TPTP Problem File: ITP163^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP163^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_374__3586996_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 : Refine_Basic/prob_374__3586996_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.2.0, 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 306 ( 67 unt; 44 typ; 0 def)
% Number of atoms : 926 ( 241 equ; 0 cnn)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 4080 ( 141 ~; 20 |; 41 &;3323 @)
% ( 0 <=>; 555 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 179 ( 179 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 43 usr; 3 con; 0-5 aty)
% Number of variables : 1218 ( 64 ^;1061 !; 45 ?;1218 :)
% ( 48 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:02.950
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
refine1665802226e_nres: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (40)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_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_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
refine1687780735RETURN:
!>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
refine1315500908_inres:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__eq__nres__rel,type,
refine1554218259es_rel:
!>[A: $tType] : ( ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oless__nres__rel,type,
refine1378444575es_rel:
!>[A: $tType] : ( ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
refine406925620_inres:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
refine1102455758nofail:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_OFAILi,type,
refine1767639642_FAILi:
!>[A: $tType] : ( refine1665802226e_nres @ A ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
refine605929679le_RES:
!>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Ocase__nres,type,
refine60818195e_nres:
!>[B: $tType,A: $tType] : ( B > ( ( set @ A ) > B ) > ( refine1665802226e_nres @ A ) > B ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_Orec__nres,type,
refine1442219249c_nres:
!>[C: $tType,A: $tType] : ( C > ( ( set @ A ) > C ) > ( refine1665802226e_nres @ A ) > C ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
refine1672542526he_RES:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v__092_060Phi_062,type,
phi: a > $o ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (245)
thf(fact_0_nres__more__simps_I4_J,axiom,
! [A: $tType,X: set @ A,Y: set @ A] :
( ( ( refine605929679le_RES @ A @ X )
= ( refine605929679le_RES @ A @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(4)
thf(fact_1_nres_Oinject,axiom,
! [A: $tType,X2: set @ A,Y2: set @ A] :
( ( ( refine605929679le_RES @ A @ X2 )
= ( refine605929679le_RES @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_2_nf__inres__RES,axiom,
! [A: $tType,X: set @ A,X3: A] :
( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X ) @ X3 )
= ( member @ A @ X3 @ X ) ) ).
% nf_inres_RES
thf(fact_3_the__RES_Osimps,axiom,
! [A: $tType,X: set @ A] :
( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
= X ) ).
% the_RES.simps
thf(fact_4_nres_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F2: ( set @ A ) > C,X2: set @ A] :
( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
= ( F2 @ X2 ) ) ).
% nres.simps(7)
thf(fact_5_nres_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: ( set @ A ) > B,X2: set @ A] :
( ( refine60818195e_nres @ B @ A @ F1 @ F2 @ ( refine605929679le_RES @ A @ X2 ) )
= ( F2 @ X2 ) ) ).
% nres.simps(5)
thf(fact_6_inres__simps_I2_J,axiom,
! [B: $tType,X: set @ B] :
( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X ) )
= ( ^ [X4: B] : ( member @ B @ X4 @ X ) ) ) ).
% inres_simps(2)
thf(fact_7_nofail__simps_I2_J,axiom,
! [B: $tType,X: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X ) ) ).
% nofail_simps(2)
thf(fact_8_nofail__RES__conv,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [M: refine1665802226e_nres @ A] :
? [M2: set @ A] :
( M
= ( refine605929679le_RES @ A @ M2 ) ) ) ) ).
% nofail_RES_conv
thf(fact_9_nres_Odistinct_I1_J,axiom,
! [A: $tType,X2: set @ A] :
( ( refine1767639642_FAILi @ A )
!= ( refine605929679le_RES @ A @ X2 ) ) ).
% nres.distinct(1)
thf(fact_10_nres_Oinduct,axiom,
! [A: $tType,P: ( refine1665802226e_nres @ A ) > $o,Nres: refine1665802226e_nres @ A] :
( ( P @ ( refine1767639642_FAILi @ A ) )
=> ( ! [X5: set @ A] : ( P @ ( refine605929679le_RES @ A @ X5 ) )
=> ( P @ Nres ) ) ) ).
% nres.induct
thf(fact_11_nres_Oexhaust,axiom,
! [A: $tType,Y3: refine1665802226e_nres @ A] :
( ( Y3
!= ( refine1767639642_FAILi @ A ) )
=> ~ ! [X22: set @ A] :
( Y3
!= ( refine605929679le_RES @ A @ X22 ) ) ) ).
% nres.exhaust
thf(fact_12_sup__nres_Oinduct,axiom,
! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
=> ( ! [V: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) )
=> ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sup_nres.induct
thf(fact_13_less__nres_Oinduct,axiom,
! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
( ! [X_1: refine1665802226e_nres @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ X_1 )
=> ( ! [Uv: set @ A] : ( P @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) )
=> ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_nres.induct
thf(fact_14_less__eq__nres_Oinduct,axiom,
! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
=> ( ! [A2: set @ A,B2: set @ A] : ( P @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
=> ( ! [Uv: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_eq_nres.induct
thf(fact_15_the__RES__inv,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ M3 )
=> ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M3 ) )
= M3 ) ) ).
% the_RES_inv
thf(fact_16_nres_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F2: ( set @ A ) > C] :
( ( refine1442219249c_nres @ C @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
= F1 ) ).
% nres.simps(6)
thf(fact_17_nres_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: ( set @ A ) > B] :
( ( refine60818195e_nres @ B @ A @ F1 @ F2 @ ( refine1767639642_FAILi @ A ) )
= F1 ) ).
% nres.simps(4)
thf(fact_18_pw__eqI,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S )
= ( refine1102455758nofail @ A @ S2 ) )
=> ( ! [X5: A] :
( ( refine1315500908_inres @ A @ S @ X5 )
= ( refine1315500908_inres @ A @ S2 @ X5 ) )
=> ( S = S2 ) ) ) ).
% pw_eqI
thf(fact_19_pw__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : Y4 = Z )
= ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S3 )
= ( refine1102455758nofail @ A @ S4 ) )
& ! [X4: A] :
( ( refine1315500908_inres @ A @ S3 @ X4 )
= ( refine1315500908_inres @ A @ S4 @ X4 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_20_not__nofail__inres,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,X3: A] :
( ~ ( refine1102455758nofail @ A @ S )
=> ( refine1315500908_inres @ A @ S @ X3 ) ) ).
% not_nofail_inres
thf(fact_21_nf__inres__def,axiom,
! [A: $tType] :
( ( refine406925620_inres @ A )
= ( ^ [M: refine1665802226e_nres @ A,X4: A] :
( ( refine1102455758nofail @ A @ M )
& ( refine1315500908_inres @ A @ M @ X4 ) ) ) ) ).
% nf_inres_def
thf(fact_22_inres__simps_I3_J,axiom,
! [C: $tType,X3: C] :
( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X3 ) )
= ( ^ [Y4: C,Z: C] : Y4 = Z
@ X3 ) ) ).
% inres_simps(3)
thf(fact_23_nofail__simps_I3_J,axiom,
! [C: $tType,X3: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X3 ) ) ).
% nofail_simps(3)
thf(fact_24_pw__leI,axiom,
! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S2 )
=> ( ( refine1102455758nofail @ A @ S )
& ! [X5: A] :
( ( refine1315500908_inres @ A @ S @ X5 )
=> ( refine1315500908_inres @ A @ S2 @ X5 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ).
% pw_leI
thf(fact_25_pw__leI_H,axiom,
! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S2 )
=> ( refine1102455758nofail @ A @ S ) )
=> ( ! [X5: A] :
( ( refine1102455758nofail @ A @ S2 )
=> ( ( refine1315500908_inres @ A @ S @ X5 )
=> ( refine1315500908_inres @ A @ S2 @ X5 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ) ).
% pw_leI'
thf(fact_26_pw__le__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
= ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ S4 )
=> ( ( refine1102455758nofail @ A @ S3 )
& ! [X4: A] :
( ( refine1315500908_inres @ A @ S3 @ X4 )
=> ( refine1315500908_inres @ A @ S4 @ X4 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_27_less__eq__nres_Osimps_I3_J,axiom,
! [A: $tType,Uv2: set @ A] :
~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) ) ).
% less_eq_nres.simps(3)
thf(fact_28_less__nres_Osimps_I2_J,axiom,
! [A: $tType,Uv2: set @ A] : ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv2 ) @ ( refine1767639642_FAILi @ A ) ) ).
% less_nres.simps(2)
thf(fact_29_less__eq__nres_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
( ! [Uu: refine1665802226e_nres @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
=> ( ! [A2: set @ A,B2: set @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
=> ~ ! [Uv: set @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% less_eq_nres.cases
thf(fact_30_less__nres_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
( ! [Uu: refine1665802226e_nres @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu ) )
=> ( ! [Uv: set @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) )
=> ~ ! [A2: set @ A,B2: set @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ).
% less_nres.cases
thf(fact_31_sup__nres_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
( ! [Uu: refine1665802226e_nres @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
=> ( ! [V: set @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) ) )
=> ~ ! [A2: set @ A,B2: set @ A] :
( X3
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ).
% sup_nres.cases
thf(fact_32_nres__more__simps_I6_J,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( ( refine1687780735RETURN @ A @ X3 )
= ( refine1687780735RETURN @ A @ Y3 ) )
= ( X3 = Y3 ) ) ).
% nres_more_simps(6)
thf(fact_33_nres__order__simps_I20_J,axiom,
! [W: $tType,X3: W,Y3: W] :
( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X3 ) @ ( refine1687780735RETURN @ W @ Y3 ) )
= ( X3 = Y3 ) ) ).
% nres_order_simps(20)
thf(fact_34_nres__order__simps_I21_J,axiom,
! [X6: $tType,X3: X6,Y: set @ X6] :
( ( ord_less_eq @ ( refine1665802226e_nres @ X6 ) @ ( refine1687780735RETURN @ X6 @ X3 ) @ ( refine605929679le_RES @ X6 @ Y ) )
= ( member @ X6 @ X3 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_35_inres__def,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A )
= ( ^ [S3: refine1665802226e_nres @ A,X4: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ S3 ) ) ) ).
% inres_def
thf(fact_36_less__nres_Osimps_I1_J,axiom,
! [A: $tType,Uu2: refine1665802226e_nres @ A] :
~ ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu2 ) ).
% less_nres.simps(1)
thf(fact_37_less__eq__nres_Osimps_I1_J,axiom,
! [A: $tType,Uu2: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Uu2 @ ( refine1767639642_FAILi @ A ) ) ).
% less_eq_nres.simps(1)
thf(fact_38_le__nofailI,axiom,
! [A: $tType,M4: refine1665802226e_nres @ A,M5: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ M4 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M5 @ M4 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M5 @ M4 ) ) ).
% le_nofailI
thf(fact_39_pwD1,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
=> ( ( refine1102455758nofail @ A @ S2 )
=> ( refine1102455758nofail @ A @ S ) ) ) ).
% pwD1
thf(fact_40_pwD2,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A,X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
=> ( ( refine1315500908_inres @ A @ S @ X3 )
=> ( refine1315500908_inres @ A @ S2 @ X3 ) ) ) ).
% pwD2
thf(fact_41_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_42_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_43_order__mono__setup_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_mono_setup.refl
thf(fact_44_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
& ( ord_less_eq @ A @ C2 @ B3 )
& ! [X7: A] :
( ( ( ord_less_eq @ A @ A3 @ X7 )
& ( ord_less @ A @ X7 @ C2 ) )
=> ( P @ X7 ) )
& ! [D: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq @ A @ D @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 != B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_50_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_51_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( A6 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_52_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less @ A @ B5 @ A6 )
| ( A6 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_53_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_54_less__eq__nres_Osimps_I2_J,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% less_eq_nres.simps(2)
thf(fact_55_nres__order__simps_I5_J,axiom,
! [E: $tType,X: set @ E,Y: set @ E] :
( ( ord_less_eq @ ( refine1665802226e_nres @ E ) @ ( refine605929679le_RES @ E @ X ) @ ( refine605929679le_RES @ E @ Y ) )
= ( ord_less_eq @ ( set @ E ) @ X @ Y ) ) ).
% nres_order_simps(5)
thf(fact_56_less__nres_Osimps_I3_J,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
= ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ).
% less_nres.simps(3)
thf(fact_57_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_58_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_59_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_60_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_61_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X5: B,Y5: B] :
( ( ord_less_eq @ B @ X5 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_62_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X5: A,Y5: A] :
( ( ord_less_eq @ A @ X5 @ Y5 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_63_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X5: B,Y5: B] :
( ( ord_less_eq @ B @ X5 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_64_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X5: A,Y5: A] :
( ( ord_less_eq @ A @ X5 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_66_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_67_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_68_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_69_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_70_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_71_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_72_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_73_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_74_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_75_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_76_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_77_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_78_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_79_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: A,B2: A] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_80_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_81_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_82_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_83_order__mono__setup_Omono__if,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [T2: A,T3: A,E2: A,E3: A,B3: $o] :
( ( ord_less_eq @ A @ T2 @ T3 )
=> ( ( ord_less_eq @ A @ E2 @ E3 )
=> ( ord_less_eq @ A @ ( if @ A @ B3 @ T2 @ E2 ) @ ( if @ A @ B3 @ T3 @ E3 ) ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_84_order__mono__setup_Omono__let,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [F: B > A,F4: B > A,X3: B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F @ X5 ) @ ( F4 @ X5 ) )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F4 @ X3 ) ) ) ) ).
% order_mono_setup.mono_let
thf(fact_85_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X5: A,Y5: B] :
( P2
= ( product_Pair @ A @ B @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_86_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_87_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_88_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A2: A,B2: B,C2: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_89_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A2: A,B2: B,C2: C,D3: D2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B2 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_90_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E4 ) ) ) ) ) ).
% prod_cases5
thf(fact_91_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E4 @ F6 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_92_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F5: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
~ ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5,G4: G3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E4 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_93_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A2: A,B2: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_94_prod__induct4,axiom,
! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
( ! [A2: A,B2: B,C2: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B2 @ ( product_Pair @ C @ D2 @ C2 @ D3 ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct4
thf(fact_95_prod__induct5,axiom,
! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) )] :
( ! [A2: A,B2: B,C2: C,D3: D2,E4: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ E ) @ C2 @ ( product_Pair @ D2 @ E @ D3 @ E4 ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct5
thf(fact_96_prod__induct6,axiom,
! [F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) )] :
( ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ F5 ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ F5 ) @ D3 @ ( product_Pair @ E @ F5 @ E4 @ F6 ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct6
thf(fact_97_prod__induct7,axiom,
! [G3: $tType,F5: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
( ! [A2: A,B2: B,C2: C,D3: D2,E4: E,F6: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D3 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E4 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct7
thf(fact_98_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A2: A,B2: B] :
( Y3
!= ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_99_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_100_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X5: B,Y5: B] :
( ( ord_less @ B @ X5 @ Y5 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_101_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X5: A,Y5: A] :
( ( ord_less @ A @ X5 @ Y5 )
=> ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_102_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X5: B,Y5: B] :
( ( ord_less @ B @ X5 @ Y5 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_103_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X5: A,Y5: A] :
( ( ord_less @ A @ X5 @ Y5 )
=> ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_104_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y5: A] : ( ord_less @ A @ Y5 @ X3 ) ) ).
% lt_ex
thf(fact_105_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_106_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% neqE
thf(fact_107_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
= ( ( ord_less @ A @ X3 @ Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_108_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order.asym
thf(fact_109_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% dense
thf(fact_110_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% less_imp_neq
thf(fact_111_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_asym
thf(fact_112_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% less_asym'
thf(fact_113_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_114_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_linear
thf(fact_115_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_116_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_117_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_118_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_119_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% less_imp_not_eq
thf(fact_120_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_not_sym
thf(fact_121_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y7: A] :
( ( ord_less @ A @ Y7 @ X5 )
=> ( P @ Y7 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_122_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X3: A] :
( ~ ( ord_less @ A @ Y3 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_123_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_124_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,P: $o] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_125_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_126_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_127_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_128_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_129_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X8: A] : ( P3 @ X8 ) )
= ( ^ [P4: A > $o] :
? [N: A] :
( ( P4 @ N )
& ! [M: A] :
( ( ord_less @ A @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_130_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: A] : ( P @ A2 @ A2 )
=> ( ! [A2: A,B2: A] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_131_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_132_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_133_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_134_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_135_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A3: A] :
? [B2: A] :
( ( ord_less @ A @ A3 @ B2 )
| ( ord_less @ A @ B2 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_136_less__eq__nres_Oelims_I3_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) )
=> ~ ( ( X3
= ( refine1767639642_FAILi @ A ) )
=> ! [Uv: set @ A] :
( Xa
!= ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% less_eq_nres.elims(3)
thf(fact_137_less__eq__nres_Oelims_I2_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ( Xa
!= ( refine1767639642_FAILi @ A ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% less_eq_nres.elims(2)
thf(fact_138_less__eq__nres_Oelims_I1_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
= Y3 )
=> ( ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ~ Y3 )
=> ( ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( Y3
= ( ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) )
=> ~ ( ( X3
= ( refine1767639642_FAILi @ A ) )
=> ( ? [Uv: set @ A] :
( Xa
= ( refine605929679le_RES @ A @ Uv ) )
=> Y3 ) ) ) ) ) ).
% less_eq_nres.elims(1)
thf(fact_139_less__nres_Oelims_I3_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ( X3
!= ( refine1767639642_FAILi @ A ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% less_nres.elims(3)
thf(fact_140_less__nres_Oelims_I2_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ( ? [Uv: set @ A] :
( X3
= ( refine605929679le_RES @ A @ Uv ) )
=> ( Xa
!= ( refine1767639642_FAILi @ A ) ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ).
% less_nres.elims(2)
thf(fact_141_less__nres_Oelims_I1_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
( ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
= Y3 )
=> ( ( ( X3
= ( refine1767639642_FAILi @ A ) )
=> Y3 )
=> ( ( ? [Uv: set @ A] :
( X3
= ( refine605929679le_RES @ A @ Uv ) )
=> ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ~ Y3 ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( Y3
= ( ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ) ).
% less_nres.elims(1)
thf(fact_142_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% leD
thf(fact_143_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% leI
thf(fact_144_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y6: A] :
( ( ord_less @ A @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ) ).
% le_less
thf(fact_145_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ) ).
% less_le
thf(fact_146_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X5: B,Y5: B] :
( ( ord_less @ B @ X5 @ Y5 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_147_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X5: A,Y5: A] :
( ( ord_less_eq @ A @ X5 @ Y5 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_148_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X5: B,Y5: B] :
( ( ord_less_eq @ B @ X5 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_149_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X5: A,Y5: A] :
( ( ord_less @ A @ X5 @ Y5 )
=> ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_150_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y3 ) )
= ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% not_le
thf(fact_151_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% not_less
thf(fact_152_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_153_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv1
thf(fact_154_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv2
thf(fact_155_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% less_imp_le
thf(fact_156_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_157_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_158_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y3: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less_eq @ A @ Y3 @ X5 ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% dense_ge
thf(fact_159_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y3: A,Z2: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% dense_le
thf(fact_160_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% le_less_linear
thf(fact_161_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_162_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ~ ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_163_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% not_le_imp_less
thf(fact_164_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_165_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_166_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_167_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_168_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_169_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_170_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X3: A,Y3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ! [W2: A] :
( ( ord_less @ A @ Z2 @ W2 )
=> ( ( ord_less @ A @ W2 @ X3 )
=> ( ord_less_eq @ A @ Y3 @ W2 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_171_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ! [W2: A] :
( ( ord_less @ A @ X3 @ W2 )
=> ( ( ord_less @ A @ W2 @ Y3 )
=> ( ord_less_eq @ A @ W2 @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_172_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_173_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_174_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_175_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_176_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B4: B,A4: B] :
( ( ~ ( ord_less_eq @ B @ B4 @ A4 ) )
= ( ord_less @ B @ A4 @ B4 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_177_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C3 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_178_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_179_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_180_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_181_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
| ~ ( ord_less_eq @ A @ A3 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_182_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_183_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_184_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_185_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( X7 != T2 ) ) ) ).
% pinf(3)
thf(fact_186_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( X7 != T2 ) ) ) ).
% pinf(4)
thf(fact_187_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ~ ( ord_less @ A @ X7 @ T2 ) ) ) ).
% pinf(5)
thf(fact_188_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ord_less @ A @ T2 @ X7 ) ) ) ).
% pinf(7)
thf(fact_189_pinf_I11_J,axiom,
! [C: $tType,D2: $tType] :
( ( ord @ C )
=> ! [F7: D2] :
? [Z3: C] :
! [X7: C] :
( ( ord_less @ C @ Z3 @ X7 )
=> ( F7 = F7 ) ) ) ).
% pinf(11)
thf(fact_190_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P5 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_191_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P5 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_192_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( X7 != T2 ) ) ) ).
% minf(3)
thf(fact_193_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( X7 != T2 ) ) ) ).
% minf(4)
thf(fact_194_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ord_less @ A @ X7 @ T2 ) ) ) ).
% minf(5)
thf(fact_195_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X7 ) ) ) ).
% minf(7)
thf(fact_196_minf_I11_J,axiom,
! [C: $tType,D2: $tType] :
( ( ord @ C )
=> ! [F7: D2] :
? [Z3: C] :
! [X7: C] :
( ( ord_less @ C @ X7 @ Z3 )
=> ( F7 = F7 ) ) ) ).
% minf(11)
thf(fact_197_psubsetI,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( A5 != B6 )
=> ( ord_less @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% psubsetI
thf(fact_198_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( A5 = B6 ) ) ) ).
% subset_antisym
thf(fact_199_subsetI,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( member @ A @ X5 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% subsetI
thf(fact_200_in__mono,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B6 ) ) ) ).
% in_mono
thf(fact_201_subsetD,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B6 ) ) ) ).
% subsetD
thf(fact_202_equalityE,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( A5 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% equalityE
thf(fact_203_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_204_equalityD1,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( A5 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% equalityD1
thf(fact_205_equalityD2,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( A5 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ).
% equalityD2
thf(fact_206_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T4: A] :
( ( member @ A @ T4 @ A7 )
=> ( member @ A @ T4 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_207_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_208_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_209_subset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_210_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = Z )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_211_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_212_psubsetD,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C3: A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B6 ) ) ) ).
% psubsetD
thf(fact_213_psubset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_trans
thf(fact_214_psubsetE,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% psubsetE
thf(fact_215_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( A7 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_216_psubset__imp__subset,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% psubset_imp_subset
thf(fact_217_psubset__subset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_218_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_219_subset__psubset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_220_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
| ( A7 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_221_less__nres_Opelims_I1_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
( ( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( refine1767639642_FAILi @ A ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Xa ) ) ) )
=> ( ! [Uv: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ Uv ) )
=> ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ( Y3
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) ) ) ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ( Y3
= ( ord_less @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) ) ) ) ) ) ).
% less_nres.pelims(1)
thf(fact_222_less__nres_Opelims_I2_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
=> ( ! [Uv: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ Uv ) )
=> ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) ) ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
=> ~ ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% less_nres.pelims(2)
thf(fact_223_less__nres_Opelims_I3_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ~ ( ord_less @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( refine1767639642_FAILi @ A ) )
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Xa ) ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1378444575es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% less_nres.pelims(3)
thf(fact_224_less__eq__nres_Opelims_I3_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
=> ( ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) )
=> ~ ( ( X3
= ( refine1767639642_FAILi @ A ) )
=> ! [Uv: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ Uv ) )
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ) ) ) ).
% less_eq_nres.pelims(3)
thf(fact_225_less__eq__nres_Opelims_I2_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
=> ( ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ ( refine1767639642_FAILi @ A ) ) ) )
=> ~ ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) )
=> ~ ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ) ) ) ) ).
% less_eq_nres.pelims(2)
thf(fact_226_less__eq__nres_Opelims_I1_J,axiom,
! [A: $tType,X3: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X3 @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ Xa ) )
=> ( ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ( Y3
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ X3 @ ( refine1767639642_FAILi @ A ) ) ) ) )
=> ( ! [A2: set @ A] :
( ( X3
= ( refine605929679le_RES @ A @ A2 ) )
=> ! [B2: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B2 ) )
=> ( ( Y3
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) ) ) ) ) )
=> ~ ( ( X3
= ( refine1767639642_FAILi @ A ) )
=> ! [Uv: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ Uv ) )
=> ( ~ Y3
=> ~ ( accp @ ( product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) ) @ ( refine1554218259es_rel @ A ) @ ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ) ) ) ) ) ).
% less_eq_nres.pelims(1)
thf(fact_227_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y7: A] :
( ( P @ Y7 )
=> ( ord_less_eq @ A @ Y7 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_228_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( ( order_Greatest @ A @ P )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_229_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_230_relprop__triggers_I6_J,axiom,
! [I2: $tType,R2: set @ I2,R3: set @ I2] :
( ( ord_less_eq @ ( set @ I2 ) @ R2 @ R3 )
=> ( ord_less_eq @ ( set @ I2 ) @ R2 @ R3 ) ) ).
% relprop_triggers(6)
thf(fact_231_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A,D4: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ( C3 = D4 )
=> ( ord_less_eq @ A @ A3 @ D4 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_232_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ S )
=> ( P @ A3 @ B3 ) )
=> ? [A2: A,B2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
& ( P @ A2 @ B2 ) ) ) ) ).
% bex2I
thf(fact_233_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F6: A > B,A2: A,B2: A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F6 @ ( product_Pair @ A @ A @ A2 @ B2 ) ) ) ).
% pairself.cases
thf(fact_234_pairself_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
( ! [F6: A > B,A2: A,B2: A] : ( P @ F6 @ ( product_Pair @ A @ A @ A2 @ B2 ) )
=> ( P @ A0 @ A1 ) ) ).
% pairself.induct
thf(fact_235_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% uncurry_apply
thf(fact_236_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y7: A] :
( ( ord_less @ B @ ( F @ Y7 ) @ ( F @ X5 ) )
=> ( P @ Y7 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_237_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y7: A] :
( ( ord_less @ B @ ( F @ Y7 ) @ ( F @ X5 ) )
=> ( P @ Y7 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_238_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_239_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X7: A] :
? [X_1: A] : ( ord_less @ A @ X7 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_240_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X7: A] :
? [Y5: A] : ( ord_less @ A @ Y5 @ X7 ) ) ).
% linordered_field_no_lb
thf(fact_241_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_12: A] : ( P @ X_12 ) ) ) ) ) ).
% all_nat_split_at
thf(fact_242_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R4: B,F6: A > B,G4: A > B,X5: A] :
( ! [Y7: A] :
( ( ord_less @ A @ Y7 @ X5 )
=> ( ( F6 @ Y7 )
= ( G4 @ Y7 ) ) )
=> ( ( P @ F6 @ X5 @ R4 )
= ( P @ G4 @ X5 @ R4 ) ) )
=> ( ! [X5: A,F6: A > B] :
( ! [Y7: A] :
( ( ord_less @ A @ Y7 @ X5 )
=> ( P @ F6 @ Y7 @ ( F6 @ Y7 ) ) )
=> ? [X_12: B] : ( P @ F6 @ X5 @ X_12 ) )
=> ? [F6: A > B] :
! [X7: A] : ( P @ F6 @ X7 @ ( F6 @ X7 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_243_Refine__Misc_Oif__mono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
( ( B3
=> ( ord_less_eq @ A @ M1 @ M12 ) )
=> ( ( ~ B3
=> ( ord_less_eq @ A @ M22 @ M23 ) )
=> ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_244_chain__subset__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( ^ [C5: set @ ( set @ A )] :
! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y6 )
| ( ord_less_eq @ ( set @ A ) @ Y6 @ X4 ) ) ) ) ) ) ).
% chain_subset_def
% Type constructors (13)
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_Set_Oset___Orderings_Opreorder_1,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_7,axiom,
! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_8,axiom,
! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_9,axiom,
! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% 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,Y3: A] :
( ( if @ A @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( refine406925620_inres @ a @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) @ x )
= ( phi @ x ) ) ).
%------------------------------------------------------------------------------