TPTP Problem File: ITP191^2.p
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%------------------------------------------------------------------------------
% File : ITP191^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_894__3417700_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_894__3417700_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 372 ( 156 unt; 66 typ; 0 def)
% Number of atoms : 697 ( 273 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 5200 ( 134 ~; 3 |; 39 &;4614 @)
% ( 0 <=>; 410 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 196 ( 196 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 56 usr; 18 con; 0-5 aty)
% Number of variables : 1252 ( 11 ^;1196 !; 16 ?;1252 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:17:53.702
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_t_Late__Semantics_Oresidual__Rep,type,
late_residual_Rep: $tType ).
thf(ty_t_Late__Semantics_Osubject__Rep,type,
late_subject_Rep: $tType ).
thf(ty_t_Late__Semantics_OfreeRes__Rep,type,
late_freeRes_Rep: $tType ).
thf(ty_t_Late__Semantics_Oresidual,type,
late_residual: $tType ).
thf(ty_t_Late__Semantics_Osubject,type,
late_subject: $tType ).
thf(ty_t_Late__Semantics_OfreeRes,type,
late_freeRes: $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Nominal_Onoption,type,
noption: $tType > $tType ).
thf(ty_t_Agent_Opi__Rep,type,
pi_Rep: $tType ).
thf(ty_t_Agent_Oname,type,
name: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Agent_Opi,type,
pi: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (52)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Agent_Ofs__name,type,
fs_name:
!>[A: $tType] : $o ).
thf(sy_cl_Agent_Opt__name,type,
pt_name:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Agent_Ocp__name__name,type,
cp_name_name:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_Agent_Opi_OInput,type,
input: name > name > pi > pi ).
thf(sy_c_Agent_Opi_OPar,type,
par: pi > pi > pi ).
thf(sy_c_Agent_Opi_ORes,type,
res: name > pi > pi ).
thf(sy_c_Agent_Orec_Oswap__name,type,
swap_name: ( product_prod @ name @ name ) > name > name ).
thf(sy_c_Agent_Osubs,type,
subs: pi > name > name > pi ).
thf(sy_c_HOL_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Late__Semantics_OfreeRes_OOutputR,type,
late_OutputR: name > name > late_freeRes ).
thf(sy_c_Late__Semantics_OfreeRes_OTauR,type,
late_TauR: late_freeRes ).
thf(sy_c_Late__Semantics_Oresidual_OBoundR,type,
late_BoundR: late_subject > name > pi > late_residual ).
thf(sy_c_Late__Semantics_Oresidual_OFreeR,type,
late_FreeR: late_freeRes > pi > late_residual ).
thf(sy_c_Late__Semantics_Oresidual__Rep_OFreeR__Rep,type,
late_r347633188eR_Rep: late_freeRes > pi > late_residual_Rep ).
thf(sy_c_Late__Semantics_Osubject_OBoundOutputS,type,
late_BoundOutputS: name > late_subject ).
thf(sy_c_Late__Semantics_Osubject_OInputS,type,
late_InputS: name > late_subject ).
thf(sy_c_Late__Semantics_Otransitions,type,
late_transitions: pi > late_residual > $o ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Nominal_Oabs__fun,type,
abs_fun:
!>[X: $tType,A: $tType] : ( X > A > X > ( noption @ A ) ) ).
thf(sy_c_Nominal_Ofresh,type,
fresh:
!>[X: $tType,A: $tType] : ( X > A > $o ) ).
thf(sy_c_Nominal_Operm,type,
perm:
!>[X: $tType,A: $tType] : ( ( list @ ( product_prod @ X @ X ) ) > A > A ) ).
thf(sy_c_Nominal_Osupports,type,
supports:
!>[X: $tType,A: $tType] : ( ( set @ X ) > A > $o ) ).
thf(sy_c_Nominal_Oswap,type,
swap:
!>[X: $tType] : ( ( product_prod @ X @ X ) > X > X ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Rel_Oeqvt,type,
eqvt:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Strong__Late__Sim_Oderivative,type,
strong2129052853vative: pi > pi > late_subject > name > ( set @ ( product_prod @ pi @ pi ) ) > $o ).
thf(sy_c_Strong__Late__Sim_Osimulation,type,
strong743114133lation: pi > ( set @ ( product_prod @ pi @ pi ) ) > pi > $o ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: pi ).
thf(sy_v_PQ____,type,
pq: pi ).
thf(sy_v_P_H____,type,
p2: pi ).
thf(sy_v_Q,type,
q: pi ).
thf(sy_v_Q_H____,type,
q2: pi ).
thf(sy_v_Rel,type,
rel: set @ ( product_prod @ pi @ pi ) ).
thf(sy_v_aa____,type,
aa: late_subject ).
thf(sy_v_ab____,type,
ab: name ).
thf(sy_v_b____,type,
b: name ).
thf(sy_v_c____,type,
c: name ).
thf(sy_v_x,type,
x: name ).
thf(sy_v_xPQ____,type,
xPQ: pi ).
thf(sy_v_y____,type,
y: name ).
thf(sy_v_ya____,type,
ya: name ).
% Relevant facts (255)
thf(fact_0_cComm1_Ohyps_I5_J,axiom,
ya != b ).
% cComm1.hyps(5)
thf(fact_1_cComm1_Ohyps_I9_J,axiom,
( ( late_OutputR @ ab @ x )
= late_TauR ) ).
% cComm1.hyps(9)
thf(fact_2_cOpen_Ohyps_I2_J,axiom,
ab != x ).
% cOpen.hyps(2)
thf(fact_3_cComm1_Ohyps_I6_J,axiom,
ya != c ).
% cComm1.hyps(6)
thf(fact_4_cOpen_Ohyps_I3_J,axiom,
( aa
= ( late_BoundOutputS @ ab ) ) ).
% cOpen.hyps(3)
thf(fact_5_assms_I3_J,axiom,
eqvt @ pi @ rel ).
% assms(3)
thf(fact_6__092_060open_062y_____A_092_060noteq_062_Ax_092_060close_062,axiom,
y != x ).
% \<open>y__ \<noteq> x\<close>
thf(fact_7_Bound_Ohyps_I1_J,axiom,
late_transitions @ ( res @ x @ ( par @ p @ q ) ) @ ( late_BoundR @ aa @ y @ xPQ ) ).
% Bound.hyps(1)
thf(fact_8_assms_I1_J,axiom,
fresh @ name @ pi @ x @ p ).
% assms(1)
thf(fact_9_cComm1_Ohyps_I3_J,axiom,
fresh @ name @ pi @ ya @ p ).
% cComm1.hyps(3)
thf(fact_10_cComm1_Ohyps_I4_J,axiom,
fresh @ name @ pi @ ya @ q ).
% cComm1.hyps(4)
thf(fact_11_cComm1_Ohyps_I7_J,axiom,
fresh @ name @ pi @ ya @ q2 ).
% cComm1.hyps(7)
thf(fact_12_Bound_Ohyps_I3_J,axiom,
fresh @ name @ pi @ y @ ( res @ x @ ( par @ p @ q ) ) ).
% Bound.hyps(3)
thf(fact_13_Bound_Ohyps_I2_J,axiom,
fresh @ name @ pi @ y @ ( par @ p @ ( res @ x @ q ) ) ).
% Bound.hyps(2)
thf(fact_14_Strong__Late__Sim__SC__Mirabelle__dtpyqonsky_OsubstRes3,axiom,
! [A2: name,P: pi,X2: name] :
( ( subs @ ( res @ A2 @ P ) @ X2 @ A2 )
= ( res @ X2 @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X2 @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P ) ) ) ).
% Strong_Late_Sim_SC_Mirabelle_dtpyqonsky.substRes3
thf(fact_15_cComm1_Ohyps_I1_J,axiom,
late_transitions @ p @ ( late_BoundR @ ( late_InputS @ b ) @ ya @ p2 ) ).
% cComm1.hyps(1)
thf(fact_16_perm__swap_I1_J,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [A2: name,B2: name,X2: X3] :
( ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 ) )
= X2 ) ) ).
% perm_swap(1)
thf(fact_17_name__id,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,X2: X] :
( ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 )
= X2 ) ) ).
% name_id
thf(fact_18_name__swap,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,B2: name,X2: X] :
( ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 )
= ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ B2 @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 ) ) ) ).
% name_swap
thf(fact_19_name__swap__bij,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,B2: name,X2: X] :
( ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 ) )
= X2 ) ) ).
% name_swap_bij
thf(fact_20_cOpen_Ohyps_I1_J,axiom,
late_transitions @ ( par @ p @ q ) @ ( late_FreeR @ ( late_OutputR @ ab @ x ) @ pq ) ).
% cOpen.hyps(1)
thf(fact_21_pi_Operm_I8_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X22: pi,X1: pi] :
( ( perm @ name @ pi @ Pi @ ( par @ X22 @ X1 ) )
= ( par @ ( perm @ name @ pi @ Pi @ X22 ) @ ( perm @ name @ pi @ Pi @ X1 ) ) ) ).
% pi.perm(8)
thf(fact_22_simps_I8_J,axiom,
! [P: pi,Q: pi,C: name,D: name] :
( ( subs @ ( par @ P @ Q ) @ C @ D )
= ( par @ ( subs @ P @ C @ D ) @ ( subs @ Q @ C @ D ) ) ) ).
% simps(8)
thf(fact_23_simps_I9_J,axiom,
! [X2: name,C: name,D: name,P: pi] :
( ( X2 != C )
=> ( ( X2 != D )
=> ( ( subs @ ( res @ X2 @ P ) @ C @ D )
= ( res @ X2 @ ( subs @ P @ C @ D ) ) ) ) ) ).
% simps(9)
thf(fact_24_pt__name1,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ! [X2: A] :
( ( perm @ name @ A @ ( nil @ ( product_prod @ name @ name ) ) @ X2 )
= X2 ) ) ).
% pt_name1
thf(fact_25_cComm1_Ohyps_I2_J,axiom,
late_transitions @ q @ ( late_FreeR @ ( late_OutputR @ b @ c ) @ q2 ) ).
% cComm1.hyps(2)
thf(fact_26__092_060open_062y_____A_092_060sharp_062_AP_A_092_060parallel_062_AQ_092_060close_062,axiom,
fresh @ name @ pi @ y @ ( par @ p @ q ) ).
% \<open>y__ \<sharp> P \<parallel> Q\<close>
thf(fact_27_subst__identity,axiom,
! [P: pi,A2: name] :
( ( subs @ P @ A2 @ A2 )
= P ) ).
% subst_identity
thf(fact_28_pi__perm__empty,axiom,
! [Pi_Rep: pi_Rep] :
( ( perm @ name @ pi_Rep @ ( nil @ ( product_prod @ name @ name ) ) @ Pi_Rep )
= Pi_Rep ) ).
% pi_perm_empty
thf(fact_29__092_060open_062y_____A_092_060sharp_062_AP_092_060close_062,axiom,
fresh @ name @ pi @ y @ p ).
% \<open>y__ \<sharp> P\<close>
thf(fact_30__092_060open_062y_____A_092_060sharp_062_AQ_092_060close_062,axiom,
fresh @ name @ pi @ y @ q ).
% \<open>y__ \<sharp> Q\<close>
thf(fact_31_pi_Ofresh_I8_J,axiom,
! [A2: name,X22: pi,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( par @ X22 @ X1 ) )
= ( ( fresh @ name @ pi @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(8)
thf(fact_32_pi_Operm_I9_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X1: name,X22: pi] :
( ( perm @ name @ pi @ Pi @ ( res @ X1 @ X22 ) )
= ( res @ ( perm @ name @ name @ Pi @ X1 ) @ ( perm @ name @ pi @ Pi @ X22 ) ) ) ).
% pi.perm(9)
thf(fact_33_perm__fresh__fresh,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [A2: name,X2: X3,B2: name] :
( ( fresh @ name @ X3 @ A2 @ X2 )
=> ( ( fresh @ name @ X3 @ B2 @ X2 )
=> ( ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 )
= X2 ) ) ) ) ).
% perm_fresh_fresh
thf(fact_34_name__fresh__fresh,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,X2: X,B2: name] :
( ( fresh @ name @ X @ A2 @ X2 )
=> ( ( fresh @ name @ X @ B2 @ X2 )
=> ( ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 )
= X2 ) ) ) ) ).
% name_fresh_fresh
thf(fact_35_Bound_Ohyps_I4_J,axiom,
fresh @ name @ late_subject @ y @ aa ).
% Bound.hyps(4)
thf(fact_36_fresh__perm__app,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [A2: name,Pi: list @ ( product_prod @ name @ name ),X2: X3] :
( ( fresh @ name @ ( list @ ( product_prod @ name @ name ) ) @ A2 @ Pi )
=> ( ( fresh @ name @ X3 @ A2 @ X2 )
=> ( fresh @ name @ X3 @ A2 @ ( perm @ name @ X3 @ Pi @ X2 ) ) ) ) ) ).
% fresh_perm_app
thf(fact_37_name__exists__fresh,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [X2: A] :
~ ! [C2: name] :
~ ( fresh @ name @ A @ C2 @ X2 ) ) ).
% name_exists_fresh
thf(fact_38_fresh__eqvt,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [Pi: list @ ( product_prod @ name @ name ),A2: name,X2: X3] :
( ( perm @ name @ $o @ Pi @ ( fresh @ name @ X3 @ A2 @ X2 ) )
= ( fresh @ name @ X3 @ ( perm @ name @ name @ Pi @ A2 ) @ ( perm @ name @ X3 @ Pi @ X2 ) ) ) ) ).
% fresh_eqvt
thf(fact_39_fresh__bij,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [Pi: list @ ( product_prod @ name @ name ),A2: name,X2: X3] :
( ( fresh @ name @ X3 @ ( perm @ name @ name @ Pi @ A2 ) @ ( perm @ name @ X3 @ Pi @ X2 ) )
= ( fresh @ name @ X3 @ A2 @ X2 ) ) ) ).
% fresh_bij
thf(fact_40_freshRes,axiom,
! [A2: name,P: pi] : ( fresh @ name @ pi @ A2 @ ( res @ A2 @ P ) ) ).
% freshRes
thf(fact_41_fresh__fact2,axiom,
! [A2: name,B2: name,P: pi] :
( ( A2 != B2 )
=> ( fresh @ name @ pi @ A2 @ ( subs @ P @ A2 @ B2 ) ) ) ).
% fresh_fact2
thf(fact_42_fresh__fact1,axiom,
! [A2: name,P: pi,C: name,B2: name] :
( ( fresh @ name @ pi @ A2 @ P )
=> ( ( A2 != C )
=> ( fresh @ name @ pi @ A2 @ ( subs @ P @ B2 @ C ) ) ) ) ).
% fresh_fact1
thf(fact_43_forget,axiom,
! [A2: name,P: pi,B2: name] :
( ( fresh @ name @ pi @ A2 @ P )
=> ( ( subs @ P @ A2 @ B2 )
= P ) ) ).
% forget
thf(fact_44_name__calc_I2_J,axiom,
! [X2: name] :
( ( perm @ name @ name @ ( nil @ ( product_prod @ name @ name ) ) @ X2 )
= X2 ) ).
% name_calc(2)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_Agent_OsubstRes3,axiom,
! [B2: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ B2 @ P )
=> ( ( subs @ ( res @ A2 @ P ) @ A2 @ B2 )
= ( res @ B2 @ ( subs @ P @ A2 @ B2 ) ) ) ) ).
% Agent.substRes3
thf(fact_50_substRes2,axiom,
! [B2: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ B2 @ P )
=> ( ( res @ A2 @ P )
= ( res @ B2 @ ( subs @ P @ A2 @ B2 ) ) ) ) ).
% substRes2
thf(fact_51_eqvt__subs,axiom,
! [P2: list @ ( product_prod @ name @ name ),P: pi,A2: name,B2: name] :
( ( perm @ name @ pi @ P2 @ ( subs @ P @ A2 @ B2 ) )
= ( subs @ ( perm @ name @ pi @ P2 @ P ) @ ( perm @ name @ name @ P2 @ A2 ) @ ( perm @ name @ name @ P2 @ B2 ) ) ) ).
% eqvt_subs
thf(fact_52_swap__simps_I1_J,axiom,
! [A2: name,B2: name] :
( ( perm @ name @ name @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ A2 )
= B2 ) ).
% swap_simps(1)
thf(fact_53_swap__simps_I2_J,axiom,
! [A2: name,B2: name] :
( ( perm @ name @ name @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ B2 )
= A2 ) ).
% swap_simps(2)
thf(fact_54_swap__simps_I3_J,axiom,
! [A2: name,C: name,B2: name] :
( ( A2 != C )
=> ( ( B2 != C )
=> ( ( perm @ name @ name @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ C )
= C ) ) ) ).
% swap_simps(3)
thf(fact_55_fresh__aux,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [C: name,A2: name,X2: X3,B2: name] :
( ( C != A2 )
=> ( ( fresh @ name @ X3 @ A2 @ X2 )
=> ( ( fresh @ name @ X3 @ C @ X2 )
=> ( fresh @ name @ X3 @ C @ ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 ) ) ) ) ) ) ).
% fresh_aux
thf(fact_56_cp__name__name1,axiom,
! [A: $tType] :
( ( cp_name_name @ A )
=> ! [Pi1: list @ ( product_prod @ name @ name ),Pi2: list @ ( product_prod @ name @ name ),X2: A] :
( ( perm @ name @ A @ Pi1 @ ( perm @ name @ A @ Pi2 @ X2 ) )
= ( perm @ name @ A @ ( perm @ name @ ( list @ ( product_prod @ name @ name ) ) @ Pi1 @ Pi2 ) @ ( perm @ name @ A @ Pi1 @ X2 ) ) ) ) ).
% cp_name_name1
thf(fact_57_name__perm__compose,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [Pi2: list @ ( product_prod @ name @ name ),Pi1: list @ ( product_prod @ name @ name ),X2: X] :
( ( perm @ name @ X @ Pi2 @ ( perm @ name @ X @ Pi1 @ X2 ) )
= ( perm @ name @ X @ ( perm @ name @ ( list @ ( product_prod @ name @ name ) ) @ Pi2 @ Pi1 ) @ ( perm @ name @ X @ Pi2 @ X2 ) ) ) ) ).
% name_perm_compose
thf(fact_58_perm__compose,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [Pi2: list @ ( product_prod @ name @ name ),Pi1: list @ ( product_prod @ name @ name ),X2: X3] :
( ( perm @ name @ X3 @ Pi2 @ ( perm @ name @ X3 @ Pi1 @ X2 ) )
= ( perm @ name @ X3 @ ( perm @ name @ ( list @ ( product_prod @ name @ name ) ) @ Pi2 @ Pi1 ) @ ( perm @ name @ X3 @ Pi2 @ X2 ) ) ) ) ).
% perm_compose
thf(fact_59_perm__bij,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [Pi: list @ ( product_prod @ name @ name ),X2: X3,Y: X3] :
( ( ( perm @ name @ X3 @ Pi @ X2 )
= ( perm @ name @ X3 @ Pi @ Y ) )
= ( X2 = Y ) ) ) ).
% perm_bij
thf(fact_60_perm__app,axiom,
! [B: $tType,X3: $tType] :
( ( pt_name @ X3 )
=> ! [Pi: list @ ( product_prod @ name @ name ),F: X3 > B,X2: X3] :
( ( perm @ name @ B @ Pi @ ( F @ X2 ) )
= ( perm @ name @ ( X3 > B ) @ Pi @ F @ ( perm @ name @ X3 @ Pi @ X2 ) ) ) ) ).
% perm_app
thf(fact_61_in__eqvt,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [Pi: list @ ( product_prod @ name @ name ),X2: X3,X6: set @ X3] :
( ( perm @ name @ $o @ Pi @ ( member @ X3 @ X2 @ X6 ) )
= ( member @ X3 @ ( perm @ name @ X3 @ Pi @ X2 ) @ ( perm @ name @ ( set @ X3 ) @ Pi @ X6 ) ) ) ) ).
% in_eqvt
thf(fact_62_eq__eqvt,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [Pi: list @ ( product_prod @ name @ name ),X2: X3,Y: X3] :
( ( perm @ name @ $o @ Pi @ ( X2 = Y ) )
= ( ( perm @ name @ X3 @ Pi @ X2 )
= ( perm @ name @ X3 @ Pi @ Y ) ) ) ) ).
% eq_eqvt
thf(fact_63_pi_Oinject_I7_J,axiom,
! [X22: pi,X1: pi,Y2: pi,Y1: pi] :
( ( ( par @ X22 @ X1 )
= ( par @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(7)
thf(fact_64_alphaRes,axiom,
! [C: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ C @ P )
=> ( ( res @ A2 @ P )
= ( res @ C @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ C ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P ) ) ) ) ).
% alphaRes
thf(fact_65_injPermSubst,axiom,
! [B2: name,P: pi,A2: name] :
( ( fresh @ name @ pi @ B2 @ P )
=> ( ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P )
= ( subs @ P @ A2 @ B2 ) ) ) ).
% injPermSubst
thf(fact_66_renaming,axiom,
! [C: name,P: pi,A2: name,B2: name] :
( ( fresh @ name @ pi @ C @ P )
=> ( ( subs @ P @ A2 @ B2 )
= ( subs @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ C @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P ) @ C @ B2 ) ) ) ).
% renaming
thf(fact_67_pi_Odistinct_I85_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi3: pi] :
( ( par @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi3 ) ) ).
% pi.distinct(85)
thf(fact_68_perm__swap_I2_J,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [A2: name,B2: name,X2: X3] :
( ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ B2 @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X2 ) )
= X2 ) ) ).
% perm_swap(2)
thf(fact_69_transitions_OClose2,axiom,
! [P: pi,A2: name,Y: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( ( fresh @ name @ pi @ Y @ Q )
=> ( ( X2 != A2 )
=> ( ( fresh @ name @ pi @ X2 @ P3 )
=> ( ( Y != A2 )
=> ( ( fresh @ name @ pi @ Y @ Q2 )
=> ( ( X2 != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P3 @ ( subs @ Q2 @ X2 @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close2
thf(fact_70_transitions_OClose1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( ( fresh @ name @ pi @ Y @ Q )
=> ( ( X2 != A2 )
=> ( ( fresh @ name @ pi @ X2 @ Q2 )
=> ( ( Y != A2 )
=> ( ( fresh @ name @ pi @ Y @ P3 )
=> ( ( X2 != Y )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P3 @ X2 @ Y ) @ Q2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Close1
thf(fact_71_Late__Semantics_OClose2,axiom,
! [P: pi,A2: name,Y: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ Y @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ P3 @ ( subs @ Q2 @ X2 @ Y ) ) ) ) ) ) ) ) ).
% Late_Semantics.Close2
thf(fact_72_Late__Semantics_OClose1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,Y: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ Y @ Q2 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( res @ Y @ ( par @ ( subs @ P3 @ X2 @ Y ) @ Q2 ) ) ) ) ) ) ) ).
% Late_Semantics.Close1
thf(fact_73_transitions_OComm2,axiom,
! [P: pi,A2: name,B2: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( X2 != A2 )
=> ( ( X2 != B2 )
=> ( ( fresh @ name @ pi @ X2 @ P3 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P3 @ ( subs @ Q2 @ X2 @ B2 ) ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm2
thf(fact_74_transitions_OComm1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,B2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( X2 != A2 )
=> ( ( X2 != B2 )
=> ( ( fresh @ name @ pi @ X2 @ Q2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P3 @ X2 @ B2 ) @ Q2 ) ) ) ) ) ) ) ) ) ) ).
% transitions.Comm1
thf(fact_75_Late__Semantics_OComm2,axiom,
! [P: pi,A2: name,B2: name,P3: pi,Q: pi,X2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ P3 @ ( subs @ Q2 @ X2 @ B2 ) ) ) ) ) ) ).
% Late_Semantics.Comm2
thf(fact_76_Late__Semantics_OComm1,axiom,
! [P: pi,A2: name,X2: name,P3: pi,Q: pi,B2: name,Q2: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P3 ) )
=> ( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ Q2 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ late_TauR @ ( par @ ( subs @ P3 @ X2 @ B2 ) @ Q2 ) ) ) ) ) ).
% Late_Semantics.Comm1
thf(fact_77_alphaBoundResidual,axiom,
! [X7: name,P: pi,A2: late_subject,X2: name] :
( ( fresh @ name @ pi @ X7 @ P )
=> ( ( late_BoundR @ A2 @ X2 @ P )
= ( late_BoundR @ A2 @ X7 @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X2 @ X7 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P ) ) ) ) ).
% alphaBoundResidual
thf(fact_78_freeRes__perm__empty,axiom,
! [FreeRes_Rep: late_freeRes_Rep] :
( ( perm @ name @ late_freeRes_Rep @ ( nil @ ( product_prod @ name @ name ) ) @ FreeRes_Rep )
= FreeRes_Rep ) ).
% freeRes_perm_empty
thf(fact_79_subject__perm__empty,axiom,
! [Subject_Rep: late_subject_Rep] :
( ( perm @ name @ late_subject_Rep @ ( nil @ ( product_prod @ name @ name ) ) @ Subject_Rep )
= Subject_Rep ) ).
% subject_perm_empty
thf(fact_80_residual__perm__empty,axiom,
! [Residual_Rep: late_residual_Rep] :
( ( perm @ name @ late_residual_Rep @ ( nil @ ( product_prod @ name @ name ) ) @ Residual_Rep )
= Residual_Rep ) ).
% residual_perm_empty
thf(fact_81_freeRes_Ofresh_I1_J,axiom,
! [A2: name,X22: name,X1: name] :
( ( fresh @ name @ late_freeRes @ A2 @ ( late_OutputR @ X22 @ X1 ) )
= ( ( fresh @ name @ name @ A2 @ X22 )
& ( fresh @ name @ name @ A2 @ X1 ) ) ) ).
% freeRes.fresh(1)
thf(fact_82_Late__Semantics_OfreeRes_Oinject,axiom,
! [X22: name,X1: name,Y2: name,Y1: name] :
( ( ( late_OutputR @ X22 @ X1 )
= ( late_OutputR @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% Late_Semantics.freeRes.inject
thf(fact_83_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_84_freeRes_Ofresh_I2_J,axiom,
! [A2: name] : ( fresh @ name @ late_freeRes @ A2 @ late_TauR ) ).
% freeRes.fresh(2)
thf(fact_85_freeRes_Operm_I2_J,axiom,
! [Pi: list @ ( product_prod @ name @ name )] :
( ( perm @ name @ late_freeRes @ Pi @ late_TauR )
= late_TauR ) ).
% freeRes.perm(2)
thf(fact_86_cComm1_Ohyps_I8_J,axiom,
fresh @ name @ product_unit @ ya @ product_Unity ).
% cComm1.hyps(8)
thf(fact_87_residual_Ofresh_I2_J,axiom,
! [A2: name,X22: late_freeRes,X1: pi] :
( ( fresh @ name @ late_residual @ A2 @ ( late_FreeR @ X22 @ X1 ) )
= ( ( fresh @ name @ late_freeRes @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% residual.fresh(2)
thf(fact_88_substTrans,axiom,
! [B2: name,P: pi,A2: name,C: name] :
( ( fresh @ name @ pi @ B2 @ P )
=> ( ( subs @ ( subs @ P @ A2 @ B2 ) @ B2 @ C )
= ( subs @ P @ A2 @ C ) ) ) ).
% substTrans
thf(fact_89_residual_Operm_I2_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X22: late_freeRes,X1: pi] :
( ( perm @ name @ late_residual @ Pi @ ( late_FreeR @ X22 @ X1 ) )
= ( late_FreeR @ ( perm @ name @ late_freeRes @ Pi @ X22 ) @ ( perm @ name @ pi @ Pi @ X1 ) ) ) ).
% residual.perm(2)
thf(fact_90_freeRes_Operm_I1_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X22: name,X1: name] :
( ( perm @ name @ late_freeRes @ Pi @ ( late_OutputR @ X22 @ X1 ) )
= ( late_OutputR @ ( perm @ name @ name @ Pi @ X22 ) @ ( perm @ name @ name @ Pi @ X1 ) ) ) ).
% freeRes.perm(1)
thf(fact_91_subject_Operm_I1_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X1: name] :
( ( perm @ name @ late_subject @ Pi @ ( late_InputS @ X1 ) )
= ( late_InputS @ ( perm @ name @ name @ Pi @ X1 ) ) ) ).
% subject.perm(1)
thf(fact_92_subject_Operm_I2_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X1: name] :
( ( perm @ name @ late_subject @ Pi @ ( late_BoundOutputS @ X1 ) )
= ( late_BoundOutputS @ ( perm @ name @ name @ Pi @ X1 ) ) ) ).
% subject.perm(2)
thf(fact_93_subject_Ofresh_I2_J,axiom,
! [A2: name,X1: name] :
( ( fresh @ name @ late_subject @ A2 @ ( late_BoundOutputS @ X1 ) )
= ( fresh @ name @ name @ A2 @ X1 ) ) ).
% subject.fresh(2)
thf(fact_94_subject_Ofresh_I1_J,axiom,
! [A2: name,X1: name] :
( ( fresh @ name @ late_subject @ A2 @ ( late_InputS @ X1 ) )
= ( fresh @ name @ name @ A2 @ X1 ) ) ).
% subject.fresh(1)
thf(fact_95_residual_Operm_I1_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X32: late_subject,X1: name,X22: pi] :
( ( perm @ name @ late_residual @ Pi @ ( late_BoundR @ X32 @ X1 @ X22 ) )
= ( late_BoundR @ ( perm @ name @ late_subject @ Pi @ X32 ) @ ( perm @ name @ name @ Pi @ X1 ) @ ( perm @ name @ pi @ Pi @ X22 ) ) ) ).
% residual.perm(1)
thf(fact_96_name__bij,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X2: name,Y: name] :
( ( ( perm @ name @ name @ Pi @ X2 )
= ( perm @ name @ name @ Pi @ Y ) )
= ( X2 = Y ) ) ).
% name_bij
thf(fact_97_freshResidual,axiom,
! [P: pi,Rs: late_residual,X2: name] :
( ( late_transitions @ P @ Rs )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( fresh @ name @ late_residual @ X2 @ Rs ) ) ) ).
% freshResidual
thf(fact_98_resCasesF,axiom,
! [X2: name,P: pi,Alpha: late_freeRes,XP: pi,F2: pi > $o] :
( ( late_transitions @ ( res @ X2 @ P ) @ ( late_FreeR @ Alpha @ XP ) )
=> ( ! [P4: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P4 ) )
=> ( ( fresh @ name @ late_freeRes @ X2 @ Alpha )
=> ( F2 @ ( res @ X2 @ P4 ) ) ) )
=> ( F2 @ XP ) ) ) ).
% resCasesF
thf(fact_99_resCasesF_H,axiom,
! [X2: name,P: pi,Alpha: late_freeRes,P3: pi] :
( ( late_transitions @ ( res @ X2 @ P ) @ ( late_FreeR @ Alpha @ P3 ) )
=> ~ ! [P5: pi,Alpha2: late_freeRes,P4: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P5 ) )
=> ( ( ( late_FreeR @ Alpha @ P3 )
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P4 ) ) )
=> ( ( late_transitions @ P5 @ ( late_FreeR @ Alpha2 @ P4 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha2 ) ) ) ) ) ).
% resCasesF'
thf(fact_100_ResF,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh @ name @ late_freeRes @ Y @ Alpha )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_FreeR @ Alpha @ ( res @ Y @ P3 ) ) ) ) ) ).
% ResF
thf(fact_101_Late__Semantics_OResB,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ late_subject @ Y @ A2 )
=> ( ( Y != X2 )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A2 @ X2 @ ( res @ Y @ P3 ) ) ) ) ) ) ).
% Late_Semantics.ResB
thf(fact_102_freshFreeDerivative_I1_J,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( fresh @ name @ late_freeRes @ Y @ Alpha ) ) ) ).
% freshFreeDerivative(1)
thf(fact_103_freshBoundDerivative_I1_J,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( fresh @ name @ late_subject @ Y @ A2 ) ) ) ).
% freshBoundDerivative(1)
thf(fact_104_transitions_OResB,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ late_subject @ Y @ A2 )
=> ( ( Y != X2 )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ late_subject @ X2 @ A2 )
=> ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A2 @ X2 @ ( res @ Y @ P3 ) ) ) ) ) ) ) ) ).
% transitions.ResB
thf(fact_105_parCasesB_H,axiom,
! [P: pi,Q: pi,B2: late_subject,Y: name,P3: pi] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ B2 @ Y @ P3 ) )
=> ( ! [P5: pi,A4: late_subject,X5: name,P4: pi,Q3: pi] :
( ( ( par @ P @ Q )
= ( par @ P5 @ Q3 ) )
=> ( ( ( late_BoundR @ B2 @ Y @ P3 )
= ( late_BoundR @ A4 @ X5 @ ( par @ P4 @ Q3 ) ) )
=> ( ( late_transitions @ P5 @ ( late_BoundR @ A4 @ X5 @ P4 ) )
=> ( ( fresh @ name @ pi @ X5 @ P5 )
=> ( ( fresh @ name @ pi @ X5 @ Q3 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) )
=> ~ ! [Q3: pi,A4: late_subject,X5: name,Q4: pi,P5: pi] :
( ( ( par @ P @ Q )
= ( par @ P5 @ Q3 ) )
=> ( ( ( late_BoundR @ B2 @ Y @ P3 )
= ( late_BoundR @ A4 @ X5 @ ( par @ P5 @ Q4 ) ) )
=> ( ( late_transitions @ Q3 @ ( late_BoundR @ A4 @ X5 @ Q4 ) )
=> ( ( fresh @ name @ pi @ X5 @ P5 )
=> ( ( fresh @ name @ pi @ X5 @ Q3 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) ) ) ).
% parCasesB'
thf(fact_106_transitions_OPar1B,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ late_subject @ X2 @ A2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P3 @ Q ) ) ) ) ) ) ) ).
% transitions.Par1B
thf(fact_107_transitions_OPar2B,axiom,
! [Q: pi,A2: late_subject,X2: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ( fresh @ name @ late_subject @ X2 @ A2 )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P @ Q2 ) ) ) ) ) ) ) ).
% transitions.Par2B
thf(fact_108_resCases_H,axiom,
! [X2: name,P: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X2 @ P ) @ Rs )
=> ( ! [P5: pi,A4: name,B3: name] :
( ( ( res @ X2 @ P )
= ( res @ B3 @ P5 ) )
=> ! [P4: pi] :
( ( Rs
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B3 @ P4 ) )
=> ( ( late_transitions @ P5 @ ( late_FreeR @ ( late_OutputR @ A4 @ B3 ) @ P4 ) )
=> ( A4 = B3 ) ) ) )
=> ( ! [P5: pi,A4: late_subject,X5: name,P4: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P5 ) )
=> ( ( Rs
= ( late_BoundR @ A4 @ X5 @ ( res @ Y3 @ P4 ) ) )
=> ( ( late_transitions @ P5 @ ( late_BoundR @ A4 @ X5 @ P4 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X5 )
=> ( ( fresh @ name @ pi @ X5 @ P5 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) )
=> ~ ! [P5: pi,Alpha2: late_freeRes,P4: pi,Y3: name] :
( ( ( res @ X2 @ P )
= ( res @ Y3 @ P5 ) )
=> ( ( Rs
= ( late_FreeR @ Alpha2 @ ( res @ Y3 @ P4 ) ) )
=> ( ( late_transitions @ P5 @ ( late_FreeR @ Alpha2 @ P4 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha2 ) ) ) ) ) ) ) ).
% resCases'
thf(fact_109_residual_Oinject_I2_J,axiom,
! [X22: late_freeRes,X1: pi,Y2: late_freeRes,Y1: pi] :
( ( ( late_FreeR @ X22 @ X1 )
= ( late_FreeR @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% residual.inject(2)
thf(fact_110_resCasesB_H,axiom,
! [X7: name,P: pi,A2: late_subject,Y4: name,P3: pi] :
( ( late_transitions @ ( res @ X7 @ P ) @ ( late_BoundR @ A2 @ Y4 @ P3 ) )
=> ( ! [P5: pi,A4: name,B3: name] :
( ( ( res @ X7 @ P )
= ( res @ B3 @ P5 ) )
=> ! [P4: pi] :
( ( ( late_BoundR @ A2 @ Y4 @ P3 )
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B3 @ P4 ) )
=> ( ( late_transitions @ P5 @ ( late_FreeR @ ( late_OutputR @ A4 @ B3 ) @ P4 ) )
=> ( A4 = B3 ) ) ) )
=> ~ ! [P5: pi,A4: late_subject,X5: name,P4: pi,Y3: name] :
( ( ( res @ X7 @ P )
= ( res @ Y3 @ P5 ) )
=> ( ( ( late_BoundR @ A2 @ Y4 @ P3 )
= ( late_BoundR @ A4 @ X5 @ ( res @ Y3 @ P4 ) ) )
=> ( ( late_transitions @ P5 @ ( late_BoundR @ A4 @ X5 @ P4 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X5 )
=> ( ( fresh @ name @ pi @ X5 @ P5 )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_111_residual_Odistinct_I1_J,axiom,
! [Subject: late_subject,Name2: name,Pi: pi,FreeRes: late_freeRes,Pi3: pi] :
( ( late_BoundR @ Subject @ Name2 @ Pi )
!= ( late_FreeR @ FreeRes @ Pi3 ) ) ).
% residual.distinct(1)
thf(fact_112_residual_Oinducts,axiom,
! [P: late_residual > $o,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi] : ( P @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) )
=> ( ! [FreeRes2: late_freeRes,Pi4: pi] : ( P @ ( late_FreeR @ FreeRes2 @ Pi4 ) )
=> ( P @ Residual ) ) ) ).
% residual.inducts
thf(fact_113_eqvt,axiom,
! [X1: pi,X22: late_residual,Pi: list @ ( product_prod @ name @ name )] :
( ( late_transitions @ X1 @ X22 )
=> ( late_transitions @ ( perm @ name @ pi @ Pi @ X1 ) @ ( perm @ name @ late_residual @ Pi @ X22 ) ) ) ).
% eqvt
thf(fact_114_freeRes_Ostrong__inducts,axiom,
! [A: $tType,P: A > late_freeRes > $o,Z: A,FreeRes3: late_freeRes] :
( ! [Name1: name,Name22: name,Z2: A] : ( P @ Z2 @ ( late_OutputR @ Name1 @ Name22 ) )
=> ( ! [Z2: A] : ( P @ Z2 @ late_TauR )
=> ( P @ Z @ FreeRes3 ) ) ) ).
% freeRes.strong_inducts
thf(fact_115_freeRes_Ostrong__induct_H,axiom,
! [N: $tType,P: N > late_freeRes > $o,Z: N,FreeRes3: late_freeRes] :
( ! [Name1: name,Name22: name,Z2: N] : ( P @ Z2 @ ( late_OutputR @ Name1 @ Name22 ) )
=> ( ! [Z2: N] : ( P @ Z2 @ late_TauR )
=> ( P @ Z @ FreeRes3 ) ) ) ).
% freeRes.strong_induct'
thf(fact_116_freeRes_Oinducts,axiom,
! [P: late_freeRes > $o,FreeRes3: late_freeRes] :
( ! [Name1: name,Name22: name] : ( P @ ( late_OutputR @ Name1 @ Name22 ) )
=> ( ( P @ late_TauR )
=> ( P @ FreeRes3 ) ) ) ).
% freeRes.inducts
thf(fact_117_Late__Semantics_OfreeRes_Odistinct_I1_J,axiom,
! [Name12: name,Name23: name] :
( ( late_OutputR @ Name12 @ Name23 )
!= late_TauR ) ).
% Late_Semantics.freeRes.distinct(1)
thf(fact_118_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name2: name,Name: name] :
( ( late_InputS @ Name2 )
!= ( late_BoundOutputS @ Name ) ) ).
% Late_Semantics.subject.distinct(1)
thf(fact_119_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_120_subject_Ostrong__induct_H,axiom,
! [N: $tType,P: N > late_subject > $o,Z: N,Subject: late_subject] :
( ! [Name3: name,Z2: N] : ( P @ Z2 @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name,Z2: N] : ( P @ Z2 @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Z @ Subject ) ) ) ).
% subject.strong_induct'
thf(fact_121_subject_Ostrong__inducts,axiom,
! [A: $tType,P: A > late_subject > $o,Z: A,Subject: late_subject] :
( ! [Name3: name,Z2: A] : ( P @ Z2 @ ( late_InputS @ Name3 ) )
=> ( ! [Name3: name,Z2: A] : ( P @ Z2 @ ( late_BoundOutputS @ Name3 ) )
=> ( P @ Z @ Subject ) ) ) ).
% subject.strong_inducts
thf(fact_122_resCasesB,axiom,
! [Y: name,P: pi,A2: late_subject,X2: name,YP: pi,F2: late_subject > pi > $o] :
( ( late_transitions @ ( res @ Y @ P ) @ ( late_BoundR @ A2 @ X2 @ YP ) )
=> ( ( X2 != Y )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ! [B3: name,P4: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ B3 @ Y ) @ P4 ) )
=> ( ( B3 != Y )
=> ( ( A2
= ( late_BoundOutputS @ B3 ) )
=> ( F2 @ ( late_BoundOutputS @ B3 ) @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X2 @ Y ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P4 ) ) ) ) )
=> ( ! [P4: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P4 ) )
=> ( ( fresh @ name @ late_subject @ Y @ A2 )
=> ( F2 @ A2 @ ( res @ Y @ P4 ) ) ) )
=> ( F2 @ A2 @ YP ) ) ) ) ) ) ).
% resCasesB
thf(fact_123_Par1F,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_FreeR @ Alpha @ ( par @ P3 @ Q ) ) ) ) ).
% Par1F
thf(fact_124_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_125_freshBoundDerivative_I2_J,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Y: name] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( ( Y != X2 )
=> ( fresh @ name @ pi @ Y @ P3 ) ) ) ) ).
% freshBoundDerivative(2)
thf(fact_126_residual_Ostrong__inducts,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [P: A > late_residual > $o,Z: A,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: A] :
( ( fresh @ name @ A @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi4: pi,Z2: A] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_inducts
thf(fact_127_residual_Ostrong__induct,axiom,
! [N: $tType] :
( ( fs_name @ N )
=> ! [P: N > late_residual > $o,Z: N,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi4: pi,Z2: N] :
( ( fresh @ name @ N @ Name3 @ Z2 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P @ Z2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi4 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi4: pi,Z2: N] : ( P @ Z2 @ ( late_FreeR @ FreeRes2 @ Pi4 ) )
=> ( P @ Z @ Residual ) ) ) ) ).
% residual.strong_induct
thf(fact_128_freshFreeDerivative_I2_J,axiom,
! [P: pi,Alpha: late_freeRes,P3: pi,Y: name] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P3 ) )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( fresh @ name @ pi @ Y @ P3 ) ) ) ).
% freshFreeDerivative(2)
thf(fact_129_Late__Semantics_OPar1B,axiom,
! [P: pi,A2: late_subject,X2: name,P3: pi,Q: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P3 ) )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P3 @ Q ) ) ) ) ) ).
% Late_Semantics.Par1B
thf(fact_130_Late__Semantics_OPar2B,axiom,
! [Q: pi,A2: late_subject,X2: name,Q2: pi,P: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ ( par @ P @ Q2 ) ) ) ) ) ).
% Late_Semantics.Par2B
thf(fact_131_parCasesB,axiom,
! [P: pi,Q: pi,A2: late_subject,X2: name,PQ: pi,Prop: pi > $o] :
( ( late_transitions @ ( par @ P @ Q ) @ ( late_BoundR @ A2 @ X2 @ PQ ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ( ( fresh @ name @ pi @ X2 @ Q )
=> ( ! [P4: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P4 ) )
=> ( Prop @ ( par @ P4 @ Q ) ) )
=> ( ! [Q4: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q4 ) )
=> ( Prop @ ( par @ P @ Q4 ) ) )
=> ( Prop @ PQ ) ) ) ) ) ) ).
% parCasesB
thf(fact_132_Open,axiom,
! [P: pi,A2: name,B2: name,P3: pi] :
( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P3 ) )
=> ( ( A2 != B2 )
=> ( late_transitions @ ( res @ B2 @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ B2 @ P3 ) ) ) ) ).
% Open
thf(fact_133_derivativeEqvtI2,axiom,
! [P: pi,Q: pi,A2: late_subject,X2: name,Rel: set @ ( product_prod @ pi @ pi ),Perm: list @ ( product_prod @ name @ name )] :
( ( strong2129052853vative @ P @ Q @ A2 @ X2 @ Rel )
=> ( ( eqvt @ pi @ Rel )
=> ( strong2129052853vative @ ( perm @ name @ pi @ Perm @ P ) @ ( perm @ name @ pi @ Perm @ Q ) @ A2 @ ( perm @ name @ name @ Perm @ X2 ) @ Rel ) ) ) ).
% derivativeEqvtI2
thf(fact_134_derivativeEqvtI,axiom,
! [P: pi,Q: pi,A2: late_subject,X2: name,Rel: set @ ( product_prod @ pi @ pi ),Perm: list @ ( product_prod @ name @ name )] :
( ( strong2129052853vative @ P @ Q @ A2 @ X2 @ Rel )
=> ( ( eqvt @ pi @ Rel )
=> ( strong2129052853vative @ ( perm @ name @ pi @ Perm @ P ) @ ( perm @ name @ pi @ Perm @ Q ) @ ( perm @ name @ late_subject @ Perm @ A2 ) @ ( perm @ name @ name @ Perm @ X2 ) @ Rel ) ) ) ).
% derivativeEqvtI
thf(fact_135_fresh__prod,axiom,
! [A: $tType,X: $tType,B: $tType,A2: X,X2: A,Y: B] :
( ( fresh @ X @ ( product_prod @ A @ B ) @ A2 @ ( product_Pair @ A @ B @ X2 @ Y ) )
= ( ( fresh @ X @ A @ A2 @ X2 )
& ( fresh @ X @ B @ A2 @ Y ) ) ) ).
% fresh_prod
thf(fact_136_Res,axiom,
! [Y: name,R: pi,S: pi] :
( ( fresh @ name @ pi @ Y @ R )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ R @ ( res @ Y @ S ) ) @ ( res @ Y @ ( par @ R @ S ) ) ) @ rel ) ) ).
% Res
thf(fact_137_ScopeExt,axiom,
! [Y: name,R: pi,Z: name,S: pi] :
( ( fresh @ name @ pi @ Y @ R )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( res @ Z @ ( par @ R @ ( res @ Y @ S ) ) ) @ ( res @ Y @ ( res @ Z @ ( par @ R @ S ) ) ) ) @ rel ) ) ).
% ScopeExt
thf(fact_138_fresh__list__cons,axiom,
! [X: $tType,A: $tType,A2: X,X2: A,Xs: list @ A] :
( ( fresh @ X @ ( list @ A ) @ A2 @ ( cons @ A @ X2 @ Xs ) )
= ( ( fresh @ X @ A @ A2 @ X2 )
& ( fresh @ X @ ( list @ A ) @ A2 @ Xs ) ) ) ).
% fresh_list_cons
thf(fact_139_Bound_Ohyps_I5_J,axiom,
fresh @ name @ ( product_prod @ name @ ( product_prod @ pi @ pi ) ) @ y @ ( product_Pair @ name @ ( product_prod @ pi @ pi ) @ x @ ( product_Pair @ pi @ pi @ p @ q ) ) ).
% Bound.hyps(5)
thf(fact_140_false__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A )] :
~ ( perm @ A @ $o @ Pi @ $false ) ).
% false_eqvt
thf(fact_141_true__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A )] : ( perm @ A @ $o @ Pi @ $true ) ).
% true_eqvt
thf(fact_142_name__fresh,axiom,
( ( fresh @ name @ name )
= ( ^ [A5: name,B4: name] : A5 != B4 ) ) ).
% name_fresh
thf(fact_143_fresh__unit,axiom,
! [A: $tType,A2: A] : ( fresh @ A @ product_unit @ A2 @ product_Unity ) ).
% fresh_unit
thf(fact_144_freshUnit,axiom,
! [Y: name] : ( fresh @ name @ product_unit @ Y @ product_Unity ) ).
% freshUnit
thf(fact_145_allE__Nil,axiom,
! [A: $tType,P: ( list @ A ) > $o] :
( ! [X_1: list @ A] : ( P @ X_1 )
=> ( P @ ( nil @ A ) ) ) ).
% allE_Nil
thf(fact_146_abs__fun__if,axiom,
! [X: $tType,A: $tType,C: $o,Pi: list @ ( product_prod @ X @ X ),X2: A,Y: A] :
( ( C
=> ( ( perm @ X @ A @ Pi @ ( if @ A @ C @ X2 @ Y ) )
= ( perm @ X @ A @ Pi @ X2 ) ) )
& ( ~ C
=> ( ( perm @ X @ A @ Pi @ ( if @ A @ C @ X2 @ Y ) )
= ( perm @ X @ A @ Pi @ Y ) ) ) ) ).
% abs_fun_if
thf(fact_147_pt__bij3,axiom,
! [X: $tType,A: $tType,X2: A,Y: A,Pi: list @ ( product_prod @ X @ X )] :
( ( X2 = Y )
=> ( ( perm @ X @ A @ Pi @ X2 )
= ( perm @ X @ A @ Pi @ Y ) ) ) ).
% pt_bij3
thf(fact_148_perm__bool__def,axiom,
! [X: $tType] :
( ( perm @ X @ $o )
= ( ^ [Pi5: list @ ( product_prod @ X @ X ),B4: $o] : B4 ) ) ).
% perm_bool_def
thf(fact_149_pt__set__bij3,axiom,
! [X: $tType,A: $tType,Pi: list @ ( product_prod @ X @ X ),X2: A,X6: set @ A] :
( ( perm @ X @ $o @ Pi @ ( member @ A @ X2 @ X6 ) )
= ( member @ A @ X2 @ X6 ) ) ).
% pt_set_bij3
thf(fact_150_perm__boolI,axiom,
! [A: $tType,P: $o,Pi: list @ ( product_prod @ A @ A )] :
( P
=> ( perm @ A @ $o @ Pi @ P ) ) ).
% perm_boolI
thf(fact_151_perm__boolE,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A )] :
~ ( perm @ A @ $o @ Pi @ $false ) ).
% perm_boolE
thf(fact_152_disj__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A3: $o,B5: $o] :
( ( perm @ A @ $o @ Pi
@ ( A3
| B5 ) )
= ( ( perm @ A @ $o @ Pi @ A3 )
| ( perm @ A @ $o @ Pi @ B5 ) ) ) ).
% disj_eqvt
thf(fact_153_conj__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A3: $o,B5: $o] :
( ( perm @ A @ $o @ Pi
@ ( A3
& B5 ) )
= ( ( perm @ A @ $o @ Pi @ A3 )
& ( perm @ A @ $o @ Pi @ B5 ) ) ) ).
% conj_eqvt
thf(fact_154_neg__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A3: $o] :
( ( perm @ A @ $o @ Pi @ ~ A3 )
= ( ~ ( perm @ A @ $o @ Pi @ A3 ) ) ) ).
% neg_eqvt
thf(fact_155_imp__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A3: $o,B5: $o] :
( ( perm @ A @ $o @ Pi
@ ( A3
=> B5 ) )
= ( ( perm @ A @ $o @ Pi @ A3 )
=> ( perm @ A @ $o @ Pi @ B5 ) ) ) ).
% imp_eqvt
thf(fact_156_perm__prod_Osimps,axiom,
! [A: $tType,X: $tType,B: $tType,Pi: list @ ( product_prod @ X @ X ),X2: A,Y: B] :
( ( perm @ X @ ( product_prod @ A @ B ) @ Pi @ ( product_Pair @ A @ B @ X2 @ Y ) )
= ( product_Pair @ A @ B @ ( perm @ X @ A @ Pi @ X2 ) @ ( perm @ X @ B @ Pi @ Y ) ) ) ).
% perm_prod.simps
thf(fact_157_fresh__prodD_I2_J,axiom,
! [B: $tType,A: $tType,C3: $tType,A2: A,X2: B,Y: C3] :
( ( fresh @ A @ ( product_prod @ B @ C3 ) @ A2 @ ( product_Pair @ B @ C3 @ X2 @ Y ) )
=> ( fresh @ A @ C3 @ A2 @ Y ) ) ).
% fresh_prodD(2)
thf(fact_158_fresh__prodD_I1_J,axiom,
! [C3: $tType,A: $tType,B: $tType,A2: A,X2: B,Y: C3] :
( ( fresh @ A @ ( product_prod @ B @ C3 ) @ A2 @ ( product_Pair @ B @ C3 @ X2 @ Y ) )
=> ( fresh @ A @ B @ A2 @ X2 ) ) ).
% fresh_prodD(1)
thf(fact_159_if__eqvt,axiom,
! [A: $tType,B: $tType,Pi: list @ ( product_prod @ A @ A ),B2: $o,C1: B,C22: B] :
( ( ( perm @ A @ $o @ Pi @ B2 )
=> ( ( perm @ A @ B @ Pi @ ( if @ B @ B2 @ C1 @ C22 ) )
= ( perm @ A @ B @ Pi @ C1 ) ) )
& ( ~ ( perm @ A @ $o @ Pi @ B2 )
=> ( ( perm @ A @ B @ Pi @ ( if @ B @ B2 @ C1 @ C22 ) )
= ( perm @ A @ B @ Pi @ C22 ) ) ) ) ).
% if_eqvt
thf(fact_160_nil__eqvt,axiom,
! [X: $tType,A: $tType,Pi: list @ ( product_prod @ X @ X )] :
( ( perm @ X @ ( list @ A ) @ Pi @ ( nil @ A ) )
= ( nil @ A ) ) ).
% nil_eqvt
thf(fact_161_fresh__list__nil,axiom,
! [X: $tType,A: $tType,A2: X] : ( fresh @ X @ ( list @ A ) @ A2 @ ( nil @ A ) ) ).
% fresh_list_nil
thf(fact_162_cons__eqvt,axiom,
! [X: $tType,A: $tType,Pi: list @ ( product_prod @ X @ X ),X2: A,Xs: list @ A] :
( ( perm @ X @ ( list @ A ) @ Pi @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ ( perm @ X @ A @ Pi @ X2 ) @ ( perm @ X @ ( list @ A ) @ Pi @ Xs ) ) ) ).
% cons_eqvt
thf(fact_163_resSimCases,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,X2: name,P: pi,C4: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [Q4: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X2 ) @ Q4 ) )
=> ( ( A4 != X2 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q4 ) @ Rel ) ) ) )
=> ( ! [Q4: pi,A4: late_subject,Y3: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y3 @ Q4 ) )
=> ( ( fresh @ name @ late_subject @ X2 @ A4 )
=> ( ( X2 != Y3 )
=> ( ( fresh @ name @ A @ Y3 @ C4 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y3 @ P6 ) )
& ( strong2129052853vative @ P6 @ ( res @ X2 @ Q4 ) @ A4 @ Y3 @ Rel ) ) ) ) ) )
=> ( ! [Q4: pi,Alpha2: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q4 ) )
=> ( ( fresh @ name @ late_freeRes @ X2 @ Alpha2 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ ( res @ X2 @ Q4 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P @ Rel @ ( res @ X2 @ Q ) ) ) ) ) ) ) ).
% resSimCases
thf(fact_164_simCasesCont,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,P: pi,C4: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [A4: late_subject,X5: name,Q4: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X5 @ Q4 ) )
=> ( ( fresh @ name @ pi @ X5 @ P )
=> ( ( fresh @ name @ pi @ X5 @ Q )
=> ( ( fresh @ name @ late_subject @ X5 @ A4 )
=> ( ( fresh @ name @ A @ X5 @ C4 )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P6 ) )
& ( strong2129052853vative @ P6 @ Q4 @ A4 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q4: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q4 ) )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q4 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ) ) ).
% simCasesCont
thf(fact_165_eqvtRelI,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ! [Rel: set @ ( product_prod @ A @ A ),P: A,Q: A,Perm: list @ ( product_prod @ name @ name )] :
( ( eqvt @ A @ Rel )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ P @ Q ) @ Rel )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( perm @ name @ A @ Perm @ P ) @ ( perm @ name @ A @ Perm @ Q ) ) @ Rel ) ) ) ) ).
% eqvtRelI
thf(fact_166_eqvtRelE,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ! [Rel: set @ ( product_prod @ A @ A ),Perm: list @ ( product_prod @ name @ name ),P: A,Q: A] :
( ( eqvt @ A @ Rel )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( perm @ name @ A @ Perm @ P ) @ ( perm @ name @ A @ Perm @ Q ) ) @ Rel )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ P @ Q ) @ Rel ) ) ) ) ).
% eqvtRelE
thf(fact_167_perm__unit_Osimps,axiom,
! [X: $tType,Pi: list @ ( product_prod @ X @ X )] :
( ( perm @ X @ product_unit @ Pi @ product_Unity )
= product_Unity ) ).
% perm_unit.simps
thf(fact_168_parSym,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi,Q: pi] :
( ! [R2: pi,S2: pi] : ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ R2 @ S2 ) @ ( par @ S2 @ R2 ) ) @ Rel )
=> ( ! [R2: pi,S2: pi,X5: name] :
( ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ R2 @ S2 ) @ Rel )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( res @ X5 @ R2 ) @ ( res @ X5 @ S2 ) ) @ Rel ) )
=> ( strong743114133lation @ ( par @ P @ Q ) @ Rel @ ( par @ Q @ P ) ) ) ) ).
% parSym
thf(fact_169_simE_I2_J,axiom,
! [P: pi,Rel: set @ ( product_prod @ pi @ pi ),Q: pi,Alpha: late_freeRes,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q2 ) )
=> ? [P4: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P4 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P4 @ Q2 ) @ Rel ) ) ) ) ).
% simE(2)
thf(fact_170_eqvt__def,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ( ( eqvt @ A )
= ( ^ [Rel2: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A,Perm2: list @ ( product_prod @ name @ name )] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ Rel2 )
=> ( member @ ( product_prod @ A @ A ) @ ( perm @ name @ ( product_prod @ A @ A ) @ Perm2 @ X4 ) @ Rel2 ) ) ) ) ) ).
% eqvt_def
thf(fact_171_parAssocLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P: pi,Q: pi,R: pi] :
( ! [S2: pi,T2: pi,U: pi] : ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ ( par @ S2 @ T2 ) @ U ) @ ( par @ S2 @ ( par @ T2 @ U ) ) ) @ Rel )
=> ( ! [S2: pi,T2: pi,X5: name] :
( ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ S2 @ T2 ) @ Rel )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( res @ X5 @ S2 ) @ ( res @ X5 @ T2 ) ) @ Rel ) )
=> ( ! [S2: pi,T2: pi,U: pi,X5: name] :
( ( fresh @ name @ pi @ X5 @ S2 )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( res @ X5 @ ( par @ ( par @ S2 @ T2 ) @ U ) ) @ ( par @ S2 @ ( res @ X5 @ ( par @ T2 @ U ) ) ) ) @ Rel ) )
=> ( ! [S2: pi,T2: pi,U: pi,X5: name] :
( ( fresh @ name @ pi @ X5 @ U )
=> ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ ( par @ ( res @ X5 @ ( par @ S2 @ T2 ) ) @ U ) @ ( res @ X5 @ ( par @ S2 @ ( par @ T2 @ U ) ) ) ) @ Rel ) )
=> ( strong743114133lation @ ( par @ ( par @ P @ Q ) @ R ) @ Rel @ ( par @ P @ ( par @ Q @ R ) ) ) ) ) ) ) ).
% parAssocLeft
thf(fact_172_simE_I1_J,axiom,
! [P: pi,Rel: set @ ( product_prod @ pi @ pi ),Q: pi,A2: late_subject,X2: name,Q2: pi] :
( ( strong743114133lation @ P @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X2 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X2 @ P )
=> ? [P4: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A2 @ X2 @ P4 ) )
& ( strong2129052853vative @ P4 @ Q2 @ A2 @ X2 @ Rel ) ) ) ) ) ).
% simE(1)
thf(fact_173_simCases,axiom,
! [Q: pi,P: pi,Rel: set @ ( product_prod @ pi @ pi )] :
( ! [A4: late_subject,Y3: name,Q4: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y3 @ Q4 ) )
=> ( ( fresh @ name @ pi @ Y3 @ P )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_BoundR @ A4 @ Y3 @ P6 ) )
& ( strong2129052853vative @ P6 @ Q4 @ A4 @ Y3 @ Rel ) ) ) )
=> ( ! [Alpha2: late_freeRes,Q4: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q4 ) )
=> ? [P6: pi] :
( ( late_transitions @ P @ ( late_FreeR @ Alpha2 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q4 ) @ Rel ) ) )
=> ( strong743114133lation @ P @ Rel @ Q ) ) ) ).
% simCases
thf(fact_174_simulation__def,axiom,
( strong743114133lation
= ( ^ [P7: pi,Rel2: set @ ( product_prod @ pi @ pi ),Q5: pi] :
( ! [A5: late_subject,X4: name,Q6: pi] :
( ( ( late_transitions @ Q5 @ ( late_BoundR @ A5 @ X4 @ Q6 ) )
& ( fresh @ name @ pi @ X4 @ P7 ) )
=> ? [P8: pi] :
( ( late_transitions @ P7 @ ( late_BoundR @ A5 @ X4 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q6 @ A5 @ X4 @ Rel2 ) ) )
& ! [Alpha3: late_freeRes,Q6: pi] :
( ( late_transitions @ Q5 @ ( late_FreeR @ Alpha3 @ Q6 ) )
=> ? [P8: pi] :
( ( late_transitions @ P7 @ ( late_FreeR @ Alpha3 @ P8 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P8 @ Q6 ) @ Rel2 ) ) ) ) ) ) ).
% simulation_def
thf(fact_175_Late__Semantics1_Osubject_Oinject_I2_J,axiom,
! [X22: name,Y2: name] :
( ( ( late_BoundOutputS @ X22 )
= ( late_BoundOutputS @ Y2 ) )
= ( X22 = Y2 ) ) ).
% Late_Semantics1.subject.inject(2)
thf(fact_176_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_177_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_178_Late__Semantics1_OfreeRes_Odistinct_I1_J,axiom,
! [X11: name,X12: name] :
( ( late_OutputR @ X11 @ X12 )
!= late_TauR ) ).
% Late_Semantics1.freeRes.distinct(1)
thf(fact_179_freeRes_Oexhaust,axiom,
! [Y: late_freeRes] :
( ! [X112: name,X122: name] :
( Y
!= ( late_OutputR @ X112 @ X122 ) )
=> ( Y = late_TauR ) ) ).
% freeRes.exhaust
thf(fact_180_subject_Oexhaust,axiom,
! [Y: late_subject] :
( ! [X13: name] :
( Y
!= ( late_InputS @ X13 ) )
=> ~ ! [X23: name] :
( Y
!= ( late_BoundOutputS @ X23 ) ) ) ).
% subject.exhaust
thf(fact_181_Late__Semantics1_Osubject_Odistinct_I1_J,axiom,
! [X1: name,X22: name] :
( ( late_InputS @ X1 )
!= ( late_BoundOutputS @ X22 ) ) ).
% Late_Semantics1.subject.distinct(1)
thf(fact_182_inputCases,axiom,
! [A2: name,X2: name,P: pi,B2: late_subject,Y: name,YP: pi,Prop: late_subject > name > pi > $o] :
( ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ B2 @ Y @ YP ) )
=> ( ( Y != A2 )
=> ( ( Y != X2 )
=> ( ( fresh @ name @ pi @ Y @ P )
=> ( ( ( B2
= ( late_InputS @ A2 ) )
=> ( ( YP
= ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X2 @ Y ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P ) )
=> ( Prop @ ( late_InputS @ A2 ) @ Y @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X2 @ Y ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P ) ) ) )
=> ( Prop @ B2 @ Y @ YP ) ) ) ) ) ) ).
% inputCases
thf(fact_183_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_184_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B6 ) )
= ( ( A2 = A6 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_185_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_186_pi_Operm_I4_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X32: name,X1: name,X22: pi] :
( ( perm @ name @ pi @ Pi @ ( input @ X32 @ X1 @ X22 ) )
= ( input @ ( perm @ name @ name @ Pi @ X32 ) @ ( perm @ name @ name @ Pi @ X1 ) @ ( perm @ name @ pi @ Pi @ X22 ) ) ) ).
% pi.perm(4)
thf(fact_187_pi_Odistinct_I57_J,axiom,
! [Name12: name,Name23: name,Pi: pi,Name: name,Pi3: pi] :
( ( input @ Name12 @ Name23 @ Pi )
!= ( res @ Name @ Pi3 ) ) ).
% pi.distinct(57)
thf(fact_188_pi_Odistinct_I55_J,axiom,
! [Name12: name,Name23: name,Pi: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name12 @ Name23 @ Pi )
!= ( par @ Pi12 @ Pi22 ) ) ).
% pi.distinct(55)
thf(fact_189_resTrans_I2_J,axiom,
! [X2: name,Y: name,P: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X2 @ ( input @ X2 @ Y @ P ) ) @ Rs ) ).
% resTrans(2)
thf(fact_190_inputFreeTrans,axiom,
! [A2: name,X2: name,P: pi,Alpha: late_freeRes,P3: pi] :
~ ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_FreeR @ Alpha @ P3 ) ) ).
% inputFreeTrans
thf(fact_191_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X5: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% surj_pair
thf(fact_192_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_193_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B6 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_194_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B3: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_195_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_196_not__Cons__self2,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( cons @ A @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_197_resInputFreeTrans,axiom,
! [X2: name,A2: name,Y: name,P: pi,Alpha: late_freeRes,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( input @ A2 @ Y @ P ) ) @ ( late_FreeR @ Alpha @ P3 ) ) ).
% resInputFreeTrans
thf(fact_198_old_Ounit_Oinducts,axiom,
! [P: product_unit > $o,Unit: product_unit] :
( ( P @ product_Unity )
=> ( P @ Unit ) ) ).
% old.unit.inducts
thf(fact_199_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
thf(fact_200_inputBoundOutputTrans,axiom,
! [A2: name,X2: name,P: pi,B2: name,Y: name,P3: pi] :
~ ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Y @ P3 ) ) ).
% inputBoundOutputTrans
thf(fact_201_Late__Semantics_OInput,axiom,
! [A2: name,X2: name,P: pi] : ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P ) ) ).
% Late_Semantics.Input
thf(fact_202_transitions_OInput,axiom,
! [X2: name,A2: name,P: pi] :
( ( X2 != A2 )
=> ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ ( late_InputS @ A2 ) @ X2 @ P ) ) ) ).
% transitions.Input
thf(fact_203_inputIneqTrans,axiom,
! [A2: name,X2: name,P: pi,B2: late_subject,Y: name,P3: pi] :
( ( late_transitions @ ( input @ A2 @ X2 @ P ) @ ( late_BoundR @ B2 @ Y @ P3 ) )
=> ~ ( fresh @ name @ late_subject @ A2 @ B2 ) ) ).
% inputIneqTrans
thf(fact_204_resInputBoundOutputTrans,axiom,
! [X2: name,A2: name,Y: name,P: pi,B2: name,Z: name,P3: pi] :
~ ( late_transitions @ ( res @ X2 @ ( input @ A2 @ Y @ P ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Z @ P3 ) ) ).
% resInputBoundOutputTrans
thf(fact_205_alphaInput,axiom,
! [C: name,P: pi,A2: name,X2: name] :
( ( fresh @ name @ pi @ C @ P )
=> ( ( input @ A2 @ X2 @ P )
= ( input @ A2 @ C @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X2 @ C ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P ) ) ) ) ).
% alphaInput
thf(fact_206_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_207_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_208_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_209_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X13: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X13 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_210_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y5: A,Ys: list @ A] :
( Xs
= ( cons @ A @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_211_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs2: list @ A] : ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys3 ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_212_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X5: A,Xs2: list @ A,Ys3: list @ A] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_213_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Y3: A,Zs: list @ A] :
( ( P @ Zs )
=> ( ( P @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_214_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: list @ A] :
( ! [X5: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X5 @ Xs2 ) )
=> ( X2
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_215_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X5: A,Xs2: list @ A] :
( ! [X213: A,X224: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X224 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_216_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
=> ( ! [X5: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X5 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_217_transpose_Ocases,axiom,
! [A: $tType,X2: list @ ( list @ A )] :
( ( X2
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X5: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_218_remdups__adj_Ocases,axiom,
! [A: $tType,X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ( ! [X5: A] :
( X2
!= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_219_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P5: A > A > $o] : ( P @ P5 @ ( nil @ A ) )
=> ( ! [P5: A > A > $o,X5: A,Ys3: list @ A] :
( ( P @ P5 @ Ys3 )
=> ( P @ P5 @ ( cons @ A @ X5 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_220_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Y3: A,Xs2: list @ A] :
( ( ( X5 = Y3 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) ) )
=> ( ( ( X5 != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_221_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F3: A > B,X5: A] : ( P @ F3 @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [F3: A > B,X5: A,Y3: A,Zs: list @ A] :
( ( P @ F3 @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ F3 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ( ! [A4: A > B] : ( P @ A4 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_222_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P5: A > A > $o] : ( P @ P5 @ ( nil @ A ) )
=> ( ! [P5: A > A > $o,X5: A] : ( P @ P5 @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [P5: A > A > $o,X5: A,Y3: A,Xs2: list @ A] :
( ( P @ P5 @ ( cons @ A @ Y3 @ Xs2 ) )
=> ( P @ P5 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_223_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_224_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F3: A > B,X_1: list @ B] : ( P @ F3 @ ( nil @ A ) @ X_1 )
=> ( ! [F3: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F3 @ As @ ( cons @ B @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_225_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ~ ! [X5: A,Ys3: list @ A] :
( X2
!= ( cons @ A @ X5 @ Ys3 ) ) ) ) ).
% strict_sorted.cases
thf(fact_226_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X5: A,Ys3: list @ A] :
( ( P @ Ys3 )
=> ( P @ ( cons @ A @ X5 @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_227_splice_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys3: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ~ ! [X5: A,Xs2: list @ A,Ys3: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_228_shuffles_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys3: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( ! [Xs2: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_229_sorted__wrt_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P5: A > A > $o] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P5 @ ( nil @ A ) ) )
=> ~ ! [P5: A > A > $o,X5: A,Ys3: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P5 @ ( cons @ A @ X5 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_230_successively_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P5: A > A > $o] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P5 @ ( nil @ A ) ) )
=> ( ! [P5: A > A > $o,X5: A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P5 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [P5: A > A > $o,X5: A,Y3: A,Xs2: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P5 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_231_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X2: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F3: A > B,X5: A] :
( X2
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ( ! [F3: A > B,X5: A,Y3: A,Zs: list @ A] :
( X2
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ~ ! [A4: A > B] :
( X2
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_232_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F3: A > B,Bs: list @ B] :
( X2
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
=> ~ ! [F3: A > B,A4: A,As: list @ A,Bs: list @ B] :
( X2
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_233_prod__cases3,axiom,
! [A: $tType,B: $tType,C3: $tType,Y: product_prod @ A @ ( product_prod @ B @ C3 )] :
~ ! [A4: A,B3: B,C2: C3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C3 ) @ A4 @ ( product_Pair @ B @ C3 @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_234_prod__cases4,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) )] :
~ ! [A4: A,B3: B,C2: C3,D3: D2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ D2 ) @ B3 @ ( product_Pair @ C3 @ D2 @ C2 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_235_prod__cases5,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) )] :
~ ! [A4: A,B3: B,C2: C3,D3: D2,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) @ B3 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_236_prod__cases6,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A4: A,B3: B,C2: C3,D3: D2,E2: E,F3: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B3 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_237_prod__cases7,axiom,
! [A: $tType,B: $tType,C3: $tType,D2: $tType,E: $tType,F4: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
~ ! [A4: A,B3: B,C2: C3,D3: D2,E2: E,F3: F4,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_238_prod__induct3,axiom,
! [C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C3 ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C3 )] :
( ! [A4: A,B3: B,C2: C3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C3 ) @ A4 @ ( product_Pair @ B @ C3 @ B3 @ C2 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_239_prod__induct4,axiom,
! [D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) )] :
( ! [A4: A,B3: B,C2: C3,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ D2 ) @ B3 @ ( product_Pair @ C3 @ D2 @ C2 @ D3 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_240_prod__induct5,axiom,
! [E: $tType,D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) )] :
( ! [A4: A,B3: B,C2: C3,D3: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ E ) ) @ B3 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_241_prod__induct6,axiom,
! [F4: $tType,E: $tType,D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A4: A,B3: B,C2: C3,D3: D2,E2: E,F3: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B3 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D3 @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_242_prod__induct7,axiom,
! [G2: $tType,F4: $tType,E: $tType,D2: $tType,C3: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
( ! [A4: A,B3: B,C2: C3,D3: D2,E2: E,F3: F4,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C3 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_243_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_244_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_245_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_246_supports__def,axiom,
! [A: $tType,X: $tType] :
( ( supports @ X @ A )
= ( ^ [S3: set @ X,X4: A] :
! [A5: X,B4: X] :
( ( ~ ( member @ X @ A5 @ S3 )
& ~ ( member @ X @ B4 @ S3 ) )
=> ( ( perm @ X @ A @ ( cons @ ( product_prod @ X @ X ) @ ( product_Pair @ X @ X @ A5 @ B4 ) @ ( nil @ ( product_prod @ X @ X ) ) ) @ X4 )
= X4 ) ) ) ) ).
% supports_def
thf(fact_247_perm__residual__Rep_Osimps_I2_J,axiom,
! [X: $tType,Pi: list @ ( product_prod @ X @ X ),FreeRes3: late_freeRes,Pia: pi] :
( ( perm @ X @ late_residual_Rep @ Pi @ ( late_r347633188eR_Rep @ FreeRes3 @ Pia ) )
= ( late_r347633188eR_Rep @ ( perm @ X @ late_freeRes @ Pi @ FreeRes3 ) @ ( perm @ X @ pi @ Pi @ Pia ) ) ) ).
% perm_residual_Rep.simps(2)
thf(fact_248_residual__Rep_Oinject_I2_J,axiom,
! [X21: late_freeRes,X222: pi,Y21: late_freeRes,Y22: pi] :
( ( ( late_r347633188eR_Rep @ X21 @ X222 )
= ( late_r347633188eR_Rep @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% residual_Rep.inject(2)
thf(fact_249_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_250_name__calc_I1_J,axiom,
! [A2: name,B2: name,Pi: list @ ( product_prod @ name @ name ),X2: name] :
( ( perm @ name @ name @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ Pi ) @ X2 )
= ( swap @ name @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( perm @ name @ name @ Pi @ X2 ) ) ) ).
% name_calc(1)
thf(fact_251_name__calc_I3_J,axiom,
! [A2: name,C: name,B2: name] :
( ( ( A2 = C )
=> ( ( swap @ name @ ( product_Pair @ name @ name @ A2 @ B2 ) @ C )
= B2 ) )
& ( ( A2 != C )
=> ( ( ( B2 = C )
=> ( ( swap @ name @ ( product_Pair @ name @ name @ A2 @ B2 ) @ C )
= A2 ) )
& ( ( B2 != C )
=> ( ( swap @ name @ ( product_Pair @ name @ name @ A2 @ B2 ) @ C )
= C ) ) ) ) ) ).
% name_calc(3)
thf(fact_252_Agent_Oswap__name__def,axiom,
( ( swap @ name )
= swap_name ) ).
% Agent.swap_name_def
thf(fact_253_alpha__fresh_H,axiom,
! [X3: $tType] :
( ( pt_name @ X3 )
=> ! [A2: name,X2: X3,B2: name,Y: X3,C: name] :
( ( ( abs_fun @ name @ X3 @ A2 @ X2 )
= ( abs_fun @ name @ X3 @ B2 @ Y ) )
=> ( ( C != A2 )
=> ( ( C != B2 )
=> ( ( fresh @ name @ X3 @ C @ X2 )
=> ( ( fresh @ name @ X3 @ C @ Y )
=> ( X2
= ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ C ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ ( perm @ name @ X3 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ B2 @ C ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ Y ) ) ) ) ) ) ) ) ) ).
% alpha_fresh'
thf(fact_254_pi_Ofresh_I9_J,axiom,
! [A2: name,X1: name,X22: pi] :
( ( fresh @ name @ pi @ A2 @ ( res @ X1 @ X22 ) )
= ( fresh @ name @ ( name > ( noption @ pi ) ) @ A2 @ ( abs_fun @ name @ pi @ X1 @ X22 ) ) ) ).
% pi.fresh(9)
% Type constructors (47)
thf(tcon_fun___Agent_Ocp__name__name,axiom,
! [A7: $tType,A8: $tType] :
( ( ( cp_name_name @ A7 )
& ( cp_name_name @ A8 ) )
=> ( cp_name_name @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Agent_Opt__name,axiom,
! [A7: $tType,A8: $tType] :
( ( ( pt_name @ A7 )
& ( pt_name @ A8 ) )
=> ( pt_name @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Agent_Ocp__name__name_1,axiom,
! [A7: $tType] :
( ( cp_name_name @ A7 )
=> ( cp_name_name @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Agent_Opt__name_3,axiom,
! [A7: $tType] :
( ( pt_name @ A7 )
=> ( pt_name @ ( set @ A7 ) ) ) ).
thf(tcon_Agent_Opi___Agent_Ocp__name__name_4,axiom,
cp_name_name @ pi ).
thf(tcon_Agent_Opi___Agent_Opt__name_5,axiom,
pt_name @ pi ).
thf(tcon_Agent_Opi___Agent_Ofs__name,axiom,
fs_name @ pi ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Agent_Ocp__name__name_6,axiom,
cp_name_name @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Agent_Opt__name_8,axiom,
pt_name @ $o ).
thf(tcon_HOL_Obool___Agent_Ofs__name_9,axiom,
fs_name @ $o ).
thf(tcon_List_Olist___Agent_Ocp__name__name_10,axiom,
! [A7: $tType] :
( ( cp_name_name @ A7 )
=> ( cp_name_name @ ( list @ A7 ) ) ) ).
thf(tcon_List_Olist___Agent_Opt__name_11,axiom,
! [A7: $tType] :
( ( pt_name @ A7 )
=> ( pt_name @ ( list @ A7 ) ) ) ).
thf(tcon_List_Olist___Agent_Ofs__name_12,axiom,
! [A7: $tType] :
( ( fs_name @ A7 )
=> ( fs_name @ ( list @ A7 ) ) ) ).
thf(tcon_Agent_Oname___Agent_Ocp__name__name_13,axiom,
cp_name_name @ name ).
thf(tcon_Agent_Oname___Agent_Opt__name_14,axiom,
pt_name @ name ).
thf(tcon_Agent_Oname___Agent_Ofs__name_15,axiom,
fs_name @ name ).
thf(tcon_Agent_Opi__Rep___Agent_Ocp__name__name_16,axiom,
cp_name_name @ pi_Rep ).
thf(tcon_Agent_Opi__Rep___Agent_Opt__name_17,axiom,
pt_name @ pi_Rep ).
thf(tcon_Nominal_Onoption___Agent_Ocp__name__name_18,axiom,
! [A7: $tType] :
( ( cp_name_name @ A7 )
=> ( cp_name_name @ ( noption @ A7 ) ) ) ).
thf(tcon_Nominal_Onoption___Agent_Opt__name_19,axiom,
! [A7: $tType] :
( ( pt_name @ A7 )
=> ( pt_name @ ( noption @ A7 ) ) ) ).
thf(tcon_Product__Type_Oprod___Agent_Ocp__name__name_20,axiom,
! [A7: $tType,A8: $tType] :
( ( ( cp_name_name @ A7 )
& ( cp_name_name @ A8 ) )
=> ( cp_name_name @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Agent_Opt__name_21,axiom,
! [A7: $tType,A8: $tType] :
( ( ( pt_name @ A7 )
& ( pt_name @ A8 ) )
=> ( pt_name @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Agent_Ofs__name_22,axiom,
! [A7: $tType,A8: $tType] :
( ( ( fs_name @ A7 )
& ( fs_name @ A8 ) )
=> ( fs_name @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_23,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Agent_Ocp__name__name_24,axiom,
cp_name_name @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_25,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Agent_Opt__name_26,axiom,
pt_name @ product_unit ).
thf(tcon_Product__Type_Ounit___Agent_Ofs__name_27,axiom,
fs_name @ product_unit ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Ocp__name__name_28,axiom,
cp_name_name @ late_freeRes ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Opt__name_29,axiom,
pt_name @ late_freeRes ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Ofs__name_30,axiom,
fs_name @ late_freeRes ).
thf(tcon_Late__Semantics_Osubject___Agent_Ocp__name__name_31,axiom,
cp_name_name @ late_subject ).
thf(tcon_Late__Semantics_Osubject___Agent_Opt__name_32,axiom,
pt_name @ late_subject ).
thf(tcon_Late__Semantics_Osubject___Agent_Ofs__name_33,axiom,
fs_name @ late_subject ).
thf(tcon_Late__Semantics_Oresidual___Agent_Ocp__name__name_34,axiom,
cp_name_name @ late_residual ).
thf(tcon_Late__Semantics_Oresidual___Agent_Opt__name_35,axiom,
pt_name @ late_residual ).
thf(tcon_Late__Semantics_Oresidual___Agent_Ofs__name_36,axiom,
fs_name @ late_residual ).
thf(tcon_Late__Semantics_OfreeRes__Rep___Agent_Ocp__name__name_37,axiom,
cp_name_name @ late_freeRes_Rep ).
thf(tcon_Late__Semantics_OfreeRes__Rep___Agent_Opt__name_38,axiom,
pt_name @ late_freeRes_Rep ).
thf(tcon_Late__Semantics_Osubject__Rep___Agent_Ocp__name__name_39,axiom,
cp_name_name @ late_subject_Rep ).
thf(tcon_Late__Semantics_Osubject__Rep___Agent_Opt__name_40,axiom,
pt_name @ late_subject_Rep ).
thf(tcon_Late__Semantics_Oresidual__Rep___Agent_Ocp__name__name_41,axiom,
cp_name_name @ late_residual_Rep ).
thf(tcon_Late__Semantics_Oresidual__Rep___Agent_Opt__name_42,axiom,
pt_name @ late_residual_Rep ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [P6: pi] :
( ( late_transitions @ ( par @ p @ ( res @ x @ q ) ) @ ( late_BoundR @ ( late_BoundOutputS @ ab ) @ y @ P6 ) )
& ( strong2129052853vative @ P6 @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ y @ x ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ ( par @ ( subs @ p2 @ ya @ c ) @ q2 ) ) @ ( late_BoundOutputS @ ab ) @ y @ rel ) ) ).
%------------------------------------------------------------------------------