TPTP Problem File: PRO022^1.p
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%------------------------------------------------------------------------------
% File : PRO022^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Processes
% Problem : Process algebra 175
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : process__175.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 311 ( 145 unt; 60 typ; 0 def)
% Number of atoms : 681 ( 319 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 5306 ( 127 ~; 20 |; 62 &;4764 @)
% ( 0 <=>; 333 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 10 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 539 ( 539 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 3 con; 0-8 aty)
% Number of variables : 1493 ( 42 ^;1349 !; 23 ?;1493 :)
% ( 79 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:54.231
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
thf(ty_t_Process__Mirabelle__xovoopgpgi_Oprocess__term,type,
proces554692349s_term: $tType > $tType > $tType ).
thf(ty_t_Process__Mirabelle__xovoopgpgi_Oprocess,type,
proces634752977rocess: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (51)
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_OCH1,type,
proces1239275103le_CH1:
!>[A: $tType,B: $tType] : ( ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > ( proces554692349s_term @ B @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_OCH2,type,
proces1239275104le_CH2:
!>[A: $tType,B: $tType] : ( ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > ( proces554692349s_term @ B @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_OCONT,type,
proces126235999e_CONT:
!>[A: $tType,B: $tType] : ( ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > ( proces554692349s_term @ B @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_OPREF,type,
proces1708129104e_PREF:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > ( proces554692349s_term @ B @ C ) ) > ( proces554692349s_term @ B @ A ) > B ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oguarded,type,
proces1121166967uarded:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > ( proces554692349s_term @ B @ C ) ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_OisACT,type,
proces687458811_isACT:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( proces554692349s_term @ B @ C ) ) > ( proces554692349s_term @ D @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OAction,type,
proces1525233512Action:
!>[A: $tType] : ( A > ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OChoice,type,
proces1915862579Choice:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Ocase__process,type,
proces1406508781rocess:
!>[A: $tType,B: $tType] : ( ( A > ( proces634752977rocess @ A ) > B ) > ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B ) > ( proces634752977rocess @ A ) > B ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Och1Of,type,
proces979765041_ch1Of:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Och2Of,type,
proces988026546_ch2Of:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OcontOf,type,
proces1778668539contOf:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OisAction,type,
proces10484146Action:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OisChoice,type,
proces401113213Choice:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Omap__process,type,
proces370407749rocess:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( proces634752977rocess @ A ) > ( proces634752977rocess @ Aa ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Opred__process,type,
proces1205983068rocess:
!>[A: $tType] : ( ( A > $o ) > ( proces634752977rocess @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OprefOf,type,
proces745025900prefOf:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > A ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Orel__process,type,
proces749077512rocess:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( proces634752977rocess @ A ) > ( proces634752977rocess @ B ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Oset__process,type,
proces1148692671rocess:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( set @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OACT,type,
proces1454156180ss_ACT:
!>[A: $tType,Pvar: $tType] : ( A > ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OCH,type,
proces89589571ess_CH:
!>[A: $tType,Pvar: $tType] : ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OPROC,type,
proces1062592052s_PROC:
!>[A: $tType,Pvar: $tType] : ( ( proces634752977rocess @ A ) > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OVAR,type,
proces1627516585ss_VAR:
!>[Pvar: $tType,A: $tType] : ( Pvar > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Ocase__process__term,type,
proces460752237s_term:
!>[Pvar: $tType,B: $tType,A: $tType] : ( ( Pvar > B ) > ( ( proces634752977rocess @ A ) > B ) > ( A > ( proces554692349s_term @ A @ Pvar ) > B ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B ) > ( proces554692349s_term @ A @ Pvar ) > B ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Opred__process__term,type,
proces2118920028s_term:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( proces554692349s_term @ A @ B ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Orec__process__term,type,
proces2117273769s_term:
!>[Pvar: $tType,D: $tType,A: $tType] : ( ( Pvar > D ) > ( ( proces634752977rocess @ A ) > D ) > ( A > ( proces554692349s_term @ A @ Pvar ) > D > D ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Orel__process__term,type,
proces2029722208s_term:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( proces554692349s_term @ A @ B ) > ( proces554692349s_term @ C @ D ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Oset1__process__term,type,
proces1264666542s_term:
!>[A: $tType,Pvar: $tType] : ( ( proces554692349s_term @ A @ Pvar ) > ( set @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Oset2__process__term,type,
proces1493547885s_term:
!>[A: $tType,Pvar: $tType] : ( ( proces554692349s_term @ A @ Pvar ) > ( set @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Osolution,type,
proces1652378886lution:
!>[A: $tType,B: $tType] : ( ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > ( proces634752977rocess @ B ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Otrace,type,
proces1971127896_trace:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( stream @ A ) > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_OSCons,type,
sCons:
!>[A: $tType] : ( A > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Opred__stream,type,
pred_stream:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_Osset,type,
sset:
!>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
thf(sy_c_Stream_Ostream__all,type,
stream_all:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Oszip,type,
szip:
!>[A: $tType,B: $tType] : ( ( stream @ A ) > ( stream @ B ) > ( stream @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_X,type,
x: a ).
thf(sy_v_sys,type,
sys: a > ( proces554692349s_term @ b @ a ) ).
%----Relevant facts (247)
thf(fact_0_g,axiom,
proces1121166967uarded @ a @ b @ a @ sys ).
% g
thf(fact_1_False,axiom,
~ ( proces687458811_isACT @ a @ b @ a @ c @ sys @ ( proces1627516585ss_VAR @ a @ c @ x ) ) ).
% False
thf(fact_2_process__term_Oinject_I1_J,axiom,
! [A: $tType,Pvar: $tType,X1: Pvar,Y1: Pvar] :
( ( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
= ( proces1627516585ss_VAR @ Pvar @ A @ Y1 ) )
= ( X1 = Y1 ) ) ).
% process_term.inject(1)
thf(fact_3_process_Oinject_I2_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A,Y21: proces634752977rocess @ A,Y22: proces634752977rocess @ A] :
( ( ( proces1915862579Choice @ A @ X21 @ X22 )
= ( proces1915862579Choice @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% process.inject(2)
thf(fact_4__092_060open_062_092_060not_062_AisACT_Asys_A_IVAR_AX_J_A_092_060Longrightarrow_062_Asolution_Asys_A_IVAR_AX_J_A_061_AChoice_A_Isolution_Asys_A_ICH1_Asys_A_IVAR_AX_J_J_J_A_Isolution_Asys_A_ICH2_Asys_A_IVAR_AX_J_J_J_092_060close_062,axiom,
( ~ ( proces687458811_isACT @ a @ b @ a @ b @ sys @ ( proces1627516585ss_VAR @ a @ b @ x ) )
=> ( ( proces1652378886lution @ a @ b @ sys @ ( proces1627516585ss_VAR @ a @ b @ x ) )
= ( proces1915862579Choice @ b @ ( proces1652378886lution @ a @ b @ sys @ ( proces1239275103le_CH1 @ a @ b @ sys @ ( proces1627516585ss_VAR @ a @ b @ x ) ) ) @ ( proces1652378886lution @ a @ b @ sys @ ( proces1239275104le_CH2 @ a @ b @ sys @ ( proces1627516585ss_VAR @ a @ b @ x ) ) ) ) ) ) ).
% \<open>\<not> isACT sys (VAR X) \<Longrightarrow> solution sys (VAR X) = Choice (solution sys (CH1 sys (VAR X))) (solution sys (CH2 sys (VAR X)))\<close>
thf(fact_5_TT,axiom,
~ ( proces687458811_isACT @ a @ b @ a @ b @ sys @ ( sys @ x ) ) ).
% TT
thf(fact_6__092_060open_062solution_Asys_A_Isys_AX_J_A_061_AChoice_A_Isolution_Asys_A_ICH1_Asys_A_Isys_AX_J_J_J_A_Isolution_Asys_A_ICH2_Asys_A_Isys_AX_J_J_J_092_060close_062,axiom,
( ( proces1652378886lution @ a @ b @ sys @ ( sys @ x ) )
= ( proces1915862579Choice @ b @ ( proces1652378886lution @ a @ b @ sys @ ( proces1239275103le_CH1 @ a @ b @ sys @ ( sys @ x ) ) ) @ ( proces1652378886lution @ a @ b @ sys @ ( proces1239275104le_CH2 @ a @ b @ sys @ ( sys @ x ) ) ) ) ) ).
% \<open>solution sys (sys X) = Choice (solution sys (CH1 sys (sys X))) (solution sys (CH2 sys (sys X)))\<close>
thf(fact_7_solution_Octr_I2_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ~ ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( ( proces1652378886lution @ A @ B @ Sys @ T )
= ( proces1915862579Choice @ B @ ( proces1652378886lution @ A @ B @ Sys @ ( proces1239275103le_CH1 @ A @ B @ Sys @ T ) ) @ ( proces1652378886lution @ A @ B @ Sys @ ( proces1239275104le_CH2 @ A @ B @ Sys @ T ) ) ) ) ) ).
% solution.ctr(2)
thf(fact_8_guarded__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( proces1121166967uarded @ A @ B @ C )
= ( ^ [Sys2: A > ( proces554692349s_term @ B @ C )] :
! [X: A,Y: C] :
( ( Sys2 @ X )
!= ( proces1627516585ss_VAR @ C @ B @ Y ) ) ) ) ).
% guarded_def
thf(fact_9_isACT__VAR,axiom,
! [C: $tType,B: $tType,A: $tType,Sys: A > ( proces554692349s_term @ B @ A ),X2: A] :
( ( proces1121166967uarded @ A @ B @ A @ Sys )
=> ( ( proces687458811_isACT @ A @ B @ A @ C @ Sys @ ( proces1627516585ss_VAR @ A @ C @ X2 ) )
= ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ ( Sys @ X2 ) ) ) ) ).
% isACT_VAR
thf(fact_10_solution_Osimps_I6_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ~ ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( ( proces988026546_ch2Of @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) )
= ( proces1652378886lution @ A @ B @ Sys @ ( proces1239275104le_CH2 @ A @ B @ Sys @ T ) ) ) ) ).
% solution.simps(6)
thf(fact_11_solution_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ~ ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( ( proces979765041_ch1Of @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) )
= ( proces1652378886lution @ A @ B @ Sys @ ( proces1239275103le_CH1 @ A @ B @ Sys @ T ) ) ) ) ).
% solution.simps(5)
thf(fact_12_process_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A2: proces634752977rocess @ A,Aa2: proces634752977rocess @ A] :
( ( proces1205983068rocess @ A @ P @ ( proces1915862579Choice @ A @ A2 @ Aa2 ) )
= ( ( proces1205983068rocess @ A @ P @ A2 )
& ( proces1205983068rocess @ A @ P @ Aa2 ) ) ) ).
% process.pred_inject(2)
thf(fact_13_process_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces1406508781rocess @ A @ B @ F1 @ F2 @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% process.simps(6)
thf(fact_14_solution_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( proces1652378886lution @ A @ B )
= ( ^ [Sys2: A > ( proces554692349s_term @ B @ A ),T2: proces554692349s_term @ B @ A] : ( if @ ( proces634752977rocess @ B ) @ ( proces687458811_isACT @ A @ B @ A @ B @ Sys2 @ T2 ) @ ( proces1525233512Action @ B @ ( proces1708129104e_PREF @ A @ B @ A @ Sys2 @ T2 ) @ ( proces1652378886lution @ A @ B @ Sys2 @ ( proces126235999e_CONT @ A @ B @ Sys2 @ T2 ) ) ) @ ( proces1915862579Choice @ B @ ( proces1652378886lution @ A @ B @ Sys2 @ ( proces1239275103le_CH1 @ A @ B @ Sys2 @ T2 ) ) @ ( proces1652378886lution @ A @ B @ Sys2 @ ( proces1239275104le_CH2 @ A @ B @ Sys2 @ T2 ) ) ) ) ) ) ).
% solution.code
thf(fact_15_process__term_Opred__inject_I1_J,axiom,
! [A: $tType,Pvar: $tType,P1: A > $o,P2: Pvar > $o,A2: Pvar] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces1627516585ss_VAR @ Pvar @ A @ A2 ) )
= ( P2 @ A2 ) ) ).
% process_term.pred_inject(1)
thf(fact_16_process__term_Osimps_I21_J,axiom,
! [A: $tType,D: $tType,Pvar: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X1: Pvar] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( F1 @ X1 ) ) ).
% process_term.simps(21)
thf(fact_17_process__term_Osimps_I17_J,axiom,
! [A: $tType,B: $tType,Pvar: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X1: Pvar] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( F1 @ X1 ) ) ).
% process_term.simps(17)
thf(fact_18_trace_Ointros_I2_J,axiom,
! [A: $tType,P3: proces634752977rocess @ A,As: stream @ A,Q: proces634752977rocess @ A] :
( ( ( proces1971127896_trace @ A @ P3 @ As )
| ( proces1971127896_trace @ A @ Q @ As ) )
=> ( proces1971127896_trace @ A @ ( proces1915862579Choice @ A @ P3 @ Q ) @ As ) ) ).
% trace.intros(2)
thf(fact_19_process_Oinject_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A,Y11: A,Y12: proces634752977rocess @ A] :
( ( ( proces1525233512Action @ A @ X11 @ X12 )
= ( proces1525233512Action @ A @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% process.inject(1)
thf(fact_20_process_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o,A2: A,Aa2: proces634752977rocess @ A] :
( ( proces1205983068rocess @ A @ P @ ( proces1525233512Action @ A @ A2 @ Aa2 ) )
= ( ( P @ A2 )
& ( proces1205983068rocess @ A @ P @ Aa2 ) ) ) ).
% process.pred_inject(1)
thf(fact_21_process_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,X11: A,X12: proces634752977rocess @ A] :
( ( proces1406508781rocess @ A @ B @ F1 @ F2 @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= ( F1 @ X11 @ X12 ) ) ).
% process.simps(5)
thf(fact_22_solution_Octr_I1_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( ( proces1652378886lution @ A @ B @ Sys @ T )
= ( proces1525233512Action @ B @ ( proces1708129104e_PREF @ A @ B @ A @ Sys @ T ) @ ( proces1652378886lution @ A @ B @ Sys @ ( proces126235999e_CONT @ A @ B @ Sys @ T ) ) ) ) ) ).
% solution.ctr(1)
thf(fact_23_process_Oexhaust,axiom,
! [A: $tType,Y2: proces634752977rocess @ A] :
( ! [X112: A,X122: proces634752977rocess @ A] :
( Y2
!= ( proces1525233512Action @ A @ X112 @ X122 ) )
=> ~ ! [X212: proces634752977rocess @ A,X222: proces634752977rocess @ A] :
( Y2
!= ( proces1915862579Choice @ A @ X212 @ X222 ) ) ) ).
% process.exhaust
thf(fact_24_process_Odistinct_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces1525233512Action @ A @ X11 @ X12 )
!= ( proces1915862579Choice @ A @ X21 @ X22 ) ) ).
% process.distinct(1)
thf(fact_25_process_Osel_I3_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces979765041_ch1Of @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= X21 ) ).
% process.sel(3)
thf(fact_26_process_Osel_I4_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces988026546_ch2Of @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= X22 ) ).
% process.sel(4)
thf(fact_27_process_Ocollapse_I2_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces401113213Choice @ A @ Process )
=> ( ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) )
= Process ) ) ).
% process.collapse(2)
thf(fact_28_solution_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( ( proces745025900prefOf @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) )
= ( proces1708129104e_PREF @ A @ B @ A @ Sys @ T ) ) ) ).
% solution.simps(3)
thf(fact_29_solution_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( ( proces1778668539contOf @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) )
= ( proces1652378886lution @ A @ B @ Sys @ ( proces126235999e_CONT @ A @ B @ Sys @ T ) ) ) ) ).
% solution.simps(4)
thf(fact_30_solution_Odisc__iff_I2_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ( proces401113213Choice @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) )
= ( ~ ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T ) ) ) ).
% solution.disc_iff(2)
thf(fact_31_trace_Ocoinduct,axiom,
! [A: $tType,X2: ( proces634752977rocess @ A ) > ( stream @ A ) > $o,X3: proces634752977rocess @ A,Xa: stream @ A] :
( ( X2 @ X3 @ Xa )
=> ( ! [X4: proces634752977rocess @ A,Xa2: stream @ A] :
( ( X2 @ X4 @ Xa2 )
=> ( ? [P4: proces634752977rocess @ A,As2: stream @ A,A3: A] :
( ( X4
= ( proces1525233512Action @ A @ A3 @ P4 ) )
& ( Xa2
= ( sCons @ A @ A3 @ As2 ) )
& ( ( X2 @ P4 @ As2 )
| ( proces1971127896_trace @ A @ P4 @ As2 ) ) )
| ? [P4: proces634752977rocess @ A,As2: stream @ A,Q2: proces634752977rocess @ A] :
( ( X4
= ( proces1915862579Choice @ A @ P4 @ Q2 ) )
& ( Xa2 = As2 )
& ( ( X2 @ P4 @ As2 )
| ( proces1971127896_trace @ A @ P4 @ As2 )
| ( X2 @ Q2 @ As2 )
| ( proces1971127896_trace @ A @ Q2 @ As2 ) ) ) ) )
=> ( proces1971127896_trace @ A @ X3 @ Xa ) ) ) ).
% trace.coinduct
thf(fact_32_trace_Osimps,axiom,
! [A: $tType] :
( ( proces1971127896_trace @ A )
= ( ^ [A1: proces634752977rocess @ A,A22: stream @ A] :
( ? [P5: proces634752977rocess @ A,As3: stream @ A,A4: A] :
( ( A1
= ( proces1525233512Action @ A @ A4 @ P5 ) )
& ( A22
= ( sCons @ A @ A4 @ As3 ) )
& ( proces1971127896_trace @ A @ P5 @ As3 ) )
| ? [P5: proces634752977rocess @ A,As3: stream @ A,Q3: proces634752977rocess @ A] :
( ( A1
= ( proces1915862579Choice @ A @ P5 @ Q3 ) )
& ( A22 = As3 )
& ( ( proces1971127896_trace @ A @ P5 @ As3 )
| ( proces1971127896_trace @ A @ Q3 @ As3 ) ) ) ) ) ) ).
% trace.simps
thf(fact_33_trace_Ocases,axiom,
! [A: $tType,A12: proces634752977rocess @ A,A23: stream @ A] :
( ( proces1971127896_trace @ A @ A12 @ A23 )
=> ( ! [P6: proces634752977rocess @ A,As4: stream @ A,A5: A] :
( ( A12
= ( proces1525233512Action @ A @ A5 @ P6 ) )
=> ( ( A23
= ( sCons @ A @ A5 @ As4 ) )
=> ~ ( proces1971127896_trace @ A @ P6 @ As4 ) ) )
=> ~ ! [P6: proces634752977rocess @ A,As4: stream @ A,Q4: proces634752977rocess @ A] :
( ( A12
= ( proces1915862579Choice @ A @ P6 @ Q4 ) )
=> ( ( A23 = As4 )
=> ~ ( ( proces1971127896_trace @ A @ P6 @ As4 )
| ( proces1971127896_trace @ A @ Q4 @ As4 ) ) ) ) ) ) ).
% trace.cases
thf(fact_34_process_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,Process: proces634752977rocess @ A] :
( ( P @ ( proces1406508781rocess @ A @ B @ F1 @ F2 @ Process ) )
= ( ( ( Process
= ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) )
=> ( P @ ( F1 @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) ) )
& ( ( Process
= ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) )
=> ( P @ ( F2 @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) ) ) ) ) ).
% process.split_sel
thf(fact_35_process_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,Process: proces634752977rocess @ A] :
( ( P @ ( proces1406508781rocess @ A @ B @ F1 @ F2 @ Process ) )
= ( ~ ( ( ( Process
= ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) )
& ~ ( P @ ( F1 @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) ) )
| ( ( Process
= ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) )
& ~ ( P @ ( F2 @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) ) ) ) ) ) ).
% process.split_sel_asm
thf(fact_36_solution_Odisc__iff_I1_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ( proces10484146Action @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) )
= ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T ) ) ).
% solution.disc_iff(1)
thf(fact_37_process_Ocollapse_I1_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces10484146Action @ A @ Process )
=> ( ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) )
= Process ) ) ).
% process.collapse(1)
thf(fact_38_not__isAction__isChoice,axiom,
! [A: $tType,P3: proces634752977rocess @ A] :
~ ( ( proces10484146Action @ A @ P3 )
& ( proces401113213Choice @ A @ P3 ) ) ).
% not_isAction_isChoice
thf(fact_39_process_Oexhaust__disc,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ~ ( proces10484146Action @ A @ Process )
=> ( proces401113213Choice @ A @ Process ) ) ).
% process.exhaust_disc
thf(fact_40_isAction__isChoice,axiom,
! [A: $tType,P3: proces634752977rocess @ A] :
( ( proces10484146Action @ A @ P3 )
| ( proces401113213Choice @ A @ P3 ) ) ).
% isAction_isChoice
thf(fact_41_process_Odistinct__disc_I1_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces10484146Action @ A @ Process )
=> ~ ( proces401113213Choice @ A @ Process ) ) ).
% process.distinct_disc(1)
thf(fact_42_process_Odistinct__disc_I2_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces401113213Choice @ A @ Process )
=> ~ ( proces10484146Action @ A @ Process ) ) ).
% process.distinct_disc(2)
thf(fact_43_process_Ocoinduct__strong,axiom,
! [A: $tType,R: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > $o,Process: proces634752977rocess @ A,Process2: proces634752977rocess @ A] :
( ( R @ Process @ Process2 )
=> ( ! [Process3: proces634752977rocess @ A,Process4: proces634752977rocess @ A] :
( ( R @ Process3 @ Process4 )
=> ( ( ( proces10484146Action @ A @ Process3 )
= ( proces10484146Action @ A @ Process4 ) )
& ( ( proces10484146Action @ A @ Process3 )
=> ( ( proces10484146Action @ A @ Process4 )
=> ( ( ( proces745025900prefOf @ A @ Process3 )
= ( proces745025900prefOf @ A @ Process4 ) )
& ( ( R @ ( proces1778668539contOf @ A @ Process3 ) @ ( proces1778668539contOf @ A @ Process4 ) )
| ( ( proces1778668539contOf @ A @ Process3 )
= ( proces1778668539contOf @ A @ Process4 ) ) ) ) ) )
& ( ( proces401113213Choice @ A @ Process3 )
=> ( ( proces401113213Choice @ A @ Process4 )
=> ( ( ( R @ ( proces979765041_ch1Of @ A @ Process3 ) @ ( proces979765041_ch1Of @ A @ Process4 ) )
| ( ( proces979765041_ch1Of @ A @ Process3 )
= ( proces979765041_ch1Of @ A @ Process4 ) ) )
& ( ( R @ ( proces988026546_ch2Of @ A @ Process3 ) @ ( proces988026546_ch2Of @ A @ Process4 ) )
| ( ( proces988026546_ch2Of @ A @ Process3 )
= ( proces988026546_ch2Of @ A @ Process4 ) ) ) ) ) ) ) )
=> ( Process = Process2 ) ) ) ).
% process.coinduct_strong
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q5: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q5 @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q5 ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_process_Ocoinduct,axiom,
! [A: $tType,R: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > $o,Process: proces634752977rocess @ A,Process2: proces634752977rocess @ A] :
( ( R @ Process @ Process2 )
=> ( ! [Process3: proces634752977rocess @ A,Process4: proces634752977rocess @ A] :
( ( R @ Process3 @ Process4 )
=> ( ( ( proces10484146Action @ A @ Process3 )
= ( proces10484146Action @ A @ Process4 ) )
& ( ( proces10484146Action @ A @ Process3 )
=> ( ( proces10484146Action @ A @ Process4 )
=> ( ( ( proces745025900prefOf @ A @ Process3 )
= ( proces745025900prefOf @ A @ Process4 ) )
& ( R @ ( proces1778668539contOf @ A @ Process3 ) @ ( proces1778668539contOf @ A @ Process4 ) ) ) ) )
& ( ( proces401113213Choice @ A @ Process3 )
=> ( ( proces401113213Choice @ A @ Process4 )
=> ( ( R @ ( proces979765041_ch1Of @ A @ Process3 ) @ ( proces979765041_ch1Of @ A @ Process4 ) )
& ( R @ ( proces988026546_ch2Of @ A @ Process3 ) @ ( proces988026546_ch2Of @ A @ Process4 ) ) ) ) ) ) )
=> ( Process = Process2 ) ) ) ).
% process.coinduct
thf(fact_49_process_Oexpand,axiom,
! [A: $tType,Process: proces634752977rocess @ A,Process2: proces634752977rocess @ A] :
( ( ( proces10484146Action @ A @ Process )
= ( proces10484146Action @ A @ Process2 ) )
=> ( ( ( proces10484146Action @ A @ Process )
=> ( ( proces10484146Action @ A @ Process2 )
=> ( ( ( proces745025900prefOf @ A @ Process )
= ( proces745025900prefOf @ A @ Process2 ) )
& ( ( proces1778668539contOf @ A @ Process )
= ( proces1778668539contOf @ A @ Process2 ) ) ) ) )
=> ( ( ( proces401113213Choice @ A @ Process )
=> ( ( proces401113213Choice @ A @ Process2 )
=> ( ( ( proces979765041_ch1Of @ A @ Process )
= ( proces979765041_ch1Of @ A @ Process2 ) )
& ( ( proces988026546_ch2Of @ A @ Process )
= ( proces988026546_ch2Of @ A @ Process2 ) ) ) ) )
=> ( Process = Process2 ) ) ) ) ).
% process.expand
thf(fact_50_process__strong__coind,axiom,
! [A: $tType,Phi: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > $o,P3: proces634752977rocess @ A,P7: proces634752977rocess @ A] :
( ( Phi @ P3 @ P7 )
=> ( ! [P6: proces634752977rocess @ A,P8: proces634752977rocess @ A] :
( ( Phi @ P6 @ P8 )
=> ( ( ( proces10484146Action @ A @ P6 )
= ( proces10484146Action @ A @ P8 ) )
& ( ( proces401113213Choice @ A @ P6 )
= ( proces401113213Choice @ A @ P8 ) ) ) )
=> ( ! [A5: A,A7: A,P6: proces634752977rocess @ A,P8: proces634752977rocess @ A] :
( ( Phi @ ( proces1525233512Action @ A @ A5 @ P6 ) @ ( proces1525233512Action @ A @ A7 @ P8 ) )
=> ( ( A5 = A7 )
& ( ( Phi @ P6 @ P8 )
| ( P6 = P8 ) ) ) )
=> ( ! [P6: proces634752977rocess @ A,Q4: proces634752977rocess @ A,P8: proces634752977rocess @ A,Q6: proces634752977rocess @ A] :
( ( Phi @ ( proces1915862579Choice @ A @ P6 @ Q4 ) @ ( proces1915862579Choice @ A @ P8 @ Q6 ) )
=> ( ( ( Phi @ P6 @ P8 )
| ( P6 = P8 ) )
& ( ( Phi @ Q4 @ Q6 )
| ( Q4 = Q6 ) ) ) )
=> ( P3 = P7 ) ) ) ) ) ).
% process_strong_coind
thf(fact_51_process__coind,axiom,
! [A: $tType,Phi: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > $o,P3: proces634752977rocess @ A,P7: proces634752977rocess @ A] :
( ( Phi @ P3 @ P7 )
=> ( ! [P6: proces634752977rocess @ A,P8: proces634752977rocess @ A] :
( ( Phi @ P6 @ P8 )
=> ( ( ( proces10484146Action @ A @ P6 )
= ( proces10484146Action @ A @ P8 ) )
& ( ( proces401113213Choice @ A @ P6 )
= ( proces401113213Choice @ A @ P8 ) ) ) )
=> ( ! [A5: A,A7: A,P6: proces634752977rocess @ A,P8: proces634752977rocess @ A] :
( ( Phi @ ( proces1525233512Action @ A @ A5 @ P6 ) @ ( proces1525233512Action @ A @ A7 @ P8 ) )
=> ( ( A5 = A7 )
& ( Phi @ P6 @ P8 ) ) )
=> ( ! [P6: proces634752977rocess @ A,Q4: proces634752977rocess @ A,P8: proces634752977rocess @ A,Q6: proces634752977rocess @ A] :
( ( Phi @ ( proces1915862579Choice @ A @ P6 @ Q4 ) @ ( proces1915862579Choice @ A @ P8 @ Q6 ) )
=> ( ( Phi @ P6 @ P8 )
& ( Phi @ Q4 @ Q6 ) ) )
=> ( P3 = P7 ) ) ) ) ) ).
% process_coind
thf(fact_52_process_Odisc_I2_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
~ ( proces10484146Action @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) ) ).
% process.disc(2)
thf(fact_53_isAction__def,axiom,
! [A: $tType] :
( ( proces10484146Action @ A )
= ( ^ [Process5: proces634752977rocess @ A] :
? [X113: A,X123: proces634752977rocess @ A] :
( Process5
= ( proces1525233512Action @ A @ X113 @ X123 ) ) ) ) ).
% isAction_def
thf(fact_54_process_OdiscI_I1_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A,X11: A,X12: proces634752977rocess @ A] :
( ( Process
= ( proces1525233512Action @ A @ X11 @ X12 ) )
=> ( proces10484146Action @ A @ Process ) ) ).
% process.discI(1)
thf(fact_55_process_Odisc_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] : ( proces10484146Action @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) ) ).
% process.disc(1)
thf(fact_56_process_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( proces1406508781rocess @ A @ B )
= ( ^ [F12: A > ( proces634752977rocess @ A ) > B,F22: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,Process5: proces634752977rocess @ A] : ( if @ B @ ( proces10484146Action @ A @ Process5 ) @ ( F12 @ ( proces745025900prefOf @ A @ Process5 ) @ ( proces1778668539contOf @ A @ Process5 ) ) @ ( F22 @ ( proces979765041_ch1Of @ A @ Process5 ) @ ( proces988026546_ch2Of @ A @ Process5 ) ) ) ) ) ).
% process.case_eq_if
thf(fact_57_process_Odisc_I4_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] : ( proces401113213Choice @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) ) ).
% process.disc(4)
thf(fact_58_process_OdiscI_I2_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( Process
= ( proces1915862579Choice @ A @ X21 @ X22 ) )
=> ( proces401113213Choice @ A @ Process ) ) ).
% process.discI(2)
thf(fact_59_isChoice__def,axiom,
! [A: $tType] :
( ( proces401113213Choice @ A )
= ( ^ [Process5: proces634752977rocess @ A] :
? [X213: proces634752977rocess @ A,X223: proces634752977rocess @ A] :
( Process5
= ( proces1915862579Choice @ A @ X213 @ X223 ) ) ) ) ).
% isChoice_def
thf(fact_60_process_Odisc_I3_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
~ ( proces401113213Choice @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) ) ).
% process.disc(3)
thf(fact_61_process_Osel_I2_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
( ( proces1778668539contOf @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= X12 ) ).
% process.sel(2)
thf(fact_62_process_Osel_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
( ( proces745025900prefOf @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= X11 ) ).
% process.sel(1)
thf(fact_63_trace_Ointros_I1_J,axiom,
! [A: $tType,P3: proces634752977rocess @ A,As: stream @ A,A2: A] :
( ( proces1971127896_trace @ A @ P3 @ As )
=> ( proces1971127896_trace @ A @ ( proces1525233512Action @ A @ A2 @ P3 ) @ ( sCons @ A @ A2 @ As ) ) ) ).
% trace.intros(1)
thf(fact_64_solution_Odisc_I1_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( proces10484146Action @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) ) ) ).
% solution.disc(1)
thf(fact_65_process_Oexhaust__sel,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( Process
!= ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) )
=> ( Process
= ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) ) ) ).
% process.exhaust_sel
thf(fact_66_solution_Odisc_I2_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T: proces554692349s_term @ B @ A] :
( ~ ( proces687458811_isACT @ A @ B @ A @ B @ Sys @ T )
=> ( proces401113213Choice @ B @ ( proces1652378886lution @ A @ B @ Sys @ T ) ) ) ).
% solution.disc(2)
thf(fact_67_stream_Oinject,axiom,
! [A: $tType,X1: A,X23: stream @ A,Y1: A,Y23: stream @ A] :
( ( ( sCons @ A @ X1 @ X23 )
= ( sCons @ A @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% stream.inject
thf(fact_68_process_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( proces749077512rocess @ A @ B )
= ( ^ [R2: A > B > $o,A4: proces634752977rocess @ A,B2: proces634752977rocess @ B] :
( ( ( proces10484146Action @ A @ A4 )
= ( proces10484146Action @ B @ B2 ) )
& ( ( proces10484146Action @ A @ A4 )
=> ( ( proces10484146Action @ B @ B2 )
=> ( ( R2 @ ( proces745025900prefOf @ A @ A4 ) @ ( proces745025900prefOf @ B @ B2 ) )
& ( proces749077512rocess @ A @ B @ R2 @ ( proces1778668539contOf @ A @ A4 ) @ ( proces1778668539contOf @ B @ B2 ) ) ) ) )
& ( ( proces401113213Choice @ A @ A4 )
=> ( ( proces401113213Choice @ B @ B2 )
=> ( ( proces749077512rocess @ A @ B @ R2 @ ( proces979765041_ch1Of @ A @ A4 ) @ ( proces979765041_ch1Of @ B @ B2 ) )
& ( proces749077512rocess @ A @ B @ R2 @ ( proces988026546_ch2Of @ A @ A4 ) @ ( proces988026546_ch2Of @ B @ B2 ) ) ) ) ) ) ) ) ).
% process.rel_sel
thf(fact_69_process_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( proces634752977rocess @ A ) > ( proces634752977rocess @ B ) > $o,X3: proces634752977rocess @ A,Y2: proces634752977rocess @ B,R: A > B > $o] :
( ( P @ X3 @ Y2 )
=> ( ! [Process3: proces634752977rocess @ A,Process4: proces634752977rocess @ B] :
( ( P @ Process3 @ Process4 )
=> ( ( ( proces10484146Action @ A @ Process3 )
= ( proces10484146Action @ B @ Process4 ) )
& ( ( proces10484146Action @ A @ Process3 )
=> ( ( proces10484146Action @ B @ Process4 )
=> ( ( R @ ( proces745025900prefOf @ A @ Process3 ) @ ( proces745025900prefOf @ B @ Process4 ) )
& ( P @ ( proces1778668539contOf @ A @ Process3 ) @ ( proces1778668539contOf @ B @ Process4 ) ) ) ) )
& ( ( proces401113213Choice @ A @ Process3 )
=> ( ( proces401113213Choice @ B @ Process4 )
=> ( ( P @ ( proces979765041_ch1Of @ A @ Process3 ) @ ( proces979765041_ch1Of @ B @ Process4 ) )
& ( P @ ( proces988026546_ch2Of @ A @ Process3 ) @ ( proces988026546_ch2Of @ B @ Process4 ) ) ) ) ) ) )
=> ( proces749077512rocess @ A @ B @ R @ X3 @ Y2 ) ) ) ).
% process.rel_coinduct
thf(fact_70_stream_Oexhaust,axiom,
! [A: $tType,Y2: stream @ A] :
~ ! [X13: A,X24: stream @ A] :
( Y2
!= ( sCons @ A @ X13 @ X24 ) ) ).
% stream.exhaust
thf(fact_71_process__term_Opred__inject_I2_J,axiom,
! [Pvar: $tType,A: $tType,P1: A > $o,P2: Pvar > $o,A2: proces634752977rocess @ A] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces1062592052s_PROC @ A @ Pvar @ A2 ) )
= ( proces1205983068rocess @ A @ P1 @ A2 ) ) ).
% process_term.pred_inject(2)
thf(fact_72_process_Oset__sel_I3_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A,Xa: A] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( member @ A @ Xa @ ( proces1148692671rocess @ A @ ( proces979765041_ch1Of @ A @ A2 ) ) )
=> ( member @ A @ Xa @ ( proces1148692671rocess @ A @ A2 ) ) ) ) ).
% process.set_sel(3)
thf(fact_73_process_Oset__sel_I4_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A,Xb: A] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( member @ A @ Xb @ ( proces1148692671rocess @ A @ ( proces988026546_ch2Of @ A @ A2 ) ) )
=> ( member @ A @ Xb @ ( proces1148692671rocess @ A @ A2 ) ) ) ) ).
% process.set_sel(4)
thf(fact_74_process_Oset__sel_I1_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A] :
( ( proces10484146Action @ A @ A2 )
=> ( member @ A @ ( proces745025900prefOf @ A @ A2 ) @ ( proces1148692671rocess @ A @ A2 ) ) ) ).
% process.set_sel(1)
thf(fact_75_process_Oset__sel_I2_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A,X3: A] :
( ( proces10484146Action @ A @ A2 )
=> ( ( member @ A @ X3 @ ( proces1148692671rocess @ A @ ( proces1778668539contOf @ A @ A2 ) ) )
=> ( member @ A @ X3 @ ( proces1148692671rocess @ A @ A2 ) ) ) ) ).
% process.set_sel(2)
thf(fact_76_process__term_Oinject_I2_J,axiom,
! [Pvar: $tType,A: $tType,X23: proces634752977rocess @ A,Y23: proces634752977rocess @ A] :
( ( ( proces1062592052s_PROC @ A @ Pvar @ X23 )
= ( proces1062592052s_PROC @ A @ Pvar @ Y23 ) )
= ( X23 = Y23 ) ) ).
% process_term.inject(2)
thf(fact_77_process_Orel__inject_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X11: A,X12: proces634752977rocess @ A,Y11: B,Y12: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ ( proces1525233512Action @ A @ X11 @ X12 ) @ ( proces1525233512Action @ B @ Y11 @ Y12 ) )
= ( ( R @ X11 @ Y11 )
& ( proces749077512rocess @ A @ B @ R @ X12 @ Y12 ) ) ) ).
% process.rel_inject(1)
thf(fact_78_process_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A,Y21: proces634752977rocess @ B,Y22: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ ( proces1915862579Choice @ A @ X21 @ X22 ) @ ( proces1915862579Choice @ B @ Y21 @ Y22 ) )
= ( ( proces749077512rocess @ A @ B @ R @ X21 @ Y21 )
& ( proces749077512rocess @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% process.rel_inject(2)
thf(fact_79_process_Orel__eq,axiom,
! [A: $tType] :
( ( proces749077512rocess @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: proces634752977rocess @ A,Z: proces634752977rocess @ A] : Y3 = Z ) ) ).
% process.rel_eq
thf(fact_80_process_Orel__cong,axiom,
! [A: $tType,B: $tType,X3: proces634752977rocess @ A,Ya: proces634752977rocess @ A,Y2: proces634752977rocess @ B,Xa: proces634752977rocess @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X3 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z2: A,Yb: B] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( proces1148692671rocess @ B @ Xa ) )
=> ( ( R @ Z2 @ Yb )
= ( Ra @ Z2 @ Yb ) ) ) )
=> ( ( proces749077512rocess @ A @ B @ R @ X3 @ Y2 )
= ( proces749077512rocess @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% process.rel_cong
thf(fact_81_process_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X3: proces634752977rocess @ B] :
( ! [X4: B] : ( Ra @ X4 @ X4 )
=> ( proces749077512rocess @ B @ B @ Ra @ X3 @ X3 ) ) ).
% process.rel_refl
thf(fact_82_process_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X3: proces634752977rocess @ A,Y2: proces634752977rocess @ B,Ra: A > B > $o] :
( ( proces749077512rocess @ A @ B @ R @ X3 @ Y2 )
=> ( ! [Z2: A,Yb: B] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ X3 ) )
=> ( ( member @ B @ Yb @ ( proces1148692671rocess @ B @ Y2 ) )
=> ( ( R @ Z2 @ Yb )
=> ( Ra @ Z2 @ Yb ) ) ) )
=> ( proces749077512rocess @ A @ B @ Ra @ X3 @ Y2 ) ) ) ).
% process.rel_mono_strong
thf(fact_83_process_Orel__refl__strong,axiom,
! [A: $tType,X3: proces634752977rocess @ A,Ra: A > A > $o] :
( ! [Z2: A] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ X3 ) )
=> ( Ra @ Z2 @ Z2 ) )
=> ( proces749077512rocess @ A @ A @ Ra @ X3 @ X3 ) ) ).
% process.rel_refl_strong
thf(fact_84_process_Orel__intros_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X11: A,Y11: B,X12: proces634752977rocess @ A,Y12: proces634752977rocess @ B] :
( ( R @ X11 @ Y11 )
=> ( ( proces749077512rocess @ A @ B @ R @ X12 @ Y12 )
=> ( proces749077512rocess @ A @ B @ R @ ( proces1525233512Action @ A @ X11 @ X12 ) @ ( proces1525233512Action @ B @ Y11 @ Y12 ) ) ) ) ).
% process.rel_intros(1)
thf(fact_85_process_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: proces634752977rocess @ A,Y21: proces634752977rocess @ B,X22: proces634752977rocess @ A,Y22: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ X21 @ Y21 )
=> ( ( proces749077512rocess @ A @ B @ R @ X22 @ Y22 )
=> ( proces749077512rocess @ A @ B @ R @ ( proces1915862579Choice @ A @ X21 @ X22 ) @ ( proces1915862579Choice @ B @ Y21 @ Y22 ) ) ) ) ).
% process.rel_intros(2)
thf(fact_86_process__term_Odistinct_I1_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar,X23: proces634752977rocess @ A] :
( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
!= ( proces1062592052s_PROC @ A @ Pvar @ X23 ) ) ).
% process_term.distinct(1)
thf(fact_87_process__term_Osimps_I18_J,axiom,
! [Pvar: $tType,B: $tType,A: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X23: proces634752977rocess @ A] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) )
= ( F2 @ X23 ) ) ).
% process_term.simps(18)
thf(fact_88_process__term_Osimps_I22_J,axiom,
! [Pvar: $tType,D: $tType,A: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X23: proces634752977rocess @ A] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) )
= ( F2 @ X23 ) ) ).
% process_term.simps(22)
thf(fact_89_process_Oset__intros_I2_J,axiom,
! [A: $tType,X3: A,A23: proces634752977rocess @ A,A12: A] :
( ( member @ A @ X3 @ ( proces1148692671rocess @ A @ A23 ) )
=> ( member @ A @ X3 @ ( proces1148692671rocess @ A @ ( proces1525233512Action @ A @ A12 @ A23 ) ) ) ) ).
% process.set_intros(2)
thf(fact_90_process_Oset__intros_I1_J,axiom,
! [A: $tType,A12: A,A23: proces634752977rocess @ A] : ( member @ A @ A12 @ ( proces1148692671rocess @ A @ ( proces1525233512Action @ A @ A12 @ A23 ) ) ) ).
% process.set_intros(1)
thf(fact_91_process_Oset__intros_I3_J,axiom,
! [A: $tType,Xa: A,A1a: proces634752977rocess @ A,A2a: proces634752977rocess @ A] :
( ( member @ A @ Xa @ ( proces1148692671rocess @ A @ A1a ) )
=> ( member @ A @ Xa @ ( proces1148692671rocess @ A @ ( proces1915862579Choice @ A @ A1a @ A2a ) ) ) ) ).
% process.set_intros(3)
thf(fact_92_process_Oset__intros_I4_J,axiom,
! [A: $tType,Xb: A,A2a: proces634752977rocess @ A,A1a: proces634752977rocess @ A] :
( ( member @ A @ Xb @ ( proces1148692671rocess @ A @ A2a ) )
=> ( member @ A @ Xb @ ( proces1148692671rocess @ A @ ( proces1915862579Choice @ A @ A1a @ A2a ) ) ) ) ).
% process.set_intros(4)
thf(fact_93_process_Opred__cong,axiom,
! [A: $tType,X3: proces634752977rocess @ A,Ya: proces634752977rocess @ A,P: A > $o,Pa: A > $o] :
( ( X3 = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( proces1205983068rocess @ A @ P @ X3 )
= ( proces1205983068rocess @ A @ Pa @ Ya ) ) ) ) ).
% process.pred_cong
thf(fact_94_process_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X3: proces634752977rocess @ A,Pa: A > $o] :
( ( proces1205983068rocess @ A @ P @ X3 )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( proces1205983068rocess @ A @ Pa @ X3 ) ) ) ).
% process.pred_mono_strong
thf(fact_95_process_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: proces634752977rocess @ A,B3: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ A2 @ B3 )
=> ( ! [X13: A,X24: proces634752977rocess @ A] :
( ( A2
= ( proces1525233512Action @ A @ X13 @ X24 ) )
=> ! [Y13: B,Y24: proces634752977rocess @ B] :
( ( B3
= ( proces1525233512Action @ B @ Y13 @ Y24 ) )
=> ( ( R @ X13 @ Y13 )
=> ~ ( proces749077512rocess @ A @ B @ R @ X24 @ Y24 ) ) ) )
=> ~ ! [X1a: proces634752977rocess @ A,X2a: proces634752977rocess @ A] :
( ( A2
= ( proces1915862579Choice @ A @ X1a @ X2a ) )
=> ! [Y1a: proces634752977rocess @ B,Y2a: proces634752977rocess @ B] :
( ( B3
= ( proces1915862579Choice @ B @ Y1a @ Y2a ) )
=> ( ( proces749077512rocess @ A @ B @ R @ X1a @ Y1a )
=> ~ ( proces749077512rocess @ A @ B @ R @ X2a @ Y2a ) ) ) ) ) ) ).
% process.rel_cases
thf(fact_96_process_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X11: A,X12: proces634752977rocess @ A,Y21: proces634752977rocess @ B,Y22: proces634752977rocess @ B] :
~ ( proces749077512rocess @ A @ B @ R @ ( proces1525233512Action @ A @ X11 @ X12 ) @ ( proces1915862579Choice @ B @ Y21 @ Y22 ) ) ).
% process.rel_distinct(1)
thf(fact_97_process_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: proces634752977rocess @ A,Y22: proces634752977rocess @ A,X11: B,X12: proces634752977rocess @ B] :
~ ( proces749077512rocess @ A @ B @ R @ ( proces1915862579Choice @ A @ Y21 @ Y22 ) @ ( proces1525233512Action @ B @ X11 @ X12 ) ) ).
% process.rel_distinct(2)
thf(fact_98_isACT_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sys: A > ( proces554692349s_term @ B @ C ),P3: proces634752977rocess @ D] :
( ( proces687458811_isACT @ A @ B @ C @ D @ Sys @ ( proces1062592052s_PROC @ D @ A @ P3 ) )
= ( proces10484146Action @ D @ P3 ) ) ).
% isACT.simps(2)
thf(fact_99_CH2_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),P3: proces634752977rocess @ B] :
( ( proces1239275104le_CH2 @ A @ B @ Sys @ ( proces1062592052s_PROC @ B @ A @ P3 ) )
= ( proces1062592052s_PROC @ B @ A @ ( proces988026546_ch2Of @ B @ P3 ) ) ) ).
% CH2.simps(2)
thf(fact_100_CH1_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),P3: proces634752977rocess @ B] :
( ( proces1239275103le_CH1 @ A @ B @ Sys @ ( proces1062592052s_PROC @ B @ A @ P3 ) )
= ( proces1062592052s_PROC @ B @ A @ ( proces979765041_ch1Of @ B @ P3 ) ) ) ).
% CH1.simps(2)
thf(fact_101_CONT_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),P3: proces634752977rocess @ B] :
( ( proces126235999e_CONT @ A @ B @ Sys @ ( proces1062592052s_PROC @ B @ A @ P3 ) )
= ( proces1062592052s_PROC @ B @ A @ ( proces1778668539contOf @ B @ P3 ) ) ) ).
% CONT.simps(2)
thf(fact_102_PREF_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ C ),P3: proces634752977rocess @ B] :
( ( proces1708129104e_PREF @ A @ B @ C @ Sys @ ( proces1062592052s_PROC @ B @ A @ P3 ) )
= ( proces745025900prefOf @ B @ P3 ) ) ).
% PREF.simps(2)
thf(fact_103_process_Oset__induct,axiom,
! [A: $tType,X3: A,A2: proces634752977rocess @ A,P: A > ( proces634752977rocess @ A ) > $o] :
( ( member @ A @ X3 @ ( proces1148692671rocess @ A @ A2 ) )
=> ( ! [Z1: A,Z22: proces634752977rocess @ A] : ( P @ Z1 @ ( proces1525233512Action @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: proces634752977rocess @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( proces1148692671rocess @ A @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( proces1525233512Action @ A @ Z1 @ Z22 ) ) ) )
=> ( ! [Z1a: proces634752977rocess @ A,Z2a: proces634752977rocess @ A,Xb2: A] :
( ( member @ A @ Xb2 @ ( proces1148692671rocess @ A @ Z1a ) )
=> ( ( P @ Xb2 @ Z1a )
=> ( P @ Xb2 @ ( proces1915862579Choice @ A @ Z1a @ Z2a ) ) ) )
=> ( ! [Z1a: proces634752977rocess @ A,Z2a: proces634752977rocess @ A,Xc: A] :
( ( member @ A @ Xc @ ( proces1148692671rocess @ A @ Z2a ) )
=> ( ( P @ Xc @ Z2a )
=> ( P @ Xc @ ( proces1915862579Choice @ A @ Z1a @ Z2a ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ) ) ).
% process.set_induct
thf(fact_104_process_Oset__cases,axiom,
! [A: $tType,E: A,A2: proces634752977rocess @ A] :
( ( member @ A @ E @ ( proces1148692671rocess @ A @ A2 ) )
=> ( ! [Z22: proces634752977rocess @ A] :
( A2
!= ( proces1525233512Action @ A @ E @ Z22 ) )
=> ( ! [Z1: A,Z22: proces634752977rocess @ A] :
( ( A2
= ( proces1525233512Action @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( proces1148692671rocess @ A @ Z22 ) ) )
=> ( ! [Z1: proces634752977rocess @ A] :
( ? [Z22: proces634752977rocess @ A] :
( A2
= ( proces1915862579Choice @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( proces1148692671rocess @ A @ Z1 ) ) )
=> ~ ! [Z1: proces634752977rocess @ A,Z22: proces634752977rocess @ A] :
( ( A2
= ( proces1915862579Choice @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( proces1148692671rocess @ A @ Z22 ) ) ) ) ) ) ) ).
% process.set_cases
thf(fact_105_smember__code,axiom,
! [A: $tType,X3: A,Y2: A,S: stream @ A] :
( ( smember @ A @ X3 @ ( sCons @ A @ Y2 @ S ) )
= ( ( X3 != Y2 )
=> ( smember @ A @ X3 @ S ) ) ) ).
% smember_code
thf(fact_106_process_Osimps_I21_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
( ( proces1148692671rocess @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= ( insert @ A @ X11 @ ( proces1148692671rocess @ A @ X12 ) ) ) ).
% process.simps(21)
thf(fact_107_stream__all__Stream,axiom,
! [A: $tType,P: A > $o,X3: A,X2: stream @ A] :
( ( stream_all @ A @ P @ ( sCons @ A @ X3 @ X2 ) )
= ( ( P @ X3 )
& ( stream_all @ A @ P @ X2 ) ) ) ).
% stream_all_Stream
thf(fact_108_process__term_Osimps_I46_J,axiom,
! [Pvar: $tType,A: $tType,X23: proces634752977rocess @ A] :
( ( proces1264666542s_term @ A @ Pvar @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) )
= ( proces1148692671rocess @ A @ X23 ) ) ).
% process_term.simps(46)
thf(fact_109_process__term_Oset__intros_I1_J,axiom,
! [Pvar: $tType,A: $tType,X3: A,Aa2: proces634752977rocess @ A] :
( ( member @ A @ X3 @ ( proces1148692671rocess @ A @ Aa2 ) )
=> ( member @ A @ X3 @ ( proces1264666542s_term @ A @ Pvar @ ( proces1062592052s_PROC @ A @ Pvar @ Aa2 ) ) ) ) ).
% process_term.set_intros(1)
thf(fact_110_process_Omap__sel_I4_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( proces988026546_ch2Of @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces370407749rocess @ A @ B @ F @ ( proces988026546_ch2Of @ A @ A2 ) ) ) ) ).
% process.map_sel(4)
thf(fact_111_process_Omap__disc__iff_I1_J,axiom,
! [B: $tType,A: $tType,F: A > B,A2: proces634752977rocess @ A] :
( ( proces10484146Action @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces10484146Action @ A @ A2 ) ) ).
% process.map_disc_iff(1)
thf(fact_112_process_Omap__disc__iff_I2_J,axiom,
! [B: $tType,A: $tType,F: A > B,A2: proces634752977rocess @ A] :
( ( proces401113213Choice @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces401113213Choice @ A @ A2 ) ) ).
% process.map_disc_iff(2)
thf(fact_113_process_Osimps_I16_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces370407749rocess @ A @ B @ F @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= ( proces1915862579Choice @ B @ ( proces370407749rocess @ A @ B @ F @ X21 ) @ ( proces370407749rocess @ A @ B @ F @ X22 ) ) ) ).
% process.simps(16)
thf(fact_114_process_Osimps_I15_J,axiom,
! [B: $tType,A: $tType,F: A > B,X11: A,X12: proces634752977rocess @ A] :
( ( proces370407749rocess @ A @ B @ F @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= ( proces1525233512Action @ B @ ( F @ X11 ) @ ( proces370407749rocess @ A @ B @ F @ X12 ) ) ) ).
% process.simps(15)
thf(fact_115_process_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X3: proces634752977rocess @ A,Xa: proces634752977rocess @ A,F: A > B,Fa: A > B] :
( ! [Z2: A,Za: A] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ X3 ) )
=> ( ( member @ A @ Za @ ( proces1148692671rocess @ A @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( proces370407749rocess @ A @ B @ F @ X3 )
= ( proces370407749rocess @ A @ B @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% process.inj_map_strong
thf(fact_116_process_Omap__cong0,axiom,
! [B: $tType,A: $tType,X3: proces634752977rocess @ A,F: A > B,G: A > B] :
( ! [Z2: A] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ X3 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( proces370407749rocess @ A @ B @ F @ X3 )
= ( proces370407749rocess @ A @ B @ G @ X3 ) ) ) ).
% process.map_cong0
thf(fact_117_process_Omap__cong,axiom,
! [B: $tType,A: $tType,X3: proces634752977rocess @ A,Ya: proces634752977rocess @ A,F: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( proces1148692671rocess @ A @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( proces370407749rocess @ A @ B @ F @ X3 )
= ( proces370407749rocess @ A @ B @ G @ Ya ) ) ) ) ).
% process.map_cong
thf(fact_118_process_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces10484146Action @ A @ A2 )
=> ( ( proces1778668539contOf @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces370407749rocess @ A @ B @ F @ ( proces1778668539contOf @ A @ A2 ) ) ) ) ).
% process.map_sel(2)
thf(fact_119_process_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces10484146Action @ A @ A2 )
=> ( ( proces745025900prefOf @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( F @ ( proces745025900prefOf @ A @ A2 ) ) ) ) ).
% process.map_sel(1)
thf(fact_120_process_Omap__sel_I3_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( proces979765041_ch1Of @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces370407749rocess @ A @ B @ F @ ( proces979765041_ch1Of @ A @ A2 ) ) ) ) ).
% process.map_sel(3)
thf(fact_121_insert__absorb2,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A6 ) )
= ( insert @ A @ X3 @ A6 ) ) ).
% insert_absorb2
thf(fact_122_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A6: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A6 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A6 ) ) ) ).
% insert_iff
thf(fact_123_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_124_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A6: set @ A] :
( ( member @ A @ A2 @ A6 )
=> ? [B5: set @ A] :
( ( A6
= ( insert @ A @ A2 @ B5 ) )
& ~ ( member @ A @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_125_insert__commute,axiom,
! [A: $tType,X3: A,Y2: A,A6: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ Y2 @ A6 ) )
= ( insert @ A @ Y2 @ ( insert @ A @ X3 @ A6 ) ) ) ).
% insert_commute
thf(fact_126_insertE,axiom,
! [A: $tType,A2: A,B3: A,A6: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A6 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A6 ) ) ) ).
% insertE
thf(fact_127_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_128_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_129_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A6: set @ A] :
( ( member @ A @ X3 @ A6 )
=> ~ ! [B5: set @ A] :
( ( A6
= ( insert @ A @ X3 @ B5 ) )
=> ( member @ A @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_130_insert__ident,axiom,
! [A: $tType,X3: A,A6: set @ A,B4: set @ A] :
( ~ ( member @ A @ X3 @ A6 )
=> ( ~ ( member @ A @ X3 @ B4 )
=> ( ( ( insert @ A @ X3 @ A6 )
= ( insert @ A @ X3 @ B4 ) )
= ( A6 = B4 ) ) ) ) ).
% insert_ident
thf(fact_131_insert__absorb,axiom,
! [A: $tType,A2: A,A6: set @ A] :
( ( member @ A @ A2 @ A6 )
=> ( ( insert @ A @ A2 @ A6 )
= A6 ) ) ).
% insert_absorb
thf(fact_132_insert__eq__iff,axiom,
! [A: $tType,A2: A,A6: set @ A,B3: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A6 )
=> ( ~ ( member @ A @ B3 @ B4 )
=> ( ( ( insert @ A @ A2 @ A6 )
= ( insert @ A @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A6 = B4 ) )
& ( ( A2 != B3 )
=> ? [C2: set @ A] :
( ( A6
= ( insert @ A @ B3 @ C2 ) )
& ~ ( member @ A @ B3 @ C2 )
& ( B4
= ( insert @ A @ A2 @ C2 ) )
& ~ ( member @ A @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_133_process__term_Opred__mono__strong,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,X3: proces554692349s_term @ A @ B,P1a: A > $o,P2a: B > $o] :
( ( proces2118920028s_term @ A @ B @ P1 @ P2 @ X3 )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X3 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z22: B] :
( ( member @ B @ Z22 @ ( proces1493547885s_term @ A @ B @ X3 ) )
=> ( ( P2 @ Z22 )
=> ( P2a @ Z22 ) ) )
=> ( proces2118920028s_term @ A @ B @ P1a @ P2a @ X3 ) ) ) ) ).
% process_term.pred_mono_strong
thf(fact_134_process__term_Opred__cong,axiom,
! [B: $tType,A: $tType,X3: proces554692349s_term @ A @ B,Ya: proces554692349s_term @ A @ B,P1: A > $o,P1a: A > $o,P2: B > $o,P2a: B > $o] :
( ( X3 = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z22: B] :
( ( member @ B @ Z22 @ ( proces1493547885s_term @ A @ B @ Ya ) )
=> ( ( P2 @ Z22 )
= ( P2a @ Z22 ) ) )
=> ( ( proces2118920028s_term @ A @ B @ P1 @ P2 @ X3 )
= ( proces2118920028s_term @ A @ B @ P1a @ P2a @ Ya ) ) ) ) ) ).
% process_term.pred_cong
thf(fact_135_stream_Oset,axiom,
! [A: $tType,X1: A,X23: stream @ A] :
( ( sset @ A @ ( sCons @ A @ X1 @ X23 ) )
= ( insert @ A @ X1 @ ( sset @ A @ X23 ) ) ) ).
% stream.set
thf(fact_136_process__term_Osimps_I47_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1264666542s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( insert @ A @ X31 @ ( proces1264666542s_term @ A @ Pvar @ X32 ) ) ) ).
% process_term.simps(47)
thf(fact_137_process__term_Oinject_I3_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar,Y31: A,Y32: proces554692349s_term @ A @ Pvar] :
( ( ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 )
= ( proces1454156180ss_ACT @ A @ Pvar @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% process_term.inject(3)
thf(fact_138_process__term_Opred__inject_I3_J,axiom,
! [Pvar: $tType,A: $tType,P1: A > $o,P2: Pvar > $o,A2: A,Aa2: proces554692349s_term @ A @ Pvar] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces1454156180ss_ACT @ A @ Pvar @ A2 @ Aa2 ) )
= ( ( P1 @ A2 )
& ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ Aa2 ) ) ) ).
% process_term.pred_inject(3)
thf(fact_139_process__term_Odistinct_I3_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
!= ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) ) ).
% process_term.distinct(3)
thf(fact_140_process__term_Oset__intros_I3_J,axiom,
! [Pvar: $tType,A: $tType,Xa: A,A23: proces554692349s_term @ A @ Pvar,A12: A] :
( ( member @ A @ Xa @ ( proces1264666542s_term @ A @ Pvar @ A23 ) )
=> ( member @ A @ Xa @ ( proces1264666542s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ A12 @ A23 ) ) ) ) ).
% process_term.set_intros(3)
thf(fact_141_process__term_Oset__intros_I2_J,axiom,
! [Pvar: $tType,A: $tType,A12: A,A23: proces554692349s_term @ A @ Pvar] : ( member @ A @ A12 @ ( proces1264666542s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ A12 @ A23 ) ) ) ).
% process_term.set_intros(2)
thf(fact_142_stream_Oset__intros_I2_J,axiom,
! [A: $tType,X3: A,A23: stream @ A,A12: A] :
( ( member @ A @ X3 @ ( sset @ A @ A23 ) )
=> ( member @ A @ X3 @ ( sset @ A @ ( sCons @ A @ A12 @ A23 ) ) ) ) ).
% stream.set_intros(2)
thf(fact_143_stream_Oset__intros_I1_J,axiom,
! [A: $tType,A12: A,A23: stream @ A] : ( member @ A @ A12 @ ( sset @ A @ ( sCons @ A @ A12 @ A23 ) ) ) ).
% stream.set_intros(1)
thf(fact_144_stream_Oset__cases,axiom,
! [A: $tType,E: A,A2: stream @ A] :
( ( member @ A @ E @ ( sset @ A @ A2 ) )
=> ( ! [Z22: stream @ A] :
( A2
!= ( sCons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: stream @ A] :
( ( A2
= ( sCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( sset @ A @ Z22 ) ) ) ) ) ).
% stream.set_cases
thf(fact_145_stream_Oset__induct,axiom,
! [A: $tType,X3: A,A2: stream @ A,P: A > ( stream @ A ) > $o] :
( ( member @ A @ X3 @ ( sset @ A @ A2 ) )
=> ( ! [Z1: A,Z22: stream @ A] : ( P @ Z1 @ ( sCons @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: stream @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( sset @ A @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( sCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% stream.set_induct
thf(fact_146_isACT_Osimps_I3_J,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,Sys: A > ( proces554692349s_term @ B @ C ),A2: D,T: proces554692349s_term @ D @ A] : ( proces687458811_isACT @ A @ B @ C @ D @ Sys @ ( proces1454156180ss_ACT @ D @ A @ A2 @ T ) ) ).
% isACT.simps(3)
thf(fact_147_process__term_Osimps_I19_J,axiom,
! [B: $tType,Pvar: $tType,A: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( F3 @ X31 @ X32 ) ) ).
% process_term.simps(19)
thf(fact_148_process__term_Oset__intros_I7_J,axiom,
! [Pvar: $tType,A: $tType,Xi: Pvar,A2b: proces554692349s_term @ A @ Pvar,A1b: A] :
( ( member @ Pvar @ Xi @ ( proces1493547885s_term @ A @ Pvar @ A2b ) )
=> ( member @ Pvar @ Xi @ ( proces1493547885s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ A1b @ A2b ) ) ) ) ).
% process_term.set_intros(7)
thf(fact_149_process__term_Osimps_I51_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1493547885s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( proces1493547885s_term @ A @ Pvar @ X32 ) ) ).
% process_term.simps(51)
thf(fact_150_process__term_Odistinct_I7_J,axiom,
! [Pvar: $tType,A: $tType,X23: proces634752977rocess @ A,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1062592052s_PROC @ A @ Pvar @ X23 )
!= ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) ) ).
% process_term.distinct(7)
thf(fact_151_process__term_Osimps_I23_J,axiom,
! [D: $tType,Pvar: $tType,A: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( F3 @ X31 @ X32 @ ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ X32 ) ) ) ).
% process_term.simps(23)
thf(fact_152_process__term_Oset__intros_I6_J,axiom,
! [A: $tType,Pvar: $tType,Ab: Pvar] : ( member @ Pvar @ Ab @ ( proces1493547885s_term @ A @ Pvar @ ( proces1627516585ss_VAR @ Pvar @ A @ Ab ) ) ) ).
% process_term.set_intros(6)
thf(fact_153_PREF_Osimps_I3_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ C ),A2: B,T: proces554692349s_term @ B @ A] :
( ( proces1708129104e_PREF @ A @ B @ C @ Sys @ ( proces1454156180ss_ACT @ B @ A @ A2 @ T ) )
= A2 ) ).
% PREF.simps(3)
thf(fact_154_CONT_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),A2: B,T: proces554692349s_term @ B @ A] :
( ( proces126235999e_CONT @ A @ B @ Sys @ ( proces1454156180ss_ACT @ B @ A @ A2 @ T ) )
= T ) ).
% CONT.simps(3)
thf(fact_155_Stream_Osmember__def,axiom,
! [A: $tType] :
( ( smember @ A )
= ( ^ [X5: A,S2: stream @ A] : ( member @ A @ X5 @ ( sset @ A @ S2 ) ) ) ) ).
% Stream.smember_def
thf(fact_156_process__term_Oset__cases_I1_J,axiom,
! [Pvar: $tType,A: $tType,E: A,A2: proces554692349s_term @ A @ Pvar] :
( ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ A2 ) )
=> ( ! [Z2: proces634752977rocess @ A] :
( ( A2
= ( proces1062592052s_PROC @ A @ Pvar @ Z2 ) )
=> ~ ( member @ A @ E @ ( proces1148692671rocess @ A @ Z2 ) ) )
=> ( ! [Z22: proces554692349s_term @ A @ Pvar] :
( A2
!= ( proces1454156180ss_ACT @ A @ Pvar @ E @ Z22 ) )
=> ( ! [Z1: A,Z22: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces1454156180ss_ACT @ A @ Pvar @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ Z22 ) ) )
=> ( ! [Z1: proces554692349s_term @ A @ Pvar] :
( ? [Z22: proces554692349s_term @ A @ Pvar] :
( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ Z1 ) ) )
=> ~ ! [Z1: proces554692349s_term @ A @ Pvar,Z22: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ Z22 ) ) ) ) ) ) ) ) ).
% process_term.set_cases(1)
thf(fact_157_process__term_Osimps_I49_J,axiom,
! [A: $tType,Pvar: $tType,X1: Pvar] :
( ( proces1493547885s_term @ A @ Pvar @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( insert @ Pvar @ X1 @ ( bot_bot @ ( set @ Pvar ) ) ) ) ).
% process_term.simps(49)
thf(fact_158_process__term_Oset__cases_I2_J,axiom,
! [Pvar: $tType,A: $tType,E: Pvar,A2: proces554692349s_term @ A @ Pvar] :
( ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ A2 ) )
=> ( ( A2
!= ( proces1627516585ss_VAR @ Pvar @ A @ E ) )
=> ( ! [Z1: A,Z22: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces1454156180ss_ACT @ A @ Pvar @ Z1 @ Z22 ) )
=> ~ ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ Z22 ) ) )
=> ( ! [Z1: proces554692349s_term @ A @ Pvar] :
( ? [Z22: proces554692349s_term @ A @ Pvar] :
( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z22 ) )
=> ~ ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ Z1 ) ) )
=> ~ ! [Z1: proces554692349s_term @ A @ Pvar,Z22: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z22 ) )
=> ~ ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ Z22 ) ) ) ) ) ) ) ).
% process_term.set_cases(2)
thf(fact_159_process__term_Opred__set,axiom,
! [B: $tType,A: $tType] :
( ( proces2118920028s_term @ A @ B )
= ( ^ [P12: A > $o,P22: B > $o,X5: proces554692349s_term @ A @ B] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ ( proces1264666542s_term @ A @ B @ X5 ) )
=> ( P12 @ Y4 ) )
& ! [Y4: B] :
( ( member @ B @ Y4 @ ( proces1493547885s_term @ A @ B @ X5 ) )
=> ( P22 @ Y4 ) ) ) ) ) ).
% process_term.pred_set
thf(fact_160_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X5: A] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_161_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_162_all__not__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ! [X5: A] :
~ ( member @ A @ X5 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_163_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_164_process__term_Oinject_I4_J,axiom,
! [Pvar: $tType,A: $tType,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ A @ Pvar,Y42: proces554692349s_term @ A @ Pvar] :
( ( ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 )
= ( proces89589571ess_CH @ A @ Pvar @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% process_term.inject(4)
thf(fact_165_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_166_ball__empty,axiom,
! [A: $tType,P: A > $o,X6: A] :
( ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X6 ) ) ).
% ball_empty
thf(fact_167_process__term_Opred__inject_I4_J,axiom,
! [Pvar: $tType,A: $tType,P1: A > $o,P2: Pvar > $o,A2: proces554692349s_term @ A @ Pvar,Aa2: proces554692349s_term @ A @ Pvar] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces89589571ess_CH @ A @ Pvar @ A2 @ Aa2 ) )
= ( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ A2 )
& ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ Aa2 ) ) ) ).
% process_term.pred_inject(4)
thf(fact_168_stream__all__iff,axiom,
! [A: $tType] :
( ( stream_all @ A )
= ( ^ [P9: A > $o,S2: stream @ A] :
! [X5: A] :
( ( member @ A @ X5 @ ( sset @ A @ S2 ) )
=> ( P9 @ X5 ) ) ) ) ).
% stream_all_iff
thf(fact_169_process__term_Oset__intros_I8_J,axiom,
! [Pvar: $tType,A: $tType,Xk: Pvar,A1c: proces554692349s_term @ A @ Pvar,A2c: proces554692349s_term @ A @ Pvar] :
( ( member @ Pvar @ Xk @ ( proces1493547885s_term @ A @ Pvar @ A1c ) )
=> ( member @ Pvar @ Xk @ ( proces1493547885s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1c @ A2c ) ) ) ) ).
% process_term.set_intros(8)
thf(fact_170_process__term_Oset__intros_I9_J,axiom,
! [Pvar: $tType,A: $tType,Xm: Pvar,A2c: proces554692349s_term @ A @ Pvar,A1c: proces554692349s_term @ A @ Pvar] :
( ( member @ Pvar @ Xm @ ( proces1493547885s_term @ A @ Pvar @ A2c ) )
=> ( member @ Pvar @ Xm @ ( proces1493547885s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1c @ A2c ) ) ) ) ).
% process_term.set_intros(9)
thf(fact_171_process__term_Odistinct_I11_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 )
!= ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) ) ).
% process_term.distinct(11)
thf(fact_172_process__term_Osimps_I24_J,axiom,
! [D: $tType,Pvar: $tType,A: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) )
= ( F4 @ X41 @ X42 @ ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ X41 ) @ ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ X42 ) ) ) ).
% process_term.simps(24)
thf(fact_173_CH1_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T1: proces554692349s_term @ B @ A,T22: proces554692349s_term @ B @ A] :
( ( proces1239275103le_CH1 @ A @ B @ Sys @ ( proces89589571ess_CH @ B @ A @ T1 @ T22 ) )
= T1 ) ).
% CH1.simps(3)
thf(fact_174_CH2_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,Sys: A > ( proces554692349s_term @ B @ A ),T1: proces554692349s_term @ B @ A,T22: proces554692349s_term @ B @ A] :
( ( proces1239275104le_CH2 @ A @ B @ Sys @ ( proces89589571ess_CH @ B @ A @ T1 @ T22 ) )
= T22 ) ).
% CH2.simps(3)
thf(fact_175_process__term_Odistinct_I9_J,axiom,
! [Pvar: $tType,A: $tType,X23: proces634752977rocess @ A,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces1062592052s_PROC @ A @ Pvar @ X23 )
!= ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) ) ).
% process_term.distinct(9)
thf(fact_176_process__term_Odistinct_I5_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
!= ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) ) ).
% process_term.distinct(5)
thf(fact_177_process__term_Osimps_I20_J,axiom,
! [B: $tType,Pvar: $tType,A: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) )
= ( F4 @ X41 @ X42 ) ) ).
% process_term.simps(20)
thf(fact_178_isACT_Osimps_I4_J,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,Sys: A > ( proces554692349s_term @ B @ C ),T1: proces554692349s_term @ D @ A,T22: proces554692349s_term @ D @ A] :
~ ( proces687458811_isACT @ A @ B @ C @ D @ Sys @ ( proces89589571ess_CH @ D @ A @ T1 @ T22 ) ) ).
% isACT.simps(4)
thf(fact_179_process__term_Oset__intros_I4_J,axiom,
! [Pvar: $tType,A: $tType,Xc2: A,A1a: proces554692349s_term @ A @ Pvar,A2a: proces554692349s_term @ A @ Pvar] :
( ( member @ A @ Xc2 @ ( proces1264666542s_term @ A @ Pvar @ A1a ) )
=> ( member @ A @ Xc2 @ ( proces1264666542s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1a @ A2a ) ) ) ) ).
% process_term.set_intros(4)
thf(fact_180_process__term_Oset__intros_I5_J,axiom,
! [Pvar: $tType,A: $tType,Xe: A,A2a: proces554692349s_term @ A @ Pvar,A1a: proces554692349s_term @ A @ Pvar] :
( ( member @ A @ Xe @ ( proces1264666542s_term @ A @ Pvar @ A2a ) )
=> ( member @ A @ Xe @ ( proces1264666542s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1a @ A2a ) ) ) ) ).
% process_term.set_intros(5)
thf(fact_181_ex__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ? [X5: A] : ( member @ A @ X5 @ A6 ) )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_182_equals0I,axiom,
! [A: $tType,A6: set @ A] :
( ! [Y5: A] :
~ ( member @ A @ Y5 @ A6 )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_183_equals0D,axiom,
! [A: $tType,A6: set @ A,A2: A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A6 ) ) ).
% equals0D
thf(fact_184_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P9: A > $o] :
! [X5: A] :
( ( member @ A @ X5 @ A8 )
=> ( P9 @ X5 ) ) ) ) ).
% Ball_def
thf(fact_185_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_186_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_187_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_188_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C3: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_189_insert__not__empty,axiom,
! [A: $tType,A2: A,A6: set @ A] :
( ( insert @ A @ A2 @ A6 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_190_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_191_process_Opred__set,axiom,
! [A: $tType] :
( ( proces1205983068rocess @ A )
= ( ^ [P9: A > $o,X5: proces634752977rocess @ A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( proces1148692671rocess @ A @ X5 ) )
=> ( P9 @ Y4 ) ) ) ) ).
% process.pred_set
thf(fact_192_process__term_Osimps_I50_J,axiom,
! [A: $tType,Pvar: $tType,X23: proces634752977rocess @ A] :
( ( proces1493547885s_term @ A @ Pvar @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) )
= ( bot_bot @ ( set @ Pvar ) ) ) ).
% process_term.simps(50)
thf(fact_193_process__term_Osimps_I45_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar] :
( ( proces1264666542s_term @ A @ Pvar @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% process_term.simps(45)
thf(fact_194_CH2_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > $o,A0: A > ( proces554692349s_term @ B @ A ),A12: proces554692349s_term @ B @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X7: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X7 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P6: proces634752977rocess @ B] : ( P @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P6 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),T12: proces554692349s_term @ B @ A,T23: proces554692349s_term @ B @ A] : ( P @ Sys3 @ ( proces89589571ess_CH @ B @ A @ T12 @ T23 ) )
=> ( ! [A5: A > ( proces554692349s_term @ B @ A ),V: B,Va: proces554692349s_term @ B @ A] : ( P @ A5 @ ( proces1454156180ss_ACT @ B @ A @ V @ Va ) )
=> ( P @ A0 @ A12 ) ) ) ) ) ).
% CH2.induct
thf(fact_195_CONT_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > $o,A0: A > ( proces554692349s_term @ B @ A ),A12: proces554692349s_term @ B @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X7: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X7 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P6: proces634752977rocess @ B] : ( P @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P6 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),A5: B,T3: proces554692349s_term @ B @ A] : ( P @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A5 @ T3 ) )
=> ( ! [A5: A > ( proces554692349s_term @ B @ A ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] : ( P @ A5 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) )
=> ( P @ A0 @ A12 ) ) ) ) ) ).
% CONT.induct
thf(fact_196_PREF_Oinduct,axiom,
! [C: $tType,A: $tType,B: $tType,P: ( A > ( proces554692349s_term @ B @ C ) ) > ( proces554692349s_term @ B @ A ) > $o,A0: A > ( proces554692349s_term @ B @ C ),A12: proces554692349s_term @ B @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X7: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X7 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P6: proces634752977rocess @ B] : ( P @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P6 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A5: B,T3: proces554692349s_term @ B @ A] : ( P @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A5 @ T3 ) )
=> ( ! [A5: A > ( proces554692349s_term @ B @ C ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] : ( P @ A5 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) )
=> ( P @ A0 @ A12 ) ) ) ) ) ).
% PREF.induct
thf(fact_197_isACT_Oinduct,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,P: ( A > ( proces554692349s_term @ B @ C ) ) > ( proces554692349s_term @ D @ A ) > $o,A0: A > ( proces554692349s_term @ B @ C ),A12: proces554692349s_term @ D @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X7: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ D @ X7 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P6: proces634752977rocess @ D] : ( P @ Sys3 @ ( proces1062592052s_PROC @ D @ A @ P6 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A5: D,T3: proces554692349s_term @ D @ A] : ( P @ Sys3 @ ( proces1454156180ss_ACT @ D @ A @ A5 @ T3 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),T12: proces554692349s_term @ D @ A,T23: proces554692349s_term @ D @ A] : ( P @ Sys3 @ ( proces89589571ess_CH @ D @ A @ T12 @ T23 ) )
=> ( P @ A0 @ A12 ) ) ) ) ) ).
% isACT.induct
thf(fact_198_process__term_Oinduct,axiom,
! [Pvar: $tType,A: $tType,P: ( proces554692349s_term @ A @ Pvar ) > $o,Process_term: proces554692349s_term @ A @ Pvar] :
( ! [X4: Pvar] : ( P @ ( proces1627516585ss_VAR @ Pvar @ A @ X4 ) )
=> ( ! [X4: proces634752977rocess @ A] : ( P @ ( proces1062592052s_PROC @ A @ Pvar @ X4 ) )
=> ( ! [X1a: A,X2a: proces554692349s_term @ A @ Pvar] :
( ( P @ X2a )
=> ( P @ ( proces1454156180ss_ACT @ A @ Pvar @ X1a @ X2a ) ) )
=> ( ! [X1a: proces554692349s_term @ A @ Pvar,X2a: proces554692349s_term @ A @ Pvar] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( proces89589571ess_CH @ A @ Pvar @ X1a @ X2a ) ) ) )
=> ( P @ Process_term ) ) ) ) ) ).
% process_term.induct
thf(fact_199_process__term_Oexhaust,axiom,
! [Pvar: $tType,A: $tType,Y2: proces554692349s_term @ A @ Pvar] :
( ! [X13: Pvar] :
( Y2
!= ( proces1627516585ss_VAR @ Pvar @ A @ X13 ) )
=> ( ! [X24: proces634752977rocess @ A] :
( Y2
!= ( proces1062592052s_PROC @ A @ Pvar @ X24 ) )
=> ( ! [X312: A,X322: proces554692349s_term @ A @ Pvar] :
( Y2
!= ( proces1454156180ss_ACT @ A @ Pvar @ X312 @ X322 ) )
=> ~ ! [X412: proces554692349s_term @ A @ Pvar,X422: proces554692349s_term @ A @ Pvar] :
( Y2
!= ( proces89589571ess_CH @ A @ Pvar @ X412 @ X422 ) ) ) ) ) ).
% process_term.exhaust
thf(fact_200_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_201_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_202_process__term_Orel__cases,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,A2: proces554692349s_term @ A @ Pvar,B3: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ A2 @ B3 )
=> ( ! [X4: Pvar] :
( ( A2
= ( proces1627516585ss_VAR @ Pvar @ A @ X4 ) )
=> ! [Y5: C] :
( ( B3
= ( proces1627516585ss_VAR @ C @ B @ Y5 ) )
=> ~ ( R22 @ X4 @ Y5 ) ) )
=> ( ! [Xa2: proces634752977rocess @ A] :
( ( A2
= ( proces1062592052s_PROC @ A @ Pvar @ Xa2 ) )
=> ! [Ya2: proces634752977rocess @ B] :
( ( B3
= ( proces1062592052s_PROC @ B @ C @ Ya2 ) )
=> ~ ( proces749077512rocess @ A @ B @ R1 @ Xa2 @ Ya2 ) ) )
=> ( ! [X1a: A,X2a: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces1454156180ss_ACT @ A @ Pvar @ X1a @ X2a ) )
=> ! [Y1a: B,Y2a: proces554692349s_term @ B @ C] :
( ( B3
= ( proces1454156180ss_ACT @ B @ C @ Y1a @ Y2a ) )
=> ( ( R1 @ X1a @ Y1a )
=> ~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X2a @ Y2a ) ) ) )
=> ~ ! [X1b: proces554692349s_term @ A @ Pvar,X2b: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces89589571ess_CH @ A @ Pvar @ X1b @ X2b ) )
=> ! [Y1b: proces554692349s_term @ B @ C,Y2b: proces554692349s_term @ B @ C] :
( ( B3
= ( proces89589571ess_CH @ B @ C @ Y1b @ Y2b ) )
=> ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X1b @ Y1b )
=> ~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X2b @ Y2b ) ) ) ) ) ) ) ) ).
% process_term.rel_cases
thf(fact_203_process__term_Orel__inject_I4_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) )
= ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X41 @ Y41 )
& ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X42 @ Y42 ) ) ) ).
% process_term.rel_inject(4)
thf(fact_204_process__term_Orel__intros_I4_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X41: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ B @ C,X42: proces554692349s_term @ A @ Pvar,Y42: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X41 @ Y41 )
=> ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X42 @ Y42 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ) ) ).
% process_term.rel_intros(4)
thf(fact_205_process__term_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X3: proces554692349s_term @ C @ D] :
( ! [X4: C] : ( R1a @ X4 @ X4 )
=> ( ! [X4: D] : ( R2a @ X4 @ X4 )
=> ( proces2029722208s_term @ C @ C @ D @ D @ R1a @ R2a @ X3 @ X3 ) ) ) ).
% process_term.rel_refl
thf(fact_206_process__term_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( proces2029722208s_term @ A @ A @ B @ B
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [Y3: B,Z: B] : Y3 = Z )
= ( ^ [Y3: proces554692349s_term @ A @ B,Z: proces554692349s_term @ A @ B] : Y3 = Z ) ) ).
% process_term.rel_eq
thf(fact_207_process__term_Orel__intros_I1_J,axiom,
! [Pvar: $tType,A: $tType,B: $tType,C: $tType,R22: Pvar > C > $o,X1: Pvar,Y1: C,R1: A > B > $o] :
( ( R22 @ X1 @ Y1 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1627516585ss_VAR @ C @ B @ Y1 ) ) ) ).
% process_term.rel_intros(1)
thf(fact_208_process__term_Orel__inject_I1_J,axiom,
! [B: $tType,A: $tType,Pvar: $tType,C: $tType,R1: A > B > $o,R22: Pvar > C > $o,X1: Pvar,Y1: C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1627516585ss_VAR @ C @ B @ Y1 ) )
= ( R22 @ X1 @ Y1 ) ) ).
% process_term.rel_inject(1)
thf(fact_209_process__term_Orel__inject_I3_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X31: A,X32: proces554692349s_term @ A @ Pvar,Y31: B,Y32: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) )
= ( ( R1 @ X31 @ Y31 )
& ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X32 @ Y32 ) ) ) ).
% process_term.rel_inject(3)
thf(fact_210_process__term_Orel__intros_I3_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,X31: A,Y31: B,R22: Pvar > C > $o,X32: proces554692349s_term @ A @ Pvar,Y32: proces554692349s_term @ B @ C] :
( ( R1 @ X31 @ Y31 )
=> ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X32 @ Y32 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) ) ) ) ).
% process_term.rel_intros(3)
thf(fact_211_process__term_Orel__intros_I2_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,X23: proces634752977rocess @ A,Y23: proces634752977rocess @ B,R22: Pvar > C > $o] :
( ( proces749077512rocess @ A @ B @ R1 @ X23 @ Y23 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) @ ( proces1062592052s_PROC @ B @ C @ Y23 ) ) ) ).
% process_term.rel_intros(2)
thf(fact_212_process__term_Orel__inject_I2_J,axiom,
! [C: $tType,Pvar: $tType,A: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X23: proces634752977rocess @ A,Y23: proces634752977rocess @ B] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) @ ( proces1062592052s_PROC @ B @ C @ Y23 ) )
= ( proces749077512rocess @ A @ B @ R1 @ X23 @ Y23 ) ) ).
% process_term.rel_inject(2)
thf(fact_213_process__term_Orel__distinct_I1_J,axiom,
! [Pvar: $tType,A: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X1: Pvar,Y23: proces634752977rocess @ B] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1062592052s_PROC @ B @ C @ Y23 ) ) ).
% process_term.rel_distinct(1)
thf(fact_214_process__term_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R22: B > D > $o,Y23: proces634752977rocess @ A,X1: D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ ( proces1062592052s_PROC @ A @ B @ Y23 ) @ ( proces1627516585ss_VAR @ D @ C @ X1 ) ) ).
% process_term.rel_distinct(2)
thf(fact_215_process__term_Orel__distinct_I10_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,Y41: proces554692349s_term @ A @ B,Y42: proces554692349s_term @ A @ B,X23: proces634752977rocess @ C] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ ( proces89589571ess_CH @ A @ B @ Y41 @ Y42 ) @ ( proces1062592052s_PROC @ C @ D @ X23 ) ) ).
% process_term.rel_distinct(10)
thf(fact_216_process__term_Orel__distinct_I9_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X23: proces634752977rocess @ A,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ).
% process_term.rel_distinct(9)
thf(fact_217_process__term_Orel__distinct_I8_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,Y31: A,Y32: proces554692349s_term @ A @ B,X23: proces634752977rocess @ C] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ ( proces1454156180ss_ACT @ A @ B @ Y31 @ Y32 ) @ ( proces1062592052s_PROC @ C @ D @ X23 ) ) ).
% process_term.rel_distinct(8)
thf(fact_218_process__term_Orel__distinct_I7_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X23: proces634752977rocess @ A,Y31: B,Y32: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1062592052s_PROC @ A @ Pvar @ X23 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) ) ).
% process_term.rel_distinct(7)
thf(fact_219_process__term_Orel__distinct_I5_J,axiom,
! [Pvar: $tType,A: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X1: Pvar,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ).
% process_term.rel_distinct(5)
thf(fact_220_process__term_Orel__distinct_I6_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R22: B > D > $o,Y41: proces554692349s_term @ A @ B,Y42: proces554692349s_term @ A @ B,X1: D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ ( proces89589571ess_CH @ A @ B @ Y41 @ Y42 ) @ ( proces1627516585ss_VAR @ D @ C @ X1 ) ) ).
% process_term.rel_distinct(6)
thf(fact_221_process__term_Orel__distinct_I3_J,axiom,
! [Pvar: $tType,A: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X1: Pvar,Y31: B,Y32: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) ) ).
% process_term.rel_distinct(3)
thf(fact_222_process__term_Orel__distinct_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R22: B > D > $o,Y31: A,Y32: proces554692349s_term @ A @ B,X1: D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ ( proces1454156180ss_ACT @ A @ B @ Y31 @ Y32 ) @ ( proces1627516585ss_VAR @ D @ C @ X1 ) ) ).
% process_term.rel_distinct(4)
thf(fact_223_process__term_Orel__distinct_I12_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,Y41: proces554692349s_term @ A @ B,Y42: proces554692349s_term @ A @ B,X31: C,X32: proces554692349s_term @ C @ D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ ( proces89589571ess_CH @ A @ B @ Y41 @ Y42 ) @ ( proces1454156180ss_ACT @ C @ D @ X31 @ X32 ) ) ).
% process_term.rel_distinct(12)
thf(fact_224_process__term_Orel__distinct_I11_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X31: A,X32: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ).
% process_term.rel_distinct(11)
thf(fact_225_process__term_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,X3: proces554692349s_term @ A @ B,Ya: proces554692349s_term @ A @ B,Y2: proces554692349s_term @ C @ D,Xa: proces554692349s_term @ C @ D,R1: A > C > $o,R1a: A > C > $o,R22: B > D > $o,R2a: B > D > $o] :
( ( X3 = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z1: A,Y13: C] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ Ya ) )
=> ( ( member @ C @ Y13 @ ( proces1264666542s_term @ C @ D @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: B,Y24: D] :
( ( member @ B @ Z22 @ ( proces1493547885s_term @ A @ B @ Ya ) )
=> ( ( member @ D @ Y24 @ ( proces1493547885s_term @ C @ D @ Xa ) )
=> ( ( R22 @ Z22 @ Y24 )
= ( R2a @ Z22 @ Y24 ) ) ) )
=> ( ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ X3 @ Y2 )
= ( proces2029722208s_term @ A @ C @ B @ D @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% process_term.rel_cong
thf(fact_226_process__term_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,X3: proces554692349s_term @ A @ B,Y2: proces554692349s_term @ C @ D,R1a: A > C > $o,R2a: B > D > $o] :
( ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R22 @ X3 @ Y2 )
=> ( ! [Z1: A,Y13: C] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X3 ) )
=> ( ( member @ C @ Y13 @ ( proces1264666542s_term @ C @ D @ Y2 ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: B,Y24: D] :
( ( member @ B @ Z22 @ ( proces1493547885s_term @ A @ B @ X3 ) )
=> ( ( member @ D @ Y24 @ ( proces1493547885s_term @ C @ D @ Y2 ) )
=> ( ( R22 @ Z22 @ Y24 )
=> ( R2a @ Z22 @ Y24 ) ) ) )
=> ( proces2029722208s_term @ A @ C @ B @ D @ R1a @ R2a @ X3 @ Y2 ) ) ) ) ).
% process_term.rel_mono_strong
thf(fact_227_process__term_Orel__refl__strong,axiom,
! [B: $tType,A: $tType,X3: proces554692349s_term @ A @ B,R1a: A > A > $o,R2a: B > B > $o] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X3 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: B] :
( ( member @ B @ Z22 @ ( proces1493547885s_term @ A @ B @ X3 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( proces2029722208s_term @ A @ A @ B @ B @ R1a @ R2a @ X3 @ X3 ) ) ) ).
% process_term.rel_refl_strong
thf(fact_228_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
( A8
= ( insert @ A @ ( the_elem @ A @ A8 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_229_is__singletonI_H,axiom,
! [A: $tType,A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y5: A] :
( ( member @ A @ X4 @ A6 )
=> ( ( member @ A @ Y5 @ A6 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton @ A @ A6 ) ) ) ).
% is_singletonI'
thf(fact_230_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
? [X5: A] :
( A8
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_231_is__singletonE,axiom,
! [A: $tType,A6: set @ A] :
( ( is_singleton @ A @ A6 )
=> ~ ! [X4: A] :
( A6
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_232_process__term_Orel__induct,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R22: Pvar > C > $o,X3: proces554692349s_term @ A @ Pvar,Y2: proces554692349s_term @ B @ C,Q5: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ B @ C ) > $o] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R22 @ X3 @ Y2 )
=> ( ! [A13: Pvar,B1: C] :
( ( R22 @ A13 @ B1 )
=> ( Q5 @ ( proces1627516585ss_VAR @ Pvar @ A @ A13 ) @ ( proces1627516585ss_VAR @ C @ B @ B1 ) ) )
=> ( ! [A24: proces634752977rocess @ A,B22: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R1 @ A24 @ B22 )
=> ( Q5 @ ( proces1062592052s_PROC @ A @ Pvar @ A24 ) @ ( proces1062592052s_PROC @ B @ C @ B22 ) ) )
=> ( ! [A31: A,A32: proces554692349s_term @ A @ Pvar,B31: B,B32: proces554692349s_term @ B @ C] :
( ( R1 @ A31 @ B31 )
=> ( ( Q5 @ A32 @ B32 )
=> ( Q5 @ ( proces1454156180ss_ACT @ A @ Pvar @ A31 @ A32 ) @ ( proces1454156180ss_ACT @ B @ C @ B31 @ B32 ) ) ) )
=> ( ! [A41: proces554692349s_term @ A @ Pvar,A42: proces554692349s_term @ A @ Pvar,B41: proces554692349s_term @ B @ C,B42: proces554692349s_term @ B @ C] :
( ( Q5 @ A41 @ B41 )
=> ( ( Q5 @ A42 @ B42 )
=> ( Q5 @ ( proces89589571ess_CH @ A @ Pvar @ A41 @ A42 ) @ ( proces89589571ess_CH @ B @ C @ B41 @ B42 ) ) ) )
=> ( Q5 @ X3 @ Y2 ) ) ) ) ) ) ).
% process_term.rel_induct
thf(fact_233_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A8: set @ A] :
( A8
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_234_isACT_Ocases,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,X3: product_prod @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X7: A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ D @ X7 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P6: proces634752977rocess @ D] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces1062592052s_PROC @ D @ A @ P6 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A5: D,T3: proces554692349s_term @ D @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces1454156180ss_ACT @ D @ A @ A5 @ T3 ) ) )
=> ~ ! [Sys3: A > ( proces554692349s_term @ B @ C ),T12: proces554692349s_term @ D @ A,T23: proces554692349s_term @ D @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces89589571ess_CH @ D @ A @ T12 @ T23 ) ) ) ) ) ) ).
% isACT.cases
thf(fact_235_CH2_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X7: A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X7 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P6: proces634752977rocess @ B] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P6 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),T12: proces554692349s_term @ B @ A,T23: proces554692349s_term @ B @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces89589571ess_CH @ B @ A @ T12 @ T23 ) ) )
=> ~ ! [A5: A > ( proces554692349s_term @ B @ A ),V: B,Va: proces554692349s_term @ B @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ A5 @ ( proces1454156180ss_ACT @ B @ A @ V @ Va ) ) ) ) ) ) ).
% CH2.cases
thf(fact_236_CONT_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X7: A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X7 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P6: proces634752977rocess @ B] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P6 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),A5: B,T3: proces554692349s_term @ B @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A5 @ T3 ) ) )
=> ~ ! [A5: A > ( proces554692349s_term @ B @ A ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ A5 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) ) ) ) ) ) ).
% CONT.cases
thf(fact_237_PREF_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X3: product_prod @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X7: A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X7 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P6: proces634752977rocess @ B] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P6 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A5: B,T3: proces554692349s_term @ B @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A5 @ T3 ) ) )
=> ~ ! [A5: A > ( proces554692349s_term @ B @ C ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] :
( X3
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ A5 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) ) ) ) ) ) ).
% PREF.cases
thf(fact_238_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A6: A] : ( pairwise @ A @ P @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_239_szip__unfold,axiom,
! [A: $tType,B: $tType,A2: A,S1: stream @ A,B3: B,S22: stream @ B] :
( ( szip @ A @ B @ ( sCons @ A @ A2 @ S1 ) @ ( sCons @ B @ B3 @ S22 ) )
= ( sCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B3 ) @ ( szip @ A @ B @ S1 @ S22 ) ) ) ).
% szip_unfold
thf(fact_240_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S3: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S3 )
=> ( ( X5 != Y4 )
=> ( R2 @ X5 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_241_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X3: A,S: set @ A] :
( ( pairwise @ A @ R3 @ ( insert @ A @ X3 @ S ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S )
& ( Y4 != X3 ) )
=> ( ( R3 @ X3 @ Y4 )
& ( R3 @ Y4 @ X3 ) ) )
& ( pairwise @ A @ R3 @ S ) ) ) ).
% pairwise_insert
thf(fact_242_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_243_stream_Opred__set,axiom,
! [A: $tType] :
( ( pred_stream @ A )
= ( ^ [P9: A > $o,X5: stream @ A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( sset @ A @ X5 ) )
=> ( P9 @ Y4 ) ) ) ) ).
% stream.pred_set
thf(fact_244_stream_Opred__inject,axiom,
! [A: $tType,P: A > $o,A2: A,Aa2: stream @ A] :
( ( pred_stream @ A @ P @ ( sCons @ A @ A2 @ Aa2 ) )
= ( ( P @ A2 )
& ( pred_stream @ A @ P @ Aa2 ) ) ) ).
% stream.pred_inject
thf(fact_245_stream_Opred__cong,axiom,
! [A: $tType,X3: stream @ A,Ya: stream @ A,P: A > $o,Pa: A > $o] :
( ( X3 = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( sset @ A @ Ya ) )
=> ( ( P @ Z2 )
= ( Pa @ Z2 ) ) )
=> ( ( pred_stream @ A @ P @ X3 )
= ( pred_stream @ A @ Pa @ Ya ) ) ) ) ).
% stream.pred_cong
thf(fact_246_stream_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X3: stream @ A,Pa: A > $o] :
( ( pred_stream @ A @ P @ X3 )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( sset @ A @ X3 ) )
=> ( ( P @ Z2 )
=> ( Pa @ Z2 ) ) )
=> ( pred_stream @ A @ Pa @ X3 ) ) ) ).
% stream.pred_mono_strong
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $true @ X3 @ Y2 )
= X3 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( proces1652378886lution @ a @ b @ sys @ ( proces1627516585ss_VAR @ a @ b @ x ) )
= ( proces1915862579Choice @ b @ ( proces1652378886lution @ a @ b @ sys @ ( proces1239275103le_CH1 @ a @ b @ sys @ ( sys @ x ) ) ) @ ( proces1652378886lution @ a @ b @ sys @ ( proces1239275104le_CH2 @ a @ b @ sys @ ( sys @ x ) ) ) ) ) ).
%------------------------------------------------------------------------------