TPTP Problem File: ITP183^1.p
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%------------------------------------------------------------------------------
% File : ITP183^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_329__3411594_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Strong_Late_Sim_SC/prob_329__3411594_1 [Des21]
% Status : Theorem
% Rating : 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 431 ( 161 unt; 117 typ; 0 def)
% Number of atoms : 906 ( 318 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 3454 ( 150 ~; 10 |; 78 &;2776 @)
% ( 0 <=>; 440 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 8 avg)
% Number of types : 32 ( 31 usr)
% Number of type conns : 303 ( 303 >; 0 *; 0 +; 0 <<)
% Number of symbols : 87 ( 86 usr; 11 con; 0-5 aty)
% Number of variables : 1290 ( 26 ^;1226 !; 38 ?;1290 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:50.938
%------------------------------------------------------------------------------
% Could-be-implicit typings (31)
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% Explicit typings (86)
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thf(sy_v_PQ____,type,
pq: pi ).
thf(sy_v_Q,type,
q: pi ).
thf(sy_v_Q_H____,type,
q2: pi ).
thf(sy_v_Rel,type,
rel: set_Pr1834926343_pi_pi ).
thf(sy_v__092_060alpha_062____,type,
alpha: late_freeRes ).
thf(sy_v_x,type,
x: name ).
% Relevant facts (313)
thf(fact_0_cRes_Ohyps_I2_J,axiom,
fresh_1641682979reeRes @ x @ alpha ).
% cRes.hyps(2)
thf(fact_1_Eqvt,axiom,
eqvt_pi @ rel ).
% Eqvt
thf(fact_2__092_060open_062P_A_092_060oplus_062_AQ_A_092_060longmapsto_062_A_092_060alpha_062_A_092_060prec_062_AQ_H_092_060close_062,axiom,
late_transitions @ ( sum @ p @ q ) @ ( late_FreeR @ alpha @ q2 ) ).
% \<open>P \<oplus> Q \<longmapsto> \<alpha> \<prec> Q'\<close>
thf(fact_3_cRes_Ohyps_I1_J,axiom,
late_transitions @ q @ ( late_FreeR @ alpha @ q2 ) ).
% cRes.hyps(1)
thf(fact_4_cSum2_Ohyps,axiom,
late_transitions @ ( res @ x @ q ) @ ( late_FreeR @ alpha @ pq ) ).
% cSum2.hyps
thf(fact_5_Free_Ohyps,axiom,
late_transitions @ ( sum @ ( res @ x @ p ) @ ( res @ x @ q ) ) @ ( late_FreeR @ alpha @ pq ) ).
% Free.hyps
thf(fact_6_pi_Odistinct_I81_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(81)
thf(fact_7_sumCases,axiom,
! [P: pi,Q: pi,Rs: late_residual] :
( ( late_transitions @ ( sum @ P @ Q ) @ Rs )
=> ( ~ ( late_transitions @ P @ Rs )
=> ( late_transitions @ Q @ Rs ) ) ) ).
% sumCases
thf(fact_8_sumCases_H,axiom,
! [P: pi,Q: pi,Rs: late_residual] :
( ( late_transitions @ ( sum @ P @ Q ) @ Rs )
=> ( ~ ( late_transitions @ P @ Rs )
=> ( late_transitions @ Q @ Rs ) ) ) ).
% sumCases'
thf(fact_9_Sum1,axiom,
! [P: pi,Rs: late_residual,Q: pi] :
( ( late_transitions @ P @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum1
thf(fact_10_Sum2,axiom,
! [Q: pi,Rs: late_residual,P: pi] :
( ( late_transitions @ Q @ Rs )
=> ( late_transitions @ ( sum @ P @ Q ) @ Rs ) ) ).
% Sum2
thf(fact_11_resCasesF,axiom,
! [X: name,P: pi,Alpha: late_freeRes,XP: pi,F: pi > $o] :
( ( late_transitions @ ( res @ X @ P ) @ ( late_FreeR @ Alpha @ XP ) )
=> ( ! [P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha )
=> ( F @ ( res @ X @ P2 ) ) ) )
=> ( F @ XP ) ) ) ).
% resCasesF
thf(fact_12_resCasesF_H,axiom,
! [X: name,P: pi,Alpha: late_freeRes,P3: pi] :
( ( late_transitions @ ( res @ X @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P2: pi,Y: name] :
( ( ( res @ X @ P )
= ( res @ Y @ P4 ) )
=> ( ( ( late_FreeR @ Alpha @ P3 )
= ( late_FreeR @ Alpha2 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P2 ) )
=> ~ ( fresh_1641682979reeRes @ Y @ Alpha2 ) ) ) ) ) ).
% resCasesF'
thf(fact_13_ResF,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh_1641682979reeRes @ Y2 @ Alpha )
=> ( late_transitions @ ( res @ Y2 @ P ) @ ( late_FreeR @ Alpha @ ( res @ Y2 @ P3 ) ) ) ) ) ).
% ResF
thf(fact_14_pi_Oinject_I6_J,axiom,
! [X2: pi,X1: pi,Y22: pi,Y1: pi] :
( ( ( sum @ X2 @ X1 )
= ( sum @ Y22 @ Y1 ) )
= ( ( X2 = Y22 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(6)
thf(fact_15_residual_Oinject_I2_J,axiom,
! [X2: late_freeRes,X1: pi,Y22: late_freeRes,Y1: pi] :
( ( ( late_FreeR @ X2 @ X1 )
= ( late_FreeR @ Y22 @ Y1 ) )
= ( ( X2 = Y22 )
& ( X1 = Y1 ) ) ) ).
% residual.inject(2)
thf(fact_16_resInputFreeTrans,axiom,
! [X: name,A: name,Y2: name,P: pi,Alpha: late_freeRes,P3: pi] :
~ ( late_transitions @ ( res @ X @ ( input @ A @ Y2 @ P ) ) @ ( late_FreeR @ Alpha @ P3 ) ) ).
% resInputFreeTrans
thf(fact_17_resZeroTrans,axiom,
! [X: name,Rs: late_residual] :
~ ( late_transitions @ ( res @ X @ piNil ) @ Rs ) ).
% resZeroTrans
thf(fact_18_pi_Odistinct_I5_J,axiom,
! [Name1: name,Name2: name,Pi: pi] :
( piNil
!= ( input @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(5)
thf(fact_19_name__exists__fresh,axiom,
! [X: late_subject] :
~ ! [C: name] :
~ ( fresh_717452603ubject @ C @ X ) ).
% name_exists_fresh
thf(fact_20_name__exists__fresh,axiom,
! [X: late_freeRes] :
~ ! [C: name] :
~ ( fresh_1641682979reeRes @ C @ X ) ).
% name_exists_fresh
thf(fact_21_name__exists__fresh,axiom,
! [X: name] :
~ ! [C: name] :
~ ( fresh_name_name @ C @ X ) ).
% name_exists_fresh
thf(fact_22_name__exists__fresh,axiom,
! [X: pi] :
~ ! [C: name] :
~ ( fresh_name_pi @ C @ X ) ).
% name_exists_fresh
thf(fact_23_pi_Odistinct_I57_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( input @ Name12 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(57)
thf(fact_24_zeroTrans,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% zeroTrans
thf(fact_25_nilCases_H,axiom,
! [Rs: late_residual] :
~ ( late_transitions @ piNil @ Rs ) ).
% nilCases'
thf(fact_26_pi_Odistinct_I53_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name12 @ Name22 @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(53)
thf(fact_27_pi_Odistinct_I15_J,axiom,
! [Name: name,Pi: pi] :
( piNil
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(15)
thf(fact_28_pi_Odistinct_I11_J,axiom,
! [Pi12: pi,Pi22: pi] :
( piNil
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(11)
thf(fact_29_resTrans_I2_J,axiom,
! [X: name,Y2: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X @ ( input @ X @ Y2 @ P ) ) @ Rs ) ).
% resTrans(2)
thf(fact_30_inputFreeTrans,axiom,
! [A: name,X: name,P: pi,Alpha: late_freeRes,P3: pi] :
~ ( late_transitions @ ( input @ A @ X @ P ) @ ( late_FreeR @ Alpha @ P3 ) ) ).
% inputFreeTrans
thf(fact_31_Id,axiom,
ord_le630093991_pi_pi @ id_pi @ rel ).
% Id
thf(fact_32_outputFreshTrans,axiom,
! [A: name,B: name,P: pi,Alpha: late_freeRes,P3: pi] :
( ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ~ ( ( fresh_1641682979reeRes @ A @ Alpha )
| ( fresh_1641682979reeRes @ B @ Alpha ) ) ) ).
% outputFreshTrans
thf(fact_33_freshFreeDerivative_I1_J,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh_name_pi @ Y2 @ P )
=> ( fresh_1641682979reeRes @ Y2 @ Alpha ) ) ) ).
% freshFreeDerivative(1)
thf(fact_34_nilSim_I2_J,axiom,
! [Rel: set_Pr1834926343_pi_pi,A: name,X: name,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( input @ A @ X @ P ) ) ).
% nilSim(2)
thf(fact_35_resTrans_I1_J,axiom,
! [X: name,B: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X @ ( output @ X @ B @ P ) ) @ Rs ) ).
% resTrans(1)
thf(fact_36_freeRes_Ofresh_I2_J,axiom,
! [A: name] : ( fresh_1641682979reeRes @ A @ late_TauR ) ).
% freeRes.fresh(2)
thf(fact_37_freshFreeDerivative_I2_J,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh_name_pi @ Y2 @ P )
=> ( fresh_name_pi @ Y2 @ P3 ) ) ) ).
% freshFreeDerivative(2)
thf(fact_38_residual_Ostrong__induct,axiom,
! [P: late_subject > late_residual > $o,Z: late_subject,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: late_subject] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_39_residual_Ostrong__induct,axiom,
! [P: late_freeRes > late_residual > $o,Z: late_freeRes,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: late_freeRes] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_40_residual_Ostrong__induct,axiom,
! [P: name > late_residual > $o,Z: name,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: name] :
( ( fresh_name_name @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: name] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_41_residual_Ostrong__induct,axiom,
! [P: pi > late_residual > $o,Z: pi,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: pi] :
( ( fresh_name_pi @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: pi] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_42_residual_Ostrong__inducts,axiom,
! [P: late_subject > late_residual > $o,Z: late_subject,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: late_subject] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_43_residual_Ostrong__inducts,axiom,
! [P: late_freeRes > late_residual > $o,Z: late_freeRes,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: late_freeRes] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_44_residual_Ostrong__inducts,axiom,
! [P: name > late_residual > $o,Z: name,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: name] :
( ( fresh_name_name @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: name] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_45_residual_Ostrong__inducts,axiom,
! [P: pi > late_residual > $o,Z: pi,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi,Z2: pi] :
( ( fresh_name_pi @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject )
=> ( P @ Z2 @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi,Z2: pi] : ( P @ Z2 @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_46_Par2F,axiom,
! [Q: pi,Alpha: late_freeRes,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha @ ( par @ P @ Q2 ) ) ) ) ).
% Par2F
thf(fact_47_pi_Ofresh_I8_J,axiom,
! [A: name,X2: pi,X1: pi] :
( ( fresh_name_pi @ A @ ( par @ X2 @ X1 ) )
= ( ( fresh_name_pi @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(8)
thf(fact_48_pi_Ofresh_I7_J,axiom,
! [A: name,X2: pi,X1: pi] :
( ( fresh_name_pi @ A @ ( sum @ X2 @ X1 ) )
= ( ( fresh_name_pi @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(7)
thf(fact_49_pi_Ofresh_I2_J,axiom,
! [A: name,X3: name,X2: name,X1: pi] :
( ( fresh_name_pi @ A @ ( output @ X3 @ X2 @ X1 ) )
= ( ( fresh_name_name @ A @ X3 )
& ( fresh_name_name @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(2)
thf(fact_50_pi_Ofresh_I1_J,axiom,
! [A: name] : ( fresh_name_pi @ A @ piNil ) ).
% pi.fresh(1)
thf(fact_51_residual_Ofresh_I2_J,axiom,
! [A: name,X2: late_freeRes,X1: pi] :
( ( fresh_81631222sidual @ A @ ( late_FreeR @ X2 @ X1 ) )
= ( ( fresh_1641682979reeRes @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% residual.fresh(2)
thf(fact_52_pi_Oinject_I7_J,axiom,
! [X2: pi,X1: pi,Y22: pi,Y1: pi] :
( ( ( par @ X2 @ X1 )
= ( par @ Y22 @ Y1 ) )
= ( ( X2 = Y22 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(7)
thf(fact_53_pi_Oinject_I1_J,axiom,
! [X3: name,X2: name,X1: pi,Y3: name,Y22: name,Y1: pi] :
( ( ( output @ X3 @ X2 @ X1 )
= ( output @ Y3 @ Y22 @ Y1 ) )
= ( ( X3 = Y3 )
& ( X2 = Y22 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(1)
thf(fact_54_mem__Collect__eq,axiom,
! [A: product_prod_pi_pi,P: product_prod_pi_pi > $o] :
( ( member558706512_pi_pi @ A @ ( collec1051790738_pi_pi @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_Pr1834926343_pi_pi] :
( ( collec1051790738_pi_pi
@ ^ [X4: product_prod_pi_pi] : ( member558706512_pi_pi @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__cong,axiom,
! [P: product_prod_pi_pi > $o,Q: product_prod_pi_pi > $o] :
( ! [X5: product_prod_pi_pi] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collec1051790738_pi_pi @ P )
= ( collec1051790738_pi_pi @ Q ) ) ) ).
% Collect_cong
thf(fact_57_pi_Odistinct_I29_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(29)
thf(fact_58_freshBoundDerivative_I2_J,axiom,
! [P: pi,A: late_subject,X: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( ( fresh_name_pi @ Y2 @ P )
=> ( ( Y2 != X )
=> ( fresh_name_pi @ Y2 @ P3 ) ) ) ) ).
% freshBoundDerivative(2)
thf(fact_59_freshBoundDerivative_I1_J,axiom,
! [P: pi,A: late_subject,X: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( ( fresh_name_pi @ Y2 @ P )
=> ( fresh_717452603ubject @ Y2 @ A ) ) ) ).
% freshBoundDerivative(1)
thf(fact_60_Late__Semantics_OPar1B,axiom,
! [P: pi,A: late_subject,X: name,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( ( fresh_name_pi @ X @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P3 @ Q ) ) ) ) ) ).
% Late_Semantics.Par1B
thf(fact_61_Late__Semantics_OPar2B,axiom,
! [Q: pi,A: late_subject,X: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P @ Q2 ) ) ) ) ) ).
% Late_Semantics.Par2B
thf(fact_62_parCasesB,axiom,
! [P: pi,Q: pi,A: late_subject,X: name,PQ: pi,Prop: pi > $o] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ PQ ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ! [P2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( Prop @ ( par @ P2 @ Q ) ) )
=> ( ! [Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q3 ) )
=> ( Prop @ ( par @ P @ Q3 ) ) )
=> ( Prop @ PQ ) ) ) ) ) ) ).
% parCasesB
thf(fact_63_parCasesB_H,axiom,
! [P: pi,Q: pi,B: late_subject,Y2: name,P3: pi] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ B @ Y2 @ P3 ) )
=> ( ! [P4: pi,A3: late_subject,X5: name,P2: pi,Q4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y2 @ P3 )
= ( late_BoundR @ A3 @ X5 @ ( par @ P2 @ Q4 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A3 @ X5 @ P2 ) )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ( ( fresh_name_pi @ X5 @ Q4 )
=> ~ ( fresh_717452603ubject @ X5 @ A3 ) ) ) ) ) )
=> ~ ! [Q4: pi,A3: late_subject,X5: name,Q3: pi,P4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y2 @ P3 )
= ( late_BoundR @ A3 @ X5 @ ( par @ P4 @ Q3 ) ) )
=> ( ( late_transitions @ Q4 @ ( late_BoundR @ A3 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ( ( fresh_name_pi @ X5 @ Q4 )
=> ~ ( fresh_717452603ubject @ X5 @ A3 ) ) ) ) ) ) ) ) ).
% parCasesB'
thf(fact_64_freshResidual,axiom,
! [P: pi,Rs: late_residual,X: name] :
( ( late_transitions @ P @ Rs )
=> ( ( fresh_name_pi @ X @ P )
=> ( fresh_81631222sidual @ X @ Rs ) ) ) ).
% freshResidual
thf(fact_65_transitions_OPar1B,axiom,
! [P: pi,A: late_subject,X: name,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P3 @ Q ) ) ) ) ) ) ) ).
% transitions.Par1B
thf(fact_66_transitions_OPar2B,axiom,
! [Q: pi,A: late_subject,X: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P @ Q2 ) ) ) ) ) ) ) ).
% transitions.Par2B
thf(fact_67_sumIdempRight,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ ( sum @ P @ P ) @ Rel @ P ) ) ).
% sumIdempRight
thf(fact_68_sumAssocRight,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi,Q: pi,R: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ ( sum @ P @ ( sum @ Q @ R ) ) @ Rel @ ( sum @ ( sum @ P @ Q ) @ R ) ) ) ).
% sumAssocRight
thf(fact_69_sumIdempLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ P @ Rel @ ( sum @ P @ P ) ) ) ).
% sumIdempLeft
thf(fact_70_sumAssocLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi,Q: pi,R: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ ( sum @ ( sum @ P @ Q ) @ R ) @ Rel @ ( sum @ P @ ( sum @ Q @ R ) ) ) ) ).
% sumAssocLeft
thf(fact_71_sumSym,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi,Q: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ ( sum @ P @ Q ) @ Rel @ ( sum @ Q @ P ) ) ) ).
% sumSym
thf(fact_72_outputBoundTrans,axiom,
! [A: name,B: name,P: pi,C2: late_subject,X: name,P3: pi] :
~ ( late_transitions @ ( output @ A @ B @ P ) @ ( late_BoundR @ C2 @ X @ P3 ) ) ).
% outputBoundTrans
thf(fact_73_transitions_OResB,axiom,
! [P: pi,A: late_subject,X: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( ( fresh_717452603ubject @ Y2 @ A )
=> ( ( Y2 != X )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( res @ Y2 @ P ) @ ( late_BoundR @ A @ X @ ( res @ Y2 @ P3 ) ) ) ) ) ) ) ) ).
% transitions.ResB
thf(fact_74_nilSim_I3_J,axiom,
! [Rel: set_Pr1834926343_pi_pi,A: name,B: name,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( output @ A @ B @ P ) ) ).
% nilSim(3)
thf(fact_75_sumZeroRight,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ P @ Rel @ ( sum @ P @ piNil ) ) ) ).
% sumZeroRight
thf(fact_76_sumZeroLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ ( sum @ P @ piNil ) @ Rel @ P ) ) ).
% sumZeroLeft
thf(fact_77_Late__Semantics_OResB,axiom,
! [P: pi,A: late_subject,X: name,P3: pi,Y2: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( ( fresh_717452603ubject @ Y2 @ A )
=> ( ( Y2 != X )
=> ( late_transitions @ ( res @ Y2 @ P ) @ ( late_BoundR @ A @ X @ ( res @ Y2 @ P3 ) ) ) ) ) ) ).
% Late_Semantics.ResB
thf(fact_78_inputIneqTrans,axiom,
! [A: name,X: name,P: pi,B: late_subject,Y2: name,P3: pi] :
( ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ B @ Y2 @ P3 ) )
=> ~ ( fresh_717452603ubject @ A @ B ) ) ).
% inputIneqTrans
thf(fact_79_pi_Odistinct_I85_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi: pi] :
( ( par @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(85)
thf(fact_80_freshRes,axiom,
! [A: name,P: pi] : ( fresh_name_pi @ A @ ( res @ A @ P ) ) ).
% freshRes
thf(fact_81_residual_Odistinct_I1_J,axiom,
! [Subject2: late_subject,Name4: name,Pi3: pi,FreeRes2: late_freeRes,Pi: pi] :
( ( late_BoundR @ Subject2 @ Name4 @ Pi3 )
!= ( late_FreeR @ FreeRes2 @ Pi ) ) ).
% residual.distinct(1)
thf(fact_82_residual_Oinducts,axiom,
! [P: late_residual > $o,Residual: late_residual] :
( ! [Subject: late_subject,Name3: name,Pi4: pi] : ( P @ ( late_BoundR @ Subject @ Name3 @ Pi4 ) )
=> ( ! [FreeRes: late_freeRes,Pi4: pi] : ( P @ ( late_FreeR @ FreeRes @ Pi4 ) )
=> ( P @ Residual ) ) ) ).
% residual.inducts
thf(fact_83_pi_Odistinct_I79_J,axiom,
! [Pi1: pi,Pi2: pi,Pi12: pi,Pi22: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(79)
thf(fact_84_pi_Odistinct_I55_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name12 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(55)
thf(fact_85_sumResLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,X: name,P: pi,Q: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( ( eqvt_pi @ Rel )
=> ( strong743114133lation @ ( sum @ ( res @ X @ P ) @ ( res @ X @ Q ) ) @ Rel @ ( res @ X @ ( sum @ P @ Q ) ) ) ) ) ).
% sumResLeft
thf(fact_86_pi_Odistinct_I13_J,axiom,
! [Pi12: pi,Pi22: pi] :
( piNil
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(13)
thf(fact_87_outputTauTrans,axiom,
! [A: name,B: name,P: pi,P3: pi] :
~ ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ late_TauR @ P3 ) ) ).
% outputTauTrans
thf(fact_88_pi_Odistinct_I31_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(31)
thf(fact_89_pi_Odistinct_I27_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(27)
thf(fact_90_pi_Odistinct_I21_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name1: name,Name2: name,Pi: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( input @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(21)
thf(fact_91_pi_Odistinct_I1_J,axiom,
! [Name1: name,Name2: name,Pi: pi] :
( piNil
!= ( output @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(1)
thf(fact_92_nilSimRight,axiom,
! [P: pi,Rel: set_Pr1834926343_pi_pi] : ( strong743114133lation @ P @ Rel @ piNil ) ).
% nilSimRight
thf(fact_93_resOutputTauTrans,axiom,
! [X: name,A: name,B: name,P: pi,P3: pi] :
~ ( late_transitions @ ( res @ X @ ( output @ A @ B @ P ) ) @ ( late_FreeR @ late_TauR @ P3 ) ) ).
% resOutputTauTrans
thf(fact_94_Par1F,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha @ ( par @ P3 @ Q ) ) ) ) ).
% Par1F
thf(fact_95_Strong__Late__Sim_Oreflexive,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong743114133lation @ P @ Rel @ P ) ) ).
% Strong_Late_Sim.reflexive
thf(fact_96_subset__antisym,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A2 @ B2 )
=> ( ( ord_le630093991_pi_pi @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_97_subsetI,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi] :
( ! [X5: product_prod_pi_pi] :
( ( member558706512_pi_pi @ X5 @ A2 )
=> ( member558706512_pi_pi @ X5 @ B2 ) )
=> ( ord_le630093991_pi_pi @ A2 @ B2 ) ) ).
% subsetI
thf(fact_98_monotonic,axiom,
! [P: pi,A2: set_Pr1834926343_pi_pi,P3: pi,B2: set_Pr1834926343_pi_pi] :
( ( strong743114133lation @ P @ A2 @ P3 )
=> ( ( ord_le630093991_pi_pi @ A2 @ B2 )
=> ( strong743114133lation @ P @ B2 @ P3 ) ) ) ).
% monotonic
thf(fact_99_order__refl,axiom,
! [X: set_Pr1834926343_pi_pi] : ( ord_le630093991_pi_pi @ X @ X ) ).
% order_refl
thf(fact_100_resOutputInputTrans,axiom,
! [X: name,A: name,B: name,P: pi,C2: name,Y2: name,P3: pi] :
~ ( late_transitions @ ( res @ X @ ( output @ A @ B @ P ) ) @ ( late_BoundR @ ( late_InputS @ C2 ) @ Y2 @ P3 ) ) ).
% resOutputInputTrans
thf(fact_101_resInputBoundOutputTrans,axiom,
! [X: name,A: name,Y2: name,P: pi,B: name,Z: name,P3: pi] :
~ ( late_transitions @ ( res @ X @ ( input @ A @ Y2 @ P ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Z @ P3 ) ) ).
% resInputBoundOutputTrans
thf(fact_102_resOutputOutputTrans,axiom,
! [X: name,A: name,P: pi,B: name,Y2: name,P3: pi] :
~ ( late_transitions @ ( res @ X @ ( output @ A @ X @ P ) ) @ ( late_FreeR @ ( late_OutputR @ B @ Y2 ) @ P3 ) ) ).
% resOutputOutputTrans
thf(fact_103_Late__Semantics_OfreeRes_Oinject,axiom,
! [X2: name,X1: name,Y22: name,Y1: name] :
( ( ( late_OutputR @ X2 @ X1 )
= ( late_OutputR @ Y22 @ Y1 ) )
= ( ( X2 = Y22 )
& ( X1 = Y1 ) ) ) ).
% Late_Semantics.freeRes.inject
thf(fact_104_Late__Semantics_Osubject_Oinject_I2_J,axiom,
! [X1: name,Y1: name] :
( ( ( late_BoundOutputS @ X1 )
= ( late_BoundOutputS @ Y1 ) )
= ( X1 = Y1 ) ) ).
% Late_Semantics.subject.inject(2)
thf(fact_105_name__fresh,axiom,
( fresh_name_name
= ( ^ [A4: name,B3: name] : ( A4 != B3 ) ) ) ).
% name_fresh
thf(fact_106_subject_Ofresh_I1_J,axiom,
! [A: name,X1: name] :
( ( fresh_717452603ubject @ A @ ( late_InputS @ X1 ) )
= ( fresh_name_name @ A @ X1 ) ) ).
% subject.fresh(1)
thf(fact_107_subject_Ofresh_I2_J,axiom,
! [A: name,X1: name] :
( ( fresh_717452603ubject @ A @ ( late_BoundOutputS @ X1 ) )
= ( fresh_name_name @ A @ X1 ) ) ).
% subject.fresh(2)
thf(fact_108_freeRes_Ofresh_I1_J,axiom,
! [A: name,X2: name,X1: name] :
( ( fresh_1641682979reeRes @ A @ ( late_OutputR @ X2 @ X1 ) )
= ( ( fresh_name_name @ A @ X2 )
& ( fresh_name_name @ A @ X1 ) ) ) ).
% freeRes.fresh(1)
thf(fact_109_subject_Oinducts,axiom,
! [P: late_subject > $o,Subject2: late_subject] :
( ! [Name3: name] : ( P @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name] : ( P @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Subject2 ) ) ) ).
% subject.inducts
thf(fact_110_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name4: name,Name: name] :
( ( late_InputS @ Name4 )
!= ( late_BoundOutputS @ Name ) ) ).
% Late_Semantics.subject.distinct(1)
thf(fact_111_freeRes_Oinducts,axiom,
! [P: late_freeRes > $o,FreeRes3: late_freeRes] :
( ! [Name13: name,Name23: name] : ( P @ ( late_OutputR @ Name13 @ Name23 ) )
=> ( ( P @ late_TauR )
=> ( P @ FreeRes3 ) ) ) ).
% freeRes.inducts
thf(fact_112_Late__Semantics_OfreeRes_Odistinct_I1_J,axiom,
! [Name12: name,Name22: name] :
( ( late_OutputR @ Name12 @ Name22 )
!= late_TauR ) ).
% Late_Semantics.freeRes.distinct(1)
thf(fact_113_Open,axiom,
! [P: pi,A: name,B: name,P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P3 ) )
=> ( ( A != B )
=> ( late_transitions @ ( res @ B @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ B @ P3 ) ) ) ) ).
% Open
thf(fact_114_order__subst1,axiom,
! [A: set_Pr1834926343_pi_pi,F2: set_Pr1834926343_pi_pi > set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A @ ( F2 @ B ) )
=> ( ( ord_le630093991_pi_pi @ B @ C2 )
=> ( ! [X5: set_Pr1834926343_pi_pi,Y: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ X5 @ Y )
=> ( ord_le630093991_pi_pi @ ( F2 @ X5 ) @ ( F2 @ Y ) ) )
=> ( ord_le630093991_pi_pi @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst2,axiom,
! [A: set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi,F2: set_Pr1834926343_pi_pi > set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A @ B )
=> ( ( ord_le630093991_pi_pi @ ( F2 @ B ) @ C2 )
=> ( ! [X5: set_Pr1834926343_pi_pi,Y: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ X5 @ Y )
=> ( ord_le630093991_pi_pi @ ( F2 @ X5 ) @ ( F2 @ Y ) ) )
=> ( ord_le630093991_pi_pi @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_116_ord__eq__le__subst,axiom,
! [A: set_Pr1834926343_pi_pi,F2: set_Pr1834926343_pi_pi > set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_le630093991_pi_pi @ B @ C2 )
=> ( ! [X5: set_Pr1834926343_pi_pi,Y: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ X5 @ Y )
=> ( ord_le630093991_pi_pi @ ( F2 @ X5 ) @ ( F2 @ Y ) ) )
=> ( ord_le630093991_pi_pi @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_117_ord__le__eq__subst,axiom,
! [A: set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi,F2: set_Pr1834926343_pi_pi > set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X5: set_Pr1834926343_pi_pi,Y: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ X5 @ Y )
=> ( ord_le630093991_pi_pi @ ( F2 @ X5 ) @ ( F2 @ Y ) ) )
=> ( ord_le630093991_pi_pi @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_eq__iff,axiom,
( ( ^ [Y4: set_Pr1834926343_pi_pi,Z3: set_Pr1834926343_pi_pi] : ( Y4 = Z3 ) )
= ( ^ [X4: set_Pr1834926343_pi_pi,Y5: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ X4 @ Y5 )
& ( ord_le630093991_pi_pi @ Y5 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_119_antisym,axiom,
! [X: set_Pr1834926343_pi_pi,Y2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ X @ Y2 )
=> ( ( ord_le630093991_pi_pi @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% antisym
thf(fact_120_eq__refl,axiom,
! [X: set_Pr1834926343_pi_pi,Y2: set_Pr1834926343_pi_pi] :
( ( X = Y2 )
=> ( ord_le630093991_pi_pi @ X @ Y2 ) ) ).
% eq_refl
thf(fact_121_order_Otrans,axiom,
! [A: set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A @ B )
=> ( ( ord_le630093991_pi_pi @ B @ C2 )
=> ( ord_le630093991_pi_pi @ A @ C2 ) ) ) ).
% order.trans
thf(fact_122_antisym__conv,axiom,
! [Y2: set_Pr1834926343_pi_pi,X: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ Y2 @ X )
=> ( ( ord_le630093991_pi_pi @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_123_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1834926343_pi_pi,Z3: set_Pr1834926343_pi_pi] : ( Y4 = Z3 ) )
= ( ^ [A4: set_Pr1834926343_pi_pi,B3: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A4 @ B3 )
& ( ord_le630093991_pi_pi @ B3 @ A4 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_124_ord__eq__le__trans,axiom,
! [A: set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( A = B )
=> ( ( ord_le630093991_pi_pi @ B @ C2 )
=> ( ord_le630093991_pi_pi @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_125_ord__le__eq__trans,axiom,
! [A: set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A @ B )
=> ( ( B = C2 )
=> ( ord_le630093991_pi_pi @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_126_order__class_Oorder_Oantisym,axiom,
! [A: set_Pr1834926343_pi_pi,B: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A @ B )
=> ( ( ord_le630093991_pi_pi @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_127_order__trans,axiom,
! [X: set_Pr1834926343_pi_pi,Y2: set_Pr1834926343_pi_pi,Z: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ X @ Y2 )
=> ( ( ord_le630093991_pi_pi @ Y2 @ Z )
=> ( ord_le630093991_pi_pi @ X @ Z ) ) ) ).
% order_trans
thf(fact_128_dual__order_Orefl,axiom,
! [A: set_Pr1834926343_pi_pi] : ( ord_le630093991_pi_pi @ A @ A ) ).
% dual_order.refl
thf(fact_129_dual__order_Otrans,axiom,
! [B: set_Pr1834926343_pi_pi,A: set_Pr1834926343_pi_pi,C2: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ B @ A )
=> ( ( ord_le630093991_pi_pi @ C2 @ B )
=> ( ord_le630093991_pi_pi @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_130_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1834926343_pi_pi,Z3: set_Pr1834926343_pi_pi] : ( Y4 = Z3 ) )
= ( ^ [A4: set_Pr1834926343_pi_pi,B3: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ B3 @ A4 )
& ( ord_le630093991_pi_pi @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_131_dual__order_Oantisym,axiom,
! [B: set_Pr1834926343_pi_pi,A: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ B @ A )
=> ( ( ord_le630093991_pi_pi @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_132_in__mono,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi,X: product_prod_pi_pi] :
( ( ord_le630093991_pi_pi @ A2 @ B2 )
=> ( ( member558706512_pi_pi @ X @ A2 )
=> ( member558706512_pi_pi @ X @ B2 ) ) ) ).
% in_mono
thf(fact_133_subsetD,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi,C2: product_prod_pi_pi] :
( ( ord_le630093991_pi_pi @ A2 @ B2 )
=> ( ( member558706512_pi_pi @ C2 @ A2 )
=> ( member558706512_pi_pi @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_134_equalityE,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi] :
( ( A2 = B2 )
=> ~ ( ( ord_le630093991_pi_pi @ A2 @ B2 )
=> ~ ( ord_le630093991_pi_pi @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_135_subset__eq,axiom,
( ord_le630093991_pi_pi
= ( ^ [A5: set_Pr1834926343_pi_pi,B4: set_Pr1834926343_pi_pi] :
! [X4: product_prod_pi_pi] :
( ( member558706512_pi_pi @ X4 @ A5 )
=> ( member558706512_pi_pi @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_136_equalityD1,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi] :
( ( A2 = B2 )
=> ( ord_le630093991_pi_pi @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_137_equalityD2,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi] :
( ( A2 = B2 )
=> ( ord_le630093991_pi_pi @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_138_subset__iff,axiom,
( ord_le630093991_pi_pi
= ( ^ [A5: set_Pr1834926343_pi_pi,B4: set_Pr1834926343_pi_pi] :
! [T: product_prod_pi_pi] :
( ( member558706512_pi_pi @ T @ A5 )
=> ( member558706512_pi_pi @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_139_subset__refl,axiom,
! [A2: set_Pr1834926343_pi_pi] : ( ord_le630093991_pi_pi @ A2 @ A2 ) ).
% subset_refl
thf(fact_140_Collect__mono,axiom,
! [P: product_prod_pi_pi > $o,Q: product_prod_pi_pi > $o] :
( ! [X5: product_prod_pi_pi] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le630093991_pi_pi @ ( collec1051790738_pi_pi @ P ) @ ( collec1051790738_pi_pi @ Q ) ) ) ).
% Collect_mono
thf(fact_141_subset__trans,axiom,
! [A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi,C3: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A2 @ B2 )
=> ( ( ord_le630093991_pi_pi @ B2 @ C3 )
=> ( ord_le630093991_pi_pi @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_142_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr1834926343_pi_pi,Z3: set_Pr1834926343_pi_pi] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr1834926343_pi_pi,B4: set_Pr1834926343_pi_pi] :
( ( ord_le630093991_pi_pi @ A5 @ B4 )
& ( ord_le630093991_pi_pi @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_143_Collect__mono__iff,axiom,
! [P: product_prod_pi_pi > $o,Q: product_prod_pi_pi > $o] :
( ( ord_le630093991_pi_pi @ ( collec1051790738_pi_pi @ P ) @ ( collec1051790738_pi_pi @ Q ) )
= ( ! [X4: product_prod_pi_pi] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_144_resCasesB_H,axiom,
! [X6: name,P: pi,A: late_subject,Y6: name,P3: pi] :
( ( late_transitions @ ( res @ X6 @ P ) @ ( late_BoundR @ A @ Y6 @ P3 ) )
=> ( ! [P4: pi,A3: name,B5: name] :
( ( ( res @ X6 @ P )
= ( res @ B5 @ P4 ) )
=> ! [P2: pi] :
( ( ( late_BoundR @ A @ Y6 @ P3 )
= ( late_BoundR @ ( late_BoundOutputS @ A3 ) @ B5 @ P2 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A3 @ B5 ) @ P2 ) )
=> ( A3 = B5 ) ) ) )
=> ~ ! [P4: pi,A3: late_subject,X5: name,P2: pi,Y: name] :
( ( ( res @ X6 @ P )
= ( res @ Y @ P4 ) )
=> ( ( ( late_BoundR @ A @ Y6 @ P3 )
= ( late_BoundR @ A3 @ X5 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A3 @ X5 @ P2 ) )
=> ( ( fresh_717452603ubject @ Y @ A3 )
=> ( ( Y != X5 )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ~ ( fresh_717452603ubject @ X5 @ A3 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_145_Output,axiom,
! [A: name,B: name,P: pi] : ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P ) ) ).
% Output
thf(fact_146_outputIneqTrans,axiom,
! [A: name,B: name,P: pi,C2: name,D: name,P3: pi] :
( ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ ( late_OutputR @ C2 @ D ) @ P3 ) )
=> ~ ( ( A != C2 )
| ( B != D ) ) ) ).
% outputIneqTrans
thf(fact_147_outputCases_H,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( output @ A @ B @ P ) @ Rs )
=> ~ ! [A3: name,B5: name,P4: pi] :
( ( ( A = A3 )
& ( B = B5 )
& ( P = P4 ) )
=> ( Rs
!= ( late_FreeR @ ( late_OutputR @ A3 @ B5 ) @ P4 ) ) ) ) ).
% outputCases'
thf(fact_148_outputCases,axiom,
! [A: name,B: name,P: pi,Alpha: late_freeRes,P3: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( output @ A @ B @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( ( Alpha
= ( late_OutputR @ A @ B ) )
=> ( ( P = P3 )
=> ( Prop @ ( late_OutputR @ A @ B ) @ P ) ) )
=> ( Prop @ Alpha @ P3 ) ) ) ).
% outputCases
thf(fact_149_Late__Semantics_OInput,axiom,
! [A: name,X: name,P: pi] : ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P ) ) ).
% Late_Semantics.Input
thf(fact_150_transitions_OInput,axiom,
! [X: name,A: name,P: pi] :
( ( X != A )
=> ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P ) ) ) ).
% transitions.Input
thf(fact_151_inputBoundOutputTrans,axiom,
! [A: name,X: name,P: pi,B: name,Y2: name,P3: pi] :
~ ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Y2 @ P3 ) ) ).
% inputBoundOutputTrans
thf(fact_152_resCases_H,axiom,
! [X: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X @ P ) @ Rs )
=> ( ! [P4: pi,A3: name,B5: name] :
( ( ( res @ X @ P )
= ( res @ B5 @ P4 ) )
=> ! [P2: pi] :
( ( Rs
= ( late_BoundR @ ( late_BoundOutputS @ A3 ) @ B5 @ P2 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A3 @ B5 ) @ P2 ) )
=> ( A3 = B5 ) ) ) )
=> ( ! [P4: pi,A3: late_subject,X5: name,P2: pi,Y: name] :
( ( ( res @ X @ P )
= ( res @ Y @ P4 ) )
=> ( ( Rs
= ( late_BoundR @ A3 @ X5 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A3 @ X5 @ P2 ) )
=> ( ( fresh_717452603ubject @ Y @ A3 )
=> ( ( Y != X5 )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ~ ( fresh_717452603ubject @ X5 @ A3 ) ) ) ) ) ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P2: pi,Y: name] :
( ( ( res @ X @ P )
= ( res @ Y @ P4 ) )
=> ( ( Rs
= ( late_FreeR @ Alpha2 @ ( res @ Y @ P2 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P2 ) )
=> ~ ( fresh_1641682979reeRes @ Y @ Alpha2 ) ) ) ) ) ) ) ).
% resCases'
thf(fact_153_transitions_OClose2,axiom,
! [P: pi,A: name,Y2: name,P3: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_name_pi @ Y2 @ P )
=> ( ( fresh_name_pi @ Y2 @ Q )
=> ( ( X != A )
=> ( ( fresh_name_pi @ X @ P3 )
=> ( ( Y2 != A )
=> ( ( fresh_name_pi @ Y2 @ Q2 )
=> ( ( X != Y2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ P3 @ ( subs @ Q2 @ X @ Y2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close2
thf(fact_154_transitions_OClose1,axiom,
! [P: pi,A: name,X: name,P3: pi,Q: pi,Y2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y2 @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_name_pi @ Y2 @ P )
=> ( ( fresh_name_pi @ Y2 @ Q )
=> ( ( X != A )
=> ( ( fresh_name_pi @ X @ Q2 )
=> ( ( Y2 != A )
=> ( ( fresh_name_pi @ Y2 @ P3 )
=> ( ( X != Y2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ ( subs @ P3 @ X @ Y2 ) @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close1
thf(fact_155_Late__Semantics_OClose2,axiom,
! [P: pi,A: name,Y2: name,P3: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ Y2 @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ P3 @ ( subs @ Q2 @ X @ Y2 ) ) ) ) ) ) ) ) ).
% Late_Semantics.Close2
thf(fact_156_Late__Semantics_OClose1,axiom,
! [P: pi,A: name,X: name,P3: pi,Q: pi,Y2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y2 @ Q2 ) )
=> ( ( fresh_name_pi @ Y2 @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y2 @ ( par @ ( subs @ P3 @ X @ Y2 ) @ Q2 ) ) ) ) ) ) ) ).
% Late_Semantics.Close1
thf(fact_157_Late__Semantics1_Osubject_Oinject_I2_J,axiom,
! [X2: name,Y22: name] :
( ( ( late_BoundOutputS @ X2 )
= ( late_BoundOutputS @ Y22 ) )
= ( X2 = Y22 ) ) ).
% Late_Semantics1.subject.inject(2)
thf(fact_158_subst__identity,axiom,
! [P: pi,A: name] :
( ( subs @ P @ A @ A )
= P ) ).
% subst_identity
thf(fact_159_Late__Semantics1_OfreeRes_Oinject,axiom,
! [X11: name,X12: name,Y11: name,Y12: name] :
( ( ( late_OutputR @ X11 @ X12 )
= ( late_OutputR @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% Late_Semantics1.freeRes.inject
thf(fact_160_Late__Semantics1_Osubject_Oinject_I1_J,axiom,
! [X1: name,Y1: name] :
( ( ( late_InputS @ X1 )
= ( late_InputS @ Y1 ) )
= ( X1 = Y1 ) ) ).
% Late_Semantics1.subject.inject(1)
thf(fact_161_simps_I9_J,axiom,
! [X: name,C2: name,D: name,P: pi] :
( ( X != C2 )
=> ( ( X != D )
=> ( ( subs @ ( res @ X @ P ) @ C2 @ D )
= ( res @ X @ ( subs @ P @ C2 @ D ) ) ) ) ) ).
% simps(9)
thf(fact_162_simps_I8_J,axiom,
! [P: pi,Q: pi,C2: name,D: name] :
( ( subs @ ( par @ P @ Q ) @ C2 @ D )
= ( par @ ( subs @ P @ C2 @ D ) @ ( subs @ Q @ C2 @ D ) ) ) ).
% simps(8)
thf(fact_163_substTrans,axiom,
! [B: name,P: pi,A: name,C2: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( subs @ ( subs @ P @ A @ B ) @ B @ C2 )
= ( subs @ P @ A @ C2 ) ) ) ).
% substTrans
thf(fact_164_simps_I7_J,axiom,
! [P: pi,Q: pi,C2: name,D: name] :
( ( subs @ ( sum @ P @ Q ) @ C2 @ D )
= ( sum @ ( subs @ P @ C2 @ D ) @ ( subs @ Q @ C2 @ D ) ) ) ).
% simps(7)
thf(fact_165_simps_I1_J,axiom,
! [C2: name,D: name] :
( ( subs @ piNil @ C2 @ D )
= piNil ) ).
% simps(1)
thf(fact_166_forget,axiom,
! [A: name,P: pi,B: name] :
( ( fresh_name_pi @ A @ P )
=> ( ( subs @ P @ A @ B )
= P ) ) ).
% forget
thf(fact_167_fresh__fact1,axiom,
! [A: name,P: pi,C2: name,B: name] :
( ( fresh_name_pi @ A @ P )
=> ( ( A != C2 )
=> ( fresh_name_pi @ A @ ( subs @ P @ B @ C2 ) ) ) ) ).
% fresh_fact1
thf(fact_168_fresh__fact2,axiom,
! [A: name,B: name,P: pi] :
( ( A != B )
=> ( fresh_name_pi @ A @ ( subs @ P @ A @ B ) ) ) ).
% fresh_fact2
thf(fact_169_substRes3,axiom,
! [B: name,P: pi,A: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( subs @ ( res @ A @ P ) @ A @ B )
= ( res @ B @ ( subs @ P @ A @ B ) ) ) ) ).
% substRes3
thf(fact_170_substRes2,axiom,
! [B: name,P: pi,A: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( res @ A @ P )
= ( res @ B @ ( subs @ P @ A @ B ) ) ) ) ).
% substRes2
thf(fact_171_freeRes_Oexhaust,axiom,
! [Y2: late_freeRes] :
( ! [X112: name,X122: name] :
( Y2
!= ( late_OutputR @ X112 @ X122 ) )
=> ( Y2 = late_TauR ) ) ).
% freeRes.exhaust
thf(fact_172_Late__Semantics1_OfreeRes_Odistinct_I1_J,axiom,
! [X11: name,X12: name] :
( ( late_OutputR @ X11 @ X12 )
!= late_TauR ) ).
% Late_Semantics1.freeRes.distinct(1)
thf(fact_173_subject_Oexhaust,axiom,
! [Y2: late_subject] :
( ! [X13: name] :
( Y2
!= ( late_InputS @ X13 ) )
=> ~ ! [X22: name] :
( Y2
!= ( late_BoundOutputS @ X22 ) ) ) ).
% subject.exhaust
thf(fact_174_Late__Semantics1_Osubject_Odistinct_I1_J,axiom,
! [X1: name,X2: name] :
( ( late_InputS @ X1 )
!= ( late_BoundOutputS @ X2 ) ) ).
% Late_Semantics1.subject.distinct(1)
thf(fact_175_Late__Semantics_OComm1,axiom,
! [P: pi,A: name,X: name,P3: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P3 @ X @ B ) @ Q2 ) ) ) ) ) ).
% Late_Semantics.Comm1
thf(fact_176_Late__Semantics_OComm2,axiom,
! [P: pi,A: name,B: name,P3: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P3 @ ( subs @ Q2 @ X @ B ) ) ) ) ) ) ).
% Late_Semantics.Comm2
thf(fact_177_transitions_OComm1,axiom,
! [P: pi,A: name,X: name,P3: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( X != A )
=> ( ( X != B )
=> ( ( fresh_name_pi @ X @ Q2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P3 @ X @ B ) @ Q2 ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm1
thf(fact_178_transitions_OComm2,axiom,
! [P: pi,A: name,B: name,P3: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( X != A )
=> ( ( X != B )
=> ( ( fresh_name_pi @ X @ P3 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P3 @ ( subs @ Q2 @ X @ B ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm2
thf(fact_179_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: late_subject] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A3: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A3 @ X ) @ Q3 ) )
=> ( ( A3 != X )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A3 ) @ X @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A3: late_subject,Y: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ Y @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A3 )
=> ( ( X != Y )
=> ( ( fresh_717452603ubject @ Y @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ Y @ P5 ) )
& ( strong2129052853vative @ P5 @ ( res @ X @ Q3 ) @ A3 @ Y @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_180_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: late_freeRes] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A3: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A3 @ X ) @ Q3 ) )
=> ( ( A3 != X )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A3 ) @ X @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A3: late_subject,Y: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ Y @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A3 )
=> ( ( X != Y )
=> ( ( fresh_1641682979reeRes @ Y @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ Y @ P5 ) )
& ( strong2129052853vative @ P5 @ ( res @ X @ Q3 ) @ A3 @ Y @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_181_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: name] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A3: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A3 @ X ) @ Q3 ) )
=> ( ( A3 != X )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A3 ) @ X @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A3: late_subject,Y: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ Y @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A3 )
=> ( ( X != Y )
=> ( ( fresh_name_name @ Y @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ Y @ P5 ) )
& ( strong2129052853vative @ P5 @ ( res @ X @ Q3 ) @ A3 @ Y @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_182_resSimCases,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,X: name,P: pi,C3: pi] :
( ( eqvt_pi @ Rel )
=> ( ! [Q3: pi,A3: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A3 @ X ) @ Q3 ) )
=> ( ( A3 != X )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A3 ) @ X @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A3: late_subject,Y: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ Y @ Q3 ) )
=> ( ( fresh_717452603ubject @ X @ A3 )
=> ( ( X != Y )
=> ( ( fresh_name_pi @ Y @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ Y @ P5 ) )
& ( strong2129052853vative @ P5 @ ( res @ X @ Q3 ) @ A3 @ Y @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha2 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ ( res @ X @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X @ Q ) ) ) ) ) ) ).
% resSimCases
thf(fact_183_simE_I1_J,axiom,
! [P: pi,Rel: set_Pr1834926343_pi_pi,Q: pi,A: late_subject,X: name,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ? [P2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
& ( strong2129052853vative @ P2 @ Q2 @ A @ X @ Rel ) ) ) ) ) ).
% simE(1)
thf(fact_184_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: late_subject] :
( ( eqvt_pi @ Rel )
=> ( ! [A3: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A3 )
=> ( ( fresh_717452603ubject @ X5 @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ X5 @ P5 ) )
& ( strong2129052853vative @ P5 @ Q3 @ A3 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_185_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: late_freeRes] :
( ( eqvt_pi @ Rel )
=> ( ! [A3: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A3 )
=> ( ( fresh_1641682979reeRes @ X5 @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ X5 @ P5 ) )
& ( strong2129052853vative @ P5 @ Q3 @ A3 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_186_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: name] :
( ( eqvt_pi @ Rel )
=> ( ! [A3: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A3 )
=> ( ( fresh_name_name @ X5 @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ X5 @ P5 ) )
& ( strong2129052853vative @ P5 @ Q3 @ A3 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_187_simCasesCont,axiom,
! [Rel: set_Pr1834926343_pi_pi,Q: pi,P: pi,C3: pi] :
( ( eqvt_pi @ Rel )
=> ( ! [A3: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P )
=> ( ( fresh_name_pi @ X5 @ Q )
=> ( ( fresh_717452603ubject @ X5 @ A3 )
=> ( ( fresh_name_pi @ X5 @ C3 )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ X5 @ P5 ) )
& ( strong2129052853vative @ P5 @ Q3 @ A3 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ).
% simCasesCont
thf(fact_188_derivativeMonotonic,axiom,
! [P: pi,Q: pi,A: late_subject,X: name,A2: set_Pr1834926343_pi_pi,B2: set_Pr1834926343_pi_pi] :
( ( strong2129052853vative @ P @ Q @ A @ X @ A2 )
=> ( ( ord_le630093991_pi_pi @ A2 @ B2 )
=> ( strong2129052853vative @ P @ Q @ A @ X @ B2 ) ) ) ).
% derivativeMonotonic
thf(fact_189_simulation__def,axiom,
( strong743114133lation
= ( ^ [P6: pi,Rel2: set_Pr1834926343_pi_pi,Q5: pi] :
( ! [A4: late_subject,X4: name,Q6: pi] :
( ( ( late_transitions @ Q5 @ ( late_BoundR @ A4 @ X4 @ Q6 ) )
& ( fresh_name_pi @ X4 @ P6 ) )
=> ? [P7: pi] :
( ( late_transitions @ P6 @ ( late_BoundR @ A4 @ X4 @ P7 ) )
& ( strong2129052853vative @ P7 @ Q6 @ A4 @ X4 @ Rel2 ) ) )
& ! [Alpha3: late_freeRes,Q6: pi] :
( ( late_transitions @ Q5 @ ( late_FreeR @ Alpha3 @ Q6 ) )
=> ? [P7: pi] :
( ( late_transitions @ P6 @ ( late_FreeR @ Alpha3 @ P7 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P7 @ Q6 ) @ Rel2 ) ) ) ) ) ) ).
% simulation_def
thf(fact_190_simCases,axiom,
! [Q: pi,P: pi,Rel: set_Pr1834926343_pi_pi] :
( ! [A3: late_subject,Y: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A3 @ Y @ Q3 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A3 @ Y @ P5 ) )
& ( strong2129052853vative @ P5 @ Q3 @ A3 @ Y @ Rel ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P5: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P5 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ).
% simCases
thf(fact_191_simE_I2_J,axiom,
! [P: pi,Rel: set_Pr1834926343_pi_pi,Q: pi,Alpha: late_freeRes,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q2 ) )
=> ? [P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P2 @ Q2 ) @ Rel ) ) ) ) ).
% simE(2)
thf(fact_192_derivativeReflexive,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi,A: late_subject,X: name] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( strong2129052853vative @ P @ P @ A @ X @ Rel ) ) ).
% derivativeReflexive
thf(fact_193_pair__in__Id__conv,axiom,
! [A: name > noption_pi,B: name > noption_pi] :
( ( member1632198138ion_pi @ ( produc1342893193ion_pi @ A @ B ) @ id_name_noption_pi )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_194_pair__in__Id__conv,axiom,
! [A: pi,B: pi] :
( ( member558706512_pi_pi @ ( product_Pair_pi_pi @ A @ B ) @ id_pi )
= ( A = B ) ) ).
% pair_in_Id_conv
thf(fact_195_IdI,axiom,
! [A: name > noption_pi] : ( member1632198138ion_pi @ ( produc1342893193ion_pi @ A @ A ) @ id_name_noption_pi ) ).
% IdI
thf(fact_196_IdI,axiom,
! [A: pi] : ( member558706512_pi_pi @ ( product_Pair_pi_pi @ A @ A ) @ id_pi ) ).
% IdI
thf(fact_197_subrelI,axiom,
! [R2: set_Pr248416762ubject,S: set_Pr248416762ubject] :
( ! [X5: name > noption_pi,Y: late_subject] :
( ( member938018755ubject @ ( produc581368530ubject @ X5 @ Y ) @ R2 )
=> ( member938018755ubject @ ( produc581368530ubject @ X5 @ Y ) @ S ) )
=> ( ord_le1705144730ubject @ R2 @ S ) ) ).
% subrelI
thf(fact_198_subrelI,axiom,
! [R2: set_Pr305743074reeRes,S: set_Pr305743074reeRes] :
( ! [X5: name > noption_pi,Y: late_freeRes] :
( ( member2081207979reeRes @ ( produc1505598906reeRes @ X5 @ Y ) @ R2 )
=> ( member2081207979reeRes @ ( produc1505598906reeRes @ X5 @ Y ) @ S ) )
=> ( ord_le1762471042reeRes @ R2 @ S ) ) ).
% subrelI
thf(fact_199_subrelI,axiom,
! [R2: set_Pr1294453099i_name,S: set_Pr1294453099i_name] :
( ! [X5: name > noption_pi,Y: name] :
( ( member1353687244i_name @ ( produc540578151i_name @ X5 @ Y ) @ R2 )
=> ( member1353687244i_name @ ( produc540578151i_name @ X5 @ Y ) @ S ) )
=> ( ord_le2082425291i_name @ R2 @ S ) ) ).
% subrelI
thf(fact_200_subrelI,axiom,
! [R2: set_Pr1694908217_pi_pi,S: set_Pr1694908217_pi_pi] :
( ! [X5: name > noption_pi,Y: pi] :
( ( member772629402_pi_pi @ ( produc1820286517_pi_pi @ X5 @ Y ) @ R2 )
=> ( member772629402_pi_pi @ ( produc1820286517_pi_pi @ X5 @ Y ) @ S ) )
=> ( ord_le1854926745_pi_pi @ R2 @ S ) ) ).
% subrelI
thf(fact_201_subrelI,axiom,
! [R2: set_Pr1569510577ion_pi,S: set_Pr1569510577ion_pi] :
( ! [X5: name > noption_pi,Y: name > noption_pi] :
( ( member1632198138ion_pi @ ( produc1342893193ion_pi @ X5 @ Y ) @ R2 )
=> ( member1632198138ion_pi @ ( produc1342893193ion_pi @ X5 @ Y ) @ S ) )
=> ( ord_le1032043601ion_pi @ R2 @ S ) ) ).
% subrelI
thf(fact_202_subrelI,axiom,
! [R2: set_Pr1834926343_pi_pi,S: set_Pr1834926343_pi_pi] :
( ! [X5: pi,Y: pi] :
( ( member558706512_pi_pi @ ( product_Pair_pi_pi @ X5 @ Y ) @ R2 )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ X5 @ Y ) @ S ) )
=> ( ord_le630093991_pi_pi @ R2 @ S ) ) ).
% subrelI
thf(fact_203_IdE,axiom,
! [P8: produc1462499409ion_pi] :
( ( member1632198138ion_pi @ P8 @ id_name_noption_pi )
=> ~ ! [X5: name > noption_pi] :
( P8
!= ( produc1342893193ion_pi @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_204_IdE,axiom,
! [P8: product_prod_pi_pi] :
( ( member558706512_pi_pi @ P8 @ id_pi )
=> ~ ! [X5: pi] :
( P8
!= ( product_Pair_pi_pi @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_205_fresh__prod,axiom,
! [A: name,X: late_subject,Y2: late_subject] :
( ( fresh_495373993ubject @ A @ ( produc1465345179ubject @ X @ Y2 ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_717452603ubject @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_206_fresh__prod,axiom,
! [A: name,X: late_subject,Y2: late_freeRes] :
( ( fresh_1638563217reeRes @ A @ ( produc242091907reeRes @ X @ Y2 ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_207_fresh__prod,axiom,
! [A: name,X: late_subject,Y2: name] :
( ( fresh_1953561446t_name @ A @ ( produc1647375838t_name @ X @ Y2 ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_name_name @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_208_fresh__prod,axiom,
! [A: name,X: late_subject,Y2: pi] :
( ( fresh_994896692ect_pi @ A @ ( produc324947500ect_pi @ X @ Y2 ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_name_pi @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_209_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y2: late_subject] :
( ( fresh_1498103185ubject @ A @ ( produc320590723ubject @ X @ Y2 ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_717452603ubject @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_210_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y2: late_freeRes] :
( ( fresh_493808761reeRes @ A @ ( produc1244821099reeRes @ X @ Y2 ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_211_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y2: name] :
( ( fresh_2007541374s_name @ A @ ( produc1701355766s_name @ X @ Y2 ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_name_name @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_212_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y2: pi] :
( ( fresh_1847501388Res_pi @ A @ ( produc1177552196Res_pi @ X @ Y2 ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_name_pi @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_213_fresh__prod,axiom,
! [A: name,X: name,Y2: late_subject] :
( ( fresh_820177506ubject @ A @ ( produc607574050ubject @ X @ Y2 ) )
= ( ( fresh_name_name @ A @ X )
& ( fresh_717452603ubject @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_214_fresh__prod,axiom,
! [A: name,X: name,Y2: late_freeRes] :
( ( fresh_1963366730reeRes @ A @ ( produc1531804426reeRes @ X @ Y2 ) )
= ( ( fresh_name_name @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y2 ) ) ) ).
% fresh_prod
thf(fact_215_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: late_subject,A6: name > noption_pi,B6: late_subject] :
( ( ( produc581368530ubject @ A @ B )
= ( produc581368530ubject @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_216_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: late_freeRes,A6: name > noption_pi,B6: late_freeRes] :
( ( ( produc1505598906reeRes @ A @ B )
= ( produc1505598906reeRes @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_217_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: name,A6: name > noption_pi,B6: name] :
( ( ( produc540578151i_name @ A @ B )
= ( produc540578151i_name @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_218_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: pi,A6: name > noption_pi,B6: pi] :
( ( ( produc1820286517_pi_pi @ A @ B )
= ( produc1820286517_pi_pi @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_219_old_Oprod_Oinject,axiom,
! [A: name > noption_pi,B: name > noption_pi,A6: name > noption_pi,B6: name > noption_pi] :
( ( ( produc1342893193ion_pi @ A @ B )
= ( produc1342893193ion_pi @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_220_old_Oprod_Oinject,axiom,
! [A: pi,B: pi,A6: pi,B6: pi] :
( ( ( product_Pair_pi_pi @ A @ B )
= ( product_Pair_pi_pi @ A6 @ B6 ) )
= ( ( A = A6 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_221_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: late_subject,Y1: name > noption_pi,Y22: late_subject] :
( ( ( produc581368530ubject @ X1 @ X2 )
= ( produc581368530ubject @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_222_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: late_freeRes,Y1: name > noption_pi,Y22: late_freeRes] :
( ( ( produc1505598906reeRes @ X1 @ X2 )
= ( produc1505598906reeRes @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_223_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: name,Y1: name > noption_pi,Y22: name] :
( ( ( produc540578151i_name @ X1 @ X2 )
= ( produc540578151i_name @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_224_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: pi,Y1: name > noption_pi,Y22: pi] :
( ( ( produc1820286517_pi_pi @ X1 @ X2 )
= ( produc1820286517_pi_pi @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_225_prod_Oinject,axiom,
! [X1: name > noption_pi,X2: name > noption_pi,Y1: name > noption_pi,Y22: name > noption_pi] :
( ( ( produc1342893193ion_pi @ X1 @ X2 )
= ( produc1342893193ion_pi @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_226_prod_Oinject,axiom,
! [X1: pi,X2: pi,Y1: pi,Y22: pi] :
( ( ( product_Pair_pi_pi @ X1 @ X2 )
= ( product_Pair_pi_pi @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X2 = Y22 ) ) ) ).
% prod.inject
thf(fact_227_surj__pair,axiom,
! [P8: produc691362330ubject] :
? [X5: name > noption_pi,Y: late_subject] :
( P8
= ( produc581368530ubject @ X5 @ Y ) ) ).
% surj_pair
thf(fact_228_surj__pair,axiom,
! [P8: produc1834551554reeRes] :
? [X5: name > noption_pi,Y: late_freeRes] :
( P8
= ( produc1505598906reeRes @ X5 @ Y ) ) ).
% surj_pair
thf(fact_229_surj__pair,axiom,
! [P8: produc405342261i_name] :
? [X5: name > noption_pi,Y: name] :
( P8
= ( produc540578151i_name @ X5 @ Y ) ) ).
% surj_pair
thf(fact_230_surj__pair,axiom,
! [P8: produc629157763_pi_pi] :
? [X5: name > noption_pi,Y: pi] :
( P8
= ( produc1820286517_pi_pi @ X5 @ Y ) ) ).
% surj_pair
thf(fact_231_surj__pair,axiom,
! [P8: produc1462499409ion_pi] :
? [X5: name > noption_pi,Y: name > noption_pi] :
( P8
= ( produc1342893193ion_pi @ X5 @ Y ) ) ).
% surj_pair
thf(fact_232_surj__pair,axiom,
! [P8: product_prod_pi_pi] :
? [X5: pi,Y: pi] :
( P8
= ( product_Pair_pi_pi @ X5 @ Y ) ) ).
% surj_pair
thf(fact_233_prod__cases,axiom,
! [P: produc691362330ubject > $o,P8: produc691362330ubject] :
( ! [A3: name > noption_pi,B5: late_subject] : ( P @ ( produc581368530ubject @ A3 @ B5 ) )
=> ( P @ P8 ) ) ).
% prod_cases
thf(fact_234_prod__cases,axiom,
! [P: produc1834551554reeRes > $o,P8: produc1834551554reeRes] :
( ! [A3: name > noption_pi,B5: late_freeRes] : ( P @ ( produc1505598906reeRes @ A3 @ B5 ) )
=> ( P @ P8 ) ) ).
% prod_cases
thf(fact_235_prod__cases,axiom,
! [P: produc405342261i_name > $o,P8: produc405342261i_name] :
( ! [A3: name > noption_pi,B5: name] : ( P @ ( produc540578151i_name @ A3 @ B5 ) )
=> ( P @ P8 ) ) ).
% prod_cases
thf(fact_236_prod__cases,axiom,
! [P: produc629157763_pi_pi > $o,P8: produc629157763_pi_pi] :
( ! [A3: name > noption_pi,B5: pi] : ( P @ ( produc1820286517_pi_pi @ A3 @ B5 ) )
=> ( P @ P8 ) ) ).
% prod_cases
thf(fact_237_prod__cases,axiom,
! [P: produc1462499409ion_pi > $o,P8: produc1462499409ion_pi] :
( ! [A3: name > noption_pi,B5: name > noption_pi] : ( P @ ( produc1342893193ion_pi @ A3 @ B5 ) )
=> ( P @ P8 ) ) ).
% prod_cases
thf(fact_238_prod__cases,axiom,
! [P: product_prod_pi_pi > $o,P8: product_prod_pi_pi] :
( ! [A3: pi,B5: pi] : ( P @ ( product_Pair_pi_pi @ A3 @ B5 ) )
=> ( P @ P8 ) ) ).
% prod_cases
thf(fact_239_Pair__inject,axiom,
! [A: name > noption_pi,B: late_subject,A6: name > noption_pi,B6: late_subject] :
( ( ( produc581368530ubject @ A @ B )
= ( produc581368530ubject @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_240_Pair__inject,axiom,
! [A: name > noption_pi,B: late_freeRes,A6: name > noption_pi,B6: late_freeRes] :
( ( ( produc1505598906reeRes @ A @ B )
= ( produc1505598906reeRes @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_241_Pair__inject,axiom,
! [A: name > noption_pi,B: name,A6: name > noption_pi,B6: name] :
( ( ( produc540578151i_name @ A @ B )
= ( produc540578151i_name @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_242_Pair__inject,axiom,
! [A: name > noption_pi,B: pi,A6: name > noption_pi,B6: pi] :
( ( ( produc1820286517_pi_pi @ A @ B )
= ( produc1820286517_pi_pi @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_243_Pair__inject,axiom,
! [A: name > noption_pi,B: name > noption_pi,A6: name > noption_pi,B6: name > noption_pi] :
( ( ( produc1342893193ion_pi @ A @ B )
= ( produc1342893193ion_pi @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_244_Pair__inject,axiom,
! [A: pi,B: pi,A6: pi,B6: pi] :
( ( ( product_Pair_pi_pi @ A @ B )
= ( product_Pair_pi_pi @ A6 @ B6 ) )
=> ~ ( ( A = A6 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_245_old_Oprod_Oexhaust,axiom,
! [Y2: produc691362330ubject] :
~ ! [A3: name > noption_pi,B5: late_subject] :
( Y2
!= ( produc581368530ubject @ A3 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_246_old_Oprod_Oexhaust,axiom,
! [Y2: produc1834551554reeRes] :
~ ! [A3: name > noption_pi,B5: late_freeRes] :
( Y2
!= ( produc1505598906reeRes @ A3 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_247_old_Oprod_Oexhaust,axiom,
! [Y2: produc405342261i_name] :
~ ! [A3: name > noption_pi,B5: name] :
( Y2
!= ( produc540578151i_name @ A3 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_248_old_Oprod_Oexhaust,axiom,
! [Y2: produc629157763_pi_pi] :
~ ! [A3: name > noption_pi,B5: pi] :
( Y2
!= ( produc1820286517_pi_pi @ A3 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_249_old_Oprod_Oexhaust,axiom,
! [Y2: produc1462499409ion_pi] :
~ ! [A3: name > noption_pi,B5: name > noption_pi] :
( Y2
!= ( produc1342893193ion_pi @ A3 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_250_old_Oprod_Oexhaust,axiom,
! [Y2: product_prod_pi_pi] :
~ ! [A3: pi,B5: pi] :
( Y2
!= ( product_Pair_pi_pi @ A3 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_251_old_Oprod_Oinducts,axiom,
! [P: produc691362330ubject > $o,Prod: produc691362330ubject] :
( ! [A3: name > noption_pi,B5: late_subject] : ( P @ ( produc581368530ubject @ A3 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_252_old_Oprod_Oinducts,axiom,
! [P: produc1834551554reeRes > $o,Prod: produc1834551554reeRes] :
( ! [A3: name > noption_pi,B5: late_freeRes] : ( P @ ( produc1505598906reeRes @ A3 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_253_old_Oprod_Oinducts,axiom,
! [P: produc405342261i_name > $o,Prod: produc405342261i_name] :
( ! [A3: name > noption_pi,B5: name] : ( P @ ( produc540578151i_name @ A3 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_254_old_Oprod_Oinducts,axiom,
! [P: produc629157763_pi_pi > $o,Prod: produc629157763_pi_pi] :
( ! [A3: name > noption_pi,B5: pi] : ( P @ ( produc1820286517_pi_pi @ A3 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_255_old_Oprod_Oinducts,axiom,
! [P: produc1462499409ion_pi > $o,Prod: produc1462499409ion_pi] :
( ! [A3: name > noption_pi,B5: name > noption_pi] : ( P @ ( produc1342893193ion_pi @ A3 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_256_old_Oprod_Oinducts,axiom,
! [P: product_prod_pi_pi > $o,Prod: product_prod_pi_pi] :
( ! [A3: pi,B5: pi] : ( P @ ( product_Pair_pi_pi @ A3 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_257_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y2: late_subject] :
( ( fresh_567880928ubject @ A @ ( produc581368530ubject @ X @ Y2 ) )
=> ( fresh_717452603ubject @ A @ Y2 ) ) ).
% fresh_prodD(2)
thf(fact_258_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y2: late_freeRes] :
( ( fresh_1711070152reeRes @ A @ ( produc1505598906reeRes @ X @ Y2 ) )
=> ( fresh_1641682979reeRes @ A @ Y2 ) ) ).
% fresh_prodD(2)
thf(fact_259_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y2: name] :
( ( fresh_453496559i_name @ A @ ( produc540578151i_name @ X @ Y2 ) )
=> ( fresh_name_name @ A @ Y2 ) ) ).
% fresh_prodD(2)
thf(fact_260_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y2: pi] :
( ( fresh_253309757_pi_pi @ A @ ( produc1820286517_pi_pi @ X @ Y2 ) )
=> ( fresh_name_pi @ A @ Y2 ) ) ).
% fresh_prodD(2)
thf(fact_261_fresh__prodD_I2_J,axiom,
! [A: name,X: name > noption_pi,Y2: name > noption_pi] :
( ( fresh_225680151ion_pi @ A @ ( produc1342893193ion_pi @ X @ Y2 ) )
=> ( fresh_543891186ion_pi @ A @ Y2 ) ) ).
% fresh_prodD(2)
thf(fact_262_fresh__prodD_I2_J,axiom,
! [A: name,X: pi,Y2: pi] :
( ( fresh_1627595501_pi_pi @ A @ ( product_Pair_pi_pi @ X @ Y2 ) )
=> ( fresh_name_pi @ A @ Y2 ) ) ).
% fresh_prodD(2)
thf(fact_263_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y2: late_subject] :
( ( fresh_567880928ubject @ A @ ( produc581368530ubject @ X @ Y2 ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_264_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y2: late_freeRes] :
( ( fresh_1711070152reeRes @ A @ ( produc1505598906reeRes @ X @ Y2 ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_265_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y2: name] :
( ( fresh_453496559i_name @ A @ ( produc540578151i_name @ X @ Y2 ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_266_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y2: pi] :
( ( fresh_253309757_pi_pi @ A @ ( produc1820286517_pi_pi @ X @ Y2 ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_267_fresh__prodD_I1_J,axiom,
! [A: name,X: name > noption_pi,Y2: name > noption_pi] :
( ( fresh_225680151ion_pi @ A @ ( produc1342893193ion_pi @ X @ Y2 ) )
=> ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_268_fresh__prodD_I1_J,axiom,
! [A: name,X: pi,Y2: pi] :
( ( fresh_1627595501_pi_pi @ A @ ( product_Pair_pi_pi @ X @ Y2 ) )
=> ( fresh_name_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_269_IdD,axiom,
! [A: name > noption_pi,B: name > noption_pi] :
( ( member1632198138ion_pi @ ( produc1342893193ion_pi @ A @ B ) @ id_name_noption_pi )
=> ( A = B ) ) ).
% IdD
thf(fact_270_IdD,axiom,
! [A: pi,B: pi] :
( ( member558706512_pi_pi @ ( product_Pair_pi_pi @ A @ B ) @ id_pi )
=> ( A = B ) ) ).
% IdD
thf(fact_271_resTauOutputTrans,axiom,
! [X: name,P: pi,A: name,B: name,P3: pi] :
~ ( late_transitions @ ( res @ X @ ( tau @ P ) ) @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P3 ) ) ).
% resTauOutputTrans
thf(fact_272_residual_Ofresh_I1_J,axiom,
! [A: name,X3: late_subject,X1: name,X2: pi] :
( ( fresh_81631222sidual @ A @ ( late_BoundR @ X3 @ X1 @ X2 ) )
= ( ( fresh_717452603ubject @ A @ X3 )
& ( fresh_543891186ion_pi @ A @ ( abs_fun_name_pi @ X1 @ X2 ) ) ) ) ).
% residual.fresh(1)
thf(fact_273_pi_Ofresh_I3_J,axiom,
! [A: name,X1: pi] :
( ( fresh_name_pi @ A @ ( tau @ X1 ) )
= ( fresh_name_pi @ A @ X1 ) ) ).
% pi.fresh(3)
thf(fact_274_simps_I2_J,axiom,
! [P: pi,C2: name,D: name] :
( ( subs @ ( tau @ P ) @ C2 @ D )
= ( tau @ ( subs @ P @ C2 @ D ) ) ) ).
% simps(2)
thf(fact_275_pi_Ofresh_I9_J,axiom,
! [A: name,X1: name,X2: pi] :
( ( fresh_name_pi @ A @ ( res @ X1 @ X2 ) )
= ( fresh_543891186ion_pi @ A @ ( abs_fun_name_pi @ X1 @ X2 ) ) ) ).
% pi.fresh(9)
thf(fact_276_pi_Ofresh_I4_J,axiom,
! [A: name,X3: name,X1: name,X2: pi] :
( ( fresh_name_pi @ A @ ( input @ X3 @ X1 @ X2 ) )
= ( ( fresh_name_name @ A @ X3 )
& ( fresh_543891186ion_pi @ A @ ( abs_fun_name_pi @ X1 @ X2 ) ) ) ) ).
% pi.fresh(4)
thf(fact_277_name__fresh__abs,axiom,
! [B: name,A: name,X: late_subject] :
( ( fresh_728038529ubject @ B @ ( abs_fu1904601852ubject @ A @ X ) )
= ( ( B = A )
| ( fresh_717452603ubject @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_278_name__fresh__abs,axiom,
! [B: name,A: name,X: late_freeRes] :
( ( fresh_785364841reeRes @ B @ ( abs_fu681348580reeRes @ A @ X ) )
= ( ( B = A )
| ( fresh_1641682979reeRes @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_279_name__fresh__abs,axiom,
! [B: name,A: name,X: name] :
( ( fresh_1093033124n_name @ B @ ( abs_fun_name_name @ A @ X ) )
= ( ( B = A )
| ( fresh_name_name @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_280_name__fresh__abs,axiom,
! [B: name,A: name,X: pi] :
( ( fresh_543891186ion_pi @ B @ ( abs_fun_name_pi @ A @ X ) )
= ( ( B = A )
| ( fresh_name_pi @ B @ X ) ) ) ).
% name_fresh_abs
thf(fact_281_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: late_subject] :
( ( fresh_728038529ubject @ B @ ( abs_fu1904601852ubject @ A @ X ) )
= ( ( B = A )
| ( fresh_717452603ubject @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_282_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: late_freeRes] :
( ( fresh_785364841reeRes @ B @ ( abs_fu681348580reeRes @ A @ X ) )
= ( ( B = A )
| ( fresh_1641682979reeRes @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_283_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: name] :
( ( fresh_1093033124n_name @ B @ ( abs_fun_name_name @ A @ X ) )
= ( ( B = A )
| ( fresh_name_name @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_284_abs__fresh_I1_J,axiom,
! [B: name,A: name,X: pi] :
( ( fresh_543891186ion_pi @ B @ ( abs_fun_name_pi @ A @ X ) )
= ( ( B = A )
| ( fresh_name_pi @ B @ X ) ) ) ).
% abs_fresh(1)
thf(fact_285_pi_Oinject_I2_J,axiom,
! [X1: pi,Y1: pi] :
( ( ( tau @ X1 )
= ( tau @ Y1 ) )
= ( X1 = Y1 ) ) ).
% pi.inject(2)
thf(fact_286_pi_Odistinct_I3_J,axiom,
! [Pi: pi] :
( piNil
!= ( tau @ Pi ) ) ).
% pi.distinct(3)
thf(fact_287_pi_Odistinct_I19_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi: pi] :
( ( output @ Name12 @ Name22 @ Pi3 )
!= ( tau @ Pi ) ) ).
% pi.distinct(19)
thf(fact_288_pi_Odistinct_I35_J,axiom,
! [Pi3: pi,Name1: name,Name2: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( input @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(35)
thf(fact_289_pi_Odistinct_I41_J,axiom,
! [Pi3: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi3 )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(41)
thf(fact_290_pi_Odistinct_I43_J,axiom,
! [Pi3: pi,Pi12: pi,Pi22: pi] :
( ( tau @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(43)
thf(fact_291_pi_Odistinct_I45_J,axiom,
! [Pi3: pi,Name: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(45)
thf(fact_292_pi_Oinject_I3_J,axiom,
! [X3: name,X1: name,X2: pi,Y3: name,Y1: name,Y22: pi] :
( ( ( input @ X3 @ X1 @ X2 )
= ( input @ Y3 @ Y1 @ Y22 ) )
= ( ( X3 = Y3 )
& ( ( abs_fun_name_pi @ X1 @ X2 )
= ( abs_fun_name_pi @ Y1 @ Y22 ) ) ) ) ).
% pi.inject(3)
thf(fact_293_residual_Oinject_I1_J,axiom,
! [X3: late_subject,X1: name,X2: pi,Y3: late_subject,Y1: name,Y22: pi] :
( ( ( late_BoundR @ X3 @ X1 @ X2 )
= ( late_BoundR @ Y3 @ Y1 @ Y22 ) )
= ( ( X3 = Y3 )
& ( ( abs_fun_name_pi @ X1 @ X2 )
= ( abs_fun_name_pi @ Y1 @ Y22 ) ) ) ) ).
% residual.inject(1)
thf(fact_294_pi_Oinject_I8_J,axiom,
! [X1: name,X2: pi,Y1: name,Y22: pi] :
( ( ( res @ X1 @ X2 )
= ( res @ Y1 @ Y22 ) )
= ( ( abs_fun_name_pi @ X1 @ X2 )
= ( abs_fun_name_pi @ Y1 @ Y22 ) ) ) ).
% pi.inject(8)
thf(fact_295_tauBoundTrans,axiom,
! [P: pi,A: late_subject,X: name,P3: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_BoundR @ A @ X @ P3 ) ) ).
% tauBoundTrans
thf(fact_296_nilSim_I1_J,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi] :
~ ( strong743114133lation @ piNil @ Rel @ ( tau @ P ) ) ).
% nilSim(1)
thf(fact_297_resTauBoundTrans,axiom,
! [X: name,P: pi,A: late_subject,Y2: name,P3: pi] :
~ ( late_transitions @ ( res @ X @ ( tau @ P ) ) @ ( late_BoundR @ A @ Y2 @ P3 ) ) ).
% resTauBoundTrans
thf(fact_298_tauOutputTrans,axiom,
! [P: pi,A: name,B: name,P3: pi] :
~ ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P3 ) ) ).
% tauOutputTrans
thf(fact_299_tauCases,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Prop: late_freeRes > pi > $o] :
( ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( ( Alpha = late_TauR )
=> ( ( P = P3 )
=> ( Prop @ late_TauR @ P ) ) )
=> ( Prop @ Alpha @ P3 ) ) ) ).
% tauCases
thf(fact_300_tauCases_H,axiom,
! [P: pi,Rs: late_residual] :
( ( late_transitions @ ( tau @ P ) @ Rs )
=> ~ ! [P4: pi] :
( ( ( tau @ P )
= ( tau @ P4 ) )
=> ( Rs
!= ( late_FreeR @ late_TauR @ P4 ) ) ) ) ).
% tauCases'
thf(fact_301_Tau,axiom,
! [P: pi] : ( late_transitions @ ( tau @ P ) @ ( late_FreeR @ late_TauR @ P ) ) ).
% Tau
thf(fact_302_inputCases_H,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( input @ A @ B @ P ) @ Rs )
=> ~ ! [X5: name,A3: name,P4: pi] :
( ( ( A = A3 )
& ( ( abs_fun_name_pi @ B @ P )
= ( abs_fun_name_pi @ X5 @ P4 ) ) )
=> ( ( Rs
= ( late_BoundR @ ( late_InputS @ A3 ) @ X5 @ P4 ) )
=> ( X5 = A3 ) ) ) ) ).
% inputCases'
thf(fact_303_mismatchIdRight,axiom,
! [Rel: set_Pr1834926343_pi_pi,A: name,B: name,P: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( ( A != B )
=> ( strong743114133lation @ P @ Rel @ ( mismatch @ A @ B @ P ) ) ) ) ).
% mismatchIdRight
thf(fact_304_mismatchIdLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,A: name,B: name,P: pi] :
( ( ord_le630093991_pi_pi @ id_pi @ Rel )
=> ( ( A != B )
=> ( strong743114133lation @ ( mismatch @ A @ B @ P ) @ Rel @ P ) ) ) ).
% mismatchIdLeft
thf(fact_305_pi_Ofresh_I6_J,axiom,
! [A: name,X3: name,X2: name,X1: pi] :
( ( fresh_name_pi @ A @ ( mismatch @ X3 @ X2 @ X1 ) )
= ( ( fresh_name_name @ A @ X3 )
& ( fresh_name_name @ A @ X2 )
& ( fresh_name_pi @ A @ X1 ) ) ) ).
% pi.fresh(6)
thf(fact_306_pi_Odistinct_I39_J,axiom,
! [Pi3: pi,Name1: name,Name2: name,Pi: pi] :
( ( tau @ Pi3 )
!= ( mismatch @ Name1 @ Name2 @ Pi ) ) ).
% pi.distinct(39)
thf(fact_307_mismatchCases,axiom,
! [A: name,B: name,P: pi,Rs: late_residual,F: name > name > $o] :
( ( late_transitions @ ( mismatch @ A @ B @ P ) @ Rs )
=> ( ( ( late_transitions @ P @ Rs )
=> ( ( A != B )
=> ( F @ A @ B ) ) )
=> ( F @ A @ B ) ) ) ).
% mismatchCases
thf(fact_308_mismatchTrans,axiom,
! [A: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( mismatch @ A @ A @ P ) @ Rs ) ).
% mismatchTrans
thf(fact_309_mismatchCases_H,axiom,
! [A: name,B: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( mismatch @ A @ B @ P ) @ Rs )
=> ~ ! [P4: pi,A3: name,B5: name] :
( ( ( A = A3 )
& ( B = B5 )
& ( P = P4 ) )
=> ( ( late_transitions @ P4 @ Rs )
=> ( A3 = B5 ) ) ) ) ).
% mismatchCases'
thf(fact_310_Mismatch,axiom,
! [P: pi,Rs: late_residual,A: name,B: name] :
( ( late_transitions @ P @ Rs )
=> ( ( A != B )
=> ( late_transitions @ ( mismatch @ A @ B @ P ) @ Rs ) ) ) ).
% Mismatch
thf(fact_311_pi_Odistinct_I75_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Name: name,Pi: pi] :
( ( mismatch @ Name12 @ Name22 @ Pi3 )
!= ( res @ Name @ Pi ) ) ).
% pi.distinct(75)
thf(fact_312_pi_Odistinct_I73_J,axiom,
! [Name12: name,Name22: name,Pi3: pi,Pi12: pi,Pi22: pi] :
( ( mismatch @ Name12 @ Name22 @ Pi3 )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(73)
% Conjectures (1)
thf(conj_0,conjecture,
late_transitions @ ( res @ x @ ( sum @ p @ q ) ) @ ( late_FreeR @ alpha @ ( res @ x @ q2 ) ) ).
%------------------------------------------------------------------------------