TPTP Problem File: DAT204^1.p
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%------------------------------------------------------------------------------
% File : DAT204^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Sorted list operations 166
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Lam09] Lammich (2009), Collections Framework
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : sorted_list_operations__166.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 343 ( 124 unt; 52 typ; 0 def)
% Number of atoms : 809 ( 353 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4330 ( 132 ~; 9 |; 90 &;3693 @)
% ( 0 <=>; 406 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 209 ( 209 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 50 usr; 4 con; 0-4 aty)
% Number of variables : 1162 ( 25 ^;1027 !; 63 ?;1162 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:57.799
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
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 ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (47)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $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_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( 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_Ocan__select,type,
can_select:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olinorder__class_Osorted,type,
linorder_sorted:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord467138063of_set:
!>[A: $tType] : ( ( set @ 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_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ 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_Osublist,type,
sublist:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List__More_Ocombinatorial__product,type,
list_c659805718roduct:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted,type,
sorted1061247458sorted:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_l1,type,
l1: list @ a ).
thf(sy_v_l2,type,
l2: list @ a ).
%----Relevant facts (255)
thf(fact_0_l1__OK,axiom,
( ( distinct @ a @ l1 )
& ( linorder_sorted @ a @ l1 ) ) ).
% l1_OK
thf(fact_1_l2__OK,axiom,
( ( distinct @ a @ l2 )
& ( linorder_sorted @ a @ l2 ) ) ).
% l2_OK
thf(fact_2_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member2 @ A @ X @ A2 )
=> ( member2 @ A @ X @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_3_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_4_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_5_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_6_set__mp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ X2 @ A2 )
=> ( member2 @ A @ X2 @ B2 ) ) ) ).
% set_mp
thf(fact_7_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ X2 @ A2 )
=> ( member2 @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_8_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ C @ A2 )
=> ( member2 @ A @ C @ B2 ) ) ) ).
% subsetD
thf(fact_9_subsetCE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ C @ A2 )
=> ( member2 @ A @ C @ B2 ) ) ) ).
% subsetCE
thf(fact_10_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_11_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [X3: A] :
( ( member2 @ A @ X3 @ A3 )
=> ( member2 @ A @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_12_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_13_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_14_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( linorder_sorted @ A @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( linorder_sorted @ A @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_15_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_16_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_17_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_18_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_19_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_20_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_21_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_22_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_23_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_24_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_25_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% order.trans
thf(fact_26_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_27_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_28_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_29_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_30_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_31_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_32_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_33_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_34_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_35_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_36_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_37_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funE
thf(fact_38_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funD
thf(fact_39_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_40_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member2 @ A @ C @ B2 )
=> ~ ( member2 @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_41_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z2: set @ A] : Y2 = Z2 )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_42_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_43_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_44_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member2 @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_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_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G2 @ X ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member2 @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member2 @ A @ C @ B2 ) ) ) ).
% rev_subsetD
thf(fact_50_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [T: A] :
( ( member2 @ A @ T @ A3 )
=> ( member2 @ A @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_51_set__rev__mp,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ( member2 @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member2 @ A @ X2 @ B2 ) ) ) ).
% set_rev_mp
thf(fact_52_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_53_mergesort__remdups__correct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L: list @ A] :
( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
& ( linorder_sorted @ A @ ( mergesort_remdups @ A @ L ) )
& ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
= ( set2 @ A @ L ) ) ) ) ).
% mergesort_remdups_correct
thf(fact_54_subset__code_I2_J,axiom,
! [B: $tType,A2: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A2 @ ( coset @ B @ Ys ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ ( set2 @ B @ Ys ) )
=> ~ ( member2 @ B @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_55_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ S )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_56_in__set__member,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( member @ A @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_57_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [X: list @ A] :
( ( ( set2 @ A @ X )
= A2 )
& ( linorder_sorted @ A @ X )
& ( distinct @ A @ X )
& ! [Y5: list @ A] :
( ( ( ( set2 @ A @ Y5 )
= A2 )
& ( linorder_sorted @ A @ Y5 )
& ( distinct @ A @ Y5 ) )
=> ( Y5 = X ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_58_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ( C = D )
=> ( ord_less_eq @ A @ A4 @ D ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_59_sorted__Cons,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Xs: list @ A] :
( ( linorder_sorted @ A @ ( cons @ A @ X2 @ Xs ) )
= ( ( linorder_sorted @ A @ Xs )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ X3 ) ) ) ) ) ).
% sorted_Cons
thf(fact_60_sorted_OCons,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,X2: A] :
( ! [X: A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ X ) )
=> ( ( linorder_sorted @ A @ Xs )
=> ( linorder_sorted @ A @ ( cons @ A @ X2 @ Xs ) ) ) ) ) ).
% sorted.Cons
thf(fact_61_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( linorder_sorted @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( linorder_sorted @ A @ Xs )
& ( linorder_sorted @ A @ Ys )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ).
% sorted_append
thf(fact_62_set__sublist__subset,axiom,
! [A: $tType,Xs: list @ A,I: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( sublist @ A @ Xs @ I ) ) @ ( set2 @ A @ Xs ) ) ).
% set_sublist_subset
thf(fact_63_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_64_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_65_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_66_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_67_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_68_sorted__many__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Zs: list @ A] :
( ( linorder_sorted @ A @ ( cons @ A @ X2 @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X2 @ Y )
& ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted_many_eq
thf(fact_69_member__rec_I1_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_70_not__Cons__self2,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( cons @ A @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_71_distinct__match,axiom,
! [A: $tType,Al: list @ A,E: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_72_Cons__eq__appendI,axiom,
! [A: $tType,X2: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_73_xy__in__set__cases,axiom,
! [A: $tType,X2: A,L: list @ A,Y: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ L ) )
=> ( ( member2 @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X2 = Y )
=> ! [L1: list @ A,L2: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ L2 ) ) ) )
=> ( ( ( X2 != Y )
=> ! [L1: list @ A,L2: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ X2 @ ( append @ A @ L2 @ ( cons @ A @ Y @ L3 ) ) ) ) ) )
=> ~ ( ( X2 != Y )
=> ! [L1: list @ A,L2: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ ( append @ A @ L2 @ ( cons @ A @ X2 @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_74_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_75_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_76_in__set__list__format,axiom,
! [A: $tType,E: A,L: list @ A] :
( ( member2 @ A @ E @ ( set2 @ A @ L ) )
=> ~ ! [L1: list @ A,L2: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ E @ L2 ) ) ) ) ).
% in_set_list_format
thf(fact_77_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_78_list__match__lel__lel,axiom,
! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
=> ( ! [C21: list @ A] :
( ( C1
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C23
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list @ A] :
( ( C12
= ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C22
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_79_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N2: A,R2: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N2 @ R2 ) )
=> ( ( N1 = N2 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
thf(fact_80_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_81_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list @ A,X3: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_82_in__set__conv__decomp__first,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs2 ) ) )
& ~ ( member2 @ A @ X2 @ ( set2 @ A @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_83_in__set__conv__decomp__last,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs2 ) ) )
& ~ ( member2 @ A @ X2 @ ( set2 @ A @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_84_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list @ A,X: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_85_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list @ A,X: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_86_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list @ A,X: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_87_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list @ A,X: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_88_in__set__conv__decomp,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_89_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list @ A,X: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_90_split__list__first,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ~ ( member2 @ A @ X2 @ ( set2 @ A @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_91_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list @ A,X: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_92_split__list__last,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ~ ( member2 @ A @ X2 @ ( set2 @ A @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_93_split__list,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_94_not__distinct__conv__prefix,axiom,
! [A: $tType,As: list @ A] :
( ( ~ ( distinct @ A @ As ) )
= ( ? [Xs2: list @ A,Y3: A,Ys2: list @ A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
& ( distinct @ A @ Xs2 )
& ( As
= ( append @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_95_finite__list,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_96_list_Oset__intros_I2_J,axiom,
! [A: $tType,X2: A,A22: list @ A,A1: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ A22 ) )
=> ( member2 @ A @ X2 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_97_list_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: list @ A] : ( member2 @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% list.set_intros(1)
thf(fact_98_set__ConsD,axiom,
! [A: $tType,Y: A,X2: A,Xs: list @ A] :
( ( member2 @ A @ Y @ ( set2 @ A @ ( cons @ A @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member2 @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_99_list_Oset__cases,axiom,
! [A: $tType,E: A,A4: list @ A] :
( ( member2 @ A @ E @ ( set2 @ A @ A4 ) )
=> ( ! [Z22: list @ A] :
( A4
!= ( cons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A4
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member2 @ A @ E @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_100_distinct__length__2__or__more,axiom,
! [A: $tType,A4: A,B4: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A4 @ ( cons @ A @ B4 @ Xs ) ) )
= ( ( A4 != B4 )
& ( distinct @ A @ ( cons @ A @ A4 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B4 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_101_in__set__sublistD,axiom,
! [A: $tType,X2: A,Xs: list @ A,I: set @ nat] :
( ( member2 @ A @ X2 @ ( set2 @ A @ ( sublist @ A @ Xs @ I ) ) )
=> ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_sublistD
thf(fact_102_notin__set__sublistI,axiom,
! [A: $tType,X2: A,Xs: list @ A,I: set @ nat] :
( ~ ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ~ ( member2 @ A @ X2 @ ( set2 @ A @ ( sublist @ A @ Xs @ I ) ) ) ) ).
% notin_set_sublistI
thf(fact_103_distinct__sublistI,axiom,
! [A: $tType,Xs: list @ A,I: set @ nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( sublist @ A @ Xs @ I ) ) ) ).
% distinct_sublistI
thf(fact_104_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X2: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_105_sorted__many,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Zs: list @ A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) )
=> ( linorder_sorted @ A @ ( cons @ A @ X2 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ) ).
% sorted_many
thf(fact_106_finite__distinct__list,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ? [Xs3: list @ A] :
( ( ( set2 @ A @ Xs3 )
= A2 )
& ( distinct @ A @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_107_distinct_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X2 @ Xs ) )
= ( ~ ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_108_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A3: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_109_finite__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% finite_subset
thf(fact_110_infinite__super,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_111_rev__finite__subset,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( finite_finite2 @ A @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_112_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_113_sorted__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( set2 @ A @ ( linord467138063of_set @ A @ A2 ) )
= A2 )
& ( linorder_sorted @ A @ ( linord467138063of_set @ A @ A2 ) )
& ( distinct @ A @ ( linord467138063of_set @ A @ A2 ) ) ) ) ) ).
% sorted_list_of_set
thf(fact_114_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Y: A] :
( ( linorder_sorted @ A @ Xs )
=> ( ! [X: A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ Y ) )
=> ( linorder_sorted @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_115_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_116_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_117_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_118_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_119_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_120_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_121_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_122_sublist__nil,axiom,
! [A: $tType,A2: set @ nat] :
( ( sublist @ A @ ( nil @ A ) @ A2 )
= ( nil @ A ) ) ).
% sublist_nil
thf(fact_123_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_124_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list @ A,E2: A,L22: list @ A,E: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E2 @ L22 ) )
= ( cons @ A @ E @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E2 = E )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_125_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E: A,L12: list @ A,E2: A,L22: list @ A] :
( ( ( cons @ A @ E @ ( nil @ A ) )
= ( append @ A @ L12 @ ( cons @ A @ E2 @ L22 ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E2 = E )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_126_list__se__match_I4_J,axiom,
! [A: $tType,L22: list @ A,A4: A,L12: list @ A] :
( ( L22
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L12 @ L22 ) )
= ( ( L12
= ( nil @ A ) )
& ( L22
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_127_list__se__match_I3_J,axiom,
! [A: $tType,L12: list @ A,A4: A,L22: list @ A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L12 @ L22 ) )
= ( ( L12
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L22
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_128_list__se__match_I2_J,axiom,
! [A: $tType,L22: list @ A,L12: list @ A,A4: A] :
( ( L22
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L22 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( L22
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_129_list__se__match_I1_J,axiom,
! [A: $tType,L12: list @ A,L22: list @ A,A4: A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L22 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L12
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L22
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_130_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list @ A,E1: A,E22: A,L22: list @ A,E12: A,E23: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E1 @ ( cons @ A @ E22 @ L22 ) ) )
= ( cons @ A @ E12 @ ( cons @ A @ E23 @ ( nil @ A ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_131_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E12: A,E23: A,L12: list @ A,E1: A,E22: A,L22: list @ A] :
( ( ( cons @ A @ E12 @ ( cons @ A @ E23 @ ( nil @ A ) ) )
= ( append @ A @ L12 @ ( cons @ A @ E1 @ ( cons @ A @ E22 @ L22 ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_132_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_133_sorted__single,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( linorder_sorted @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% sorted_single
thf(fact_134_sorted__list__of__set_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ~ ( finite_finite2 @ A @ A2 )
=> ( ( linord467138063of_set @ A @ A2 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.infinite
thf(fact_135_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,Xs3: list @ A,Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_136_revg_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [A5: A,As2: list @ A,B5: list @ A] :
( ( P @ As2 @ ( cons @ A @ A5 @ B5 ) )
=> ( P @ ( cons @ A @ A5 @ As2 ) @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_137_zipf_Oinduct,axiom,
! [A: $tType,C2: $tType,B: $tType,P: ( A > B > C2 ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > C2,A1: list @ A,A22: list @ B] :
( ! [F3: A > B > C2] : ( P @ F3 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [F3: A > B > C2,A5: A,As2: list @ A,B5: B,Bs: list @ B] :
( ( P @ F3 @ As2 @ Bs )
=> ( P @ F3 @ ( cons @ A @ A5 @ As2 ) @ ( cons @ B @ B5 @ Bs ) ) )
=> ( ! [A5: A > B > C2,V: A,Va: list @ A] : ( P @ A5 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
=> ( ! [A5: A > B > C2,V: B,Va: list @ B] : ( P @ A5 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% zipf.induct
thf(fact_138_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E3: A,W12: list @ A,W22: list @ B] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons @ A @ E3 @ W12 ) @ W22 ) )
=> ( ! [E3: B,W13: list @ A,W23: list @ B] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons @ B @ E3 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_139_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X1: A,X23: A,Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_140_list__all__zip_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > $o,A1: list @ A,A22: list @ B] :
( ! [P2: A > B > $o] : ( P @ P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,A5: A,As2: list @ A,B5: B,Bs: list @ B] :
( ( P @ P2 @ As2 @ Bs )
=> ( P @ P2 @ ( cons @ A @ A5 @ As2 ) @ ( cons @ B @ B5 @ Bs ) ) )
=> ( ! [P2: A > B > $o,V: A,Va: list @ A] : ( P @ P2 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,V: B,Va: list @ B] : ( P @ P2 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_141_mergesort__by__rel__merge_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A,A22: list @ A] :
( ! [R: A > A > $o,X: A,Xs3: list @ A,Y4: A,Ys3: list @ A] :
( ( ( R @ X @ Y4 )
=> ( P @ R @ Xs3 @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ( ~ ( R @ X @ Y4 )
=> ( P @ R @ ( cons @ A @ X @ Xs3 ) @ Ys3 ) )
=> ( P @ R @ ( cons @ A @ X @ Xs3 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) )
=> ( ! [R: A > A > $o,Xs3: list @ A] : ( P @ R @ Xs3 @ ( nil @ A ) )
=> ( ! [R: A > A > $o,V: A,Va: list @ A] : ( P @ R @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_142_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R3: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ B ) )
=> ( ! [X1: list @ B] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [X: A,Xs3: list @ A,Y4: B,Ys3: list @ B] :
( ( R3 @ X @ Y4 )
=> ( ( P @ Xs3 @ ( cons @ B @ Y4 @ Ys3 ) )
=> ( P @ ( cons @ A @ X @ Xs3 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
=> ( ! [X: A,Xs3: list @ A,Y4: B,Ys3: list @ B] :
( ~ ( R3 @ X @ Y4 )
=> ( ( P @ ( cons @ A @ X @ Xs3 ) @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs3 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_143_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F3: A > B,X1: list @ B] : ( P @ F3 @ ( nil @ A ) @ X1 )
=> ( ! [F3: A > B,A5: A,As2: list @ A,Bs: list @ B] :
( ( P @ F3 @ As2 @ ( cons @ B @ ( F3 @ A5 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A5 @ As2 ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_144_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,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_145_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,Y4: A,Xs3: list @ A] :
( ( ( X = Y4 )
=> ( P @ ( cons @ A @ X @ Xs3 ) ) )
=> ( ( ( X != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs3 ) ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_146_remdups__adj_Ocases,axiom,
! [A: $tType,X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ( ! [X: A] :
( X2
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y4: A,Xs3: list @ A] :
( X2
!= ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_147_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X: A,Xs3: list @ A,Y4: A,Ys3: list @ A] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs3 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_148_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,Xs3: list @ A] : ( P @ ( cons @ A @ X @ Xs3 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys3 ) )
=> ( ! [X: A,Xs3: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs3 ) @ ( cons @ B @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_149_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys2: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_150_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_151_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_152_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_153_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_154_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_155_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_156_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_157_sorted_ONil,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_sorted @ A @ ( nil @ A ) ) ) ).
% sorted.Nil
thf(fact_158_subset__sorted_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L22: list @ A] : ( sorted1061247458sorted @ A @ ( nil @ A ) @ L22 ) ) ).
% subset_sorted.simps(1)
thf(fact_159_distinct__sorted__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] : ( distinct @ A @ ( linord467138063of_set @ A @ A2 ) ) ) ).
% distinct_sorted_list_of_set
thf(fact_160_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_161_rev__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,Xs3: list @ A] : ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X: A,Xs3: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_162_list__rev__decomp,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ? [Ll: list @ A,E3: A] :
( L
= ( append @ A @ Ll @ ( cons @ A @ E3 @ ( nil @ A ) ) ) ) ) ).
% list_rev_decomp
thf(fact_163_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs2: list @ A,X3: A] :
( L
= ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_164_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ! [X: A,Y4: B] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) )
=> ( ! [X: A,Xs3: list @ A,Y4: B] :
( ( Xs3
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X: A,Y4: B,Ys3: list @ B] :
( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X @ ( nil @ A ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( ! [X: A,Xs3: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs3 @ Ys3 )
=> ( ( Xs3
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_165_list__Cons__eq__append__cases,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X2 @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_166_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X2: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X2 @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X2 @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_167_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,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_168_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X2: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X2 @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X2 @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_169_Cons__eq__append__conv,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_170_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y4: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_171_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_172_distinct__singleton,axiom,
! [A: $tType,X2: A] : ( distinct @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_173_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F: A > ( list @ B ),G2: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( bind @ A @ B @ Xs @ F )
= ( bind @ A @ B @ Ys @ G2 ) ) ) ) ).
% list_bind_cong
thf(fact_174_subset__sorted_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X12: A,L12: list @ A] :
~ ( sorted1061247458sorted @ A @ ( cons @ A @ X12 @ L12 ) @ ( nil @ A ) ) ) ).
% subset_sorted.simps(2)
thf(fact_175_finite__set__choice,axiom,
! [B: $tType,A: $tType,A2: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A2 )
=> ( ! [X: A] :
( ( member2 @ A @ X @ A2 )
=> ? [X13: B] : ( P @ X @ X13 ) )
=> ? [F3: A > B] :
! [X4: A] :
( ( member2 @ A @ X4 @ A2 )
=> ( P @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_176_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] : ( finite_finite2 @ A @ A2 ) ) ).
% finite
thf(fact_177_not__distinct__decomp,axiom,
! [A: $tType,Ws: list @ A] :
( ~ ( distinct @ A @ Ws )
=> ? [Xs3: list @ A,Ys3: list @ A,Zs3: list @ A,Y4: A] :
( Ws
= ( append @ A @ Xs3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_178_subset__code_I3_J,axiom,
! [C2: $tType] :
~ ( ord_less_eq @ ( set @ C2 ) @ ( coset @ C2 @ ( nil @ C2 ) ) @ ( set2 @ C2 @ ( nil @ C2 ) ) ) ).
% subset_code(3)
thf(fact_179_sorted_Oinducts,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: list @ A,P: ( list @ A ) > $o] :
( ( linorder_sorted @ A @ X2 )
=> ( ( P @ ( nil @ A ) )
=> ( ! [Xs3: list @ A,X: A] :
( ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Xs3 ) )
=> ( ord_less_eq @ A @ X @ Xa ) )
=> ( ( linorder_sorted @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X @ Xs3 ) ) ) ) )
=> ( P @ X2 ) ) ) ) ) ).
% sorted.inducts
thf(fact_180_sorted_Osimps,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( linorder_sorted @ A )
= ( ^ [A6: list @ A] :
( ( A6
= ( nil @ A ) )
| ? [Xs2: list @ A,X3: A] :
( ( A6
= ( cons @ A @ X3 @ Xs2 ) )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) )
& ( linorder_sorted @ A @ Xs2 ) ) ) ) ) ) ).
% sorted.simps
thf(fact_181_sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: list @ A] :
( ( linorder_sorted @ A @ A4 )
=> ( ( A4
!= ( nil @ A ) )
=> ~ ! [Xs3: list @ A,X: A] :
( ( A4
= ( cons @ A @ X @ Xs3 ) )
=> ( ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Xs3 ) )
=> ( ord_less_eq @ A @ X @ Xa ) )
=> ~ ( linorder_sorted @ A @ Xs3 ) ) ) ) ) ) ).
% sorted.cases
thf(fact_182_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y4: A,Ys3: list @ A] :
( ( distinct @ A @ Ys3 )
=> ( ( member2 @ A @ Y4 @ ( set2 @ A @ Ys3 ) )
=> ! [Zs3: list @ A] :
( Xs
!= ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_183_the__elem__set,axiom,
! [A: $tType,X2: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= X2 ) ).
% the_elem_set
thf(fact_184_quicksort_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ~ ! [X: A,Xs3: list @ A] :
( X2
!= ( cons @ A @ X @ Xs3 ) ) ) ) ).
% quicksort.cases
thf(fact_185_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_186_not__in__set__insert,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ~ ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X2 @ Xs )
= ( cons @ A @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_187_in__set__insert,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_188_distinct__insert,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X2 @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_189_insert__Nil,axiom,
! [A: $tType,X2: A] :
( ( insert @ A @ X2 @ ( nil @ A ) )
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_190_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_191_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X3: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) ) @ Xs2 @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_192_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_193_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_194_distinct__product__lists,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ! [X: list @ A] :
( ( member2 @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( distinct @ A @ X ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_195_remove__code_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( remove @ A @ X2 @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( insert @ A @ X2 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_196_combinatorial__product_Osimps_I1_J,axiom,
! [A: $tType] :
( ( list_c659805718roduct @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% combinatorial_product.simps(1)
thf(fact_197_member__remove,axiom,
! [A: $tType,X2: A,Y: A,A2: set @ A] :
( ( member2 @ A @ X2 @ ( remove @ A @ Y @ A2 ) )
= ( ( member2 @ A @ X2 @ A2 )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_198_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_199_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_200_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_201_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_202_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_203_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_204_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P3: A > $o,Xs2: list @ A] :
? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs2 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_205_can__select__set__list__ex1,axiom,
! [A: $tType,P: A > $o,A2: list @ A] :
( ( can_select @ A @ P @ ( set2 @ A @ A2 ) )
= ( list_ex1 @ A @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_206_sublist__singleton,axiom,
! [A: $tType,A2: set @ nat,X2: A] :
( ( ( member2 @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ( sublist @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ A2 )
= ( cons @ A @ X2 @ ( nil @ A ) ) ) )
& ( ~ ( member2 @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ( sublist @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ A2 )
= ( nil @ A ) ) ) ) ).
% sublist_singleton
thf(fact_207_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P3: A > $o,A3: set @ A] :
? [X3: A] :
( ( member2 @ A @ X3 @ A3 )
& ( P3 @ X3 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ A3 )
& ( P3 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_208_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_209_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_210_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_211_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_212_distinct__n__lists,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_213_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_214_count__notin,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ~ ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X2 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_215_sorted__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( ( linord467138063of_set @ A @ A2 )
= ( nil @ A ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set_eq_Nil_iff
thf(fact_216_bot__apply,axiom,
! [C2: $tType,D2: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D2 > C2 ) )
= ( ^ [X3: D2] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_217_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_218_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_219_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X3: A] :
~ ( member2 @ A @ X3 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_220_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member2 @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_221_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_222_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_223_sublist__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( sublist @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% sublist_empty
thf(fact_224_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_225_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_226_sorted__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( linord467138063of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set_empty
thf(fact_227_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_228_memb__imp__not__empty,axiom,
! [A: $tType,X2: A,S: set @ A] :
( ( member2 @ A @ X2 @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_229_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X: A] :
~ ( member2 @ A @ X @ S ) ) ).
% set_notEmptyE
thf(fact_230_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_231_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X3: A] : ( member2 @ A @ X3 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_232_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y4: A] :
~ ( member2 @ A @ Y4 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_233_equals0D,axiom,
! [A: $tType,A2: set @ A,A4: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member2 @ A @ A4 @ A2 ) ) ).
% equals0D
thf(fact_234_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_235_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% bot.extremum
thf(fact_236_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
= ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_237_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
=> ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_238_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_239_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_240_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_241_subset__emptyI,axiom,
! [A: $tType,A2: set @ A] :
( ! [X: A] :
~ ( member2 @ A @ X @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_242_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_243_IntI,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member2 @ A @ C @ A2 )
=> ( ( member2 @ A @ C @ B2 )
=> ( member2 @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_244_Int__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member2 @ A @ C @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member2 @ A @ C @ A2 )
& ( member2 @ A @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_245_finite__Int,axiom,
! [A: $tType,F4: set @ A,G3: set @ A] :
( ( ( finite_finite2 @ A @ F4 )
| ( finite_finite2 @ A @ G3 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F4 @ G3 ) ) ) ).
% finite_Int
thf(fact_246_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A2 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_247_disjoint__mono,axiom,
! [A: $tType,A4: set @ A,A7: set @ A,B4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ A7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A7 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_248_disjointI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X: A] :
( ( member2 @ A @ X @ A4 )
=> ~ ( member2 @ A @ X @ B4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_249_disjoint__iff__not__equal,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A2 )
=> ! [Y3: A] :
( ( member2 @ A @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_250_Int__empty__right,axiom,
! [A: $tType,A2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_251_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_252_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_253_Int__emptyI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member2 @ A @ X @ A2 )
=> ~ ( member2 @ A @ X @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_254_Int__Collect__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ! [X: A] :
( ( member2 @ A @ X @ A2 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A2 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
%----Subclasses (4)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( type @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ord @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( order @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( preorder @ A @ ( type2 @ A ) ) ) ).
%----Type constructors (27)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 @ ( type2 @ A9 ) )
=> ( order_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_12,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_14,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_15,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
bot @ $o @ ( type2 @ $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 ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type2 @ a ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( sorted1061247458sorted @ a @ l1 @ l2 )
= ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ l1 ) @ ( set2 @ a @ l2 ) ) ) ).
%------------------------------------------------------------------------------