TPTP Problem File: ITP187^2.p
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%------------------------------------------------------------------------------
% File : ITP187^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_617__3414780_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_617__3414780_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.00 v7.5.0
% Syntax : Number of formulae : 320 ( 167 unt; 53 typ; 0 def)
% Number of atoms : 634 ( 262 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 3959 ( 147 ~; 5 |; 49 &;3422 @)
% ( 0 <=>; 336 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 142 ( 142 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 48 usr; 16 con; 0-5 aty)
% Number of variables : 1276 ( 9 ^;1234 !; 14 ?;1276 :)
% ( 19 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:17:08.690
%------------------------------------------------------------------------------
% 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 (45)
thf(sy_cl_Agent_Ofs__name,type,
fs_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_Osubs,type,
subs: pi > name > name > pi ).
thf(sy_c_Agent_Osubst__name,type,
subst_name: name > name > name > name ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_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_PQR____,type,
pqr: pi ).
thf(sy_v_P_H____,type,
p2: 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_Rel,type,
rel: set @ ( product_prod @ pi @ 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_I6_J,axiom,
x != y ).
% cClose2.hyps(6)
thf(fact_2_cClose2_Ohyps_I10_J,axiom,
y != a ).
% cClose2.hyps(10)
thf(fact_3_cClose2_Ohyps_I5_J,axiom,
x != a ).
% cClose2.hyps(5)
thf(fact_4_Ass,axiom,
! [S: pi,T2: pi,U: pi] : ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ ( par @ S @ T2 ) @ U ) @ ( par @ S @ ( par @ T2 @ U ) ) ) @ rel ) ).
% Ass
thf(fact_5_Res,axiom,
! [S: pi,T2: pi,X2: name] :
( ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ S @ T2 ) @ rel )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( res @ X2 @ S ) @ ( res @ X2 @ T2 ) ) @ rel ) ) ).
% Res
thf(fact_6__092_060open_062_I_060_092_060nu_062y_062_IP_H_A_092_060parallel_062_AQ_H_091x_058_058_061y_093_J_A_092_060parallel_062_AR_M_A_060_092_060nu_062y_062_IP_H_A_092_060parallel_062_A_IQ_H_A_092_060parallel_062_AR_J_091x_058_058_061y_093_J_J_A_092_060in_062_ARel_092_060close_062,axiom,
member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ ( res @ y @ ( par @ p2 @ ( subs @ q2 @ x @ y ) ) ) @ r ) @ ( res @ y @ ( par @ p2 @ ( subs @ ( par @ q2 @ r ) @ x @ y ) ) ) ) @ rel ).
% \<open>(<\<nu>y>(P' \<parallel> Q'[x::=y]) \<parallel> R, <\<nu>y>(P' \<parallel> (Q' \<parallel> R)[x::=y])) \<in> Rel\<close>
thf(fact_7__092_060open_062_IP_A_092_060parallel_062_AQ_J_A_092_060parallel_062_AR_A_092_060longmapsto_062_A_092_060tau_062_A_092_060prec_062_A_060_092_060nu_062y_062_IP_H_A_092_060parallel_062_AQ_H_091x_058_058_061y_093_J_A_092_060parallel_062_AR_092_060close_062,axiom,
late_transitions @ ( par @ ( par @ p @ q ) @ r ) @ ( late_FreeR @ late_TauR @ ( par @ ( res @ y @ ( par @ p2 @ ( subs @ q2 @ x @ y ) ) ) @ r ) ) ).
% \<open>(P \<parallel> Q) \<parallel> R \<longmapsto> \<tau> \<prec> <\<nu>y>(P' \<parallel> Q'[x::=y]) \<parallel> R\<close>
thf(fact_8_Free_Ohyps,axiom,
late_transitions @ ( par @ p @ ( par @ q @ r ) ) @ ( late_FreeR @ alpha @ pqr ) ).
% Free.hyps
thf(fact_9__092_060open_062_I_060_092_060nu_062y_062_IP_H_A_092_060parallel_062_AQ_H_091x_058_058_061y_093_J_A_092_060parallel_062_AR_M_A_060_092_060nu_062y_062_IP_H_A_092_060parallel_062_AQ_H_091x_058_058_061y_093_A_092_060parallel_062_AR_J_J_A_092_060in_062_ARel_092_060close_062,axiom,
member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ ( res @ y @ ( par @ p2 @ ( subs @ q2 @ x @ y ) ) ) @ r ) @ ( res @ y @ ( par @ p2 @ ( par @ ( subs @ q2 @ x @ y ) @ r ) ) ) ) @ rel ).
% \<open>(<\<nu>y>(P' \<parallel> Q'[x::=y]) \<parallel> R, <\<nu>y>(P' \<parallel> Q'[x::=y] \<parallel> R)) \<in> Rel\<close>
thf(fact_10__092_060open_062P_A_092_060parallel_062_AQ_A_092_060longmapsto_062_A_092_060tau_062_A_092_060prec_062_A_060_092_060nu_062y_062_IP_H_A_092_060parallel_062_AQ_H_091x_058_058_061y_093_J_092_060close_062,axiom,
late_transitions @ ( par @ p @ q ) @ ( late_FreeR @ late_TauR @ ( res @ y @ ( par @ p2 @ ( subs @ q2 @ x @ y ) ) ) ) ).
% \<open>P \<parallel> Q \<longmapsto> \<tau> \<prec> <\<nu>y>(P' \<parallel> Q'[x::=y])\<close>
thf(fact_11_cClose2_Ohyps_I13_J,axiom,
fresh @ name @ pi @ y @ q ).
% cClose2.hyps(13)
thf(fact_12_cClose2_Ohyps_I12_J,axiom,
fresh @ name @ pi @ x @ q ).
% cClose2.hyps(12)
thf(fact_13_cClose2_Ohyps_I7_J,axiom,
fresh @ name @ pi @ x @ p2 ).
% cClose2.hyps(7)
thf(fact_14_cClose2_Ohyps_I8_J,axiom,
fresh @ name @ pi @ y @ p ).
% cClose2.hyps(8)
thf(fact_15_cClose2_Ohyps_I3_J,axiom,
fresh @ name @ pi @ x @ p ).
% cClose2.hyps(3)
thf(fact_16_cClose2_Ohyps_I9_J,axiom,
fresh @ name @ pi @ y @ ( par @ q @ r ) ).
% cClose2.hyps(9)
thf(fact_17_cClose2_Ohyps_I4_J,axiom,
fresh @ name @ pi @ x @ ( par @ q @ r ) ).
% cClose2.hyps(4)
thf(fact_18_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_19_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_20_FreshExt,axiom,
! [X2: name,S: pi,T2: pi,U: pi] :
( ( fresh @ name @ pi @ X2 @ S )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( res @ X2 @ ( par @ ( par @ S @ T2 ) @ U ) ) @ ( par @ S @ ( res @ X2 @ ( par @ T2 @ U ) ) ) ) @ rel ) ) ).
% FreshExt
thf(fact_21_FreshExt_H,axiom,
! [X2: name,U: pi,S: pi,T2: pi] :
( ( fresh @ name @ pi @ X2 @ U )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ ( res @ X2 @ ( par @ S @ T2 ) ) @ U ) @ ( res @ X2 @ ( par @ S @ ( par @ T2 @ U ) ) ) ) @ rel ) ) ).
% FreshExt'
thf(fact_22_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_23_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_24_subst__identity,axiom,
! [P: pi,A2: name] :
( ( subs @ P @ A2 @ A2 )
= P ) ).
% subst_identity
thf(fact_25__092_060open_062x_A_092_060sharp_062_AR_092_060close_062,axiom,
fresh @ name @ pi @ x @ r ).
% \<open>x \<sharp> R\<close>
thf(fact_26__092_060open_062y_A_092_060sharp_062_AR_092_060close_062,axiom,
fresh @ name @ pi @ y @ r ).
% \<open>y \<sharp> R\<close>
thf(fact_27_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_29_cClose2_Ohyps_I11_J,axiom,
fresh @ name @ pi @ y @ qr ).
% cClose2.hyps(11)
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,
! [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_32_name__exists__fresh,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [X2: A] :
~ ! [C3: name] :
~ ( fresh @ name @ A @ C3 @ X2 ) ) ).
% name_exists_fresh
thf(fact_33_freshRes,axiom,
! [A2: name,P: pi] : ( fresh @ name @ pi @ A2 @ ( res @ A2 @ P ) ) ).
% freshRes
thf(fact_34_fresh__fact2,axiom,
! [A2: name,B2: name,P: pi] :
( ( A2 != B2 )
=> ( fresh @ name @ pi @ A2 @ ( subs @ P @ A2 @ B2 ) ) ) ).
% fresh_fact2
thf(fact_35_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_36_forget,axiom,
! [A2: name,P: pi,B2: name] :
( ( fresh @ name @ pi @ A2 @ P )
=> ( ( subs @ P @ A2 @ B2 )
= P ) ) ).
% forget
thf(fact_37_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_38_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_39_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_40_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_41_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_42_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C3: C,D3: D2,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_43_prod__induct6,axiom,
! [F: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C3: C,D3: D2,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F ) @ D3 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_44_prod__induct5,axiom,
! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
( ! [A4: A,B4: B,C3: C,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
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,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
= A5 ) ).
% 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_prod__induct4,axiom,
! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
( ! [A4: A,B4: B,C3: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C3 @ D3 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_49_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_50_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C3: C,D3: D2,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_51_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C3: C,D3: D2,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F ) @ D3 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_52_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
~ ! [A4: A,B4: B,C3: C,D3: D2,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_53_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A4: A,B4: B,C3: C,D3: D2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C3 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_54_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% prod_cases3
thf(fact_55_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_56_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_57_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X4: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_58_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_59_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_60_pi_Odistinct_I85_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( par @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(85)
thf(fact_61_cClose2_Ohyps_I2_J,axiom,
late_transitions @ ( par @ q @ r ) @ ( late_BoundR @ ( late_InputS @ a ) @ x @ qr ) ).
% cClose2.hyps(2)
thf(fact_62_fresh__prod,axiom,
! [A: $tType,X: $tType,B: $tType,A2: X,X2: A,Y: B] :
( ( fresh @ X @ ( product_prod @ A @ B ) @ A2 @ ( product_Pair @ A @ B @ X2 @ Y ) )
= ( ( fresh @ X @ A @ A2 @ X2 )
& ( fresh @ X @ B @ A2 @ Y ) ) ) ).
% fresh_prod
thf(fact_63_cClose2_Ohyps_I1_J,axiom,
late_transitions @ p @ ( late_BoundR @ ( late_BoundOutputS @ a ) @ y @ p2 ) ).
% cClose2.hyps(1)
thf(fact_64_cPar1_Ohyps,axiom,
late_transitions @ q @ ( late_BoundR @ ( late_InputS @ a ) @ x @ q2 ) ).
% cPar1.hyps
thf(fact_65_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_66_fresh__prodD_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,A2: A,X2: B,Y: C] :
( ( fresh @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ X2 @ Y ) )
=> ( fresh @ A @ C @ A2 @ Y ) ) ).
% fresh_prodD(2)
thf(fact_67_fresh__prodD_I1_J,axiom,
! [C: $tType,A: $tType,B: $tType,A2: A,X2: B,Y: C] :
( ( fresh @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ X2 @ Y ) )
=> ( fresh @ A @ B @ A2 @ X2 ) ) ).
% fresh_prodD(1)
thf(fact_68_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_69_resOutputTauTrans,axiom,
! [X2: name,A2: name,B2: name,P: pi,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ B2 @ P ) ) @ ( late_FreeR @ late_TauR @ P2 ) ) ).
% resOutputTauTrans
thf(fact_70_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_71_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_72_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_73_freeRes_Ofresh_I2_J,axiom,
! [A2: name] : ( fresh @ name @ late_freeRes @ A2 @ late_TauR ) ).
% freeRes.fresh(2)
thf(fact_74_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_75_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_76_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_77_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name2: name,Name: name] :
( ( late_InputS @ Name2 )
!= ( late_BoundOutputS @ Name ) ) ).
% Late_Semantics.subject.distinct(1)
thf(fact_78_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_79_outputBoundTrans,axiom,
! [A2: name,B2: name,P: pi,C2: late_subject,X2: name,P2: pi] :
~ ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_BoundR @ C2 @ X2 @ P2 ) ) ).
% outputBoundTrans
thf(fact_80_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_81_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_82_resOutputInputTrans,axiom,
! [X2: name,A2: name,B2: name,P: pi,C2: name,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ B2 @ P ) ) @ ( late_BoundR @ ( late_InputS @ C2 ) @ Y @ P2 ) ) ).
% resOutputInputTrans
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_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_85_outputFreshTrans,axiom,
! [A2: name,B2: name,P: pi,Alpha: late_freeRes,P2: pi] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ Alpha @ P2 ) )
=> ~ ( ( fresh @ name @ late_freeRes @ A2 @ Alpha )
| ( fresh @ name @ late_freeRes @ B2 @ Alpha ) ) ) ).
% outputFreshTrans
thf(fact_86_residual_Odistinct_I1_J,axiom,
! [Subject: late_subject,Name2: name,Pi3: pi,FreeRes: late_freeRes,Pi: pi] :
( ( late_BoundR @ Subject @ Name2 @ Pi3 )
!= ( late_FreeR @ FreeRes @ Pi ) ) ).
% residual.distinct(1)
thf(fact_87_residual_Oinducts,axiom,
! [P: late_residual > $o,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi] : ( P @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) )
=> ( ! [FreeRes2: late_freeRes,Pi4: pi] : ( P @ ( late_FreeR @ FreeRes2 @ Pi4 ) )
=> ( P @ Residual ) ) ) ).
% residual.inducts
thf(fact_88_pi_Odistinct_I29_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(29)
thf(fact_89_pi_Odistinct_I31_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(31)
thf(fact_90_parCasesB_H,axiom,
! [P: pi,Q: pi,B2: late_subject,Y: name,P2: pi] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ B2 @ Y @ P2 ) )
=> ( ! [P4: pi,A4: late_subject,X4: name,P5: pi,Q3: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q3 ) )
=> ( ( ( late_BoundR @ B2 @ Y @ P2 )
= ( late_BoundR @ A4 @ X4 @ ( par @ P5 @ Q3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X4 @ P5 ) )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ( ( fresh @ name @ pi @ X4 @ Q3 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) )
=> ~ ! [Q3: pi,A4: late_subject,X4: name,Q4: pi,P4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q3 ) )
=> ( ( ( late_BoundR @ B2 @ Y @ P2 )
= ( late_BoundR @ A4 @ X4 @ ( par @ P4 @ Q4 ) ) )
=> ( ( late_transitions @ Q3 @ ( late_BoundR @ A4 @ X4 @ Q4 ) )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ( ( fresh @ name @ pi @ X4 @ Q3 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) ) ) ) ).
% parCasesB'
thf(fact_91_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_92_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_93_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_94_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_95_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_96_residual_Ostrong__inducts,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [P: A > late_residual > $o,Z: A,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: A] :
( ( fresh @ name @ A @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi4: pi,Z2: A] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_inducts
thf(fact_97_residual_Ostrong__induct,axiom,
! [N: $tType] :
( ( fs_name @ N )
=> ! [P: N > late_residual > $o,Z: N,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: N] :
( ( fresh @ name @ N @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi4: pi,Z2: N] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_induct
thf(fact_98_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_99_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_100_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_101_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_102_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_103_resCasesF,axiom,
! [X2: name,P: pi,Alpha: late_freeRes,XP: pi,F3: pi > $o] :
( ( late_transitions @ ( res @ X2 @ P ) @ ( late_FreeR @ Alpha @ XP ) )
=> ( ! [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P5 ) )
=> ( ( fresh @ name @ late_freeRes @ X2 @ Alpha )
=> ( F3 @ ( res @ X2 @ P5 ) ) ) )
=> ( F3 @ XP ) ) ) ).
% resCasesF
thf(fact_104_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,P5: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( ( late_FreeR @ Alpha @ P2 )
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P5 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha2 ) ) ) ) ) ).
% resCasesF'
thf(fact_105_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_106_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_107_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_108_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 )
=> ( ! [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P5 ) )
=> ( Prop @ ( par @ P5 @ Q ) ) )
=> ( ! [Q4: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q4 ) )
=> ( Prop @ ( par @ P @ Q4 ) ) )
=> ( Prop @ PQ ) ) ) ) ) ) ).
% parCasesB
thf(fact_109_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_110_outputTauTrans,axiom,
! [A2: name,B2: name,P: pi,P2: pi] :
~ ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ late_TauR @ P2 ) ) ).
% outputTauTrans
thf(fact_111_transitions_OComm1,axiom,
! [P: pi,A2: name,X2: name,P2: pi,Q: pi,B2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P2 ) )
=> ( ( 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 @ P2 @ X2 @ B2 ) @ Q2 ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm1
thf(fact_112_transitions_OComm2,axiom,
! [P: pi,A2: name,B2: name,P2: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P2 ) )
=> ( ( 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 @ P2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P2 @ ( subs @ Q2 @ X2 @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm2
thf(fact_113_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_114_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_115_Late__Semantics_OComm1,axiom,
! [P: pi,A2: name,X2: name,P2: pi,Q: pi,B2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P2 @ X2 @ B2 ) @ Q2 ) ) ) ) ) ).
% Late_Semantics.Comm1
thf(fact_116_Late__Semantics_OComm2,axiom,
! [P: pi,A2: name,B2: name,P2: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ 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 @ B2 ) ) ) ) ) ) ).
% Late_Semantics.Comm2
thf(fact_117_subject_Oexhaust,axiom,
! [Y: late_subject] :
( ! [X12: name] :
( Y
!= ( late_InputS @ X12 ) )
=> ~ ! [X23: name] :
( Y
!= ( late_BoundOutputS @ X23 ) ) ) ).
% subject.exhaust
thf(fact_118_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_119_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_120_name__fresh,axiom,
( ( fresh @ name @ name )
= ( ^ [A6: name,B5: name] : A6 != B5 ) ) ).
% name_fresh
thf(fact_121_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_122_Late__Semantics1_OfreeRes_Odistinct_I1_J,axiom,
! [X11: name,X122: name] :
( ( late_OutputR @ X11 @ X122 )
!= late_TauR ) ).
% Late_Semantics1.freeRes.distinct(1)
thf(fact_123_freeRes_Oexhaust,axiom,
! [Y: late_freeRes] :
( ! [X112: name,X123: name] :
( Y
!= ( late_OutputR @ X112 @ X123 ) )
=> ( Y = late_TauR ) ) ).
% freeRes.exhaust
thf(fact_124_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_125_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_126_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_127_Late__Semantics_OfreeRes_Odistinct_I1_J,axiom,
! [Name1: name,Name22: name] :
( ( late_OutputR @ Name1 @ Name22 )
!= late_TauR ) ).
% Late_Semantics.freeRes.distinct(1)
thf(fact_128_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_129_Output,axiom,
! [A2: name,B2: name,P: pi] : ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P ) ) ).
% Output
thf(fact_130_outputIneqTrans,axiom,
! [A2: name,B2: name,P: pi,C2: name,D: name,P2: pi] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ ( late_OutputR @ C2 @ D ) @ P2 ) )
=> ~ ( ( A2 != C2 )
| ( B2 != D ) ) ) ).
% outputIneqTrans
thf(fact_131_outputCases_H,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ Rs )
=> ~ ! [A4: name,B4: name,P4: pi] :
( ( ( A2 = A4 )
& ( B2 = B4 )
& ( P = P4 ) )
=> ( Rs
!= ( late_FreeR @ ( late_OutputR @ A4 @ B4 ) @ P4 ) ) ) ) ).
% outputCases'
thf(fact_132_outputCases,axiom,
! [A2: name,B2: name,P: pi,Alpha: late_freeRes,P2: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( output @ A2 @ B2 @ P ) @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( ( Alpha
= ( late_OutputR @ A2 @ B2 ) )
=> ( ( P = P2 )
=> ( Prop @ ( late_OutputR @ A2 @ B2 ) @ P ) ) )
=> ( Prop @ Alpha @ P2 ) ) ) ).
% outputCases
thf(fact_133_resOutputOutputTrans,axiom,
! [X2: name,A2: name,P: pi,B2: name,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( output @ A2 @ X2 @ P ) ) @ ( late_FreeR @ ( late_OutputR @ B2 @ Y ) @ P2 ) ) ).
% resOutputOutputTrans
thf(fact_134_Open,axiom,
! [P: pi,A2: name,B2: name,P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P2 ) )
=> ( ( A2 != B2 )
=> ( late_transitions @ ( res @ B2 @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ B2 @ P2 ) ) ) ) ).
% Open
thf(fact_135_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,B4: name] :
( ( ( res @ X5 @ P )
= ( res @ B4 @ P4 ) )
=> ! [P5: pi] :
( ( ( late_BoundR @ A2 @ Y4 @ P2 )
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B4 @ P5 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A4 @ B4 ) @ P5 ) )
=> ( A4 = B4 ) ) ) )
=> ~ ! [P4: pi,A4: late_subject,X4: name,P5: pi,Y3: name] :
( ( ( res @ X5 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( ( late_BoundR @ A2 @ Y4 @ P2 )
= ( late_BoundR @ A4 @ X4 @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X4 @ P5 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X4 )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_136_resCases_H,axiom,
! [X2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X2 @ P ) @ Rs )
=> ( ! [P4: pi,A4: name,B4: name] :
( ( ( res @ X2 @ P )
= ( res @ B4 @ P4 ) )
=> ! [P5: pi] :
( ( Rs
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B4 @ P5 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A4 @ B4 ) @ P5 ) )
=> ( A4 = B4 ) ) ) )
=> ( ! [P4: pi,A4: late_subject,X4: name,P5: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( Rs
= ( late_BoundR @ A4 @ X4 @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X4 @ P5 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X4 )
=> ( ( fresh @ name @ pi @ X4 @ P4 )
=> ~ ( fresh @ name @ late_subject @ X4 @ A4 ) ) ) ) ) ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P5: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( Rs
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P5 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha2 ) ) ) ) ) ) ) ).
% resCases'
thf(fact_137_Late__Semantics1_Osubject_Odistinct_I1_J,axiom,
! [X1: name,X22: name] :
( ( late_InputS @ X1 )
!= ( late_BoundOutputS @ X22 ) ) ).
% Late_Semantics1.subject.distinct(1)
thf(fact_138_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_139_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_140_resTauOutputTrans,axiom,
! [X2: name,P: pi,A2: name,B2: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( tau @ P ) ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P2 ) ) ).
% resTauOutputTrans
thf(fact_141_resInputBoundOutputTrans,axiom,
! [X2: name,A2: name,Y: name,P: pi,B2: name,Z: name,P2: pi] :
~ ( late_transitions @ ( res @ X2 @ ( input @ A2 @ Y @ P ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Z @ P2 ) ) ).
% resInputBoundOutputTrans
thf(fact_142_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_143_simps_I2_J,axiom,
! [P: pi,C2: name,D: name] :
( ( subs @ ( tau @ P ) @ C2 @ D )
= ( tau @ ( subs @ P @ C2 @ D ) ) ) ).
% simps(2)
thf(fact_144_pi_Oinject_I2_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( tau @ X1 )
= ( tau @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(2)
thf(fact_145_pi_Odistinct_I35_J,axiom,
! [Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( input @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(35)
thf(fact_146_pi_Odistinct_I55_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name1 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(55)
thf(fact_147_pi_Odistinct_I57_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(57)
thf(fact_148_pi_Odistinct_I43_J,axiom,
! [Pi3: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(43)
thf(fact_149_pi_Odistinct_I45_J,axiom,
! [Pi3: pi,Name: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(45)
thf(fact_150_pi_Odistinct_I21_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( input @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(21)
thf(fact_151_pi_Odistinct_I19_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( tau @ Pi ) ) ).
% pi.distinct(19)
thf(fact_152_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_153_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_154_tauBoundTrans,axiom,
! [P: pi,A2: late_subject,X2: name,P2: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_BoundR @ A2 @ X2 @ P2 ) ) ).
% tauBoundTrans
thf(fact_155_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_156_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_157_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_158_inputBoundOutputTrans,axiom,
! [A2: name,X2: name,P: pi,B2: name,Y: name,P2: pi] :
~ ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Y @ P2 ) ) ).
% inputBoundOutputTrans
thf(fact_159_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_160_inputIneqTrans,axiom,
! [A2: name,X2: name,P: pi,B2: late_subject,Y: name,P2: pi] :
( ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ B2 @ Y @ P2 ) )
=> ~ ( fresh @ name @ late_subject @ A2 @ B2 ) ) ).
% inputIneqTrans
thf(fact_161_tauOutputTrans,axiom,
! [P: pi,A2: name,B2: name,P2: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P2 ) ) ).
% tauOutputTrans
thf(fact_162_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_163_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_164_Tau,axiom,
! [P: pi] : ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ late_TauR @ P ) ) ).
% Tau
thf(fact_165_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_166_simps_I3_J,axiom,
! [A2: name,B2: name,P: pi,C2: name,D: name] :
( ( subs @ ( output @ A2 @ B2 @ P ) @ C2 @ D )
= ( output @ ( subst_name @ A2 @ C2 @ D ) @ ( subst_name @ B2 @ C2 @ D ) @ ( subs @ P @ C2 @ D ) ) ) ).
% simps(3)
thf(fact_167_substInput,axiom,
! [X2: name,B2: name,C2: name,A2: name,P: pi] :
( ( X2 != B2 )
=> ( ( X2 != C2 )
=> ( ( subs @ ( input @ A2 @ X2 @ P ) @ B2 @ C2 )
= ( input @ ( subst_name @ A2 @ B2 @ C2 ) @ X2 @ ( subs @ P @ B2 @ C2 ) ) ) ) ) ).
% substInput
thf(fact_168_simps_I4_J,axiom,
! [X2: name,A2: name,C2: name,D: name,P: pi] :
( ( X2 != A2 )
=> ( ( X2 != C2 )
=> ( ( X2 != D )
=> ( ( subs @ ( input @ A2 @ X2 @ P ) @ C2 @ D )
= ( input @ ( subst_name @ A2 @ C2 @ D ) @ X2 @ ( subs @ P @ C2 @ D ) ) ) ) ) ) ).
% simps(4)
thf(fact_169_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_170_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_171_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_172_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_173_subst__name__def,axiom,
( subst_name
= ( ^ [A6: name,B5: name,C4: name] : ( if @ name @ ( A6 = B5 ) @ C4 @ A6 ) ) ) ).
% subst_name_def
thf(fact_174_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_175_abs__fresh_I1_J,axiom,
! [X6: $tType] :
( ( fs_name @ X6 )
=> ! [B2: name,A2: name,X2: X6] :
( ( fresh @ name @ ( name > ( noption @ X6 ) ) @ B2 @ ( abs_fun @ name @ X6 @ A2 @ X2 ) )
= ( ( B2 = A2 )
| ( fresh @ name @ X6 @ B2 @ X2 ) ) ) ) ).
% abs_fresh(1)
thf(fact_176_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_177_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_178_inputCases_H,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( input @ A2 @ B2 @ P ) @ Rs )
=> ~ ! [X4: name,A4: name,P4: pi] :
( ( ( A2 = A4 )
& ( ( abs_fun @ name @ pi @ B2 @ P )
= ( abs_fun @ name @ pi @ X4 @ P4 ) ) )
=> ( ( Rs
= ( late_BoundR @ ( late_InputS @ A4 ) @ X4 @ P4 ) )
=> ( X4 = A4 ) ) ) ) ).
% inputCases'
thf(fact_179_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_180_resSimCases,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,X2: name,P: pi,C5: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [Q4: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X2 ) @ Q4 ) )
=> ( ( A4 != X2 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q4 ) @ Rel ) ) ) )
=> ( ! [Q4: pi,A4: late_subject,Y3: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y3 @ Q4 ) )
=> ( ( fresh @ name @ late_subject @ X2 @ A4 )
=> ( ( X2 != Y3 )
=> ( ( fresh @ name @ A @ Y3 @ C5 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y3 @ P6 ) )
& ( strong2129052853vative @ P6 @ ( res @ X2 @ Q4 ) @ A4 @ Y3 @ Rel ) ) ) ) ) )
=> ( ! [Q4: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q4 ) )
=> ( ( fresh @ name @ late_freeRes @ X2 @ Alpha2 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ ( res @ X2 @ Q4 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X2 @ Q ) ) ) ) ) ) ) ).
% resSimCases
thf(fact_181_parSym,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi,Q: pi] :
( ! [R: pi,S2: pi] : ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ R @ S2 ) @ ( par @ S2 @ R ) ) @ Rel )
=> ( ! [R: pi,S2: pi,X4: name] :
( ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ R @ S2 ) @ Rel )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( res @ X4 @ R ) @ ( res @ X4 @ S2 ) ) @ Rel ) )
=> ( strong743114133lation @ ( par @ P @ Q ) @ Rel @ ( par @ Q @ P ) ) ) ) ).
% parSym
thf(fact_182_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 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P5 ) )
& ( strong2129052853vative @ P5 @ Q2 @ A2 @ X2 @ Rel ) ) ) ) ) ).
% simE(1)
thf(fact_183_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 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P5 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P5 @ Q2 ) @ Rel ) ) ) ) ).
% simE(2)
thf(fact_184_simCasesCont,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,P: pi,C5: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [A4: late_subject,X4: name,Q4: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X4 @ Q4 ) )
=> ( ( fresh @ name @ pi @ X4 @ P )
=> ( ( fresh @ name @ pi @ X4 @ Q )
=> ( ( fresh @ name @ late_subject @ X4 @ A4 )
=> ( ( fresh @ name @ A @ X4 @ C5 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ X4 @ P6 ) )
& ( strong2129052853vative @ P6 @ Q4 @ A4 @ X4 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q4: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q4 ) )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q4 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ) ).
% simCasesCont
thf(fact_185_simCases,axiom,
! [Q: pi,P: pi,Rel: set @ ( product_prod @ pi @ pi )] :
( ! [A4: late_subject,Y3: name,Q4: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y3 @ Q4 ) )
=> ( ( fresh @ name @ pi @ Y3 @ P )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y3 @ P6 ) )
& ( strong2129052853vative @ P6 @ Q4 @ A4 @ Y3 @ Rel ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q4: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q4 ) )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q4 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ).
% simCases
thf(fact_186_simulation__def,axiom,
( strong743114133lation
= ( ^ [P7: pi,Rel2: set @ ( product_prod @ pi @ pi ),Q5: pi] :
( ! [A6: late_subject,X3: name,Q6: pi] :
( ( ( late_transitions @ Q5 @ ( late_BoundR @ A6 @ X3 @ Q6 ) )
& ( fresh @ name @ pi @ X3 @ P7 ) )
=> ? [P8: pi] :
( ( late_transitions @ P7 @ ( late_BoundR @ A6 @ X3 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q6 @ A6 @ X3 @ Rel2 ) ) )
& ! [Alpha3: late_freeRes,Q6: pi] :
( ( late_transitions @ Q5 @ ( late_FreeR @ Alpha3 @ Q6 ) )
=> ? [P8: pi] :
( ( late_transitions @ P7 @ ( late_FreeR @ Alpha3 @ P8 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P8 @ Q6 ) @ Rel2 ) ) ) ) ) ) ).
% simulation_def
thf(fact_187_parZeroRight,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ! [Q3: pi] : ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ Q3 @ ( par @ Q3 @ piNil ) ) @ Rel )
=> ( strong743114133lation @ P @ Rel @ ( par @ P @ piNil ) ) ) ).
% parZeroRight
thf(fact_188_parZeroLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ! [Q3: pi] : ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ Q3 @ piNil ) @ Q3 ) @ Rel )
=> ( strong743114133lation @ ( par @ P @ piNil ) @ Rel @ P ) ) ).
% parZeroLeft
thf(fact_189_simps_I5_J,axiom,
! [A2: name,B2: name,P: pi,C2: name,D: name] :
( ( subs @ ( match @ A2 @ B2 @ P ) @ C2 @ D )
= ( match @ ( subst_name @ A2 @ C2 @ D ) @ ( subst_name @ B2 @ C2 @ D ) @ ( subs @ P @ C2 @ D ) ) ) ).
% simps(5)
thf(fact_190_pi_Ofresh_I1_J,axiom,
! [A2: name] : ( fresh @ name @ pi @ A2 @ piNil ) ).
% pi.fresh(1)
thf(fact_191_simps_I1_J,axiom,
! [C2: name,D: name] :
( ( subs @ piNil @ C2 @ D )
= piNil ) ).
% simps(1)
thf(fact_192_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_193_matchNilLeft,axiom,
! [A2: name,B2: name,Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
( ( A2 != B2 )
=> ( strong743114133lation @ piNil @ Rel @ ( match @ A2 @ B2 @ P ) ) ) ).
% matchNilLeft
thf(fact_194_nilSimRight,axiom,
! [P: pi,Rel: set @ ( product_prod @ pi @ pi )] : ( strong743114133lation @ P @ Rel @ piNil ) ).
% nilSimRight
thf(fact_195_pi_Odistinct_I49_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi3 )
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(49)
thf(fact_196_pi_Odistinct_I5_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( input @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(5)
thf(fact_197_matchCases,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual,F3: name > name > $o] :
( ( late_transitions @ ( match @ A2 @ B2 @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A2 = B2 )
=> ( F3 @ A2 @ A2 ) ) )
=> ( F3 @ A2 @ B2 ) ) ) ).
% matchCases
thf(fact_198_matchTrans,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( match @ A2 @ B2 @ P ) @ Rs )
=> ( A2 = B2 ) ) ).
% matchTrans
thf(fact_199_matchCases_H,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( match @ A2 @ B2 @ P ) @ Rs )
=> ~ ! [P4: pi] :
( ( ( B2 = A2 )
& ( P = P4 ) )
=> ~ ( late_transitions @ P4 @ Rs ) ) ) ).
% matchCases'
thf(fact_200_Match,axiom,
! [P: pi,Rs: late_residual,B2: name] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( match @ B2 @ B2 @ P ) @ Rs ) ) ).
% Match
thf(fact_201_nilCases_H,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% nilCases'
thf(fact_202_zeroTrans,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% zeroTrans
thf(fact_203_pi_Odistinct_I65_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( match @ Name1 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(65)
thf(fact_204_pi_Odistinct_I67_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( match @ Name1 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(67)
thf(fact_205_pi_Odistinct_I13_J,axiom,
! [Pi12: pi,Pi22: pi] :
( piNil
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(13)
thf(fact_206_pi_Odistinct_I15_J,axiom,
! [Name: name,Pi: pi] :
( piNil
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(15)
thf(fact_207_pi_Odistinct_I7_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(7)
thf(fact_208_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_209_pi_Odistinct_I3_J,axiom,
! [Pi: pi] :
( piNil
!= ( tau @ Pi ) ) ).
% pi.distinct(3)
thf(fact_210_pi_Odistinct_I37_J,axiom,
! [Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(37)
thf(fact_211_pi_Odistinct_I1_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( output @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(1)
thf(fact_212_pi_Odistinct_I23_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( match @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(23)
thf(fact_213_resZeroTrans,axiom,
! [X2: name,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ piNil ) @ Rs ) ).
% resZeroTrans
thf(fact_214_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_215_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_216_nilSim_I1_J,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( tau @ P ) ) ).
% nilSim(1)
thf(fact_217_simps_I6_J,axiom,
! [A2: name,B2: name,P: pi,C2: name,D: name] :
( ( subs @ ( mismatch @ A2 @ B2 @ P ) @ C2 @ D )
= ( mismatch @ ( subst_name @ A2 @ C2 @ D ) @ ( subst_name @ B2 @ C2 @ D ) @ ( subs @ P @ C2 @ D ) ) ) ).
% simps(6)
thf(fact_218_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_219_pi_Odistinct_I9_J,axiom,
! [Name13: name,Name24: name,Pi: pi] :
( piNil
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(9)
thf(fact_220_pi_Odistinct_I61_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( match @ Name1 @ Name22 @ Pi3 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(61)
thf(fact_221_pi_Odistinct_I25_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(25)
thf(fact_222_pi_Odistinct_I39_J,axiom,
! [Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(39)
thf(fact_223_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_224_pi_Odistinct_I75_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(75)
thf(fact_225_pi_Odistinct_I73_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(73)
thf(fact_226_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_227_mismatchCases_H,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( mismatch @ A2 @ B2 @ P ) @ Rs )
=> ~ ! [P4: pi,A4: name,B4: name] :
( ( ( A2 = A4 )
& ( B2 = B4 )
& ( P = P4 ) )
=> ( ( late_transitions @ P4 @ Rs )
=> ( A4 = B4 ) ) ) ) ).
% mismatchCases'
thf(fact_228_mismatchTrans,axiom,
! [A2: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( mismatch @ A2 @ A2 @ P ) @ Rs ) ).
% mismatchTrans
thf(fact_229_mismatchCases,axiom,
! [A2: name,B2: name,P: pi,Rs: late_residual,F3: name > name > $o] :
( ( late_transitions @ ( mismatch @ A2 @ B2 @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A2 != B2 )
=> ( F3 @ A2 @ B2 ) ) )
=> ( F3 @ A2 @ B2 ) ) ) ).
% mismatchCases
thf(fact_230_pi_Odistinct_I51_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Name13: name,Name24: name,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi3 )
!= ( mismatch @ Name13 @ Name24 @ Pi ) ) ).
% pi.distinct(51)
thf(fact_231_mismatchNilLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),A2: name,P: pi] : ( strong743114133lation @ piNil @ Rel @ ( mismatch @ A2 @ A2 @ P ) ) ).
% mismatchNilLeft
thf(fact_232_pi_Ostrong__induct,axiom,
! [N: $tType] :
( ( fs_name @ N )
=> ! [P: N > pi > $o,Z: N,Pi3: pi] :
( ! [Z2: N] : ( P @ Z2 @ piNil )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( output @ Name12 @ Name23 @ Pi4 ) ) )
=> ( ! [Pi4: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( tau @ Pi4 ) ) )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: N] :
( ( fresh @ name @ N @ Name23 @ Z2 )
=> ( ( fresh @ name @ name @ Name23 @ Name12 )
=> ( ! [Za: N] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( input @ Name12 @ Name23 @ Pi4 ) ) ) ) )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( match @ Name12 @ Name23 @ Pi4 ) ) )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( mismatch @ Name12 @ Name23 @ Pi4 ) ) )
=> ( ! [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,Pi4: pi,Z2: N] :
( ( fresh @ name @ N @ Name3 @ Z2 )
=> ( ! [Za: N] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( res @ Name3 @ Pi4 ) ) ) )
=> ( ! [Pi4: pi,Z2: N] :
( ! [Za: N] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( bang @ Pi4 ) ) )
=> ( P @ Z @ Pi3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_induct
thf(fact_233_pi_Ostrong__inducts,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [P: A > pi > $o,Z: A,Pi3: pi] :
( ! [Z2: A] : ( P @ Z2 @ piNil )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( output @ Name12 @ Name23 @ Pi4 ) ) )
=> ( ! [Pi4: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( tau @ Pi4 ) ) )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: A] :
( ( fresh @ name @ A @ Name23 @ Z2 )
=> ( ( fresh @ name @ name @ Name23 @ Name12 )
=> ( ! [Za: A] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( input @ Name12 @ Name23 @ Pi4 ) ) ) ) )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( match @ Name12 @ Name23 @ Pi4 ) ) )
=> ( ! [Name12: name,Name23: name,Pi4: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( mismatch @ Name12 @ Name23 @ Pi4 ) ) )
=> ( ! [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,Pi4: pi,Z2: A] :
( ( fresh @ name @ A @ Name3 @ Z2 )
=> ( ! [Za: A] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( res @ Name3 @ Pi4 ) ) ) )
=> ( ! [Pi4: pi,Z2: A] :
( ! [Za: A] : ( P @ Za @ Pi4 )
=> ( P @ Z2 @ ( bang @ Pi4 ) ) )
=> ( P @ Z @ Pi3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_inducts
thf(fact_234_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_235_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_236_simps_I10_J,axiom,
! [P: pi,C2: name,D: name] :
( ( subs @ ( bang @ P ) @ C2 @ D )
= ( bang @ ( subs @ P @ C2 @ D ) ) ) ).
% simps(10)
thf(fact_237_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_238_pi_Odistinct_I77_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi3 )
!= ( bang @ Pi ) ) ).
% pi.distinct(77)
thf(fact_239_pi_Odistinct_I71_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( mismatch @ Name1 @ Name22 @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(71)
thf(fact_240_pi_Odistinct_I59_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi: pi] :
( ( input @ Name1 @ Name22 @ Pi3 )
!= ( bang @ Pi ) ) ).
% pi.distinct(59)
thf(fact_241_pi_Odistinct_I53_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name1 @ Name22 @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(53)
thf(fact_242_pi_Odistinct_I47_J,axiom,
! [Pi3: pi,Pi: pi] :
( ( tau @ Pi3 )
!= ( bang @ Pi ) ) ).
% pi.distinct(47)
thf(fact_243_pi_Odistinct_I41_J,axiom,
! [Pi3: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(41)
thf(fact_244_pi_Odistinct_I79_J,axiom,
! [Pi1: pi,Pi2: pi,Pi12: pi,Pi22: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(79)
thf(fact_245_pi_Odistinct_I81_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(81)
thf(fact_246_pi_Odistinct_I27_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(27)
thf(fact_247_pi_Odistinct_I83_J,axiom,
! [Pi1: pi,Pi2: pi,Pi: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( bang @ Pi ) ) ).
% pi.distinct(83)
thf(fact_248_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_249_pi_Oinject_I9_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( bang @ X1 )
= ( bang @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(9)
thf(fact_250_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_251_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_252_Sum1,axiom,
! [P: pi,Rs: late_residual,Q: pi] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum1
thf(fact_253_Sum2,axiom,
! [Q: pi,Rs: late_residual,P: pi] :
( ( late_transitions @ Q @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum2
thf(fact_254_pi_Odistinct_I33_J,axiom,
! [Name1: name,Name22: name,Pi3: pi,Pi: pi] :
( ( output @ Name1 @ Name22 @ Pi3 )
!= ( bang @ Pi ) ) ).
% pi.distinct(33)
thf(fact_255_pi_Odistinct_I89_J,axiom,
! [Name2: name,Pi3: pi,Pi: pi] :
( ( res @ Name2 @ Pi3 )
!= ( bang @ Pi ) ) ).
% pi.distinct(89)
% Type constructors (7)
thf(tcon_Agent_Opi___Agent_Ofs__name,axiom,
fs_name @ pi ).
thf(tcon_HOL_Obool___Agent_Ofs__name_1,axiom,
fs_name @ $o ).
thf(tcon_Agent_Oname___Agent_Ofs__name_2,axiom,
fs_name @ name ).
thf(tcon_Product__Type_Oprod___Agent_Ofs__name_3,axiom,
! [A7: $tType,A8: $tType] :
( ( ( fs_name @ A7 )
& ( fs_name @ A8 ) )
=> ( fs_name @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Ofs__name_4,axiom,
fs_name @ late_freeRes ).
thf(tcon_Late__Semantics_Osubject___Agent_Ofs__name_5,axiom,
fs_name @ late_subject ).
thf(tcon_Late__Semantics_Oresidual___Agent_Ofs__name_6,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,
? [P6: pi] :
( ( late_transitions @ ( par @ ( par @ p @ q ) @ r ) @ ( late_FreeR @ late_TauR @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ ( res @ y @ ( par @ p2 @ ( subs @ ( par @ q2 @ r ) @ x @ y ) ) ) ) @ rel ) ) ).
%------------------------------------------------------------------------------