TPTP Problem File: ITP158^2.p
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%------------------------------------------------------------------------------
% File : ITP158^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Prover problem prob_480__3258466_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 : Prover/prob_480__3258466_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 329 ( 136 unt; 62 typ; 0 def)
% Number of atoms : 659 ( 401 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 6098 ( 145 ~; 26 |; 82 &;5557 @)
% ( 0 <=>; 288 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 259 ( 259 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 59 usr; 6 con; 0-5 aty)
% Number of variables : 1312 ( 14 ^;1171 !; 77 ?;1312 :)
% ( 50 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:20:06.928
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Prover__Mirabelle__xkffrtyxot_Oform,type,
prover2006609834e_form: $tType ).
thf(ty_t_Prover__Mirabelle__xkffrtyxot_OU,type,
prover_Mirabelle_U: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (56)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ocan__select,type,
can_select:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
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_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[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_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_Prover__Mirabelle__xkffrtyxot_OSEval,type,
prover1899965912_SEval: ( product_prod @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ) ) > ( nat > prover_Mirabelle_U ) > ( list @ prover2006609834e_form ) > $o ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_OSvalid,type,
prover1090191840Svalid: ( list @ prover2006609834e_form ) > $o ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_Ofinst,type,
prover48307765_finst: prover2006609834e_form > nat > prover2006609834e_form ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_Oflatten,type,
prover12291693latten:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_Oform_OFAll,type,
prover946642470e_FAll: prover2006609834e_form > prover2006609834e_form ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_Ofv,type,
prover_Mirabelle_fv: prover2006609834e_form > ( list @ nat ) ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_Ois__env,type,
prover1043414700is_env: ( product_prod @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ) ) > ( nat > prover_Mirabelle_U ) > $o ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_Omaxvar,type,
prover572158330maxvar: ( list @ nat ) > nat ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_OpreSuc,type,
prover1646808149preSuc: ( list @ nat ) > ( list @ nat ) ).
thf(sy_c_Prover__Mirabelle__xkffrtyxot_Osfv,type,
prover_Mirabelle_sfv: ( list @ prover2006609834e_form ) > ( list @ nat ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a: set @ prover_Mirabelle_U ).
thf(sy_v_b,type,
b: nat > ( list @ prover_Mirabelle_U ) > $o ).
thf(sy_v_f,type,
f: prover2006609834e_form ).
thf(sy_v_s,type,
s: list @ prover2006609834e_form ).
thf(sy_v_u,type,
u: nat ).
% Relevant facts (255)
thf(fact_0_form_Oinject_I5_J,axiom,
! [X5: prover2006609834e_form,Y5: prover2006609834e_form] :
( ( ( prover946642470e_FAll @ X5 )
= ( prover946642470e_FAll @ Y5 ) )
= ( X5 = Y5 ) ) ).
% form.inject(5)
thf(fact_1_SEval_Osimps_I1_J,axiom,
! [M: product_prod @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ),E: nat > prover_Mirabelle_U] :
~ ( prover1899965912_SEval @ M @ E @ ( nil @ prover2006609834e_form ) ) ).
% SEval.simps(1)
thf(fact_2_SEval__cong,axiom,
! [S: list @ prover2006609834e_form,E1: nat > prover_Mirabelle_U,E2: nat > prover_Mirabelle_U,M: product_prod @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o )] :
( ! [X: nat] :
( ( member @ nat @ X @ ( set2 @ nat @ ( prover_Mirabelle_sfv @ S ) ) )
=> ( ( E1 @ X )
= ( E2 @ X ) ) )
=> ( ( prover1899965912_SEval @ M @ E1 @ S )
= ( prover1899965912_SEval @ M @ E2 @ S ) ) ) ).
% SEval_cong
thf(fact_3_SEval__append,axiom,
! [M: product_prod @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ),E: nat > prover_Mirabelle_U,Xs: list @ prover2006609834e_form,Ys: list @ prover2006609834e_form] :
( ( prover1899965912_SEval @ M @ E @ ( append @ prover2006609834e_form @ Xs @ Ys ) )
= ( ( prover1899965912_SEval @ M @ E @ Xs )
| ( prover1899965912_SEval @ M @ E @ Ys ) ) ) ).
% SEval_append
thf(fact_4_sound__FAll,axiom,
! [U: nat,F: prover2006609834e_form,S: list @ prover2006609834e_form] :
( ~ ( member @ nat @ U @ ( set2 @ nat @ ( prover_Mirabelle_sfv @ ( cons @ prover2006609834e_form @ ( prover946642470e_FAll @ F ) @ S ) ) ) )
=> ( ( prover1090191840Svalid @ ( append @ prover2006609834e_form @ S @ ( cons @ prover2006609834e_form @ ( prover48307765_finst @ F @ U ) @ ( nil @ prover2006609834e_form ) ) ) )
=> ( prover1090191840Svalid @ ( cons @ prover2006609834e_form @ ( prover946642470e_FAll @ F ) @ S ) ) ) ) ).
% sound_FAll
thf(fact_5_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_6_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_7_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_8_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_9_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_10_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_11_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_12_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_13_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_14_Svalid__def,axiom,
( prover1090191840Svalid
= ( ^ [S2: list @ prover2006609834e_form] :
! [MI: product_prod @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ),E3: nat > prover_Mirabelle_U] :
( ( prover1043414700is_env @ MI @ E3 )
=> ( prover1899965912_SEval @ MI @ E3 @ S2 ) ) ) ) ).
% Svalid_def
thf(fact_15_split__list,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs ) ) ) ) ).
% split_list
thf(fact_16_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_17_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_18_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_19_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs2 )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_20_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C2 )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_21_not__Cons__self2,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( cons @ A @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_22_sfv__nil,axiom,
( ( prover_Mirabelle_sfv @ ( nil @ prover2006609834e_form ) )
= ( nil @ nat ) ) ).
% sfv_nil
thf(fact_23_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs2 @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs2 )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_24_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_25_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Ys2: list @ A] :
( ( P @ Ys2 )
=> ( P @ ( cons @ A @ X @ Ys2 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_26_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ~ ! [X: A,Ys2: list @ A] :
( X2
!= ( cons @ A @ X @ Ys2 ) ) ) ) ).
% strict_sorted.cases
thf(fact_27_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] :
( ! [F2: A > B,X_1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X_1 )
=> ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_28_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_29_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X: A] : ( P @ P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X: A,Y2: A,Xs2: list @ A] :
( ( P @ P2 @ ( cons @ A @ Y2 @ Xs2 ) )
=> ( P @ P2 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_30_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F2: A > B,X: A] : ( P @ F2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [F2: A > B,X: A,Y2: A,Zs: list @ A] :
( ( P @ F2 @ ( cons @ A @ Y2 @ Zs ) )
=> ( P @ F2 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs ) ) ) )
=> ( ! [A3: A > B] : ( P @ A3 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_31_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y2: A,Xs2: list @ A] :
( ( ( X = Y2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y2 )
=> ( P @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_32_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X: A,Ys2: list @ A] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons @ A @ X @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_33_remdups__adj_Ocases,axiom,
! [A: $tType,X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ( ! [X: A] :
( X2
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_34_transpose_Ocases,axiom,
! [A: $tType,X2: list @ ( list @ A )] :
( ( X2
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_35_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 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( P @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) )
=> ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_36_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X: A,Xs2: list @ A] :
( ! [X212: A,X222: list @ A] :
( ( Xs2
= ( cons @ A @ X212 @ X222 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_37_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: list @ A] :
( ! [X: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X @ Xs2 ) )
=> ( X2
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_38_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y2: A,Zs: list @ A] :
( ( P @ Zs )
=> ( ( P @ ( cons @ A @ Y2 @ Zs ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_39_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 )
=> ( ! [X: A,Xs2: list @ A,Ys2: list @ A] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_40_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_41_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_42_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_43_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X213: A,X223: list @ A] :
( Y
!= ( cons @ A @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_49_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_50_list_Oset__cases,axiom,
! [A: $tType,E: A,A2: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A2 ) )
=> ( ! [Z2: list @ A] :
( A2
!= ( cons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A2
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_51_set__ConsD,axiom,
! [A: $tType,Y: A,X2: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_52_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_53_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X22: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X22 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_54_append__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X2 @ Xs ) @ Ys )
= ( cons @ A @ X2 @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_55_Cons__eq__appendI,axiom,
! [A: $tType,X2: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs2 ) )
=> ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_56_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_57_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_58_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_59_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_60_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,X2: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs2 )
= ( cons @ A @ X2 @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs2
= ( cons @ A @ X2 @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_61_Cons__eq__append__conv,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs2 ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X2 @ Xs )
= Zs2 ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_62_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_63_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_64_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list @ A,X3: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_65_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list @ A,X3: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_66_in__set__conv__decomp__first,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ~ ( member @ A @ X2 @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_67_in__set__conv__decomp__last,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ~ ( member @ A @ X2 @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_68_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_69_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X: A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_70_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs ) ) )
& ( P @ X )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_71_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X: A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs ) ) )
& ( P @ X )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_72_in__set__conv__decomp,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_73_append__Cons__eq__iff,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,Xs3: list @ A,Ys5: list @ A] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X2 @ Ys ) )
= ( append @ A @ Xs3 @ ( cons @ A @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_74_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_75_split__list__first,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs ) ) )
& ~ ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_76_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X: A] :
( ? [Zs: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_77_split__list__last,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs ) ) )
& ~ ( member @ A @ X2 @ ( set2 @ A @ Zs ) ) ) ) ).
% split_list_last
thf(fact_78_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_79_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X24: B,Y1: A,Y23: B] :
( ( ( product_Pair @ A @ B @ X12 @ X24 )
= ( product_Pair @ A @ B @ Y1 @ Y23 ) )
= ( ( X12 = Y1 )
& ( X24 = Y23 ) ) ) ).
% prod.inject
thf(fact_80_the__elem__set,axiom,
! [A: $tType,X2: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= X2 ) ).
% the_elem_set
thf(fact_81_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X2: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X2 @ Xs ) @ F )
= ( append @ A @ ( F @ X2 ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_82_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X2: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X2 @ Xs ) )
= ( append @ A @ ( F @ X2 ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_83_not__in__set__insert,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X2 @ Xs )
= ( cons @ A @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_84_insert__Nil,axiom,
! [A: $tType,X2: A] :
( ( insert @ A @ X2 @ ( nil @ A ) )
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_85_sfv__cons,axiom,
! [A2: prover2006609834e_form,List: list @ prover2006609834e_form] :
( ( prover_Mirabelle_sfv @ ( cons @ prover2006609834e_form @ A2 @ List ) )
= ( append @ nat @ ( prover_Mirabelle_fv @ A2 ) @ ( prover_Mirabelle_sfv @ List ) ) ) ).
% sfv_cons
thf(fact_86_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X2 @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_87_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_88_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_89_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_90_in__set__insert,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_91_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_92_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F2: A > B,Bs: list @ B] :
( X2
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
=> ~ ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
( X2
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A3 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_93_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_94_in__set__butlastD,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_95_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_96_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X2: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F2: A > B,X: A] :
( X2
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ( ! [F2: A > B,X: A,Y2: A,Zs: list @ A] :
( X2
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs ) ) ) )
=> ~ ! [A3: A > B] :
( X2
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_97_sorted__wrt_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P2: A > A > $o] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( nil @ A ) ) )
=> ~ ! [P2: A > A > $o,X: A,Ys2: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_98_successively_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P2: A > A > $o] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X: A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ~ ! [P2: A > A > $o,X: A,Y2: A,Xs2: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_99_splice_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys2: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ~ ! [X: A,Xs2: list @ A,Ys2: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_100_shuffles_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys2: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ! [Xs2: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_101_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F: A > ( list @ B ),G: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( bind @ A @ B @ Xs @ F )
= ( bind @ A @ B @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_102_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X2 @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_103_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_104_in__set__butlast__appendI,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X2 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member @ A @ X2 @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X2 @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_105_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_106_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X: A,Y2: B] :
( P3
= ( product_Pair @ A @ B @ X @ Y2 ) ) ).
% surj_pair
thf(fact_107_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A3: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B4 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_108_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_109_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A3: A,B4: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% prod_cases3
thf(fact_110_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_111_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E4 ) ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D,E5: E4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E4 ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E4 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E4 ) @ C3 @ ( product_Pair @ D @ E4 @ D2 @ E5 ) ) ) ) ) ).
% prod_cases5
thf(fact_112_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E4: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D,E5: E4,F2: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E4 @ F3 ) @ D2 @ ( product_Pair @ E4 @ F3 @ E5 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_113_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E4: $tType,F3: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
~ ! [A3: A,B4: B,C3: C,D2: D,E5: E4,F2: F3,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E4 @ ( product_prod @ F3 @ G2 ) @ E5 @ ( product_Pair @ F3 @ G2 @ F2 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_114_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A3: A,B4: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A3 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_115_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A3: A,B4: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_116_prod__induct5,axiom,
! [E4: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E4 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E4 ) ) )] :
( ! [A3: A,B4: B,C3: C,D2: D,E5: E4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E4 ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E4 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E4 ) @ C3 @ ( product_Pair @ D @ E4 @ D2 @ E5 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_117_prod__induct6,axiom,
! [F3: $tType,E4: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) ) )] :
( ! [A3: A,B4: B,C3: C,D2: D,E5: E4,F2: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E4 @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E4 @ F3 ) @ D2 @ ( product_Pair @ E4 @ F3 @ E5 @ F2 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_118_prod__induct7,axiom,
! [G2: $tType,F3: $tType,E4: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
( ! [A3: A,B4: B,C3: C,D2: D,E5: E4,F2: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A3 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E4 @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E4 @ ( product_prod @ F3 @ G2 ) @ E5 @ ( product_Pair @ F3 @ G2 @ F2 @ G3 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_119_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A3: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A3 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_120_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A3: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A3 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_121_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X3: A,Xs4: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X3 @ ( set2 @ A @ Xs4 ) ) @ Xs4 @ ( cons @ A @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_122_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_123_flatten_Osimps_I2_J,axiom,
! [A: $tType,A2: list @ A,List: list @ ( list @ A )] :
( ( prover12291693latten @ A @ ( cons @ ( list @ A ) @ A2 @ List ) )
= ( append @ A @ A2 @ ( prover12291693latten @ A @ List ) ) ) ).
% flatten.simps(2)
thf(fact_124_flatten_Osimps_I1_J,axiom,
! [A: $tType] :
( ( prover12291693latten @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% flatten.simps(1)
thf(fact_125_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_126_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_127_fv_Osimps_I5_J,axiom,
! [F: prover2006609834e_form] :
( ( prover_Mirabelle_fv @ ( prover946642470e_FAll @ F ) )
= ( prover1646808149preSuc @ ( prover_Mirabelle_fv @ F ) ) ) ).
% fv.simps(5)
thf(fact_128_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_129_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_130_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= X2 ) ).
% last_snoc
thf(fact_131_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_132_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_133_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_134_last__in__set,axiom,
! [A: $tType,As2: list @ A] :
( ( As2
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% last_in_set
thf(fact_135_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs5: list @ A,Ys6: list @ A] :
( ( Xs
= ( append @ A @ Xs5 @ Ss ) )
& ( Ys
= ( append @ A @ Ys6 @ Ss ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys6
= ( nil @ A ) )
| ( ( last @ A @ Xs5 )
!= ( last @ A @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_136_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_137_preSuc_Osimps_I1_J,axiom,
( ( prover1646808149preSuc @ ( nil @ nat ) )
= ( nil @ nat ) ) ).
% preSuc.simps(1)
thf(fact_138_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_139_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( X2 = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_140_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_141_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys @ Zs2 ) )
= ( ( ( Xss2
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs2
= ( nil @ A ) ) ) )
& ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs4: list @ A,Xs6: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs4 ) )
& ( Zs2
= ( append @ A @ Xs6 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_142_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lexord @ A @ R ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_143_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( concat @ A @ Xss2 )
= ( nil @ A ) )
= ( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_144_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss2 ) )
= ( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_145_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% concat_append
thf(fact_146_Cons__listrel1__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X2 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_147_lexord__cons__cons,axiom,
! [A: $tType,A2: A,X2: list @ A,B2: A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A2 @ X2 ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
| ( ( A2 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_148_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R ) )
= ( ? [A6: A,X3: list @ A] :
( Y
= ( cons @ A @ A6 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_149_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_150_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_151_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),X2: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ X2 @ Ys ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_152_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_153_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_154_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Us2: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( Us2 = Vs ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us2 @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_155_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_156_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: list @ A,Y: list @ A] :
( ! [A3: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B4 ) @ R )
| ( A3 = B4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A3 ) @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( lexord @ A @ R ) )
| ( X2 = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_157_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_158_concat_Osimps_I2_J,axiom,
! [A: $tType,X2: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X2 @ Xs ) )
= ( append @ A @ X2 @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_159_lexord__Nil__right,axiom,
! [A: $tType,X2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_160_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A ),X2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X2 @ U ) @ ( append @ A @ X2 @ V ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_161_listrel1I1,axiom,
! [A: $tType,X2: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_162_Cons__listrel1E1,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y2: A] :
( ( Ys
= ( cons @ A @ Y2 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R ) )
=> ~ ! [Zs: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_163_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X: A] :
( ( Xs
= ( cons @ A @ X @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) )
=> ~ ! [Zs: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_164_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs2: list @ A] :
( ! [X: A,Y2: A,Z: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_165_lexord__append__leftD,axiom,
! [A: $tType,X2: list @ A,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X2 @ U ) @ ( append @ A @ X2 @ V ) ) @ ( lexord @ A @ R ) )
=> ( ! [A3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_166_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B5: A,Z3: list @ A] :
( Y
= ( cons @ A @ B5 @ Z3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ ( append @ A @ X2 @ Y ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_167_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys @ Zs2 ) )
=> ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_168_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
=> ! [Us3: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_169_listrel1I,axiom,
! [A: $tType,X2: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us2: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ X2 @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_170_lexord__append__left__rightI,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X2: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A2 @ X2 ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_171_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lexord @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_172_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A22
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_173_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X3: A,Y3: B,Xs4: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X3 @ Xs4 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_174_listrel_Oinducts,axiom,
! [A: $tType,B: $tType,X12: list @ A,X24: list @ B,R: set @ ( product_prod @ A @ B ),P: ( list @ A ) > ( list @ B ) > $o] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X12 @ X24 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Y2: B,Xs2: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ).
% listrel.inducts
thf(fact_175_lexord__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irrefl
thf(fact_176_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% listrel.Nil
thf(fact_177_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_178_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_179_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_180_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y2: B,Ys2: list @ B] :
( ( Xs
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y2 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_181_listrel_OCons,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,R: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_182_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R3 ) ) ) ).
% irrefl_def
thf(fact_183_irreflI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 )
=> ( irrefl @ A @ R2 ) ) ).
% irreflI
thf(fact_184_lenlex__append2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Us2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us2 @ Xs ) @ ( append @ A @ Us2 @ Ys ) ) @ ( lenlex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append2
thf(fact_185_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_186_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_187_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_188_irrefl__lex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R ) ) ) ).
% irrefl_lex
thf(fact_189_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_190_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_191_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P4: A > $o,Xs4: list @ A] :
? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs4 ) )
& ( P4 @ X3 )
& ! [Y3: A] :
( ( ( member @ A @ Y3 @ ( set2 @ A @ Xs4 ) )
& ( P4 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_192_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R ) ) ).
% Nil_notin_lex
thf(fact_193_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% Nil2_notin_lex
thf(fact_194_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R ) ) ) ).
% lex_append_leftI
thf(fact_195_lexl__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: list @ A] :
( ( irrefl @ A @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ X2 ) @ ( lex @ A @ R ) ) ) ).
% lexl_not_refl
thf(fact_196_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_197_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_198_can__select__set__list__ex1,axiom,
! [A: $tType,P: A > $o,A4: list @ A] :
( ( can_select @ A @ P @ ( set2 @ A @ A4 ) )
= ( list_ex1 @ A @ P @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_199_Cons__in__lex,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X2 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_200_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us2: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us2 )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_201_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_202_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_203_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_204_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P4: A > $o,A7: set @ A] :
? [X3: A] :
( ( member @ A @ X3 @ A7 )
& ( P4 @ X3 )
& ! [Y3: A] :
( ( ( member @ A @ Y3 @ A7 )
& ( P4 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_205_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_206_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_207_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys2: list @ B,Z: C,Zs: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) @ ( cons @ C @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_208_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_209_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_210_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X: A,Xs5: list @ A,Y2: A,Ys6: list @ A] :
( ( X != Y2 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Xs5 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_211_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs2: list @ A,Ys: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs2 ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_212_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_213_lenlex__append1,axiom,
! [A: $tType,Us2: list @ A,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us2 @ Xs ) @ ( lenlex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us2 @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_214_lexord__lex,axiom,
! [A: $tType,X2: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( lex @ A @ R ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X2 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_215_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R ) )
| ( ( M = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_216_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_217_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys7: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys7 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys7 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_218_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_219_in__measures_I2_J,axiom,
! [A: $tType,X2: A,Y: A,F: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
= ( ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y ) )
| ( ( ( F @ X2 )
= ( F @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_220_in__measures_I1_J,axiom,
! [A: $tType,X2: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_221_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_222_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_223_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_224_length__pos__if__in__set,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_225_measures__less,axiom,
! [A: $tType,F: A > nat,X2: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_226_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_227_maxvar_Osimps_I1_J,axiom,
( ( prover572158330maxvar @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% maxvar.simps(1)
thf(fact_228_count__notin,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X2 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_229_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_230_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_231_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N2 ) @ ( nth @ B @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_232_nth__Cons__0,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ ( zero_zero @ nat ) )
= X2 ) ).
% nth_Cons_0
thf(fact_233_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_234_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_235_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X2: A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_236_in__set__conv__nth,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_237_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_238_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_239_gen__length__code_I1_J,axiom,
! [A: $tType,N: nat] :
( ( gen_length @ A @ N @ ( nil @ A ) )
= N ) ).
% gen_length_code(1)
thf(fact_240_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_241_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I: nat] :
( ( ord_less @ nat @ I @ K )
=> ? [X6: A] : ( P @ I @ X6 ) ) )
= ( ? [Xs4: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs4 )
= K )
& ! [I: nat] :
( ( ord_less @ nat @ I @ K )
=> ( P @ I @ ( nth @ A @ Xs4 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_242_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z4: list @ A] : Y4 = Z4 )
= ( ^ [Xs4: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs4 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( nth @ A @ Xs4 @ I )
= ( nth @ A @ Ys3 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_243_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_244_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
= ( ? [Y3: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ Y3 ) @ R )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N2 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_245_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I2 @ Xs )
= ( take @ A @ I2 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Ys @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_246_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I3: nat,X2: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ I3 @ Y )
= ( list_update @ A @ Xs @ I3 @ Y ) ) ).
% list_update_overwrite
thf(fact_247_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K: nat,X2: A] :
( ( ( list_update @ A @ Xs @ K @ X2 )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_248_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I3: nat,X2: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I3 @ X2 ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_249_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I3: nat] :
( ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_250_nth__list__update__neq,axiom,
! [A: $tType,I3: nat,J: nat,Xs: list @ A,X2: A] :
( ( I3 != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X2 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_251_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_252_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs4: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_253_nth__take,axiom,
! [A: $tType,I3: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I3 )
= ( nth @ A @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_254_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y )
= ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
% Type constructors (6)
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o ).
% 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 (3)
thf(conj_0,hypothesis,
~ ( member @ nat @ u @ ( set2 @ nat @ ( prover_Mirabelle_sfv @ ( cons @ prover2006609834e_form @ ( prover946642470e_FAll @ f ) @ s ) ) ) ) ).
thf(conj_1,hypothesis,
! [A10: set @ prover_Mirabelle_U,B5: nat > ( list @ prover_Mirabelle_U ) > $o,E6: nat > prover_Mirabelle_U] :
( ( prover1043414700is_env @ ( product_Pair @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ) @ A10 @ B5 ) @ E6 )
=> ( prover1899965912_SEval @ ( product_Pair @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ) @ A10 @ B5 ) @ E6 @ ( append @ prover2006609834e_form @ s @ ( cons @ prover2006609834e_form @ ( prover48307765_finst @ f @ u ) @ ( nil @ prover2006609834e_form ) ) ) ) ) ).
thf(conj_2,conjecture,
! [E5: nat > prover_Mirabelle_U] :
( ~ ( prover1043414700is_env @ ( product_Pair @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ) @ a @ b ) @ E5 )
| ( prover1899965912_SEval @ ( product_Pair @ ( set @ prover_Mirabelle_U ) @ ( nat > ( list @ prover_Mirabelle_U ) > $o ) @ a @ b ) @ E5 @ ( cons @ prover2006609834e_form @ ( prover946642470e_FAll @ f ) @ s ) ) ) ).
%------------------------------------------------------------------------------