fresh @ name @ name @ A2 @ X3 )
& ( fresh @ name @ name @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(6)
thf(fact_249_pi_Odistinct_I39_J,axiom,
! [Pi3: pi,Name1: name,Name2: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( mismatch @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(39)
thf(fact_250_mismatchCases,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual,F: name > name > $o] :
( ( late_transitions @ ( mismatch @ A2 @ B2 @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A2 != B2 )
=> ( F @ A2 @ B2 ) ) )
=> ( F @ A2 @ B2 ) ) ) ).
% mismatchCases
thf(fact_251_mismatchTrans,axiom,
! [A2: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( mismatch @ A2 @ A2 @ P ) @ Rs ) ).
% mismatchTrans
thf(fact_252_mismatchCases_H,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( mismatch @ A2 @ B2 @ P ) @ Rs )
=> ~ ! [P4: pi,A4: name,B5: name] :
( ( ( A2 = A4 )
& ( B2 = B5 )
& ( P = P4 ) )
=> ( ( late_transitions @ P4 @ Rs )
=> ( A4 = B5 ) ) ) ) ).
% mismatchCases'
thf(fact_253_Mismatch,axiom,
! [P: pi,Rs: late_residual,A2: name,B2: name] :
( ( late_transitions @ P @ Rs )
=> ( ( A2 != B2 )
=> ( late_transitions @ ( mismatch @ A2 @ B2 @ P ) @ Rs ) ) ) ).
% Mismatch
thf(fact_254_pi_Odistinct_I75_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( mismatch @ Name12 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(75)
thf(fact_255_pi_Odistinct_I73_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( mismatch @ Name12 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(73)
% Type constructors (17)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Agent_Opi___Agent_Ofs__name,axiom,
fs_name @ pi ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Agent_Ofs__name_7,axiom,
fs_name @ $o ).
thf(tcon_Agent_Oname___Agent_Ofs__name_8,axiom,
fs_name @ name ).
thf(tcon_Product__Type_Oprod___Agent_Ofs__name_9,axiom,
! [A8: $tType,A9: $tType] :
( ( ( fs_name @ A8 )
& ( fs_name @ A9 ) )
=> ( fs_name @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Ofs__name_10,axiom,
fs_name @ late_freeRes ).
thf(tcon_Late__Semantics_Osubject___Agent_Ofs__name_11,axiom,
fs_name @ late_subject ).
thf(tcon_Late__Semantics_Oresidual___Agent_Ofs__name_12,axiom,
fs_name @ late_residual ).
% Conjectures (1)
thf(conj_0,conjecture,
late_transitions @ ( res @ x @ ( sum @ p @ q ) ) @ ( late_FreeR @ alpha @ ( res @ x @ q2 ) ) ).
%------------------------------------------------------------------------------