TPTP Problem File: COM195^1.p
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%------------------------------------------------------------------------------
% File : COM195^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Grammars and languages 1319
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : gram_lang__1319.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 320 ( 37 unt; 43 typ; 0 def)
% Number of atoms : 1012 ( 202 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3896 ( 78 ~; 25 |; 65 &;3228 @)
% ( 0 <=>; 500 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 42 usr; 2 con; 0-3 aty)
% Number of variables : 1060 ( 52 ^; 936 !; 32 ?;1060 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:48:25.615
%------------------------------------------------------------------------------
%----Could-be-implicit typings (3)
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 (40)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[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_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[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_Groups_Oordered__ab__group__add,type,
ordered_ab_group_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_Gram__Lang__Mirabelle__ojxrtuoybn_Oleqv,type,
gram_L1456083582e_leqv:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_Osubs,type,
gram_L608943123e_subs:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > 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_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $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_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or331188842AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or1433837966ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or1361889807AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or578182835ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_L1,type,
l1: set @ ( set @ a ) ).
thf(sy_v_L2,type,
l2: set @ ( set @ a ) ).
%----Relevant facts (256)
thf(fact_0_subs__refl,axiom,
! [A: $tType,L1: set @ ( set @ A )] : ( gram_L608943123e_subs @ A @ L1 @ L1 ) ).
% subs_refl
thf(fact_1_subs__trans,axiom,
! [A: $tType,L1: set @ ( set @ A ),L2: set @ ( set @ A ),L3: set @ ( set @ A )] :
( ( gram_L608943123e_subs @ A @ L1 @ L2 )
=> ( ( gram_L608943123e_subs @ A @ L2 @ L3 )
=> ( gram_L608943123e_subs @ A @ L1 @ L3 ) ) ) ).
% subs_trans
thf(fact_2_leqv__def,axiom,
! [A: $tType] :
( ( gram_L1456083582e_leqv @ A )
= ( ^ [L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
( ( gram_L608943123e_subs @ A @ L12 @ L22 )
& ( gram_L608943123e_subs @ A @ L22 @ L12 ) ) ) ) ).
% leqv_def
thf(fact_3_incl__subs,axiom,
! [A: $tType,L2: set @ ( set @ A ),L1: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ L2 @ L1 )
=> ( gram_L608943123e_subs @ A @ L1 @ L2 ) ) ).
% incl_subs
thf(fact_4_subs__def,axiom,
! [A: $tType] :
( ( gram_L608943123e_subs @ A )
= ( ^ [L12: set @ ( set @ A ),L22: set @ ( set @ A )] :
! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ L22 )
=> ? [Y: set @ A] :
( ( member @ ( set @ A ) @ Y @ L12 )
& ( ord_less_eq @ ( set @ A ) @ Y @ X ) ) ) ) ) ).
% subs_def
thf(fact_5_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_6_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_7_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_8_set__mp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) ) ) ).
% set_mp
thf(fact_9_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_10_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% subsetD
thf(fact_11_subsetCE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% subsetCE
thf(fact_12_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_13_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_14_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_15_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_16_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_17_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_18_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_19_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_20_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% order_trans
thf(fact_21_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_22_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_23_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_24_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_25_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_26_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_27_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% le_cases
thf(fact_28_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( X3 = Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% eq_refl
thf(fact_29_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% linear
thf(fact_30_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ) ).
% antisym
thf(fact_31_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% eq_iff
thf(fact_32_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_33_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_34_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_35_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_36_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_37_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_38_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funE
thf(fact_39_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G2: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funD
thf(fact_40_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_41_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member @ A @ C @ B2 )
=> ~ ( member @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_42_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z2: set @ A] : Y3 = Z2 )
= ( ^ [A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_43_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_44_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_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
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G2 @ X2 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_50_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% rev_subsetD
thf(fact_51_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [T: A] :
( ( member @ A @ T @ A3 )
=> ( member @ A @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_52_set__rev__mp,axiom,
! [A: $tType,X3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ X3 @ B2 ) ) ) ).
% set_rev_mp
thf(fact_53_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F2 @ Y ) @ ( F2 @ X ) ) ) ) ) ) ).
% antimono_def
thf(fact_54_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_55_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% antimonoE
thf(fact_56_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% antimonoD
thf(fact_57_Pow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_58_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) )
= ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ).
% strict_mono_less_eq
thf(fact_59_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T2: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( pairwise @ A @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_60_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X3 ) @ ( set_ord_greaterThan @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% greaterThan_subset_iff
thf(fact_61_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X3 ) @ ( set_ord_lessThan @ A @ Y2 ) )
= ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% lessThan_subset_iff
thf(fact_62_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X3 ) @ ( set_ord_atLeast @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% atLeast_subset_iff
thf(fact_63_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_lessThan @ A @ X3 )
= ( set_ord_lessThan @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% lessThan_eq_iff
thf(fact_64_atLeast__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_atLeast @ A @ X3 )
= ( set_ord_atLeast @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% atLeast_eq_iff
thf(fact_65_greaterThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_greaterThan @ A @ X3 )
= ( set_ord_greaterThan @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% greaterThan_eq_iff
thf(fact_66_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_67_PowI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% PowI
thf(fact_68_Pow__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_69_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A4 ) @ ( set_ord_atLeast @ A @ A4 ) ) ) ).
% Ioi_le_Ico
thf(fact_70_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R: A > A > $o,S2: set @ A] :
! [X: A] :
( ( member @ A @ X @ S2 )
=> ! [Y: A] :
( ( member @ A @ Y @ S2 )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ) ).
% pairwise_def
thf(fact_71_Pow__top,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% Pow_top
thf(fact_72_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% strict_mono_eq
thf(fact_73_PowD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% PowD
thf(fact_74_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [L: A,H: A,L4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_75_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A4 ) @ ( set_ord_greaterThan @ A @ B4 ) )
= ( ord_less @ A @ B4 @ A4 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_76_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_77_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_78_atMost__subset__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X3 ) @ ( set_ord_atMost @ A @ Y2 ) )
= ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% atMost_subset_iff
thf(fact_79_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,M: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ I @ J ) @ ( set_or1433837966ssThan @ A @ M @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_80_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [H: A,L4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_le_Ici
thf(fact_81_psubsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_82_ComplI,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ~ ( member @ A @ C @ A2 )
=> ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_83_Compl__iff,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_84_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A2 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_85_atMost__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_atMost @ A @ X3 )
= ( set_ord_atMost @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% atMost_eq_iff
thf(fact_86_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or331188842AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_87_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [L: A,H: A,L4: A,H2: A] :
( ( ( set_or331188842AtMost @ A @ L @ H )
= ( set_or331188842AtMost @ A @ L4 @ H2 ) )
= ( ( ( L = L4 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L4 @ H2 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_88_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_89_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_90_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_91_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_92_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_93_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or1433837966ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_94_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B4 )
| ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_95_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_96_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_97_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [L: A,H: A,H2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H2 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ H @ H2 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_98_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [H: A,L4: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or331188842AtMost @ A @ L4 @ H2 ) ) ) ).
% not_Iic_le_Icc
thf(fact_99_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
= ( ( ~ ( ord_less_eq @ A @ A4 @ B4 )
| ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D )
& ( ( ord_less @ A @ C @ A4 )
| ( ord_less @ A @ B4 @ D ) ) ) )
& ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_100_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
= ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less @ A @ B4 @ D ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_101_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
= ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_102_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
= ( set_or1433837966ssThan @ A @ C @ D ) )
=> ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ C @ D )
=> ( B4 = D ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_103_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
= ( set_or1433837966ssThan @ A @ C @ D ) )
=> ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ C @ D )
=> ( A4 = C ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_104_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ C @ D )
=> ( ( ( set_or1433837966ssThan @ A @ A4 @ B4 )
= ( set_or1433837966ssThan @ A @ C @ D ) )
= ( ( A4 = C )
& ( B4 = D ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_105_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_106_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [H2: A,L: A,H: A] :
( ( set_ord_atMost @ A @ H2 )
!= ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% not_Iic_eq_Icc
thf(fact_107_ComplD,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C @ A2 ) ) ).
% ComplD
thf(fact_108_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_109_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less @ B @ X2 @ Y4 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_110_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C: B] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_111_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less @ B @ X2 @ Y4 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_112_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_113_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_114_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% gt_ex
thf(fact_115_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neqE
thf(fact_116_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
= ( ( ord_less @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_117_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_118_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_119_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_120_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_asym
thf(fact_121_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_122_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z )
=> ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% less_trans
thf(fact_123_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_linear
thf(fact_124_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_125_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_126_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_127_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_128_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_129_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_not_sym
thf(fact_130_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A4: A] :
( ! [X2: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_131_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less @ A @ Y2 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_132_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_133_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,P: $o] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_134_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_135_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_136_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_137_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_138_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_139_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_140_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_141_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_142_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A4 ) @ ( set_ord_lessThan @ A @ B4 ) )
= ( ord_less @ A @ A4 @ B4 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_143_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_144_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_145_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_146_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_147_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_148_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_149_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X3: A,Y2: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_150_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_151_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_152_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_153_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_154_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_155_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_156_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_157_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ~ ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_158_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_159_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% le_less_linear
thf(fact_160_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,Z: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% dense_le
thf(fact_161_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y2: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z ) ) ) ).
% dense_ge
thf(fact_162_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% less_le_trans
thf(fact_163_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z )
=> ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% le_less_trans
thf(fact_164_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_165_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_166_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% less_imp_le
thf(fact_167_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_168_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% not_less
thf(fact_169_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% not_le
thf(fact_170_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_171_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_172_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_173_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less @ B @ X2 @ Y4 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_174_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( X != Y ) ) ) ) ) ).
% less_le
thf(fact_175_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ) ).
% le_less
thf(fact_176_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% leI
thf(fact_177_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% leD
thf(fact_178_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_179_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_180_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_181_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_182_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_183_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_184_psubsetE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_185_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [L4: A,L: A,H: A] :
( ( set_ord_atLeast @ A @ L4 )
!= ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% not_Ici_eq_Icc
thf(fact_186_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_187_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [H: A,L4: A] :
( ( set_ord_atMost @ A @ H )
!= ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_eq_Ici
thf(fact_188_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) ).
% strict_monoD
thf(fact_189_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( order_strict_mono @ A @ B @ F ) ) ) ).
% strict_monoI
thf(fact_190_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_191_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) )
= ( ord_less @ A @ X3 @ Y2 ) ) ) ) ).
% strict_mono_less
thf(fact_192_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [L: A,L4: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or331188842AtMost @ A @ L4 @ H2 ) ) ) ).
% not_Ici_le_Icc
thf(fact_193_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1433837966ssThan @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
| ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_194_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [L: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_Ici_le_Iic
thf(fact_195_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less @ A @ A4 @ B4 ) ) ) ).
% neg_less_iff_less
thf(fact_196_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% neg_le_iff_le
thf(fact_197_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_198_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ( uminus_uminus @ A @ X3 )
= ( uminus_uminus @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% compl_eq_compl_iff
thf(fact_199_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
= X3 ) ) ).
% double_compl
thf(fact_200_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B4 ) )
= ( A4 = B4 ) ) ) ).
% neg_equal_iff_equal
thf(fact_201_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% uminus_apply
thf(fact_202_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
= A4 ) ) ).
% add.inverse_inverse
thf(fact_203_psubsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_204_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_205_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A3: A > B,X: A] : ( uminus_uminus @ B @ ( A3 @ X ) ) ) ) ) ).
% fun_Compl_def
thf(fact_206_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( A4
= ( uminus_uminus @ A @ B4 ) )
= ( B4
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% equation_minus_iff
thf(fact_207_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ( uminus_uminus @ A @ A4 )
= B4 )
= ( ( uminus_uminus @ A @ B4 )
= A4 ) ) ) ).
% minus_equation_iff
thf(fact_208_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_imp_neg_le
thf(fact_209_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% minus_le_iff
thf(fact_210_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
= ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_minus_iff
thf(fact_211_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% compl_le_swap2
thf(fact_212_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% compl_le_swap1
thf(fact_213_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% compl_mono
thf(fact_214_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ( ord_less @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% compl_less_swap1
thf(fact_215_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y2: A,X3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% compl_less_swap2
thf(fact_216_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ B4 ) )
= ( ord_less @ A @ B4 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% less_minus_iff
thf(fact_217_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ B4 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ A4 ) ) ) ).
% minus_less_iff
thf(fact_218_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
= ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% compl_less_compl_iff
thf(fact_219_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 ) )
& ! [D2: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A4 @ X2 )
& ( ord_less @ A @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_220_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T3 ) ) ) ).
% pinf(6)
thf(fact_221_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_222_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_223_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_224_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( X4 != T3 ) ) ) ).
% pinf(3)
thf(fact_225_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( X4 != T3 ) ) ) ).
% pinf(4)
thf(fact_226_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T3 ) ) ) ).
% pinf(5)
thf(fact_227_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ord_less @ A @ T3 @ X4 ) ) ) ).
% pinf(7)
thf(fact_228_pinf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D3] :
? [Z3: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ Z3 @ X4 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_229_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_230_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_231_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( X4 != T3 ) ) ) ).
% minf(3)
thf(fact_232_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( X4 != T3 ) ) ) ).
% minf(4)
thf(fact_233_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ord_less @ A @ X4 @ T3 ) ) ) ).
% minf(5)
thf(fact_234_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ~ ( ord_less @ A @ T3 @ X4 ) ) ) ).
% minf(7)
thf(fact_235_minf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D3] :
? [Z3: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ X4 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_236_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ~ ( ord_less_eq @ A @ T3 @ X4 ) ) ) ).
% minf(8)
thf(fact_237_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z3 )
=> ( ord_less_eq @ A @ X4 @ T3 ) ) ) ).
% minf(6)
thf(fact_238_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T3: A] :
? [Z3: A] :
! [X4: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ord_less_eq @ A @ T3 @ X4 ) ) ) ).
% pinf(8)
thf(fact_239_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or1433837966ssThan @ A @ C @ D ) )
= ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_240_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C @ D ) )
= ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_241_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or578182835ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_242_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or578182835ssThan @ A @ C @ D ) )
= ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_243_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ M ) @ ( set_or578182835ssThan @ A @ M @ U ) )
= ( set_or1433837966ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_244_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or1361889807AtMost @ A @ C @ D ) )
= ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_245_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% sup_apply
thf(fact_246_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ A4 )
= A4 ) ) ).
% sup.idem
thf(fact_247_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ X3 )
= X3 ) ) ).
% sup_idem
thf(fact_248_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B4 ) )
= ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% sup.left_idem
thf(fact_249_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% sup_left_idem
thf(fact_250_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B4 ) @ B4 )
= ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% sup.right_idem
thf(fact_251_UnCI,axiom,
! [A: $tType,C: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C @ B2 )
=> ( member @ A @ C @ A2 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_252_Un__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C @ A2 )
| ( member @ A @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_253_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B4: A,C: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C ) @ A4 )
= ( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% sup.bounded_iff
thf(fact_254_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
= ( ( ord_less_eq @ A @ X3 @ Z )
& ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_255_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_subset_iff
%----Type constructors (19)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
=> ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A7: $tType,A8: $tType] :
( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
=> ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
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_Ouminus,axiom,
! [A7: $tType,A8: $tType] :
( ( uminus @ A8 @ ( type2 @ A8 ) )
=> ( uminus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_6,axiom,
! [A7: $tType] : ( uminus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_7,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_8,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_10,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_11,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_12,axiom,
uminus @ $o @ ( type2 @ $o ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( ( gram_L608943123e_subs @ a @ l1 @ l2 )
& ( gram_L608943123e_subs @ a @ l2 @ l1 ) ) ).
thf(conj_1,conjecture,
gram_L608943123e_subs @ a @ l1 @ l2 ).
%------------------------------------------------------------------------------