TPTP Problem File: ITP185^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP185^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_529__3413750_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_529__3413750_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 330 ( 180 unt; 55 typ; 0 def)
% Number of atoms : 646 ( 279 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 3623 ( 142 ~; 7 |; 35 &;3041 @)
% ( 0 <=>; 398 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 287 ( 287 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 50 usr; 15 con; 0-12 aty)
% Number of variables : 1265 ( 39 ^;1202 !; 4 ?;1265 :)
% ( 20 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:49.259
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
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_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 (48)
thf(sy_cl_Agent_Ofs__name,type,
fs_name:
!>[A: $tType] : $o ).
thf(sy_cl_Agent_Opt__name,type,
pt_name:
!>[A: $tType] : $o ).
thf(sy_c_Agent_Opi_OBang,type,
bang: pi > pi ).
thf(sy_c_Agent_Opi_OInput,type,
input: name > name > pi > pi ).
thf(sy_c_Agent_Opi_OMatch,type,
match: 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_Opi_Opi__rec,type,
pi_rec:
!>[T: $tType] : ( T > ( name > name > pi > T > T ) > ( pi > T > T ) > ( name > name > pi > T > T ) > ( name > name > pi > T > T ) > ( name > name > pi > T > T ) > ( pi > pi > T > T > T ) > ( pi > pi > T > T > T ) > ( name > pi > T > T ) > ( pi > T > T ) > pi > T ) ).
thf(sy_c_Agent_Osubs,type,
subs: pi > name > name > pi ).
thf(sy_c_Agent_Osubst__name,type,
subst_name: name > name > name > name ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Late__Semantics1_OfreeRes_Ocase__freeRes,type,
late_case_freeRes:
!>[A: $tType] : ( ( name > name > A ) > A > late_freeRes > A ) ).
thf(sy_c_Late__Semantics1_Osubject_Ocase__subject,type,
late_case_subject:
!>[A: $tType] : ( ( name > A ) > ( name > A ) > late_subject > A ) ).
thf(sy_c_Late__Semantics_ObangPred,type,
late_bangPred: pi > pi > $o ).
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_Oresidual_Oresidual__rec,type,
late_residual_rec:
!>[T: $tType] : ( ( late_subject > name > pi > T ) > ( late_freeRes > pi > T ) > late_residual > T ) ).
thf(sy_c_Late__Semantics_Oresidual_Oresidual__rec__set,type,
late_r582136743ec_set:
!>[T: $tType] : ( ( late_subject > name > pi > T ) > ( late_freeRes > pi > T ) > late_residual > T > $o ) ).
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_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
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_Nominal_Osupp,type,
supp:
!>[A: $tType,X: $tType] : ( A > ( set @ X ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: pi ).
thf(sy_v_PQR____,type,
pqr: pi ).
thf(sy_v_Q,type,
q: pi ).
thf(sy_v_QR____,type,
qr: pi ).
thf(sy_v_Q_H____,type,
q2: pi ).
thf(sy_v_R,type,
r: pi ).
thf(sy_v_R_H____,type,
r2: pi ).
thf(sy_v__092_060alpha_062____,type,
alpha: late_freeRes ).
thf(sy_v_a____,type,
a: name ).
thf(sy_v_x____,type,
x: name ).
thf(sy_v_y____,type,
y: name ).
% Relevant facts (256)
thf(fact_0_cClose2_Ohyps_I14_J,axiom,
alpha = late_TauR ).
% cClose2.hyps(14)
thf(fact_1_cClose2_Ohyps_I10_J,axiom,
y != a ).
% cClose2.hyps(10)
thf(fact_2_cClose2_Ohyps_I5_J,axiom,
x != a ).
% cClose2.hyps(5)
thf(fact_3_cClose2_Ohyps_I9_J,axiom,
fresh @ name @ pi @ y @ r ).
% cClose2.hyps(9)
thf(fact_4_cClose2_Ohyps_I6_J,axiom,
x != y ).
% cClose2.hyps(6)
thf(fact_5_cClose2_Ohyps_I8_J,axiom,
fresh @ name @ pi @ y @ q ).
% cClose2.hyps(8)
thf(fact_6_cClose2_Ohyps_I3_J,axiom,
fresh @ name @ pi @ x @ q ).
% cClose2.hyps(3)
thf(fact_7_cClose2_Ohyps_I13_J,axiom,
fresh @ name @ pi @ y @ p ).
% cClose2.hyps(13)
thf(fact_8_cClose2_Ohyps_I12_J,axiom,
fresh @ name @ pi @ x @ p ).
% cClose2.hyps(12)
thf(fact_9_cClose2_Ohyps_I4_J,axiom,
fresh @ name @ pi @ x @ r ).
% cClose2.hyps(4)
thf(fact_10_cClose2_Ohyps_I7_J,axiom,
fresh @ name @ pi @ x @ q2 ).
% cClose2.hyps(7)
thf(fact_11_cClose2_Ohyps_I11_J,axiom,
fresh @ name @ pi @ y @ r2 ).
% cClose2.hyps(11)
thf(fact_12_cClose2_Ohyps_I2_J,axiom,
late_transitions @ r @ ( late_BoundR @ ( late_InputS @ a ) @ x @ r2 ) ).
% cClose2.hyps(2)
thf(fact_13__092_060open_062P_A_092_060parallel_062_AQ_A_092_060longmapsto_062_Aa_060_092_060nu_062y_062_A_092_060prec_062_AP_A_092_060parallel_062_AQ_H_092_060close_062,axiom,
late_transitions @ ( par @ p @ q ) @ ( late_BoundR @ ( late_BoundOutputS @ a ) @ y @ ( par @ p @ q2 ) ) ).
% \<open>P \<parallel> Q \<longmapsto> a<\<nu>y> \<prec> P \<parallel> Q'\<close>
thf(fact_14_cClose2_Ohyps_I1_J,axiom,
late_transitions @ q @ ( late_BoundR @ ( late_BoundOutputS @ a ) @ y @ q2 ) ).
% cClose2.hyps(1)
thf(fact_15_cPar2_Ohyps,axiom,
late_transitions @ ( par @ q @ r ) @ ( late_FreeR @ alpha @ qr ) ).
% cPar2.hyps
thf(fact_16_Free_Ohyps,axiom,
late_transitions @ ( par @ p @ ( par @ q @ r ) ) @ ( late_FreeR @ alpha @ pqr ) ).
% Free.hyps
thf(fact_17_simps_I9_J,axiom,
! [X2: name,C: name,D: name,P: pi] :
( ( X2 != C )
=> ( ( X2 != D )
=> ( ( subs @ ( res @ X2 @ P ) @ C @ D )
= ( res @ X2 @ ( subs @ P @ C @ D ) ) ) ) ) ).
% simps(9)
thf(fact_18_simps_I8_J,axiom,
! [P: pi,Q: pi,C: name,D: name] :
( ( subs @ ( par @ P @ Q ) @ C @ D )
= ( par @ ( subs @ P @ C @ D ) @ ( subs @ Q @ C @ D ) ) ) ).
% simps(8)
thf(fact_19_Par1F,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha @ ( par @ P2 @ Q ) ) ) ) ).
% Par1F
thf(fact_20_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_21_subst__identity,axiom,
! [P: pi,A2: name] :
( ( subs @ P @ A2 @ A2 )
= P ) ).
% subst_identity
thf(fact_22_pi_Odistinct_I85_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( par @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(85)
thf(fact_23_Late__Semantics_OClose1,axiom,
! [P: pi,A2: name,X2: name,P2: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ Q2 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P2 @ X2 @ Y ) @ Q2 ) ) ) ) ) ) ) ).
% Late_Semantics.Close1
thf(fact_24_Late__Semantics_OClose2,axiom,
! [P: pi,A2: name,Y: name,P2: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ Y @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P2 @ ( subs @ Q2 @ X2 @ Y ) ) ) ) ) ) ) ) ).
% Late_Semantics.Close2
thf(fact_25_transitions_OClose1,axiom,
! [P: pi,A2: name,X2: name,P2: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( ( fresh @ name @ pi @ Y @ Q )
=> ( ( X2 != A2 )
=> ( ( fresh @ name @ pi @ X2 @ Q2 )
=> ( ( Y != A2 )
=> ( ( fresh @ name @ pi @ Y @ P2 )
=> ( ( X2 != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P2 @ X2 @ Y ) @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close1
thf(fact_26_transitions_OClose2,axiom,
! [P: pi,A2: name,Y: name,P2: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( ( fresh @ name @ pi @ Y @ Q )
=> ( ( X2 != A2 )
=> ( ( fresh @ name @ pi @ X2 @ P2 )
=> ( ( Y != A2 )
=> ( ( fresh @ name @ pi @ Y @ Q2 )
=> ( ( X2 != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P2 @ ( subs @ Q2 @ X2 @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close2
thf(fact_27_substRes2,axiom,
! [B: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ B @ P )
=> ( ( res @ A2 @ P )
= ( res @ B @ ( subs @ P @ A2 @ B ) ) ) ) ).
% substRes2
thf(fact_28_substRes3,axiom,
! [B: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ B @ P )
=> ( ( subs @ ( res @ A2 @ P ) @ A2 @ B )
= ( res @ B @ ( subs @ P @ A2 @ B ) ) ) ) ).
% substRes3
thf(fact_29_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_30_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_31_substTrans,axiom,
! [B: name,P: pi,A2: name,C: name] :
( ( fresh @ name @ pi @ B @ P )
=> ( ( subs @ ( subs @ P @ A2 @ B ) @ B @ C )
= ( subs @ P @ A2 @ C ) ) ) ).
% substTrans
thf(fact_32_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name2: name,Name: name] :
( ( late_InputS @ Name2 )
!= ( late_BoundOutputS @ Name ) ) ).
% Late_Semantics.subject.distinct(1)
thf(fact_33_name__exists__fresh,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [X2: A] :
~ ! [C2: name] :
~ ( fresh @ name @ A @ C2 @ X2 ) ) ).
% name_exists_fresh
thf(fact_34_subject_Oinducts,axiom,
! [P: late_subject > $o,Subject: late_subject] :
( ! [Name3: name] : ( P @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name] : ( P @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Subject ) ) ) ).
% subject.inducts
thf(fact_35_subject_Ostrong__induct_H,axiom,
! [N: $tType,P: N > late_subject > $o,Z: N,Subject: late_subject] :
( ! [Name3: name,Z2: N] : ( P @ Z2 @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name,Z2: N] : ( P @ Z2 @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Z @ Subject ) ) ) ).
% subject.strong_induct'
thf(fact_36_subject_Ostrong__inducts,axiom,
! [A: $tType,P: A > late_subject > $o,Z: A,Subject: late_subject] :
( ! [Name3: name,Z2: A] : ( P @ Z2 @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name,Z2: A] : ( P @ Z2 @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Z @ Subject ) ) ) ).
% subject.strong_inducts
thf(fact_37_residual_Ostrong__inducts,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [P: A > late_residual > $o,Z: A,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi3: pi,Z2: A] :
( ( fresh @ name @ A @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi3 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi3: pi,Z2: A] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_inducts
thf(fact_38_residual_Ostrong__induct,axiom,
! [N: $tType] :
( ( fs_name @ N )
=> ! [P: N > late_residual > $o,Z: N,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi3: pi,Z2: N] :
( ( fresh @ name @ N @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi3 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi3: pi,Z2: N] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_induct
thf(fact_39_freshBoundDerivative_I2_J,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( ( Y != X2 )
=> ( fresh @ name @ pi @ Y @ P2 ) ) ) ) ).
% freshBoundDerivative(2)
thf(fact_40_freshRes,axiom,
! [A2: name,P: pi] : ( fresh @ name @ pi @ A2 @ ( res @ A2 @ P ) ) ).
% freshRes
thf(fact_41_residual_Oinducts,axiom,
! [P: late_residual > $o,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi3: pi] : ( P @ ( late_BoundR @ Subject2 @ Name3 @ Pi3 ) )
=> ( ! [FreeRes: late_freeRes,Pi3: pi] : ( P @ ( late_FreeR @ FreeRes @ Pi3 ) )
=> ( P @ Residual ) ) ) ).
% residual.inducts
thf(fact_42_residual_Odistinct_I1_J,axiom,
! [Subject: late_subject,Name2: name,Pi4: pi,FreeRes2: late_freeRes,Pi: pi] :
( ( late_BoundR @ Subject @ Name2 @ Pi4 )
!= ( late_FreeR @ FreeRes2 @ Pi ) ) ).
% residual.distinct(1)
thf(fact_43_fresh__fact2,axiom,
! [A2: name,B: name,P: pi] :
( ( A2 != B )
=> ( fresh @ name @ pi @ A2 @ ( subs @ P @ A2 @ B ) ) ) ).
% fresh_fact2
thf(fact_44_fresh__fact1,axiom,
! [A2: name,P: pi,C: name,B: name] :
( ( fresh @ name @ pi @ A2 @ P )
=> ( ( A2 != C )
=> ( fresh @ name @ pi @ A2 @ ( subs @ P @ B @ C ) ) ) ) ).
% fresh_fact1
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
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_forget,axiom,
! [A2: name,P: pi,B: name] :
( ( fresh @ name @ pi @ A2 @ P )
=> ( ( subs @ P @ A2 @ B )
= P ) ) ).
% forget
thf(fact_50_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 )
=> ( ! [P3: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( Prop @ ( par @ P3 @ Q ) ) )
=> ( ! [Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q3 ) )
=> ( Prop @ ( par @ P @ Q3 ) ) )
=> ( Prop @ PQ ) ) ) ) ) ) ).
% parCasesB
thf(fact_51_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_52_Late__Semantics_OPar1B,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P2 @ Q ) ) ) ) ) ).
% Late_Semantics.Par1B
thf(fact_53_pi_Oinject_I7_J,axiom,
! [X22: pi,X1: pi,Y2: pi,Y1: pi] :
( ( ( par @ X22 @ X1 )
= ( par @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(7)
thf(fact_54_freshFreeDerivative_I2_J,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( fresh @ name @ pi @ Y @ P2 ) ) ) ).
% freshFreeDerivative(2)
thf(fact_55_residual_Oinject_I2_J,axiom,
! [X22: late_freeRes,X1: pi,Y2: late_freeRes,Y1: pi] :
( ( ( late_FreeR @ X22 @ X1 )
= ( late_FreeR @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% residual.inject(2)
thf(fact_56_transitions_OComm2,axiom,
! [P: pi,A2: name,B: name,P2: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( X2 != A2 )
=> ( ( X2 != B )
=> ( ( fresh @ name @ pi @ X2 @ P2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P2 @ ( subs @ Q2 @ X2 @ B ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm2
thf(fact_57_transitions_OComm1,axiom,
! [P: pi,A2: name,X2: name,P2: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( X2 != A2 )
=> ( ( X2 != B )
=> ( ( fresh @ name @ pi @ X2 @ Q2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P2 @ X2 @ B ) @ Q2 ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm1
thf(fact_58_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_59_Late__Semantics1_Osubject_Oinject_I2_J,axiom,
! [X22: name,Y2: name] :
( ( ( late_BoundOutputS @ X22 )
= ( late_BoundOutputS @ Y2 ) )
= ( X22 = Y2 ) ) ).
% Late_Semantics1.subject.inject(2)
thf(fact_60_Late__Semantics_OComm2,axiom,
! [P: pi,A2: name,B: name,P2: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P2 @ ( subs @ Q2 @ X2 @ B ) ) ) ) ) ) ).
% Late_Semantics.Comm2
thf(fact_61_Late__Semantics_OComm1,axiom,
! [P: pi,A2: name,X2: name,P2: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P2 @ X2 @ B ) @ Q2 ) ) ) ) ) ).
% Late_Semantics.Comm1
thf(fact_62_Late__Semantics1_Osubject_Odistinct_I1_J,axiom,
! [X1: name,X22: name] :
( ( late_InputS @ X1 )
!= ( late_BoundOutputS @ X22 ) ) ).
% Late_Semantics1.subject.distinct(1)
thf(fact_63_subject_Oexhaust,axiom,
! [Y: late_subject] :
( ! [X12: name] :
( Y
!= ( late_InputS @ X12 ) )
=> ~ ! [X23: name] :
( Y
!= ( late_BoundOutputS @ X23 ) ) ) ).
% subject.exhaust
thf(fact_64_Open,axiom,
! [P: pi,A2: name,B: name,P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ P2 ) )
=> ( ( A2 != B )
=> ( late_transitions @ ( res @ B @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ B @ P2 ) ) ) ) ).
% Open
thf(fact_65_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_66_Late__Semantics_OfreeRes_Oinject,axiom,
! [X22: name,X1: name,Y2: name,Y1: name] :
( ( ( late_OutputR @ X22 @ X1 )
= ( late_OutputR @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% Late_Semantics.freeRes.inject
thf(fact_67_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_68_freeRes_Ofresh_I2_J,axiom,
! [A2: name] : ( fresh @ name @ late_freeRes @ A2 @ late_TauR ) ).
% freeRes.fresh(2)
thf(fact_69_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_70_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_71_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_72_Late__Semantics1_OfreeRes_Odistinct_I1_J,axiom,
! [X11: name,X122: name] :
( ( late_OutputR @ X11 @ X122 )
!= late_TauR ) ).
% Late_Semantics1.freeRes.distinct(1)
thf(fact_73_freeRes_Oexhaust,axiom,
! [Y: late_freeRes] :
( ! [X112: name,X123: name] :
( Y
!= ( late_OutputR @ X112 @ X123 ) )
=> ( Y = late_TauR ) ) ).
% freeRes.exhaust
thf(fact_74_Late__Semantics_OfreeRes_Odistinct_I1_J,axiom,
! [Name1: name,Name22: name] :
( ( late_OutputR @ Name1 @ Name22 )
!= late_TauR ) ).
% Late_Semantics.freeRes.distinct(1)
thf(fact_75_freeRes_Oinducts,axiom,
! [P: late_freeRes > $o,FreeRes3: late_freeRes] :
( ! [Name12: name,Name23: name] : ( P @ ( late_OutputR @ Name12 @ Name23 ) )
=> ( ( P @ late_TauR )
=> ( P @ FreeRes3 ) ) ) ).
% freeRes.inducts
thf(fact_76_freeRes_Ostrong__induct_H,axiom,
! [N: $tType,P: N > late_freeRes > $o,Z: N,FreeRes3: late_freeRes] :
( ! [Name12: name,Name23: name,Z2: N] : ( P @ Z2 @ ( late_OutputR @ Name12 @ Name23 ) )
=> ( ! [Z2: N] : ( P @ Z2 @ late_TauR )
=> ( P @ Z @ FreeRes3 ) ) ) ).
% freeRes.strong_induct'
thf(fact_77_freeRes_Ostrong__inducts,axiom,
! [A: $tType,P: A > late_freeRes > $o,Z: A,FreeRes3: late_freeRes] :
( ! [Name12: name,Name23: name,Z2: A] : ( P @ Z2 @ ( late_OutputR @ Name12 @ Name23 ) )
=> ( ! [Z2: A] : ( P @ Z2 @ late_TauR )
=> ( P @ Z @ FreeRes3 ) ) ) ).
% freeRes.strong_inducts
thf(fact_78_resCases_H,axiom,
! [X2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X2 @ P ) @ Rs )
=> ( ! [P4: pi,A4: name,B3: name] :
( ( ( res @ X2 @ P )
= ( res @ B3 @ P4 ) )
=> ! [P3: pi] :
( ( Rs
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B3 @ P3 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A4 @ B3 ) @ P3 ) )
=> ( A4 = B3 ) ) ) )
=> ( ! [P4: pi,A4: late_subject,X4: name,P3: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( Rs
= ( late_BoundR @ A4 @ X4 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X4 @ P3 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X4 )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P3: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( Rs
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P3 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha2 ) ) ) ) ) ) ) ).
% resCases'
thf(fact_79_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_80_resCasesF,axiom,
! [X2: name,P: pi,Alpha: late_freeRes,XP: pi,F2: pi > $o] :
( ( late_transitions @ ( res @ X2 @ P ) @ ( late_FreeR @ Alpha @ XP ) )
=> ( ! [P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh @ name @ late_freeRes @ X2 @ Alpha )
=> ( F2 @ ( res @ X2 @ P3 ) ) ) )
=> ( F2 @ XP ) ) ) ).
% resCasesF
thf(fact_81_resCasesF_H,axiom,
! [X2: name,P: pi,Alpha: late_freeRes,P2: pi] :
( ( late_transitions @ ( res @ X2 @ P ) @ ( late_FreeR @ Alpha @ P2 ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P3: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( ( late_FreeR @ Alpha @ P2 )
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P3 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha2 ) ) ) ) ) ).
% resCasesF'
thf(fact_82_ResF,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh @ name @ late_freeRes @ Y @ Alpha )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_FreeR @ Alpha @ ( res @ Y @ P2 ) ) ) ) ) ).
% ResF
thf(fact_83_Late__Semantics_OResB,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
=> ( ( fresh @ name @ late_subject @ Y @ A2 )
=> ( ( Y != X2 )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A2 @ X2 @ ( res @ Y @ P2 ) ) ) ) ) ) ).
% Late_Semantics.ResB
thf(fact_84_freshFreeDerivative_I1_J,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( fresh @ name @ late_freeRes @ Y @ Alpha ) ) ) ).
% freshFreeDerivative(1)
thf(fact_85_freshBoundDerivative_I1_J,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( fresh @ name @ late_subject @ Y @ A2 ) ) ) ).
% freshBoundDerivative(1)
thf(fact_86_resCasesB_H,axiom,
! [X5: name,P: pi,A2: late_subject,Y4: name,P2: pi] :
( ( late_transitions @ ( res @ X5 @ P ) @ ( late_BoundR @ A2 @ Y4 @ P2 ) )
=> ( ! [P4: pi,A4: name,B3: name] :
( ( ( res @ X5 @ P )
= ( res @ B3 @ P4 ) )
=> ! [P3: pi] :
( ( ( late_BoundR @ A2 @ Y4 @ P2 )
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B3 @ P3 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A4 @ B3 ) @ P3 ) )
=> ( A4 = B3 ) ) ) )
=> ~ ! [P4: pi,A4: late_subject,X4: name,P3: pi,Y3: name] :
( ( ( res @ X5 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( ( late_BoundR @ A2 @ Y4 @ P2 )
= ( late_BoundR @ A4 @ X4 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X4 @ P3 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X4 )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_87_parCasesB_H,axiom,
! [P: pi,Q: pi,B: late_subject,Y: name,P2: pi] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ B @ Y @ P2 ) )
=> ( ! [P4: pi,A4: late_subject,X4: name,P3: pi,Q4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y @ P2 )
= ( late_BoundR @ A4 @ X4 @ ( par @ P3 @ Q4 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X4 @ P3 ) )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ( ( fresh @ name @ pi @ X4 @ Q4 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) )
=> ~ ! [Q4: pi,A4: late_subject,X4: name,Q3: pi,P4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y @ P2 )
= ( late_BoundR @ A4 @ X4 @ ( par @ P4 @ Q3 ) ) )
=> ( ( late_transitions @ Q4 @ ( late_BoundR @ A4 @ X4 @ Q3 ) )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ( ( fresh @ name @ pi @ X4 @ Q4 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) ) ) ) ).
% parCasesB'
thf(fact_88_transitions_OPar1B,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
=> ( ( 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 @ P2 @ Q ) ) ) ) ) ) ) ).
% transitions.Par1B
thf(fact_89_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_90_transitions_OResB,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P2 ) )
=> ( ( fresh @ name @ late_subject @ Y @ A2 )
=> ( ( Y != X2 )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ late_subject @ X2 @ A2 )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A2 @ X2 @ ( res @ Y @ P2 ) ) ) ) ) ) ) ) ).
% transitions.ResB
thf(fact_91_freeRes_Orecs_I2_J,axiom,
! [T: $tType,P: T > $o,F1: name > name > T,F22: T] :
( ! [X12: name,X23: name] : ( P @ ( F1 @ X12 @ X23 ) )
=> ( ( P @ F22 )
=> ( ( late_freeRes_rec @ T @ F1 @ F22 @ late_TauR )
= F22 ) ) ) ).
% freeRes.recs(2)
thf(fact_92_subject_Orecs_I1_J,axiom,
! [T: $tType,P: T > $o,F1: name > T,F22: name > T,Name2: name] :
( ! [X12: name] : ( P @ ( F1 @ X12 ) )
=> ( ! [X12: name] : ( P @ ( F22 @ X12 ) )
=> ( ( late_subject_rec @ T @ F1 @ F22 @ ( late_InputS @ Name2 ) )
= ( F1 @ Name2 ) ) ) ) ).
% subject.recs(1)
thf(fact_93_subject_Orecs_I2_J,axiom,
! [T: $tType,P: T > $o,F1: name > T,F22: name > T,Name2: name] :
( ! [X12: name] : ( P @ ( F1 @ X12 ) )
=> ( ! [X12: name] : ( P @ ( F22 @ X12 ) )
=> ( ( late_subject_rec @ T @ F1 @ F22 @ ( late_BoundOutputS @ Name2 ) )
= ( F22 @ Name2 ) ) ) ) ).
% subject.recs(2)
thf(fact_94_freeRes_Orecs_I1_J,axiom,
! [T: $tType,P: T > $o,F1: name > name > T,F22: T,Name1: name,Name22: name] :
( ! [X12: name,X23: name] : ( P @ ( F1 @ X12 @ X23 ) )
=> ( ( P @ F22 )
=> ( ( late_freeRes_rec @ T @ F1 @ F22 @ ( late_OutputR @ Name1 @ Name22 ) )
= ( F1 @ Name1 @ Name22 ) ) ) ) ).
% freeRes.recs(1)
thf(fact_95_resTauOutputTrans,axiom,
! [X2: name,P: pi,A2: name,B: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( tau @ P ) ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ P2 ) ) ).
% resTauOutputTrans
thf(fact_96_resOutputTauTrans,axiom,
! [X2: name,A2: name,B: name,P: pi,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ B @ P ) ) @ ( late_FreeR @ late_TauR @ P2 ) ) ).
% resOutputTauTrans
thf(fact_97_resOutputInputTrans,axiom,
! [X2: name,A2: name,B: name,P: pi,C: name,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ B @ P ) ) @ ( late_BoundR @ ( late_InputS @ C ) @ Y @ P2 ) ) ).
% resOutputInputTrans
thf(fact_98_name__fresh,axiom,
( ( fresh @ name @ name )
= ( ^ [A5: name,B4: name] : A5 != B4 ) ) ).
% name_fresh
thf(fact_99_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_100_simps_I2_J,axiom,
! [P: pi,C: name,D: name] :
( ( subs @ ( tau @ P ) @ C @ D )
= ( tau @ ( subs @ P @ C @ D ) ) ) ).
% simps(2)
thf(fact_101_pi_Ofresh_I2_J,axiom,
! [A2: name,X32: name,X22: name,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( output @ X32 @ X22 @ X1 ) )
= ( ( fresh @ name @ name @ A2 @ X32 )
& ( fresh @ name @ name @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(2)
thf(fact_102_pi_Odistinct_I19_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( tau @ Pi ) ) ).
% pi.distinct(19)
thf(fact_103_pi_Oinject_I1_J,axiom,
! [X32: name,X22: name,X1: pi,Y32: name,Y2: name,Y1: pi] :
( ( ( output @ X32 @ X22 @ X1 )
= ( output @ Y32 @ Y2 @ Y1 ) )
= ( ( X32 = Y32 )
& ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(1)
thf(fact_104_pi_Oinject_I2_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( tau @ X1 )
= ( tau @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(2)
thf(fact_105_pi_Odistinct_I29_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(29)
thf(fact_106_pi_Odistinct_I31_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(31)
thf(fact_107_pi_Odistinct_I43_J,axiom,
! [Pi4: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi4 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(43)
thf(fact_108_pi_Odistinct_I45_J,axiom,
! [Pi4: pi,Name: name,Pi: pi] :
( ( tau @ Pi4 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(45)
thf(fact_109_resTrans_I1_J,axiom,
! [X2: name,B: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ ( output @ X2 @ B @ P ) ) @ Rs ) ).
% resTrans(1)
thf(fact_110_outputBoundTrans,axiom,
! [A2: name,B: name,P: pi,C: late_subject,X2: name,P2: pi] :
~ ( late_transitions @ ( output @ A2 @ B @ P ) @ ( late_BoundR @ C @ X2 @ P2 ) ) ).
% outputBoundTrans
thf(fact_111_outputFreshTrans,axiom,
! [A2: name,B: name,P: pi,Alpha: late_freeRes,P2: pi] :
( ( late_transitions @ ( output @ A2 @ B @ P ) @ ( late_FreeR @ Alpha @ P2 ) )
=> ~ ( ( fresh @ name @ late_freeRes @ A2 @ Alpha )
| ( fresh @ name @ late_freeRes @ B @ Alpha ) ) ) ).
% outputFreshTrans
thf(fact_112_tauBoundTrans,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_BoundR @ A2 @ X2 @ P2 ) ) ).
% tauBoundTrans
thf(fact_113_resTauBoundTrans,axiom,
! [X2: name,P: pi,A2: late_subject,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( tau @ P ) ) @ ( late_BoundR @ A2 @ Y @ P2 ) ) ).
% resTauBoundTrans
thf(fact_114_Output,axiom,
! [A2: name,B: name,P: pi] : ( late_transitions @ ( output @ A2 @ B @ P ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ P ) ) ).
% Output
thf(fact_115_outputIneqTrans,axiom,
! [A2: name,B: name,P: pi,C: name,D: name,P2: pi] :
( ( late_transitions @ ( output @ A2 @ B @ P ) @ ( late_FreeR @ ( late_OutputR @ C @ D ) @ P2 ) )
=> ~ ( ( A2 != C )
| ( B != D ) ) ) ).
% outputIneqTrans
thf(fact_116_outputCases_H,axiom,
! [A2: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( output @ A2 @ B @ P ) @ Rs )
=> ~ ! [A4: name,B3: name,P4: pi] :
( ( ( A2 = A4 )
& ( B = B3 )
& ( P = P4 ) )
=> ( Rs
!= ( late_FreeR @ ( late_OutputR @ A4 @ B3 ) @ P4 ) ) ) ) ).
% outputCases'
thf(fact_117_outputCases,axiom,
! [A2: name,B: name,P: pi,Alpha: late_freeRes,P2: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( output @ A2 @ B @ P ) @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( ( Alpha
= ( late_OutputR @ A2 @ B ) )
=> ( ( P = P2 )
=> ( Prop @ ( late_OutputR @ A2 @ B ) @ P ) ) )
=> ( Prop @ Alpha @ P2 ) ) ) ).
% outputCases
thf(fact_118_outputTauTrans,axiom,
! [A2: name,B: name,P: pi,P2: pi] :
~ ( late_transitions @ ( output @ A2 @ B @ P ) @ ( late_FreeR @ late_TauR @ P2 ) ) ).
% outputTauTrans
thf(fact_119_tauOutputTrans,axiom,
! [P: pi,A2: name,B: name,P2: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B ) @ P2 ) ) ).
% tauOutputTrans
thf(fact_120_Tau,axiom,
! [P: pi] : ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ late_TauR @ P ) ) ).
% Tau
thf(fact_121_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_122_tauCases,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( ( Alpha = late_TauR )
=> ( ( P = P2 )
=> ( Prop @ late_TauR @ P ) ) )
=> ( Prop @ Alpha @ P2 ) ) ) ).
% tauCases
thf(fact_123_resOutputOutputTrans,axiom,
! [X2: name,A2: name,P: pi,B: name,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ X2 @ P ) ) @ ( late_FreeR @ ( late_OutputR @ B @ Y ) @ P2 ) ) ).
% resOutputOutputTrans
thf(fact_124_resInputBoundOutputTrans,axiom,
! [X2: name,A2: name,Y: name,P: pi,B: name,Z: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( input @ A2 @ Y @ P ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Z @ P2 ) ) ).
% resInputBoundOutputTrans
thf(fact_125_residual_Ofresh_I1_J,axiom,
! [A2: name,X32: late_subject,X1: name,X22: pi] :
( ( fresh @ name @ late_residual @ A2 @ ( late_BoundR @ X32 @ X1 @ X22 ) )
= ( ( fresh @ name @ late_subject @ A2 @ X32 )
& ( fresh @ name @ ( name > ( noption @ pi ) ) @ A2 @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ) ).
% residual.fresh(1)
thf(fact_126_simps_I3_J,axiom,
! [A2: name,B: name,P: pi,C: name,D: name] :
( ( subs @ ( output @ A2 @ B @ P ) @ C @ D )
= ( output @ ( subst_name @ A2 @ C @ D ) @ ( subst_name @ B @ C @ D ) @ ( subs @ P @ C @ D ) ) ) ).
% simps(3)
thf(fact_127_inputIneqTrans,axiom,
! [A2: name,X2: name,P: pi,B: late_subject,Y: name,P2: pi] :
( ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ B @ Y @ P2 ) )
=> ~ ( fresh @ name @ late_subject @ A2 @ B ) ) ).
% inputIneqTrans
thf(fact_128_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_129_substInput,axiom,
! [X2: name,B: name,C: name,A2: name,P: pi] :
( ( X2 != B )
=> ( ( X2 != C )
=> ( ( subs @ ( input @ A2 @ X2 @ P ) @ B @ C )
= ( input @ ( subst_name @ A2 @ B @ C ) @ X2 @ ( subs @ P @ B @ C ) ) ) ) ) ).
% substInput
thf(fact_130_simps_I4_J,axiom,
! [X2: name,A2: name,C: name,D: name,P: pi] :
( ( X2 != A2 )
=> ( ( X2 != C )
=> ( ( X2 != D )
=> ( ( subs @ ( input @ A2 @ X2 @ P ) @ C @ D )
= ( input @ ( subst_name @ A2 @ C @ D ) @ X2 @ ( subs @ P @ C @ D ) ) ) ) ) ) ).
% simps(4)
thf(fact_131_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_132_pi_Ofresh_I4_J,axiom,
! [A2: name,X32: name,X1: name,X22: pi] :
( ( fresh @ name @ pi @ A2 @ ( input @ X32 @ X1 @ X22 ) )
= ( ( fresh @ name @ name @ A2 @ X32 )
& ( fresh @ name @ ( name > ( noption @ pi ) ) @ A2 @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ) ).
% pi.fresh(4)
thf(fact_133_pi_Oinject_I3_J,axiom,
! [X32: name,X1: name,X22: pi,Y32: name,Y1: name,Y2: pi] :
( ( ( input @ X32 @ X1 @ X22 )
= ( input @ Y32 @ Y1 @ Y2 ) )
= ( ( X32 = Y32 )
& ( ( abs_fun @ name @ pi @ X1 @ X22 )
= ( abs_fun @ name @ pi @ Y1 @ Y2 ) ) ) ) ).
% pi.inject(3)
thf(fact_134_subst__name__def,axiom,
( subst_name
= ( ^ [A5: name,B4: name,C3: name] : ( if @ name @ ( A5 = B4 ) @ C3 @ A5 ) ) ) ).
% subst_name_def
thf(fact_135_pi_Oinject_I8_J,axiom,
! [X1: name,X22: pi,Y1: name,Y2: pi] :
( ( ( res @ X1 @ X22 )
= ( res @ Y1 @ Y2 ) )
= ( ( abs_fun @ name @ pi @ X1 @ X22 )
= ( abs_fun @ name @ pi @ Y1 @ Y2 ) ) ) ).
% pi.inject(8)
thf(fact_136_residual_Oinject_I1_J,axiom,
! [X32: late_subject,X1: name,X22: pi,Y32: late_subject,Y1: name,Y2: pi] :
( ( ( late_BoundR @ X32 @ X1 @ X22 )
= ( late_BoundR @ Y32 @ Y1 @ Y2 ) )
= ( ( X32 = Y32 )
& ( ( abs_fun @ name @ pi @ X1 @ X22 )
= ( abs_fun @ name @ pi @ Y1 @ Y2 ) ) ) ) ).
% residual.inject(1)
thf(fact_137_name__fresh__abs,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [B: name,A2: name,X2: A] :
( ( fresh @ name @ ( name > ( noption @ A ) ) @ B @ ( abs_fun @ name @ A @ A2 @ X2 ) )
= ( ( B = A2 )
| ( fresh @ name @ A @ B @ X2 ) ) ) ) ).
% name_fresh_abs
thf(fact_138_abs__fresh_I1_J,axiom,
! [X6: $tType] :
( ( fs_name @ X6 )
=> ! [B: name,A2: name,X2: X6] :
( ( fresh @ name @ ( name > ( noption @ X6 ) ) @ B @ ( abs_fun @ name @ X6 @ A2 @ X2 ) )
= ( ( B = A2 )
| ( fresh @ name @ X6 @ B @ X2 ) ) ) ) ).
% abs_fresh(1)
thf(fact_139_pi_Odistinct_I55_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name1 @ Name22 @ Pi4 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(55)
thf(fact_140_pi_Odistinct_I57_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name: name,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi4 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(57)
thf(fact_141_pi_Odistinct_I21_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( input @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(21)
thf(fact_142_pi_Odistinct_I35_J,axiom,
! [Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( tau @ Pi4 )
!= ( input @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(35)
thf(fact_143_inputCases_H,axiom,
! [A2: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( input @ A2 @ B @ P ) @ Rs )
=> ~ ! [X4: name,A4: name,P4: pi] :
( ( ( A2 = A4 )
& ( ( abs_fun @ name @ pi @ B @ P )
= ( abs_fun @ name @ pi @ X4 @ P4 ) ) )
=> ( ( Rs
= ( late_BoundR @ ( late_InputS @ A4 ) @ X4 @ P4 ) )
=> ( X4 = A4 ) ) ) ) ).
% inputCases'
thf(fact_144_resTrans_I2_J,axiom,
! [X2: name,Y: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ ( input @ X2 @ Y @ P ) ) @ Rs ) ).
% resTrans(2)
thf(fact_145_inputFreeTrans,axiom,
! [A2: name,X2: name,P: pi,Alpha: late_freeRes,P2: pi] :
~ ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_FreeR @ Alpha @ P2 ) ) ).
% inputFreeTrans
thf(fact_146_resInputFreeTrans,axiom,
! [X2: name,A2: name,Y: name,P: pi,Alpha: late_freeRes,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( input @ A2 @ Y @ P ) ) @ ( late_FreeR @ Alpha @ P2 ) ) ).
% resInputFreeTrans
thf(fact_147_inputBoundOutputTrans,axiom,
! [A2: name,X2: name,P: pi,B: name,Y: name,P2: pi] :
~ ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Y @ P2 ) ) ).
% inputBoundOutputTrans
thf(fact_148_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_149_simps_I5_J,axiom,
! [A2: name,B: name,P: pi,C: name,D: name] :
( ( subs @ ( match @ A2 @ B @ P ) @ C @ D )
= ( match @ ( subst_name @ A2 @ C @ D ) @ ( subst_name @ B @ C @ D ) @ ( subs @ P @ C @ D ) ) ) ).
% simps(5)
thf(fact_150_simps_I6_J,axiom,
! [A2: name,B: name,P: pi,C: name,D: name] :
( ( subs @ ( mismatch @ A2 @ B @ P ) @ C @ D )
= ( mismatch @ ( subst_name @ A2 @ C @ D ) @ ( subst_name @ B @ C @ D ) @ ( subs @ P @ C @ D ) ) ) ).
% simps(6)
thf(fact_151_pi_Ofresh_I5_J,axiom,
! [A2: name,X32: name,X22: name,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( match @ X32 @ X22 @ X1 ) )
= ( ( fresh @ name @ name @ A2 @ X32 )
& ( fresh @ name @ name @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(5)
thf(fact_152_pi_Ofresh_I6_J,axiom,
! [A2: name,X32: name,X22: name,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( mismatch @ X32 @ X22 @ X1 ) )
= ( ( fresh @ name @ name @ A2 @ X32 )
& ( fresh @ name @ name @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(6)
thf(fact_153_pi_Odistinct_I61_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( match @ Name1 @ Name22 @ Pi4 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(61)
thf(fact_154_pi_Oinject_I4_J,axiom,
! [X32: name,X22: name,X1: pi,Y32: name,Y2: name,Y1: pi] :
( ( ( match @ X32 @ X22 @ X1 )
= ( match @ Y32 @ Y2 @ Y1 ) )
= ( ( X32 = Y32 )
& ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(4)
thf(fact_155_pi_Oinject_I5_J,axiom,
! [X32: name,X22: name,X1: pi,Y32: name,Y2: name,Y1: pi] :
( ( ( mismatch @ X32 @ X22 @ X1 )
= ( mismatch @ Y32 @ Y2 @ Y1 ) )
= ( ( X32 = Y32 )
& ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(5)
thf(fact_156_Mismatch,axiom,
! [P: pi,Rs: late_residual,A2: name,B: name] :
( ( late_transitions @ P @ Rs )
=> ( ( A2 != B )
=> ( late_transitions @ ( mismatch @ A2 @ B @ P ) @ Rs ) ) ) ).
% Mismatch
thf(fact_157_Match,axiom,
! [P: pi,Rs: late_residual,B: name] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( match @ B @ B @ P ) @ Rs ) ) ).
% Match
thf(fact_158_mismatchCases_H,axiom,
! [A2: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( mismatch @ A2 @ B @ P ) @ Rs )
=> ~ ! [P4: pi,A4: name,B3: name] :
( ( ( A2 = A4 )
& ( B = B3 )
& ( P = P4 ) )
=> ( ( late_transitions @ P4 @ Rs )
=> ( A4 = B3 ) ) ) ) ).
% mismatchCases'
thf(fact_159_mismatchTrans,axiom,
! [A2: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( mismatch @ A2 @ A2 @ P ) @ Rs ) ).
% mismatchTrans
thf(fact_160_mismatchCases,axiom,
! [A2: name,B: name,P: pi,Rs: late_residual,F2: name > name > $o] :
( ( late_transitions @ ( mismatch @ A2 @ B @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A2 != B )
=> ( F2 @ A2 @ B ) ) )
=> ( F2 @ A2 @ B ) ) ) ).
% mismatchCases
thf(fact_161_matchCases_H,axiom,
! [A2: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( match @ A2 @ B @ P ) @ Rs )
=> ~ ! [P4: pi] :
( ( ( B = A2 )
& ( P = P4 ) )
=> ~ ( late_transitions @ P4 @ Rs ) ) ) ).
% matchCases'
thf(fact_162_matchTrans,axiom,
! [A2: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( match @ A2 @ B @ P ) @ Rs )
=> ( A2 = B ) ) ).
% matchTrans
thf(fact_163_matchCases,axiom,
! [A2: name,B: name,P: pi,Rs: late_residual,F2: name > name > $o] :
( ( late_transitions @ ( match @ A2 @ B @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A2 = B )
=> ( F2 @ A2 @ A2 ) ) )
=> ( F2 @ A2 @ B ) ) ) ).
% matchCases
thf(fact_164_pi_Odistinct_I65_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( match @ Name1 @ Name22 @ Pi4 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(65)
thf(fact_165_pi_Odistinct_I73_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi4 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(73)
thf(fact_166_pi_Odistinct_I67_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name: name,Pi: pi] :
( ( match @ Name1 @ Name22 @ Pi4 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(67)
thf(fact_167_pi_Odistinct_I75_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name: name,Pi: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi4 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(75)
thf(fact_168_pi_Odistinct_I49_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi4 )
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(49)
thf(fact_169_pi_Odistinct_I51_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi4 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(51)
thf(fact_170_pi_Odistinct_I23_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(23)
thf(fact_171_pi_Odistinct_I25_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(25)
thf(fact_172_pi_Odistinct_I37_J,axiom,
! [Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( tau @ Pi4 )
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(37)
thf(fact_173_pi_Odistinct_I39_J,axiom,
! [Pi4: pi,Name13: name,Name24: name,Pi: pi] :
( ( tau @ Pi4 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(39)
thf(fact_174_abs__fun__eq1,axiom,
! [X: $tType,A: $tType,A2: X,X2: A,Y: A] :
( ( ( abs_fun @ X @ A @ A2 @ X2 )
= ( abs_fun @ X @ A @ A2 @ Y ) )
= ( X2 = Y ) ) ).
% abs_fun_eq1
thf(fact_175_freeRes_Ocase_I2_J,axiom,
! [A: $tType,F1: name > name > A,F22: A] :
( ( late_case_freeRes @ A @ F1 @ F22 @ late_TauR )
= F22 ) ).
% freeRes.case(2)
thf(fact_176_subject_Ocase_I1_J,axiom,
! [A: $tType,F1: name > A,F22: name > A,X1: name] :
( ( late_case_subject @ A @ F1 @ F22 @ ( late_InputS @ X1 ) )
= ( F1 @ X1 ) ) ).
% subject.case(1)
thf(fact_177_freeRes_Ocase_I1_J,axiom,
! [A: $tType,F1: name > name > A,F22: A,X11: name,X122: name] :
( ( late_case_freeRes @ A @ F1 @ F22 @ ( late_OutputR @ X11 @ X122 ) )
= ( F1 @ X11 @ X122 ) ) ).
% freeRes.case(1)
thf(fact_178_subject_Ocase_I2_J,axiom,
! [A: $tType,F1: name > A,F22: name > A,X22: name] :
( ( late_case_subject @ A @ F1 @ F22 @ ( late_BoundOutputS @ X22 ) )
= ( F22 @ X22 ) ) ).
% subject.case(2)
thf(fact_179_pi_Ostrong__inducts,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [P: A > pi > $o,Z: A,Pi4: pi] :
( ! [Z2: A] : ( P @ Z2 @ piNil )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: A] :
( ( fresh @ name @ A @ Name23 @ Z2 )
=> ( ( fresh @ name @ name @ Name23 @ Name12 )
=> ( ! [Za: A] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name12 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi23: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi13 )
=> ( ! [Za: A] : ( P @ Za @ Pi23 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Pi13: pi,Pi23: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi13 )
=> ( ! [Za: A] : ( P @ Za @ Pi23 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Name3: name,Pi3: pi,Z2: A] :
( ( fresh @ name @ A @ Name3 @ Z2 )
=> ( ! [Za: A] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name3 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_inducts
thf(fact_180_pi_Ostrong__induct,axiom,
! [N: $tType] :
( ( fs_name @ N )
=> ! [P: N > pi > $o,Z: N,Pi4: pi] :
( ! [Z2: N] : ( P @ Z2 @ piNil )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ( fresh @ name @ N @ Name23 @ Z2 )
=> ( ( fresh @ name @ name @ Name23 @ Name12 )
=> ( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name12 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi23: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi13 )
=> ( ! [Za: N] : ( P @ Za @ Pi23 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Pi13: pi,Pi23: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi13 )
=> ( ! [Za: N] : ( P @ Za @ Pi23 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Name3: name,Pi3: pi,Z2: N] :
( ( fresh @ name @ N @ Name3 @ Z2 )
=> ( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name3 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_induct
thf(fact_181_residual_Ostrong__induct_H,axiom,
! [N: $tType,F: N > ( set @ name ),P: N > late_residual > $o,Z: N,Residual: late_residual] :
( ! [X4: N] : ( finite_finite @ name @ ( F @ X4 ) )
=> ( ! [Subject2: late_subject,Name3: name,Pi3: pi,Z2: N] :
( ~ ( member @ name @ Name3 @ ( F @ Z2 ) )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi3 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi3: pi,Z2: N] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_induct'
thf(fact_182_pi_Ofresh_I10_J,axiom,
! [A2: name,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( bang @ X1 ) )
= ( fresh @ name @ pi @ A2 @ X1 ) ) ).
% pi.fresh(10)
thf(fact_183_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_184_pi_Ofresh_I1_J,axiom,
! [A2: name] : ( fresh @ name @ pi @ A2 @ piNil ) ).
% pi.fresh(1)
thf(fact_185_simps_I10_J,axiom,
! [P: pi,C: name,D: name] :
( ( subs @ ( bang @ P ) @ C @ D )
= ( bang @ ( subs @ P @ C @ D ) ) ) ).
% simps(10)
thf(fact_186_simps_I7_J,axiom,
! [P: pi,Q: pi,C: name,D: name] :
( ( subs @ ( sum @ P @ Q ) @ C @ D )
= ( sum @ ( subs @ P @ C @ D ) @ ( subs @ Q @ C @ D ) ) ) ).
% simps(7)
thf(fact_187_simps_I1_J,axiom,
! [C: name,D: name] :
( ( subs @ piNil @ C @ D )
= piNil ) ).
% simps(1)
thf(fact_188_pi_Odistinct_I59_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi4 )
!= ( bang @ Pi ) ) ).
% pi.distinct(59)
thf(fact_189_pi_Odistinct_I53_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name1 @ Name22 @ Pi4 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(53)
thf(fact_190_pi_Odistinct_I87_J,axiom,
! [Pi1: pi,Pi2: pi,Pi: pi] :
( ( par @ Pi1 @ Pi2 )
!= ( bang @ Pi ) ) ).
% pi.distinct(87)
thf(fact_191_pi_Odistinct_I89_J,axiom,
! [Name2: name,Pi4: pi,Pi: pi] :
( ( res @ Name2 @ Pi4 )
!= ( bang @ Pi ) ) ).
% pi.distinct(89)
thf(fact_192_pi_Odistinct_I33_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( bang @ Pi ) ) ).
% pi.distinct(33)
thf(fact_193_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_194_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_195_Sum1,axiom,
! [P: pi,Rs: late_residual,Q: pi] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum1
thf(fact_196_Sum2,axiom,
! [Q: pi,Rs: late_residual,P: pi] :
( ( late_transitions @ Q @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum2
thf(fact_197_pi_Odistinct_I47_J,axiom,
! [Pi4: pi,Pi: pi] :
( ( tau @ Pi4 )
!= ( bang @ Pi ) ) ).
% pi.distinct(47)
thf(fact_198_pi_Odistinct_I79_J,axiom,
! [Pi1: pi,Pi2: pi,Pi12: pi,Pi22: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(79)
thf(fact_199_pi_Odistinct_I81_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(81)
thf(fact_200_pi_Odistinct_I11_J,axiom,
! [Pi12: pi,Pi22: pi] :
( piNil
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(11)
thf(fact_201_pi_Odistinct_I17_J,axiom,
! [Pi: pi] :
( piNil
!= ( bang @ Pi ) ) ).
% pi.distinct(17)
thf(fact_202_pi_Odistinct_I83_J,axiom,
! [Pi1: pi,Pi2: pi,Pi: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( bang @ Pi ) ) ).
% pi.distinct(83)
thf(fact_203_pi_Oinject_I6_J,axiom,
! [X22: pi,X1: pi,Y2: pi,Y1: pi] :
( ( ( sum @ X22 @ X1 )
= ( sum @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(6)
thf(fact_204_pi_Oinject_I9_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( bang @ X1 )
= ( bang @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(9)
thf(fact_205_pi_Odistinct_I3_J,axiom,
! [Pi: pi] :
( piNil
!= ( tau @ Pi ) ) ).
% pi.distinct(3)
thf(fact_206_pi_Odistinct_I1_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( output @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(1)
thf(fact_207_pi_Odistinct_I5_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( input @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(5)
thf(fact_208_pi_Odistinct_I15_J,axiom,
! [Name: name,Pi: pi] :
( piNil
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(15)
thf(fact_209_pi_Odistinct_I13_J,axiom,
! [Pi12: pi,Pi22: pi] :
( piNil
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(13)
thf(fact_210_nilCases_H,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% nilCases'
thf(fact_211_zeroTrans,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% zeroTrans
thf(fact_212_pi_Odistinct_I41_J,axiom,
! [Pi4: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi4 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(41)
thf(fact_213_pi_Odistinct_I27_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name1 @ Name22 @ Pi4 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(27)
thf(fact_214_pi_Odistinct_I77_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi4 )
!= ( bang @ Pi ) ) ).
% pi.distinct(77)
thf(fact_215_pi_Odistinct_I71_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi4 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(71)
thf(fact_216_pi_Odistinct_I69_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi: pi] :
( ( match @ Name1 @ Name22 @ Pi4 )
!= ( bang @ Pi ) ) ).
% pi.distinct(69)
thf(fact_217_pi_Odistinct_I63_J,axiom,
! [Name1: name,Name22: name,Pi4: pi,Pi12: pi,Pi22: pi] :
( ( match @ Name1 @ Name22 @ Pi4 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(63)
thf(fact_218_pi_Odistinct_I9_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(9)
thf(fact_219_pi_Odistinct_I7_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(7)
thf(fact_220_Bang,axiom,
! [P: pi,Rs: late_residual] :
( ( late_transitions @ ( par @ P @ ( bang @ P ) ) @ Rs )
=> ( late_transitions @ ( bang @ P ) @ Rs ) ) ).
% Bang
thf(fact_221_bangCases,axiom,
! [P: pi,Rs: late_residual] :
( ( late_transitions @ ( bang @ P ) @ Rs )
=> ~ ! [P4: pi] :
( ( ( bang @ P )
= ( bang @ P4 ) )
=> ~ ( late_transitions @ ( par @ P4 @ ( bang @ P4 ) ) @ Rs ) ) ) ).
% bangCases
thf(fact_222_pi_Ostrong__induct_H,axiom,
! [N: $tType,F: N > ( set @ name ),P: N > pi > $o,Z: N,Pi4: pi] :
( ! [X4: N] : ( finite_finite @ name @ ( F @ X4 ) )
=> ( ! [Z2: N] : ( P @ Z2 @ piNil )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ~ ( member @ name @ Name23 @ ( F @ Z2 ) )
=> ( ( fresh @ name @ name @ Name23 @ Name12 )
=> ( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name12 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi23: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi13 )
=> ( ! [Za: N] : ( P @ Za @ Pi23 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Pi13: pi,Pi23: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi13 )
=> ( ! [Za: N] : ( P @ Za @ Pi23 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Name3: name,Pi3: pi,Z2: N] :
( ~ ( member @ name @ Name3 @ ( F @ Z2 ) )
=> ( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name3 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_induct'
thf(fact_223_resZeroTrans,axiom,
! [X2: name,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ piNil ) @ Rs ) ).
% resZeroTrans
thf(fact_224_pi_Oinducts,axiom,
! [P: pi > $o,Pi4: pi] :
( ( P @ piNil )
=> ( ! [Name12: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( output @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( tau @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( input @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( match @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Name12: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( mismatch @ Name12 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi23: pi] :
( ( P @ Pi13 )
=> ( ( P @ Pi23 )
=> ( P @ ( sum @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Pi13: pi,Pi23: pi] :
( ( P @ Pi13 )
=> ( ( P @ Pi23 )
=> ( P @ ( par @ Pi13 @ Pi23 ) ) ) )
=> ( ! [Name3: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( res @ Name3 @ Pi3 ) ) )
=> ( ! [Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( bang @ Pi3 ) ) )
=> ( P @ Pi4 ) ) ) ) ) ) ) ) ) ) ) ).
% pi.inducts
thf(fact_225_aux2,axiom,
! [P: pi] : ( late_bangPred @ P @ ( par @ P @ ( bang @ P ) ) ) ).
% aux2
thf(fact_226_bangPred_Ocases,axiom,
! [A1: pi,A22: pi] :
( ( late_bangPred @ A1 @ A22 )
=> ( ( A22
!= ( bang @ A1 ) )
=> ( A22
= ( par @ A1 @ ( bang @ A1 ) ) ) ) ) ).
% bangPred.cases
thf(fact_227_bangPred_Osimps,axiom,
( late_bangPred
= ( ^ [A12: pi,A23: pi] :
( ? [P5: pi] :
( ( A12 = P5 )
& ( A23
= ( bang @ P5 ) ) )
| ? [P5: pi] :
( ( A12 = P5 )
& ( A23
= ( par @ P5 @ ( bang @ P5 ) ) ) ) ) ) ) ).
% bangPred.simps
thf(fact_228_aux1,axiom,
! [P: pi] : ( late_bangPred @ P @ ( bang @ P ) ) ).
% aux1
thf(fact_229_bangPred_Oinducts,axiom,
! [X1: pi,X22: pi,P: pi > pi > $o] :
( ( late_bangPred @ X1 @ X22 )
=> ( ! [P4: pi] : ( P @ P4 @ ( bang @ P4 ) )
=> ( ! [P4: pi] : ( P @ P4 @ ( par @ P4 @ ( bang @ P4 ) ) )
=> ( P @ X1 @ X22 ) ) ) ) ).
% bangPred.inducts
thf(fact_230_subs__def,axiom,
( subs
= ( ^ [X3: pi,C3: name,D2: name] :
( pi_rec @ pi @ piNil
@ ^ [A5: name,B4: name,P5: pi] : ( output @ ( subst_name @ A5 @ C3 @ D2 ) @ ( subst_name @ B4 @ C3 @ D2 ) )
@ ^ [P5: pi] : tau
@ ^ [A5: name,Y5: name,P5: pi] : ( input @ ( subst_name @ A5 @ C3 @ D2 ) @ Y5 )
@ ^ [A5: name,B4: name,P5: pi] : ( match @ ( subst_name @ A5 @ C3 @ D2 ) @ ( subst_name @ B4 @ C3 @ D2 ) )
@ ^ [A5: name,B4: name,P5: pi] : ( mismatch @ ( subst_name @ A5 @ C3 @ D2 ) @ ( subst_name @ B4 @ C3 @ D2 ) )
@ ^ [P5: pi,Q5: pi] : sum
@ ^ [P5: pi,Q5: pi] : par
@ ^ [Y5: name,P5: pi] : ( res @ Y5 )
@ ^ [P5: pi] : bang
@ X3 ) ) ) ).
% subs_def
thf(fact_231_abs__fresh_I2_J,axiom,
! [X7: $tType] :
( ( pt_name @ X7 )
=> ! [X2: X7,B: name,A2: name] :
( ( finite_finite @ name @ ( supp @ X7 @ name @ X2 ) )
=> ( ( fresh @ name @ ( name > ( noption @ X7 ) ) @ B @ ( abs_fun @ name @ X7 @ A2 @ X2 ) )
= ( ( B = A2 )
| ( fresh @ name @ X7 @ B @ X2 ) ) ) ) ) ).
% abs_fresh(2)
thf(fact_232_exists__fresh,axiom,
! [A: $tType,X2: A] :
( ( finite_finite @ name @ ( supp @ A @ name @ X2 ) )
=> ~ ! [C2: name] :
~ ( fresh @ name @ A @ C2 @ X2 ) ) ).
% exists_fresh
thf(fact_233_exists__fresh_H,axiom,
! [A: $tType,X2: A] :
( ( finite_finite @ name @ ( supp @ A @ name @ X2 ) )
=> ? [C2: name] : ( fresh @ name @ A @ C2 @ X2 ) ) ).
% exists_fresh'
thf(fact_234_supp__fresh__iff,axiom,
! [X: $tType,A: $tType] :
( ( supp @ A @ X )
= ( ^ [X3: A] :
( collect @ X
@ ^ [A5: X] :
~ ( fresh @ X @ A @ A5 @ X3 ) ) ) ) ).
% supp_fresh_iff
thf(fact_235_fresh__def,axiom,
! [A: $tType,X: $tType] :
( ( fresh @ X @ A )
= ( ^ [A5: X,X3: A] :
~ ( member @ X @ A5 @ ( supp @ A @ X @ X3 ) ) ) ) ).
% fresh_def
thf(fact_236_pi_Osupp_I9_J,axiom,
! [X1: name,X22: pi] :
( ( supp @ pi @ name @ ( res @ X1 @ X22 ) )
= ( supp @ ( name > ( noption @ pi ) ) @ name @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ).
% pi.supp(9)
thf(fact_237_freeRes_Ocase__distrib,axiom,
! [A: $tType,B2: $tType,H: A > B2,F1: name > name > A,F22: A,FreeRes3: late_freeRes] :
( ( H @ ( late_case_freeRes @ A @ F1 @ F22 @ FreeRes3 ) )
= ( late_case_freeRes @ B2
@ ^ [X13: name,X24: name] : ( H @ ( F1 @ X13 @ X24 ) )
@ ( H @ F22 )
@ FreeRes3 ) ) ).
% freeRes.case_distrib
thf(fact_238_subject_Ocase__distrib,axiom,
! [B2: $tType,A: $tType,H: A > B2,F1: name > A,F22: name > A,Subject: late_subject] :
( ( H @ ( late_case_subject @ A @ F1 @ F22 @ Subject ) )
= ( late_case_subject @ B2
@ ^ [X3: name] : ( H @ ( F1 @ X3 ) )
@ ^ [X3: name] : ( H @ ( F22 @ X3 ) )
@ Subject ) ) ).
% subject.case_distrib
thf(fact_239_pi_Osupp_I3_J,axiom,
! [X1: pi] :
( ( supp @ pi @ name @ ( tau @ X1 ) )
= ( supp @ pi @ name @ X1 ) ) ).
% pi.supp(3)
thf(fact_240_subject_Osupp_I2_J,axiom,
! [X1: name] :
( ( supp @ late_subject @ name @ ( late_BoundOutputS @ X1 ) )
= ( supp @ name @ name @ X1 ) ) ).
% subject.supp(2)
thf(fact_241_subject_Osupp_I1_J,axiom,
! [X1: name] :
( ( supp @ late_subject @ name @ ( late_InputS @ X1 ) )
= ( supp @ name @ name @ X1 ) ) ).
% subject.supp(1)
thf(fact_242_pi_Osupp_I10_J,axiom,
! [X1: pi] :
( ( supp @ pi @ name @ ( bang @ X1 ) )
= ( supp @ pi @ name @ X1 ) ) ).
% pi.supp(10)
thf(fact_243_fs__name1,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [X2: A] : ( finite_finite @ name @ ( supp @ A @ name @ X2 ) ) ) ).
% fs_name1
thf(fact_244_residual_Orecs_I2_J,axiom,
! [N: $tType,T: $tType] :
( ( pt_name @ T )
=> ! [F1: late_subject > name > pi > T,F22: late_freeRes > pi > T,Z: N,P: T > $o,FreeRes3: late_freeRes,Pi4: pi] :
( ( finite_finite @ name @ ( supp @ ( late_subject > name > pi > T ) @ name @ F1 ) )
=> ( ( finite_finite @ name @ ( supp @ ( late_freeRes > pi > T ) @ name @ F22 ) )
=> ( ( finite_finite @ name @ ( supp @ N @ name @ Z ) )
=> ( ! [X12: late_subject,X23: name,X33: pi] : ( P @ ( F1 @ X12 @ X23 @ X33 ) )
=> ( ! [X12: late_freeRes,X23: pi] : ( P @ ( F22 @ X12 @ X23 ) )
=> ( ! [X12: late_subject,X23: name,X33: pi] :
( ( fresh @ name @ ( late_subject > name > pi > T ) @ X23 @ F1 )
=> ( ( fresh @ name @ late_subject @ X23 @ X12 )
=> ( ( fresh @ name @ N @ X23 @ Z )
=> ( ( P @ ( F1 @ X12 @ X23 @ X33 ) )
=> ( fresh @ name @ T @ X23 @ ( F1 @ X12 @ X23 @ X33 ) ) ) ) ) )
=> ( ( late_residual_rec @ T @ F1 @ F22 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
= ( F22 @ FreeRes3 @ Pi4 ) ) ) ) ) ) ) ) ) ).
% residual.recs(2)
thf(fact_245_residual_Orecs_I1_J,axiom,
! [N: $tType,T: $tType] :
( ( pt_name @ T )
=> ! [F1: late_subject > name > pi > T,F22: late_freeRes > pi > T,Z: N,P: T > $o,Name2: name,Subject: late_subject,Pi4: pi] :
( ( finite_finite @ name @ ( supp @ ( late_subject > name > pi > T ) @ name @ F1 ) )
=> ( ( finite_finite @ name @ ( supp @ ( late_freeRes > pi > T ) @ name @ F22 ) )
=> ( ( finite_finite @ name @ ( supp @ N @ name @ Z ) )
=> ( ! [X12: late_subject,X23: name,X33: pi] : ( P @ ( F1 @ X12 @ X23 @ X33 ) )
=> ( ! [X12: late_freeRes,X23: pi] : ( P @ ( F22 @ X12 @ X23 ) )
=> ( ! [X12: late_subject,X23: name,X33: pi] :
( ( fresh @ name @ ( late_subject > name > pi > T ) @ X23 @ F1 )
=> ( ( fresh @ name @ late_subject @ X23 @ X12 )
=> ( ( fresh @ name @ N @ X23 @ Z )
=> ( ( P @ ( F1 @ X12 @ X23 @ X33 ) )
=> ( fresh @ name @ T @ X23 @ ( F1 @ X12 @ X23 @ X33 ) ) ) ) ) )
=> ( ( fresh @ name @ ( late_subject > name > pi > T ) @ Name2 @ F1 )
=> ( ( fresh @ name @ ( late_freeRes > pi > T ) @ Name2 @ F22 )
=> ( ( fresh @ name @ late_subject @ Name2 @ Subject )
=> ( ( late_residual_rec @ T @ F1 @ F22 @ ( late_BoundR @ Subject @ Name2 @ Pi4 ) )
= ( F1 @ Subject @ Name2 @ Pi4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% residual.recs(1)
thf(fact_246_residual_Orec__fresh,axiom,
! [T: $tType] :
( ( pt_name @ T )
=> ! [F1: late_subject > name > pi > T,F22: late_freeRes > pi > T,X2: late_residual,Y: T,A2: name] :
( ( finite_finite @ name @ ( supp @ ( late_subject > name > pi > T ) @ name @ F1 ) )
=> ( ( finite_finite @ name @ ( supp @ ( late_freeRes > pi > T ) @ name @ F22 ) )
=> ( ( late_r582136743ec_set @ T @ F1 @ F22 @ X2 @ Y )
=> ( ! [Y6: T] :
( ( late_r582136743ec_set @ T @ F1 @ F22 @ X2 @ Y6 )
=> ( Y6 = Y ) )
=> ( ( fresh @ name @ late_residual @ A2 @ X2 )
=> ( ( fresh @ name @ ( late_subject > name > pi > T ) @ A2 @ F1 )
=> ( ( fresh @ name @ ( late_freeRes > pi > T ) @ A2 @ F22 )
=> ( fresh @ name @ T @ A2 @ Y ) ) ) ) ) ) ) ) ) ).
% residual.rec_fresh
thf(fact_247_residual_Orec__unique,axiom,
! [N: $tType,T: $tType] :
( ( pt_name @ T )
=> ! [F1: late_subject > name > pi > T,F22: late_freeRes > pi > T,Z: N,P: T > $o,X2: late_residual] :
( ( finite_finite @ name @ ( supp @ ( late_subject > name > pi > T ) @ name @ F1 ) )
=> ( ( finite_finite @ name @ ( supp @ ( late_freeRes > pi > T ) @ name @ F22 ) )
=> ( ( finite_finite @ name @ ( supp @ N @ name @ Z ) )
=> ( ! [X12: late_subject,X23: name,X33: pi] : ( P @ ( F1 @ X12 @ X23 @ X33 ) )
=> ( ! [X12: late_freeRes,X23: pi] : ( P @ ( F22 @ X12 @ X23 ) )
=> ( ! [X12: late_subject,X23: name] :
( ( fresh @ name @ ( late_subject > name > pi > T ) @ X23 @ F1 )
=> ( ( fresh @ name @ late_subject @ X23 @ X12 )
=> ( ( fresh @ name @ N @ X23 @ Z )
=> ! [X33: pi] :
( ( P @ ( F1 @ X12 @ X23 @ X33 ) )
=> ( fresh @ name @ T @ X23 @ ( F1 @ X12 @ X23 @ X33 ) ) ) ) ) )
=> ? [X4: T] :
( ( late_r582136743ec_set @ T @ F1 @ F22 @ X2 @ X4 )
& ! [Y7: T] :
( ( late_r582136743ec_set @ T @ F1 @ F22 @ X2 @ Y7 )
=> ( Y7 = X4 ) ) ) ) ) ) ) ) ) ) ).
% residual.rec_unique
thf(fact_248_residual_Orec__fin__supp,axiom,
! [T: $tType] :
( ( pt_name @ T )
=> ! [F1: late_subject > name > pi > T,F22: late_freeRes > pi > T,X1: late_residual,Y1: T] :
( ( finite_finite @ name @ ( supp @ ( late_subject > name > pi > T ) @ name @ F1 ) )
=> ( ( finite_finite @ name @ ( supp @ ( late_freeRes > pi > T ) @ name @ F22 ) )
=> ( ( late_r582136743ec_set @ T @ F1 @ F22 @ X1 @ Y1 )
=> ( finite_finite @ name @ ( supp @ T @ name @ Y1 ) ) ) ) ) ) ).
% residual.rec_fin_supp
thf(fact_249_pi_Osupp_I4_J,axiom,
! [X32: name,X1: name,X22: pi] :
( ( supp @ pi @ name @ ( input @ X32 @ X1 @ X22 ) )
= ( sup_sup @ ( set @ name ) @ ( supp @ name @ name @ X32 ) @ ( supp @ ( name > ( noption @ pi ) ) @ name @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ) ).
% pi.supp(4)
thf(fact_250_residual_Osupp_I1_J,axiom,
! [X32: late_subject,X1: name,X22: pi] :
( ( supp @ late_residual @ name @ ( late_BoundR @ X32 @ X1 @ X22 ) )
= ( sup_sup @ ( set @ name ) @ ( supp @ late_subject @ name @ X32 ) @ ( supp @ ( name > ( noption @ pi ) ) @ name @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ) ).
% residual.supp(1)
thf(fact_251_pi_Osupp_I7_J,axiom,
! [X22: pi,X1: pi] :
( ( supp @ pi @ name @ ( sum @ X22 @ X1 ) )
= ( sup_sup @ ( set @ name ) @ ( supp @ pi @ name @ X22 ) @ ( supp @ pi @ name @ X1 ) ) ) ).
% pi.supp(7)
thf(fact_252_pi_Osupp_I8_J,axiom,
! [X22: pi,X1: pi] :
( ( supp @ pi @ name @ ( par @ X22 @ X1 ) )
= ( sup_sup @ ( set @ name ) @ ( supp @ pi @ name @ X22 ) @ ( supp @ pi @ name @ X1 ) ) ) ).
% pi.supp(8)
thf(fact_253_residual_Osupp_I2_J,axiom,
! [X22: late_freeRes,X1: pi] :
( ( supp @ late_residual @ name @ ( late_FreeR @ X22 @ X1 ) )
= ( sup_sup @ ( set @ name ) @ ( supp @ late_freeRes @ name @ X22 ) @ ( supp @ pi @ name @ X1 ) ) ) ).
% residual.supp(2)
thf(fact_254_freeRes_Osupp_I1_J,axiom,
! [X22: name,X1: name] :
( ( supp @ late_freeRes @ name @ ( late_OutputR @ X22 @ X1 ) )
= ( sup_sup @ ( set @ name ) @ ( supp @ name @ name @ X22 ) @ ( supp @ name @ name @ X1 ) ) ) ).
% freeRes.supp(1)
thf(fact_255_pi_Osupp_I2_J,axiom,
! [X32: name,X22: name,X1: pi] :
( ( supp @ pi @ name @ ( output @ X32 @ X22 @ X1 ) )
= ( sup_sup @ ( set @ name ) @ ( supp @ name @ name @ X32 ) @ ( sup_sup @ ( set @ name ) @ ( supp @ name @ name @ X22 ) @ ( supp @ pi @ name @ X1 ) ) ) ) ).
% pi.supp(2)
% Type constructors (15)
thf(tcon_fun___Agent_Opt__name,axiom,
! [A6: $tType,A7: $tType] :
( ( ( pt_name @ A6 )
& ( pt_name @ A7 ) )
=> ( pt_name @ ( A6 > A7 ) ) ) ).
thf(tcon_Set_Oset___Agent_Opt__name_1,axiom,
! [A6: $tType] :
( ( pt_name @ A6 )
=> ( pt_name @ ( set @ A6 ) ) ) ).
thf(tcon_Agent_Opi___Agent_Opt__name_2,axiom,
pt_name @ pi ).
thf(tcon_Agent_Opi___Agent_Ofs__name,axiom,
fs_name @ pi ).
thf(tcon_HOL_Obool___Agent_Opt__name_3,axiom,
pt_name @ $o ).
thf(tcon_HOL_Obool___Agent_Ofs__name_4,axiom,
fs_name @ $o ).
thf(tcon_Agent_Oname___Agent_Opt__name_5,axiom,
pt_name @ name ).
thf(tcon_Agent_Oname___Agent_Ofs__name_6,axiom,
fs_name @ name ).
thf(tcon_Nominal_Onoption___Agent_Opt__name_7,axiom,
! [A6: $tType] :
( ( pt_name @ A6 )
=> ( pt_name @ ( noption @ A6 ) ) ) ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Opt__name_8,axiom,
pt_name @ late_freeRes ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Ofs__name_9,axiom,
fs_name @ late_freeRes ).
thf(tcon_Late__Semantics_Osubject___Agent_Opt__name_10,axiom,
pt_name @ late_subject ).
thf(tcon_Late__Semantics_Osubject___Agent_Ofs__name_11,axiom,
fs_name @ late_subject ).
thf(tcon_Late__Semantics_Oresidual___Agent_Opt__name_12,axiom,
pt_name @ late_residual ).
thf(tcon_Late__Semantics_Oresidual___Agent_Ofs__name_13,axiom,
fs_name @ late_residual ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
late_transitions @ ( par @ ( par @ p @ q ) @ r ) @ ( late_FreeR @ late_TauR @ ( res @ y @ ( par @ ( par @ p @ q2 ) @ ( subs @ r2 @ x @ y ) ) ) ) ).
%------------------------------------------------------------------------------