TPTP Problem File: DAT208^1.p
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%------------------------------------------------------------------------------
% File : DAT208^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Sorted list operations 293
% 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__293.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.67 v7.3.0, 1.00 v7.1.0
% Syntax : Number of formulae : 329 ( 46 unt; 45 typ; 0 def)
% Number of atoms : 984 ( 188 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3964 ( 87 ~; 28 |; 70 &;3248 @)
% ( 0 <=>; 531 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 149 ( 149 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 44 usr; 4 con; 0-4 aty)
% Number of variables : 1070 ( 64 ^; 927 !; 41 ?;1070 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:38.793
%------------------------------------------------------------------------------
%----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 (41)
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_Lattices_Olattice,type,
lattice:
!>[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_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_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_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[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_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > 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_Olinorder__class_Osorted,type,
linorder_sorted:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord467138063of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge,type,
merges2133357844_merge:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Osorted__by__rel,type,
sorted_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
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_Sorted__List__Operations__Mirabelle__fineeiboro_Odelete__sorted,type,
sorted763967275sorted:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Omemb__sorted,type,
sorted873716653sorted:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
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_a,type,
a2: a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xsa,type,
xsa: list @ a ).
%----Relevant facts (256)
thf(fact_0_set__eq__sorted__correct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L1: list @ A,L2: list @ A] :
( ( ( distinct @ A @ L1 )
& ( linorder_sorted @ A @ L1 ) )
=> ( ( ( distinct @ A @ L2 )
& ( linorder_sorted @ A @ L2 ) )
=> ( ( L1 = L2 )
= ( ( set2 @ A @ L1 )
= ( set2 @ A @ L2 ) ) ) ) ) ) ).
% set_eq_sorted_correct
thf(fact_1_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_2_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_3_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% minf(8)
thf(fact_4_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% minf(6)
thf(fact_5_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ T @ X2 ) ) ) ).
% pinf(8)
thf(fact_6_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ T ) ) ) ).
% pinf(6)
thf(fact_7_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_8_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_9_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ) ).
% le_less
thf(fact_10_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ( X3 != Y2 ) ) ) ) ) ).
% less_le
thf(fact_11_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_12_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_13_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_14_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_15_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_16_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( P @ A3 @ B4 ) )
=> ( ! [A3: A,B4: A] :
( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_17_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_18_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_19_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_20_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_21_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_22_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_23_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_24_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_25_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_26_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_27_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_28_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_29_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_30_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_31_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_32_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_33_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_34_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_35_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_36_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_37_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_38_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_39_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_40_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_41_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_42_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_43_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_44_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member2 @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G2 @ X4 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_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_50_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_51_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_52_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_53_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_54_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_55_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_56_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_57_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_58_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_59_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_60_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_61_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_62_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_63_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_64_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_65_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_66_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_67_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_68_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_69_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_70_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_71_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_72_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_73_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_74_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_75_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_76_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_77_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T ) ) ) ).
% pinf(3)
thf(fact_78_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T ) ) ) ).
% pinf(4)
thf(fact_79_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less @ A @ X2 @ T ) ) ) ).
% pinf(5)
thf(fact_80_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less @ A @ T @ X2 ) ) ) ).
% pinf(7)
thf(fact_81_pinf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D] :
? [Z: C2] :
! [X2: C2] :
( ( ord_less @ C2 @ Z @ X2 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_82_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_83_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_84_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T ) ) ) ).
% minf(3)
thf(fact_85_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T ) ) ) ).
% minf(4)
thf(fact_86_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less @ A @ X2 @ T ) ) ) ).
% minf(5)
thf(fact_87_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less @ A @ T @ X2 ) ) ) ).
% minf(7)
thf(fact_88_minf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D] :
? [Z: C2] :
! [X2: C2] :
( ( ord_less @ C2 @ X2 @ Z )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_89_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_90_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_91_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_92_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_93_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_94_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_95_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_96_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_97_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_98_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_99_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_100_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_101_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_102_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_103_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_104_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_105_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_106_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z2: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_107_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_108_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_109_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_110_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_111_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_112_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_113_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_114_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_115_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_116_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_117_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_118_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_119_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ C3 ) )
=> ( P @ X2 ) )
& ! [D2: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_120_subset__sorted__correct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L1: list @ A,L2: list @ A] :
( ( ( distinct @ A @ L1 )
& ( linorder_sorted @ A @ L1 ) )
=> ( ( ( distinct @ A @ L2 )
& ( linorder_sorted @ A @ L2 ) )
=> ( ( sorted1061247458sorted @ A @ L1 @ L2 )
= ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ).
% subset_sorted_correct
thf(fact_121_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_122_memb__sorted__correct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,X: A] :
( ( linorder_sorted @ A @ Xs )
=> ( ( sorted873716653sorted @ A @ Xs @ X )
= ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% memb_sorted_correct
thf(fact_123_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_124_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [X4: list @ A] :
( ( ( set2 @ A @ X4 )
= A4 )
& ( linorder_sorted @ A @ X4 )
& ( distinct @ A @ X4 )
& ! [Y5: list @ A] :
( ( ( ( set2 @ A @ Y5 )
= A4 )
& ( linorder_sorted @ A @ Y5 )
& ( distinct @ A @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_125_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B4: A] :
( ( ord_less @ A @ A2 @ B4 )
| ( ord_less @ A @ B4 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_126_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_127_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X2: A] :
? [X1: A] : ( ord_less @ A @ X2 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_128_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X2: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X2 ) ) ).
% linordered_field_no_lb
thf(fact_129_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_130_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
& ~ ( ord_less_eq @ ( A > B ) @ G @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_131_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_132_finite__list,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_133_finite__distinct__list,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A4 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_134_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ( C = D3 )
=> ( ord_less_eq @ A @ A2 @ D3 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_135_subsetI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X4: A] :
( ( member2 @ A @ X4 @ A4 )
=> ( member2 @ A @ X4 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% subsetI
thf(fact_136_psubsetI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% psubsetI
thf(fact_137_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_138_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_139_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C4 )
=> ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% psubset_trans
thf(fact_140_psubsetD,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ( ( member2 @ A @ C @ A4 )
=> ( member2 @ A @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_141_finite__set__choice,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X4: A] :
( ( member2 @ A @ X4 @ A4 )
=> ? [X12: B] : ( P @ X4 @ X12 ) )
=> ? [F4: A > B] :
! [X2: A] :
( ( member2 @ A @ X2 @ A4 )
=> ( P @ X2 @ ( F4 @ X2 ) ) ) ) ) ).
% finite_set_choice
thf(fact_142_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_143_finite__psubset__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [B6: set @ A] :
( ( ord_less @ ( set @ A ) @ B6 @ A7 )
=> ( P @ B6 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_144_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
| ( A6 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_145_subset__psubset__trans,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C4 )
=> ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_146_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_147_psubset__subset__trans,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
=> ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_148_psubset__imp__subset,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_149_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_150_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ~ ( member2 @ A @ C @ B3 )
=> ~ ( member2 @ A @ C @ A4 ) ) ) ).
% contra_subsetD
thf(fact_151_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z3: set @ A] : Y4 = Z3 )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_152_subset__trans,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_153_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_154_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_155_rev__subsetD,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member2 @ A @ C @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( member2 @ A @ C @ B3 ) ) ) ).
% rev_subsetD
thf(fact_156_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [T2: A] :
( ( member2 @ A @ T2 @ A6 )
=> ( member2 @ A @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_157_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B3: set @ A] :
( ( member2 @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( member2 @ A @ X @ B3 ) ) ) ).
% set_rev_mp
thf(fact_158_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( A6 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_159_equalityD2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( A4 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ).
% equalityD2
thf(fact_160_equalityD1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( A4 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% equalityD1
thf(fact_161_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [X3: A] :
( ( member2 @ A @ X3 @ A6 )
=> ( member2 @ A @ X3 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_162_equalityE,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( A4 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% equalityE
thf(fact_163_subsetCE,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( member2 @ A @ C @ A4 )
=> ( member2 @ A @ C @ B3 ) ) ) ).
% subsetCE
thf(fact_164_psubsetE,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% psubsetE
thf(fact_165_subsetD,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( member2 @ A @ C @ A4 )
=> ( member2 @ A @ C @ B3 ) ) ) ).
% subsetD
thf(fact_166_in__mono,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( member2 @ A @ X @ A4 )
=> ( member2 @ A @ X @ B3 ) ) ) ).
% in_mono
thf(fact_167_set__mp,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( member2 @ A @ X @ A4 )
=> ( member2 @ A @ X @ B3 ) ) ) ).
% set_mp
thf(fact_168_rev__finite__subset,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_169_infinite__super,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% infinite_super
thf(fact_170_finite__subset,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( finite_finite2 @ A @ B3 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_subset
thf(fact_171_sorted__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( set2 @ A @ ( linord467138063of_set @ A @ A4 ) )
= A4 )
& ( linorder_sorted @ A @ ( linord467138063of_set @ A @ A4 ) )
& ( distinct @ A @ ( linord467138063of_set @ A @ A4 ) ) ) ) ) ).
% sorted_list_of_set
thf(fact_172_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_173_subset__code_I2_J,axiom,
! [B: $tType,A4: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A4 @ ( coset @ B @ Ys ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ ( set2 @ B @ Ys ) )
=> ~ ( member2 @ B @ X3 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_174_sorted__by__rel__linord,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L: list @ A] :
( ( sorted_by_rel @ A @ ( ord_less_eq @ A ) @ L )
= ( linorder_sorted @ A @ L ) ) ) ).
% sorted_by_rel_linord
thf(fact_175_UnCI,axiom,
! [A: $tType,C: A,B3: set @ A,A4: set @ A] :
( ( ~ ( member2 @ A @ C @ B3 )
=> ( member2 @ A @ C @ A4 ) )
=> ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnCI
thf(fact_176_Un__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member2 @ A @ C @ A4 )
| ( member2 @ A @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_177_finite__Un,axiom,
! [A: $tType,F3: set @ A,G3: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F3 @ G3 ) )
= ( ( finite_finite2 @ A @ F3 )
& ( finite_finite2 @ A @ G3 ) ) ) ).
% finite_Un
thf(fact_178_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_179_Un__mono,axiom,
! [A: $tType,A4: set @ A,C4: set @ A,B3: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_180_Un__least,axiom,
! [A: $tType,A4: set @ A,C4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 ) ) ) ).
% Un_least
thf(fact_181_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_upper1
thf(fact_182_Un__upper2,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_upper2
thf(fact_183_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_184_Un__absorb2,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B3 )
= A4 ) ) ).
% Un_absorb2
thf(fact_185_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_186_infinite__Un,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% infinite_Un
thf(fact_187_Un__infinite,axiom,
! [A: $tType,S: set @ A,T3: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T3 ) ) ) ).
% Un_infinite
thf(fact_188_finite__UnI,axiom,
! [A: $tType,F3: set @ A,G3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( finite_finite2 @ A @ G3 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F3 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_189_UnE,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( ~ ( member2 @ A @ C @ A4 )
=> ( member2 @ A @ C @ B3 ) ) ) ).
% UnE
thf(fact_190_UnI1,axiom,
! [A: $tType,C: A,A4: set @ A,B3: set @ A] :
( ( member2 @ A @ C @ A4 )
=> ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnI1
thf(fact_191_UnI2,axiom,
! [A: $tType,C: A,B3: set @ A,A4: set @ A] :
( ( member2 @ A @ C @ B3 )
=> ( member2 @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ) ).
% UnI2
thf(fact_192_bex__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ A4 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member2 @ A @ X3 @ B3 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_193_ball__Un,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A4 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_194_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Un_assoc
thf(fact_195_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_196_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% Un_commute
thf(fact_197_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B3 ) ) ).
% Un_left_absorb
thf(fact_198_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_199_sorted__by__rel__weaken,axiom,
! [A: $tType,L0: list @ A,R: A > A > $o,R2: A > A > $o] :
( ! [X4: A,Y3: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ L0 ) )
=> ( ( member2 @ A @ Y3 @ ( set2 @ A @ L0 ) )
=> ( ( R @ X4 @ Y3 )
=> ( R2 @ X4 @ Y3 ) ) ) )
=> ( ( sorted_by_rel @ A @ R @ L0 )
=> ( sorted_by_rel @ A @ R2 @ L0 ) ) ) ).
% sorted_by_rel_weaken
thf(fact_200_distinct__sorted__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] : ( distinct @ A @ ( linord467138063of_set @ A @ A4 ) ) ) ).
% distinct_sorted_list_of_set
thf(fact_201_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_202_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_203_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_204_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% sup_apply
thf(fact_205_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_206_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_207_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_208_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_209_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_210_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_211_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_212_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_213_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% sup_fun_def
thf(fact_214_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% sup.assoc
thf(fact_215_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_216_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_217_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y2: A] : ( sup_sup @ A @ Y2 @ X3 ) ) ) ) ).
% sup_commute
thf(fact_218_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% sup.left_commute
thf(fact_219_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_220_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_221_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_222_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_223_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_224_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_225_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_226_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_227_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_228_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D3 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_229_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_230_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X ) ) ) ) ).
% sup_least
thf(fact_231_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ Y2 )
= Y2 ) ) ) ) ).
% le_iff_sup
thf(fact_232_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_233_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_234_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [F: A > A > A,X: A,Y: A] :
( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ X4 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: A,Y3: A,Z: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ( ord_less_eq @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ ( F @ Y3 @ Z ) @ X4 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_235_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_236_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_237_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_238_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_239_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_240_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_241_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_242_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_243_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_244_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_245_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_246_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_247_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_248_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_249_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ C @ A2 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_250_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_251_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_252_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_253_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_254_merge__correct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L1: list @ A,L2: list @ A] :
( ( ( distinct @ A @ L1 )
& ( linorder_sorted @ A @ L1 ) )
=> ( ( ( distinct @ A @ L2 )
& ( linorder_sorted @ A @ L2 ) )
=> ( ( distinct @ A @ ( merge @ A @ L1 @ L2 ) )
& ( linorder_sorted @ A @ ( merge @ A @ L1 @ L2 ) )
& ( ( set2 @ A @ ( merge @ A @ L1 @ L2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ) ).
% merge_correct
thf(fact_255_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( merges2133357844_merge @ A @ R @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_mergesort_by_rel_merge
%----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 (19)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_9,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_10,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type2 @ a ) ).
%----Conjectures (4)
thf(conj_0,hypothesis,
( ( distinct @ a @ ( sorted763967275sorted @ a @ x @ xsa ) )
& ( linorder_sorted @ a @ ( sorted763967275sorted @ a @ x @ xsa ) )
& ! [X2: a] :
( ( member2 @ a @ X2 @ ( set2 @ a @ ( sorted763967275sorted @ a @ x @ xsa ) ) )
= ( ( member2 @ a @ X2 @ ( set2 @ a @ xsa ) )
& ( X2 != x ) ) ) ) ).
thf(conj_1,hypothesis,
( ( linorder_sorted @ a @ xsa )
& ! [X2: a] :
( ( member2 @ a @ X2 @ ( set2 @ a @ xsa ) )
=> ( ord_less_eq @ a @ a2 @ X2 ) ) ) ).
thf(conj_2,hypothesis,
( ~ ( member2 @ a @ a2 @ ( set2 @ a @ xsa ) )
& ( distinct @ a @ xsa ) ) ).
thf(conj_3,conjecture,
( ( ( x != a2 )
| ! [X4: a] :
( ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) )
= ( ( ( X4 = a2 )
| ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
& ( X4 != a2 ) ) ) )
& ( ( x = a2 )
| ( ( ~ ( ord_less @ a @ a2 @ x )
| ! [X4: a] :
( ( ( X4 = a2 )
| ( ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) )
& ( X4 != x ) ) )
= ( ( ( X4 = a2 )
| ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
& ( X4 != x ) ) ) )
& ( ( ord_less @ a @ a2 @ x )
| ! [X4: a] :
( ( ( X4 = a2 )
| ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
= ( ( ( X4 = a2 )
| ( member2 @ a @ X4 @ ( set2 @ a @ xsa ) ) )
& ( X4 != x ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------