TPTP Problem File: DAT203^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT203^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Sorted list operations 132
% 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__132.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 326 ( 58 unt; 49 typ; 0 def)
% Number of atoms : 906 ( 210 equ; 0 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 4321 ( 113 ~; 19 |; 64 &;3567 @)
% ( 0 <=>; 558 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 195 ( 195 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 48 usr; 8 con; 0-4 aty)
% Number of variables : 1137 ( 46 ^;1016 !; 36 ?;1137 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:48.668
%------------------------------------------------------------------------------
%----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 (45)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_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_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_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( 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,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
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_Odiff__sorted,type,
sorted1267110213sorted:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_member,type,
member:
!>[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_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_x1__l1__props,axiom,
( ( distinct @ a @ ( cons @ a @ x1 @ l1a ) )
& ( linorder_sorted @ a @ ( cons @ a @ x1 @ l1a ) ) ) ).
% x1_l1_props
thf(fact_3_x2__l2__props,axiom,
( ( distinct @ a @ ( cons @ a @ x2 @ l2b ) )
& ( linorder_sorted @ a @ ( cons @ a @ x2 @ l2b ) ) ) ).
% x2_l2_props
thf(fact_4_l1__props,axiom,
( ( distinct @ a @ l1a )
& ( linorder_sorted @ a @ l1a ) ) ).
% l1_props
thf(fact_5_l2__props,axiom,
( ( distinct @ a @ l2b )
& ( linorder_sorted @ a @ l2b ) ) ).
% l2_props
thf(fact_6_x1__nin__l1,axiom,
~ ( member @ a @ x1 @ ( set2 @ a @ l1a ) ) ).
% x1_nin_l1
thf(fact_7_x2__nin__l2,axiom,
~ ( member @ a @ x2 @ ( set2 @ a @ l2b ) ) ).
% x2_nin_l2
thf(fact_8_Cons_Ohyps,axiom,
( ( ( distinct @ a @ l2b )
& ( linorder_sorted @ a @ l2b ) )
=> ( ( distinct @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b ) )
& ( linorder_sorted @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b ) )
& ( ( set2 @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b ) )
= ( minus_minus @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ) ) ).
% Cons.hyps
thf(fact_9_ind__hyp__l1,axiom,
! [L2: list @ a] :
( ( ( distinct @ a @ L2 )
& ( linorder_sorted @ a @ L2 ) )
=> ( ( distinct @ a @ ( sorted1267110213sorted @ a @ l1a @ L2 ) )
& ( linorder_sorted @ a @ ( sorted1267110213sorted @ a @ l1a @ L2 ) )
& ( ( set2 @ a @ ( sorted1267110213sorted @ a @ l1a @ L2 ) )
= ( minus_minus @ ( set @ a ) @ ( set2 @ a @ l1a ) @ ( set2 @ a @ L2 ) ) ) ) ) ).
% ind_hyp_l1
thf(fact_10_ind__hyp__l2,axiom,
( ( distinct @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b ) )
& ( linorder_sorted @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b ) )
& ( ( set2 @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ l2b ) )
= ( minus_minus @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ l2b ) ) ) ) ).
% ind_hyp_l2
thf(fact_11_x2__le__x1,axiom,
~ ( ord_less @ a @ x1 @ x2 ) ).
% x2_le_x1
thf(fact_12_x2__le,axiom,
! [X: a] :
( ( member @ a @ X @ ( set2 @ a @ l2b ) )
=> ( ord_less_eq @ a @ x2 @ X ) ) ).
% x2_le
thf(fact_13_x1__le,axiom,
! [X: a] :
( ( member @ a @ X @ ( set2 @ a @ l1a ) )
=> ( ord_less_eq @ a @ x1 @ X ) ) ).
% x1_le
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_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_16_DiffI,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ~ ( member @ A @ C @ B )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_17_Diff__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
= ( ( member @ A @ C @ A2 )
& ~ ( member @ A @ C @ B ) ) ) ).
% Diff_iff
thf(fact_18_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B ) @ B )
= ( minus_minus @ ( set @ A ) @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_19_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_20_minus__apply,axiom,
! [B2: $tType,A: $tType] :
( ( minus @ B2 @ ( type2 @ B2 ) )
=> ( ( minus_minus @ ( A > B2 ) )
= ( ^ [A3: A > B2,B3: A > B2,X2: A] : ( minus_minus @ B2 @ ( A3 @ X2 ) @ ( B3 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_21_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_22_list_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: list @ A,A1: A] :
( ( member @ A @ X @ ( set2 @ A @ A22 ) )
=> ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_23_list_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% list.set_intros(1)
thf(fact_24_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_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_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 )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ).
% sorted_Cons
thf(fact_27_sorted_OCons,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ X3 ) )
=> ( ( linorder_sorted @ A @ Xs )
=> ( linorder_sorted @ A @ ( cons @ A @ X @ Xs ) ) ) ) ) ).
% sorted.Cons
thf(fact_28_diff__sorted_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X1: A,X23: A,L1: list @ A,L2: list @ A] :
( ( ( ord_less @ A @ X1 @ X23 )
=> ( ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X23 @ L2 ) )
= ( cons @ A @ X1 @ ( sorted1267110213sorted @ A @ L1 @ ( cons @ A @ X23 @ L2 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X23 )
=> ( ( ( X1 = X23 )
=> ( ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X23 @ L2 ) )
= ( sorted1267110213sorted @ A @ L1 @ L2 ) ) )
& ( ( X1 != X23 )
=> ( ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X23 @ L2 ) )
= ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ).
% diff_sorted.simps(3)
thf(fact_29_fun__diff__def,axiom,
! [B2: $tType,A: $tType] :
( ( minus @ B2 @ ( type2 @ B2 ) )
=> ( ( minus_minus @ ( A > B2 ) )
= ( ^ [A3: A > B2,B3: A > B2,X2: A] : ( minus_minus @ B2 @ ( A3 @ X2 ) @ ( B3 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_30_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_31_DiffD2,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
=> ~ ( member @ A @ C @ B ) ) ).
% DiffD2
thf(fact_32_DiffD1,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
=> ( member @ A @ C @ A2 ) ) ).
% DiffD1
thf(fact_33_DiffE,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% DiffE
thf(fact_34_list_Oset__cases,axiom,
! [A: $tType,E: A,A4: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A4 ) )
=> ( ! [Z2: list @ A] :
( A4
!= ( cons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A4
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_35_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_36_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_37_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_38_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B4 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_39_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B4 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_40_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ( minus_minus @ A @ A4 @ B4 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A4 @ B4 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_41_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,D: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B4 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_42_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ( minus_minus @ A @ A4 @ B4 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
= ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_43_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B4 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_44_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B4 ) ) ) ) ).
% diff_left_mono
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,D: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ D @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B4 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_50_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_51_psubset__imp__ex__mem,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_52_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,D2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ D2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B ) @ ( minus_minus @ ( set @ A ) @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_53_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_54_double__diff,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C2 )
=> ( ( minus_minus @ ( set @ A ) @ B @ ( minus_minus @ ( set @ A ) @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_55_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_56_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_57_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_58_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_59_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z3: A] : Y2 = Z3 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_72_ord__le__eq__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 @ ( type2 @ B2 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B2,C: B2] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 @ ( type2 @ B2 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B2 > A,B4: B2,C: B2] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B2 @ B4 @ C )
=> ( ! [X3: B2,Y4: B2] :
( ( ord_less_eq @ B2 @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C3 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C3 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C3 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_75_order__subst1,axiom,
! [A: $tType,B2: $tType] :
( ( ( order @ B2 @ ( type2 @ B2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B2 > A,B4: B2,C: B2] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B2 @ B4 @ C )
=> ( ! [X3: B2,Y4: B2] :
( ( ord_less_eq @ B2 @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_76_le__fun__def,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ( ( ord_less_eq @ ( A > B2 ) )
= ( ^ [F2: A > B2,G2: A > B2] :
! [X2: A] : ( ord_less_eq @ B2 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_77_le__funI,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ! [F: A > B2,G: A > B2] :
( ! [X3: A] : ( ord_less_eq @ B2 @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_78_le__funE,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ! [F: A > B2,G: A > B2,X: A] :
( ( ord_less_eq @ ( A > B2 ) @ F @ G )
=> ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_79_le__funD,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ! [F: A > B2,G: A > B2,X: A] :
( ( ord_less_eq @ ( A > B2 ) @ F @ G )
=> ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_80_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( A4 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_81_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( A4 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_82_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_83_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_84_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_85_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_86_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_87_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_88_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_89_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_90_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_91_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_92_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_93_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_94_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_95_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_96_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( A4 = B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_97_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_98_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_99_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_trans
thf(fact_100_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% less_asym'
thf(fact_101_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_102_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_103_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z4: A] :
( ( ord_less @ A @ X @ Z4 )
& ( ord_less @ A @ Z4 @ Y ) ) ) ) ).
% dense
thf(fact_104_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% order.asym
thf(fact_105_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_106_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_107_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% gt_ex
thf(fact_108_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_109_order__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ C3 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C3 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_110_order__less__subst1,axiom,
! [A: $tType,B2: $tType] :
( ( ( order @ B2 @ ( type2 @ B2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B2 > A,B4: B2,C: B2] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B2 @ B4 @ C )
=> ( ! [X3: B2,Y4: B2] :
( ( ord_less @ B2 @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_111_ord__less__eq__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 @ ( type2 @ B2 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B2,C: B2] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ B2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_112_ord__eq__less__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 @ ( type2 @ B2 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B2 > A,B4: B2,C: B2] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less @ B2 @ B4 @ C )
=> ( ! [X3: B2,Y4: B2] :
( ( ord_less @ B2 @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_113_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ( minus_minus @ A @ A4 @ B4 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A4 = B4 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_114_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,C: A,B4: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B4 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_115_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( A4 != B4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_116_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_117_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A6: A] :
( ( ord_less_eq @ A @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_118_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( ( ord_less @ A @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_119_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% order.strict_implies_order
thf(fact_120_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_121_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_122_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_123_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_124_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_125_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_126_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_127_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_128_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_129_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_130_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_131_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_132_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_133_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_134_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_135_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_136_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_137_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_138_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_139_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% le_neq_trans
thf(fact_140_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_141_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_142_order__less__le__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C3 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C3 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_143_order__less__le__subst1,axiom,
! [A: $tType,B2: $tType] :
( ( ( order @ B2 @ ( type2 @ B2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B2 > A,B4: B2,C: B2] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B2 @ B4 @ C )
=> ( ! [X3: B2,Y4: B2] :
( ( ord_less_eq @ B2 @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_144_order__le__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ C3 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C3 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_145_order__le__less__subst1,axiom,
! [A: $tType,B2: $tType] :
( ( ( order @ B2 @ ( type2 @ B2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B2 > A,B4: B2,C: B2] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B2 @ B4 @ C )
=> ( ! [X3: B2,Y4: B2] :
( ( ord_less @ B2 @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_146_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_147_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_148_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_149_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_150_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,P: A > $o] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B4 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A4 @ C4 )
& ( ord_less_eq @ A @ C4 @ B4 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A4 @ X4 )
& ( ord_less @ A @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D3: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A4 @ X3 )
& ( ord_less @ A @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D3 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_151_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% pinf(6)
thf(fact_152_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% pinf(8)
thf(fact_153_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% minf(6)
thf(fact_154_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% minf(8)
thf(fact_155_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_156_sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: list @ A] :
( ( linorder_sorted @ A @ A4 )
=> ( ( A4
!= ( nil @ A ) )
=> ~ ! [Xs2: list @ A,X3: A] :
( ( A4
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X3 @ Xa ) )
=> ~ ( linorder_sorted @ A @ Xs2 ) ) ) ) ) ) ).
% sorted.cases
thf(fact_157_sorted_Osimps,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( linorder_sorted @ A )
= ( ^ [A6: list @ A] :
( ( A6
= ( nil @ A ) )
| ? [Xs3: list @ A,X2: A] :
( ( A6
= ( cons @ A @ X2 @ Xs3 ) )
& ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs3 ) )
=> ( ord_less_eq @ A @ X2 @ Y3 ) )
& ( linorder_sorted @ A @ Xs3 ) ) ) ) ) ) ).
% sorted.simps
thf(fact_158_subsetI,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% subsetI
thf(fact_159_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_160_psubsetI,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less @ ( set @ A ) @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_161_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_162_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_163_sorted__single,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( linorder_sorted @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted_single
thf(fact_164_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_165_set__mp,axiom,
! [A: $tType,A2: set @ A,B: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B ) ) ) ).
% set_mp
thf(fact_166_in__mono,axiom,
! [A: $tType,A2: set @ A,B: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B ) ) ) ).
% in_mono
thf(fact_167_subsetD,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% subsetD
thf(fact_168_psubsetD,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% psubsetD
thf(fact_169_psubsetE,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_170_subsetCE,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B ) ) ) ).
% subsetCE
thf(fact_171_equalityE,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ~ ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ) ).
% equalityE
thf(fact_172_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_173_equalityD1,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% equalityD1
thf(fact_174_equalityD2,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( A2 = B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A2 ) ) ).
% equalityD2
thf(fact_175_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_176_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( member @ A @ X @ B ) ) ) ).
% set_rev_mp
thf(fact_177_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A3 )
=> ( member @ A @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_178_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( member @ A @ C @ B ) ) ) ).
% rev_subsetD
thf(fact_179_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_180_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_181_subset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_182_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less @ ( set @ A ) @ B @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_183_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z3: set @ A] : Y2 = Z3 )
= ( ^ [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_184_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ~ ( member @ A @ C @ B )
=> ~ ( member @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_185_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_186_less__fun__def,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 @ ( type2 @ B2 ) )
=> ( ( ord_less @ ( A > B2 ) )
= ( ^ [F2: A > B2,G2: A > B2] :
( ( ord_less_eq @ ( A > B2 ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B2 ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_187_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_188_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_189_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_190_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B )
=> ( ( ord_less @ ( set @ A ) @ B @ C2 )
=> ( ord_less @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_191_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_192_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_193_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B2: $tType,P: ( A > B2 ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2,A1: list @ A,A22: list @ B2] :
( ! [F3: A > B2,X12: list @ B2] : ( P @ F3 @ ( nil @ A ) @ X12 )
=> ( ! [F3: A > B2,A5: A,As: list @ A,Bs: list @ B2] :
( ( P @ F3 @ As @ ( cons @ B2 @ ( F3 @ A5 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A5 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_194_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_195_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y4: A,Xs2: list @ A] :
( ( ( X3 = Y4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_196_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y4: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_197_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A,Y4: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_198_list__induct2_H,axiom,
! [A: $tType,B2: $tType,P: ( list @ A ) > ( list @ B2 ) > $o,Xs: list @ A,Ys: list @ B2] :
( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( nil @ B2 ) )
=> ( ! [Y4: B2,Ys2: list @ B2] : ( P @ ( nil @ A ) @ ( cons @ B2 @ Y4 @ Ys2 ) )
=> ( ! [X3: A,Xs2: list @ A,Y4: B2,Ys2: list @ B2] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B2 @ Y4 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_199_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_200_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X24: list @ A] :
( ( P @ X24 )
=> ( P @ ( cons @ A @ X12 @ X24 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_201_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_202_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_203_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_204_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_205_sorted_ONil,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_sorted @ A @ ( nil @ A ) ) ) ).
% sorted.Nil
thf(fact_206_diff__sorted_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: list @ A] :
( ( sorted1267110213sorted @ A @ ( nil @ A ) @ L2 )
= ( nil @ A ) ) ) ).
% diff_sorted.simps(1)
thf(fact_207_inter__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X12: A,L12: list @ A,X24: A,L22: list @ A] :
( ( ( ord_less @ A @ X12 @ X24 )
=> ( P @ L12 @ ( cons @ A @ X24 @ L22 ) ) )
=> ( ( ~ ( ord_less @ A @ X12 @ X24 )
=> ( ( X12 = X24 )
=> ( P @ L12 @ L22 ) ) )
=> ( ( ~ ( ord_less @ A @ X12 @ X24 )
=> ( ( X12 != X24 )
=> ( P @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) )
=> ( P @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X24 @ L22 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% inter_sorted.induct
thf(fact_208_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_209_diff__sorted_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [V2: A,Va2: list @ A] :
( ( sorted1267110213sorted @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V2 @ Va2 ) ) ) ).
% diff_sorted.simps(2)
thf(fact_210_minf_I11_J,axiom,
! [C3: $tType,D4: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F4: D4] :
? [Z4: C3] :
! [X4: C3] :
( ( ord_less @ C3 @ X4 @ Z4 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_211_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less @ A @ T @ X4 ) ) ) ).
% minf(7)
thf(fact_212_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less @ A @ X4 @ T ) ) ) ).
% minf(5)
thf(fact_213_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T ) ) ) ).
% minf(4)
thf(fact_214_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T ) ) ) ).
% minf(3)
thf(fact_215_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_216_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_217_pinf_I11_J,axiom,
! [C3: $tType,D4: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F4: D4] :
? [Z4: C3] :
! [X4: C3] :
( ( ord_less @ C3 @ Z4 @ X4 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_218_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less @ A @ T @ X4 ) ) ) ).
% pinf(7)
thf(fact_219_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T ) ) ) ).
% pinf(5)
thf(fact_220_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T ) ) ) ).
% pinf(4)
thf(fact_221_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T ) ) ) ).
% pinf(3)
thf(fact_222_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_223_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_224_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A4: A] :
? [B5: A] :
( ( ord_less @ A @ A4 @ B5 )
| ( ord_less @ A @ B5 @ A4 ) ) ) ).
% ex_gt_or_lt
thf(fact_225_diff__sorted_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,Xa2: list @ A,Y: list @ A] :
( ( ( sorted1267110213sorted @ A @ X @ Xa2 )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [V: A,Va: list @ A] :
( ( X
= ( cons @ A @ V @ Va ) )
=> ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X24: A,L22: list @ A] :
( ( Xa2
= ( cons @ A @ X24 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X12 @ X24 )
=> ( Y
= ( cons @ A @ X12 @ ( sorted1267110213sorted @ A @ L12 @ ( cons @ A @ X24 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X24 )
=> ( ( ( X12 = X24 )
=> ( Y
= ( sorted1267110213sorted @ A @ L12 @ L22 ) ) )
& ( ( X12 != X24 )
=> ( Y
= ( sorted1267110213sorted @ A @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ).
% diff_sorted.elims
thf(fact_226_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X2: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X2 @ ( set2 @ A @ Xs3 ) ) @ Xs3 @ ( cons @ A @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_227_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,X3: A] :
( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X3 @ Xa ) )
=> ( ( linorder_sorted @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) ) ) )
=> ( P @ X ) ) ) ) ) ).
% sorted.inducts
thf(fact_228_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_229_revg_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [A5: A,As: list @ A,B5: list @ A] :
( ( P @ As @ ( cons @ A @ A5 @ B5 ) )
=> ( P @ ( cons @ A @ A5 @ As ) @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_230_zipf_Oinduct,axiom,
! [A: $tType,C3: $tType,B2: $tType,P: ( A > B2 > C3 ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2 > C3,A1: list @ A,A22: list @ B2] :
( ! [F3: A > B2 > C3] : ( P @ F3 @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [F3: A > B2 > C3,A5: A,As: list @ A,B5: B2,Bs: list @ B2] :
( ( P @ F3 @ As @ Bs )
=> ( P @ F3 @ ( cons @ A @ A5 @ As ) @ ( cons @ B2 @ B5 @ Bs ) ) )
=> ( ! [A5: A > B2 > C3,V: A,Va: list @ A] : ( P @ A5 @ ( cons @ A @ V @ Va ) @ ( nil @ B2 ) )
=> ( ! [A5: A > B2 > C3,V: B2,Va: list @ B2] : ( P @ A5 @ ( nil @ A ) @ ( cons @ B2 @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% zipf.induct
thf(fact_231_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_232_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_233_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( P @ X2 ) ) ) ) ).
% subset_Collect_conv
thf(fact_234_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B2: $tType,P: ( list @ A ) > ( list @ B2 ) > $o,R: A > B2 > $o,Xs: list @ A,Ys: list @ B2] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B2 ) )
=> ( ! [X12: list @ B2] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X3: A,Xs2: list @ A,Y4: B2,Ys2: list @ B2] :
( ( R @ X3 @ Y4 )
=> ( ( P @ Xs2 @ ( cons @ B2 @ Y4 @ Ys2 ) )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B2 @ Y4 @ Ys2 ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y4: B2,Ys2: list @ B2] :
( ~ ( R @ X3 @ Y4 )
=> ( ( P @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B2 @ Y4 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_235_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] :
( ! [R3: A > A > $o,X3: A,Xs2: list @ A,Y4: A,Ys2: list @ A] :
( ( ( R3 @ X3 @ Y4 )
=> ( P @ R3 @ Xs2 @ ( cons @ A @ Y4 @ Ys2 ) ) )
=> ( ( ~ ( R3 @ X3 @ Y4 )
=> ( P @ R3 @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 ) )
=> ( P @ R3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) ) )
=> ( ! [R3: A > A > $o,Xs2: list @ A] : ( P @ R3 @ Xs2 @ ( nil @ A ) )
=> ( ! [R3: A > A > $o,V: A,Va: list @ A] : ( P @ R3 @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_236_list__all__zip_Oinduct,axiom,
! [A: $tType,B2: $tType,P: ( A > B2 > $o ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2 > $o,A1: list @ A,A22: list @ B2] :
( ! [P3: A > B2 > $o] : ( P @ P3 @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [P3: A > B2 > $o,A5: A,As: list @ A,B5: B2,Bs: list @ B2] :
( ( P @ P3 @ As @ Bs )
=> ( P @ P3 @ ( cons @ A @ A5 @ As ) @ ( cons @ B2 @ B5 @ Bs ) ) )
=> ( ! [P3: A > B2 > $o,V: A,Va: list @ A] : ( P @ P3 @ ( cons @ A @ V @ Va ) @ ( nil @ B2 ) )
=> ( ! [P3: A > B2 > $o,V: B2,Va: list @ B2] : ( P @ P3 @ ( nil @ A ) @ ( cons @ B2 @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_237_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X12: A,X24: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X12 @ ( cons @ A @ X24 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_238_list__2pre__induct,axiom,
! [A: $tType,B2: $tType,P: ( list @ A ) > ( list @ B2 ) > $o,W1: list @ A,W2: list @ B2] :
( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [E2: A,W12: list @ A,W22: list @ B2] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons @ A @ E2 @ W12 ) @ W22 ) )
=> ( ! [E2: B2,W13: list @ A,W23: list @ B2] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons @ B2 @ E2 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_239_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_240_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_241_quicksort_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ Xs2 ) ) ) ) ).
% quicksort.cases
thf(fact_242_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_243_distinct__product__lists,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( distinct @ A @ X3 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_244_subset__code_I3_J,axiom,
! [C3: $tType] :
~ ( ord_less_eq @ ( set @ C3 ) @ ( coset @ C3 @ ( nil @ C3 ) ) @ ( set2 @ C3 @ ( nil @ C3 ) ) ) ).
% subset_code(3)
thf(fact_245_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_246_subset__code_I2_J,axiom,
! [B2: $tType,A2: set @ B2,Ys: list @ B2] :
( ( ord_less_eq @ ( set @ B2 ) @ A2 @ ( coset @ B2 @ Ys ) )
= ( ! [X2: B2] :
( ( member @ B2 @ X2 @ ( set2 @ B2 @ Ys ) )
=> ~ ( member @ B2 @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_247_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_248_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Y: A] :
( ( linorder_sorted @ A @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ Y ) )
=> ( linorder_sorted @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_249_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_250_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_251_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_252_member__remove,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A2 ) )
= ( ( member @ A @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_253_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_254_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
%----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 (13)
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_fun___Groups_Ominus,axiom,
! [A7: $tType,A8: $tType] :
( ( minus @ A8 @ ( type2 @ A8 ) )
=> ( minus @ ( 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_Set_Oset___Groups_Ominus_4,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_8,axiom,
minus @ $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,
( ( distinct @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ ( cons @ a @ x2 @ l2b ) ) )
& ( linorder_sorted @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ ( cons @ a @ x2 @ l2b ) ) )
& ( ( set2 @ a @ ( sorted1267110213sorted @ a @ ( cons @ a @ x1 @ l1a ) @ ( cons @ a @ x2 @ l2b ) ) )
= ( minus_minus @ ( set @ a ) @ ( set2 @ a @ ( cons @ a @ x1 @ l1a ) ) @ ( set2 @ a @ ( cons @ a @ x2 @ l2b ) ) ) ) ) ).
%------------------------------------------------------------------------------