TPTP Problem File: ITP184^1.p
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%------------------------------------------------------------------------------
% File : ITP184^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_470__3413054_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_470__3413054_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 426 ( 188 unt; 105 typ; 0 def)
% Number of atoms : 875 ( 354 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 4294 ( 173 ~; 11 |; 82 &;3438 @)
% ( 0 <=>; 590 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 9 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 290 ( 290 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 79 usr; 13 con; 0-5 aty)
% Number of variables : 1693 ( 9 ^;1645 !; 39 ?;1693 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:15.857
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_M_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_J,type,
produc1462499409ion_pi: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Late____Semantics__Osubject_J,type,
produc691362330ubject: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Late____Semantics__OfreeRes_J,type,
produc1834551554reeRes: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Agent__Oname_J,type,
produc405342261i_name: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Agent__Opi_J,type,
produc629157763_pi_pi: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Late____Semantics__Osubject_J,type,
produc1359399907ubject: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Late____Semantics__OfreeRes_J,type,
produc355105483reeRes: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Late____Semantics__Osubject_J,type,
produc214645451ubject: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Late____Semantics__OfreeRes_J,type,
produc1357834675reeRes: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Agent__Oname_J,type,
produc1998742700t_name: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Agent__Oname_J,type,
produc2052722628s_name: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Agent__Oname_Mt__Late____Semantics__Osubject_J,type,
produc865358760ubject: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Agent__Oname_Mt__Late____Semantics__OfreeRes_J,type,
produc2008547984reeRes: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Agent__Opi_J,type,
produc140253050ect_pi: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Agent__Opi_J,type,
produc992857746Res_pi: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J_J,type,
set_Pr1834926343_pi_pi: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J,type,
product_prod_pi_pi: $tType ).
thf(ty_n_t__Nominal__Onoption_It__Late____Semantics__Osubject_J,type,
noption_Late_subject: $tType ).
thf(ty_n_t__Nominal__Onoption_It__Late____Semantics__OfreeRes_J,type,
noption_Late_freeRes: $tType ).
thf(ty_n_t__Nominal__Onoption_It__Agent__Oname_J,type,
noption_name: $tType ).
thf(ty_n_t__Nominal__Onoption_It__Agent__Opi_J,type,
noption_pi: $tType ).
thf(ty_n_t__Late____Semantics__Oresidual,type,
late_residual: $tType ).
thf(ty_n_t__Late____Semantics__Osubject,type,
late_subject: $tType ).
thf(ty_n_t__Late____Semantics__OfreeRes,type,
late_freeRes: $tType ).
thf(ty_n_t__Agent__Oname,type,
name: $tType ).
thf(ty_n_t__Agent__Opi,type,
pi: $tType ).
% Explicit typings (79)
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_001t__Agent__Oname,type,
if_name: $o > name > name > name ).
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_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_Otransitions,type,
late_transitions: pi > late_residual > $o ).
thf(sy_c_Nominal_Oabs__fun_001t__Agent__Oname_001t__Agent__Oname,type,
abs_fun_name_name: name > name > name > noption_name ).
thf(sy_c_Nominal_Oabs__fun_001t__Agent__Oname_001t__Agent__Opi,type,
abs_fun_name_pi: name > pi > name > noption_pi ).
thf(sy_c_Nominal_Oabs__fun_001t__Agent__Oname_001t__Late____Semantics__OfreeRes,type,
abs_fu681348580reeRes: name > late_freeRes > name > noption_Late_freeRes ).
thf(sy_c_Nominal_Oabs__fun_001t__Agent__Oname_001t__Late____Semantics__Osubject,type,
abs_fu1904601852ubject: name > late_subject > name > noption_Late_subject ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Oname_J_J,type,
fresh_1093033124n_name: name > ( name > noption_name ) > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J,type,
fresh_543891186ion_pi: name > ( name > noption_pi ) > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Late____Semantics__OfreeRes_J_J,type,
fresh_785364841reeRes: name > ( name > noption_Late_freeRes ) > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Late____Semantics__Osubject_J_J,type,
fresh_728038529ubject: name > ( name > noption_Late_subject ) > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Agent__Oname,type,
fresh_name_name: name > name > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Agent__Opi,type,
fresh_name_pi: name > pi > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Late____Semantics__OfreeRes,type,
fresh_1641682979reeRes: name > late_freeRes > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Late____Semantics__Oresidual,type,
fresh_81631222sidual: name > late_residual > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Late____Semantics__Osubject,type,
fresh_717452603ubject: name > late_subject > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_M_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_J,type,
fresh_225680151ion_pi: name > produc1462499409ion_pi > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Agent__Oname_J,type,
fresh_453496559i_name: name > produc405342261i_name > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Agent__Opi_J,type,
fresh_253309757_pi_pi: name > produc629157763_pi_pi > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Late____Semantics__OfreeRes_J,type,
fresh_1711070152reeRes: name > produc1834551554reeRes > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_I_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_Mt__Late____Semantics__Osubject_J,type,
fresh_567880928ubject: name > produc691362330ubject > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Agent__Oname_Mt__Late____Semantics__OfreeRes_J,type,
fresh_1963366730reeRes: name > produc2008547984reeRes > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Agent__Oname_Mt__Late____Semantics__Osubject_J,type,
fresh_820177506ubject: name > produc865358760ubject > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J,type,
fresh_1627595501_pi_pi: name > product_prod_pi_pi > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Agent__Oname_J,type,
fresh_2007541374s_name: name > produc2052722628s_name > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Agent__Opi_J,type,
fresh_1847501388Res_pi: name > produc992857746Res_pi > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Late____Semantics__OfreeRes_J,type,
fresh_493808761reeRes: name > produc1357834675reeRes > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Late____Semantics__Osubject_J,type,
fresh_1498103185ubject: name > produc214645451ubject > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Agent__Oname_J,type,
fresh_1953561446t_name: name > produc1998742700t_name > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Agent__Opi_J,type,
fresh_994896692ect_pi: name > produc140253050ect_pi > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Late____Semantics__OfreeRes_J,type,
fresh_1638563217reeRes: name > produc355105483reeRes > $o ).
thf(sy_c_Nominal_Ofresh_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Late____Semantics__Osubject_J,type,
fresh_495373993ubject: name > produc1359399907ubject > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J,type,
produc1342893193ion_pi: ( name > noption_pi ) > ( name > noption_pi ) > produc1462499409ion_pi ).
thf(sy_c_Product__Type_OPair_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_001t__Agent__Oname,type,
produc540578151i_name: ( name > noption_pi ) > name > produc405342261i_name ).
thf(sy_c_Product__Type_OPair_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_001t__Agent__Opi,type,
produc1820286517_pi_pi: ( name > noption_pi ) > pi > produc629157763_pi_pi ).
thf(sy_c_Product__Type_OPair_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_001t__Late____Semantics__OfreeRes,type,
produc1505598906reeRes: ( name > noption_pi ) > late_freeRes > produc1834551554reeRes ).
thf(sy_c_Product__Type_OPair_001_062_It__Agent__Oname_Mt__Nominal__Onoption_It__Agent__Opi_J_J_001t__Late____Semantics__Osubject,type,
produc581368530ubject: ( name > noption_pi ) > late_subject > produc691362330ubject ).
thf(sy_c_Product__Type_OPair_001t__Agent__Oname_001t__Late____Semantics__OfreeRes,type,
produc1531804426reeRes: name > late_freeRes > produc2008547984reeRes ).
thf(sy_c_Product__Type_OPair_001t__Agent__Oname_001t__Late____Semantics__Osubject,type,
produc607574050ubject: name > late_subject > produc865358760ubject ).
thf(sy_c_Product__Type_OPair_001t__Agent__Opi_001t__Agent__Opi,type,
product_Pair_pi_pi: pi > pi > product_prod_pi_pi ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Agent__Oname,type,
produc1701355766s_name: late_freeRes > name > produc2052722628s_name ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Agent__Opi,type,
produc1177552196Res_pi: late_freeRes > pi > produc992857746Res_pi ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Late____Semantics__OfreeRes,type,
produc1244821099reeRes: late_freeRes > late_freeRes > produc1357834675reeRes ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Late____Semantics__Osubject,type,
produc320590723ubject: late_freeRes > late_subject > produc214645451ubject ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Agent__Oname,type,
produc1647375838t_name: late_subject > name > produc1998742700t_name ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Agent__Opi,type,
produc324947500ect_pi: late_subject > pi > produc140253050ect_pi ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Late____Semantics__OfreeRes,type,
produc242091907reeRes: late_subject > late_freeRes > produc355105483reeRes ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Late____Semantics__Osubject,type,
produc1465345179ubject: late_subject > late_subject > produc1359399907ubject ).
thf(sy_c_Rel_Oeqvt_001t__Agent__Opi,type,
eqvt_pi: set_Pr1834926343_pi_pi > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J,type,
collec1051790738_pi_pi: ( product_prod_pi_pi > $o ) > set_Pr1834926343_pi_pi ).
thf(sy_c_Strong__Late__Sim_Oderivative,type,
strong2129052853vative: pi > pi > late_subject > name > set_Pr1834926343_pi_pi > $o ).
thf(sy_c_Strong__Late__Sim_Osimulation,type,
strong743114133lation: pi > set_Pr1834926343_pi_pi > pi > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J,type,
member558706512_pi_pi: product_prod_pi_pi > set_Pr1834926343_pi_pi > $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_Rel,type,
rel: set_Pr1834926343_pi_pi ).
thf(sy_v_a____,type,
a: late_subject ).
thf(sy_v_x____,type,
x: name ).
% Relevant facts (317)
thf(fact_0__092_060open_062derivative_A_I_IP_A_092_060parallel_062_AQ_H_J_A_092_060parallel_062_AR_J_A_IP_A_092_060parallel_062_AQ_H_A_092_060parallel_062_AR_J_Aa_Ax_ARel_092_060close_062,axiom,
strong2129052853vative @ ( par @ ( par @ p @ q2 ) @ r ) @ ( par @ p @ ( par @ q2 @ r ) ) @ a @ x @ rel ).
% \<open>derivative ((P \<parallel> Q') \<parallel> R) (P \<parallel> Q' \<parallel> R) a x Rel\<close>
thf(fact_1_cPar1_Ohyps,axiom,
late_transitions @ q @ ( late_BoundR @ a @ x @ q2 ) ).
% cPar1.hyps
thf(fact_2__092_060open_062_IP_A_092_060parallel_062_AQ_J_A_092_060parallel_062_AR_A_092_060longmapsto_062_Aa_092_060guillemotleft_062x_092_060guillemotright_062_A_092_060prec_062_A_IP_A_092_060parallel_062_AQ_H_J_A_092_060parallel_062_AR_092_060close_062,axiom,
late_transitions @ ( par @ ( par @ p @ q ) @ r ) @ ( late_BoundR @ a @ x @ ( par @ ( par @ p @ q2 ) @ r ) ) ).
% \<open>(P \<parallel> Q) \<parallel> R \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> (P \<parallel> Q') \<parallel> R\<close>
thf(fact_3_cPar2_Ohyps,axiom,
late_transitions @ ( par @ q @ r ) @ ( late_BoundR @ a @ x @ qr ) ).
% cPar2.hyps
thf(fact_4_Bound_Ohyps_I1_J,axiom,
late_transitions @ ( par @ p @ ( par @ q @ r ) ) @ ( late_BoundR @ a @ x @ pqr ) ).
% Bound.hyps(1)
thf(fact_5__092_060open_062P_A_092_060parallel_062_AQ_A_092_060longmapsto_062_Aa_092_060guillemotleft_062x_092_060guillemotright_062_A_092_060prec_062_AP_A_092_060parallel_062_AQ_H_092_060close_062,axiom,
late_transitions @ ( par @ p @ q ) @ ( late_BoundR @ a @ x @ ( par @ p @ q2 ) ) ).
% \<open>P \<parallel> Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P \<parallel> Q'\<close>
thf(fact_6_Bound_Ohyps_I2_J,axiom,
fresh_name_pi @ x @ ( par @ ( par @ p @ q ) @ r ) ).
% Bound.hyps(2)
thf(fact_7__092_060open_062x_A_092_060sharp_062_AQ_A_092_060parallel_062_AR_092_060close_062,axiom,
fresh_name_pi @ x @ ( par @ q @ r ) ).
% \<open>x \<sharp> Q \<parallel> R\<close>
thf(fact_8__092_060open_062x_A_092_060sharp_062_AQ_092_060close_062,axiom,
fresh_name_pi @ x @ q ).
% \<open>x \<sharp> Q\<close>
thf(fact_9__092_060open_062x_A_092_060sharp_062_AP_092_060close_062,axiom,
fresh_name_pi @ x @ p ).
% \<open>x \<sharp> P\<close>
thf(fact_10__092_060open_062x_A_092_060sharp_062_AR_092_060close_062,axiom,
fresh_name_pi @ x @ r ).
% \<open>x \<sharp> R\<close>
thf(fact_11_Ass,axiom,
! [S: pi,T: pi,U: pi] : ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( par @ ( par @ S @ T ) @ U ) @ ( par @ S @ ( par @ T @ U ) ) ) @ rel ) ).
% Ass
thf(fact_12_pi_Oinject_I7_J,axiom,
! [X2: pi,X1: pi,Y2: pi,Y1: pi] :
( ( ( par @ X2 @ X1 )
= ( par @ Y2 @ Y1 ) )
= ( ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(7)
thf(fact_13_Late__Semantics_OPar1B,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ X @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P2 @ Q ) ) ) ) ) ).
% Late_Semantics.Par1B
thf(fact_14_Late__Semantics_OPar2B,axiom,
! [Q: pi,A: late_subject,X: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P @ Q2 ) ) ) ) ) ).
% Late_Semantics.Par2B
thf(fact_15_parCasesB,axiom,
! [P: pi,Q: pi,A: late_subject,X: name,PQ: pi,Prop: pi > $o] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ PQ ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ! [P3: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( Prop @ ( par @ P3 @ Q ) ) )
=> ( ! [Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q3 ) )
=> ( Prop @ ( par @ P @ Q3 ) ) )
=> ( Prop @ PQ ) ) ) ) ) ) ).
% parCasesB
thf(fact_16_tauBoundTrans,axiom,
! [P: pi,A: late_subject,X: name,P2: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_BoundR @ A @ X @ P2 ) ) ).
% tauBoundTrans
thf(fact_17_outputBoundTrans,axiom,
! [A: name,B: name,P: pi,C: late_subject,X: name,P2: pi] :
~ ( late_transitions @ ( output @ A @ B @ P ) @ ( late_BoundR @ C @ X @ P2 ) ) ).
% outputBoundTrans
thf(fact_18_Res,axiom,
! [S: pi,T: pi,X: name] :
( ( member558706512_pi_pi @ ( product_Pair_pi_pi @ S @ T ) @ rel )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ X @ S ) @ ( res @ X @ T ) ) @ rel ) ) ).
% Res
thf(fact_19_FreshExt_H,axiom,
! [X: name,U: pi,S: pi,T: pi] :
( ( fresh_name_pi @ X @ U )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( par @ ( res @ X @ ( par @ S @ T ) ) @ U ) @ ( res @ X @ ( par @ S @ ( par @ T @ U ) ) ) ) @ rel ) ) ).
% FreshExt'
thf(fact_20_FreshExt,axiom,
! [X: name,S: pi,T: pi,U: pi] :
( ( fresh_name_pi @ X @ S )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ X @ ( par @ ( par @ S @ T ) @ U ) ) @ ( par @ S @ ( res @ X @ ( par @ T @ U ) ) ) ) @ rel ) ) ).
% FreshExt
thf(fact_21_pi_Ofresh_I8_J,axiom,
! [A: name,X2: pi,X1: pi] :
( ( fresh_name_pi @ A @ ( par @ X2 @ X1 ) )
= ( ( fresh_name_pi @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(8)
thf(fact_22_pi_Ofresh_I3_J,axiom,
! [A: name,X1: pi] :
( ( fresh_name_pi @ A @ ( tau @ X1 ) )
= ( fresh_name_pi @ A @ X1 ) ) ).
% pi.fresh(3)
thf(fact_23_pi_Oinject_I2_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( tau @ X1 )
= ( tau @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(2)
thf(fact_24_pi_Oinject_I1_J,axiom,
! [X3: name,X2: name,X1: pi,Y3: name,Y2: name,Y1: pi] :
( ( ( output @ X3 @ X2 @ X1 )
= ( output @ Y3 @ Y2 @ Y1 ) )
= ( ( X3 = Y3 )
& ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(1)
thf(fact_25_pi_Odistinct_I45_J,axiom,
! [Pi: pi,Name: name,Pi2: pi] :
( ( tau @ Pi )
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(45)
thf(fact_26_pi_Odistinct_I31_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name: name,Pi2: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(31)
thf(fact_27_pi_Odistinct_I19_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi2: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( tau @ Pi2 ) ) ).
% pi.distinct(19)
thf(fact_28_resTrans_I1_J,axiom,
! [X: name,B: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X @ ( output @ X @ B @ P ) ) @ Rs ) ).
% resTrans(1)
thf(fact_29_freshRes,axiom,
! [A: name,P: pi] : ( fresh_name_pi @ A @ ( res @ A @ P ) ) ).
% freshRes
thf(fact_30_name__exists__fresh,axiom,
! [X: late_subject] :
~ ! [C2: name] :
~ ( fresh_717452603ubject @ C2 @ X ) ).
% name_exists_fresh
thf(fact_31_name__exists__fresh,axiom,
! [X: late_freeRes] :
~ ! [C2: name] :
~ ( fresh_1641682979reeRes @ C2 @ X ) ).
% name_exists_fresh
thf(fact_32_name__exists__fresh,axiom,
! [X: name] :
~ ! [C2: name] :
~ ( fresh_name_name @ C2 @ X ) ).
% name_exists_fresh
thf(fact_33_name__exists__fresh,axiom,
! [X: pi] :
~ ! [C2: name] :
~ ( fresh_name_pi @ C2 @ X ) ).
% name_exists_fresh
thf(fact_34_pi_Odistinct_I85_J,axiom,
! [Pi1: pi,Pi22: pi,Name: name,Pi2: pi] :
( ( par @ Pi1 @ Pi22 )
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(85)
thf(fact_35_pi_Odistinct_I29_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( par @ Pi12 @ Pi23 ) ) ).
% pi.distinct(29)
thf(fact_36_pi_Odistinct_I43_J,axiom,
! [Pi: pi,Pi12: pi,Pi23: pi] :
( ( tau @ Pi )
!= ( par @ Pi12 @ Pi23 ) ) ).
% pi.distinct(43)
thf(fact_37_resTauBoundTrans,axiom,
! [X: name,P: pi,A: late_subject,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( tau @ P ) ) @ ( late_BoundR @ A @ Y @ P2 ) ) ).
% resTauBoundTrans
thf(fact_38_freshBoundDerivative_I2_J,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( ( Y != X )
=> ( fresh_name_pi @ Y @ P2 ) ) ) ) ).
% freshBoundDerivative(2)
thf(fact_39_fresh__prod,axiom,
! [A: name,X: late_subject,Y: late_subject] :
( ( fresh_495373993ubject @ A @ ( produc1465345179ubject @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_717452603ubject @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_40_fresh__prod,axiom,
! [A: name,X: late_subject,Y: late_freeRes] :
( ( fresh_1638563217reeRes @ A @ ( produc242091907reeRes @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_41_fresh__prod,axiom,
! [A: name,X: late_subject,Y: name] :
( ( fresh_1953561446t_name @ A @ ( produc1647375838t_name @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_name_name @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_42_fresh__prod,axiom,
! [A: name,X: late_subject,Y: pi] :
( ( fresh_994896692ect_pi @ A @ ( produc324947500ect_pi @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_name_pi @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_43_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: late_subject] :
( ( fresh_1498103185ubject @ A @ ( produc320590723ubject @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_717452603ubject @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_44_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: late_freeRes] :
( ( fresh_493808761reeRes @ A @ ( produc1244821099reeRes @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_45_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: name] :
( ( fresh_2007541374s_name @ A @ ( produc1701355766s_name @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_name_name @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_46_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: pi] :
( ( fresh_1847501388Res_pi @ A @ ( produc1177552196Res_pi @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_name_pi @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_47_fresh__prod,axiom,
! [A: name,X: name,Y: late_subject] :
( ( fresh_820177506ubject @ A @ ( produc607574050ubject @ X @ Y ) )
= ( ( fresh_name_name @ A @ X )
& ( fresh_717452603ubject @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_48_fresh__prod,axiom,
! [A: name,X: name,Y: late_freeRes] :
( ( fresh_1963366730reeRes @ A @ ( produc1531804426reeRes @ X @ Y ) )
= ( ( fresh_name_name @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_49_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: late_subject,Y1: name > noption_pi,Y2: late_subject] :
( ( ( produc581368530ubject @ X1 @ X2 )
= ( produc581368530ubject @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_50_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: late_freeRes,Y1: name > noption_pi,Y2: late_freeRes] :
( ( ( produc1505598906reeRes @ X1 @ X2 )
= ( produc1505598906reeRes @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_51_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: name,Y1: name > noption_pi,Y2: name] :
( ( ( produc540578151i_name @ X1 @ X2 )
= ( produc540578151i_name @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_52_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: pi,Y1: name > noption_pi,Y2: pi] :
( ( ( produc1820286517_pi_pi @ X1 @ X2 )
= ( produc1820286517_pi_pi @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_53_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: name > noption_pi,Y1: name > noption_pi,Y2: name > noption_pi] :
( ( ( produc1342893193ion_pi @ X1 @ X2 )
= ( produc1342893193ion_pi @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_54_prod_Oinject,axiom,
! [X1: pi,X2: pi,Y1: pi,Y2: pi] :
( ( ( product_Pair_pi_pi @ X1 @ X2 )
= ( product_Pair_pi_pi @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_55_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: late_subject,A2: name > noption_pi,B2: late_subject] :
( ( ( produc581368530ubject @ A @ B )
= ( produc581368530ubject @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_56_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: late_freeRes,A2: name > noption_pi,B2: late_freeRes] :
( ( ( produc1505598906reeRes @ A @ B )
= ( produc1505598906reeRes @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_57_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: name,A2: name > noption_pi,B2: name] :
( ( ( produc540578151i_name @ A @ B )
= ( produc540578151i_name @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_58_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: pi,A2: name > noption_pi,B2: pi] :
( ( ( produc1820286517_pi_pi @ A @ B )
= ( produc1820286517_pi_pi @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_59_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: name > noption_pi,A2: name > noption_pi,B2: name > noption_pi] :
( ( ( produc1342893193ion_pi @ A @ B )
= ( produc1342893193ion_pi @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_60_old_Oprod_Oinject,axiom,
! [A: pi,B: pi,A2: pi,B2: pi] :
( ( ( product_Pair_pi_pi @ A @ B )
= ( product_Pair_pi_pi @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_61_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y: late_subject] :
( ( fresh_567880928ubject @ A @ ( produc581368530ubject @ X @ Y ) )
=> ( fresh_717452603ubject @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_62_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y: late_freeRes] :
( ( fresh_1711070152reeRes @ A @ ( produc1505598906reeRes @ X @ Y ) )
=> ( fresh_1641682979reeRes @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_63_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y: name] :
( ( fresh_453496559i_name @ A @ ( produc540578151i_name @ X @ Y ) )
=> ( fresh_name_name @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_64_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y: pi] :
( ( fresh_253309757_pi_pi @ A @ ( produc1820286517_pi_pi @ X @ Y ) )
=> ( fresh_name_pi @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_65_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y: name > noption_pi] :
( ( fresh_225680151ion_pi @ A @ ( produc1342893193ion_pi @ X @ Y ) )
=> ( fresh_543891186ion_pi @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_66_fresh__prodD_I2_J,axiom,
! [A: name,X: pi,Y: pi] :
( ( fresh_1627595501_pi_pi @ A @ ( product_Pair_pi_pi @ X @ Y ) )
=> ( fresh_name_pi @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_67_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y: late_subject] :
( ( fresh_567880928ubject @ A @ ( produc581368530ubject @ X @ Y ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_68_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y: late_freeRes] :
( ( fresh_1711070152reeRes @ A @ ( produc1505598906reeRes @ X @ Y ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_69_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y: name] :
( ( fresh_453496559i_name @ A @ ( produc540578151i_name @ X @ Y ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_70_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y: pi] :
( ( fresh_253309757_pi_pi @ A @ ( produc1820286517_pi_pi @ X @ Y ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_71_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y: name > noption_pi] :
( ( fresh_225680151ion_pi @ A @ ( produc1342893193ion_pi @ X @ Y ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_72_fresh__prodD_I1_J,axiom,
! [A: name,X: pi,Y: pi] :
( ( fresh_1627595501_pi_pi @ A @ ( product_Pair_pi_pi @ X @ Y ) )
=> ( fresh_name_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_73_resOutputInputTrans,axiom,
! [X: name,A: name,B: name,P: pi,C: name,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( output @ A @ B @ P ) ) @ ( late_BoundR @ ( late_InputS @ C ) @ Y @ P2 ) ) ).
% resOutputInputTrans
thf(fact_74_transitions_OResB,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_717452603ubject @ Y @ A )
=> ( ( Y != X )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A @ X @ ( res @ Y @ P2 ) ) ) ) ) ) ) ) ).
% transitions.ResB
thf(fact_75_transitions_OPar2B,axiom,
! [Q: pi,A: late_subject,X: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P @ Q2 ) ) ) ) ) ) ) ).
% transitions.Par2B
thf(fact_76_transitions_OPar1B,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P2 @ Q ) ) ) ) ) ) ) ).
% transitions.Par1B
thf(fact_77_mem__Collect__eq,axiom,
! [A: product_prod_pi_pi,P: product_prod_pi_pi > $o] :
( ( member558706512_pi_pi @ A @ ( collec1051790738_pi_pi @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A3: set_Pr1834926343_pi_pi] :
( ( collec1051790738_pi_pi
@ ^ [X4: product_prod_pi_pi] : ( member558706512_pi_pi @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_79_Collect__cong,axiom,
! [P: product_prod_pi_pi > $o,Q: product_prod_pi_pi > $o] :
( ! [X5: product_prod_pi_pi] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collec1051790738_pi_pi @ P )
= ( collec1051790738_pi_pi @ Q ) ) ) ).
% Collect_cong
thf(fact_80_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,X5: name,P3: pi,Q4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y @ P2 )
= ( late_BoundR @ A4 @ X5 @ ( par @ P3 @ Q4 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X5 @ P3 ) )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ( ( fresh_name_pi @ X5 @ Q4 )
=> ~ ( fresh_717452603ubject @ X5 @ A4 ) ) ) ) ) )
=> ~ ! [Q4: pi,A4: late_subject,X5: name,Q3: pi,P4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y @ P2 )
= ( late_BoundR @ A4 @ X5 @ ( par @ P4 @ Q3 ) ) )
=> ( ( late_transitions @ Q4 @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ( ( fresh_name_pi @ X5 @ Q4 )
=> ~ ( fresh_717452603ubject @ X5 @ A4 ) ) ) ) ) ) ) ) ).
% parCasesB'
thf(fact_81_pi_Ofresh_I2_J,axiom,
! [A: name,X3: name,X2: name,X1: pi] :
( ( fresh_name_pi @ A @ ( output @ X3 @ X2 @ X1 ) )
= ( ( fresh_name_name @ A @ X3 )
& ( fresh_name_name @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(2)
thf(fact_82_subject_Ofresh_I1_J,axiom,
! [A: name,X1: name] :
( ( fresh_717452603ubject @ A @ ( late_InputS @ X1 ) )
= ( fresh_name_name @ A @ X1 ) ) ).
% subject.fresh(1)
thf(fact_83_freshResidual,axiom,
! [P: pi,Rs: late_residual,X: name] :
( ( late_transitions @ P @ Rs )
=> ( ( fresh_name_pi @ X @ P )
=> ( fresh_81631222sidual @ X @ Rs ) ) ) ).
% freshResidual
thf(fact_84_Late__Semantics_OResB,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_717452603ubject @ Y @ A )
=> ( ( Y != X )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A @ X @ ( res @ Y @ P2 ) ) ) ) ) ) ).
% Late_Semantics.ResB
thf(fact_85_freshBoundDerivative_I1_J,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( fresh_717452603ubject @ Y @ A ) ) ) ).
% freshBoundDerivative(1)
thf(fact_86_old_Oprod_Oinducts,axiom,
! [P: produc691362330ubject > $o,Prod: produc691362330ubject] :
( ! [A4: name > noption_pi,B3: late_subject] : ( P @ ( produc581368530ubject @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_87_old_Oprod_Oinducts,axiom,
! [P: produc1834551554reeRes > $o,Prod: produc1834551554reeRes] :
( ! [A4: name > noption_pi,B3: late_freeRes] : ( P @ ( produc1505598906reeRes @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_88_old_Oprod_Oinducts,axiom,
! [P: produc405342261i_name > $o,Prod: produc405342261i_name] :
( ! [A4: name > noption_pi,B3: name] : ( P @ ( produc540578151i_name @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_89_old_Oprod_Oinducts,axiom,
! [P: produc629157763_pi_pi > $o,Prod: produc629157763_pi_pi] :
( ! [A4: name > noption_pi,B3: pi] : ( P @ ( produc1820286517_pi_pi @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_90_old_Oprod_Oinducts,axiom,
! [P: produc1462499409ion_pi > $o,Prod: produc1462499409ion_pi] :
( ! [A4: name > noption_pi,B3: name > noption_pi] : ( P @ ( produc1342893193ion_pi @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_91_old_Oprod_Oinducts,axiom,
! [P: product_prod_pi_pi > $o,Prod: product_prod_pi_pi] :
( ! [A4: pi,B3: pi] : ( P @ ( product_Pair_pi_pi @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_92_old_Oprod_Oexhaust,axiom,
! [Y: produc691362330ubject] :
~ ! [A4: name > noption_pi,B3: late_subject] :
( Y
!= ( produc581368530ubject @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_93_old_Oprod_Oexhaust,axiom,
! [Y: produc1834551554reeRes] :
~ ! [A4: name > noption_pi,B3: late_freeRes] :
( Y
!= ( produc1505598906reeRes @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_94_old_Oprod_Oexhaust,axiom,
! [Y: produc405342261i_name] :
~ ! [A4: name > noption_pi,B3: name] :
( Y
!= ( produc540578151i_name @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_95_old_Oprod_Oexhaust,axiom,
! [Y: produc629157763_pi_pi] :
~ ! [A4: name > noption_pi,B3: pi] :
( Y
!= ( produc1820286517_pi_pi @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_96_old_Oprod_Oexhaust,axiom,
! [Y: produc1462499409ion_pi] :
~ ! [A4: name > noption_pi,B3: name > noption_pi] :
( Y
!= ( produc1342893193ion_pi @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_97_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_pi_pi] :
~ ! [A4: pi,B3: pi] :
( Y
!= ( product_Pair_pi_pi @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_98_Pair__inject,axiom,
! [A: name > noption_pi,B: late_subject,A2: name > noption_pi,B2: late_subject] :
( ( ( produc581368530ubject @ A @ B )
= ( produc581368530ubject @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_99_Pair__inject,axiom,
! [A: name > noption_pi,B: late_freeRes,A2: name > noption_pi,B2: late_freeRes] :
( ( ( produc1505598906reeRes @ A @ B )
= ( produc1505598906reeRes @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_100_Pair__inject,axiom,
! [A: name > noption_pi,B: name,A2: name > noption_pi,B2: name] :
( ( ( produc540578151i_name @ A @ B )
= ( produc540578151i_name @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_101_Pair__inject,axiom,
! [A: name > noption_pi,B: pi,A2: name > noption_pi,B2: pi] :
( ( ( produc1820286517_pi_pi @ A @ B )
= ( produc1820286517_pi_pi @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_102_Pair__inject,axiom,
! [A: name > noption_pi,B: name > noption_pi,A2: name > noption_pi,B2: name > noption_pi] :
( ( ( produc1342893193ion_pi @ A @ B )
= ( produc1342893193ion_pi @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_103_Pair__inject,axiom,
! [A: pi,B: pi,A2: pi,B2: pi] :
( ( ( product_Pair_pi_pi @ A @ B )
= ( product_Pair_pi_pi @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_104_prod__cases,axiom,
! [P: produc691362330ubject > $o,P5: produc691362330ubject] :
( ! [A4: name > noption_pi,B3: late_subject] : ( P @ ( produc581368530ubject @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_105_prod__cases,axiom,
! [P: produc1834551554reeRes > $o,P5: produc1834551554reeRes] :
( ! [A4: name > noption_pi,B3: late_freeRes] : ( P @ ( produc1505598906reeRes @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_106_prod__cases,axiom,
! [P: produc405342261i_name > $o,P5: produc405342261i_name] :
( ! [A4: name > noption_pi,B3: name] : ( P @ ( produc540578151i_name @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_107_prod__cases,axiom,
! [P: produc629157763_pi_pi > $o,P5: produc629157763_pi_pi] :
( ! [A4: name > noption_pi,B3: pi] : ( P @ ( produc1820286517_pi_pi @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_108_prod__cases,axiom,
! [P: produc1462499409ion_pi > $o,P5: produc1462499409ion_pi] :
( ! [A4: name > noption_pi,B3: name > noption_pi] : ( P @ ( produc1342893193ion_pi @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_109_prod__cases,axiom,
! [P: product_prod_pi_pi > $o,P5: product_prod_pi_pi] :
( ! [A4: pi,B3: pi] : ( P @ ( product_Pair_pi_pi @ A4 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_110_surj__pair,axiom,
! [P5: produc691362330ubject] :
? [X5: name > noption_pi,Y4: late_subject] :
( P5
= ( produc581368530ubject @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_111_surj__pair,axiom,
! [P5: produc1834551554reeRes] :
? [X5: name > noption_pi,Y4: late_freeRes] :
( P5
= ( produc1505598906reeRes @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_112_surj__pair,axiom,
! [P5: produc405342261i_name] :
? [X5: name > noption_pi,Y4: name] :
( P5
= ( produc540578151i_name @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_113_surj__pair,axiom,
! [P5: produc629157763_pi_pi] :
? [X5: name > noption_pi,Y4: pi] :
( P5
= ( produc1820286517_pi_pi @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_114_surj__pair,axiom,
! [P5: produc1462499409ion_pi] :
? [X5: name > noption_pi,Y4: name > noption_pi] :
( P5
= ( produc1342893193ion_pi @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_115_surj__pair,axiom,
! [P5: product_prod_pi_pi] :
? [X5: pi,Y4: pi] :
( P5
= ( product_Pair_pi_pi @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_116_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_117_parSym,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi,Q: pi] :
( ! [R: pi,S2: pi] : ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( par @ R @ S2 ) @ ( par @ S2 @ R ) ) @ Rel )
=> ( ! [R: pi,S2: pi,X5: name] :
( ( member558706512_pi_pi @ ( product_Pair_pi_pi @ R @ S2 ) @ Rel )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ X5 @ R ) @ ( res @ X5 @ S2 ) ) @ Rel ) )
=> ( strong743114133lation @ ( par @ P @ Q ) @ Rel @ ( par @ Q @ P ) ) ) ) ).
% parSym
thf(fact_118_inputIneqTrans,axiom,
! [A: name,X: name,P: pi,B: late_subject,Y: name,P2: pi] :
( ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ B @ Y @ P2 ) )
=> ~ ( fresh_717452603ubject @ A @ B ) ) ).
% inputIneqTrans
thf(fact_119_Late__Semantics_OInput,axiom,
! [A: name,X: name,P: pi] : ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P ) ) ).
% Late_Semantics.Input
thf(fact_120_transitions_OInput,axiom,
! [X: name,A: name,P: pi] :
( ( X != A )
=> ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P ) ) ) ).
% transitions.Input
thf(fact_121_name__fresh,axiom,
( fresh_name_name
= ( ^ [A5: name,B4: name] : A5 != B4 ) ) ).
% name_fresh
thf(fact_122_pi_Odistinct_I55_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( input @ Name1 @ Name2 @ Pi )
!= ( par @ Pi12 @ Pi23 ) ) ).
% pi.distinct(55)
thf(fact_123_pi_Odistinct_I57_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name: name,Pi2: pi] :
( ( input @ Name1 @ Name2 @ Pi )
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(57)
thf(fact_124_pi_Odistinct_I21_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( input @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(21)
thf(fact_125_pi_Odistinct_I35_J,axiom,
! [Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( tau @ Pi )
!= ( input @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(35)
thf(fact_126_resTrans_I2_J,axiom,
! [X: name,Y: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X @ ( input @ X @ Y @ P ) ) @ Rs ) ).
% resTrans(2)
thf(fact_127_simE_I1_J,axiom,
! [P: pi,Rel: set_Pr1834926343_pi_pi,Q: pi,A: late_subject,X: name,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ? [P3: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
& ( strong2129052853vative @ P3 @ Q2 @ A @ X @ Rel ) ) ) ) ) ).
% simE(1)
thf(fact_128_resInputBoundOutputTrans,axiom,
! [X: name,A: name,Y: name,P: pi,B: name,Z: name,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( input @ A @ Y @ P ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Z @ P2 ) ) ).
% resInputBoundOutputTrans
thf(fact_129_simulation__def,axiom,
( strong743114133lation
= ( ^ [P6: pi,Rel2: set_Pr1834926343_pi_pi,Q5: pi] :
( ! [A5: late_subject,X4: name,Q6: pi] :
( ( ( late_transitions @ Q5 @ ( late_BoundR @ A5 @ X4 @ Q6 ) )
& ( fresh_name_pi @ X4 @ P6 ) )
=> ? [P7: pi] :
( ( late_transitions @ P6 @ ( late_BoundR @ A5 @ X4 @ P7 ) )
& ( strong2129052853vative @ P7 @ Q6 @ A5 @ X4 @ Rel2 ) ) )
& ! [Alpha: late_freeRes,Q6: pi] :
( ( late_transitions @ Q5 @ ( late_FreeR @ Alpha @ Q6 ) )
=> ? [P7: pi] :
( ( late_transitions @ P6 @ ( late_FreeR @ Alpha @ P7 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P7 @ Q6 ) @ Rel2 ) ) ) ) ) ) ).
% simulation_def
thf(fact_130_simCases,axiom,
! [Q: pi,P: pi,Rel: set_Pr1834926343_pi_pi] :
( ! [A4: late_subject,Y4: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y4 @ Q3 ) )
=> ( ( fresh_name_pi @ Y4 @ P )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y4 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q3 @ A4 @ Y4 @ Rel ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ).
% simCases
thf(fact_131_residual_Ofresh_I1_J,axiom,
! [A: name,X3: late_subject,X1: name,X2: pi] :
( ( fresh_81631222sidual @ A @ ( late_BoundR @ X3 @ X1 @ X2 ) )
= ( ( fresh_717452603ubject @ A @ X3 )
& ( fresh_543891186ion_pi @ A @ ( abs_fun_name_pi @ X1 @ X2 ) ) ) ) ).
% residual.fresh(1)
thf(fact_132_inputCases_H,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( input @ A @ B @ P ) @ Rs )
=> ~ ! [X5: name,A4: name,P4: pi] :
( ( ( A = A4 )
& ( ( abs_fun_name_pi @ B @ P )
= ( abs_fun_name_pi @ X5 @ P4 ) ) )
=> ( ( Rs
= ( late_BoundR @ ( late_InputS @ A4 ) @ X5 @ P4 ) )
=> ( X5 = A4 ) ) ) ) ).
% inputCases'
thf(fact_133_Late__Semantics1_Osubject_Oinject_I2_J,axiom,
! [X2: name,Y2: name] :
( ( ( late_BoundOutputS @ X2 )
= ( late_BoundOutputS @ Y2 ) )
= ( X2 = Y2 ) ) ).
% Late_Semantics1.subject.inject(2)
thf(fact_134_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_135_subject_Ofresh_I2_J,axiom,
! [A: name,X1: name] :
( ( fresh_717452603ubject @ A @ ( late_BoundOutputS @ X1 ) )
= ( fresh_name_name @ A @ X1 ) ) ).
% subject.fresh(2)
thf(fact_136_residual_Ofresh_I2_J,axiom,
! [A: name,X2: late_freeRes,X1: pi] :
( ( fresh_81631222sidual @ A @ ( late_FreeR @ X2 @ X1 ) )
= ( ( fresh_1641682979reeRes @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% residual.fresh(2)
thf(fact_137_pi_Ofresh_I9_J,axiom,
! [A: name,X1: name,X2: pi] :
( ( fresh_name_pi @ A @ ( res @ X1 @ X2 ) )
= ( fresh_543891186ion_pi @ A @ ( abs_fun_name_pi @ X1 @ X2 ) ) ) ).
% pi.fresh(9)
thf(fact_138_pi_Ofresh_I4_J,axiom,
! [A: name,X3: name,X1: name,X2: pi] :
( ( fresh_name_pi @ A @ ( input @ X3 @ X1 @ X2 ) )
= ( ( fresh_name_name @ A @ X3 )
& ( fresh_543891186ion_pi @ A @ ( abs_fun_name_pi @ X1 @ X2 ) ) ) ) ).
% pi.fresh(4)
thf(fact_139_residual_Oinject_I2_J,axiom,
! [X2: late_freeRes,X1: pi,Y2: late_freeRes,Y1: pi] :
( ( ( late_FreeR @ X2 @ X1 )
= ( late_FreeR @ Y2 @ Y1 ) )
= ( ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% residual.inject(2)
thf(fact_140_abs__fun__eq1,axiom,
! [A: name,X: name,Y: name] :
( ( ( abs_fun_name_name @ A @ X )
= ( abs_fun_name_name @ A @ Y ) )
= ( X = Y ) ) ).
% abs_fun_eq1
thf(fact_141_abs__fun__eq1,axiom,
! [A: name,X: late_freeRes,Y: late_freeRes] :
( ( ( abs_fu681348580reeRes @ A @ X )
= ( abs_fu681348580reeRes @ A @ Y ) )
= ( X = Y ) ) ).
% abs_fun_eq1
thf(fact_142_abs__fun__eq1,axiom,
! [A: name,X: late_subject,Y: late_subject] :
( ( ( abs_fu1904601852ubject @ A @ X )
= ( abs_fu1904601852ubject @ A @ Y ) )
= ( X = Y ) ) ).
% abs_fun_eq1
thf(fact_143_abs__fun__eq1,axiom,
! [A: name,X: pi,Y: pi] :
( ( ( abs_fun_name_pi @ A @ X )
= ( abs_fun_name_pi @ A @ Y ) )
= ( X = Y ) ) ).
% abs_fun_eq1
thf(fact_144_pi_Oinject_I8_J,axiom,
! [X1: name,X2: pi,Y1: name,Y2: pi] :
( ( ( res @ X1 @ X2 )
= ( res @ Y1 @ Y2 ) )
= ( ( abs_fun_name_pi @ X1 @ X2 )
= ( abs_fun_name_pi @ Y1 @ Y2 ) ) ) ).
% pi.inject(8)
thf(fact_145_residual_Oinject_I1_J,axiom,
! [X3: late_subject,X1: name,X2: pi,Y3: late_subject,Y1: name,Y2: pi] :
( ( ( late_BoundR @ X3 @ X1 @ X2 )
= ( late_BoundR @ Y3 @ Y1 @ Y2 ) )
= ( ( X3 = Y3 )
& ( ( abs_fun_name_pi @ X1 @ X2 )
= ( abs_fun_name_pi @ Y1 @ Y2 ) ) ) ) ).
% residual.inject(1)
thf(fact_146_residual_Odistinct_I1_J,axiom,
! [Subject: late_subject,Name3: name,Pi: pi,FreeRes: late_freeRes,Pi2: pi] :
( ( late_BoundR @ Subject @ Name3 @ Pi )
!= ( late_FreeR @ FreeRes @ Pi2 ) ) ).
% residual.distinct(1)
thf(fact_147_residual_Oinducts,axiom,
! [P: late_residual > $o,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi] : ( P @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi] : ( P @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Residual ) ) ) ).
% residual.inducts
thf(fact_148_name__fresh__abs,axiom,
! [B: name,A: name,X: late_subject] :
( ( fresh_728038529ubject @ B @ ( abs_fu1904601852ubject @ A @ X ) )
= ( ( B = A )
| ( fresh_717452603ubject @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_149_name__fresh__abs,axiom,
! [B: name,A: name,X: late_freeRes] :
( ( fresh_785364841reeRes @ B @ ( abs_fu681348580reeRes @ A @ X ) )
= ( ( B = A )
| ( fresh_1641682979reeRes @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_150_name__fresh__abs,axiom,
! [B: name,A: name,X: name] :
( ( fresh_1093033124n_name @ B @ ( abs_fun_name_name @ A @ X ) )
= ( ( B = A )
| ( fresh_name_name @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_151_name__fresh__abs,axiom,
! [B: name,A: name,X: pi] :
( ( fresh_543891186ion_pi @ B @ ( abs_fun_name_pi @ A @ X ) )
= ( ( B = A )
| ( fresh_name_pi @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_152_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: late_subject] :
( ( fresh_728038529ubject @ B @ ( abs_fu1904601852ubject @ A @ X ) )
= ( ( B = A )
| ( fresh_717452603ubject @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_153_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: late_freeRes] :
( ( fresh_785364841reeRes @ B @ ( abs_fu681348580reeRes @ A @ X ) )
= ( ( B = A )
| ( fresh_1641682979reeRes @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_154_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: name] :
( ( fresh_1093033124n_name @ B @ ( abs_fun_name_name @ A @ X ) )
= ( ( B = A )
| ( fresh_name_name @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_155_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: pi] :
( ( fresh_543891186ion_pi @ B @ ( abs_fun_name_pi @ A @ X ) )
= ( ( B = A )
| ( fresh_name_pi @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_156_pi_Oinject_I3_J,axiom,
! [X3: name,X1: name,X2: pi,Y3: name,Y1: name,Y2: pi] :
( ( ( input @ X3 @ X1 @ X2 )
= ( input @ Y3 @ Y1 @ Y2 ) )
= ( ( X3 = Y3 )
& ( ( abs_fun_name_pi @ X1 @ X2 )
= ( abs_fun_name_pi @ Y1 @ Y2 ) ) ) ) ).
% pi.inject(3)
thf(fact_157_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name3: name,Name: name] :
( ( late_InputS @ Name3 )
!= ( late_BoundOutputS @ Name ) ) ).
% Late_Semantics.subject.distinct(1)
thf(fact_158_subject_Oinducts,axiom,
! [P: late_subject > $o,Subject: late_subject] :
( ! [Name4: name] : ( P @ ( late_InputS @ Name4 ) )
=> ( ! [Name4: name] : ( P @ ( late_BoundOutputS @ Name4 ) )
=> ( P @ Subject ) ) ) ).
% subject.inducts
thf(fact_159_subject_Oexhaust,axiom,
! [Y: late_subject] :
( ! [X12: name] :
( Y
!= ( late_InputS @ X12 ) )
=> ~ ! [X22: name] :
( Y
!= ( late_BoundOutputS @ X22 ) ) ) ).
% subject.exhaust
thf(fact_160_Late__Semantics1_Osubject_Odistinct_I1_J,axiom,
! [X1: name,X2: name] :
( ( late_InputS @ X1 )
!= ( late_BoundOutputS @ X2 ) ) ).
% Late_Semantics1.subject.distinct(1)
thf(fact_161_Par1F,axiom,
! [P: pi,Alpha3: late_freeRes,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha3 @ ( par @ P2 @ Q ) ) ) ) ).
% Par1F
thf(fact_162_Par2F,axiom,
! [Q: pi,Alpha3: late_freeRes,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha3 @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha3 @ ( par @ P @ Q2 ) ) ) ) ).
% Par2F
thf(fact_163_ResF,axiom,
! [P: pi,Alpha3: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ( ( fresh_1641682979reeRes @ Y @ Alpha3 )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_FreeR @ Alpha3 @ ( res @ Y @ P2 ) ) ) ) ) ).
% ResF
thf(fact_164_resCasesF_H,axiom,
! [X: name,P: pi,Alpha3: late_freeRes,P2: pi] :
( ( late_transitions @ ( res @ X @ P ) @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P3: pi,Y4: name] :
( ( ( res @ X @ P )
= ( res @ Y4 @ P4 ) )
=> ( ( ( late_FreeR @ Alpha3 @ P2 )
= ( late_FreeR @ Alpha2 @ ( res @ Y4 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P3 ) )
=> ~ ( fresh_1641682979reeRes @ Y4 @ Alpha2 ) ) ) ) ) ).
% resCasesF'
thf(fact_165_resCasesF,axiom,
! [X: name,P: pi,Alpha3: late_freeRes,XP: pi,F: pi > $o] :
( ( late_transitions @ ( res @ X @ P ) @ ( late_FreeR @ Alpha3 @ XP ) )
=> ( ! [P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha3 @ P3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha3 )
=> ( F @ ( res @ X @ P3 ) ) ) )
=> ( F @ XP ) ) ) ).
% resCasesF
thf(fact_166_freshFreeDerivative_I1_J,axiom,
! [P: pi,Alpha3: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( fresh_1641682979reeRes @ Y @ Alpha3 ) ) ) ).
% freshFreeDerivative(1)
thf(fact_167_freshFreeDerivative_I2_J,axiom,
! [P: pi,Alpha3: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( fresh_name_pi @ Y @ P2 ) ) ) ).
% freshFreeDerivative(2)
thf(fact_168_inputFreeTrans,axiom,
! [A: name,X: name,P: pi,Alpha3: late_freeRes,P2: pi] :
~ ( late_transitions @ ( input @ A @ X @ P ) @ ( late_FreeR @ Alpha3 @ P2 ) ) ).
% inputFreeTrans
thf(fact_169_outputFreshTrans,axiom,
! [A: name,B: name,P: pi,Alpha3: late_freeRes,P2: pi] :
( ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ~ ( ( fresh_1641682979reeRes @ A @ Alpha3 )
| ( fresh_1641682979reeRes @ B @ Alpha3 ) ) ) ).
% outputFreshTrans
thf(fact_170_simE_I2_J,axiom,
! [P: pi,Rel: set_Pr1834926343_pi_pi,Q: pi,Alpha3: late_freeRes,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_FreeR @ Alpha3 @ Q2 ) )
=> ? [P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha3 @ P3 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P3 @ Q2 ) @ Rel ) ) ) ) ).
% simE(2)
thf(fact_171_resInputFreeTrans,axiom,
! [X: name,A: name,Y: name,P: pi,Alpha3: late_freeRes,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( input @ A @ Y @ P ) ) @ ( late_FreeR @ Alpha3 @ P2 ) ) ).
% resInputFreeTrans
thf(fact_172_residual_Ostrong__inducts,axiom,
! [P: late_subject > late_residual > $o,Z: late_subject,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: late_subject] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_173_residual_Ostrong__inducts,axiom,
! [P: late_freeRes > late_residual > $o,Z: late_freeRes,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: late_freeRes] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_174_residual_Ostrong__inducts,axiom,
! [P: name > late_residual > $o,Z: name,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: name] :
( ( fresh_name_name @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: name] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_175_residual_Ostrong__inducts,axiom,
! [P: pi > late_residual > $o,Z: pi,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: pi] :
( ( fresh_name_pi @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: pi] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_176_residual_Ostrong__induct,axiom,
! [P: late_subject > late_residual > $o,Z: late_subject,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: late_subject] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_177_residual_Ostrong__induct,axiom,
! [P: late_freeRes > late_residual > $o,Z: late_freeRes,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: late_freeRes] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_178_residual_Ostrong__induct,axiom,
! [P: name > late_residual > $o,Z: name,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: name] :
( ( fresh_name_name @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: name] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_179_residual_Ostrong__induct,axiom,
! [P: pi > late_residual > $o,Z: pi,Residual: late_residual] :
( ! [Subject2: late_subject,Name4: name,Pi3: pi,Z2: pi] :
( ( fresh_name_pi @ Name4 @ Z2 )
=> ( ( fresh_717452603ubject @ Name4 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name4 @ Pi3 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi3: pi,Z2: pi] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi3 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_180_inputBoundOutputTrans,axiom,
! [A: name,X: name,P: pi,B: name,Y: name,P2: pi] :
~ ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Y @ P2 ) ) ).
% inputBoundOutputTrans
thf(fact_181_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: late_subject] :
( ( eqvt_pi @ Rel )
=> ( ! [A4: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A4 )
=> ( ( fresh_717452603ubject @ X5 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q3 @ A4 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_182_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: late_freeRes] :
( ( eqvt_pi @ Rel )
=> ( ! [A4: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A4 )
=> ( ( fresh_1641682979reeRes @ X5 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q3 @ A4 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_183_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: name] :
( ( eqvt_pi @ Rel )
=> ( ! [A4: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A4 )
=> ( ( fresh_name_name @ X5 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q3 @ A4 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_184_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: pi] :
( ( eqvt_pi @ Rel )
=> ( ! [A4: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A4 )
=> ( ( fresh_name_pi @ X5 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q3 @ A4 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_185_resCases_H,axiom,
! [X: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X @ P ) @ Rs )
=> ( ! [P4: pi,A4: name,B3: name] :
( ( ( res @ X @ 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,X5: name,P3: pi,Y4: name] :
( ( ( res @ X @ P )
= ( res @ Y4 @ P4 ) )
=> ( ( Rs
= ( late_BoundR @ A4 @ X5 @ ( res @ Y4 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X5 @ P3 ) )
=> ( ( fresh_717452603ubject @ Y4 @ A4 )
=> ( ( Y4 != X5 )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ~ ( fresh_717452603ubject @ X5 @ A4 ) ) ) ) ) ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P3: pi,Y4: name] :
( ( ( res @ X @ P )
= ( res @ Y4 @ P4 ) )
=> ( ( Rs
= ( late_FreeR @ Alpha2 @ ( res @ Y4 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P3 ) )
=> ~ ( fresh_1641682979reeRes @ Y4 @ Alpha2 ) ) ) ) ) ) ) ).
% resCases'
thf(fact_186_resCasesB_H,axiom,
! [X6: name,P: pi,A: late_subject,Y5: name,P2: pi] :
( ( late_transitions @ ( res @ X6 @ P ) @ ( late_BoundR @ A @ Y5 @ P2 ) )
=> ( ! [P4: pi,A4: name,B3: name] :
( ( ( res @ X6 @ P )
= ( res @ B3 @ P4 ) )
=> ! [P3: pi] :
( ( ( late_BoundR @ A @ Y5 @ 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,X5: name,P3: pi,Y4: name] :
( ( ( res @ X6 @ P )
= ( res @ Y4 @ P4 ) )
=> ( ( ( late_BoundR @ A @ Y5 @ P2 )
= ( late_BoundR @ A4 @ X5 @ ( res @ Y4 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A4 @ X5 @ P3 ) )
=> ( ( fresh_717452603ubject @ Y4 @ A4 )
=> ( ( Y4 != X5 )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ~ ( fresh_717452603ubject @ X5 @ A4 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_187_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: late_subject] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X ) @ Q3 ) )
=> ( ( A4 != X )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A4: late_subject,Y4: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y4 @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A4 )
=> ( ( X != Y4 )
=> ( ( fresh_717452603ubject @ Y4 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y4 @ P8 ) )
& ( strong2129052853vative @ P8 @ ( res @ X @ Q3 ) @ A4 @ Y4 @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_188_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: late_freeRes] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X ) @ Q3 ) )
=> ( ( A4 != X )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A4: late_subject,Y4: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y4 @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A4 )
=> ( ( X != Y4 )
=> ( ( fresh_1641682979reeRes @ Y4 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y4 @ P8 ) )
& ( strong2129052853vative @ P8 @ ( res @ X @ Q3 ) @ A4 @ Y4 @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_189_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: name] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X ) @ Q3 ) )
=> ( ( A4 != X )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A4: late_subject,Y4: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y4 @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A4 )
=> ( ( X != Y4 )
=> ( ( fresh_name_name @ Y4 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y4 @ P8 ) )
& ( strong2129052853vative @ P8 @ ( res @ X @ Q3 ) @ A4 @ Y4 @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_190_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: pi] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X ) @ Q3 ) )
=> ( ( A4 != X )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A4: late_subject,Y4: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y4 @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A4 )
=> ( ( X != Y4 )
=> ( ( fresh_name_pi @ Y4 @ C3 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y4 @ P8 ) )
& ( strong2129052853vative @ P8 @ ( res @ X @ Q3 ) @ A4 @ Y4 @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_191_Open,axiom,
! [P: pi,A: name,B: name,P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) )
=> ( ( A != B )
=> ( late_transitions @ ( res @ B @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ B @ P2 ) ) ) ) ).
% Open
thf(fact_192_resOutputTauTrans,axiom,
! [X: name,A: name,B: name,P: pi,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( output @ A @ B @ P ) ) @ ( late_FreeR @ late_TauR @ P2 ) ) ).
% resOutputTauTrans
thf(fact_193_Late__Semantics_OfreeRes_Oinject,axiom,
! [X2: name,X1: name,Y2: name,Y1: name] :
( ( ( late_OutputR @ X2 @ X1 )
= ( late_OutputR @ Y2 @ Y1 ) )
= ( ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% Late_Semantics.freeRes.inject
thf(fact_194_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_195_freeRes_Ofresh_I2_J,axiom,
! [A: name] : ( fresh_1641682979reeRes @ A @ late_TauR ) ).
% freeRes.fresh(2)
thf(fact_196_freeRes_Ofresh_I1_J,axiom,
! [A: name,X2: name,X1: name] :
( ( fresh_1641682979reeRes @ A @ ( late_OutputR @ X2 @ X1 ) )
= ( ( fresh_name_name @ A @ X2 )
& ( fresh_name_name @ A @ X1 ) ) ) ).
% freeRes.fresh(1)
thf(fact_197_freeRes_Oinducts,axiom,
! [P: late_freeRes > $o,FreeRes3: late_freeRes] :
( ! [Name13: name,Name23: name] : ( P @ ( late_OutputR @ Name13 @ Name23 ) )
=> ( ( P @ late_TauR )
=> ( P @ FreeRes3 ) ) ) ).
% freeRes.inducts
thf(fact_198_Late__Semantics_OfreeRes_Odistinct_I1_J,axiom,
! [Name1: name,Name2: name] :
( ( late_OutputR @ Name1 @ Name2 )
!= late_TauR ) ).
% Late_Semantics.freeRes.distinct(1)
thf(fact_199_freeRes_Oexhaust,axiom,
! [Y: late_freeRes] :
( ! [X112: name,X123: name] :
( Y
!= ( late_OutputR @ X112 @ X123 ) )
=> ( Y = late_TauR ) ) ).
% freeRes.exhaust
thf(fact_200_Late__Semantics1_OfreeRes_Odistinct_I1_J,axiom,
! [X11: name,X122: name] :
( ( late_OutputR @ X11 @ X122 )
!= late_TauR ) ).
% Late_Semantics1.freeRes.distinct(1)
thf(fact_201_outputCases,axiom,
! [A: name,B: name,P: pi,Alpha3: late_freeRes,P2: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ( ( ( Alpha3
= ( late_OutputR @ A @ B ) )
=> ( ( P = P2 )
=> ( Prop @ ( late_OutputR @ A @ B ) @ P ) ) )
=> ( Prop @ Alpha3 @ P2 ) ) ) ).
% outputCases
thf(fact_202_outputCases_H,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( output @ A @ B @ P ) @ Rs )
=> ~ ! [A4: name,B3: name,P4: pi] :
( ( ( A = A4 )
& ( B = B3 )
& ( P = P4 ) )
=> ( Rs
!= ( late_FreeR @ ( late_OutputR @ A4 @ B3 ) @ P4 ) ) ) ) ).
% outputCases'
thf(fact_203_outputIneqTrans,axiom,
! [A: name,B: name,P: pi,C: name,D: name,P2: pi] :
( ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ ( late_OutputR @ C @ D ) @ P2 ) )
=> ~ ( ( A != C )
| ( B != D ) ) ) ).
% outputIneqTrans
thf(fact_204_Output,axiom,
! [A: name,B: name,P: pi] : ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P ) ) ).
% Output
thf(fact_205_tauOutputTrans,axiom,
! [P: pi,A: name,B: name,P2: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) ) ).
% tauOutputTrans
thf(fact_206_outputTauTrans,axiom,
! [A: name,B: name,P: pi,P2: pi] :
~ ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ late_TauR @ P2 ) ) ).
% outputTauTrans
thf(fact_207_tauCases,axiom,
! [P: pi,Alpha3: late_freeRes,P2: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ Alpha3 @ P2 ) )
=> ( ( ( Alpha3 = late_TauR )
=> ( ( P = P2 )
=> ( Prop @ late_TauR @ P ) ) )
=> ( Prop @ Alpha3 @ P2 ) ) ) ).
% tauCases
thf(fact_208_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_209_Tau,axiom,
! [P: pi] : ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ late_TauR @ P ) ) ).
% Tau
thf(fact_210_resOutputOutputTrans,axiom,
! [X: name,A: name,P: pi,B: name,Y: name,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( output @ A @ X @ P ) ) @ ( late_FreeR @ ( late_OutputR @ B @ Y ) @ P2 ) ) ).
% resOutputOutputTrans
thf(fact_211_resTauOutputTrans,axiom,
! [X: name,P: pi,A: name,B: name,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( tau @ P ) ) @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) ) ).
% resTauOutputTrans
thf(fact_212_Late__Semantics_OClose1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ Q2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P2 @ X @ Y ) @ Q2 ) ) ) ) ) ) ) ).
% Late_Semantics.Close1
thf(fact_213_Late__Semantics_OClose2,axiom,
! [P: pi,A: name,Y: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ Y @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P2 @ ( subs @ Q2 @ X @ Y ) ) ) ) ) ) ) ) ).
% Late_Semantics.Close2
thf(fact_214_transitions_OClose1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_name_pi @ Y @ P )
=> ( ( fresh_name_pi @ Y @ Q )
=> ( ( X != A )
=> ( ( fresh_name_pi @ X @ Q2 )
=> ( ( Y != A )
=> ( ( fresh_name_pi @ Y @ P2 )
=> ( ( X != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P2 @ X @ Y ) @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close1
thf(fact_215_transitions_OClose2,axiom,
! [P: pi,A: name,Y: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_name_pi @ Y @ P )
=> ( ( fresh_name_pi @ Y @ Q )
=> ( ( X != A )
=> ( ( fresh_name_pi @ X @ P2 )
=> ( ( Y != A )
=> ( ( fresh_name_pi @ Y @ Q2 )
=> ( ( X != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P2 @ ( subs @ Q2 @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close2
thf(fact_216_subst__identity,axiom,
! [P: pi,A: name] :
( ( subs @ P @ A @ A )
= P ) ).
% subst_identity
thf(fact_217_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_218_simps_I9_J,axiom,
! [X: name,C: name,D: name,P: pi] :
( ( X != C )
=> ( ( X != D )
=> ( ( subs @ ( res @ X @ P ) @ C @ D )
= ( res @ X @ ( subs @ P @ C @ D ) ) ) ) ) ).
% simps(9)
thf(fact_219_substTrans,axiom,
! [B: name,P: pi,A: name,C: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( subs @ ( subs @ P @ A @ B ) @ B @ C )
= ( subs @ P @ A @ C ) ) ) ).
% substTrans
thf(fact_220_simps_I2_J,axiom,
! [P: pi,C: name,D: name] :
( ( subs @ ( tau @ P ) @ C @ D )
= ( tau @ ( subs @ P @ C @ D ) ) ) ).
% simps(2)
thf(fact_221_forget,axiom,
! [A: name,P: pi,B: name] :
( ( fresh_name_pi @ A @ P )
=> ( ( subs @ P @ A @ B )
= P ) ) ).
% forget
thf(fact_222_fresh__fact1,axiom,
! [A: name,P: pi,C: name,B: name] :
( ( fresh_name_pi @ A @ P )
=> ( ( A != C )
=> ( fresh_name_pi @ A @ ( subs @ P @ B @ C ) ) ) ) ).
% fresh_fact1
thf(fact_223_fresh__fact2,axiom,
! [A: name,B: name,P: pi] :
( ( A != B )
=> ( fresh_name_pi @ A @ ( subs @ P @ A @ B ) ) ) ).
% fresh_fact2
thf(fact_224_substRes3,axiom,
! [B: name,P: pi,A: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( subs @ ( res @ A @ P ) @ A @ B )
= ( res @ B @ ( subs @ P @ A @ B ) ) ) ) ).
% substRes3
thf(fact_225_substRes2,axiom,
! [B: name,P: pi,A: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( res @ A @ P )
= ( res @ B @ ( subs @ P @ A @ B ) ) ) ) ).
% substRes2
thf(fact_226_Late__Semantics_OComm1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P2 @ X @ B ) @ Q2 ) ) ) ) ) ).
% Late_Semantics.Comm1
thf(fact_227_Late__Semantics_OComm2,axiom,
! [P: pi,A: name,B: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P2 @ ( subs @ Q2 @ X @ B ) ) ) ) ) ) ).
% Late_Semantics.Comm2
thf(fact_228_transitions_OComm2,axiom,
! [P: pi,A: name,B: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( X != A )
=> ( ( X != B )
=> ( ( fresh_name_pi @ X @ P2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P2 @ ( subs @ Q2 @ X @ B ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm2
thf(fact_229_transitions_OComm1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( X != A )
=> ( ( X != B )
=> ( ( fresh_name_pi @ X @ Q2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P2 @ X @ B ) @ Q2 ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm1
thf(fact_230_simps_I3_J,axiom,
! [A: name,B: name,P: pi,C: name,D: name] :
( ( subs @ ( output @ A @ B @ P ) @ C @ D )
= ( output @ ( subst_name @ A @ C @ D ) @ ( subst_name @ B @ C @ D ) @ ( subs @ P @ C @ D ) ) ) ).
% simps(3)
thf(fact_231_simps_I4_J,axiom,
! [X: name,A: name,C: name,D: name,P: pi] :
( ( X != A )
=> ( ( X != C )
=> ( ( X != D )
=> ( ( subs @ ( input @ A @ X @ P ) @ C @ D )
= ( input @ ( subst_name @ A @ C @ D ) @ X @ ( subs @ P @ C @ D ) ) ) ) ) ) ).
% simps(4)
thf(fact_232_substInput,axiom,
! [X: name,B: name,C: name,A: name,P: pi] :
( ( X != B )
=> ( ( X != C )
=> ( ( subs @ ( input @ A @ X @ P ) @ B @ C )
= ( input @ ( subst_name @ A @ B @ C ) @ X @ ( subs @ P @ B @ C ) ) ) ) ) ).
% substInput
thf(fact_233_subst__name__def,axiom,
( subst_name
= ( ^ [A5: name,B4: name,C4: name] : ( if_name @ ( A5 = B4 ) @ C4 @ A5 ) ) ) ).
% subst_name_def
thf(fact_234_parZeroLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
( ! [Q4: pi] : ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( par @ Q4 @ piNil ) @ Q4 ) @ Rel )
=> ( strong743114133lation @ ( par @ P @ piNil ) @ Rel @ P ) ) ).
% parZeroLeft
thf(fact_235_parZeroRight,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
( ! [Q4: pi] : ( member558706512_pi_pi @ ( product_Pair_pi_pi @ Q4 @ ( par @ Q4 @ piNil ) ) @ Rel )
=> ( strong743114133lation @ P @ Rel @ ( par @ P @ piNil ) ) ) ).
% parZeroRight
thf(fact_236_simps_I6_J,axiom,
! [A: name,B: name,P: pi,C: name,D: name] :
( ( subs @ ( mismatch @ A @ B @ P ) @ C @ D )
= ( mismatch @ ( subst_name @ A @ C @ D ) @ ( subst_name @ B @ C @ D ) @ ( subs @ P @ C @ D ) ) ) ).
% simps(6)
thf(fact_237_pi_Ofresh_I1_J,axiom,
! [A: name] : ( fresh_name_pi @ A @ piNil ) ).
% pi.fresh(1)
thf(fact_238_simps_I1_J,axiom,
! [C: name,D: name] :
( ( subs @ piNil @ C @ D )
= piNil ) ).
% simps(1)
thf(fact_239_pi_Ofresh_I6_J,axiom,
! [A: name,X3: name,X2: name,X1: pi] :
( ( fresh_name_pi @ A @ ( mismatch @ X3 @ X2 @ X1 ) )
= ( ( fresh_name_name @ A @ X3 )
& ( fresh_name_name @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(6)
thf(fact_240_mismatchNilLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,A: name,P: pi] : ( strong743114133lation @ piNil @ Rel @ ( mismatch @ A @ A @ P ) ) ).
% mismatchNilLeft
thf(fact_241_pi_Oinject_I5_J,axiom,
! [X3: name,X2: name,X1: pi,Y3: name,Y2: name,Y1: pi] :
( ( ( mismatch @ X3 @ X2 @ X1 )
= ( mismatch @ Y3 @ Y2 @ Y1 ) )
= ( ( X3 = Y3 )
& ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(5)
thf(fact_242_pi_Odistinct_I9_J,axiom,
! [Name12: name,Name22: name,Pi2: pi] :
( piNil
!= ( mismatch @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(9)
thf(fact_243_nilSimRight,axiom,
! [P: pi,Rel: set_Pr1834926343_pi_pi] : ( strong743114133lation @ P @ Rel @ piNil ) ).
% nilSimRight
thf(fact_244_pi_Odistinct_I3_J,axiom,
! [Pi2: pi] :
( piNil
!= ( tau @ Pi2 ) ) ).
% pi.distinct(3)
thf(fact_245_pi_Odistinct_I39_J,axiom,
! [Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( tau @ Pi )
!= ( mismatch @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(39)
thf(fact_246_pi_Odistinct_I1_J,axiom,
! [Name12: name,Name22: name,Pi2: pi] :
( piNil
!= ( output @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(1)
thf(fact_247_pi_Odistinct_I25_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( mismatch @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(25)
thf(fact_248_pi_Odistinct_I5_J,axiom,
! [Name12: name,Name22: name,Pi2: pi] :
( piNil
!= ( input @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(5)
thf(fact_249_pi_Odistinct_I51_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( input @ Name1 @ Name2 @ Pi )
!= ( mismatch @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(51)
thf(fact_250_pi_Odistinct_I15_J,axiom,
! [Name: name,Pi2: pi] :
( piNil
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(15)
thf(fact_251_pi_Odistinct_I75_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name: name,Pi2: pi] :
( ( mismatch @ Name1 @ Name2 @ Pi )
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(75)
thf(fact_252_pi_Odistinct_I13_J,axiom,
! [Pi12: pi,Pi23: pi] :
( piNil
!= ( par @ Pi12 @ Pi23 ) ) ).
% pi.distinct(13)
thf(fact_253_pi_Odistinct_I73_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( mismatch @ Name1 @ Name2 @ Pi )
!= ( par @ Pi12 @ Pi23 ) ) ).
% pi.distinct(73)
thf(fact_254_zeroTrans,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% zeroTrans
thf(fact_255_nilCases_H,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% nilCases'
thf(fact_256_Mismatch,axiom,
! [P: pi,Rs: late_residual,A: name,B: name] :
( ( late_transitions @ P @ Rs )
=> ( ( A != B )
=> ( late_transitions @ ( mismatch @ A @ B @ P ) @ Rs ) ) ) ).
% Mismatch
thf(fact_257_mismatchCases_H,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( mismatch @ A @ B @ P ) @ Rs )
=> ~ ! [P4: pi,A4: name,B3: name] :
( ( ( A = A4 )
& ( B = B3 )
& ( P = P4 ) )
=> ( ( late_transitions @ P4 @ Rs )
=> ( A4 = B3 ) ) ) ) ).
% mismatchCases'
thf(fact_258_mismatchTrans,axiom,
! [A: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( mismatch @ A @ A @ P ) @ Rs ) ).
% mismatchTrans
thf(fact_259_mismatchCases,axiom,
! [A: name,B: name,P: pi,Rs: late_residual,F: name > name > $o] :
( ( late_transitions @ ( mismatch @ A @ B @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A != B )
=> ( F @ A @ B ) ) )
=> ( F @ A @ B ) ) ) ).
% mismatchCases
thf(fact_260_resZeroTrans,axiom,
! [X: name,Rs: late_residual] :
~ ( late_transitions @ ( res @ X @ piNil ) @ Rs ) ).
% resZeroTrans
thf(fact_261_nilSim_I2_J,axiom,
! [Rel: set_Pr1834926343_pi_pi,A: name,X: name,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( input @ A @ X @ P ) ) ).
% nilSim(2)
thf(fact_262_nilSim_I3_J,axiom,
! [Rel: set_Pr1834926343_pi_pi,A: name,B: name,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( output @ A @ B @ P ) ) ).
% nilSim(3)
thf(fact_263_nilSim_I1_J,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( tau @ P ) ) ).
% nilSim(1)
thf(fact_264_simps_I5_J,axiom,
! [A: name,B: name,P: pi,C: name,D: name] :
( ( subs @ ( match @ A @ B @ P ) @ C @ D )
= ( match @ ( subst_name @ A @ C @ D ) @ ( subst_name @ B @ C @ D ) @ ( subs @ P @ C @ D ) ) ) ).
% simps(5)
thf(fact_265_pi_Ofresh_I5_J,axiom,
! [A: name,X3: name,X2: name,X1: pi] :
( ( fresh_name_pi @ A @ ( match @ X3 @ X2 @ X1 ) )
= ( ( fresh_name_name @ A @ X3 )
& ( fresh_name_name @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(5)
thf(fact_266_pi_Odistinct_I61_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( match @ Name1 @ Name2 @ Pi )
!= ( mismatch @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(61)
thf(fact_267_pi_Odistinct_I7_J,axiom,
! [Name12: name,Name22: name,Pi2: pi] :
( piNil
!= ( match @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(7)
thf(fact_268_matchCases,axiom,
! [A: name,B: name,P: pi,Rs: late_residual,F: name > name > $o] :
( ( late_transitions @ ( match @ A @ B @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A = B )
=> ( F @ A @ A ) ) )
=> ( F @ A @ B ) ) ) ).
% matchCases
thf(fact_269_matchTrans,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( match @ A @ B @ P ) @ Rs )
=> ( A = B ) ) ).
% matchTrans
thf(fact_270_matchCases_H,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( match @ A @ B @ P ) @ Rs )
=> ~ ! [P4: pi] :
( ( ( B = A )
& ( P = P4 ) )
=> ~ ( late_transitions @ P4 @ Rs ) ) ) ).
% matchCases'
thf(fact_271_Match,axiom,
! [P: pi,Rs: late_residual,B: name] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( match @ B @ B @ P ) @ Rs ) ) ).
% Match
thf(fact_272_pi_Odistinct_I65_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( match @ Name1 @ Name2 @ Pi )
!= ( par @ Pi12 @ Pi23 ) ) ).
% pi.distinct(65)
thf(fact_273_pi_Odistinct_I67_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name: name,Pi2: pi] :
( ( match @ Name1 @ Name2 @ Pi )
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(67)
thf(fact_274_pi_Odistinct_I49_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( input @ Name1 @ Name2 @ Pi )
!= ( match @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(49)
thf(fact_275_pi_Odistinct_I23_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( match @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(23)
thf(fact_276_pi_Odistinct_I37_J,axiom,
! [Pi: pi,Name12: name,Name22: name,Pi2: pi] :
( ( tau @ Pi )
!= ( match @ Name12 @ Name22 @ Pi2 ) ) ).
% pi.distinct(37)
thf(fact_277_pi_Oinject_I4_J,axiom,
! [X3: name,X2: name,X1: pi,Y3: name,Y2: name,Y1: pi] :
( ( ( match @ X3 @ X2 @ X1 )
= ( match @ Y3 @ Y2 @ Y1 ) )
= ( ( X3 = Y3 )
& ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(4)
thf(fact_278_matchNilLeft,axiom,
! [A: name,B: name,Rel: set_Pr1834926343_pi_pi,P: pi] :
( ( A != B )
=> ( strong743114133lation @ piNil @ Rel @ ( match @ A @ B @ P ) ) ) ).
% matchNilLeft
thf(fact_279_pi_Ostrong__inducts,axiom,
! [P: late_subject > pi > $o,Z: late_subject,Pi: pi] :
( ! [Z2: late_subject] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name4 @ Z2 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_inducts
thf(fact_280_pi_Ostrong__inducts,axiom,
! [P: late_freeRes > pi > $o,Z: late_freeRes,Pi: pi] :
( ! [Z2: late_freeRes] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name4 @ Z2 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_inducts
thf(fact_281_pi_Ostrong__inducts,axiom,
! [P: name > pi > $o,Z: name,Pi: pi] :
( ! [Z2: name] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ( fresh_name_name @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi13 )
=> ( ! [Za: name] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi13 )
=> ( ! [Za: name] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: name] :
( ( fresh_name_name @ Name4 @ Z2 )
=> ( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_inducts
thf(fact_282_pi_Ostrong__inducts,axiom,
! [P: pi > pi > $o,Z: pi,Pi: pi] :
( ! [Z2: pi] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ( fresh_name_pi @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi13 )
=> ( ! [Za: pi] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi13 )
=> ( ! [Za: pi] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: pi] :
( ( fresh_name_pi @ Name4 @ Z2 )
=> ( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_inducts
thf(fact_283_pi_Ostrong__induct,axiom,
! [P: late_subject > pi > $o,Z: late_subject,Pi: pi] :
( ! [Z2: late_subject] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name4 @ Z2 )
=> ( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: late_subject] :
( ! [Za: late_subject] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_induct
thf(fact_284_pi_Ostrong__induct,axiom,
! [P: late_freeRes > pi > $o,Z: late_freeRes,Pi: pi] :
( ! [Z2: late_freeRes] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi13 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name4 @ Z2 )
=> ( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: late_freeRes] :
( ! [Za: late_freeRes] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_induct
thf(fact_285_pi_Ostrong__induct,axiom,
! [P: name > pi > $o,Z: name,Pi: pi] :
( ! [Z2: name] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ( fresh_name_name @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi13 )
=> ( ! [Za: name] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi13 )
=> ( ! [Za: name] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: name] :
( ( fresh_name_name @ Name4 @ Z2 )
=> ( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: name] :
( ! [Za: name] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_induct
thf(fact_286_pi_Ostrong__induct,axiom,
! [P: pi > pi > $o,Z: pi,Pi: pi] :
( ! [Z2: pi] : ( P @ Z2 @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ( fresh_name_pi @ Name23 @ Z2 )
=> ( ( fresh_name_name @ Name23 @ Name13 )
=> ( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( input @ Name13 @ Name23 @ Pi3 ) ) ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi13 )
=> ( ! [Za: pi] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi13 )
=> ( ! [Za: pi] : ( P @ Za @ Pi24 )
=> ( P @ Z2 @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi,Z2: pi] :
( ( fresh_name_pi @ Name4 @ Z2 )
=> ( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( res @ Name4 @ Pi3 ) ) ) )
=> ( ! [Pi3: pi,Z2: pi] :
( ! [Za: pi] : ( P @ Za @ Pi3 )
=> ( P @ Z2 @ ( bang @ Pi3 ) ) )
=> ( P @ Z @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.strong_induct
thf(fact_287_pi_Ofresh_I10_J,axiom,
! [A: name,X1: pi] :
( ( fresh_name_pi @ A @ ( bang @ X1 ) )
= ( fresh_name_pi @ A @ X1 ) ) ).
% pi.fresh(10)
thf(fact_288_pi_Ofresh_I7_J,axiom,
! [A: name,X2: pi,X1: pi] :
( ( fresh_name_pi @ A @ ( sum @ X2 @ X1 ) )
= ( ( fresh_name_pi @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(7)
thf(fact_289_simps_I10_J,axiom,
! [P: pi,C: name,D: name] :
( ( subs @ ( bang @ P ) @ C @ D )
= ( bang @ ( subs @ P @ C @ D ) ) ) ).
% simps(10)
thf(fact_290_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_291_pi_Odistinct_I63_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( match @ Name1 @ Name2 @ Pi )
!= ( sum @ Pi12 @ Pi23 ) ) ).
% pi.distinct(63)
thf(fact_292_pi_Odistinct_I69_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi2: pi] :
( ( match @ Name1 @ Name2 @ Pi )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(69)
thf(fact_293_Bang,axiom,
! [P: pi,Rs: late_residual] :
( ( late_transitions @ ( par @ P @ ( bang @ P ) ) @ Rs )
=> ( late_transitions @ ( bang @ P ) @ Rs ) ) ).
% Bang
thf(fact_294_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_295_pi_Odistinct_I87_J,axiom,
! [Pi1: pi,Pi22: pi,Pi2: pi] :
( ( par @ Pi1 @ Pi22 )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(87)
thf(fact_296_pi_Odistinct_I89_J,axiom,
! [Name3: name,Pi: pi,Pi2: pi] :
( ( res @ Name3 @ Pi )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(89)
thf(fact_297_pi_Odistinct_I59_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi2: pi] :
( ( input @ Name1 @ Name2 @ Pi )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(59)
thf(fact_298_pi_Odistinct_I33_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi2: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(33)
thf(fact_299_Sum2,axiom,
! [Q: pi,Rs: late_residual,P: pi] :
( ( late_transitions @ Q @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum2
thf(fact_300_Sum1,axiom,
! [P: pi,Rs: late_residual,Q: pi] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum1
thf(fact_301_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_302_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_303_pi_Odistinct_I47_J,axiom,
! [Pi: pi,Pi2: pi] :
( ( tau @ Pi )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(47)
thf(fact_304_pi_Odistinct_I79_J,axiom,
! [Pi1: pi,Pi22: pi,Pi12: pi,Pi23: pi] :
( ( sum @ Pi1 @ Pi22 )
!= ( par @ Pi12 @ Pi23 ) ) ).
% pi.distinct(79)
thf(fact_305_pi_Odistinct_I81_J,axiom,
! [Pi1: pi,Pi22: pi,Name: name,Pi2: pi] :
( ( sum @ Pi1 @ Pi22 )
!= ( res @ Name @ Pi2 ) ) ).
% pi.distinct(81)
thf(fact_306_pi_Odistinct_I53_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( input @ Name1 @ Name2 @ Pi )
!= ( sum @ Pi12 @ Pi23 ) ) ).
% pi.distinct(53)
thf(fact_307_pi_Odistinct_I27_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( output @ Name1 @ Name2 @ Pi )
!= ( sum @ Pi12 @ Pi23 ) ) ).
% pi.distinct(27)
thf(fact_308_pi_Odistinct_I41_J,axiom,
! [Pi: pi,Pi12: pi,Pi23: pi] :
( ( tau @ Pi )
!= ( sum @ Pi12 @ Pi23 ) ) ).
% pi.distinct(41)
thf(fact_309_pi_Oinject_I9_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( bang @ X1 )
= ( bang @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(9)
thf(fact_310_pi_Oinject_I6_J,axiom,
! [X2: pi,X1: pi,Y2: pi,Y1: pi] :
( ( ( sum @ X2 @ X1 )
= ( sum @ Y2 @ Y1 ) )
= ( ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(6)
thf(fact_311_pi_Odistinct_I83_J,axiom,
! [Pi1: pi,Pi22: pi,Pi2: pi] :
( ( sum @ Pi1 @ Pi22 )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(83)
thf(fact_312_pi_Odistinct_I11_J,axiom,
! [Pi12: pi,Pi23: pi] :
( piNil
!= ( sum @ Pi12 @ Pi23 ) ) ).
% pi.distinct(11)
thf(fact_313_pi_Odistinct_I17_J,axiom,
! [Pi2: pi] :
( piNil
!= ( bang @ Pi2 ) ) ).
% pi.distinct(17)
thf(fact_314_pi_Odistinct_I71_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi12: pi,Pi23: pi] :
( ( mismatch @ Name1 @ Name2 @ Pi )
!= ( sum @ Pi12 @ Pi23 ) ) ).
% pi.distinct(71)
thf(fact_315_pi_Odistinct_I77_J,axiom,
! [Name1: name,Name2: name,Pi: pi,Pi2: pi] :
( ( mismatch @ Name1 @ Name2 @ Pi )
!= ( bang @ Pi2 ) ) ).
% pi.distinct(77)
thf(fact_316_pi_Oinducts,axiom,
! [P: pi > $o,Pi: pi] :
( ( P @ piNil )
=> ( ! [Name13: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( output @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( tau @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( input @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( match @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Name13: name,Name23: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( mismatch @ Name13 @ Name23 @ Pi3 ) ) )
=> ( ! [Pi13: pi,Pi24: pi] :
( ( P @ Pi13 )
=> ( ( P @ Pi24 )
=> ( P @ ( sum @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Pi13: pi,Pi24: pi] :
( ( P @ Pi13 )
=> ( ( P @ Pi24 )
=> ( P @ ( par @ Pi13 @ Pi24 ) ) ) )
=> ( ! [Name4: name,Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( res @ Name4 @ Pi3 ) ) )
=> ( ! [Pi3: pi] :
( ( P @ Pi3 )
=> ( P @ ( bang @ Pi3 ) ) )
=> ( P @ Pi ) ) ) ) ) ) ) ) ) ) ) ).
% pi.inducts
% Helper facts (3)
thf(help_If_3_1_If_001t__Agent__Oname_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Agent__Oname_T,axiom,
! [X: name,Y: name] :
( ( if_name @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Agent__Oname_T,axiom,
! [X: name,Y: name] :
( ( if_name @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [P8: pi] :
( ( late_transitions @ ( par @ ( par @ p @ q ) @ r ) @ ( late_BoundR @ a @ x @ P8 ) )
& ( strong2129052853vative @ P8 @ ( par @ p @ ( par @ q2 @ r ) ) @ a @ x @ rel ) ) ).
%------------------------------------------------------------------------------