TPTP Problem File: ITP079^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP079^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_209__6623758_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Irreducible/prob_209__6623758_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 337 ( 49 unt; 48 typ; 0 def)
% Number of atoms : 1027 ( 179 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3347 ( 79 ~; 21 |; 72 &;2652 @)
% ( 0 <=>; 523 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 185 ( 185 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 4 con; 0-6 aty)
% Number of variables : 1105 ( 68 ^; 974 !; 10 ?;1105 :)
% ( 53 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:39.550
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_val,type,
val: $tType ).
% Explicit typings (45)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_Misc_Oswap__args2,type,
swap_args2:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C > A ) > C > B > 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_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
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_RefineG__Transfer_OREFINEG__TRANSFER__ALIGN,type,
refine111645177_ALIGN:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_RefineG__Transfer_OREFINEG__TRANSFER__POST__SIMP,type,
refine574149253T_SIMP:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Refine__Misc_Ocont,type,
refine_cont:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Refine__Misc_Ogalois__connection,type,
refine1150083786ection:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Refine__Misc_Oinf__distrib,type,
refine_inf_distrib:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Refine__Misc_Ostrict,type,
refine_strict:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Refine__Mono__Prover_Omono__setup__loc,type,
refine551993528up_loc:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
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_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_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_P,type,
p: set @ val ).
thf(sy_v_scc,type,
scc: set @ val ).
% Relevant facts (255)
thf(fact_0__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Ascc_A_092_060Longrightarrow_062_Ax_A_092_060in_062_AP_092_060close_062,axiom,
! [X: val] :
( ( member @ val @ X @ scc )
=> ( member @ val @ X @ p ) ) ).
% \<open>\<And>x. x \<in> scc \<Longrightarrow> x \<in> P\<close>
thf(fact_1_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_2_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_3_order__mono__setup_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_mono_setup.refl
thf(fact_4_relprop__triggers_I6_J,axiom,
! [I: $tType,R: set @ I,R2: set @ I] :
( ( ord_less_eq @ ( set @ I ) @ R @ R2 )
=> ( ord_less_eq @ ( set @ I ) @ R @ R2 ) ) ).
% relprop_triggers(6)
thf(fact_5_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_6_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_7_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_8_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B3: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_9_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_10_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_11_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_12_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_13_order__mono__setup_Omono__let,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [F: B > A,F2: B > A,X3: B] :
( ! [X2: B] : ( ord_less_eq @ A @ ( F @ X2 ) @ ( F2 @ X2 ) )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F2 @ X3 ) ) ) ) ).
% order_mono_setup.mono_let
thf(fact_14_order__mono__setup_Omono__if,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [T2: A,T3: A,E: A,E2: A,B4: $o] :
( ( ord_less_eq @ A @ T2 @ T3 )
=> ( ( ord_less_eq @ A @ E @ E2 )
=> ( ord_less_eq @ A @ ( if @ A @ B4 @ T2 @ E ) @ ( if @ A @ B4 @ T3 @ E2 ) ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_15_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_16_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [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_17_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_18_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_19_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( P @ X4 ) ) ) ) ).
% subset_Collect_conv
thf(fact_20_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_21_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_22_transfer_Otransfer__Let,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [Alpha: C > A,F: B > C,F3: B > A,X3: B] :
( ! [X2: B] : ( ord_less_eq @ A @ ( Alpha @ ( F @ X2 ) ) @ ( F3 @ X2 ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( F @ X3 ) ) @ ( F3 @ X3 ) ) ) ) ).
% transfer.transfer_Let
thf(fact_23_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_24_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_25_transfer_Otransfer__if,axiom,
! [C: $tType,A: $tType] :
( ( comple187826305attice @ A )
=> ! [B4: $o,Alpha: C > A,S1: C,S12: A,S2: C,S22: A] :
( ( B4
=> ( ord_less_eq @ A @ ( Alpha @ S1 ) @ S12 ) )
=> ( ( ~ B4
=> ( ord_less_eq @ A @ ( Alpha @ S2 ) @ S22 ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B4 @ S1 @ S2 ) ) @ ( if @ A @ B4 @ S12 @ S22 ) ) ) ) ) ).
% transfer.transfer_if
thf(fact_26_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_27_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_28_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ 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_29_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_30_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_31_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_32_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_33_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_34_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_35_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_36_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A,D: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ( C2 = D )
=> ( ord_less_eq @ A @ A4 @ D ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_37_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_38_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_39_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_40_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ) ).
% antisym
thf(fact_41_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% linear
thf(fact_42_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 = Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% eq_refl
thf(fact_43_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% le_cases
thf(fact_44_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
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
@ ^ [X4: A] : ( member @ A @ X4 @ 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,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_50_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_51_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_52_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_53_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_54_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( order_Greatest @ A @ P )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_55_Refine__Misc_Oif__mono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B4: $o,M1: A,M12: A,M2: A,M22: A] :
( ( B4
=> ( ord_less_eq @ A @ M1 @ M12 ) )
=> ( ( ~ B4
=> ( ord_less_eq @ A @ M2 @ M22 ) )
=> ( ord_less_eq @ A @ ( if @ A @ B4 @ M1 @ M2 ) @ ( if @ A @ B4 @ M12 @ M22 ) ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_56_lhs__step__If,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B4: $o,T2: A,M: A,E: A] :
( ( B4
=> ( ord_less_eq @ A @ T2 @ M ) )
=> ( ( ~ B4
=> ( ord_less_eq @ A @ E @ M ) )
=> ( ord_less_eq @ A @ ( if @ A @ B4 @ T2 @ E ) @ M ) ) ) ) ).
% lhs_step_If
thf(fact_57_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B4: A] :
( ( A4 = B4 )
| ~ ( ord_less_eq @ A @ A4 @ B4 )
| ~ ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% verit_la_disequality
thf(fact_58_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X5: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_59_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ 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_60_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ 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_61_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F4 @ Y4 ) @ ( F4 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_62_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_63_order__mono__setup_Omono__setup__loc__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( refine551993528up_loc @ A @ ( ord_less_eq @ A ) ) ) ).
% order_mono_setup.mono_setup_loc_axioms
thf(fact_64_swap__args2__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( swap_args2 @ B @ C @ A )
= ( ^ [F4: B > C > A,X4: C,Y4: B] : ( F4 @ Y4 @ X4 ) ) ) ).
% swap_args2_def
thf(fact_65_galois__connection_O_092_060gamma_062_092_060alpha_062__infl,axiom,
! [B: $tType,A: $tType] :
( ( ( comple187826305attice @ A )
& ( comple187826305attice @ B ) )
=> ! [Alpha: A > B,Gamma: B > A,X3: A] :
( ( refine1150083786ection @ A @ B @ Alpha @ Gamma )
=> ( ord_less_eq @ A @ X3 @ ( Gamma @ ( Alpha @ X3 ) ) ) ) ) ).
% galois_connection.\<gamma>\<alpha>_infl
thf(fact_66_galois__connection_O_092_060alpha_062_092_060gamma_062__defl,axiom,
! [A: $tType,B: $tType] :
( ( ( comple187826305attice @ B )
& ( comple187826305attice @ A ) )
=> ! [Alpha: A > B,Gamma: B > A,X3: B] :
( ( refine1150083786ection @ A @ B @ Alpha @ Gamma )
=> ( ord_less_eq @ B @ ( Alpha @ ( Gamma @ X3 ) ) @ X3 ) ) ) ).
% galois_connection.\<alpha>\<gamma>_defl
thf(fact_67_galois__connection_Ogalois,axiom,
! [A: $tType,B: $tType] :
( ( ( comple187826305attice @ B )
& ( comple187826305attice @ A ) )
=> ! [Alpha: A > B,Gamma: B > A,C2: A,A4: B] :
( ( refine1150083786ection @ A @ B @ Alpha @ Gamma )
=> ( ( ord_less_eq @ A @ C2 @ ( Gamma @ A4 ) )
= ( ord_less_eq @ B @ ( Alpha @ C2 ) @ A4 ) ) ) ) ).
% galois_connection.galois
thf(fact_68_galois__connection_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( ( comple187826305attice @ B )
& ( comple187826305attice @ A ) )
=> ! [Gamma: B > A,Alpha: A > B] :
( ! [C4: A,A6: B] :
( ( ord_less_eq @ A @ C4 @ ( Gamma @ A6 ) )
= ( ord_less_eq @ B @ ( Alpha @ C4 ) @ A6 ) )
=> ( refine1150083786ection @ A @ B @ Alpha @ Gamma ) ) ) ).
% galois_connection.intro
thf(fact_69_galois__connection__def,axiom,
! [B: $tType,A: $tType] :
( ( ( comple187826305attice @ A )
& ( comple187826305attice @ B ) )
=> ( ( refine1150083786ection @ A @ B )
= ( ^ [Alpha2: A > B,Gamma2: B > A] :
! [C5: A,A5: B] :
( ( ord_less_eq @ A @ C5 @ ( Gamma2 @ A5 ) )
= ( ord_less_eq @ B @ ( Alpha2 @ C5 ) @ A5 ) ) ) ) ) ).
% galois_connection_def
thf(fact_70_mono__setup__loc__def,axiom,
! [A: $tType] :
( ( refine551993528up_loc @ A )
= ( ^ [Le: A > A > $o] :
! [X4: A] : ( Le @ X4 @ X4 ) ) ) ).
% mono_setup_loc_def
thf(fact_71_mono__setup__loc_Orefl,axiom,
! [A: $tType,Le2: A > A > $o,X3: A] :
( ( refine551993528up_loc @ A @ Le2 )
=> ( Le2 @ X3 @ X3 ) ) ).
% mono_setup_loc.refl
thf(fact_72_mono__setup__loc_Ointro,axiom,
! [A: $tType,Le2: A > A > $o] :
( ! [X2: A] : ( Le2 @ X2 @ X2 )
=> ( refine551993528up_loc @ A @ Le2 ) ) ).
% mono_setup_loc.intro
thf(fact_73_mono__setup__loc_Omono__if,axiom,
! [A: $tType,Le2: A > A > $o,T2: A,T3: A,E: A,E2: A,B4: $o] :
( ( refine551993528up_loc @ A @ Le2 )
=> ( ( Le2 @ T2 @ T3 )
=> ( ( Le2 @ E @ E2 )
=> ( Le2 @ ( if @ A @ B4 @ T2 @ E ) @ ( if @ A @ B4 @ T3 @ E2 ) ) ) ) ) ).
% mono_setup_loc.mono_if
thf(fact_74_mono__setup__loc_Omono__let,axiom,
! [A: $tType,B: $tType,Le2: A > A > $o,F: B > A,F2: B > A,X3: B] :
( ( refine551993528up_loc @ A @ Le2 )
=> ( ! [X2: B] : ( Le2 @ ( F @ X2 ) @ ( F2 @ X2 ) )
=> ( Le2 @ ( F @ X3 ) @ ( F2 @ X3 ) ) ) ) ).
% mono_setup_loc.mono_let
thf(fact_75_galois__connection_Oinf__dist___092_060alpha_062,axiom,
! [B: $tType,A: $tType] :
( ( ( comple187826305attice @ A )
& ( comple187826305attice @ B ) )
=> ! [Alpha: A > B,Gamma: B > A] :
( ( refine1150083786ection @ A @ B @ Alpha @ Gamma )
=> ( refine_inf_distrib @ A @ B @ Alpha ) ) ) ).
% galois_connection.inf_dist_\<alpha>
thf(fact_76_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ 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_77_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ 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_78_START__REFINEG__TRANSFER,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [D: A,C2: A,A4: A] :
( ( refine111645177_ALIGN @ A @ A @ D @ C2 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ( refine574149253T_SIMP @ A @ C2 @ D )
=> ( ord_less_eq @ A @ D @ A4 ) ) ) ) ) ).
% START_REFINEG_TRANSFER
thf(fact_79_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ 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_80_transfer_Otransfer__rec__list,axiom,
! [C: $tType,D2: $tType,A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [Alpha: C > A,Fn: B > C,Fn2: B > A,Fc: D2 > ( list @ D2 ) > ( B > C ) > B > C,Fc2: D2 > ( list @ D2 ) > ( B > A ) > B > A,L: list @ D2,S3: B] :
( ! [S4: B] : ( ord_less_eq @ A @ ( Alpha @ ( Fn @ S4 ) ) @ ( Fn2 @ S4 ) )
=> ( ! [X2: D2,L2: list @ D2,Rec: B > C,Rec2: B > A,S4: B] :
( ! [Sa: B] : ( ord_less_eq @ A @ ( Alpha @ ( Rec @ Sa ) ) @ ( Rec2 @ Sa ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( Fc @ X2 @ L2 @ Rec @ S4 ) ) @ ( Fc2 @ X2 @ L2 @ Rec2 @ S4 ) ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( rec_list @ ( B > C ) @ D2 @ Fn @ Fc @ L @ S3 ) ) @ ( rec_list @ ( B > A ) @ D2 @ Fn2 @ Fc2 @ L @ S3 ) ) ) ) ) ).
% transfer.transfer_rec_list
thf(fact_81_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ 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_82_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_lessThan @ A @ X3 )
= ( set_ord_lessThan @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% lessThan_eq_iff
thf(fact_83_atLeast__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_atLeast @ A @ X3 )
= ( set_ord_atLeast @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% atLeast_eq_iff
thf(fact_84_greaterThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_greaterThan @ A @ X3 )
= ( set_ord_greaterThan @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% greaterThan_eq_iff
thf(fact_85_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I2 ) ) ) ).
% atLeast_iff
thf(fact_86_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ 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_87_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A4 ) @ ( set_ord_atLeast @ A @ A4 ) ) ) ).
% Ioi_le_Ico
thf(fact_88_STOP__REFINEG__TRANSFER,axiom,
! [A: $tType,C2: A] : ( refine574149253T_SIMP @ A @ C2 @ C2 ) ).
% STOP_REFINEG_TRANSFER
thf(fact_89_REFINEG__TRANSFER__ALIGNI,axiom,
! [A: $tType,B: $tType,X3: A,Y2: B] : ( refine111645177_ALIGN @ A @ B @ X3 @ Y2 ) ).
% REFINEG_TRANSFER_ALIGNI
thf(fact_90_REFINEG__TRANSFER__ALIGN__def,axiom,
! [B: $tType,A: $tType] :
( ( refine111645177_ALIGN @ A @ B )
= ( ^ [X4: A,Y4: B] : $true ) ) ).
% REFINEG_TRANSFER_ALIGN_def
thf(fact_91_REFINEG__TRANSFER__POST__SIMP__def,axiom,
! [A: $tType] :
( ( refine574149253T_SIMP @ A )
= ( ^ [Y: A,Z: A] : Y = Z ) ) ).
% REFINEG_TRANSFER_POST_SIMP_def
thf(fact_92_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ 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_93_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,L3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ L3 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_94_inf__distribD_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ( complete_Sup @ A )
& ( bot @ A )
& ( complete_Sup @ B )
& ( bot @ B ) )
=> ! [F: A > B] :
( ( refine_inf_distrib @ A @ B @ F )
=> ( refine_strict @ A @ B @ F ) ) ) ).
% inf_distribD(1)
thf(fact_95_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_96_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_97_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ 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_98_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_99_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B4: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B4 ) )
= B4 ) ) ).
% verit_minus_simplify(4)
thf(fact_100_atMost__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ( set_ord_atMost @ A @ X3 )
= ( set_ord_atMost @ A @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% atMost_eq_iff
thf(fact_101_ComplI,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ~ ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_102_Compl__iff,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_103_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_104_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or331188842AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_105_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H: A,L3: A,H2: A] :
( ( ( set_or331188842AtMost @ A @ L @ H )
= ( set_or331188842AtMost @ A @ L3 @ H2 ) )
= ( ( ( L = L3 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L3 @ H2 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_106_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I2 @ K ) ) ) ).
% atMost_iff
thf(fact_107_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I2 @ K ) ) ) ).
% lessThan_iff
thf(fact_108_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_109_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_110_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I2 ) ) ) ).
% greaterThan_iff
thf(fact_111_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B4: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C2 @ D ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B4 )
| ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_112_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_113_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_114_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ 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_115_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ 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_116_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L: A,H: A] :
( ( set_ord_atMost @ A @ H2 )
!= ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% not_Iic_eq_Icc
thf(fact_117_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L3: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or331188842AtMost @ A @ L3 @ H2 ) ) ) ).
% not_Iic_le_Icc
thf(fact_118_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B4: A,C2: A,D: A] :
( ( ord_less @ ( set @ A ) @ ( set_or331188842AtMost @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C2 @ D ) )
= ( ( ~ ( ord_less_eq @ A @ A4 @ B4 )
| ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B4 @ D )
& ( ( ord_less @ A @ C2 @ A4 )
| ( ord_less @ A @ B4 @ D ) ) ) )
& ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_119_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% verit_comp_simplify1(1)
thf(fact_120_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( A4 = B4 )
=> ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B4 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_121_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_122_all__nat__split__at,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A,P: A > $o] :
( ! [I3: A] :
( ( ord_less @ A @ I3 @ K )
=> ( P @ I3 ) )
=> ( ( P @ K )
=> ( ! [I3: A] :
( ( ord_less @ A @ K @ I3 )
=> ( P @ I3 ) )
=> ! [X_1: A] : ( P @ X_1 ) ) ) ) ) ).
% all_nat_split_at
thf(fact_123_ComplD,axiom,
! [A: $tType,C2: A,A2: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C2 @ A2 ) ) ).
% ComplD
thf(fact_124_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_125_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_126_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B4: A,F: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_127_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_128_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_129_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% lt_ex
thf(fact_130_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_12: A] : ( ord_less @ A @ X3 @ X_12 ) ) ).
% gt_ex
thf(fact_131_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neqE
thf(fact_132_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
= ( ( ord_less @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_133_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% order.asym
thf(fact_134_dense,axiom,
! [A: $tType] :
( ( dense_order @ 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_135_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_136_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_asym
thf(fact_137_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% less_asym'
thf(fact_138_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_139_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_linear
thf(fact_140_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_141_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( A4 = B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_142_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_143_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_144_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_145_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_not_sym
thf(fact_146_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ 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_147_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less @ A @ Y2 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_148_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_149_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,P: $o] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_150_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_151_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_152_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_153_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_154_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X6: A] : ( P2 @ X6 ) )
= ( ^ [P3: A > $o] :
? [N2: A] :
( ( P3 @ N2 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N2 )
=> ~ ( P3 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_155_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B4: A] :
( ! [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A4 @ B4 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_156_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_157_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ 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_158_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( A4 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_159_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( A4 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_160_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ 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_161_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% leD
thf(fact_162_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% leI
thf(fact_163_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ) ).
% le_less
thf(fact_164_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ) ).
% less_le
thf(fact_165_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_166_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_167_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_168_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B4: A,F: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_169_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% not_le
thf(fact_170_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% not_less
thf(fact_171_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% le_neq_trans
thf(fact_172_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_173_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_174_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% less_imp_le
thf(fact_175_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_176_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_177_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y2: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_178_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y2: A,Z2: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_179_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% le_less_linear
thf(fact_180_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ 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_181_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_182_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_183_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_184_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A,C2: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_185_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_186_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_187_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_188_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_189_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X3: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_190_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_191_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% order.strict_implies_order
thf(fact_192_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_193_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_194_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_195_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ 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_196_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B7: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
= ( ord_less @ B @ A7 @ B7 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_197_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_198_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_199_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_200_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_201_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_202_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_203_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_204_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_205_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L3: A,L: A,H: A] :
( ( set_ord_atLeast @ A @ L3 )
!= ( set_or331188842AtMost @ A @ L @ H ) ) ) ).
% not_Ici_eq_Icc
thf(fact_206_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H: A,L3: A] :
( ( set_ord_atMost @ A @ H )
!= ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_eq_Ici
thf(fact_207_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ 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_208_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( order_strict_mono @ A @ B @ F ) ) ) ).
% strict_monoI
thf(fact_209_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_210_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ 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_211_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L3: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or331188842AtMost @ A @ L3 @ H2 ) ) ) ).
% not_Ici_le_Icc
thf(fact_212_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ 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_213_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_le_Ici
thf(fact_214_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ 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_215_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ 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_216_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ 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_217_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ 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_218_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ 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_219_psubsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_220_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_221_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ 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_222_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ 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_223_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ 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_224_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ 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 )
& ! [X7: A] :
( ( ( ord_less_eq @ A @ A4 @ X7 )
& ( ord_less @ A @ X7 @ C4 ) )
=> ( P @ X7 ) )
& ! [D3: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A4 @ X2 )
& ( ord_less @ A @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D3 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_225_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ~ ( ord_less_eq @ A @ X7 @ T2 ) ) ) ).
% pinf(6)
thf(fact_226_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X7 ) ) ) ).
% minf(8)
thf(fact_227_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ord_less_eq @ A @ X7 @ T2 ) ) ) ).
% minf(6)
thf(fact_228_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ord_less_eq @ A @ T2 @ X7 ) ) ) ).
% pinf(8)
thf(fact_229_inf__distribI,axiom,
! [B: $tType,A: $tType] :
( ( ( complete_Sup @ A )
& ( bot @ A )
& ( complete_Sup @ B )
& ( bot @ B ) )
=> ! [F: A > B] :
( ( refine_strict @ A @ B @ F )
=> ( ( refine_cont @ A @ B @ F )
=> ( refine_inf_distrib @ A @ B @ F ) ) ) ) ).
% inf_distribI
thf(fact_230_inf__distrib__def,axiom,
! [B: $tType,A: $tType] :
( ( ( complete_Sup @ A )
& ( bot @ A )
& ( complete_Sup @ B )
& ( bot @ B ) )
=> ( ( refine_inf_distrib @ A @ B )
= ( ^ [F4: A > B] :
( ( refine_strict @ A @ B @ F4 )
& ( refine_cont @ A @ B @ F4 ) ) ) ) ) ).
% inf_distrib_def
thf(fact_231_inf__distribD_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ( complete_Sup @ A )
& ( bot @ A )
& ( complete_Sup @ B )
& ( bot @ B ) )
=> ! [F: A > B] :
( ( refine_inf_distrib @ A @ B @ F )
=> ( refine_cont @ A @ B @ F ) ) ) ).
% inf_distribD(2)
thf(fact_232_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B4: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A4 @ B4 ) @ ( set_or331188842AtMost @ A @ C2 @ D ) )
= ( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B4 @ D ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_233_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or331188842AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_234_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A2 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_235_Un__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C2 @ A2 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_236_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
& ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_237_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C2 ) @ A4 )
= ( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% sup.bounded_iff
thf(fact_238_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
thf(fact_239_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or578182835ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_240_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B4: A,A4: A] :
( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ) ).
% sup.coboundedI2
thf(fact_241_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A4: A,B4: A] :
( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ) ).
% sup.coboundedI1
thf(fact_242_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_243_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_244_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A4: A] : ( ord_less_eq @ A @ B4 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% sup.cobounded2
thf(fact_245_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B4: A] : ( ord_less_eq @ A @ A4 @ ( sup_sup @ A @ A4 @ B4 ) ) ) ).
% sup.cobounded1
thf(fact_246_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_247_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C2 ) @ A4 ) ) ) ) ).
% sup.boundedI
thf(fact_248_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B4 @ C2 ) @ A4 )
=> ~ ( ( ord_less_eq @ A @ B4 @ A4 )
=> ~ ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% sup.boundedE
thf(fact_249_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( sup_sup @ A @ X3 @ Y2 )
= Y2 ) ) ) ).
% sup_absorb2
thf(fact_250_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( sup_sup @ A @ X3 @ Y2 )
= X3 ) ) ) ).
% sup_absorb1
thf(fact_251_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( sup_sup @ A @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb2
thf(fact_252_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( sup_sup @ A @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb1
thf(fact_253_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X3: A,Y2: A] :
( ! [X2: A,Y3: A] : ( ord_less_eq @ A @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ( ord_less_eq @ A @ ( F @ Y3 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup @ A @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ) ).
% sup_unique
thf(fact_254_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B4: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B4 ) )
=> ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% sup.orderI
% Subclasses (4)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ord @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( order @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( preorder @ A ) ) ).
% Type constructors (25)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A8: $tType,A9: $tType] :
( ( comple187826305attice @ A9 )
=> ( comple187826305attice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 )
=> ( semilattice_sup @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A8: $tType,A9: $tType] :
( ( boolean_algebra @ A9 )
=> ( boolean_algebra @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A8: $tType,A9: $tType] :
( ( complete_Sup @ A9 )
=> ( complete_Sup @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
! [A8: $tType] : ( comple187826305attice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_2,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_3,axiom,
! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_4,axiom,
! [A8: $tType] : ( complete_Sup @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_9,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_10,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_12,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_13,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $true @ X3 @ Y2 )
= X3 ) ).
% Free types (1)
thf(tfree_0,hypothesis,
linorder @ val ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ ( set @ val ) @ scc @ p ).
%------------------------------------------------------------------------------