TPTP Problem File: ITP182^2.p
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%------------------------------------------------------------------------------
% File : ITP182^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Strong_Late_Sim_SC problem prob_279__3411076_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_279__3411076_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 362 ( 142 unt; 57 typ; 0 def)
% Number of atoms : 737 ( 247 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 4949 ( 117 ~; 4 |; 44 &;4374 @)
% ( 0 <=>; 410 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 203 ( 203 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 48 usr; 11 con; 0-5 aty)
% Number of variables : 1249 ( 36 ^;1167 !; 15 ?;1249 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:02.313
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
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_Oprod,type,
product_prod: $tType > $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 (45)
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_Orderings_Oorder,type,
order:
!>[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_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_c_Agent_Opi_OInput,type,
input: name > name > pi > pi ).
thf(sy_c_Agent_Opi_ORes,type,
res: name > pi > pi ).
thf(sy_c_Agent_Opi_OSum,type,
sum: pi > pi > pi ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
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_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_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_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
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_Rel_Oeqvt,type,
eqvt:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_OId,type,
id:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
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_Rel,type,
rel: set @ ( product_prod @ pi @ pi ) ).
thf(sy_v_a____,type,
a: late_subject ).
thf(sy_v_aa____,type,
aa: name ).
thf(sy_v_x,type,
x: name ).
thf(sy_v_y____,type,
y: name ).
% Relevant facts (255)
thf(fact_0_Id,axiom,
ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ rel ).
% Id
thf(fact_1_cOpen_Ohyps_I2_J,axiom,
aa != x ).
% cOpen.hyps(2)
thf(fact_2_cOpen_Ohyps_I3_J,axiom,
( a
= ( late_BoundOutputS @ aa ) ) ).
% cOpen.hyps(3)
thf(fact_3__092_060open_062y_A_092_060noteq_062_Ax_092_060close_062,axiom,
y != x ).
% \<open>y \<noteq> x\<close>
thf(fact_4_Eqvt,axiom,
eqvt @ pi @ rel ).
% Eqvt
thf(fact_5_perm__swap_I1_J,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [A2: name,B2: name,X3: X2] :
( ( perm @ name @ X2 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ ( perm @ name @ X2 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 ) )
= X3 ) ) ).
% perm_swap(1)
thf(fact_6_name__id,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,X3: X] :
( ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 )
= X3 ) ) ).
% name_id
thf(fact_7_name__swap,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,B2: name,X3: X] :
( ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 )
= ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ B2 @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 ) ) ) ).
% name_swap
thf(fact_8_name__swap__bij,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,B2: name,X3: 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 ) ) ) @ X3 ) )
= X3 ) ) ).
% name_swap_bij
thf(fact_9_pt__name1,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ! [X3: A] :
( ( perm @ name @ A @ ( nil @ ( product_prod @ name @ name ) ) @ X3 )
= X3 ) ) ).
% pt_name1
thf(fact_10_perm__swap_I2_J,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [A2: name,B2: name,X3: X2] :
( ( perm @ name @ X2 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ ( perm @ name @ X2 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ B2 @ A2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 ) )
= X3 ) ) ).
% perm_swap(2)
thf(fact_11_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_12_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_13__092_060open_062y_A_092_060sharp_062_AP_H_092_060close_062,axiom,
fresh @ name @ pi @ y @ p2 ).
% \<open>y \<sharp> P'\<close>
thf(fact_14_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_15_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_16_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_17_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_18_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_19_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_20_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_21_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_22_perm__fresh__fresh,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [A2: name,X3: X2,B2: name] :
( ( fresh @ name @ X2 @ A2 @ X3 )
=> ( ( fresh @ name @ X2 @ B2 @ X3 )
=> ( ( perm @ name @ X2 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 )
= X3 ) ) ) ) ).
% perm_fresh_fresh
thf(fact_23_name__fresh__fresh,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [A2: name,X3: X,B2: name] :
( ( fresh @ name @ X @ A2 @ X3 )
=> ( ( fresh @ name @ X @ B2 @ X3 )
=> ( ( perm @ name @ X @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 )
= X3 ) ) ) ) ).
% name_fresh_fresh
thf(fact_24_Bound_Ohyps_I4_J,axiom,
fresh @ name @ late_subject @ y @ a ).
% Bound.hyps(4)
thf(fact_25_subseteq__eqvt,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi: list @ ( product_prod @ name @ name ),X4: set @ X2,Y: set @ X2] :
( ( perm @ name @ $o @ Pi @ ( ord_less_eq @ ( set @ X2 ) @ X4 @ Y ) )
= ( ord_less_eq @ ( set @ X2 ) @ ( perm @ name @ ( set @ X2 ) @ Pi @ X4 ) @ ( perm @ name @ ( set @ X2 ) @ Pi @ Y ) ) ) ) ).
% subseteq_eqvt
thf(fact_26_fresh__perm__app,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [A2: name,Pi: list @ ( product_prod @ name @ name ),X3: X2] :
( ( fresh @ name @ ( list @ ( product_prod @ name @ name ) ) @ A2 @ Pi )
=> ( ( fresh @ name @ X2 @ A2 @ X3 )
=> ( fresh @ name @ X2 @ A2 @ ( perm @ name @ X2 @ Pi @ X3 ) ) ) ) ) ).
% fresh_perm_app
thf(fact_27_name__exists__fresh,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [X3: A] :
~ ! [C: name] :
~ ( fresh @ name @ A @ C @ X3 ) ) ).
% name_exists_fresh
thf(fact_28_fresh__eqvt,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi: list @ ( product_prod @ name @ name ),A2: name,X3: X2] :
( ( perm @ name @ $o @ Pi @ ( fresh @ name @ X2 @ A2 @ X3 ) )
= ( fresh @ name @ X2 @ ( perm @ name @ name @ Pi @ A2 ) @ ( perm @ name @ X2 @ Pi @ X3 ) ) ) ) ).
% fresh_eqvt
thf(fact_29_fresh__bij,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi: list @ ( product_prod @ name @ name ),A2: name,X3: X2] :
( ( fresh @ name @ X2 @ ( perm @ name @ name @ Pi @ A2 ) @ ( perm @ name @ X2 @ Pi @ X3 ) )
= ( fresh @ name @ X2 @ A2 @ X3 ) ) ) ).
% fresh_bij
thf(fact_30_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F: A > B,Bs: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
=> ~ ! [F: A > B,A4: A,As: list @ A,Bs: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_31_name__calc_I2_J,axiom,
! [X3: name] :
( ( perm @ name @ name @ ( nil @ ( product_prod @ name @ name ) ) @ X3 )
= X3 ) ).
% name_calc(2)
thf(fact_32_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F: A > B,X5: A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ( ! [F: A > B,X5: A,Y3: A,Zs: list @ A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ~ ! [A4: A > B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_33_successively_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P @ ( nil @ A ) ) )
=> ( ! [P: A > A > $o,X5: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [P: A > A > $o,X5: A,Y3: A,Xs: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_34_sorted__wrt_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P @ ( nil @ A ) ) )
=> ~ ! [P: A > A > $o,X5: A,Ys: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P @ ( cons @ A @ X5 @ Ys ) ) ) ) ).
% sorted_wrt.cases
thf(fact_35_shuffles_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ! [Xs: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ~ ! [X5: A,Xs: list @ A,Y3: A,Ys: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) ) ) ) ).
% shuffles.cases
thf(fact_36_splice_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ~ ! [X5: A,Xs: list @ A,Ys: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs ) @ Ys ) ) ) ).
% splice.cases
thf(fact_37_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_38_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_39_swap__simps_I3_J,axiom,
! [A2: name,C2: name,B2: name] :
( ( A2 != C2 )
=> ( ( B2 != C2 )
=> ( ( perm @ name @ name @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ C2 )
= C2 ) ) ) ).
% swap_simps(3)
thf(fact_40_fresh__aux,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [C2: name,A2: name,X3: X2,B2: name] :
( ( C2 != A2 )
=> ( ( fresh @ name @ X2 @ A2 @ X3 )
=> ( ( fresh @ name @ X2 @ C2 @ X3 )
=> ( fresh @ name @ X2 @ C2 @ ( perm @ name @ X2 @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ X3 ) ) ) ) ) ) ).
% fresh_aux
thf(fact_41_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_42_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y4: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y4
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_43_prod__induct7,axiom,
! [G: $tType,F2: $tType,E: $tType,D: $tType,C3: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C: C3,D2: D,E2: E,F: F2,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B4 @ ( product_Pair @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F @ G2 ) ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct7
thf(fact_44_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C3: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A4: A,B4: B,C: C3,D2: D,E2: E,F: F2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct6
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X6: A] : ( member @ A @ X6 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P2 @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X5: A] :
( ( F3 @ X5 )
= ( G3 @ X5 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_49_prod__induct5,axiom,
! [E: $tType,D: $tType,C3: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C: C3,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C3 @ ( product_prod @ D @ E ) @ C @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct5
thf(fact_50_prod__induct4,axiom,
! [D: $tType,C3: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D ) )] :
( ! [A4: A,B4: B,C: C3,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ D ) @ B4 @ ( product_Pair @ C3 @ D @ C @ D2 ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct4
thf(fact_51_prod__induct3,axiom,
! [C3: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C3 ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C3 )] :
( ! [A4: A,B4: B,C: C3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C3 ) @ A4 @ ( product_Pair @ B @ C3 @ B4 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_52_prod__cases7,axiom,
! [A: $tType,B: $tType,C3: $tType,D: $tType,E: $tType,F2: $tType,G: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C: C3,D2: D,E2: E,F: F2,G2: G] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) ) @ B4 @ ( product_Pair @ C3 @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) ) @ C @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G ) @ E2 @ ( product_Pair @ F2 @ G @ F @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_53_prod__cases6,axiom,
! [A: $tType,B: $tType,C3: $tType,D: $tType,E: $tType,F2: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A4: A,B4: B,C: C3,D2: D,E2: E,F: F2] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C3 @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F ) ) ) ) ) ) ).
% prod_cases6
thf(fact_54_prod__cases5,axiom,
! [A: $tType,B: $tType,C3: $tType,D: $tType,E: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C: C3,D2: D,E2: E] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C3 @ ( product_prod @ D @ E ) @ C @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_55_prod__cases4,axiom,
! [A: $tType,B: $tType,C3: $tType,D: $tType,Y4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D ) )] :
~ ! [A4: A,B4: B,C: C3,D2: D] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C3 @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C3 @ D ) @ B4 @ ( product_Pair @ C3 @ D @ C @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_56_prod__cases3,axiom,
! [A: $tType,B: $tType,C3: $tType,Y4: product_prod @ A @ ( product_prod @ B @ C3 )] :
~ ! [A4: A,B4: B,C: C3] :
( Y4
!= ( product_Pair @ A @ ( product_prod @ B @ C3 ) @ A4 @ ( product_Pair @ B @ C3 @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_57_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_58_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_59_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X5: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% surj_pair
thf(fact_60_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( cons @ A @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_61_cp__name__name1,axiom,
! [A: $tType] :
( ( cp_name_name @ A )
=> ! [Pi1: list @ ( product_prod @ name @ name ),Pi2: list @ ( product_prod @ name @ name ),X3: A] :
( ( perm @ name @ A @ Pi1 @ ( perm @ name @ A @ Pi2 @ X3 ) )
= ( perm @ name @ A @ ( perm @ name @ ( list @ ( product_prod @ name @ name ) ) @ Pi1 @ Pi2 ) @ ( perm @ name @ A @ Pi1 @ X3 ) ) ) ) ).
% cp_name_name1
thf(fact_62_name__perm__compose,axiom,
! [X: $tType] :
( ( pt_name @ X )
=> ! [Pi2: list @ ( product_prod @ name @ name ),Pi1: list @ ( product_prod @ name @ name ),X3: X] :
( ( perm @ name @ X @ Pi2 @ ( perm @ name @ X @ Pi1 @ X3 ) )
= ( perm @ name @ X @ ( perm @ name @ ( list @ ( product_prod @ name @ name ) ) @ Pi2 @ Pi1 ) @ ( perm @ name @ X @ Pi2 @ X3 ) ) ) ) ).
% name_perm_compose
thf(fact_63_perm__compose,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi2: list @ ( product_prod @ name @ name ),Pi1: list @ ( product_prod @ name @ name ),X3: X2] :
( ( perm @ name @ X2 @ Pi2 @ ( perm @ name @ X2 @ Pi1 @ X3 ) )
= ( perm @ name @ X2 @ ( perm @ name @ ( list @ ( product_prod @ name @ name ) ) @ Pi2 @ Pi1 ) @ ( perm @ name @ X2 @ Pi2 @ X3 ) ) ) ) ).
% perm_compose
thf(fact_64_perm__bij,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi: list @ ( product_prod @ name @ name ),X3: X2,Y4: X2] :
( ( ( perm @ name @ X2 @ Pi @ X3 )
= ( perm @ name @ X2 @ Pi @ Y4 ) )
= ( X3 = Y4 ) ) ) ).
% perm_bij
thf(fact_65_perm__app,axiom,
! [B: $tType,X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi: list @ ( product_prod @ name @ name ),F3: X2 > B,X3: X2] :
( ( perm @ name @ B @ Pi @ ( F3 @ X3 ) )
= ( perm @ name @ ( X2 > B ) @ Pi @ F3 @ ( perm @ name @ X2 @ Pi @ X3 ) ) ) ) ).
% perm_app
thf(fact_66_in__eqvt,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi: list @ ( product_prod @ name @ name ),X3: X2,X4: set @ X2] :
( ( perm @ name @ $o @ Pi @ ( member @ X2 @ X3 @ X4 ) )
= ( member @ X2 @ ( perm @ name @ X2 @ Pi @ X3 ) @ ( perm @ name @ ( set @ X2 ) @ Pi @ X4 ) ) ) ) ).
% in_eqvt
thf(fact_67_eq__eqvt,axiom,
! [X2: $tType] :
( ( pt_name @ X2 )
=> ! [Pi: list @ ( product_prod @ name @ name ),X3: X2,Y4: X2] :
( ( perm @ name @ $o @ Pi @ ( X3 = Y4 ) )
= ( ( perm @ name @ X2 @ Pi @ X3 )
= ( perm @ name @ X2 @ Pi @ Y4 ) ) ) ) ).
% eq_eqvt
thf(fact_68_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X5: A,Ys: list @ A] :
( ( P2 @ Ys )
=> ( P2 @ ( cons @ A @ X5 @ Ys ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_69_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ~ ! [X5: A,Ys: list @ A] :
( X3
!= ( cons @ A @ X5 @ Ys ) ) ) ) ).
% strict_sorted.cases
thf(fact_70_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P2: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F: A > B,X_1: list @ B] : ( P2 @ F @ ( nil @ A ) @ X_1 )
=> ( ! [F: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P2 @ F @ As @ ( cons @ B @ ( F @ A4 ) @ Bs ) )
=> ( P2 @ F @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_71_list__nonempty__induct,axiom,
! [A: $tType,Xs2: list @ A,P2: ( list @ A ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P2 @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons @ A @ X5 @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_72_successively_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P: A > A > $o] : ( P2 @ P @ ( nil @ A ) )
=> ( ! [P: A > A > $o,X5: A] : ( P2 @ P @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [P: A > A > $o,X5: A,Y3: A,Xs: list @ A] :
( ( P2 @ P @ ( cons @ A @ Y3 @ Xs ) )
=> ( P2 @ P @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_73_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P2: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F: A > B,X5: A] : ( P2 @ F @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [F: A > B,X5: A,Y3: A,Zs: list @ A] :
( ( P2 @ F @ ( cons @ A @ Y3 @ Zs ) )
=> ( P2 @ F @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ( ! [A4: A > B] : ( P2 @ A4 @ ( nil @ A ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_74_remdups__adj_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X5: A] : ( P2 @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Y3: A,Xs: list @ A] :
( ( ( X5 = Y3 )
=> ( P2 @ ( cons @ A @ X5 @ Xs ) ) )
=> ( ( ( X5 != Y3 )
=> ( P2 @ ( cons @ A @ Y3 @ Xs ) ) )
=> ( P2 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs ) ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_75_sorted__wrt_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P: A > A > $o] : ( P2 @ P @ ( nil @ A ) )
=> ( ! [P: A > A > $o,X5: A,Ys: list @ A] :
( ( P2 @ P @ Ys )
=> ( P2 @ P @ ( cons @ A @ X5 @ Ys ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_76_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X5: A] :
( X3
!= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Y3: A,Xs: list @ A] :
( X3
!= ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_77_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X5: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_78_shuffles_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [Xs: list @ A] : ( P2 @ Xs @ ( nil @ A ) )
=> ( ! [X5: A,Xs: list @ A,Y3: A,Ys: list @ A] :
( ( P2 @ Xs @ ( cons @ A @ Y3 @ Ys ) )
=> ( ( P2 @ ( cons @ A @ X5 @ Xs ) @ Ys )
=> ( P2 @ ( cons @ A @ X5 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_79_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ! [X5: A,Xs: list @ A] :
( ! [X212: A,X223: list @ A] :
( ( Xs
= ( cons @ A @ X212 @ X223 ) )
=> ( P2 @ Xs ) )
=> ( P2 @ ( cons @ A @ X5 @ Xs ) ) )
=> ( ( P2 @ ( nil @ A ) )
=> ( P2 @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_80_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X5: A,Xs: list @ A] :
( X3
!= ( cons @ A @ X5 @ Xs ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_81_induct__list012,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs2: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X5: A] : ( P2 @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Y3: A,Zs: list @ A] :
( ( P2 @ Zs )
=> ( ( P2 @ ( cons @ A @ Y3 @ Zs ) )
=> ( P2 @ ( cons @ A @ X5 @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% induct_list012
thf(fact_82_splice_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [X5: A,Xs: list @ A,Ys: list @ A] :
( ( P2 @ Ys @ Xs )
=> ( P2 @ ( cons @ A @ X5 @ Xs ) @ Ys ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_83_list__induct2_H,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,Xs2: list @ A,Ys2: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs: list @ A] : ( P2 @ ( cons @ A @ X5 @ Xs ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys: list @ B] : ( P2 @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys ) )
=> ( ! [X5: A,Xs: list @ A,Y3: B,Ys: list @ B] :
( ( P2 @ Xs @ Ys )
=> ( P2 @ ( cons @ A @ X5 @ Xs ) @ ( cons @ B @ Y3 @ Ys ) ) )
=> ( P2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_84_neq__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
= ( ? [Y5: A,Ys3: list @ A] :
( Xs2
= ( cons @ A @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_85_list_Oinducts,axiom,
! [A: $tType,P2: ( list @ A ) > $o,List: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X12: A,X23: list @ A] :
( ( P2 @ X23 )
=> ( P2 @ ( cons @ A @ X12 @ X23 ) ) )
=> ( P2 @ List ) ) ) ).
% list.inducts
thf(fact_86_list_Oexhaust,axiom,
! [A: $tType,Y4: list @ A] :
( ( Y4
!= ( nil @ A ) )
=> ~ ! [X213: A,X224: list @ A] :
( Y4
!= ( cons @ A @ X213 @ X224 ) ) ) ).
% list.exhaust
thf(fact_87_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_88_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_89__092_060open_062y_A_092_060sharp_062_AQ_092_060close_062,axiom,
fresh @ name @ pi @ y @ q ).
% \<open>y \<sharp> Q\<close>
thf(fact_90_pair__in__Id__conv,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
= ( A2 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_91_IdI,axiom,
! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id @ A ) ) ).
% IdI
thf(fact_92_fresh__prod,axiom,
! [A: $tType,X: $tType,B: $tType,A2: X,X3: A,Y4: B] :
( ( fresh @ X @ ( product_prod @ A @ B ) @ A2 @ ( product_Pair @ A @ B @ X3 @ Y4 ) )
= ( ( fresh @ X @ A @ A2 @ X3 )
& ( fresh @ X @ B @ A2 @ Y4 ) ) ) ).
% fresh_prod
thf(fact_93__092_060open_062y_A_092_060sharp_062_AP_092_060close_062,axiom,
fresh @ name @ pi @ y @ p ).
% \<open>y \<sharp> P\<close>
thf(fact_94_derivativeReflexive,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P2: pi,A2: late_subject,X3: name] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong2129052853vative @ P2 @ P2 @ A2 @ X3 @ Rel ) ) ).
% derivativeReflexive
thf(fact_95_derivativeEqvtI,axiom,
! [P2: pi,Q: pi,A2: late_subject,X3: name,Rel: set @ ( product_prod @ pi @ pi ),Perm: list @ ( product_prod @ name @ name )] :
( ( strong2129052853vative @ P2 @ Q @ A2 @ X3 @ Rel )
=> ( ( eqvt @ pi @ Rel )
=> ( strong2129052853vative @ ( perm @ name @ pi @ Perm @ P2 ) @ ( perm @ name @ pi @ Perm @ Q ) @ ( perm @ name @ late_subject @ Perm @ A2 ) @ ( perm @ name @ name @ Perm @ X3 ) @ Rel ) ) ) ).
% derivativeEqvtI
thf(fact_96_derivativeEqvtI2,axiom,
! [P2: pi,Q: pi,A2: late_subject,X3: name,Rel: set @ ( product_prod @ pi @ pi ),Perm: list @ ( product_prod @ name @ name )] :
( ( strong2129052853vative @ P2 @ Q @ A2 @ X3 @ Rel )
=> ( ( eqvt @ pi @ Rel )
=> ( strong2129052853vative @ ( perm @ name @ pi @ Perm @ P2 ) @ ( perm @ name @ pi @ Perm @ Q ) @ A2 @ ( perm @ name @ name @ Perm @ X3 ) @ Rel ) ) ) ).
% derivativeEqvtI2
thf(fact_97_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_98_eqvtRelI,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ! [Rel: set @ ( product_prod @ A @ A ),P2: A,Q: A,Perm: list @ ( product_prod @ name @ name )] :
( ( eqvt @ A @ Rel )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ P2 @ Q ) @ Rel )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( perm @ name @ A @ Perm @ P2 ) @ ( perm @ name @ A @ Perm @ Q ) ) @ Rel ) ) ) ) ).
% eqvtRelI
thf(fact_99_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_100_true__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A )] : ( perm @ A @ $o @ Pi @ $true ) ).
% true_eqvt
thf(fact_101_false__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A )] :
~ ( perm @ A @ $o @ Pi @ $false ) ).
% false_eqvt
thf(fact_102_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_103_name__bij,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X3: name,Y4: name] :
( ( ( perm @ name @ name @ Pi @ X3 )
= ( perm @ name @ name @ Pi @ Y4 ) )
= ( X3 = Y4 ) ) ).
% name_bij
thf(fact_104_imp__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A5: $o,B5: $o] :
( ( perm @ A @ $o @ Pi
@ ( A5
=> B5 ) )
= ( ( perm @ A @ $o @ Pi @ A5 )
=> ( perm @ A @ $o @ Pi @ B5 ) ) ) ).
% imp_eqvt
thf(fact_105_neg__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A5: $o] :
( ( perm @ A @ $o @ Pi @ ~ A5 )
= ( ~ ( perm @ A @ $o @ Pi @ A5 ) ) ) ).
% neg_eqvt
thf(fact_106_conj__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A5: $o,B5: $o] :
( ( perm @ A @ $o @ Pi
@ ( A5
& B5 ) )
= ( ( perm @ A @ $o @ Pi @ A5 )
& ( perm @ A @ $o @ Pi @ B5 ) ) ) ).
% conj_eqvt
thf(fact_107_disj__eqvt,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A ),A5: $o,B5: $o] :
( ( perm @ A @ $o @ Pi
@ ( A5
| B5 ) )
= ( ( perm @ A @ $o @ Pi @ A5 )
| ( perm @ A @ $o @ Pi @ B5 ) ) ) ).
% disj_eqvt
thf(fact_108_perm__boolE,axiom,
! [A: $tType,Pi: list @ ( product_prod @ A @ A )] :
~ ( perm @ A @ $o @ Pi @ $false ) ).
% perm_boolE
thf(fact_109_perm__boolI,axiom,
! [A: $tType,P2: $o,Pi: list @ ( product_prod @ A @ A )] :
( P2
=> ( perm @ A @ $o @ Pi @ P2 ) ) ).
% perm_boolI
thf(fact_110_pt__set__bij3,axiom,
! [X: $tType,A: $tType,Pi: list @ ( product_prod @ X @ X ),X3: A,X4: set @ A] :
( ( perm @ X @ $o @ Pi @ ( member @ A @ X3 @ X4 ) )
= ( member @ A @ X3 @ X4 ) ) ).
% pt_set_bij3
thf(fact_111_perm__bool__def,axiom,
! [X: $tType] :
( ( perm @ X @ $o )
= ( ^ [Pi3: list @ ( product_prod @ X @ X ),B6: $o] : B6 ) ) ).
% perm_bool_def
thf(fact_112_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_113_fresh__list__nil,axiom,
! [X: $tType,A: $tType,A2: X] : ( fresh @ X @ ( list @ A ) @ A2 @ ( nil @ A ) ) ).
% fresh_list_nil
thf(fact_114_allE__Nil,axiom,
! [A: $tType,P2: ( list @ A ) > $o] :
( ! [X_1: list @ A] : ( P2 @ X_1 )
=> ( P2 @ ( nil @ A ) ) ) ).
% allE_Nil
thf(fact_115_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_116_pt__bij3,axiom,
! [X: $tType,A: $tType,X3: A,Y4: A,Pi: list @ ( product_prod @ X @ X )] :
( ( X3 = Y4 )
=> ( ( perm @ X @ A @ Pi @ X3 )
= ( perm @ X @ A @ Pi @ Y4 ) ) ) ).
% pt_bij3
thf(fact_117_abs__fun__if,axiom,
! [X: $tType,A: $tType,C2: $o,Pi: list @ ( product_prod @ X @ X ),X3: A,Y4: A] :
( ( C2
=> ( ( perm @ X @ A @ Pi @ ( if @ A @ C2 @ X3 @ Y4 ) )
= ( perm @ X @ A @ Pi @ X3 ) ) )
& ( ~ C2
=> ( ( perm @ X @ A @ Pi @ ( if @ A @ C2 @ X3 @ Y4 ) )
= ( perm @ X @ A @ Pi @ Y4 ) ) ) ) ).
% abs_fun_if
thf(fact_118_eqvt__def,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ( ( eqvt @ A )
= ( ^ [Rel2: set @ ( product_prod @ A @ A )] :
! [X6: product_prod @ A @ A,Perm2: list @ ( product_prod @ name @ name )] :
( ( member @ ( product_prod @ A @ A ) @ X6 @ Rel2 )
=> ( member @ ( product_prod @ A @ A ) @ ( perm @ name @ ( product_prod @ A @ A ) @ Perm2 @ X6 ) @ Rel2 ) ) ) ) ) ).
% eqvt_def
thf(fact_119_perm__prod_Osimps,axiom,
! [A: $tType,X: $tType,B: $tType,Pi: list @ ( product_prod @ X @ X ),X3: A,Y4: B] :
( ( perm @ X @ ( product_prod @ A @ B ) @ Pi @ ( product_Pair @ A @ B @ X3 @ Y4 ) )
= ( product_Pair @ A @ B @ ( perm @ X @ A @ Pi @ X3 ) @ ( perm @ X @ B @ Pi @ Y4 ) ) ) ).
% perm_prod.simps
thf(fact_120_cons__eqvt,axiom,
! [X: $tType,A: $tType,Pi: list @ ( product_prod @ X @ X ),X3: A,Xs2: list @ A] :
( ( perm @ X @ ( list @ A ) @ Pi @ ( cons @ A @ X3 @ Xs2 ) )
= ( cons @ A @ ( perm @ X @ A @ Pi @ X3 ) @ ( perm @ X @ ( list @ A ) @ Pi @ Xs2 ) ) ) ).
% cons_eqvt
thf(fact_121_fresh__prodD_I1_J,axiom,
! [C3: $tType,A: $tType,B: $tType,A2: A,X3: B,Y4: C3] :
( ( fresh @ A @ ( product_prod @ B @ C3 ) @ A2 @ ( product_Pair @ B @ C3 @ X3 @ Y4 ) )
=> ( fresh @ A @ B @ A2 @ X3 ) ) ).
% fresh_prodD(1)
thf(fact_122_fresh__prodD_I2_J,axiom,
! [B: $tType,A: $tType,C3: $tType,A2: A,X3: B,Y4: C3] :
( ( fresh @ A @ ( product_prod @ B @ C3 ) @ A2 @ ( product_Pair @ B @ C3 @ X3 @ Y4 ) )
=> ( fresh @ A @ C3 @ A2 @ Y4 ) ) ).
% fresh_prodD(2)
thf(fact_123_fresh__list__cons,axiom,
! [X: $tType,A: $tType,A2: X,X3: A,Xs2: list @ A] :
( ( fresh @ X @ ( list @ A ) @ A2 @ ( cons @ A @ X3 @ Xs2 ) )
= ( ( fresh @ X @ A @ A2 @ X3 )
& ( fresh @ X @ ( list @ A ) @ A2 @ Xs2 ) ) ) ).
% fresh_list_cons
thf(fact_124_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_125_IdE,axiom,
! [A: $tType,P3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( id @ A ) )
=> ~ ! [X5: A] :
( P3
!= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ).
% IdE
thf(fact_126_derivativeMonotonic,axiom,
! [P2: pi,Q: pi,A2: late_subject,X3: name,A5: set @ ( product_prod @ pi @ pi ),B5: set @ ( product_prod @ pi @ pi )] :
( ( strong2129052853vative @ P2 @ Q @ A2 @ X3 @ A5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ A5 @ B5 )
=> ( strong2129052853vative @ P2 @ Q @ A2 @ X3 @ B5 ) ) ) ).
% derivativeMonotonic
thf(fact_127_eqvtRelE,axiom,
! [A: $tType] :
( ( pt_name @ A )
=> ! [Rel: set @ ( product_prod @ A @ A ),Perm: list @ ( product_prod @ name @ name ),P2: A,Q: A] :
( ( eqvt @ A @ Rel )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( perm @ name @ A @ Perm @ P2 ) @ ( perm @ name @ A @ Perm @ Q ) ) @ Rel )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ P2 @ Q ) @ Rel ) ) ) ) ).
% eqvtRelE
thf(fact_128_Bound_Ohyps_I3_J,axiom,
fresh @ name @ pi @ y @ ( sum @ ( res @ x @ p ) @ ( res @ x @ q ) ) ).
% Bound.hyps(3)
thf(fact_129_Bound_Ohyps_I2_J,axiom,
fresh @ name @ pi @ y @ ( res @ x @ ( sum @ p @ q ) ) ).
% Bound.hyps(2)
thf(fact_130_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_131_subsetI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( member @ A @ X5 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% subsetI
thf(fact_132_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_133_name__fresh,axiom,
( ( fresh @ name @ name )
= ( ^ [A6: name,B6: name] : A6 != B6 ) ) ).
% name_fresh
thf(fact_134_pi_Ofresh_I7_J,axiom,
! [A2: name,X22: pi,X1: pi] :
( ( fresh @ name @ pi @ A2 @ ( sum @ X22 @ X1 ) )
= ( ( fresh @ name @ pi @ A2 @ X22 )
& ( fresh @ name @ pi @ A2 @ X1 ) ) ) ).
% pi.fresh(7)
thf(fact_135_pi_Operm_I7_J,axiom,
! [Pi: list @ ( product_prod @ name @ name ),X22: pi,X1: pi] :
( ( perm @ name @ pi @ Pi @ ( sum @ X22 @ X1 ) )
= ( sum @ ( perm @ name @ pi @ Pi @ X22 ) @ ( perm @ name @ pi @ Pi @ X1 ) ) ) ).
% pi.perm(7)
thf(fact_136_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_137_pi_Odistinct_I81_J,axiom,
! [Pi1: pi,Pi2: pi,Name: name,Pi4: pi] :
( ( sum @ Pi1 @ Pi2 )
!= ( res @ Name @ Pi4 ) ) ).
% pi.distinct(81)
thf(fact_138_pi_Oinject_I6_J,axiom,
! [X22: pi,X1: pi,Y2: pi,Y1: pi] :
( ( ( sum @ X22 @ X1 )
= ( sum @ Y2 @ Y1 ) )
= ( ( X22 = Y2 )
& ( X1 = Y1 ) ) ) ).
% pi.inject(6)
thf(fact_139_freshRes,axiom,
! [A2: name,P2: pi] : ( fresh @ name @ pi @ A2 @ ( res @ A2 @ P2 ) ) ).
% freshRes
thf(fact_140_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_funD
thf(fact_141_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) ) ) ).
% le_funE
thf(fact_142_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_143_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X6: A] : ( ord_less_eq @ B @ ( F4 @ X6 ) @ ( G4 @ X6 ) ) ) ) ) ).
% le_fun_def
thf(fact_144_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C3,C2: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ C3 @ ( F3 @ X5 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ C3 @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_146_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_147_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C2 )
=> ( ! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_148_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z: A] : Y6 = Z )
= ( ^ [X6: A,Y5: A] :
( ( ord_less_eq @ A @ X6 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X6 ) ) ) ) ) ).
% eq_iff
thf(fact_149_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) ).
% antisym
thf(fact_150_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% linear
thf(fact_151_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y4: A] :
( ( X3 = Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ).
% eq_refl
thf(fact_152_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% le_cases
thf(fact_153_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_154_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y4: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_155_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y4: A,X3: A] :
( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_156_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z: A] : Y6 = Z )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
& ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_157_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_158_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_159_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_160_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y4: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_161_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_162_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_163_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_164_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y6: A,Z: A] : Y6 = Z )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ B6 @ A6 )
& ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_165_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_166_in__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_167_subsetD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_168_equalityE,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_169_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X6: A] :
( ( member @ A @ X6 @ A7 )
=> ( member @ A @ X6 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_170_equalityD1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_171_equalityD2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_172_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A7 )
=> ( member @ A @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_173_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_174_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P2 @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_175_subset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_176_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y6: set @ A,Z: set @ A] : Y6 = Z )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_177_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X6: A] :
( ( P2 @ X6 )
=> ( Q @ X6 ) ) ) ) ).
% Collect_mono_iff
thf(fact_178_alphaRes,axiom,
! [C2: name,P2: pi,A2: name] :
( ( fresh @ name @ pi @ C2 @ P2 )
=> ( ( res @ A2 @ P2 )
= ( res @ C2 @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ C2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P2 ) ) ) ) ).
% alphaRes
thf(fact_179_calculation,axiom,
late_transitions @ ( res @ x @ ( sum @ p @ q ) ) @ ( late_BoundR @ ( late_BoundOutputS @ aa ) @ y @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ y @ x ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ p2 ) ) ).
% calculation
thf(fact_180_Bound_Ohyps_I1_J,axiom,
late_transitions @ ( sum @ ( res @ x @ p ) @ ( res @ x @ q ) ) @ ( late_BoundR @ a @ y @ pq ) ).
% Bound.hyps(1)
thf(fact_181__092_060open_062_060_092_060nu_062x_062_IP_A_092_060oplus_062_AQ_J_A_092_060longmapsto_062_Aa_060_092_060nu_062x_062_A_092_060prec_062_AP_H_092_060close_062,axiom,
late_transitions @ ( res @ x @ ( sum @ p @ q ) ) @ ( late_BoundR @ ( late_BoundOutputS @ aa ) @ x @ p2 ) ).
% \<open><\<nu>x>(P \<oplus> Q) \<longmapsto> a<\<nu>x> \<prec> P'\<close>
thf(fact_182_sumResLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),X3: name,P2: pi,Q: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( ( eqvt @ pi @ Rel )
=> ( strong743114133lation @ ( sum @ ( res @ X3 @ P2 ) @ ( res @ X3 @ Q ) ) @ Rel @ ( res @ X3 @ ( sum @ P2 @ Q ) ) ) ) ) ).
% sumResLeft
thf(fact_183_cSum1_Ohyps,axiom,
late_transitions @ ( res @ x @ p ) @ ( late_BoundR @ a @ y @ pq ) ).
% cSum1.hyps
thf(fact_184_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_185_freshBoundDerivative_I1_J,axiom,
! [P2: pi,A2: late_subject,X3: name,P4: pi,Y4: name] :
( ( late_transitions @ P2 @ ( late_BoundR @ A2 @ X3 @ P4 ) )
=> ( ( fresh @ name @ pi @ Y4 @ P2 )
=> ( fresh @ name @ late_subject @ Y4 @ A2 ) ) ) ).
% freshBoundDerivative(1)
thf(fact_186_Late__Semantics_OResB,axiom,
! [P2: pi,A2: late_subject,X3: name,P4: pi,Y4: name] :
( ( late_transitions @ P2 @ ( late_BoundR @ A2 @ X3 @ P4 ) )
=> ( ( fresh @ name @ late_subject @ Y4 @ A2 )
=> ( ( Y4 != X3 )
=> ( late_transitions @ ( res @ Y4 @ P2 ) @ ( late_BoundR @ A2 @ X3 @ ( res @ Y4 @ P4 ) ) ) ) ) ) ).
% Late_Semantics.ResB
thf(fact_187_freshBoundDerivative_I2_J,axiom,
! [P2: pi,A2: late_subject,X3: name,P4: pi,Y4: name] :
( ( late_transitions @ P2 @ ( late_BoundR @ A2 @ X3 @ P4 ) )
=> ( ( fresh @ name @ pi @ Y4 @ P2 )
=> ( ( Y4 != X3 )
=> ( fresh @ name @ pi @ Y4 @ P4 ) ) ) ) ).
% freshBoundDerivative(2)
thf(fact_188_simE_I1_J,axiom,
! [P2: pi,Rel: set @ ( product_prod @ pi @ pi ),Q: pi,A2: late_subject,X3: name,Q2: pi] :
( ( strong743114133lation @ P2 @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_BoundR @ A2 @ X3 @ Q2 ) )
=> ( ( fresh @ name @ pi @ X3 @ P2 )
=> ? [P5: pi] :
( ( late_transitions @ P2 @ ( late_BoundR @ A2 @ X3 @ P5 ) )
& ( strong2129052853vative @ P5 @ Q2 @ A2 @ X3 @ Rel ) ) ) ) ) ).
% simE(1)
thf(fact_189_Sum2,axiom,
! [Q: pi,Rs: late_residual,P2: pi] :
( ( late_transitions @ Q @ Rs )
=> ( late_transitions @ ( sum @ P2 @ Q ) @ Rs ) ) ).
% Sum2
thf(fact_190_Sum1,axiom,
! [P2: pi,Rs: late_residual,Q: pi] :
( ( late_transitions @ P2 @ Rs )
=> ( late_transitions @ ( sum @ P2 @ Q ) @ Rs ) ) ).
% Sum1
thf(fact_191_sumCases_H,axiom,
! [P2: pi,Q: pi,Rs: late_residual] :
( ( late_transitions @ ( sum @ P2 @ Q ) @ Rs )
=> ( ~ ( late_transitions @ P2 @ Rs )
=> ( late_transitions @ Q @ Rs ) ) ) ).
% sumCases'
thf(fact_192_sumCases,axiom,
! [P2: pi,Q: pi,Rs: late_residual] :
( ( late_transitions @ ( sum @ P2 @ Q ) @ Rs )
=> ( ~ ( late_transitions @ P2 @ Rs )
=> ( late_transitions @ Q @ Rs ) ) ) ).
% sumCases
thf(fact_193_freshResidual,axiom,
! [P2: pi,Rs: late_residual,X3: name] :
( ( late_transitions @ P2 @ Rs )
=> ( ( fresh @ name @ pi @ X3 @ P2 )
=> ( fresh @ name @ late_residual @ X3 @ Rs ) ) ) ).
% freshResidual
thf(fact_194_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_195_transitions_OResB,axiom,
! [P2: pi,A2: late_subject,X3: name,P4: pi,Y4: name] :
( ( late_transitions @ P2 @ ( late_BoundR @ A2 @ X3 @ P4 ) )
=> ( ( fresh @ name @ late_subject @ Y4 @ A2 )
=> ( ( Y4 != X3 )
=> ( ( fresh @ name @ pi @ X3 @ P2 )
=> ( ( fresh @ name @ late_subject @ X3 @ A2 )
=> ( late_transitions @ ( res @ Y4 @ P2 ) @ ( late_BoundR @ A2 @ X3 @ ( res @ Y4 @ P4 ) ) ) ) ) ) ) ) ).
% transitions.ResB
thf(fact_196_monotonic,axiom,
! [P2: pi,A5: set @ ( product_prod @ pi @ pi ),P4: pi,B5: set @ ( product_prod @ pi @ pi )] :
( ( strong743114133lation @ P2 @ A5 @ P4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ A5 @ B5 )
=> ( strong743114133lation @ P2 @ B5 @ P4 ) ) ) ).
% monotonic
thf(fact_197_Strong__Late__Sim_Oreflexive,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P2: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ P2 @ Rel @ P2 ) ) ).
% Strong_Late_Sim.reflexive
thf(fact_198_sumIdempRight,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P2: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ P2 @ P2 ) @ Rel @ P2 ) ) ).
% sumIdempRight
thf(fact_199_sumAssocRight,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P2: pi,Q: pi,R2: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ P2 @ ( sum @ Q @ R2 ) ) @ Rel @ ( sum @ ( sum @ P2 @ Q ) @ R2 ) ) ) ).
% sumAssocRight
thf(fact_200_sumIdempLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P2: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ P2 @ Rel @ ( sum @ P2 @ P2 ) ) ) ).
% sumIdempLeft
thf(fact_201_sumAssocLeft,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P2: pi,Q: pi,R2: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ ( sum @ P2 @ Q ) @ R2 ) @ Rel @ ( sum @ P2 @ ( sum @ Q @ R2 ) ) ) ) ).
% sumAssocLeft
thf(fact_202_sumSym,axiom,
! [Rel: set @ ( product_prod @ pi @ pi ),P2: pi,Q: pi] :
( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ ( id @ pi ) @ Rel )
=> ( strong743114133lation @ ( sum @ P2 @ Q ) @ Rel @ ( sum @ Q @ P2 ) ) ) ).
% sumSym
thf(fact_203_eqvtI,axiom,
! [P2: pi,Rel: set @ ( product_prod @ pi @ pi ),Q: pi,Rel3: set @ ( product_prod @ pi @ pi ),Perm: list @ ( product_prod @ name @ name )] :
( ( strong743114133lation @ P2 @ Rel @ Q )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ pi @ pi ) ) @ Rel @ Rel3 )
=> ( ( eqvt @ pi @ Rel3 )
=> ( strong743114133lation @ ( perm @ name @ pi @ Perm @ P2 ) @ Rel3 @ ( perm @ name @ pi @ Perm @ Q ) ) ) ) ) ).
% eqvtI
thf(fact_204_alphaBoundResidual,axiom,
! [X7: name,P2: pi,A2: late_subject,X3: name] :
( ( fresh @ name @ pi @ X7 @ P2 )
=> ( ( late_BoundR @ A2 @ X3 @ P2 )
= ( late_BoundR @ A2 @ X7 @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X3 @ X7 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P2 ) ) ) ) ).
% alphaBoundResidual
thf(fact_205_cOpen_Ohyps_I1_J,axiom,
late_transitions @ p @ ( late_FreeR @ ( late_OutputR @ aa @ x ) @ p2 ) ).
% cOpen.hyps(1)
thf(fact_206_simCasesCont,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,P2: pi,C4: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [A4: late_subject,X5: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ X5 @ Q3 ) )
=> ( ( fresh @ name @ pi @ X5 @ P2 )
=> ( ( fresh @ name @ pi @ X5 @ Q )
=> ( ( fresh @ name @ late_subject @ X5 @ A4 )
=> ( ( fresh @ name @ A @ X5 @ C4 )
=> ? [P6: pi] :
( ( late_transitions @ P2 @ ( late_BoundR @ A4 @ X5 @ P6 ) )
& ( strong2129052853vative @ P6 @ Q3 @ A4 @ X5 @ Rel ) ) ) ) ) ) )
=> ( ! [Alpha: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q3 ) )
=> ? [P6: pi] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P2 @ Rel @ Q ) ) ) ) ) ).
% simCasesCont
thf(fact_207__092_060open_062P_A_092_060oplus_062_AQ_A_092_060longmapsto_062_Aa_091x_093_A_092_060prec_062_AP_H_092_060close_062,axiom,
late_transitions @ ( sum @ p @ q ) @ ( late_FreeR @ ( late_OutputR @ aa @ x ) @ p2 ) ).
% \<open>P \<oplus> Q \<longmapsto> a[x] \<prec> P'\<close>
thf(fact_208_Late__Semantics1_OfreeRes_Oinject,axiom,
! [X11: name,X122: name,Y11: name,Y12: name] :
( ( ( late_OutputR @ X11 @ X122 )
= ( late_OutputR @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 ) ) ) ).
% Late_Semantics1.freeRes.inject
thf(fact_209_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_210_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_211_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_212_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_213_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_214_residual_Odistinct_I1_J,axiom,
! [Subject: late_subject,Name2: name,Pi: pi,FreeRes: late_freeRes,Pi4: pi] :
( ( late_BoundR @ Subject @ Name2 @ Pi )
!= ( late_FreeR @ FreeRes @ Pi4 ) ) ).
% residual.distinct(1)
thf(fact_215_residual_Oinducts,axiom,
! [P2: late_residual > $o,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi5: pi] : ( P2 @ ( late_BoundR @ Subject2 @ Name3 @ Pi5 ) )
=> ( ! [FreeRes2: late_freeRes,Pi5: pi] : ( P2 @ ( late_FreeR @ FreeRes2 @ Pi5 ) )
=> ( P2 @ Residual ) ) ) ).
% residual.inducts
thf(fact_216_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_217_ResF,axiom,
! [P2: pi,Alpha2: late_freeRes,P4: pi,Y4: name] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha2 @ P4 ) )
=> ( ( fresh @ name @ late_freeRes @ Y4 @ Alpha2 )
=> ( late_transitions @ ( res @ Y4 @ P2 ) @ ( late_FreeR @ Alpha2 @ ( res @ Y4 @ P4 ) ) ) ) ) ).
% ResF
thf(fact_218_resCasesF_H,axiom,
! [X3: name,P2: pi,Alpha2: late_freeRes,P4: pi] :
( ( late_transitions @ ( res @ X3 @ P2 ) @ ( late_FreeR @ Alpha2 @ P4 ) )
=> ~ ! [P: pi,Alpha: late_freeRes,P5: pi,Y3: name] :
( ( ( res @ X3 @ P2 )
= ( res @ Y3 @ P ) )
=> ( ( ( late_FreeR @ Alpha2 @ P4 )
= ( late_FreeR @ Alpha @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P5 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha ) ) ) ) ) ).
% resCasesF'
thf(fact_219_resCasesF,axiom,
! [X3: name,P2: pi,Alpha2: late_freeRes,XP: pi,F5: pi > $o] :
( ( late_transitions @ ( res @ X3 @ P2 ) @ ( late_FreeR @ Alpha2 @ XP ) )
=> ( ! [P5: pi] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha2 @ P5 ) )
=> ( ( fresh @ name @ late_freeRes @ X3 @ Alpha2 )
=> ( F5 @ ( res @ X3 @ P5 ) ) ) )
=> ( F5 @ XP ) ) ) ).
% resCasesF
thf(fact_220_freshFreeDerivative_I2_J,axiom,
! [P2: pi,Alpha2: late_freeRes,P4: pi,Y4: name] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha2 @ P4 ) )
=> ( ( fresh @ name @ pi @ Y4 @ P2 )
=> ( fresh @ name @ pi @ Y4 @ P4 ) ) ) ).
% freshFreeDerivative(2)
thf(fact_221_freshFreeDerivative_I1_J,axiom,
! [P2: pi,Alpha2: late_freeRes,P4: pi,Y4: name] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha2 @ P4 ) )
=> ( ( fresh @ name @ pi @ Y4 @ P2 )
=> ( fresh @ name @ late_freeRes @ Y4 @ Alpha2 ) ) ) ).
% freshFreeDerivative(1)
thf(fact_222_Open,axiom,
! [P2: pi,A2: name,B2: name,P4: pi] :
( ( late_transitions @ P2 @ ( late_FreeR @ ( late_OutputR @ A2 @ B2 ) @ P4 ) )
=> ( ( A2 != B2 )
=> ( late_transitions @ ( res @ B2 @ P2 ) @ ( late_BoundR @ ( late_BoundOutputS @ A2 ) @ B2 @ P4 ) ) ) ) ).
% Open
thf(fact_223_residual_Ostrong__inducts,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [P2: A > late_residual > $o,Z2: A,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi5: pi,Z3: A] :
( ( fresh @ name @ A @ Name3 @ Z3 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P2 @ Z3 @ ( late_BoundR @ Subject2 @ Name3 @ Pi5 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi5: pi,Z3: A] : ( P2 @ Z3 @ ( late_FreeR @ FreeRes2 @ Pi5 ) )
=> ( P2 @ Z2 @ Residual ) ) ) ) ).
% residual.strong_inducts
thf(fact_224_residual_Ostrong__induct,axiom,
! [N: $tType] :
( ( fs_name @ N )
=> ! [P2: N > late_residual > $o,Z2: N,Residual: late_residual] :
( ! [Subject2: late_subject,Name3: name,Pi5: pi,Z3: N] :
( ( fresh @ name @ N @ Name3 @ Z3 )
=> ( ( fresh @ name @ late_subject @ Name3 @ Subject2 )
=> ( P2 @ Z3 @ ( late_BoundR @ Subject2 @ Name3 @ Pi5 ) ) ) )
=> ( ! [FreeRes2: late_freeRes,Pi5: pi,Z3: N] : ( P2 @ Z3 @ ( late_FreeR @ FreeRes2 @ Pi5 ) )
=> ( P2 @ Z2 @ Residual ) ) ) ) ).
% residual.strong_induct
thf(fact_225_simE_I2_J,axiom,
! [P2: pi,Rel: set @ ( product_prod @ pi @ pi ),Q: pi,Alpha2: late_freeRes,Q2: pi] :
( ( strong743114133lation @ P2 @ Rel @ Q )
=> ( ( late_transitions @ Q @ ( late_FreeR @ Alpha2 @ Q2 ) )
=> ? [P5: pi] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha2 @ P5 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P5 @ Q2 ) @ Rel ) ) ) ) ).
% simE(2)
thf(fact_226_resCasesB_H,axiom,
! [X7: name,P2: pi,A2: late_subject,Y7: name,P4: pi] :
( ( late_transitions @ ( res @ X7 @ P2 ) @ ( late_BoundR @ A2 @ Y7 @ P4 ) )
=> ( ! [P: pi,A4: name,B4: name] :
( ( ( res @ X7 @ P2 )
= ( res @ B4 @ P ) )
=> ! [P5: pi] :
( ( ( late_BoundR @ A2 @ Y7 @ P4 )
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B4 @ P5 ) )
=> ( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A4 @ B4 ) @ P5 ) )
=> ( A4 = B4 ) ) ) )
=> ~ ! [P: pi,A4: late_subject,X5: name,P5: pi,Y3: name] :
( ( ( res @ X7 @ P2 )
= ( res @ Y3 @ P ) )
=> ( ( ( late_BoundR @ A2 @ Y7 @ P4 )
= ( late_BoundR @ A4 @ X5 @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P5 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X5 )
=> ( ( fresh @ name @ pi @ X5 @ P )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) ) ) ) ).
% resCasesB'
thf(fact_227_resCases_H,axiom,
! [X3: name,P2: pi,Rs: late_residual] :
( ( late_transitions @ ( res @ X3 @ P2 ) @ Rs )
=> ( ! [P: pi,A4: name,B4: name] :
( ( ( res @ X3 @ P2 )
= ( res @ B4 @ P ) )
=> ! [P5: pi] :
( ( Rs
= ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ B4 @ P5 ) )
=> ( ( late_transitions @ P @ ( late_FreeR @ ( late_OutputR @ A4 @ B4 ) @ P5 ) )
=> ( A4 = B4 ) ) ) )
=> ( ! [P: pi,A4: late_subject,X5: name,P5: pi,Y3: name] :
( ( ( res @ X3 @ P2 )
= ( res @ Y3 @ P ) )
=> ( ( Rs
= ( late_BoundR @ A4 @ X5 @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P @ ( late_BoundR @ A4 @ X5 @ P5 ) )
=> ( ( fresh @ name @ late_subject @ Y3 @ A4 )
=> ( ( Y3 != X5 )
=> ( ( fresh @ name @ pi @ X5 @ P )
=> ~ ( fresh @ name @ late_subject @ X5 @ A4 ) ) ) ) ) ) )
=> ~ ! [P: pi,Alpha: late_freeRes,P5: pi,Y3: name] :
( ( ( res @ X3 @ P2 )
= ( res @ Y3 @ P ) )
=> ( ( Rs
= ( late_FreeR @ Alpha @ ( res @ Y3 @ P5 ) ) )
=> ( ( late_transitions @ P @ ( late_FreeR @ Alpha @ P5 ) )
=> ~ ( fresh @ name @ late_freeRes @ Y3 @ Alpha ) ) ) ) ) ) ) ).
% resCases'
thf(fact_228_resCasesB,axiom,
! [Y4: name,P2: pi,A2: late_subject,X3: name,YP: pi,F5: late_subject > pi > $o] :
( ( late_transitions @ ( res @ Y4 @ P2 ) @ ( late_BoundR @ A2 @ X3 @ YP ) )
=> ( ( X3 != Y4 )
=> ( ( fresh @ name @ pi @ X3 @ P2 )
=> ( ! [B4: name,P5: pi] :
( ( late_transitions @ P2 @ ( late_FreeR @ ( late_OutputR @ B4 @ Y4 ) @ P5 ) )
=> ( ( B4 != Y4 )
=> ( ( A2
= ( late_BoundOutputS @ B4 ) )
=> ( F5 @ ( late_BoundOutputS @ B4 ) @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X3 @ Y4 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P5 ) ) ) ) )
=> ( ! [P5: pi] :
( ( late_transitions @ P2 @ ( late_BoundR @ A2 @ X3 @ P5 ) )
=> ( ( fresh @ name @ late_subject @ Y4 @ A2 )
=> ( F5 @ A2 @ ( res @ Y4 @ P5 ) ) ) )
=> ( F5 @ A2 @ YP ) ) ) ) ) ) ).
% resCasesB
thf(fact_229_resSimCases,axiom,
! [A: $tType] :
( ( fs_name @ A )
=> ! [Rel: set @ ( product_prod @ pi @ pi ),Q: pi,X3: name,P2: pi,C4: A] :
( ( eqvt @ pi @ Rel )
=> ( ! [Q3: pi,A4: name] :
( ( late_transitions @ Q @ ( late_FreeR @ ( late_OutputR @ A4 @ X3 ) @ Q3 ) )
=> ( ( A4 != X3 )
=> ? [P6: pi] :
( ( late_transitions @ P2 @ ( late_BoundR @ ( late_BoundOutputS @ A4 ) @ X3 @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q3 ) @ Rel ) ) ) )
=> ( ! [Q3: pi,A4: late_subject,Y3: name] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y3 @ Q3 ) )
=> ( ( fresh @ name @ late_subject @ X3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( fresh @ name @ A @ Y3 @ C4 )
=> ? [P6: pi] :
( ( late_transitions @ P2 @ ( late_BoundR @ A4 @ Y3 @ P6 ) )
& ( strong2129052853vative @ P6 @ ( res @ X3 @ Q3 ) @ A4 @ Y3 @ Rel ) ) ) ) ) )
=> ( ! [Q3: pi,Alpha: late_freeRes] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q3 ) )
=> ( ( fresh @ name @ late_freeRes @ X3 @ Alpha )
=> ? [P6: pi] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ ( res @ X3 @ Q3 ) ) @ Rel ) ) ) )
=> ( strong743114133lation @ P2 @ Rel @ ( res @ X3 @ Q ) ) ) ) ) ) ) ).
% resSimCases
thf(fact_230_simulation__def,axiom,
( strong743114133lation
= ( ^ [P7: pi,Rel2: set @ ( product_prod @ pi @ pi ),Q4: pi] :
( ! [A6: late_subject,X6: name,Q5: pi] :
( ( ( late_transitions @ Q4 @ ( late_BoundR @ A6 @ X6 @ Q5 ) )
& ( fresh @ name @ pi @ X6 @ P7 ) )
=> ? [P8: pi] :
( ( late_transitions @ P7 @ ( late_BoundR @ A6 @ X6 @ P8 ) )
& ( strong2129052853vative @ P8 @ Q5 @ A6 @ X6 @ Rel2 ) ) )
& ! [Alpha3: late_freeRes,Q5: pi] :
( ( late_transitions @ Q4 @ ( late_FreeR @ Alpha3 @ Q5 ) )
=> ? [P8: pi] :
( ( late_transitions @ P7 @ ( late_FreeR @ Alpha3 @ P8 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P8 @ Q5 ) @ Rel2 ) ) ) ) ) ) ).
% simulation_def
thf(fact_231_simCases,axiom,
! [Q: pi,P2: pi,Rel: set @ ( product_prod @ pi @ pi )] :
( ! [A4: late_subject,Y3: name,Q3: pi] :
( ( late_transitions @ Q @ ( late_BoundR @ A4 @ Y3 @ Q3 ) )
=> ( ( fresh @ name @ pi @ Y3 @ P2 )
=> ? [P6: pi] :
( ( late_transitions @ P2 @ ( late_BoundR @ A4 @ Y3 @ P6 ) )
& ( strong2129052853vative @ P6 @ Q3 @ A4 @ Y3 @ Rel ) ) ) )
=> ( ! [Alpha: late_freeRes,Q3: pi] :
( ( late_transitions @ Q @ ( late_FreeR @ Alpha @ Q3 ) )
=> ? [P6: pi] :
( ( late_transitions @ P2 @ ( late_FreeR @ Alpha @ P6 ) )
& ( member @ ( product_prod @ pi @ pi ) @ ( product_Pair @ pi @ pi @ P6 @ Q3 ) @ Rel ) ) )
=> ( strong743114133lation @ P2 @ Rel @ Q ) ) ) ).
% simCases
thf(fact_232_supports__def,axiom,
! [A: $tType,X: $tType] :
( ( supports @ X @ A )
= ( ^ [S2: set @ X,X6: A] :
! [A6: X,B6: X] :
( ( ~ ( member @ X @ A6 @ S2 )
& ~ ( member @ X @ B6 @ S2 ) )
=> ( ( perm @ X @ A @ ( cons @ ( product_prod @ X @ X ) @ ( product_Pair @ X @ X @ A6 @ B6 ) @ ( nil @ ( product_prod @ X @ X ) ) ) @ X6 )
= X6 ) ) ) ) ).
% supports_def
thf(fact_233_IdD,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id @ A ) )
=> ( A2 = B2 ) ) ).
% IdD
thf(fact_234_supports__subset,axiom,
! [X: $tType,A: $tType,S1: set @ X,X3: A,S22: set @ X] :
( ( supports @ X @ A @ S1 @ X3 )
=> ( ( ord_less_eq @ ( set @ X ) @ S1 @ S22 )
=> ( supports @ X @ A @ S22 @ X3 ) ) ) ).
% supports_subset
thf(fact_235_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K: A,Kl: list @ A] : ( Lab @ ( cons @ A @ K @ Kl ) ) ) ) ).
% shift_def
thf(fact_236_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_237_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_238_alphaInput,axiom,
! [C2: name,P2: pi,A2: name,X3: name] :
( ( fresh @ name @ pi @ C2 @ P2 )
=> ( ( input @ A2 @ X3 @ P2 )
= ( input @ A2 @ C2 @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X3 @ C2 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P2 ) ) ) ) ).
% alphaInput
thf(fact_239_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_240_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_241_resTrans_I2_J,axiom,
! [X3: name,Y4: name,P2: pi,Rs: late_residual] :
~ ( late_transitions @ ( res @ X3 @ ( input @ X3 @ Y4 @ P2 ) ) @ Rs ) ).
% resTrans(2)
thf(fact_242_pi_Odistinct_I53_J,axiom,
! [Name1: name,Name22: name,Pi: pi,Pi12: pi,Pi22: pi] :
( ( input @ Name1 @ Name22 @ Pi )
!= ( sum @ Pi12 @ Pi22 ) ) ).
% pi.distinct(53)
thf(fact_243_pi_Odistinct_I57_J,axiom,
! [Name1: name,Name22: name,Pi: pi,Name: name,Pi4: pi] :
( ( input @ Name1 @ Name22 @ Pi )
!= ( res @ Name @ Pi4 ) ) ).
% pi.distinct(57)
thf(fact_244_inputFreeTrans,axiom,
! [A2: name,X3: name,P2: pi,Alpha2: late_freeRes,P4: pi] :
~ ( late_transitions @ ( input @ A2 @ X3 @ P2 ) @ ( late_FreeR @ Alpha2 @ P4 ) ) ).
% inputFreeTrans
thf(fact_245_resInputFreeTrans,axiom,
! [X3: name,A2: name,Y4: name,P2: pi,Alpha2: late_freeRes,P4: pi] :
~ ( late_transitions @ ( res @ X3 @ ( input @ A2 @ Y4 @ P2 ) ) @ ( late_FreeR @ Alpha2 @ P4 ) ) ).
% resInputFreeTrans
thf(fact_246_inputBoundOutputTrans,axiom,
! [A2: name,X3: name,P2: pi,B2: name,Y4: name,P4: pi] :
~ ( late_transitions @ ( input @ A2 @ X3 @ P2 ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Y4 @ P4 ) ) ).
% inputBoundOutputTrans
thf(fact_247_inputIneqTrans,axiom,
! [A2: name,X3: name,P2: pi,B2: late_subject,Y4: name,P4: pi] :
( ( late_transitions @ ( input @ A2 @ X3 @ P2 ) @ ( late_BoundR @ B2 @ Y4 @ P4 ) )
=> ~ ( fresh @ name @ late_subject @ A2 @ B2 ) ) ).
% inputIneqTrans
thf(fact_248_resInputBoundOutputTrans,axiom,
! [X3: name,A2: name,Y4: name,P2: pi,B2: name,Z2: name,P4: pi] :
~ ( late_transitions @ ( res @ X3 @ ( input @ A2 @ Y4 @ P2 ) ) @ ( late_BoundR @ ( late_BoundOutputS @ B2 ) @ Z2 @ P4 ) ) ).
% resInputBoundOutputTrans
thf(fact_249_inputCases,axiom,
! [A2: name,X3: name,P2: pi,B2: late_subject,Y4: name,YP: pi,Prop: late_subject > name > pi > $o] :
( ( late_transitions @ ( input @ A2 @ X3 @ P2 ) @ ( late_BoundR @ B2 @ Y4 @ YP ) )
=> ( ( Y4 != A2 )
=> ( ( Y4 != X3 )
=> ( ( fresh @ name @ pi @ Y4 @ P2 )
=> ( ( ( B2
= ( late_InputS @ A2 ) )
=> ( ( YP
= ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X3 @ Y4 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P2 ) )
=> ( Prop @ ( late_InputS @ A2 ) @ Y4 @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ X3 @ Y4 ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ P2 ) ) ) )
=> ( Prop @ B2 @ Y4 @ YP ) ) ) ) ) ) ).
% inputCases
thf(fact_250_name__calc_I1_J,axiom,
! [A2: name,B2: name,Pi: list @ ( product_prod @ name @ name ),X3: name] :
( ( perm @ name @ name @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ A2 @ B2 ) @ Pi ) @ X3 )
= ( swap @ name @ ( product_Pair @ name @ name @ A2 @ B2 ) @ ( perm @ name @ name @ Pi @ X3 ) ) ) ).
% name_calc(1)
thf(fact_251_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_252_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_253_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_254_Late__Semantics_Osubject_Odistinct_I1_J,axiom,
! [Name2: name,Name: name] :
( ( late_InputS @ Name2 )
!= ( late_BoundOutputS @ Name ) ) ).
% Late_Semantics.subject.distinct(1)
% Type constructors (46)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Agent_Ocp__name__name,axiom,
! [A8: $tType,A9: $tType] :
( ( ( cp_name_name @ A8 )
& ( cp_name_name @ A9 ) )
=> ( cp_name_name @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Agent_Opt__name,axiom,
! [A8: $tType,A9: $tType] :
( ( ( pt_name @ A8 )
& ( pt_name @ A9 ) )
=> ( pt_name @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Agent_Ocp__name__name_2,axiom,
! [A8: $tType] :
( ( cp_name_name @ A8 )
=> ( cp_name_name @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Agent_Opt__name_5,axiom,
! [A8: $tType] :
( ( pt_name @ A8 )
=> ( pt_name @ ( set @ A8 ) ) ) ).
thf(tcon_Agent_Opi___Agent_Ocp__name__name_6,axiom,
cp_name_name @ pi ).
thf(tcon_Agent_Opi___Agent_Opt__name_7,axiom,
pt_name @ pi ).
thf(tcon_Agent_Opi___Agent_Ofs__name,axiom,
fs_name @ pi ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Agent_Ocp__name__name_9,axiom,
cp_name_name @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_11,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Agent_Opt__name_12,axiom,
pt_name @ $o ).
thf(tcon_HOL_Obool___Agent_Ofs__name_13,axiom,
fs_name @ $o ).
thf(tcon_List_Olist___Agent_Ocp__name__name_14,axiom,
! [A8: $tType] :
( ( cp_name_name @ A8 )
=> ( cp_name_name @ ( list @ A8 ) ) ) ).
thf(tcon_List_Olist___Agent_Opt__name_15,axiom,
! [A8: $tType] :
( ( pt_name @ A8 )
=> ( pt_name @ ( list @ A8 ) ) ) ).
thf(tcon_List_Olist___Agent_Ofs__name_16,axiom,
! [A8: $tType] :
( ( fs_name @ A8 )
=> ( fs_name @ ( list @ A8 ) ) ) ).
thf(tcon_Agent_Oname___Agent_Ocp__name__name_17,axiom,
cp_name_name @ name ).
thf(tcon_Agent_Oname___Agent_Opt__name_18,axiom,
pt_name @ name ).
thf(tcon_Agent_Oname___Agent_Ofs__name_19,axiom,
fs_name @ name ).
thf(tcon_Agent_Opi__Rep___Agent_Ocp__name__name_20,axiom,
cp_name_name @ pi_Rep ).
thf(tcon_Agent_Opi__Rep___Agent_Opt__name_21,axiom,
pt_name @ pi_Rep ).
thf(tcon_Product__Type_Oprod___Agent_Ocp__name__name_22,axiom,
! [A8: $tType,A9: $tType] :
( ( ( cp_name_name @ A8 )
& ( cp_name_name @ A9 ) )
=> ( cp_name_name @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Agent_Opt__name_23,axiom,
! [A8: $tType,A9: $tType] :
( ( ( pt_name @ A8 )
& ( pt_name @ A9 ) )
=> ( pt_name @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Product__Type_Oprod___Agent_Ofs__name_24,axiom,
! [A8: $tType,A9: $tType] :
( ( ( fs_name @ A8 )
& ( fs_name @ A9 ) )
=> ( fs_name @ ( product_prod @ A8 @ A9 ) ) ) ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Ocp__name__name_25,axiom,
cp_name_name @ late_freeRes ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Opt__name_26,axiom,
pt_name @ late_freeRes ).
thf(tcon_Late__Semantics_OfreeRes___Agent_Ofs__name_27,axiom,
fs_name @ late_freeRes ).
thf(tcon_Late__Semantics_Osubject___Agent_Ocp__name__name_28,axiom,
cp_name_name @ late_subject ).
thf(tcon_Late__Semantics_Osubject___Agent_Opt__name_29,axiom,
pt_name @ late_subject ).
thf(tcon_Late__Semantics_Osubject___Agent_Ofs__name_30,axiom,
fs_name @ late_subject ).
thf(tcon_Late__Semantics_Oresidual___Agent_Ocp__name__name_31,axiom,
cp_name_name @ late_residual ).
thf(tcon_Late__Semantics_Oresidual___Agent_Opt__name_32,axiom,
pt_name @ late_residual ).
thf(tcon_Late__Semantics_Oresidual___Agent_Ofs__name_33,axiom,
fs_name @ late_residual ).
thf(tcon_Late__Semantics_OfreeRes__Rep___Agent_Ocp__name__name_34,axiom,
cp_name_name @ late_freeRes_Rep ).
thf(tcon_Late__Semantics_OfreeRes__Rep___Agent_Opt__name_35,axiom,
pt_name @ late_freeRes_Rep ).
thf(tcon_Late__Semantics_Osubject__Rep___Agent_Ocp__name__name_36,axiom,
cp_name_name @ late_subject_Rep ).
thf(tcon_Late__Semantics_Osubject__Rep___Agent_Opt__name_37,axiom,
pt_name @ late_subject_Rep ).
thf(tcon_Late__Semantics_Oresidual__Rep___Agent_Ocp__name__name_38,axiom,
cp_name_name @ late_residual_Rep ).
thf(tcon_Late__Semantics_Oresidual__Rep___Agent_Opt__name_39,axiom,
pt_name @ late_residual_Rep ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y4: A] :
( ( if @ A @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y4: A] :
( ( if @ A @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
strong2129052853vative @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ y @ x ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ p2 ) @ ( perm @ name @ pi @ ( cons @ ( product_prod @ name @ name ) @ ( product_Pair @ name @ name @ y @ x ) @ ( nil @ ( product_prod @ name @ name ) ) ) @ p2 ) @ ( late_BoundOutputS @ aa ) @ y @ rel ).
%------------------------------------------------------------------------------