TPTP Problem File: ITP190^1.p
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%------------------------------------------------------------------------------
% File : ITP190^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_808__3416684_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_808__3416684_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 540 ( 225 unt; 179 typ; 0 def)
% Number of atoms : 861 ( 411 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 3945 ( 127 ~; 1 |; 51 &;3390 @)
% ( 0 <=>; 376 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 7 avg)
% Number of types : 41 ( 40 usr)
% Number of type conns : 359 ( 359 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 139 usr; 18 con; 0-5 aty)
% Number of variables : 1267 ( 6 ^;1249 !; 12 ?;1267 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:32:25.375
%------------------------------------------------------------------------------
% Could-be-implicit typings (40)
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% Explicit typings (139)
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perm_n1415585007_pi_pi: list_P227219501e_name > product_prod_pi_pi > product_prod_pi_pi ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Agent__Oname_J,type,
perm_n258463228s_name: list_P227219501e_name > produc2052722628s_name > produc2052722628s_name ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Agent__Opi_J,type,
perm_n1211707594Res_pi: list_P227219501e_name > produc992857746Res_pi > produc992857746Res_pi ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Agent__Opi____Rep_J,type,
perm_n840427369pi_Rep: list_P227219501e_name > produc1777919905pi_Rep > produc1777919905pi_Rep ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Late____Semantics__OfreeRes_J,type,
perm_n169932155reeRes: list_P227219501e_name > produc1357834675reeRes > produc1357834675reeRes ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__OfreeRes_Mt__Late____Semantics__Osubject_J,type,
perm_n1174226579ubject: list_P227219501e_name > produc214645451ubject > produc214645451ubject ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Agent__Oname_J,type,
perm_n204483300t_name: list_P227219501e_name > produc1998742700t_name > produc1998742700t_name ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Agent__Opi_J,type,
perm_n359102898ect_pi: list_P227219501e_name > produc140253050ect_pi > produc140253050ect_pi ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Agent__Opi____Rep_J,type,
perm_n1467004033pi_Rep: list_P227219501e_name > produc257012921pi_Rep > produc257012921pi_Rep ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Late____Semantics__OfreeRes_J,type,
perm_n1314686611reeRes: list_P227219501e_name > produc355105483reeRes > produc355105483reeRes ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Product____Type__Oprod_It__Late____Semantics__Osubject_Mt__Late____Semantics__Osubject_J,type,
perm_n171497387ubject: list_P227219501e_name > produc1359399907ubject > produc1359399907ubject ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Set__Oset_It__Agent__Oname_J,type,
perm_name_set_name: list_P227219501e_name > set_name > set_name ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Set__Oset_It__Agent__Opi_J,type,
perm_name_set_pi: list_P227219501e_name > set_pi > set_pi ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Set__Oset_It__Agent__Opi____Rep_J,type,
perm_name_set_pi_Rep: list_P227219501e_name > set_pi_Rep > set_pi_Rep ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Set__Oset_It__Late____Semantics__OfreeRes_J,type,
perm_n1555914327reeRes: list_P227219501e_name > set_Late_freeRes > set_Late_freeRes ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Set__Oset_It__Late____Semantics__Osubject_J,type,
perm_n412725103ubject: list_P227219501e_name > set_Late_subject > set_Late_subject ).
thf(sy_c_Nominal_Operm_001t__Agent__Oname_001t__Set__Oset_It__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J_J,type,
perm_n675238095_pi_pi: list_P227219501e_name > set_Pr1834926343_pi_pi > set_Pr1834926343_pi_pi ).
thf(sy_c_Nominal_Oswap_001t__Agent__Oname,type,
swap_name2: produc367475239e_name > name > name ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Agent__Oname_Mt__Agent__Oname_J_M_062_It__Product____Type__Oprod_It__Agent__Oname_Mt__Agent__Oname_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Agent__Oname_Mt__Agent__Oname_J_J,type,
produc1069608854e_name: ( produc367475239e_name > produc367475239e_name > $o ) > list_P227219501e_name > produc1872410278e_name ).
thf(sy_c_Product__Type_OPair_001t__Agent__Oname_001t__Agent__Oname,type,
produc1570949143e_name: name > name > produc367475239e_name ).
thf(sy_c_Product__Type_OPair_001t__Agent__Oname_001t__Late____Semantics__OfreeRes,type,
produc1531804426reeRes: name > late_freeRes > produc2008547984reeRes ).
thf(sy_c_Product__Type_OPair_001t__Agent__Oname_001t__Late____Semantics__Osubject,type,
produc607574050ubject: name > late_subject > produc865358760ubject ).
thf(sy_c_Product__Type_OPair_001t__Agent__Opi_001t__Agent__Opi,type,
product_Pair_pi_pi: pi > pi > product_prod_pi_pi ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Agent__Oname,type,
produc1701355766s_name: late_freeRes > name > produc2052722628s_name ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Agent__Opi,type,
produc1177552196Res_pi: late_freeRes > pi > produc992857746Res_pi ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Agent__Opi____Rep,type,
produc216486617pi_Rep: late_freeRes > pi_Rep > produc1777919905pi_Rep ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Late____Semantics__OfreeRes,type,
produc1244821099reeRes: late_freeRes > late_freeRes > produc1357834675reeRes ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__OfreeRes_001t__Late____Semantics__Osubject,type,
produc320590723ubject: late_freeRes > late_subject > produc214645451ubject ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Agent__Oname,type,
produc1647375838t_name: late_subject > name > produc1998742700t_name ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Agent__Opi,type,
produc324947500ect_pi: late_subject > pi > produc140253050ect_pi ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Agent__Opi____Rep,type,
produc843063281pi_Rep: late_subject > pi_Rep > produc257012921pi_Rep ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Late____Semantics__OfreeRes,type,
produc242091907reeRes: late_subject > late_freeRes > produc355105483reeRes ).
thf(sy_c_Product__Type_OPair_001t__Late____Semantics__Osubject_001t__Late____Semantics__Osubject,type,
produc1465345179ubject: late_subject > late_subject > produc1359399907ubject ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Agent__Oname_Mt__Agent__Oname_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Agent__Oname_Mt__Agent__Oname_J_J,type,
produc208131031e_name: list_P227219501e_name > list_P227219501e_name > produc890461031e_name ).
thf(sy_c_Rel_Oeqvt_001t__Agent__Opi,type,
eqvt_pi: set_Pr1834926343_pi_pi > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J,type,
collec1051790738_pi_pi: ( product_prod_pi_pi > $o ) > set_Pr1834926343_pi_pi ).
thf(sy_c_Strong__Late__Sim_Oderivative,type,
strong2129052853vative: pi > pi > late_subject > name > set_Pr1834926343_pi_pi > $o ).
thf(sy_c_Strong__Late__Sim_Osimulation,type,
strong743114133lation: pi > set_Pr1834926343_pi_pi > pi > $o ).
thf(sy_c_member_001t__Agent__Oname,type,
member_name: name > set_name > $o ).
thf(sy_c_member_001t__Agent__Opi,type,
member_pi: pi > set_pi > $o ).
thf(sy_c_member_001t__Agent__Opi____Rep,type,
member_pi_Rep: pi_Rep > set_pi_Rep > $o ).
thf(sy_c_member_001t__Late____Semantics__OfreeRes,type,
member_Late_freeRes: late_freeRes > set_Late_freeRes > $o ).
thf(sy_c_member_001t__Late____Semantics__Osubject,type,
member_Late_subject: late_subject > set_Late_subject > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Agent__Opi_Mt__Agent__Opi_J,type,
member558706512_pi_pi: product_prod_pi_pi > set_Pr1834926343_pi_pi > $o ).
thf(sy_v_P,type,
p: pi ).
thf(sy_v_P_H____,type,
p2: pi ).
thf(sy_v_PxQ____,type,
pxQ: pi ).
thf(sy_v_Q,type,
q: pi ).
thf(sy_v_Q_H____,type,
q2: pi ).
thf(sy_v_Q_Ha____,type,
q_a: pi ).
thf(sy_v_Rel,type,
rel: set_Pr1834926343_pi_pi ).
thf(sy_v__092_060alpha_062____,type,
alpha: late_freeRes ).
thf(sy_v_aa____,type,
aa: name ).
thf(sy_v_b____,type,
b: name ).
thf(sy_v_x,type,
x: name ).
thf(sy_v_y____,type,
y: name ).
thf(sy_v_z____,type,
z: name ).
% Relevant facts (349)
thf(fact_0__092_060open_062a_A_061_Ab_092_060close_062,axiom,
aa = b ).
% \<open>a = b\<close>
thf(fact_1_cOpen_Ohyps_I2_J,axiom,
b != x ).
% cOpen.hyps(2)
thf(fact_2_cClose1_Ohyps_I10_J,axiom,
z != aa ).
% cClose1.hyps(10)
thf(fact_3_cClose1_Ohyps_I6_J,axiom,
y != z ).
% cClose1.hyps(6)
thf(fact_4_cClose1_Ohyps_I5_J,axiom,
y != aa ).
% cClose1.hyps(5)
thf(fact_5__092_060open_062z_A_092_060noteq_062_Ax_092_060close_062,axiom,
z != x ).
% \<open>z \<noteq> x\<close>
thf(fact_6_cOpen_Ohyps_I1_J,axiom,
late_transitions @ q @ ( late_FreeR @ ( late_OutputR @ b @ x ) @ q_a ) ).
% cOpen.hyps(1)
thf(fact_7_cOpen_Ohyps_I3_J,axiom,
( ( late_BoundOutputS @ aa )
= ( late_BoundOutputS @ b ) ) ).
% cOpen.hyps(3)
thf(fact_8_cClose1_Ohyps_I14_J,axiom,
alpha = late_TauR ).
% cClose1.hyps(14)
thf(fact_9_cClose1_Ohyps_I13_J,axiom,
fresh_name_name @ z @ x ).
% cClose1.hyps(13)
thf(fact_10_perm__swap_I1_J,axiom,
! [A: name,B: name,X: late_subject] :
( ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(1)
thf(fact_11_perm__swap_I1_J,axiom,
! [A: name,B: name,X: late_freeRes] :
( ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(1)
thf(fact_12_perm__swap_I1_J,axiom,
! [A: name,B: name,X: pi_Rep] :
( ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(1)
thf(fact_13_perm__swap_I1_J,axiom,
! [A: name,B: name,X: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(1)
thf(fact_14_perm__swap_I1_J,axiom,
! [A: name,B: name,X: pi] :
( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(1)
thf(fact_15_name__id,axiom,
! [A: name,X: late_subject] :
( ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ A ) @ nil_Pr743626285e_name ) @ X )
= X ) ).
% name_id
thf(fact_16_name__id,axiom,
! [A: name,X: late_freeRes] :
( ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ A ) @ nil_Pr743626285e_name ) @ X )
= X ) ).
% name_id
thf(fact_17_name__id,axiom,
! [A: name,X: pi_Rep] :
( ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ A ) @ nil_Pr743626285e_name ) @ X )
= X ) ).
% name_id
thf(fact_18_name__id,axiom,
! [A: name,X: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ A ) @ nil_Pr743626285e_name ) @ X )
= X ) ).
% name_id
thf(fact_19_name__id,axiom,
! [A: name,X: pi] :
( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ A ) @ nil_Pr743626285e_name ) @ X )
= X ) ).
% name_id
thf(fact_20_name__swap,axiom,
! [A: name,B: name,X: late_subject] :
( ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) ) ).
% name_swap
thf(fact_21_name__swap,axiom,
! [A: name,B: name,X: late_freeRes] :
( ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) ) ).
% name_swap
thf(fact_22_name__swap,axiom,
! [A: name,B: name,X: pi_Rep] :
( ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) ) ).
% name_swap
thf(fact_23_name__swap,axiom,
! [A: name,B: name,X: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) ) ).
% name_swap
thf(fact_24_name__swap,axiom,
! [A: name,B: name,X: pi] :
( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) ) ).
% name_swap
thf(fact_25_name__swap__bij,axiom,
! [A: name,B: name,X: late_subject] :
( ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% name_swap_bij
thf(fact_26_name__swap__bij,axiom,
! [A: name,B: name,X: late_freeRes] :
( ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% name_swap_bij
thf(fact_27_name__swap__bij,axiom,
! [A: name,B: name,X: pi_Rep] :
( ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% name_swap_bij
thf(fact_28_name__swap__bij,axiom,
! [A: name,B: name,X: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% name_swap_bij
thf(fact_29_name__swap__bij,axiom,
! [A: name,B: name,X: pi] :
( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% name_swap_bij
thf(fact_30_pt__name1,axiom,
! [X: late_subject] :
( ( perm_n1559866809ubject @ nil_Pr743626285e_name @ X )
= X ) ).
% pt_name1
thf(fact_31_pt__name1,axiom,
! [X: late_freeRes] :
( ( perm_n336613537reeRes @ nil_Pr743626285e_name @ X )
= X ) ).
% pt_name1
thf(fact_32_pt__name1,axiom,
! [X: pi_Rep] :
( ( perm_name_pi_Rep @ nil_Pr743626285e_name @ X )
= X ) ).
% pt_name1
thf(fact_33_pt__name1,axiom,
! [X: name] :
( ( perm_name_name @ nil_Pr743626285e_name @ X )
= X ) ).
% pt_name1
thf(fact_34_pt__name1,axiom,
! [X: pi] :
( ( perm_name_pi @ nil_Pr743626285e_name @ X )
= X ) ).
% pt_name1
thf(fact_35__092_060open_062y_A_092_060noteq_062_Ax_092_060close_062,axiom,
y != x ).
% \<open>y \<noteq> x\<close>
thf(fact_36_perm__swap_I2_J,axiom,
! [A: name,B: name,X: late_subject] :
( ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(2)
thf(fact_37_perm__swap_I2_J,axiom,
! [A: name,B: name,X: late_freeRes] :
( ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(2)
thf(fact_38_perm__swap_I2_J,axiom,
! [A: name,B: name,X: pi_Rep] :
( ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(2)
thf(fact_39_perm__swap_I2_J,axiom,
! [A: name,B: name,X: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(2)
thf(fact_40_perm__swap_I2_J,axiom,
! [A: name,B: name,X: pi] :
( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ B @ A ) @ nil_Pr743626285e_name ) @ X ) )
= X ) ).
% perm_swap(2)
thf(fact_41_eqvt,axiom,
! [X1: pi,X2: late_residual,Pi: list_P227219501e_name] :
( ( late_transitions @ X1 @ X2 )
=> ( late_transitions @ ( perm_name_pi @ Pi @ X1 ) @ ( perm_n414416632sidual @ Pi @ X2 ) ) ) ).
% eqvt
thf(fact_42__092_060open_062z_A_092_060sharp_062_AQ_092_060close_062,axiom,
fresh_name_pi @ z @ q ).
% \<open>z \<sharp> Q\<close>
thf(fact_43_Late__Semantics_OfreeRes_Oinject,axiom,
! [X2: name,X1: name,Y2: name,Y1: name] :
( ( ( late_OutputR @ X2 @ X1 )
= ( late_OutputR @ Y2 @ Y1 ) )
= ( ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% Late_Semantics.freeRes.inject
thf(fact_44_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_45_cClose1_Ohyps_I11_J,axiom,
fresh_name_pi @ z @ p2 ).
% cClose1.hyps(11)
thf(fact_46_cClose1_Ohyps_I7_J,axiom,
fresh_name_pi @ y @ q2 ).
% cClose1.hyps(7)
thf(fact_47_cClose1_Ohyps_I12_J,axiom,
fresh_name_name @ y @ x ).
% cClose1.hyps(12)
thf(fact_48_assms_I1_J,axiom,
fresh_name_pi @ x @ p ).
% assms(1)
thf(fact_49_cClose1_Ohyps_I8_J,axiom,
fresh_name_pi @ z @ p ).
% cClose1.hyps(8)
thf(fact_50_cClose1_Ohyps_I3_J,axiom,
fresh_name_pi @ y @ p ).
% cClose1.hyps(3)
thf(fact_51_residual__perm__empty,axiom,
! [Residual_Rep: late_residual_Rep] :
( ( perm_n890803237al_Rep @ nil_Pr743626285e_name @ Residual_Rep )
= Residual_Rep ) ).
% residual_perm_empty
thf(fact_52_subject__perm__empty,axiom,
! [Subject_Rep: late_subject_Rep] :
( ( perm_n697952036ct_Rep @ nil_Pr743626285e_name @ Subject_Rep )
= Subject_Rep ) ).
% subject_perm_empty
thf(fact_53_freeRes__perm__empty,axiom,
! [FreeRes_Rep: late_freeRes_Rep] :
( ( perm_n1938283068es_Rep @ nil_Pr743626285e_name @ FreeRes_Rep )
= FreeRes_Rep ) ).
% freeRes_perm_empty
thf(fact_54_pi__perm__empty,axiom,
! [Pi_Rep: pi_Rep] :
( ( perm_name_pi_Rep @ nil_Pr743626285e_name @ Pi_Rep )
= Pi_Rep ) ).
% pi_perm_empty
thf(fact_55_freeRes_Operm_I1_J,axiom,
! [Pi: list_P227219501e_name,X2: name,X1: name] :
( ( perm_n336613537reeRes @ Pi @ ( late_OutputR @ X2 @ X1 ) )
= ( late_OutputR @ ( perm_name_name @ Pi @ X2 ) @ ( perm_name_name @ Pi @ X1 ) ) ) ).
% freeRes.perm(1)
thf(fact_56_subject_Operm_I2_J,axiom,
! [Pi: list_P227219501e_name,X1: name] :
( ( perm_n1559866809ubject @ Pi @ ( late_BoundOutputS @ X1 ) )
= ( late_BoundOutputS @ ( perm_name_name @ Pi @ X1 ) ) ) ).
% subject.perm(2)
thf(fact_57_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_58_Late__Semantics1_Osubject_Oinject_I2_J,axiom,
! [X2: name,Y2: name] :
( ( ( late_BoundOutputS @ X2 )
= ( late_BoundOutputS @ Y2 ) )
= ( X2 = Y2 ) ) ).
% Late_Semantics1.subject.inject(2)
thf(fact_59_freeRes_Operm_I2_J,axiom,
! [Pi: list_P227219501e_name] :
( ( perm_n336613537reeRes @ Pi @ late_TauR )
= late_TauR ) ).
% freeRes.perm(2)
thf(fact_60_name__fresh,axiom,
( fresh_name_name
= ( ^ [A2: name,B2: name] : A2 != B2 ) ) ).
% name_fresh
thf(fact_61_residual_Operm_I2_J,axiom,
! [Pi: list_P227219501e_name,X2: late_freeRes,X1: pi] :
( ( perm_n414416632sidual @ Pi @ ( late_FreeR @ X2 @ X1 ) )
= ( late_FreeR @ ( perm_n336613537reeRes @ Pi @ X2 ) @ ( perm_name_pi @ Pi @ X1 ) ) ) ).
% residual.perm(2)
thf(fact_62_perm__fresh__fresh,axiom,
! [A: name,X: pi_Rep,B: name] :
( ( fresh_name_pi_Rep @ A @ X )
=> ( ( fresh_name_pi_Rep @ B @ X )
=> ( ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% perm_fresh_fresh
thf(fact_63_perm__fresh__fresh,axiom,
! [A: name,X: late_subject,B: name] :
( ( fresh_717452603ubject @ A @ X )
=> ( ( fresh_717452603ubject @ B @ X )
=> ( ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% perm_fresh_fresh
thf(fact_64_perm__fresh__fresh,axiom,
! [A: name,X: late_freeRes,B: name] :
( ( fresh_1641682979reeRes @ A @ X )
=> ( ( fresh_1641682979reeRes @ B @ X )
=> ( ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% perm_fresh_fresh
thf(fact_65_perm__fresh__fresh,axiom,
! [A: name,X: name,B: name] :
( ( fresh_name_name @ A @ X )
=> ( ( fresh_name_name @ B @ X )
=> ( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% perm_fresh_fresh
thf(fact_66_perm__fresh__fresh,axiom,
! [A: name,X: pi,B: name] :
( ( fresh_name_pi @ A @ X )
=> ( ( fresh_name_pi @ B @ X )
=> ( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% perm_fresh_fresh
thf(fact_67_perm__fresh__fresh,axiom,
! [A: name,X: name > noption_pi,B: name] :
( ( fresh_543891186ion_pi @ A @ X )
=> ( ( fresh_543891186ion_pi @ B @ X )
=> ( ( perm_n1667196784ion_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% perm_fresh_fresh
thf(fact_68_name__fresh__fresh,axiom,
! [A: name,X: pi_Rep,B: name] :
( ( fresh_name_pi_Rep @ A @ X )
=> ( ( fresh_name_pi_Rep @ B @ X )
=> ( ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% name_fresh_fresh
thf(fact_69_name__fresh__fresh,axiom,
! [A: name,X: late_subject,B: name] :
( ( fresh_717452603ubject @ A @ X )
=> ( ( fresh_717452603ubject @ B @ X )
=> ( ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% name_fresh_fresh
thf(fact_70_name__fresh__fresh,axiom,
! [A: name,X: late_freeRes,B: name] :
( ( fresh_1641682979reeRes @ A @ X )
=> ( ( fresh_1641682979reeRes @ B @ X )
=> ( ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% name_fresh_fresh
thf(fact_71_name__fresh__fresh,axiom,
! [A: name,X: name,B: name] :
( ( fresh_name_name @ A @ X )
=> ( ( fresh_name_name @ B @ X )
=> ( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% name_fresh_fresh
thf(fact_72_name__fresh__fresh,axiom,
! [A: name,X: pi,B: name] :
( ( fresh_name_pi @ A @ X )
=> ( ( fresh_name_pi @ B @ X )
=> ( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% name_fresh_fresh
thf(fact_73_name__fresh__fresh,axiom,
! [A: name,X: name > noption_pi,B: name] :
( ( fresh_543891186ion_pi @ A @ X )
=> ( ( fresh_543891186ion_pi @ B @ X )
=> ( ( perm_n1667196784ion_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X )
= X ) ) ) ).
% name_fresh_fresh
thf(fact_74_cClose1_Ohyps_I9_J,axiom,
fresh_name_pi @ z @ ( res @ x @ q ) ).
% cClose1.hyps(9)
thf(fact_75_cClose1_Ohyps_I4_J,axiom,
fresh_name_pi @ y @ ( res @ x @ q ) ).
% cClose1.hyps(4)
thf(fact_76_fresh__perm__app,axiom,
! [A: name,Pi: list_P227219501e_name,X: pi_Rep] :
( ( fresh_922791155e_name @ A @ Pi )
=> ( ( fresh_name_pi_Rep @ A @ X )
=> ( fresh_name_pi_Rep @ A @ ( perm_name_pi_Rep @ Pi @ X ) ) ) ) ).
% fresh_perm_app
thf(fact_77_fresh__perm__app,axiom,
! [A: name,Pi: list_P227219501e_name,X: late_subject] :
( ( fresh_922791155e_name @ A @ Pi )
=> ( ( fresh_717452603ubject @ A @ X )
=> ( fresh_717452603ubject @ A @ ( perm_n1559866809ubject @ Pi @ X ) ) ) ) ).
% fresh_perm_app
thf(fact_78_fresh__perm__app,axiom,
! [A: name,Pi: list_P227219501e_name,X: late_freeRes] :
( ( fresh_922791155e_name @ A @ Pi )
=> ( ( fresh_1641682979reeRes @ A @ X )
=> ( fresh_1641682979reeRes @ A @ ( perm_n336613537reeRes @ Pi @ X ) ) ) ) ).
% fresh_perm_app
thf(fact_79_fresh__perm__app,axiom,
! [A: name,Pi: list_P227219501e_name,X: name] :
( ( fresh_922791155e_name @ A @ Pi )
=> ( ( fresh_name_name @ A @ X )
=> ( fresh_name_name @ A @ ( perm_name_name @ Pi @ X ) ) ) ) ).
% fresh_perm_app
thf(fact_80_fresh__perm__app,axiom,
! [A: name,Pi: list_P227219501e_name,X: pi] :
( ( fresh_922791155e_name @ A @ Pi )
=> ( ( fresh_name_pi @ A @ X )
=> ( fresh_name_pi @ A @ ( perm_name_pi @ Pi @ X ) ) ) ) ).
% fresh_perm_app
thf(fact_81_fresh__perm__app,axiom,
! [A: name,Pi: list_P227219501e_name,X: name > noption_pi] :
( ( fresh_922791155e_name @ A @ Pi )
=> ( ( fresh_543891186ion_pi @ A @ X )
=> ( fresh_543891186ion_pi @ A @ ( perm_n1667196784ion_pi @ Pi @ X ) ) ) ) ).
% fresh_perm_app
thf(fact_82_name__exists__fresh,axiom,
! [X: late_subject] :
~ ! [C: name] :
~ ( fresh_717452603ubject @ C @ X ) ).
% name_exists_fresh
thf(fact_83_name__exists__fresh,axiom,
! [X: late_freeRes] :
~ ! [C: name] :
~ ( fresh_1641682979reeRes @ C @ X ) ).
% name_exists_fresh
thf(fact_84_name__exists__fresh,axiom,
! [X: name] :
~ ! [C: name] :
~ ( fresh_name_name @ C @ X ) ).
% name_exists_fresh
thf(fact_85_name__exists__fresh,axiom,
! [X: pi] :
~ ! [C: name] :
~ ( fresh_name_pi @ C @ X ) ).
% name_exists_fresh
thf(fact_86_fresh__eqvt,axiom,
! [Pi: list_P227219501e_name,A: name,X: pi_Rep] :
( ( perm_name_o @ Pi @ ( fresh_name_pi_Rep @ A @ X ) )
= ( fresh_name_pi_Rep @ ( perm_name_name @ Pi @ A ) @ ( perm_name_pi_Rep @ Pi @ X ) ) ) ).
% fresh_eqvt
thf(fact_87_fresh__eqvt,axiom,
! [Pi: list_P227219501e_name,A: name,X: late_subject] :
( ( perm_name_o @ Pi @ ( fresh_717452603ubject @ A @ X ) )
= ( fresh_717452603ubject @ ( perm_name_name @ Pi @ A ) @ ( perm_n1559866809ubject @ Pi @ X ) ) ) ).
% fresh_eqvt
thf(fact_88_fresh__eqvt,axiom,
! [Pi: list_P227219501e_name,A: name,X: late_freeRes] :
( ( perm_name_o @ Pi @ ( fresh_1641682979reeRes @ A @ X ) )
= ( fresh_1641682979reeRes @ ( perm_name_name @ Pi @ A ) @ ( perm_n336613537reeRes @ Pi @ X ) ) ) ).
% fresh_eqvt
thf(fact_89_fresh__eqvt,axiom,
! [Pi: list_P227219501e_name,A: name,X: name] :
( ( perm_name_o @ Pi @ ( fresh_name_name @ A @ X ) )
= ( fresh_name_name @ ( perm_name_name @ Pi @ A ) @ ( perm_name_name @ Pi @ X ) ) ) ).
% fresh_eqvt
thf(fact_90_fresh__eqvt,axiom,
! [Pi: list_P227219501e_name,A: name,X: pi] :
( ( perm_name_o @ Pi @ ( fresh_name_pi @ A @ X ) )
= ( fresh_name_pi @ ( perm_name_name @ Pi @ A ) @ ( perm_name_pi @ Pi @ X ) ) ) ).
% fresh_eqvt
thf(fact_91_fresh__eqvt,axiom,
! [Pi: list_P227219501e_name,A: name,X: name > noption_pi] :
( ( perm_name_o @ Pi @ ( fresh_543891186ion_pi @ A @ X ) )
= ( fresh_543891186ion_pi @ ( perm_name_name @ Pi @ A ) @ ( perm_n1667196784ion_pi @ Pi @ X ) ) ) ).
% fresh_eqvt
thf(fact_92_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_93_Collect__mem__eq,axiom,
! [A3: set_Pr1834926343_pi_pi] :
( ( collec1051790738_pi_pi
@ ^ [X3: product_prod_pi_pi] : ( member558706512_pi_pi @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_94_fresh__bij,axiom,
! [Pi: list_P227219501e_name,A: name,X: pi_Rep] :
( ( fresh_name_pi_Rep @ ( perm_name_name @ Pi @ A ) @ ( perm_name_pi_Rep @ Pi @ X ) )
= ( fresh_name_pi_Rep @ A @ X ) ) ).
% fresh_bij
thf(fact_95_fresh__bij,axiom,
! [Pi: list_P227219501e_name,A: name,X: late_subject] :
( ( fresh_717452603ubject @ ( perm_name_name @ Pi @ A ) @ ( perm_n1559866809ubject @ Pi @ X ) )
= ( fresh_717452603ubject @ A @ X ) ) ).
% fresh_bij
thf(fact_96_fresh__bij,axiom,
! [Pi: list_P227219501e_name,A: name,X: late_freeRes] :
( ( fresh_1641682979reeRes @ ( perm_name_name @ Pi @ A ) @ ( perm_n336613537reeRes @ Pi @ X ) )
= ( fresh_1641682979reeRes @ A @ X ) ) ).
% fresh_bij
thf(fact_97_fresh__bij,axiom,
! [Pi: list_P227219501e_name,A: name,X: name] :
( ( fresh_name_name @ ( perm_name_name @ Pi @ A ) @ ( perm_name_name @ Pi @ X ) )
= ( fresh_name_name @ A @ X ) ) ).
% fresh_bij
thf(fact_98_fresh__bij,axiom,
! [Pi: list_P227219501e_name,A: name,X: pi] :
( ( fresh_name_pi @ ( perm_name_name @ Pi @ A ) @ ( perm_name_pi @ Pi @ X ) )
= ( fresh_name_pi @ A @ X ) ) ).
% fresh_bij
thf(fact_99_fresh__bij,axiom,
! [Pi: list_P227219501e_name,A: name,X: name > noption_pi] :
( ( fresh_543891186ion_pi @ ( perm_name_name @ Pi @ A ) @ ( perm_n1667196784ion_pi @ Pi @ X ) )
= ( fresh_543891186ion_pi @ A @ X ) ) ).
% fresh_bij
thf(fact_100_name__calc_I2_J,axiom,
! [X: name] :
( ( perm_name_name @ nil_Pr743626285e_name @ X )
= X ) ).
% name_calc(2)
thf(fact_101_freeRes_Oexhaust,axiom,
! [Y: late_freeRes] :
( ! [X112: name,X122: name] :
( Y
!= ( late_OutputR @ X112 @ X122 ) )
=> ( Y = late_TauR ) ) ).
% freeRes.exhaust
thf(fact_102_freeRes_Oinducts,axiom,
! [P: late_freeRes > $o,FreeRes: late_freeRes] :
( ! [Name1: name,Name2: name] : ( P @ ( late_OutputR @ Name1 @ Name2 ) )
=> ( ( P @ late_TauR )
=> ( P @ FreeRes ) ) ) ).
% freeRes.inducts
thf(fact_103_Late__Semantics1_OfreeRes_Odistinct_I1_J,axiom,
! [X11: name,X12: name] :
( ( late_OutputR @ X11 @ X12 )
!= late_TauR ) ).
% Late_Semantics1.freeRes.distinct(1)
thf(fact_104_Late__Semantics_OfreeRes_Odistinct_I1_J,axiom,
! [Name12: name,Name22: name] :
( ( late_OutputR @ Name12 @ Name22 )
!= late_TauR ) ).
% Late_Semantics.freeRes.distinct(1)
thf(fact_105_freshFreeDerivative_I2_J,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( fresh_name_pi @ Y @ P2 ) ) ) ).
% freshFreeDerivative(2)
thf(fact_106_swap__simps_I1_J,axiom,
! [A: name,B: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ A )
= B ) ).
% swap_simps(1)
thf(fact_107_swap__simps_I2_J,axiom,
! [A: name,B: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ B )
= A ) ).
% swap_simps(2)
thf(fact_108_swap__simps_I3_J,axiom,
! [A: name,C2: name,B: name] :
( ( A != C2 )
=> ( ( B != C2 )
=> ( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ C2 )
= C2 ) ) ) ).
% swap_simps(3)
thf(fact_109_fresh__aux,axiom,
! [C2: name,A: name,X: pi_Rep,B: name] :
( ( C2 != A )
=> ( ( fresh_name_pi_Rep @ A @ X )
=> ( ( fresh_name_pi_Rep @ C2 @ X )
=> ( fresh_name_pi_Rep @ C2 @ ( perm_name_pi_Rep @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) ) ) ) ) ).
% fresh_aux
thf(fact_110_fresh__aux,axiom,
! [C2: name,A: name,X: late_subject,B: name] :
( ( C2 != A )
=> ( ( fresh_717452603ubject @ A @ X )
=> ( ( fresh_717452603ubject @ C2 @ X )
=> ( fresh_717452603ubject @ C2 @ ( perm_n1559866809ubject @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) ) ) ) ) ).
% fresh_aux
thf(fact_111_fresh__aux,axiom,
! [C2: name,A: name,X: late_freeRes,B: name] :
( ( C2 != A )
=> ( ( fresh_1641682979reeRes @ A @ X )
=> ( ( fresh_1641682979reeRes @ C2 @ X )
=> ( fresh_1641682979reeRes @ C2 @ ( perm_n336613537reeRes @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) ) ) ) ) ).
% fresh_aux
thf(fact_112_fresh__aux,axiom,
! [C2: name,A: name,X: name,B: name] :
( ( C2 != A )
=> ( ( fresh_name_name @ A @ X )
=> ( ( fresh_name_name @ C2 @ X )
=> ( fresh_name_name @ C2 @ ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) ) ) ) ) ).
% fresh_aux
thf(fact_113_fresh__aux,axiom,
! [C2: name,A: name,X: pi,B: name] :
( ( C2 != A )
=> ( ( fresh_name_pi @ A @ X )
=> ( ( fresh_name_pi @ C2 @ X )
=> ( fresh_name_pi @ C2 @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) ) ) ) ) ).
% fresh_aux
thf(fact_114_fresh__aux,axiom,
! [C2: name,A: name,X: name > noption_pi,B: name] :
( ( C2 != A )
=> ( ( fresh_543891186ion_pi @ A @ X )
=> ( ( fresh_543891186ion_pi @ C2 @ X )
=> ( fresh_543891186ion_pi @ C2 @ ( perm_n1667196784ion_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ X ) ) ) ) ) ).
% fresh_aux
thf(fact_115_cp__name__name1,axiom,
! [Pi1: list_P227219501e_name,Pi2: list_P227219501e_name,X: late_subject] :
( ( perm_n1559866809ubject @ Pi1 @ ( perm_n1559866809ubject @ Pi2 @ X ) )
= ( perm_n1559866809ubject @ ( perm_n1160739829e_name @ Pi1 @ Pi2 ) @ ( perm_n1559866809ubject @ Pi1 @ X ) ) ) ).
% cp_name_name1
thf(fact_116_cp__name__name1,axiom,
! [Pi1: list_P227219501e_name,Pi2: list_P227219501e_name,X: late_freeRes] :
( ( perm_n336613537reeRes @ Pi1 @ ( perm_n336613537reeRes @ Pi2 @ X ) )
= ( perm_n336613537reeRes @ ( perm_n1160739829e_name @ Pi1 @ Pi2 ) @ ( perm_n336613537reeRes @ Pi1 @ X ) ) ) ).
% cp_name_name1
thf(fact_117_cp__name__name1,axiom,
! [Pi1: list_P227219501e_name,Pi2: list_P227219501e_name,X: pi_Rep] :
( ( perm_name_pi_Rep @ Pi1 @ ( perm_name_pi_Rep @ Pi2 @ X ) )
= ( perm_name_pi_Rep @ ( perm_n1160739829e_name @ Pi1 @ Pi2 ) @ ( perm_name_pi_Rep @ Pi1 @ X ) ) ) ).
% cp_name_name1
thf(fact_118_cp__name__name1,axiom,
! [Pi1: list_P227219501e_name,Pi2: list_P227219501e_name,X: name] :
( ( perm_name_name @ Pi1 @ ( perm_name_name @ Pi2 @ X ) )
= ( perm_name_name @ ( perm_n1160739829e_name @ Pi1 @ Pi2 ) @ ( perm_name_name @ Pi1 @ X ) ) ) ).
% cp_name_name1
thf(fact_119_cp__name__name1,axiom,
! [Pi1: list_P227219501e_name,Pi2: list_P227219501e_name,X: pi] :
( ( perm_name_pi @ Pi1 @ ( perm_name_pi @ Pi2 @ X ) )
= ( perm_name_pi @ ( perm_n1160739829e_name @ Pi1 @ Pi2 ) @ ( perm_name_pi @ Pi1 @ X ) ) ) ).
% cp_name_name1
thf(fact_120_name__perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: late_subject] :
( ( perm_n1559866809ubject @ Pi2 @ ( perm_n1559866809ubject @ Pi1 @ X ) )
= ( perm_n1559866809ubject @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_n1559866809ubject @ Pi2 @ X ) ) ) ).
% name_perm_compose
thf(fact_121_name__perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: late_freeRes] :
( ( perm_n336613537reeRes @ Pi2 @ ( perm_n336613537reeRes @ Pi1 @ X ) )
= ( perm_n336613537reeRes @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_n336613537reeRes @ Pi2 @ X ) ) ) ).
% name_perm_compose
thf(fact_122_name__perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: pi_Rep] :
( ( perm_name_pi_Rep @ Pi2 @ ( perm_name_pi_Rep @ Pi1 @ X ) )
= ( perm_name_pi_Rep @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_name_pi_Rep @ Pi2 @ X ) ) ) ).
% name_perm_compose
thf(fact_123_name__perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: name] :
( ( perm_name_name @ Pi2 @ ( perm_name_name @ Pi1 @ X ) )
= ( perm_name_name @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_name_name @ Pi2 @ X ) ) ) ).
% name_perm_compose
thf(fact_124_name__perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: pi] :
( ( perm_name_pi @ Pi2 @ ( perm_name_pi @ Pi1 @ X ) )
= ( perm_name_pi @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_name_pi @ Pi2 @ X ) ) ) ).
% name_perm_compose
thf(fact_125_perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: late_subject] :
( ( perm_n1559866809ubject @ Pi2 @ ( perm_n1559866809ubject @ Pi1 @ X ) )
= ( perm_n1559866809ubject @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_n1559866809ubject @ Pi2 @ X ) ) ) ).
% perm_compose
thf(fact_126_perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: late_freeRes] :
( ( perm_n336613537reeRes @ Pi2 @ ( perm_n336613537reeRes @ Pi1 @ X ) )
= ( perm_n336613537reeRes @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_n336613537reeRes @ Pi2 @ X ) ) ) ).
% perm_compose
thf(fact_127_perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: pi_Rep] :
( ( perm_name_pi_Rep @ Pi2 @ ( perm_name_pi_Rep @ Pi1 @ X ) )
= ( perm_name_pi_Rep @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_name_pi_Rep @ Pi2 @ X ) ) ) ).
% perm_compose
thf(fact_128_perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: name] :
( ( perm_name_name @ Pi2 @ ( perm_name_name @ Pi1 @ X ) )
= ( perm_name_name @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_name_name @ Pi2 @ X ) ) ) ).
% perm_compose
thf(fact_129_perm__compose,axiom,
! [Pi2: list_P227219501e_name,Pi1: list_P227219501e_name,X: pi] :
( ( perm_name_pi @ Pi2 @ ( perm_name_pi @ Pi1 @ X ) )
= ( perm_name_pi @ ( perm_n1160739829e_name @ Pi2 @ Pi1 ) @ ( perm_name_pi @ Pi2 @ X ) ) ) ).
% perm_compose
thf(fact_130_perm__bij,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Y: late_subject] :
( ( ( perm_n1559866809ubject @ Pi @ X )
= ( perm_n1559866809ubject @ Pi @ Y ) )
= ( X = Y ) ) ).
% perm_bij
thf(fact_131_perm__bij,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Y: late_freeRes] :
( ( ( perm_n336613537reeRes @ Pi @ X )
= ( perm_n336613537reeRes @ Pi @ Y ) )
= ( X = Y ) ) ).
% perm_bij
thf(fact_132_perm__bij,axiom,
! [Pi: list_P227219501e_name,X: pi_Rep,Y: pi_Rep] :
( ( ( perm_name_pi_Rep @ Pi @ X )
= ( perm_name_pi_Rep @ Pi @ Y ) )
= ( X = Y ) ) ).
% perm_bij
thf(fact_133_perm__bij,axiom,
! [Pi: list_P227219501e_name,X: name,Y: name] :
( ( ( perm_name_name @ Pi @ X )
= ( perm_name_name @ Pi @ Y ) )
= ( X = Y ) ) ).
% perm_bij
thf(fact_134_perm__bij,axiom,
! [Pi: list_P227219501e_name,X: pi,Y: pi] :
( ( ( perm_name_pi @ Pi @ X )
= ( perm_name_pi @ Pi @ Y ) )
= ( X = Y ) ) ).
% perm_bij
thf(fact_135_perm__app,axiom,
! [Pi: list_P227219501e_name,F: late_subject > late_subject,X: late_subject] :
( ( perm_n1559866809ubject @ Pi @ ( F @ X ) )
= ( perm_n1447856217ubject @ Pi @ F @ ( perm_n1559866809ubject @ Pi @ X ) ) ) ).
% perm_app
thf(fact_136_perm__app,axiom,
! [Pi: list_P227219501e_name,F: late_freeRes > late_subject,X: late_freeRes] :
( ( perm_n1559866809ubject @ Pi @ ( F @ X ) )
= ( perm_n303101761ubject @ Pi @ F @ ( perm_n336613537reeRes @ Pi @ X ) ) ) ).
% perm_app
thf(fact_137_perm__app,axiom,
! [Pi: list_P227219501e_name,F: pi_Rep > late_subject,X: pi_Rep] :
( ( perm_n1559866809ubject @ Pi @ ( F @ X ) )
= ( perm_n698371375ubject @ Pi @ F @ ( perm_name_pi_Rep @ Pi @ X ) ) ) ).
% perm_app
thf(fact_138_perm__app,axiom,
! [Pi: list_P227219501e_name,F: name > late_subject,X: name] :
( ( perm_n1559866809ubject @ Pi @ ( F @ X ) )
= ( perm_n1387060402ubject @ Pi @ F @ ( perm_name_name @ Pi @ X ) ) ) ).
% perm_app
thf(fact_139_perm__app,axiom,
! [Pi: list_P227219501e_name,F: pi > late_subject,X: pi] :
( ( perm_n1559866809ubject @ Pi @ ( F @ X ) )
= ( perm_n190055296ubject @ Pi @ F @ ( perm_name_pi @ Pi @ X ) ) ) ).
% perm_app
thf(fact_140_perm__app,axiom,
! [Pi: list_P227219501e_name,F: late_subject > late_freeRes,X: late_subject] :
( ( perm_n336613537reeRes @ Pi @ ( F @ X ) )
= ( perm_n443561793reeRes @ Pi @ F @ ( perm_n1559866809ubject @ Pi @ X ) ) ) ).
% perm_app
thf(fact_141_perm__app,axiom,
! [Pi: list_P227219501e_name,F: late_freeRes > late_freeRes,X: late_freeRes] :
( ( perm_n336613537reeRes @ Pi @ ( F @ X ) )
= ( perm_n1446290985reeRes @ Pi @ F @ ( perm_n336613537reeRes @ Pi @ X ) ) ) ).
% perm_app
thf(fact_142_perm__app,axiom,
! [Pi: list_P227219501e_name,F: pi_Rep > late_freeRes,X: pi_Rep] :
( ( perm_n336613537reeRes @ Pi @ ( F @ X ) )
= ( perm_n1841560599reeRes @ Pi @ F @ ( perm_name_pi_Rep @ Pi @ X ) ) ) ).
% perm_app
thf(fact_143_perm__app,axiom,
! [Pi: list_P227219501e_name,F: name > late_freeRes,X: name] :
( ( perm_n336613537reeRes @ Pi @ ( F @ X ) )
= ( perm_n382765978reeRes @ Pi @ F @ ( perm_name_name @ Pi @ X ) ) ) ).
% perm_app
thf(fact_144_perm__app,axiom,
! [Pi: list_P227219501e_name,F: pi > late_freeRes,X: pi] :
( ( perm_n336613537reeRes @ Pi @ ( F @ X ) )
= ( perm_n1333244520reeRes @ Pi @ F @ ( perm_name_pi @ Pi @ X ) ) ) ).
% perm_app
thf(fact_145_in__eqvt,axiom,
! [Pi: list_P227219501e_name,X: product_prod_pi_pi,X4: set_Pr1834926343_pi_pi] :
( ( perm_name_o @ Pi @ ( member558706512_pi_pi @ X @ X4 ) )
= ( member558706512_pi_pi @ ( perm_n1415585007_pi_pi @ Pi @ X ) @ ( perm_n675238095_pi_pi @ Pi @ X4 ) ) ) ).
% in_eqvt
thf(fact_146_in__eqvt,axiom,
! [Pi: list_P227219501e_name,X: late_subject,X4: set_Late_subject] :
( ( perm_name_o @ Pi @ ( member_Late_subject @ X @ X4 ) )
= ( member_Late_subject @ ( perm_n1559866809ubject @ Pi @ X ) @ ( perm_n412725103ubject @ Pi @ X4 ) ) ) ).
% in_eqvt
thf(fact_147_in__eqvt,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,X4: set_Late_freeRes] :
( ( perm_name_o @ Pi @ ( member_Late_freeRes @ X @ X4 ) )
= ( member_Late_freeRes @ ( perm_n336613537reeRes @ Pi @ X ) @ ( perm_n1555914327reeRes @ Pi @ X4 ) ) ) ).
% in_eqvt
thf(fact_148_in__eqvt,axiom,
! [Pi: list_P227219501e_name,X: pi_Rep,X4: set_pi_Rep] :
( ( perm_name_o @ Pi @ ( member_pi_Rep @ X @ X4 ) )
= ( member_pi_Rep @ ( perm_name_pi_Rep @ Pi @ X ) @ ( perm_name_set_pi_Rep @ Pi @ X4 ) ) ) ).
% in_eqvt
thf(fact_149_in__eqvt,axiom,
! [Pi: list_P227219501e_name,X: name,X4: set_name] :
( ( perm_name_o @ Pi @ ( member_name @ X @ X4 ) )
= ( member_name @ ( perm_name_name @ Pi @ X ) @ ( perm_name_set_name @ Pi @ X4 ) ) ) ).
% in_eqvt
thf(fact_150_in__eqvt,axiom,
! [Pi: list_P227219501e_name,X: pi,X4: set_pi] :
( ( perm_name_o @ Pi @ ( member_pi @ X @ X4 ) )
= ( member_pi @ ( perm_name_pi @ Pi @ X ) @ ( perm_name_set_pi @ Pi @ X4 ) ) ) ).
% in_eqvt
thf(fact_151_eq__eqvt,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Y: late_subject] :
( ( perm_name_o @ Pi @ ( X = Y ) )
= ( ( perm_n1559866809ubject @ Pi @ X )
= ( perm_n1559866809ubject @ Pi @ Y ) ) ) ).
% eq_eqvt
thf(fact_152_eq__eqvt,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Y: late_freeRes] :
( ( perm_name_o @ Pi @ ( X = Y ) )
= ( ( perm_n336613537reeRes @ Pi @ X )
= ( perm_n336613537reeRes @ Pi @ Y ) ) ) ).
% eq_eqvt
thf(fact_153_eq__eqvt,axiom,
! [Pi: list_P227219501e_name,X: pi_Rep,Y: pi_Rep] :
( ( perm_name_o @ Pi @ ( X = Y ) )
= ( ( perm_name_pi_Rep @ Pi @ X )
= ( perm_name_pi_Rep @ Pi @ Y ) ) ) ).
% eq_eqvt
thf(fact_154_eq__eqvt,axiom,
! [Pi: list_P227219501e_name,X: name,Y: name] :
( ( perm_name_o @ Pi @ ( X = Y ) )
= ( ( perm_name_name @ Pi @ X )
= ( perm_name_name @ Pi @ Y ) ) ) ).
% eq_eqvt
thf(fact_155_eq__eqvt,axiom,
! [Pi: list_P227219501e_name,X: pi,Y: pi] :
( ( perm_name_o @ Pi @ ( X = Y ) )
= ( ( perm_name_pi @ Pi @ X )
= ( perm_name_pi @ Pi @ Y ) ) ) ).
% eq_eqvt
thf(fact_156_residual_Oinject_I2_J,axiom,
! [X2: late_freeRes,X1: pi,Y2: late_freeRes,Y1: pi] :
( ( ( late_FreeR @ X2 @ X1 )
= ( late_FreeR @ Y2 @ Y1 ) )
= ( ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% residual.inject(2)
thf(fact_157_fresh__prod,axiom,
! [A: name,X: late_subject,Y: late_subject] :
( ( fresh_495373993ubject @ A @ ( produc1465345179ubject @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_717452603ubject @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_158_fresh__prod,axiom,
! [A: name,X: late_subject,Y: late_freeRes] :
( ( fresh_1638563217reeRes @ A @ ( produc242091907reeRes @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_159_fresh__prod,axiom,
! [A: name,X: late_subject,Y: name] :
( ( fresh_1953561446t_name @ A @ ( produc1647375838t_name @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_name_name @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_160_fresh__prod,axiom,
! [A: name,X: late_subject,Y: pi] :
( ( fresh_994896692ect_pi @ A @ ( produc324947500ect_pi @ X @ Y ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_name_pi @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_161_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: late_subject] :
( ( fresh_1498103185ubject @ A @ ( produc320590723ubject @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_717452603ubject @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_162_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: late_freeRes] :
( ( fresh_493808761reeRes @ A @ ( produc1244821099reeRes @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_163_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: name] :
( ( fresh_2007541374s_name @ A @ ( produc1701355766s_name @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_name_name @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_164_fresh__prod,axiom,
! [A: name,X: late_freeRes,Y: pi] :
( ( fresh_1847501388Res_pi @ A @ ( produc1177552196Res_pi @ X @ Y ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_name_pi @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_165_fresh__prod,axiom,
! [A: name,X: name,Y: late_subject] :
( ( fresh_820177506ubject @ A @ ( produc607574050ubject @ X @ Y ) )
= ( ( fresh_name_name @ A @ X )
& ( fresh_717452603ubject @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_166_fresh__prod,axiom,
! [A: name,X: name,Y: late_freeRes] :
( ( fresh_1963366730reeRes @ A @ ( produc1531804426reeRes @ X @ Y ) )
= ( ( fresh_name_name @ A @ X )
& ( fresh_1641682979reeRes @ A @ Y ) ) ) ).
% fresh_prod
thf(fact_167_list_Oinject,axiom,
! [X21: produc367475239e_name,X22: list_P227219501e_name,Y21: produc367475239e_name,Y22: list_P227219501e_name] :
( ( ( cons_P1213805021e_name @ X21 @ X22 )
= ( cons_P1213805021e_name @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_168_old_Oprod_Oinject,axiom,
! [A: name,B: name,A4: name,B3: name] :
( ( ( produc1570949143e_name @ A @ B )
= ( produc1570949143e_name @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_169_old_Oprod_Oinject,axiom,
! [A: pi,B: pi,A4: pi,B3: pi] :
( ( ( product_Pair_pi_pi @ A @ B )
= ( product_Pair_pi_pi @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_170_prod_Oinject,axiom,
! [X1: name,X2: name,Y1: name,Y2: name] :
( ( ( produc1570949143e_name @ X1 @ X2 )
= ( produc1570949143e_name @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_171_prod_Oinject,axiom,
! [X1: pi,X2: pi,Y1: pi,Y2: pi] :
( ( ( product_Pair_pi_pi @ X1 @ X2 )
= ( product_Pair_pi_pi @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_172_cClose1_Ohyps_I2_J,axiom,
late_transitions @ ( res @ x @ q ) @ ( late_BoundR @ ( late_BoundOutputS @ aa ) @ z @ q2 ) ).
% cClose1.hyps(2)
thf(fact_173_fresh__list__cons,axiom,
! [A: name,X: late_subject,Xs: list_Late_subject] :
( ( fresh_838462539ubject @ A @ ( cons_Late_subject @ X @ Xs ) )
= ( ( fresh_717452603ubject @ A @ X )
& ( fresh_838462539ubject @ A @ Xs ) ) ) ).
% fresh_list_cons
thf(fact_174_fresh__list__cons,axiom,
! [A: name,X: late_freeRes,Xs: list_Late_freeRes] :
( ( fresh_1981651763reeRes @ A @ ( cons_Late_freeRes @ X @ Xs ) )
= ( ( fresh_1641682979reeRes @ A @ X )
& ( fresh_1981651763reeRes @ A @ Xs ) ) ) ).
% fresh_list_cons
thf(fact_175_fresh__list__cons,axiom,
! [A: name,X: name,Xs: list_name] :
( ( fresh_name_list_name @ A @ ( cons_name @ X @ Xs ) )
= ( ( fresh_name_name @ A @ X )
& ( fresh_name_list_name @ A @ Xs ) ) ) ).
% fresh_list_cons
thf(fact_176_fresh__list__cons,axiom,
! [A: name,X: pi,Xs: list_pi] :
( ( fresh_name_list_pi @ A @ ( cons_pi @ X @ Xs ) )
= ( ( fresh_name_pi @ A @ X )
& ( fresh_name_list_pi @ A @ Xs ) ) ) ).
% fresh_list_cons
thf(fact_177_fresh__list__cons,axiom,
! [A: name,X: name > noption_pi,Xs: list_name_noption_pi] :
( ( fresh_1135477378ion_pi @ A @ ( cons_name_noption_pi @ X @ Xs ) )
= ( ( fresh_543891186ion_pi @ A @ X )
& ( fresh_1135477378ion_pi @ A @ Xs ) ) ) ).
% fresh_list_cons
thf(fact_178_fresh__prodD_I2_J,axiom,
! [A: name,X: name,Y: name] :
( ( fresh_441084397e_name @ A @ ( produc1570949143e_name @ X @ Y ) )
=> ( fresh_name_name @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_179_fresh__prodD_I2_J,axiom,
! [A: name,X: pi,Y: pi] :
( ( fresh_1627595501_pi_pi @ A @ ( product_Pair_pi_pi @ X @ Y ) )
=> ( fresh_name_pi @ A @ Y ) ) ).
% fresh_prodD(2)
thf(fact_180_fresh__prodD_I1_J,axiom,
! [A: name,X: name,Y: name] :
( ( fresh_441084397e_name @ A @ ( produc1570949143e_name @ X @ Y ) )
=> ( fresh_name_name @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_181_fresh__prodD_I1_J,axiom,
! [A: name,X: pi,Y: pi] :
( ( fresh_1627595501_pi_pi @ A @ ( product_Pair_pi_pi @ X @ Y ) )
=> ( fresh_name_pi @ A @ X ) ) ).
% fresh_prodD(1)
thf(fact_182_cons__eqvt,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Xs: list_Late_subject] :
( ( perm_n1812647369ubject @ Pi @ ( cons_Late_subject @ X @ Xs ) )
= ( cons_Late_subject @ ( perm_n1559866809ubject @ Pi @ X ) @ ( perm_n1812647369ubject @ Pi @ Xs ) ) ) ).
% cons_eqvt
thf(fact_183_cons__eqvt,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Xs: list_Late_freeRes] :
( ( perm_n808352945reeRes @ Pi @ ( cons_Late_freeRes @ X @ Xs ) )
= ( cons_Late_freeRes @ ( perm_n336613537reeRes @ Pi @ X ) @ ( perm_n808352945reeRes @ Pi @ Xs ) ) ) ).
% cons_eqvt
thf(fact_184_cons__eqvt,axiom,
! [Pi: list_P227219501e_name,X: pi_Rep,Xs: list_pi_Rep] :
( ( perm_n729617823pi_Rep @ Pi @ ( cons_pi_Rep @ X @ Xs ) )
= ( cons_pi_Rep @ ( perm_name_pi_Rep @ Pi @ X ) @ ( perm_n729617823pi_Rep @ Pi @ Xs ) ) ) ).
% cons_eqvt
thf(fact_185_cons__eqvt,axiom,
! [Pi: list_P227219501e_name,X: name,Xs: list_name] :
( ( perm_name_list_name @ Pi @ ( cons_name @ X @ Xs ) )
= ( cons_name @ ( perm_name_name @ Pi @ X ) @ ( perm_name_list_name @ Pi @ Xs ) ) ) ).
% cons_eqvt
thf(fact_186_cons__eqvt,axiom,
! [Pi: list_P227219501e_name,X: pi,Xs: list_pi] :
( ( perm_name_list_pi @ Pi @ ( cons_pi @ X @ Xs ) )
= ( cons_pi @ ( perm_name_pi @ Pi @ X ) @ ( perm_name_list_pi @ Pi @ Xs ) ) ) ).
% cons_eqvt
thf(fact_187_freeRes_Ofresh_I2_J,axiom,
! [A: name] : ( fresh_1641682979reeRes @ A @ late_TauR ) ).
% freeRes.fresh(2)
thf(fact_188_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_189_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_190_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_191_pi_Operm_I9_J,axiom,
! [Pi: list_P227219501e_name,X1: name,X2: pi] :
( ( perm_name_pi @ Pi @ ( res @ X1 @ X2 ) )
= ( res @ ( perm_name_name @ Pi @ X1 ) @ ( perm_name_pi @ Pi @ X2 ) ) ) ).
% pi.perm(9)
thf(fact_192_cClose1_Ohyps_I1_J,axiom,
late_transitions @ p @ ( late_BoundR @ ( late_InputS @ aa ) @ y @ p2 ) ).
% cClose1.hyps(1)
thf(fact_193_residual_Operm_I1_J,axiom,
! [Pi: list_P227219501e_name,X32: late_subject,X1: name,X2: pi] :
( ( perm_n414416632sidual @ Pi @ ( late_BoundR @ X32 @ X1 @ X2 ) )
= ( late_BoundR @ ( perm_n1559866809ubject @ Pi @ X32 ) @ ( perm_name_name @ Pi @ X1 ) @ ( perm_name_pi @ Pi @ X2 ) ) ) ).
% residual.perm(1)
thf(fact_194_name__bij,axiom,
! [Pi: list_P227219501e_name,X: name,Y: name] :
( ( ( perm_name_name @ Pi @ X )
= ( perm_name_name @ Pi @ Y ) )
= ( X = Y ) ) ).
% name_bij
thf(fact_195_Late__Semantics_OResB,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_717452603ubject @ Y @ A )
=> ( ( Y != X )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A @ X @ ( res @ Y @ P2 ) ) ) ) ) ) ).
% Late_Semantics.ResB
thf(fact_196_transitions_OResB,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_717452603ubject @ Y @ A )
=> ( ( Y != X )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A @ X @ ( res @ Y @ P2 ) ) ) ) ) ) ) ) ).
% transitions.ResB
thf(fact_197_resCasesF,axiom,
! [X: name,P: pi,Alpha: late_freeRes,XP: pi,F2: pi > $o] :
( ( late_transitions @ ( res @ X @ P ) @ ( late_FreeR @ Alpha @ XP ) )
=> ( ! [P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh_1641682979reeRes @ X @ Alpha )
=> ( F2 @ ( res @ X @ P3 ) ) ) )
=> ( F2 @ XP ) ) ) ).
% resCasesF
thf(fact_198_resCasesF_H,axiom,
! [X: name,P: pi,Alpha: late_freeRes,P2: pi] :
( ( late_transitions @ ( res @ X @ P ) @ ( late_FreeR @ Alpha @ P2 ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P3: pi,Y3: name] :
( ( ( res @ X @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( ( late_FreeR @ Alpha @ P2 )
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P3 ) )
=> ~ ( fresh_1641682979reeRes @ Y3 @ Alpha2 ) ) ) ) ) ).
% resCasesF'
thf(fact_199_ResF,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh_1641682979reeRes @ Y @ Alpha )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_FreeR @ Alpha @ ( res @ Y @ P2 ) ) ) ) ) ).
% ResF
thf(fact_200_freshBoundDerivative_I1_J,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( fresh_717452603ubject @ Y @ A ) ) ) ).
% freshBoundDerivative(1)
thf(fact_201_freshRes,axiom,
! [A: name,P: pi] : ( fresh_name_pi @ A @ ( res @ A @ P ) ) ).
% freshRes
thf(fact_202_residual_Odistinct_I1_J,axiom,
! [Subject: late_subject,Name: name,Pi: pi,FreeRes2: late_freeRes,Pi3: pi] :
( ( late_BoundR @ Subject @ Name @ Pi )
!= ( late_FreeR @ FreeRes2 @ Pi3 ) ) ).
% residual.distinct(1)
thf(fact_203_residual_Oinducts,axiom,
! [P: late_residual > $o,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi] : ( P @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi] : ( P @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Residual ) ) ) ).
% residual.inducts
thf(fact_204_resCases_H,axiom,
! [X: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X @ P ) @ Rs )
=> ( ! [P4: pi,A5: name,B4: name] :
( ( ( res @ X @ P )
= ( res @ B4 @ P4 ) )
=> ! [P3: pi] :
( ( Rs
= ( late_BoundR @ ( late_BoundOutputS @ A5 ) @ B4 @ P3 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A5 @ B4 ) @ P3 ) )
=> ( A5 = B4 ) ) ) )
=> ( ! [P4: pi,A5: late_subject,X5: name,P3: pi,Y3: name] :
( ( ( res @ X @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( Rs
= ( late_BoundR @ A5 @ X5 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A5 @ X5 @ P3 ) )
=> ( ( fresh_717452603ubject @ Y3 @ A5 )
=> ( ( Y3 != X5 )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ~ ( fresh_717452603ubject @ X5 @ A5 ) ) ) ) ) ) )
=> ~ ! [P4: pi,Alpha2: late_freeRes,P3: pi,Y3: name] :
( ( ( res @ X @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( Rs
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ Alpha2 @ P3 ) )
=> ~ ( fresh_1641682979reeRes @ Y3 @ Alpha2 ) ) ) ) ) ) ) ).
% resCases'
thf(fact_205_sorted__wrt_Ocases,axiom,
! [X: produc1872410278e_name] :
( ! [P4: produc367475239e_name > produc367475239e_name > $o] :
( X
!= ( produc1069608854e_name @ P4 @ nil_Pr743626285e_name ) )
=> ~ ! [P4: produc367475239e_name > produc367475239e_name > $o,X5: produc367475239e_name,Ys: list_P227219501e_name] :
( X
!= ( produc1069608854e_name @ P4 @ ( cons_P1213805021e_name @ X5 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_206_successively_Ocases,axiom,
! [X: produc1872410278e_name] :
( ! [P4: produc367475239e_name > produc367475239e_name > $o] :
( X
!= ( produc1069608854e_name @ P4 @ nil_Pr743626285e_name ) )
=> ( ! [P4: produc367475239e_name > produc367475239e_name > $o,X5: produc367475239e_name] :
( X
!= ( produc1069608854e_name @ P4 @ ( cons_P1213805021e_name @ X5 @ nil_Pr743626285e_name ) ) )
=> ~ ! [P4: produc367475239e_name > produc367475239e_name > $o,X5: produc367475239e_name,Y3: produc367475239e_name,Xs2: list_P227219501e_name] :
( X
!= ( produc1069608854e_name @ P4 @ ( cons_P1213805021e_name @ X5 @ ( cons_P1213805021e_name @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_207_splice_Ocases,axiom,
! [X: produc890461031e_name] :
( ! [Ys: list_P227219501e_name] :
( X
!= ( produc208131031e_name @ nil_Pr743626285e_name @ Ys ) )
=> ~ ! [X5: produc367475239e_name,Xs2: list_P227219501e_name,Ys: list_P227219501e_name] :
( X
!= ( produc208131031e_name @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ Ys ) ) ) ).
% splice.cases
thf(fact_208_shuffles_Ocases,axiom,
! [X: produc890461031e_name] :
( ! [Ys: list_P227219501e_name] :
( X
!= ( produc208131031e_name @ nil_Pr743626285e_name @ Ys ) )
=> ( ! [Xs2: list_P227219501e_name] :
( X
!= ( produc208131031e_name @ Xs2 @ nil_Pr743626285e_name ) )
=> ~ ! [X5: produc367475239e_name,Xs2: list_P227219501e_name,Y3: produc367475239e_name,Ys: list_P227219501e_name] :
( X
!= ( produc208131031e_name @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ ( cons_P1213805021e_name @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_209_resCasesB_H,axiom,
! [X6: name,P: pi,A: late_subject,Y4: name,P2: pi] :
( ( late_transitions @ ( res @ X6 @ P ) @ ( late_BoundR @ A @ Y4 @ P2 ) )
=> ( ! [P4: pi,A5: name,B4: name] :
( ( ( res @ X6 @ P )
= ( res @ B4 @ P4 ) )
=> ! [P3: pi] :
( ( ( late_BoundR @ A @ Y4 @ P2 )
= ( late_BoundR @ ( late_BoundOutputS @ A5 ) @ B4 @ P3 ) )
=> ( ( late_transitions @ P4 @ ( late_FreeR @ ( late_OutputR @ A5 @ B4 ) @ P3 ) )
=> ( A5 = B4 ) ) ) )
=> ~ ! [P4: pi,A5: late_subject,X5: name,P3: pi,Y3: name] :
( ( ( res @ X6 @ P )
= ( res @ Y3 @ P4 ) )
=> ( ( ( late_BoundR @ A @ Y4 @ P2 )
= ( late_BoundR @ A5 @ X5 @ ( res @ Y3 @ P3 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A5 @ X5 @ P3 ) )
=> ( ( fresh_717452603ubject @ Y3 @ A5 )
=> ( ( Y3 != X5 )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ~ ( fresh_717452603ubject @ X5 @ A5 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_210_Open,axiom,
! [P: pi,A: name,B: name,P2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) )
=> ( ( A != B )
=> ( late_transitions @ ( res @ B @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ B @ P2 ) ) ) ) ).
% Open
thf(fact_211_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_212_freshBoundDerivative_I2_J,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( ( Y != X )
=> ( fresh_name_pi @ Y @ P2 ) ) ) ) ).
% freshBoundDerivative(2)
thf(fact_213_residual_Ostrong__inducts,axiom,
! [P: late_subject > late_residual > $o,Z: late_subject,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: late_subject] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_214_residual_Ostrong__inducts,axiom,
! [P: late_freeRes > late_residual > $o,Z: late_freeRes,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: late_freeRes] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_215_residual_Ostrong__inducts,axiom,
! [P: name > late_residual > $o,Z: name,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: name] :
( ( fresh_name_name @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: name] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_216_residual_Ostrong__inducts,axiom,
! [P: pi > late_residual > $o,Z: pi,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: pi] :
( ( fresh_name_pi @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: pi] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_inducts
thf(fact_217_residual_Ostrong__induct,axiom,
! [P: late_subject > late_residual > $o,Z: late_subject,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: late_subject] :
( ( fresh_717452603ubject @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: late_subject] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_218_residual_Ostrong__induct,axiom,
! [P: late_freeRes > late_residual > $o,Z: late_freeRes,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: late_freeRes] :
( ( fresh_1641682979reeRes @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: late_freeRes] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_219_residual_Ostrong__induct,axiom,
! [P: name > late_residual > $o,Z: name,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: name] :
( ( fresh_name_name @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: name] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_220_residual_Ostrong__induct,axiom,
! [P: pi > late_residual > $o,Z: pi,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: pi] :
( ( fresh_name_pi @ Name3 @ Z2 )
=> ( ( fresh_717452603ubject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes3: late_freeRes,Pi4: pi,Z2: pi] : ( P @ Z2 @ ( late_FreeR @ FreeRes3 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ).
% residual.strong_induct
thf(fact_221_resCasesB,axiom,
! [Y: name,P: pi,A: late_subject,X: name,YP: pi,F2: late_subject > pi > $o] :
( ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A @ X @ YP ) )
=> ( ( X != Y )
=> ( ( fresh_name_pi @ X @ P )
=> ( ! [B4: name,P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ B4 @ Y ) @ P3 ) )
=> ( ( B4 != Y )
=> ( ( A
= ( late_BoundOutputS @ B4 ) )
=> ( F2 @ ( late_BoundOutputS @ B4 ) @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ X @ Y ) @ nil_Pr743626285e_name ) @ P3 ) ) ) ) )
=> ( ! [P3: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( ( fresh_717452603ubject @ Y @ A )
=> ( F2 @ A @ ( res @ Y @ P3 ) ) ) )
=> ( F2 @ A @ YP ) ) ) ) ) ) ).
% resCasesB
thf(fact_222_freshFreeDerivative_I1_J,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( fresh_1641682979reeRes @ Y @ Alpha ) ) ) ).
% freshFreeDerivative(1)
thf(fact_223_surj__pair,axiom,
! [P5: produc367475239e_name] :
? [X5: name,Y3: name] :
( P5
= ( produc1570949143e_name @ X5 @ Y3 ) ) ).
% surj_pair
thf(fact_224_surj__pair,axiom,
! [P5: product_prod_pi_pi] :
? [X5: pi,Y3: pi] :
( P5
= ( product_Pair_pi_pi @ X5 @ Y3 ) ) ).
% surj_pair
thf(fact_225_prod__cases,axiom,
! [P: produc367475239e_name > $o,P5: produc367475239e_name] :
( ! [A5: name,B4: name] : ( P @ ( produc1570949143e_name @ A5 @ B4 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_226_prod__cases,axiom,
! [P: product_prod_pi_pi > $o,P5: product_prod_pi_pi] :
( ! [A5: pi,B4: pi] : ( P @ ( product_Pair_pi_pi @ A5 @ B4 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_227_Pair__inject,axiom,
! [A: name,B: name,A4: name,B3: name] :
( ( ( produc1570949143e_name @ A @ B )
= ( produc1570949143e_name @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_228_Pair__inject,axiom,
! [A: pi,B: pi,A4: pi,B3: pi] :
( ( ( product_Pair_pi_pi @ A @ B )
= ( product_Pair_pi_pi @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_229_old_Oprod_Oexhaust,axiom,
! [Y: produc367475239e_name] :
~ ! [A5: name,B4: name] :
( Y
!= ( produc1570949143e_name @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_230_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_pi_pi] :
~ ! [A5: pi,B4: pi] :
( Y
!= ( product_Pair_pi_pi @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_231_old_Oprod_Oinducts,axiom,
! [P: produc367475239e_name > $o,Prod: produc367475239e_name] :
( ! [A5: name,B4: name] : ( P @ ( produc1570949143e_name @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_232_old_Oprod_Oinducts,axiom,
! [P: product_prod_pi_pi > $o,Prod: product_prod_pi_pi] :
( ! [A5: pi,B4: pi] : ( P @ ( product_Pair_pi_pi @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_233_not__Cons__self2,axiom,
! [X: produc367475239e_name,Xs: list_P227219501e_name] :
( ( cons_P1213805021e_name @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_234_allE__Nil,axiom,
! [P: list_P227219501e_name > $o] :
( ! [X_1: list_P227219501e_name] : ( P @ X_1 )
=> ( P @ nil_Pr743626285e_name ) ) ).
% allE_Nil
thf(fact_235_if__eqvt,axiom,
! [Pi: list_P227219501e_name,B: $o,C1: late_subject,C22: late_subject] :
( ( ( perm_name_o @ Pi @ B )
=> ( ( perm_n1559866809ubject @ Pi @ ( if_Late_subject @ B @ C1 @ C22 ) )
= ( perm_n1559866809ubject @ Pi @ C1 ) ) )
& ( ~ ( perm_name_o @ Pi @ B )
=> ( ( perm_n1559866809ubject @ Pi @ ( if_Late_subject @ B @ C1 @ C22 ) )
= ( perm_n1559866809ubject @ Pi @ C22 ) ) ) ) ).
% if_eqvt
thf(fact_236_if__eqvt,axiom,
! [Pi: list_P227219501e_name,B: $o,C1: late_freeRes,C22: late_freeRes] :
( ( ( perm_name_o @ Pi @ B )
=> ( ( perm_n336613537reeRes @ Pi @ ( if_Late_freeRes @ B @ C1 @ C22 ) )
= ( perm_n336613537reeRes @ Pi @ C1 ) ) )
& ( ~ ( perm_name_o @ Pi @ B )
=> ( ( perm_n336613537reeRes @ Pi @ ( if_Late_freeRes @ B @ C1 @ C22 ) )
= ( perm_n336613537reeRes @ Pi @ C22 ) ) ) ) ).
% if_eqvt
thf(fact_237_if__eqvt,axiom,
! [Pi: list_P227219501e_name,B: $o,C1: pi_Rep,C22: pi_Rep] :
( ( ( perm_name_o @ Pi @ B )
=> ( ( perm_name_pi_Rep @ Pi @ ( if_pi_Rep @ B @ C1 @ C22 ) )
= ( perm_name_pi_Rep @ Pi @ C1 ) ) )
& ( ~ ( perm_name_o @ Pi @ B )
=> ( ( perm_name_pi_Rep @ Pi @ ( if_pi_Rep @ B @ C1 @ C22 ) )
= ( perm_name_pi_Rep @ Pi @ C22 ) ) ) ) ).
% if_eqvt
thf(fact_238_if__eqvt,axiom,
! [Pi: list_P227219501e_name,B: $o,C1: name,C22: name] :
( ( ( perm_name_o @ Pi @ B )
=> ( ( perm_name_name @ Pi @ ( if_name @ B @ C1 @ C22 ) )
= ( perm_name_name @ Pi @ C1 ) ) )
& ( ~ ( perm_name_o @ Pi @ B )
=> ( ( perm_name_name @ Pi @ ( if_name @ B @ C1 @ C22 ) )
= ( perm_name_name @ Pi @ C22 ) ) ) ) ).
% if_eqvt
thf(fact_239_if__eqvt,axiom,
! [Pi: list_P227219501e_name,B: $o,C1: pi,C22: pi] :
( ( ( perm_name_o @ Pi @ B )
=> ( ( perm_name_pi @ Pi @ ( if_pi @ B @ C1 @ C22 ) )
= ( perm_name_pi @ Pi @ C1 ) ) )
& ( ~ ( perm_name_o @ Pi @ B )
=> ( ( perm_name_pi @ Pi @ ( if_pi @ B @ C1 @ C22 ) )
= ( perm_name_pi @ Pi @ C22 ) ) ) ) ).
% if_eqvt
thf(fact_240_pt__bij3,axiom,
! [X: late_subject,Y: late_subject,Pi: list_P227219501e_name] :
( ( X = Y )
=> ( ( perm_n1559866809ubject @ Pi @ X )
= ( perm_n1559866809ubject @ Pi @ Y ) ) ) ).
% pt_bij3
thf(fact_241_pt__bij3,axiom,
! [X: late_freeRes,Y: late_freeRes,Pi: list_P227219501e_name] :
( ( X = Y )
=> ( ( perm_n336613537reeRes @ Pi @ X )
= ( perm_n336613537reeRes @ Pi @ Y ) ) ) ).
% pt_bij3
thf(fact_242_pt__bij3,axiom,
! [X: pi_Rep,Y: pi_Rep,Pi: list_P227219501e_name] :
( ( X = Y )
=> ( ( perm_name_pi_Rep @ Pi @ X )
= ( perm_name_pi_Rep @ Pi @ Y ) ) ) ).
% pt_bij3
thf(fact_243_pt__bij3,axiom,
! [X: name,Y: name,Pi: list_P227219501e_name] :
( ( X = Y )
=> ( ( perm_name_name @ Pi @ X )
= ( perm_name_name @ Pi @ Y ) ) ) ).
% pt_bij3
thf(fact_244_pt__bij3,axiom,
! [X: pi,Y: pi,Pi: list_P227219501e_name] :
( ( X = Y )
=> ( ( perm_name_pi @ Pi @ X )
= ( perm_name_pi @ Pi @ Y ) ) ) ).
% pt_bij3
thf(fact_245_abs__fun__if,axiom,
! [C2: $o,Pi: list_P227219501e_name,X: late_subject,Y: late_subject] :
( ( C2
=> ( ( perm_n1559866809ubject @ Pi @ ( if_Late_subject @ C2 @ X @ Y ) )
= ( perm_n1559866809ubject @ Pi @ X ) ) )
& ( ~ C2
=> ( ( perm_n1559866809ubject @ Pi @ ( if_Late_subject @ C2 @ X @ Y ) )
= ( perm_n1559866809ubject @ Pi @ Y ) ) ) ) ).
% abs_fun_if
thf(fact_246_abs__fun__if,axiom,
! [C2: $o,Pi: list_P227219501e_name,X: late_freeRes,Y: late_freeRes] :
( ( C2
=> ( ( perm_n336613537reeRes @ Pi @ ( if_Late_freeRes @ C2 @ X @ Y ) )
= ( perm_n336613537reeRes @ Pi @ X ) ) )
& ( ~ C2
=> ( ( perm_n336613537reeRes @ Pi @ ( if_Late_freeRes @ C2 @ X @ Y ) )
= ( perm_n336613537reeRes @ Pi @ Y ) ) ) ) ).
% abs_fun_if
thf(fact_247_abs__fun__if,axiom,
! [C2: $o,Pi: list_P227219501e_name,X: pi_Rep,Y: pi_Rep] :
( ( C2
=> ( ( perm_name_pi_Rep @ Pi @ ( if_pi_Rep @ C2 @ X @ Y ) )
= ( perm_name_pi_Rep @ Pi @ X ) ) )
& ( ~ C2
=> ( ( perm_name_pi_Rep @ Pi @ ( if_pi_Rep @ C2 @ X @ Y ) )
= ( perm_name_pi_Rep @ Pi @ Y ) ) ) ) ).
% abs_fun_if
thf(fact_248_abs__fun__if,axiom,
! [C2: $o,Pi: list_P227219501e_name,X: name,Y: name] :
( ( C2
=> ( ( perm_name_name @ Pi @ ( if_name @ C2 @ X @ Y ) )
= ( perm_name_name @ Pi @ X ) ) )
& ( ~ C2
=> ( ( perm_name_name @ Pi @ ( if_name @ C2 @ X @ Y ) )
= ( perm_name_name @ Pi @ Y ) ) ) ) ).
% abs_fun_if
thf(fact_249_abs__fun__if,axiom,
! [C2: $o,Pi: list_P227219501e_name,X: pi,Y: pi] :
( ( C2
=> ( ( perm_name_pi @ Pi @ ( if_pi @ C2 @ X @ Y ) )
= ( perm_name_pi @ Pi @ X ) ) )
& ( ~ C2
=> ( ( perm_name_pi @ Pi @ ( if_pi @ C2 @ X @ Y ) )
= ( perm_name_pi @ Pi @ Y ) ) ) ) ).
% abs_fun_if
thf(fact_250_alphaRes,axiom,
! [C2: name,P: pi,A: name] :
( ( fresh_name_pi @ C2 @ P )
=> ( ( res @ A @ P )
= ( res @ C2 @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ C2 ) @ nil_Pr743626285e_name ) @ P ) ) ) ) ).
% alphaRes
thf(fact_251_alphaBoundResidual,axiom,
! [X6: name,P: pi,A: late_subject,X: name] :
( ( fresh_name_pi @ X6 @ P )
=> ( ( late_BoundR @ A @ X @ P )
= ( late_BoundR @ A @ X6 @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ X @ X6 ) @ nil_Pr743626285e_name ) @ P ) ) ) ) ).
% alphaBoundResidual
thf(fact_252_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Y: late_subject] :
( ( perm_n171497387ubject @ Pi @ ( produc1465345179ubject @ X @ Y ) )
= ( produc1465345179ubject @ ( perm_n1559866809ubject @ Pi @ X ) @ ( perm_n1559866809ubject @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_253_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Y: late_freeRes] :
( ( perm_n1314686611reeRes @ Pi @ ( produc242091907reeRes @ X @ Y ) )
= ( produc242091907reeRes @ ( perm_n1559866809ubject @ Pi @ X ) @ ( perm_n336613537reeRes @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_254_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Y: pi_Rep] :
( ( perm_n1467004033pi_Rep @ Pi @ ( produc843063281pi_Rep @ X @ Y ) )
= ( produc843063281pi_Rep @ ( perm_n1559866809ubject @ Pi @ X ) @ ( perm_name_pi_Rep @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_255_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Y: name] :
( ( perm_n204483300t_name @ Pi @ ( produc1647375838t_name @ X @ Y ) )
= ( produc1647375838t_name @ ( perm_n1559866809ubject @ Pi @ X ) @ ( perm_name_name @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_256_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_subject,Y: pi] :
( ( perm_n359102898ect_pi @ Pi @ ( produc324947500ect_pi @ X @ Y ) )
= ( produc324947500ect_pi @ ( perm_n1559866809ubject @ Pi @ X ) @ ( perm_name_pi @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_257_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Y: late_subject] :
( ( perm_n1174226579ubject @ Pi @ ( produc320590723ubject @ X @ Y ) )
= ( produc320590723ubject @ ( perm_n336613537reeRes @ Pi @ X ) @ ( perm_n1559866809ubject @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_258_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Y: late_freeRes] :
( ( perm_n169932155reeRes @ Pi @ ( produc1244821099reeRes @ X @ Y ) )
= ( produc1244821099reeRes @ ( perm_n336613537reeRes @ Pi @ X ) @ ( perm_n336613537reeRes @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_259_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Y: pi_Rep] :
( ( perm_n840427369pi_Rep @ Pi @ ( produc216486617pi_Rep @ X @ Y ) )
= ( produc216486617pi_Rep @ ( perm_n336613537reeRes @ Pi @ X ) @ ( perm_name_pi_Rep @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_260_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Y: name] :
( ( perm_n258463228s_name @ Pi @ ( produc1701355766s_name @ X @ Y ) )
= ( produc1701355766s_name @ ( perm_n336613537reeRes @ Pi @ X ) @ ( perm_name_name @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_261_perm__prod_Osimps,axiom,
! [Pi: list_P227219501e_name,X: late_freeRes,Y: pi] :
( ( perm_n1211707594Res_pi @ Pi @ ( produc1177552196Res_pi @ X @ Y ) )
= ( produc1177552196Res_pi @ ( perm_n336613537reeRes @ Pi @ X ) @ ( perm_name_pi @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_262_list_Odistinct_I1_J,axiom,
! [X21: produc367475239e_name,X22: list_P227219501e_name] :
( nil_Pr743626285e_name
!= ( cons_P1213805021e_name @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_263_list_OdiscI,axiom,
! [List: list_P227219501e_name,X21: produc367475239e_name,X22: list_P227219501e_name] :
( ( List
= ( cons_P1213805021e_name @ X21 @ X22 ) )
=> ( List != nil_Pr743626285e_name ) ) ).
% list.discI
thf(fact_264_list_Oexhaust,axiom,
! [Y: list_P227219501e_name] :
( ( Y != nil_Pr743626285e_name )
=> ~ ! [X212: produc367475239e_name,X222: list_P227219501e_name] :
( Y
!= ( cons_P1213805021e_name @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_265_list_Oinducts,axiom,
! [P: list_P227219501e_name > $o,List: list_P227219501e_name] :
( ( P @ nil_Pr743626285e_name )
=> ( ! [X13: produc367475239e_name,X23: list_P227219501e_name] :
( ( P @ X23 )
=> ( P @ ( cons_P1213805021e_name @ X13 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_266_neq__Nil__conv,axiom,
! [Xs: list_P227219501e_name] :
( ( Xs != nil_Pr743626285e_name )
= ( ? [Y5: produc367475239e_name,Ys2: list_P227219501e_name] :
( Xs
= ( cons_P1213805021e_name @ Y5 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_267_list__induct2_H,axiom,
! [P: list_P227219501e_name > list_P227219501e_name > $o,Xs: list_P227219501e_name,Ys3: list_P227219501e_name] :
( ( P @ nil_Pr743626285e_name @ nil_Pr743626285e_name )
=> ( ! [X5: produc367475239e_name,Xs2: list_P227219501e_name] : ( P @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ nil_Pr743626285e_name )
=> ( ! [Y3: produc367475239e_name,Ys: list_P227219501e_name] : ( P @ nil_Pr743626285e_name @ ( cons_P1213805021e_name @ Y3 @ Ys ) )
=> ( ! [X5: produc367475239e_name,Xs2: list_P227219501e_name,Y3: produc367475239e_name,Ys: list_P227219501e_name] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ ( cons_P1213805021e_name @ Y3 @ Ys ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% list_induct2'
thf(fact_268_splice_Oinduct,axiom,
! [P: list_P227219501e_name > list_P227219501e_name > $o,A0: list_P227219501e_name,A1: list_P227219501e_name] :
( ! [X_1: list_P227219501e_name] : ( P @ nil_Pr743626285e_name @ X_1 )
=> ( ! [X5: produc367475239e_name,Xs2: list_P227219501e_name,Ys: list_P227219501e_name] :
( ( P @ Ys @ Xs2 )
=> ( P @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ Ys ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_269_induct__list012,axiom,
! [P: list_P227219501e_name > $o,Xs: list_P227219501e_name] :
( ( P @ nil_Pr743626285e_name )
=> ( ! [X5: produc367475239e_name] : ( P @ ( cons_P1213805021e_name @ X5 @ nil_Pr743626285e_name ) )
=> ( ! [X5: produc367475239e_name,Y3: produc367475239e_name,Zs: list_P227219501e_name] :
( ( P @ Zs )
=> ( ( P @ ( cons_P1213805021e_name @ Y3 @ Zs ) )
=> ( P @ ( cons_P1213805021e_name @ X5 @ ( cons_P1213805021e_name @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_270_shuffles_Oinduct,axiom,
! [P: list_P227219501e_name > list_P227219501e_name > $o,A0: list_P227219501e_name,A1: list_P227219501e_name] :
( ! [X_1: list_P227219501e_name] : ( P @ nil_Pr743626285e_name @ X_1 )
=> ( ! [Xs2: list_P227219501e_name] : ( P @ Xs2 @ nil_Pr743626285e_name )
=> ( ! [X5: produc367475239e_name,Xs2: list_P227219501e_name,Y3: produc367475239e_name,Ys: list_P227219501e_name] :
( ( P @ Xs2 @ ( cons_P1213805021e_name @ Y3 @ Ys ) )
=> ( ( P @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ Ys )
=> ( P @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ ( cons_P1213805021e_name @ Y3 @ Ys ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_271_transpose_Ocases,axiom,
! [X: list_l1406151091e_name] :
( ( X != nil_li29751603e_name )
=> ( ! [Xss: list_l1406151091e_name] :
( X
!= ( cons_l798145763e_name @ nil_Pr743626285e_name @ Xss ) )
=> ~ ! [X5: produc367475239e_name,Xs2: list_P227219501e_name,Xss: list_l1406151091e_name] :
( X
!= ( cons_l798145763e_name @ ( cons_P1213805021e_name @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_272_remdups__adj_Ocases,axiom,
! [X: list_P227219501e_name] :
( ( X != nil_Pr743626285e_name )
=> ( ! [X5: produc367475239e_name] :
( X
!= ( cons_P1213805021e_name @ X5 @ nil_Pr743626285e_name ) )
=> ~ ! [X5: produc367475239e_name,Y3: produc367475239e_name,Xs2: list_P227219501e_name] :
( X
!= ( cons_P1213805021e_name @ X5 @ ( cons_P1213805021e_name @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_273_sorted__wrt_Oinduct,axiom,
! [P: ( produc367475239e_name > produc367475239e_name > $o ) > list_P227219501e_name > $o,A0: produc367475239e_name > produc367475239e_name > $o,A1: list_P227219501e_name] :
( ! [P4: produc367475239e_name > produc367475239e_name > $o] : ( P @ P4 @ nil_Pr743626285e_name )
=> ( ! [P4: produc367475239e_name > produc367475239e_name > $o,X5: produc367475239e_name,Ys: list_P227219501e_name] :
( ( P @ P4 @ Ys )
=> ( P @ P4 @ ( cons_P1213805021e_name @ X5 @ Ys ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_274_Free_Ohyps,axiom,
late_transitions @ ( par @ p @ ( res @ x @ q ) ) @ ( late_FreeR @ alpha @ pxQ ) ).
% Free.hyps
thf(fact_275_alphaInput,axiom,
! [C2: name,P: pi,A: name,X: name] :
( ( fresh_name_pi @ C2 @ P )
=> ( ( input @ A @ X @ P )
= ( input @ A @ C2 @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ X @ C2 ) @ nil_Pr743626285e_name ) @ P ) ) ) ) ).
% alphaInput
thf(fact_276_name__calc_I1_J,axiom,
! [A: name,B: name,Pi: list_P227219501e_name,X: name] :
( ( perm_name_name @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ Pi ) @ X )
= ( swap_name2 @ ( produc1570949143e_name @ A @ B ) @ ( perm_name_name @ Pi @ X ) ) ) ).
% name_calc(1)
thf(fact_277_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_278_residual__Rep_Oinject_I2_J,axiom,
! [X21: late_freeRes,X22: pi,Y21: late_freeRes,Y22: pi] :
( ( ( late_r347633188eR_Rep @ X21 @ X22 )
= ( late_r347633188eR_Rep @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% residual_Rep.inject(2)
thf(fact_279_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_280_pi_Operm_I8_J,axiom,
! [Pi: list_P227219501e_name,X2: pi,X1: pi] :
( ( perm_name_pi @ Pi @ ( par @ X2 @ X1 ) )
= ( par @ ( perm_name_pi @ Pi @ X2 ) @ ( perm_name_pi @ Pi @ X1 ) ) ) ).
% pi.perm(8)
thf(fact_281_pi_Operm_I4_J,axiom,
! [Pi: list_P227219501e_name,X32: name,X1: name,X2: pi] :
( ( perm_name_pi @ Pi @ ( input @ X32 @ X1 @ X2 ) )
= ( input @ ( perm_name_name @ Pi @ X32 ) @ ( perm_name_name @ Pi @ X1 ) @ ( perm_name_pi @ Pi @ X2 ) ) ) ).
% pi.perm(4)
thf(fact_282_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_283_subject_Operm_I1_J,axiom,
! [Pi: list_P227219501e_name,X1: name] :
( ( perm_n1559866809ubject @ Pi @ ( late_InputS @ X1 ) )
= ( late_InputS @ ( perm_name_name @ Pi @ X1 ) ) ) ).
% subject.perm(1)
thf(fact_284_ScopeExt,axiom,
! [Y: name,R: pi,Z: name,S: pi] :
( ( fresh_name_pi @ Y @ R )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ Y @ ( res @ Z @ ( par @ R @ S ) ) ) @ ( res @ Z @ ( par @ R @ ( res @ Y @ S ) ) ) ) @ rel ) ) ).
% ScopeExt
thf(fact_285_Res,axiom,
! [Y: name,R: pi,S: pi] :
( ( fresh_name_pi @ Y @ R )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ Y @ ( par @ R @ S ) ) @ ( par @ R @ ( res @ Y @ S ) ) ) @ rel ) ) ).
% Res
thf(fact_286_pi_Odistinct_I55_J,axiom,
! [Name12: name,Name22: name,Pi: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name12 @ Name22 @ Pi )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(55)
thf(fact_287_pi_Oinject_I7_J,axiom,
! [X2: pi,X1: pi,Y2: pi,Y1: pi] :
( ( ( par @ X2 @ X1 )
= ( par @ Y2 @ Y1 ) )
= ( ( X2 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(7)
thf(fact_288_pi_Odistinct_I85_J,axiom,
! [Pi1: pi,Pi2: pi,Name4: name,Pi3: pi] :
( ( par @ Pi1 @ Pi2 )
!= ( res @ Name4 @ Pi3 ) ) ).
% pi.distinct(85)
thf(fact_289_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_290_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_291_pi_Odistinct_I57_J,axiom,
! [Name12: name,Name22: name,Pi: pi,Name4: name,Pi3: pi] :
( ( input @ Name12 @ Name22 @ Pi )
!= ( res @ Name4 @ Pi3 ) ) ).
% pi.distinct(57)
thf(fact_292_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name: name,Name4: name] :
( ( late_InputS @ Name )
!= ( late_BoundOutputS @ Name4 ) ) ).
% Late_Semantics.subject.distinct(1)
thf(fact_293_Late__Semantics1_Osubject_Odistinct_I1_J,axiom,
! [X1: name,X2: name] :
( ( late_InputS @ X1 )
!= ( late_BoundOutputS @ X2 ) ) ).
% Late_Semantics1.subject.distinct(1)
thf(fact_294_subject_Oinducts,axiom,
! [P: late_subject > $o,Subject: late_subject] :
( ! [Name3: name] : ( P @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name] : ( P @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Subject ) ) ) ).
% subject.inducts
thf(fact_295_subject_Oexhaust,axiom,
! [Y: late_subject] :
( ! [X13: name] :
( Y
!= ( late_InputS @ X13 ) )
=> ~ ! [X23: name] :
( Y
!= ( late_BoundOutputS @ X23 ) ) ) ).
% subject.exhaust
thf(fact_296_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_297_Par1F,axiom,
! [P: pi,Alpha: late_freeRes,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha @ ( par @ P2 @ Q ) ) ) ) ).
% Par1F
thf(fact_298_name__calc_I3_J,axiom,
! [A: name,C2: name,B: name] :
( ( ( A = C2 )
=> ( ( swap_name2 @ ( produc1570949143e_name @ A @ B ) @ C2 )
= B ) )
& ( ( A != C2 )
=> ( ( ( B = C2 )
=> ( ( swap_name2 @ ( produc1570949143e_name @ A @ B ) @ C2 )
= A ) )
& ( ( B != C2 )
=> ( ( swap_name2 @ ( produc1570949143e_name @ A @ B ) @ C2 )
= C2 ) ) ) ) ) ).
% name_calc(3)
thf(fact_299_resTrans_I2_J,axiom,
! [X: name,Y: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X @ ( input @ X @ Y @ P ) ) @ Rs ) ).
% resTrans(2)
thf(fact_300_inputIneqTrans,axiom,
! [A: name,X: name,P: pi,B: late_subject,Y: name,P2: pi] :
( ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ B @ Y @ P2 ) )
=> ~ ( fresh_717452603ubject @ A @ B ) ) ).
% inputIneqTrans
thf(fact_301_inputFreeTrans,axiom,
! [A: name,X: name,P: pi,Alpha: late_freeRes,P2: pi] :
~ ( late_transitions @ ( input @ A @ X @ P ) @ ( late_FreeR @ Alpha @ P2 ) ) ).
% inputFreeTrans
thf(fact_302_Late__Semantics_OPar1B,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ X @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P2 @ Q ) ) ) ) ) ).
% Late_Semantics.Par1B
thf(fact_303_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_304_parCasesB,axiom,
! [P: pi,Q: pi,A: late_subject,X: name,PQ: pi,Prop: pi > $o] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ PQ ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ! [P3: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
=> ( Prop @ ( par @ P3 @ Q ) ) )
=> ( ! [Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q3 ) )
=> ( Prop @ ( par @ P @ Q3 ) ) )
=> ( Prop @ PQ ) ) ) ) ) ) ).
% parCasesB
thf(fact_305_parCasesB_H,axiom,
! [P: pi,Q: pi,B: late_subject,Y: name,P2: pi] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ B @ Y @ P2 ) )
=> ( ! [P4: pi,A5: late_subject,X5: name,P3: pi,Q4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y @ P2 )
= ( late_BoundR @ A5 @ X5 @ ( par @ P3 @ Q4 ) ) )
=> ( ( late_transitions @ P4 @ ( late_BoundR @ A5 @ X5 @ P3 ) )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ( ( fresh_name_pi @ X5 @ Q4 )
=> ~ ( fresh_717452603ubject @ X5 @ A5 ) ) ) ) ) )
=> ~ ! [Q4: pi,A5: late_subject,X5: name,Q3: pi,P4: pi] :
( ( ( par @ P @ Q )
= ( par @ P4 @ Q4 ) )
=> ( ( ( late_BoundR @ B @ Y @ P2 )
= ( late_BoundR @ A5 @ X5 @ ( par @ P4 @ Q3 ) ) )
=> ( ( late_transitions @ Q4 @ ( late_BoundR @ A5 @ X5 @ Q3 ) )
=> ( ( fresh_name_pi @ X5 @ P4 )
=> ( ( fresh_name_pi @ X5 @ Q4 )
=> ~ ( fresh_717452603ubject @ X5 @ A5 ) ) ) ) ) ) ) ) ).
% parCasesB'
thf(fact_306_transitions_OPar1B,axiom,
! [P: pi,A: late_subject,X: name,P2: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_717452603ubject @ X @ A )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A @ X @ ( par @ P2 @ Q ) ) ) ) ) ) ) ).
% transitions.Par1B
thf(fact_307_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_308_resInputFreeTrans,axiom,
! [X: name,A: name,Y: name,P: pi,Alpha: late_freeRes,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( input @ A @ Y @ P ) ) @ ( late_FreeR @ Alpha @ P2 ) ) ).
% resInputFreeTrans
thf(fact_309_inputBoundOutputTrans,axiom,
! [A: name,X: name,P: pi,B: name,Y: name,P2: pi] :
~ ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Y @ P2 ) ) ).
% inputBoundOutputTrans
thf(fact_310_inputCases,axiom,
! [A: name,X: name,P: pi,B: late_subject,Y: name,YP: pi,Prop: late_subject > name > pi > $o] :
( ( late_transitions @ ( input @ A @ X @ P ) @ ( late_BoundR @ B @ Y @ YP ) )
=> ( ( Y != A )
=> ( ( Y != X )
=> ( ( fresh_name_pi @ Y @ P )
=> ( ( ( B
= ( late_InputS @ A ) )
=> ( ( YP
= ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ X @ Y ) @ nil_Pr743626285e_name ) @ P ) )
=> ( Prop @ ( late_InputS @ A ) @ Y @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ X @ Y ) @ nil_Pr743626285e_name ) @ P ) ) ) )
=> ( Prop @ B @ Y @ YP ) ) ) ) ) ) ).
% inputCases
thf(fact_311_resInputBoundOutputTrans,axiom,
! [X: name,A: name,Y: name,P: pi,B: name,Z: name,P2: pi] :
~ ( late_transitions @ ( res @ X @ ( input @ A @ Y @ P ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B ) @ Z @ P2 ) ) ).
% resInputBoundOutputTrans
thf(fact_312_Late__Semantics_OClose1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ Q2 ) )
=> ( ( fresh_name_pi @ Y @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P2 @ X @ Y ) @ Q2 ) ) ) ) ) ) ) ).
% Late_Semantics.Close1
thf(fact_313_Late__Semantics_OClose2,axiom,
! [P: pi,A: name,Y: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ Y @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P2 @ ( subs @ Q2 @ X @ Y ) ) ) ) ) ) ) ) ).
% Late_Semantics.Close2
thf(fact_314_transitions_OClose1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_name_pi @ Y @ P )
=> ( ( fresh_name_pi @ Y @ Q )
=> ( ( X != A )
=> ( ( fresh_name_pi @ X @ Q2 )
=> ( ( Y != A )
=> ( ( fresh_name_pi @ Y @ P2 )
=> ( ( X != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P2 @ X @ Y ) @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close1
thf(fact_315_EqvtRel,axiom,
eqvt_pi @ rel ).
% EqvtRel
thf(fact_316_subst__identity,axiom,
! [P: pi,A: name] :
( ( subs @ P @ A @ A )
= P ) ).
% subst_identity
thf(fact_317_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_318_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_319_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_320_forget,axiom,
! [A: name,P: pi,B: name] :
( ( fresh_name_pi @ A @ P )
=> ( ( subs @ P @ A @ B )
= P ) ) ).
% forget
thf(fact_321_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_322_fresh__fact2,axiom,
! [A: name,B: name,P: pi] :
( ( A != B )
=> ( fresh_name_pi @ A @ ( subs @ P @ A @ B ) ) ) ).
% fresh_fact2
thf(fact_323_Agent_OsubstRes3,axiom,
! [B: name,P: pi,A: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( subs @ ( res @ A @ P ) @ A @ B )
= ( res @ B @ ( subs @ P @ A @ B ) ) ) ) ).
% Agent.substRes3
thf(fact_324_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_325_eqvt__subs,axiom,
! [P5: list_P227219501e_name,P: pi,A: name,B: name] :
( ( perm_name_pi @ P5 @ ( subs @ P @ A @ B ) )
= ( subs @ ( perm_name_pi @ P5 @ P ) @ ( perm_name_name @ P5 @ A ) @ ( perm_name_name @ P5 @ B ) ) ) ).
% eqvt_subs
thf(fact_326_Strong__Late__Sim__SC__Mirabelle__ygjvazaixl_OsubstRes3,axiom,
! [A: name,P: pi,X: name] :
( ( subs @ ( res @ A @ P ) @ X @ A )
= ( res @ X @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ X @ A ) @ nil_Pr743626285e_name ) @ P ) ) ) ).
% Strong_Late_Sim_SC_Mirabelle_ygjvazaixl.substRes3
thf(fact_327_renaming,axiom,
! [C2: name,P: pi,A: name,B: name] :
( ( fresh_name_pi @ C2 @ P )
=> ( ( subs @ P @ A @ B )
= ( subs @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ C2 @ A ) @ nil_Pr743626285e_name ) @ P ) @ C2 @ B ) ) ) ).
% renaming
thf(fact_328_injPermSubst,axiom,
! [B: name,P: pi,A: name] :
( ( fresh_name_pi @ B @ P )
=> ( ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ A @ B ) @ nil_Pr743626285e_name ) @ P )
= ( subs @ P @ A @ B ) ) ) ).
% injPermSubst
thf(fact_329_Late__Semantics_OComm1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P2 @ X @ B ) @ Q2 ) ) ) ) ) ).
% Late_Semantics.Comm1
thf(fact_330_Late__Semantics_OComm2,axiom,
! [P: pi,A: name,B: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P2 @ ( subs @ Q2 @ X @ B ) ) ) ) ) ) ).
% Late_Semantics.Comm2
thf(fact_331_transitions_OComm2,axiom,
! [P: pi,A: name,B: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( X != A )
=> ( ( X != B )
=> ( ( fresh_name_pi @ X @ P2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P2 @ ( subs @ Q2 @ X @ B ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm2
thf(fact_332_transitions_OComm1,axiom,
! [P: pi,A: name,X: name,P2: pi,Q: pi,B: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A ) @ X @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A @ B ) @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( X != A )
=> ( ( X != B )
=> ( ( fresh_name_pi @ X @ Q2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P2 @ X @ B ) @ Q2 ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm1
thf(fact_333_transitions_OClose2,axiom,
! [P: pi,A: name,Y: name,P2: pi,Q: pi,X: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A ) @ Y @ P2 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A ) @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ( ( fresh_name_pi @ X @ Q )
=> ( ( fresh_name_pi @ Y @ P )
=> ( ( fresh_name_pi @ Y @ Q )
=> ( ( X != A )
=> ( ( fresh_name_pi @ X @ P2 )
=> ( ( Y != A )
=> ( ( fresh_name_pi @ Y @ Q2 )
=> ( ( X != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P2 @ ( subs @ Q2 @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close2
thf(fact_334_Agent_Oswap__name__def,axiom,
swap_name2 = swap_name ).
% Agent.swap_name_def
thf(fact_335_swap__name_Osimps,axiom,
! [A: name,C2: name,B: name] :
( ( ( A = C2 )
=> ( ( swap_name @ ( produc1570949143e_name @ A @ B ) @ C2 )
= B ) )
& ( ( A != C2 )
=> ( ( ( B = C2 )
=> ( ( swap_name @ ( produc1570949143e_name @ A @ B ) @ C2 )
= A ) )
& ( ( B != C2 )
=> ( ( swap_name @ ( produc1570949143e_name @ A @ B ) @ C2 )
= C2 ) ) ) ) ) ).
% swap_name.simps
thf(fact_336_name__prm__name_Osimps_I2_J,axiom,
! [X: produc367475239e_name,Xs: list_P227219501e_name,A: name] :
( ( name_prm_name @ ( cons_P1213805021e_name @ X @ Xs ) @ A )
= ( swap_name @ X @ ( name_prm_name @ Xs @ A ) ) ) ).
% name_prm_name.simps(2)
thf(fact_337_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 ) )
=> ? [P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P3 @ Q2 ) @ Rel ) ) ) ) ).
% simE(2)
thf(fact_338_simE_I1_J,axiom,
! [P: pi,Rel: set_Pr1834926343_pi_pi,Q: pi,A: late_subject,X: name,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_BoundR @ A @ X @ Q2 ) )
=> ( ( fresh_name_pi @ X @ P )
=> ? [P3: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A @ X @ P3 ) )
& ( strong2129052853vative @ P3 @ Q2 @ A @ X @ Rel ) ) ) ) ) ).
% simE(1)
thf(fact_339_simulation__def,axiom,
( strong743114133lation
= ( ^ [P6: pi,Rel2: set_Pr1834926343_pi_pi,Q5: pi] :
( ! [A2: late_subject,X3: name,Q6: pi] :
( ( ( late_transitions @ Q5 @ ( late_BoundR @ A2 @ X3 @ Q6 ) )
& ( fresh_name_pi @ X3 @ P6 ) )
=> ? [P7: pi] :
( ( late_transitions @ P6 @ ( late_BoundR @ A2 @ X3 @ P7 ) )
& ( strong2129052853vative @ P7 @ Q6 @ A2 @ X3 @ 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_340_simCases,axiom,
! [Q: pi,P: pi,Rel: set_Pr1834926343_pi_pi] :
( ! [A5: late_subject,Y3: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A5 @ Y3 @ Q3 ) )
=> ( ( fresh_name_pi @ Y3 @ P )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A5 @ Y3 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q3 @ A5 @ Y3 @ Rel ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q3 ) )
=> ? [P8: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P8 ) )
& ( member558706512_pi_pi @ ( product_Pair_pi_pi @ P8 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ).
% simCases
thf(fact_341_name__prm__name_Osimps_I1_J,axiom,
! [A: name] :
( ( name_prm_name @ nil_Pr743626285e_name @ A )
= A ) ).
% name_prm_name.simps(1)
thf(fact_342_derivativeEqvtI2,axiom,
! [P: pi,Q: pi,A: late_subject,X: name,Rel: set_Pr1834926343_pi_pi,Perm: list_P227219501e_name] :
( ( strong2129052853vative @ P @ Q @ A @ X @ Rel )
=> ( ( eqvt_pi @ Rel )
=> ( strong2129052853vative @ ( perm_name_pi @ Perm @ P ) @ ( perm_name_pi @ Perm @ Q ) @ A @ ( perm_name_name @ Perm @ X ) @ Rel ) ) ) ).
% derivativeEqvtI2
thf(fact_343_Agent_Oname__prm__name__def,axiom,
perm_name_name = name_prm_name ).
% Agent.name_prm_name_def
thf(fact_344_parSym,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi,Q: pi] :
( ! [R2: pi,S2: pi] : ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( par @ R2 @ S2 ) @ ( par @ S2 @ R2 ) ) @ Rel )
=> ( ! [R2: pi,S2: pi,X5: name] :
( ( member558706512_pi_pi @ ( product_Pair_pi_pi @ R2 @ S2 ) @ Rel )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ X5 @ R2 ) @ ( res @ X5 @ S2 ) ) @ Rel ) )
=> ( strong743114133lation @ ( par @ P @ Q ) @ Rel @ ( par @ Q @ P ) ) ) ) ).
% parSym
thf(fact_345_parAssocLeft,axiom,
! [Rel: set_Pr1834926343_pi_pi,P: pi,Q: pi,R: pi] :
( ! [S2: pi,T: pi,U: pi] : ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( par @ ( par @ S2 @ T ) @ U ) @ ( par @ S2 @ ( par @ T @ U ) ) ) @ Rel )
=> ( ! [S2: pi,T: pi,X5: name] :
( ( member558706512_pi_pi @ ( product_Pair_pi_pi @ S2 @ T ) @ Rel )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ X5 @ S2 ) @ ( res @ X5 @ T ) ) @ Rel ) )
=> ( ! [S2: pi,T: pi,U: pi,X5: name] :
( ( fresh_name_pi @ X5 @ S2 )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( res @ X5 @ ( par @ ( par @ S2 @ T ) @ U ) ) @ ( par @ S2 @ ( res @ X5 @ ( par @ T @ U ) ) ) ) @ Rel ) )
=> ( ! [S2: pi,T: pi,U: pi,X5: name] :
( ( fresh_name_pi @ X5 @ U )
=> ( member558706512_pi_pi @ ( product_Pair_pi_pi @ ( par @ ( res @ X5 @ ( par @ S2 @ T ) ) @ U ) @ ( res @ X5 @ ( par @ S2 @ ( par @ T @ U ) ) ) ) @ Rel ) )
=> ( strong743114133lation @ ( par @ ( par @ P @ Q ) @ R ) @ Rel @ ( par @ P @ ( par @ Q @ R ) ) ) ) ) ) ) ).
% parAssocLeft
thf(fact_346_derivativeEqvtI,axiom,
! [P: pi,Q: pi,A: late_subject,X: name,Rel: set_Pr1834926343_pi_pi,Perm: list_P227219501e_name] :
( ( strong2129052853vative @ P @ Q @ A @ X @ Rel )
=> ( ( eqvt_pi @ Rel )
=> ( strong2129052853vative @ ( perm_name_pi @ Perm @ P ) @ ( perm_name_pi @ Perm @ Q ) @ ( perm_n1559866809ubject @ Perm @ A ) @ ( perm_name_name @ Perm @ X ) @ Rel ) ) ) ).
% derivativeEqvtI
thf(fact_347_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_348_pi_Ofresh_I4_J,axiom,
! [A: name,X32: name,X1: name,X2: pi] :
( ( fresh_name_pi @ A @ ( input @ X32 @ X1 @ X2 ) )
= ( ( fresh_name_name @ A @ X32 )
& ( fresh_543891186ion_pi @ A @ ( abs_fun_name_pi @ X1 @ X2 ) ) ) ) ).
% pi.fresh(4)
% Helper facts (11)
thf(help_If_2_1_If_001t__Agent__Opi_T,axiom,
! [X: pi,Y: pi] :
( ( if_pi @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Agent__Opi_T,axiom,
! [X: pi,Y: pi] :
( ( if_pi @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Agent__Oname_T,axiom,
! [X: name,Y: name] :
( ( if_name @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Agent__Oname_T,axiom,
! [X: name,Y: name] :
( ( if_name @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Agent__Opi____Rep_T,axiom,
! [X: pi_Rep,Y: pi_Rep] :
( ( if_pi_Rep @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Agent__Opi____Rep_T,axiom,
! [X: pi_Rep,Y: pi_Rep] :
( ( if_pi_Rep @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Late____Semantics__OfreeRes_T,axiom,
! [X: late_freeRes,Y: late_freeRes] :
( ( if_Late_freeRes @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Late____Semantics__OfreeRes_T,axiom,
! [X: late_freeRes,Y: late_freeRes] :
( ( if_Late_freeRes @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Late____Semantics__Osubject_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Late____Semantics__Osubject_T,axiom,
! [X: late_subject,Y: late_subject] :
( ( if_Late_subject @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Late____Semantics__Osubject_T,axiom,
! [X: late_subject,Y: late_subject] :
( ( if_Late_subject @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
late_transitions @ ( perm_name_pi @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ z @ x ) @ nil_Pr743626285e_name ) @ q ) @ ( perm_n414416632sidual @ ( cons_P1213805021e_name @ ( produc1570949143e_name @ z @ x ) @ nil_Pr743626285e_name ) @ ( late_FreeR @ ( late_OutputR @ aa @ x ) @ q_a ) ) ).
%------------------------------------------------------------------------------