TPTP Problem File: DAT205^1.p
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%------------------------------------------------------------------------------
% File : DAT205^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Sorted list operations 194
% 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__194.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 320 ( 101 unt; 46 typ; 0 def)
% Number of atoms : 802 ( 405 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4422 ( 150 ~; 17 |; 95 &;3753 @)
% ( 0 <=>; 407 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 180 ( 180 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 45 usr; 8 con; 0-4 aty)
% Number of variables : 1118 ( 22 ^; 992 !; 66 ?;1118 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:06.808
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
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_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olinorder__class_Osorted,type,
linorder_sorted:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ 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_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_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_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_l1a____,type,
l1a: list @ a ).
thf(sy_v_l2,type,
l2: list @ a ).
thf(sy_v_l2b____,type,
l2b: list @ a ).
thf(sy_v_x1____,type,
x1: a ).
thf(sy_v_x2____,type,
x2: a ).
%----Relevant facts (255)
thf(fact_0_x1__nin__l1,axiom,
~ ( member2 @ a @ x1 @ ( set2 @ a @ l1a ) ) ).
% x1_nin_l1
thf(fact_1_x2__nin__l2,axiom,
~ ( member2 @ a @ x2 @ ( set2 @ a @ l2b ) ) ).
% x2_nin_l2
thf(fact_2_ind__hyp__l2,axiom,
( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b )
= ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ).
% ind_hyp_l2
thf(fact_3_x2__le,axiom,
! [X: a] :
( ( member2 @ a @ X @ ( set2 @ a @ l2b ) )
=> ( ord_less_eq @ a @ x2 @ X ) ) ).
% x2_le
thf(fact_4_x1__le,axiom,
! [X: a] :
( ( member2 @ a @ X @ ( set2 @ a @ l1a ) )
=> ( ord_less_eq @ a @ x1 @ X ) ) ).
% x1_le
thf(fact_5_Cons_Ohyps,axiom,
( ( ( distinct @ a @ l2b )
& ( linorder_sorted @ a @ l2b ) )
=> ( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b )
= ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ) ).
% Cons.hyps
thf(fact_6_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_7_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ A2 )
=> ( member2 @ A @ X2 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_8_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_9_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_10_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_11_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_12_list_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: list @ A,A1: A] :
( ( member2 @ A @ X @ ( set2 @ A @ A22 ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_13_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_14_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member2 @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member2 @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_15_l1__OK,axiom,
( ( distinct @ a @ l1 )
& ( linorder_sorted @ a @ l1 ) ) ).
% l1_OK
thf(fact_16_l2__OK,axiom,
( ( distinct @ a @ l2 )
& ( linorder_sorted @ a @ l2 ) ) ).
% l2_OK
thf(fact_17_x1__l1__props,axiom,
( ( distinct @ a @ ( cons @ a @ x1 @ l1a ) )
& ( linorder_sorted @ a @ ( cons @ a @ x1 @ l1a ) ) ) ).
% x1_l1_props
thf(fact_18_x2__l2__props,axiom,
( ( distinct @ a @ ( cons @ a @ x2 @ l2b ) )
& ( linorder_sorted @ a @ ( cons @ a @ x2 @ l2b ) ) ) ).
% x2_l2_props
thf(fact_19_l2__props,axiom,
( ( distinct @ a @ l2b )
& ( linorder_sorted @ a @ l2b ) ) ).
% l2_props
thf(fact_20_l1__props,axiom,
( ( distinct @ a @ l1a )
& ( linorder_sorted @ a @ l1a ) ) ).
% l1_props
thf(fact_21_ind__hyp__l1,axiom,
! [L2: list @ a] :
( ( ( distinct @ a @ L2 )
& ( linorder_sorted @ a @ L2 ) )
=> ( ( sorted1061247458sorted @ a @ l1a @ L2 )
= ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ l1a ) @ ( set2 @ a @ L2 ) ) ) ) ).
% ind_hyp_l1
thf(fact_22_sorted__many__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( linorder_sorted @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted_many_eq
thf(fact_23_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_24_distinct__length__2__or__more,axiom,
! [A: $tType,A3: A,B3: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A3 @ ( cons @ A @ B3 @ Xs ) ) )
= ( ( A3 != B3 )
& ( distinct @ A @ ( cons @ A @ A3 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B3 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_25_sorted__many,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( linorder_sorted @ A @ ( cons @ A @ Y @ Zs ) )
=> ( linorder_sorted @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) ) ) ) ) ).
% sorted_many
thf(fact_26_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_27_sorted__Cons,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A] :
( ( linorder_sorted @ A @ ( cons @ A @ X @ Xs ) )
= ( ( linorder_sorted @ A @ Xs )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ).
% sorted_Cons
thf(fact_28_sorted_OCons,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,X: A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ X2 ) )
=> ( ( linorder_sorted @ A @ Xs )
=> ( linorder_sorted @ A @ ( cons @ A @ X @ Xs ) ) ) ) ) ).
% sorted.Cons
thf(fact_29_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_30_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_31_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_32_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_33_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_34_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_35_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_36_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_37_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_38_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_39_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_40_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_41_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_42_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_43_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member2 @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% 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] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_49_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_50_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_51_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_52_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_53_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_54_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_55_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_56_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_57_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_58_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_59_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z2: set @ A] : Y2 = Z2 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_60_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_61_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_62_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_63_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_64_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T: A] :
( ( member2 @ A @ T @ A5 )
=> ( member2 @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_65_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( member2 @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member2 @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_66_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_67_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_68_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( member2 @ A @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_69_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_70_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_71_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_72_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ X @ A2 )
=> ( member2 @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_73_set__mp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ X @ A2 )
=> ( member2 @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_74_list_Oset__cases,axiom,
! [A: $tType,E: A,A3: list @ A] :
( ( member2 @ A @ E @ ( set2 @ A @ A3 ) )
=> ( ! [Z22: list @ A] :
( A3
!= ( cons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A3
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member2 @ A @ E @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_75_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_76_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_77_sorted_Oinducts,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,P: ( list @ A ) > $o] :
( ( linorder_sorted @ A @ X )
=> ( ( P @ ( nil @ A ) )
=> ( ! [Xs2: list @ A,X2: A] :
( ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X2 @ Xa ) )
=> ( ( linorder_sorted @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) ) )
=> ( P @ X ) ) ) ) ) ).
% sorted.inducts
thf(fact_78_sorted_Osimps,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( linorder_sorted @ A )
= ( ^ [A6: list @ A] :
( ( A6
= ( nil @ A ) )
| ? [Xs3: list @ A,X3: A] :
( ( A6
= ( cons @ A @ X3 @ Xs3 ) )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs3 ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) )
& ( linorder_sorted @ A @ Xs3 ) ) ) ) ) ) ).
% sorted.simps
thf(fact_79_sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A3: list @ A] :
( ( linorder_sorted @ A @ A3 )
=> ( ( A3
!= ( nil @ A ) )
=> ~ ! [Xs2: list @ A,X2: A] :
( ( A3
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X2 @ Xa ) )
=> ~ ( linorder_sorted @ A @ Xs2 ) ) ) ) ) ) ).
% sorted.cases
thf(fact_80_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_81_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X3: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( member2 @ A @ X3 @ ( set2 @ A @ Xs3 ) ) @ Xs3 @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_82_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_83_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_84_in__set__member,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( member @ A @ Xs @ X ) ) ).
% in_set_member
thf(fact_85_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_86_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_87_sorted__single,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( linorder_sorted @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted_single
thf(fact_88_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_89_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_90_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_91_revg_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [A4: A,As: list @ A,B4: list @ A] :
( ( P @ As @ ( cons @ A @ A4 @ B4 ) )
=> ( P @ ( cons @ A @ A4 @ As ) @ B4 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_92_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,A4: A,As: list @ A,B4: B,Bs: list @ B] :
( ( P @ F3 @ As @ Bs )
=> ( P @ F3 @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B4 @ Bs ) ) )
=> ( ! [A4: A > B > C2,V: A,Va: list @ A] : ( P @ A4 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
=> ( ! [A4: A > B > C2,V: B,Va: list @ B] : ( P @ A4 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% zipf.induct
thf(fact_93_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 ) )
=> ( ! [E2: A,W12: list @ A,W22: list @ B] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons @ A @ E2 @ W12 ) @ W22 ) )
=> ( ! [E2: B,W13: list @ A,W23: list @ B] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons @ B @ E2 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_94_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X1: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_95_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,A4: A,As: list @ A,B4: B,Bs: list @ B] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B4 @ 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_96_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,X2: A,Xs2: list @ A,Y4: A,Ys2: list @ A] :
( ( ( R @ X2 @ Y4 )
=> ( P @ R @ Xs2 @ ( cons @ A @ Y4 @ Ys2 ) ) )
=> ( ( ~ ( R @ X2 @ Y4 )
=> ( P @ R @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 ) )
=> ( P @ R @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) ) )
=> ( ! [R: A > A > $o,Xs2: list @ A] : ( P @ R @ Xs2 @ ( 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_97_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R2: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X1: list @ B] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( R2 @ X2 @ Y4 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y4 @ Ys2 ) )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ~ ( R2 @ X2 @ Y4 )
=> ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_98_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_99_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_100_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_101_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_102_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_103_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 ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys2 ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_104_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 ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_105_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y4: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_106_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y4: A,Xs2: list @ A] :
( ( ( X2 = Y4 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_107_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_108_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,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F3 @ As @ ( cons @ B @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_109_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_110_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_111_sorted_ONil,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_sorted @ A @ ( nil @ A ) ) ) ).
% sorted.Nil
thf(fact_112_subset__sorted_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: list @ A] : ( sorted1061247458sorted @ A @ ( nil @ A ) @ L2 ) ) ).
% subset_sorted.simps(1)
thf(fact_113_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_114_subset__sorted_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X12: A,L1: list @ A] :
~ ( sorted1061247458sorted @ A @ ( cons @ A @ X12 @ L1 ) @ ( nil @ A ) ) ) ).
% subset_sorted.simps(2)
thf(fact_115_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A,D: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ( C = D )
=> ( ord_less_eq @ A @ A3 @ D ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_116_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_117_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N2: A,R22: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N2 @ R22 ) )
=> ( ( N1 = N2 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
thf(fact_118_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_119_quicksort_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ Xs2 ) ) ) ) ).
% quicksort.cases
thf(fact_120_remove__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( insert @ A @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_121_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Y: A] :
( ( linorder_sorted @ A @ Xs )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ Y ) )
=> ( linorder_sorted @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_122_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_123_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( linorder_sorted @ A @ Xs )
=> ! [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X4 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_124_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_125_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_126_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_127_member__remove,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( member2 @ A @ X @ ( remove @ A @ Y @ A2 ) )
= ( ( member2 @ A @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_128_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_129_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_130_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_131_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_132_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_133_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_134_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_135_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_136_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L1: list @ A,E12: A,E23: A,L2: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
= ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_137_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list @ A,E12: A,E23: A,L2: list @ A,E1: A,E22: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_138_list__se__match_I1_J,axiom,
! [A: $tType,L1: list @ A,L2: list @ A,A3: A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L1
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_139_list__se__match_I2_J,axiom,
! [A: $tType,L2: list @ A,L1: list @ A,A3: A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_140_list__se__match_I3_J,axiom,
! [A: $tType,L1: list @ A,A3: A,L2: list @ A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_141_list__se__match_I4_J,axiom,
! [A: $tType,L2: list @ A,A3: A,L1: list @ A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_142_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E: A,L1: list @ A,E3: A,L2: list @ A] :
( ( ( cons @ A @ E @ ( nil @ A ) )
= ( append @ A @ L1 @ ( cons @ A @ E3 @ L2 ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E3 = E )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_143_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list @ A,E3: A,L2: list @ A,E: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E3 @ L2 ) )
= ( cons @ A @ E @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E3 = E )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_144_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_145_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_146_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps: list @ A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs4 ) )
& ( Ys
= ( append @ A @ Ps @ Ys4 ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( hd @ A @ Xs4 )
!= ( hd @ A @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_147_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 ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_148_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_149_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_150_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_151_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_152_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_153_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_154_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_155_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_156_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y4: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_157_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_158_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_159_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_160_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_161_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_162_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 ) )
=> ( ! [X2: A,Y4: B] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Y4: B,Ys2: list @ B] :
( ( Ys2
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_163_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X3: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_164_list__rev__decomp,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ? [Ll: list @ A,E2: A] :
( L
= ( append @ A @ Ll @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% list_rev_decomp
thf(fact_165_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 ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_166_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_167_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list @ A,X3: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_168_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_169_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_170_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_171_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X2: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_172_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_173_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X2: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_174_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_175_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_176_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_177_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_178_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_179_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_180_in__set__list__format,axiom,
! [A: $tType,E: A,L: list @ A] :
( ( member2 @ A @ E @ ( set2 @ A @ L ) )
=> ~ ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ E @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_181_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y: A] :
( ( member2 @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member2 @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X = Y )
=> ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L12: list @ A,L22: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L3 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L12: list @ A,L22: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_182_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_183_list_Oset__sel_I1_J,axiom,
! [A: $tType,A3: list @ A] :
( ( A3
!= ( nil @ A ) )
=> ( member2 @ A @ ( hd @ A @ A3 ) @ ( set2 @ A @ A3 ) ) ) ).
% list.set_sel(1)
thf(fact_184_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member2 @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_185_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs3 ) )
& ( P3 @ X3 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_186_not__distinct__decomp,axiom,
! [A: $tType,Ws: list @ A] :
( ~ ( distinct @ A @ Ws )
=> ? [Xs2: list @ A,Ys2: list @ A,Zs3: list @ A,Y4: A] :
( Ws
= ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ ( append @ A @ Ys2 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_187_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_188_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list @ A] :
( ( ~ ( distinct @ A @ As2 ) )
= ( ? [Xs3: list @ A,Y3: A,Ys3: list @ A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs3 ) )
& ( distinct @ A @ Xs3 )
& ( As2
= ( append @ A @ Xs3 @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_189_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y4: A,Ys2: list @ A] :
( ( distinct @ A @ Ys2 )
=> ( ( member2 @ A @ Y4 @ ( set2 @ A @ Ys2 ) )
=> ! [Zs3: list @ A] :
( Xs
!= ( append @ A @ Ys2 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_190_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_191_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_192_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F )
= ( append @ A @ ( F @ X ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
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_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_195_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P3: A > $o,A5: set @ A] :
? [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( P3 @ X3 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ A5 )
& ( P3 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_196_distinct__product__lists,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ! [X2: list @ A] :
( ( member2 @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( distinct @ A @ X2 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_197_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F: A > ( list @ B ),G2: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( bind @ A @ B @ Xs @ F )
= ( bind @ A @ B @ Ys @ G2 ) ) ) ) ).
% list_bind_cong
thf(fact_198_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( append @ A @ ( F @ X ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_199_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_200_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_201_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_202_hd__last__singletonI,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( hd @ A @ Xs )
= ( last @ A @ Xs ) )
=> ( ( distinct @ A @ Xs )
=> ( Xs
= ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ) ).
% hd_last_singletonI
thf(fact_203_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_204_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_205_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_206_Misc_Olast__in__set,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( member2 @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% Misc.last_in_set
thf(fact_207_List_Olast__in__set,axiom,
! [A: $tType,As2: list @ A] :
( ( As2
!= ( nil @ A ) )
=> ( member2 @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% List.last_in_set
thf(fact_208_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_209_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_210_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_211_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_212_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys
= ( append @ A @ Ys4 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_213_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_214_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_215_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_216_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_217_sorted__hd__last,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L: list @ A] :
( ( linorder_sorted @ A @ L )
=> ( ( L
!= ( nil @ A ) )
=> ( ord_less_eq @ A @ ( hd @ A @ L ) @ ( last @ A @ L ) ) ) ) ) ).
% sorted_hd_last
thf(fact_218_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_219_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_220_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_221_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_222_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_223_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_224_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X
= ( hd @ A @ L ) )
| ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_225_last__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( ( tl @ A @ Xs )
!= ( nil @ A ) ) )
=> ( ( last @ A @ ( tl @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_tl
thf(fact_226_tl__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( last @ A @ ( tl @ A @ Xs ) ) ) ) ).
% tl_last
thf(fact_227_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E2: A] :
( Xs
!= ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_228_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_229_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_230_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_231_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_232_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_233_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_234_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_235_sorted__tl,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ( linorder_sorted @ A @ Xs )
=> ( linorder_sorted @ A @ ( tl @ A @ Xs ) ) ) ) ).
% sorted_tl
thf(fact_236_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_237_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_238_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( linorder_sorted @ A @ Xs )
=> ( linorder_sorted @ A @ ( butlast @ A @ Xs ) ) ) ) ) ).
% sorted_butlast
thf(fact_239_in__set__tlD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_tlD
thf(fact_240_List_Odistinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% List.distinct_tl
thf(fact_241_in__set__butlastD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_242_distinct__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% distinct_butlast
thf(fact_243_in__set__butlast__appendI,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_244_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( X
!= ( hd @ A @ Xs ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_245_Misc_Odistinct__tl,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( ( distinct @ A @ L )
=> ( distinct @ A @ ( tl @ A @ L ) ) ) ) ).
% Misc.distinct_tl
thf(fact_246_list_Oset__sel_I2_J,axiom,
! [A: $tType,A3: list @ A,X: A] :
( ( A3
!= ( nil @ A ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ A3 ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ A3 ) ) ) ) ).
% list.set_sel(2)
thf(fact_247_tl__subset,axiom,
! [A: $tType,Xs: list @ A,A2: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A2 ) ) ) ).
% tl_subset
thf(fact_248_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_249_distinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( X
= ( hd @ A @ Xs ) )
=> ~ ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% distinct_hd_tl
thf(fact_250_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A2: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A2 ) ) ) ).
% butlast_subset
thf(fact_251_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
=> ~ ! [Xl: A] :
( L
!= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_252_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
= ( ? [Xl2: A] :
( L
= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_253_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,X: A] :
( ( Ys
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast'
thf(fact_254_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
%----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 (10)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $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,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type2 @ a ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1a ) @ ( cons @ a @ x2 @ l2b ) )
= ( ord_less_eq @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ ( cons @ a @ x2 @ l2b ) ) ) ) ).
%------------------------------------------------------------------------------