TPTP Problem File: ITP159^2.p
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%------------------------------------------------------------------------------
% File : ITP159^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_1005__3594596_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_1005__3594596_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 344 ( 140 unt; 45 typ; 0 def)
% Number of atoms : 756 ( 180 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4384 ( 56 ~; 6 |; 34 &;3949 @)
% ( 0 <=>; 339 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 160 ( 160 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 42 usr; 4 con; 0-5 aty)
% Number of variables : 1086 ( 53 ^; 970 !; 15 ?;1086 :)
% ( 48 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:56.949
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
refine1665802226e_nres: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $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_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (39)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[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_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Orel__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ 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__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_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
refine1687780735RETURN:
!>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
refine81118332bs_fun:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
refine463715084e_bind:
!>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
refine1073749519nc_fun:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( 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_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_ORES,type,
refine605929679le_RES:
!>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
refine1672542526he_RES:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_R,type,
r: set @ ( product_prod @ a @ b ) ).
thf(sy_v_X,type,
x: refine1665802226e_nres @ a ).
% Relevant facts (253)
thf(fact_0_nres__order__simps_I4_J,axiom,
! [D: $tType,M: refine1665802226e_nres @ D] :
( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( top_top @ ( refine1665802226e_nres @ D ) ) @ M )
= ( M
= ( top_top @ ( refine1665802226e_nres @ D ) ) ) ) ).
% nres_order_simps(4)
thf(fact_1_conc__fun__fail__iff_I2_J,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( ( top_top @ ( refine1665802226e_nres @ A ) )
= ( refine1073749519nc_fun @ A @ B @ R @ S ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% conc_fun_fail_iff(2)
thf(fact_2_conc__fun__fail__iff_I1_J,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
= ( top_top @ ( refine1665802226e_nres @ A ) ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% conc_fun_fail_iff(1)
thf(fact_3_conc__fun__FAIL,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% conc_fun_FAIL
thf(fact_4_nres__order__simps_I3_J,axiom,
! [C: $tType,M: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% nres_order_simps(3)
thf(fact_5_conc__trans__additional_I1_J,axiom,
! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_6_conc__trans,axiom,
! [A: $tType,B: $tType,C: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A2: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) ) ) ) ) ).
% conc_trans
thf(fact_7_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_8_order__mono__setup_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_mono_setup.refl
thf(fact_9_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_10_meta__le__everything__if__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [M2: A,X2: A] :
( ( M2
= ( top_top @ A ) )
=> ( ord_less_eq @ A @ X2 @ M2 ) ) ) ).
% meta_le_everything_if_top
thf(fact_11_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_12_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_13_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_14_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_15_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_16_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_17_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_18_conc__fun__R__mono,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_19_order__mono__setup_Omono__let,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [F: B > A,F3: B > A,X2: B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F3 @ X3 ) )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% order_mono_setup.mono_let
thf(fact_20_order__mono__setup_Omono__if,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [T: A,T2: A,E: A,E2: A,B3: $o] :
( ( ord_less_eq @ A @ T @ T2 )
=> ( ( ord_less_eq @ A @ E @ E2 )
=> ( ord_less_eq @ A @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_21_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_22_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_23_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_24_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_25_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_26_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_27_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_28_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_29_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_30_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_31_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_32_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_33_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_34_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_35_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_36_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_37_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_38_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_39_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 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_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 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_41_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 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_42_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 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_43_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_49_transfer_Otransfer__Let,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [Alpha: C > A,F: B > C,F4: B > A,X2: B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( Alpha @ ( F @ X3 ) ) @ ( F4 @ X3 ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( F @ X2 ) ) @ ( F4 @ X2 ) ) ) ) ).
% transfer.transfer_Let
thf(fact_50_transfer_Otransfer__if,axiom,
! [C: $tType,A: $tType] :
( ( comple187826305attice @ A )
=> ! [B3: $o,Alpha: C > A,S1: C,S12: A,S2: C,S22: A] :
( ( B3
=> ( ord_less_eq @ A @ ( Alpha @ S1 ) @ S12 ) )
=> ( ( ~ B3
=> ( ord_less_eq @ A @ ( Alpha @ S2 ) @ S22 ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B3 @ S1 @ S2 ) ) @ ( if @ A @ B3 @ S12 @ S22 ) ) ) ) ) ).
% transfer.transfer_if
thf(fact_51_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X4: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_52_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_53_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C3: A,D2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ( C3 = D2 )
=> ( ord_less_eq @ A @ A3 @ D2 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_54_Id__refine,axiom,
! [A: $tType,S: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ S ) ) ).
% Id_refine
thf(fact_55_conc__trans__additional_I2_J,axiom,
! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ B2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% conc_trans_additional(2)
thf(fact_56_conc__trans__additional_I3_J,axiom,
! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ B @ ( id @ B ) @ C2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% conc_trans_additional(3)
thf(fact_57_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X: A] :
( ( member @ A @ X @ S )
=> ( P @ X ) ) ) ) ).
% subset_Collect_conv
thf(fact_58_conc__trans__additional_I6_J,axiom,
! [E3: $tType,A2: refine1665802226e_nres @ E3,B2: refine1665802226e_nres @ E3,C2: refine1665802226e_nres @ E3] :
( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ B2 @ ( refine1073749519nc_fun @ E3 @ E3 @ ( id @ E3 ) @ C2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ C2 ) ) ) ).
% conc_trans_additional(6)
thf(fact_59_conc__trans__additional_I5_J,axiom,
! [D: $tType,A2: refine1665802226e_nres @ D,B2: refine1665802226e_nres @ D,C2: refine1665802226e_nres @ D] :
( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ ( refine1073749519nc_fun @ D @ D @ ( id @ D ) @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ B2 @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ C2 ) ) ) ).
% conc_trans_additional(5)
thf(fact_60_conc__trans__additional_I4_J,axiom,
! [C: $tType,A2: refine1665802226e_nres @ C,B2: refine1665802226e_nres @ C,C2: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ ( refine1073749519nc_fun @ C @ C @ ( id @ C ) @ B2 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ B2 @ ( refine1073749519nc_fun @ C @ C @ ( id @ C ) @ C2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ C2 ) ) ) ).
% conc_trans_additional(4)
thf(fact_61_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_62_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_63_bijective__Id,axiom,
! [A: $tType] : ( bijective @ A @ A @ ( id @ A ) ) ).
% bijective_Id
thf(fact_64_relprop__id__orient,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( R
= ( id @ A ) )
=> ( ( id @ A )
= R ) ) ).
% relprop_id_orient
thf(fact_65_relprop__triggers_I2_J,axiom,
! [C: $tType,R: set @ ( product_prod @ C @ C )] :
( ( R
= ( id @ C ) )
=> ( R
= ( id @ C ) ) ) ).
% relprop_triggers(2)
thf(fact_66_relprop__triggers_I3_J,axiom,
! [D: $tType,R: set @ ( product_prod @ D @ D )] :
( ( R
= ( id @ D ) )
=> ( ( id @ D )
= R ) ) ).
% relprop_triggers(3)
thf(fact_67_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_68_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_69_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : Y = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_70_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_71_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_72_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_73_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_74_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_75_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_76_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_77_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_78_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_79_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% subsetD
thf(fact_80_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_81_Id__SPEC__refine,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,Phi: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ A @ ( id @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% Id_SPEC_refine
thf(fact_82_abs__trans__additional_I2_J,axiom,
! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ A @ A @ ( id @ A ) @ A2 ) @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% abs_trans_additional(2)
thf(fact_83_abs__trans__additional_I3_J,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ B @ B @ ( id @ B ) @ B2 ) @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% abs_trans_additional(3)
thf(fact_84_abs__trans__additional_I4_J,axiom,
! [C: $tType,A2: refine1665802226e_nres @ C,B2: refine1665802226e_nres @ C,C2: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ C @ C @ ( id @ C ) @ A2 ) @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ C @ C @ ( id @ C ) @ B2 ) @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ A2 @ C2 ) ) ) ).
% abs_trans_additional(4)
thf(fact_85_abs__trans__additional_I5_J,axiom,
! [D: $tType,A2: refine1665802226e_nres @ D,B2: refine1665802226e_nres @ D,C2: refine1665802226e_nres @ D] :
( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( refine81118332bs_fun @ D @ D @ ( id @ D ) @ A2 ) @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ B2 @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ A2 @ C2 ) ) ) ).
% abs_trans_additional(5)
thf(fact_86_abs__trans__additional_I6_J,axiom,
! [E3: $tType,A2: refine1665802226e_nres @ E3,B2: refine1665802226e_nres @ E3,C2: refine1665802226e_nres @ E3] :
( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine81118332bs_fun @ E3 @ E3 @ ( id @ E3 ) @ B2 ) @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ A2 @ C2 ) ) ) ).
% abs_trans_additional(6)
thf(fact_87_rel__restrict__mono2,axiom,
! [A: $tType,R: set @ A,S: set @ A,A2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ R @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A2 @ S ) @ ( rel_restrict @ A @ A2 @ R ) ) ) ).
% rel_restrict_mono2
thf(fact_88_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_89_nres_Oinject,axiom,
! [A: $tType,X22: set @ A,Y22: set @ A] :
( ( ( refine605929679le_RES @ A @ X22 )
= ( refine605929679le_RES @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nres.inject
thf(fact_90_nres__more__simps_I4_J,axiom,
! [A: $tType,X5: set @ A,Y7: set @ A] :
( ( ( refine605929679le_RES @ A @ X5 )
= ( refine605929679le_RES @ A @ Y7 ) )
= ( X5 = Y7 ) ) ).
% nres_more_simps(4)
thf(fact_91_abs__fun__simps_I1_J,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% abs_fun_simps(1)
thf(fact_92_relprop__UNIV__orient,axiom,
! [A: $tType,R: set @ A] :
( ( R
= ( top_top @ ( set @ A ) ) )
=> ( ( top_top @ ( set @ A ) )
= R ) ) ).
% relprop_UNIV_orient
thf(fact_93_eq__UNIV__iff,axiom,
! [A: $tType,S: set @ A] :
( ( S
= ( top_top @ ( set @ A ) ) )
= ( ! [X: A] : ( member @ A @ X @ S ) ) ) ).
% eq_UNIV_iff
thf(fact_94_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_95_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_96_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_97_if__rule,axiom,
! [A: $tType,B3: $o,S12: refine1665802226e_nres @ A,Phi: A > $o,S22: refine1665802226e_nres @ A] :
( ( B3
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S12 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
=> ( ( ~ B3
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S22 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B3 @ S12 @ S22 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% if_rule
thf(fact_98_RES__rule,axiom,
! [A: $tType,S: set @ A,Phi: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% RES_rule
thf(fact_99_SPEC__rule,axiom,
! [A: $tType,Phi: A > $o,Phi2: A > $o] :
( ! [X3: A] :
( ( Phi @ X3 )
=> ( Phi2 @ X3 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_100_SPEC__trans,axiom,
! [A: $tType,X2: refine1665802226e_nres @ A,Y2: refine1665802226e_nres @ A,Postcond: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ Y2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_101_SPEC__cons__rule,axiom,
! [A: $tType,M2: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
=> ( ! [X3: A] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_102_nres__inequalities_I1_J,axiom,
! [A: $tType,X5: set @ A] :
( ( top_top @ ( refine1665802226e_nres @ A ) )
!= ( refine605929679le_RES @ A @ X5 ) ) ).
% nres_inequalities(1)
thf(fact_103_nres__cases,axiom,
! [A: $tType,M: refine1665802226e_nres @ A] :
( ( M
!= ( top_top @ ( refine1665802226e_nres @ A ) ) )
=> ~ ! [X6: set @ A] :
( M
!= ( refine605929679le_RES @ A @ X6 ) ) ) ).
% nres_cases
thf(fact_104_rel__restrict__sub,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A2 ) @ R ) ).
% rel_restrict_sub
thf(fact_105_rel__restrict__mono,axiom,
! [A: $tType,A2: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A2 @ R ) @ ( rel_restrict @ A @ B2 @ R ) ) ) ).
% rel_restrict_mono
thf(fact_106_abs__trans__additional_I1_J,axiom,
! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% abs_trans_additional(1)
thf(fact_107_abs__trans,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C2: refine1665802226e_nres @ B,B2: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A2: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) @ B2 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B2 ) @ A2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) ) @ A2 ) ) ) ).
% abs_trans
thf(fact_108_nres__order__simps_I5_J,axiom,
! [E3: $tType,X5: set @ E3,Y7: set @ E3] :
( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine605929679le_RES @ E3 @ X5 ) @ ( refine605929679le_RES @ E3 @ Y7 ) )
= ( ord_less_eq @ ( set @ E3 ) @ X5 @ Y7 ) ) ).
% nres_order_simps(5)
thf(fact_109_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_110_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_111_nf__inres__SPEC,axiom,
! [A: $tType,Phi: A > $o,X2: A] :
( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X2 )
= ( Phi @ X2 ) ) ).
% nf_inres_SPEC
thf(fact_112_nf__inres__RES,axiom,
! [A: $tType,X5: set @ A,X2: A] :
( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X5 ) @ X2 )
= ( member @ A @ X2 @ X5 ) ) ).
% nf_inres_RES
thf(fact_113_abs__fun__simps_I3_J,axiom,
! [B: $tType,A: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
( ~ ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
=> ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% abs_fun_simps(3)
thf(fact_114_the__RES_Osimps,axiom,
! [A: $tType,X5: set @ A] :
( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X5 ) )
= X5 ) ).
% the_RES.simps
thf(fact_115_nres__order__simps_I21_J,axiom,
! [X7: $tType,X2: X7,Y7: set @ X7] :
( ( ord_less_eq @ ( refine1665802226e_nres @ X7 ) @ ( refine1687780735RETURN @ X7 @ X2 ) @ ( refine605929679le_RES @ X7 @ Y7 ) )
= ( member @ X7 @ X2 @ Y7 ) ) ).
% nres_order_simps(21)
thf(fact_116_nres__more__simps_I6_J,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( ( refine1687780735RETURN @ A @ X2 )
= ( refine1687780735RETURN @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres_more_simps(6)
thf(fact_117_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_118_nres__order__simps_I20_J,axiom,
! [W: $tType,X2: W,Y2: W] :
( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X2 ) @ ( refine1687780735RETURN @ W @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres_order_simps(20)
thf(fact_119_nres__inequalities_I3_J,axiom,
! [C: $tType,X2: C] :
( ( top_top @ ( refine1665802226e_nres @ C ) )
!= ( refine1687780735RETURN @ C @ X2 ) ) ).
% nres_inequalities(3)
thf(fact_120_RETURN__to__SPEC__rule,axiom,
! [A: $tType,M2: refine1665802226e_nres @ A,V: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ( ^ [Y: A,Z: A] : Y = Z
@ V ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M2 @ ( refine1687780735RETURN @ A @ V ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_121_RETURN__rule,axiom,
! [A: $tType,Phi: A > $o,X2: A] :
( ( Phi @ X2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X2 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_122_Domain__Id,axiom,
! [A: $tType] :
( ( domain @ A @ A @ ( id @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_Id
thf(fact_123_Domain__mono,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S3 ) ) ) ).
% Domain_mono
thf(fact_124_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_125_abs__fun__simps_I2_J,axiom,
! [A: $tType,B: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
=> ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
= ( refine605929679le_RES @ A @ ( image @ B @ A @ R @ X5 ) ) ) ) ).
% abs_fun_simps(2)
thf(fact_126_Image__Id,axiom,
! [A: $tType,A2: set @ A] :
( ( image @ A @ A @ ( id @ A ) @ A2 )
= A2 ) ).
% Image_Id
thf(fact_127_Image__mono,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),A7: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A7 ) @ ( image @ A @ B @ R3 @ A2 ) ) ) ) ).
% Image_mono
thf(fact_128_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_129_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X2: A] :
( ( P
& ( top_top @ ( A > $o ) @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_130_RETURN__SPEC__refine,axiom,
! [B: $tType,A: $tType,X2: B,R: set @ ( product_prod @ B @ A ),Phi: A > $o] :
( ? [X8: A] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ X8 ) @ R )
& ( Phi @ X8 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X2 ) @ ( refine1073749519nc_fun @ B @ A @ R @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_131_the__RES__inv,axiom,
! [A: $tType,M2: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ M2 )
=> ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M2 ) )
= M2 ) ) ).
% the_RES_inv
thf(fact_132_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id @ A ) )
= ( A3 = B3 ) ) ).
% pair_in_Id_conv
thf(fact_133_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id @ A ) ) ).
% IdI
thf(fact_134_nofail__simps_I2_J,axiom,
! [B: $tType,X5: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X5 ) ) ).
% nofail_simps(2)
thf(fact_135_nofail__simps_I1_J,axiom,
! [A: $tType] :
~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% nofail_simps(1)
thf(fact_136_nofail__simps_I3_J,axiom,
! [C: $tType,X2: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X2 ) ) ).
% nofail_simps(3)
thf(fact_137_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% DomainE
thf(fact_138_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_139_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% Domain.cases
thf(fact_140_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
= ( ? [A4: A,B4: B] :
( ( A3 = A4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_141_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_142_Domain_Oinducts,axiom,
! [B: $tType,A: $tType,X2: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
( ( member @ A @ X2 @ ( domain @ A @ B @ R3 ) )
=> ( ! [A5: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R3 )
=> ( P @ A5 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_143_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S3 ) ) ).
% subrelI
thf(fact_144_IdE,axiom,
! [A: $tType,P2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P2 @ ( id @ A ) )
=> ~ ! [X3: A] :
( P2
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_145_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X2: A,Y2: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ A2 @ R ) )
=> ~ ( member @ A @ Y2 @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_146_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X2: A,Y2: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ A2 @ R ) )
=> ~ ( member @ A @ X2 @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_147_rel__restrictI,axiom,
! [A: $tType,X2: A,R: set @ A,Y2: A,E4: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X2 @ R )
=> ( ~ ( member @ A @ Y2 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ E4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_148_rel__restrict__lift,axiom,
! [A: $tType,X2: A,Y2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( rel_restrict @ A @ E4 @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ E4 ) ) ).
% rel_restrict_lift
thf(fact_149_pairself_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
( ! [F5: A > B,A5: A,B5: A] : ( P @ F5 @ ( product_Pair @ A @ A @ A5 @ B5 ) )
=> ( P @ A0 @ A1 ) ) ).
% pairself.induct
thf(fact_150_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F5: A > B,A5: A,B5: A] :
( X2
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% pairself.cases
thf(fact_151_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 ) )
=> ? [A5: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ S )
& ( P @ A5 @ B5 ) ) ) ) ).
% bex2I
thf(fact_152_rel__cong,axiom,
! [A: $tType,B: $tType,F: A > B,G: A > B,X2: A,Y2: A] :
( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G ) @ ( id @ ( A > B ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( id @ A ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( G @ Y2 ) ) @ ( id @ B ) ) ) ) ).
% rel_cong
thf(fact_153_rel__fun__cong,axiom,
! [A: $tType,B: $tType,F: A > B,G: A > B,X2: A] :
( ( member @ ( product_prod @ ( A > B ) @ ( A > B ) ) @ ( product_Pair @ ( A > B ) @ ( A > B ) @ F @ G ) @ ( id @ ( A > B ) ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( G @ X2 ) ) @ ( id @ B ) ) ) ).
% rel_fun_cong
thf(fact_154_nofail__antimono__fun,axiom,
! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X2: A] :
( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
=> ( ( refine1102455758nofail @ B @ ( G @ X2 ) )
=> ( refine1102455758nofail @ B @ ( F @ X2 ) ) ) ) ).
% nofail_antimono_fun
thf(fact_155_pw__conc__nofail,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
= ( refine1102455758nofail @ B @ S ) ) ).
% pw_conc_nofail
thf(fact_156_intro__nofail_I2_J,axiom,
! [A: $tType,S: refine1665802226e_nres @ A] :
( ( ( top_top @ ( refine1665802226e_nres @ A ) )
!= S )
= ( refine1102455758nofail @ A @ S ) ) ).
% intro_nofail(2)
thf(fact_157_nofail__def,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [S4: refine1665802226e_nres @ A] :
( S4
!= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% nofail_def
thf(fact_158_not__nofail__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A] :
( ( ~ ( refine1102455758nofail @ A @ S ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% not_nofail_iff
thf(fact_159_rel__arg__cong,axiom,
! [A: $tType,B: $tType,X2: A,Y2: A,F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( id @ A ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) @ ( id @ B ) ) ) ).
% rel_arg_cong
thf(fact_160_pwD1,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 )
=> ( ( refine1102455758nofail @ A @ S5 )
=> ( refine1102455758nofail @ A @ S ) ) ) ).
% pwD1
thf(fact_161_le__nofailI,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A,M: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ M3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 ) ) ).
% le_nofailI
thf(fact_162_nofail__RES__conv,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [M4: refine1665802226e_nres @ A] :
? [M5: set @ A] :
( M4
= ( refine605929679le_RES @ A @ M5 ) ) ) ) ).
% nofail_RES_conv
thf(fact_163_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y3: B,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R5 ) )
=> ( Y3 = Z3 ) )
& ! [X: A,Y3: A,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z3 ) @ R5 ) )
=> ( X = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_164_RES__refine,axiom,
! [A: $tType,B: $tType,S: set @ A,S5: set @ B,R: set @ ( product_prod @ A @ B )] :
( ! [S6: A] :
( ( member @ A @ S6 @ S )
=> ? [X9: B] :
( ( member @ B @ X9 @ S5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S6 @ X9 ) @ R ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ S5 ) ) ) ) ).
% RES_refine
thf(fact_165_RETURN__refine,axiom,
! [A: $tType,B: $tType,X2: A,X10: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ X10 ) @ R )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ X10 ) ) ) ) ).
% RETURN_refine
thf(fact_166_for__in__RI,axiom,
! [B: $tType,A: $tType,X2: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X2 @ ( domain @ A @ B @ R ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ ( fun_of_rel @ A @ B @ R @ X2 ) ) @ R ) ) ).
% for_in_RI
thf(fact_167_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A3: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( id @ A ) )
=> ( A3 = B3 ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_168_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R ) @ R )
=> ( ~ ( member @ A @ V @ R )
=> ~ ( ~ ( member @ A @ U @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_169_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ A ),As: A > B] :
! [I2: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R6 )
=> ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_170_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_171_rtrancl__mono__rightI,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ S5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S5 ) ) ) ).
% rtrancl_mono_rightI
thf(fact_172_rtrancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
=> ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ U2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ V2 ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_173_r__le__rtrancl,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S ) ) ).
% r_le_rtrancl
thf(fact_174_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q2: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_175_rtrancl__image__advance,axiom,
! [A: $tType,Q2: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ R )
=> ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_176_rtrancl__reachable__induct,axiom,
! [A: $tType,I3: set @ A,INV: set @ A,E4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ I3 @ INV )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ INV ) @ INV )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I3 ) @ INV ) ) ) ).
% rtrancl_reachable_induct
thf(fact_177_rtrancl__image__unfold__right,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),V2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V2 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V2 ) ) ).
% rtrancl_image_unfold_right
thf(fact_178_reachable__mono,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A @ A ),X5: set @ A,X11: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ X11 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ X5 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ X11 ) ) ) ) ).
% reachable_mono
thf(fact_179_Domain__rtrancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( domain @ A @ A @ ( transitive_rtrancl @ A @ R ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_rtrancl
thf(fact_180_Not__Domain__rtrancl,axiom,
! [A: $tType,X2: A,R: set @ ( product_prod @ A @ A ),Y2: A] :
( ~ ( member @ A @ X2 @ ( domain @ A @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_rtrancl @ A @ R ) )
= ( X2 = Y2 ) ) ) ).
% Not_Domain_rtrancl
thf(fact_181_Image__closed__trancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ X5 ) @ X5 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ X5 )
= X5 ) ) ).
% Image_closed_trancl
thf(fact_182_rtrancl__subset__rtrancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( transitive_rtrancl @ A @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_183_rtrancl__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ R ) )
=> ( ( transitive_rtrancl @ A @ S )
= ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl_subset
thf(fact_184_rtrancl__mono,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% rtrancl_mono
thf(fact_185_rel__restrict__tranclI,axiom,
! [A: $tType,X2: A,Y2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ E4 ) )
=> ( ~ ( member @ A @ X2 @ R )
=> ( ~ ( member @ A @ Y2 @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_186_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_187_pw__abs__nofail,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B] :
( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
= ( ( refine1102455758nofail @ B @ M )
& ! [X: B] :
( ( refine1315500908_inres @ B @ M @ X )
=> ( member @ B @ X @ ( domain @ B @ A @ R ) ) ) ) ) ).
% pw_abs_nofail
thf(fact_188_inres__simps_I2_J,axiom,
! [B: $tType,X5: set @ B] :
( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X5 ) )
= ( ^ [X: B] : ( member @ B @ X @ X5 ) ) ) ).
% inres_simps(2)
thf(fact_189_trancl__domain,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( domain @ A @ A @ ( transitive_trancl @ A @ R3 ) )
= ( domain @ A @ A @ R3 ) ) ).
% trancl_domain
thf(fact_190_inres__simps_I1_J,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
= ( ^ [Uu: A] : $true ) ) ).
% inres_simps(1)
thf(fact_191_inres__simps_I3_J,axiom,
! [C: $tType,X2: C] :
( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X2 ) )
= ( ^ [Y: C,Z: C] : Y = Z
@ X2 ) ) ).
% inres_simps(3)
thf(fact_192_trancl__mono,axiom,
! [A: $tType,P2: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P2 @ ( transitive_trancl @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 )
=> ( member @ ( product_prod @ A @ A ) @ P2 @ ( transitive_trancl @ A @ S3 ) ) ) ) ).
% trancl_mono
thf(fact_193_trancl__Image__unfold__left,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( image @ A @ A @ E4 @ S ) ) ) ).
% trancl_Image_unfold_left
thf(fact_194_trancl__Image__unfold__right,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S )
= ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ S ) ) ) ).
% trancl_Image_unfold_right
thf(fact_195_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
=> ~ ( member @ A @ W2 @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_196_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R ) ) )
=> ~ ( member @ A @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_197_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B3: A,A2: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A2 @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A2 ) @ R ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_198_rel__restrict__trancl__sub,axiom,
! [A: $tType,A2: set @ ( product_prod @ A @ A ),R: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A2 @ R ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A2 ) @ R ) ) ).
% rel_restrict_trancl_sub
thf(fact_199_pw__eqI,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S )
= ( refine1102455758nofail @ A @ S5 ) )
=> ( ! [X3: A] :
( ( refine1315500908_inres @ A @ S @ X3 )
= ( refine1315500908_inres @ A @ S5 @ X3 ) )
=> ( S = S5 ) ) ) ).
% pw_eqI
thf(fact_200_pw__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : Y = Z )
= ( ^ [S4: refine1665802226e_nres @ A,S7: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S4 )
= ( refine1102455758nofail @ A @ S7 ) )
& ! [X: A] :
( ( refine1315500908_inres @ A @ S4 @ X )
= ( refine1315500908_inres @ A @ S7 @ X ) ) ) ) ) ).
% pw_eq_iff
thf(fact_201_not__nofail__inres,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,X2: A] :
( ~ ( refine1102455758nofail @ A @ S )
=> ( refine1315500908_inres @ A @ S @ X2 ) ) ).
% not_nofail_inres
thf(fact_202_pwD2,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A,X2: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 )
=> ( ( refine1315500908_inres @ A @ S @ X2 )
=> ( refine1315500908_inres @ A @ S5 @ X2 ) ) ) ).
% pwD2
thf(fact_203_trancl__sub,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( transitive_trancl @ A @ R ) ) ).
% trancl_sub
thf(fact_204_trancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V2: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V2 )
=> ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ U2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ V2 ) ) ) ) ).
% trancl_mono_mp
thf(fact_205_pw__leI,axiom,
! [A: $tType,S5: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S5 )
=> ( ( refine1102455758nofail @ A @ S )
& ! [X3: A] :
( ( refine1315500908_inres @ A @ S @ X3 )
=> ( refine1315500908_inres @ A @ S5 @ X3 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 ) ) ).
% pw_leI
thf(fact_206_pw__leI_H,axiom,
! [A: $tType,S5: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S5 )
=> ( refine1102455758nofail @ A @ S ) )
=> ( ! [X3: A] :
( ( refine1102455758nofail @ A @ S5 )
=> ( ( refine1315500908_inres @ A @ S @ X3 )
=> ( refine1315500908_inres @ A @ S5 @ X3 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S5 ) ) ) ).
% pw_leI'
thf(fact_207_pw__le__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
= ( ^ [S4: refine1665802226e_nres @ A,S7: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ S7 )
=> ( ( refine1102455758nofail @ A @ S4 )
& ! [X: A] :
( ( refine1315500908_inres @ A @ S4 @ X )
=> ( refine1315500908_inres @ A @ S7 @ X ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_208_inres__def,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A )
= ( ^ [S4: refine1665802226e_nres @ A,X: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X ) @ S4 ) ) ) ).
% inres_def
thf(fact_209_nf__inres__def,axiom,
! [A: $tType] :
( ( refine406925620_inres @ A )
= ( ^ [M4: refine1665802226e_nres @ A,X: A] :
( ( refine1102455758nofail @ A @ M4 )
& ( refine1315500908_inres @ A @ M4 @ X ) ) ) ) ).
% nf_inres_def
thf(fact_210_pw__abs__inres,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B,A3: A] :
( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) @ A3 )
= ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
=> ? [C4: B] :
( ( refine1315500908_inres @ B @ M @ C4 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C4 @ A3 ) @ R ) ) ) ) ).
% pw_abs_inres
thf(fact_211_trancl__reflcl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id @ A ) ) )
= ( transitive_rtrancl @ A @ R3 ) ) ).
% trancl_reflcl
thf(fact_212_pw__bind__leI,axiom,
! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M ) )
=> ( ! [X3: B] :
( ( refine1102455758nofail @ B @ M )
=> ( ( refine1315500908_inres @ B @ M @ X3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_213_Un__iff,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( ( member @ A @ C3 @ A2 )
| ( member @ A @ C3 @ B2 ) ) ) ).
% Un_iff
thf(fact_214_UnCI,axiom,
! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ A2 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_215_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_216_bind__FAIL,axiom,
! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% bind_FAIL
thf(fact_217_nres__monad2,axiom,
! [A: $tType,M: refine1665802226e_nres @ A] :
( ( refine463715084e_bind @ A @ A @ M @ ( refine1687780735RETURN @ A ) )
= M ) ).
% nres_monad2
thf(fact_218_nres__monad1,axiom,
! [A: $tType,B: $tType,X2: B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X2 ) @ F )
= ( F @ X2 ) ) ).
% nres_monad1
thf(fact_219_rtrancl__reflcl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id @ A ) ) )
= ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl_reflcl
thf(fact_220_rtrancl__reflcl__absorb,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( id @ A ) )
= ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl_reflcl_absorb
thf(fact_221_rtrancl__Un__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ).
% rtrancl_Un_subset
thf(fact_222_Un__mono,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_223_Un__least,axiom,
! [A: $tType,A2: set @ A,C2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_224_Un__upper1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_225_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_226_Un__absorb1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_227_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_228_subset__UnE,axiom,
! [A: $tType,C2: set @ A,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ~ ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ A2 )
=> ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ B2 )
=> ( C2
!= ( sup_sup @ ( set @ A ) @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_229_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_230_Domain__Un__eq,axiom,
! [B: $tType,A: $tType,A2: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A2 ) @ ( domain @ A @ B @ B2 ) ) ) ).
% Domain_Un_eq
thf(fact_231_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_232_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_233_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_234_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_235_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_236_ball__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( P @ X ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( P @ X ) )
& ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_237_bex__Un,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
& ( P @ X ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A2 )
& ( P @ X ) )
| ? [X: A] :
( ( member @ A @ X @ B2 )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_238_UnI2,axiom,
! [A: $tType,C3: A,B2: set @ A,A2: set @ A] :
( ( member @ A @ C3 @ B2 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_239_UnI1,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_240_UnE,axiom,
! [A: $tType,C3: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
=> ( ~ ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B2 ) ) ) ).
% UnE
thf(fact_241_rel__restrict__union,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A2: set @ A,B2: set @ A] :
( ( rel_restrict @ A @ R @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R @ A2 ) @ B2 ) ) ).
% rel_restrict_union
thf(fact_242_ibind__strict_I2_J,axiom,
! [A: $tType,F: product_unit > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ product_unit @ A @ ( top_top @ ( refine1665802226e_nres @ product_unit ) ) @ F )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% ibind_strict(2)
thf(fact_243_Un__UNIV__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_244_Un__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_245_pw__bind__nofail,axiom,
! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M @ F ) )
= ( ( refine1102455758nofail @ B @ M )
& ! [X: B] :
( ( refine1315500908_inres @ B @ M @ X )
=> ( refine1102455758nofail @ A @ ( F @ X ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_246_bind__cong,axiom,
! [B: $tType,A: $tType,M2: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
( ( M2 = M6 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M6 )
=> ( ( F @ X3 )
= ( F3 @ X3 ) ) )
=> ( ( refine463715084e_bind @ A @ B @ M2 @ F )
= ( refine463715084e_bind @ A @ B @ M6 @ F3 ) ) ) ) ).
% bind_cong
thf(fact_247_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M3 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M3 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
thf(fact_248_trancl__union__outside,axiom,
! [A: $tType,V: A,W2: A,E4: set @ ( product_prod @ A @ A ),U2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W2 ) @ ( transitive_trancl @ A @ E4 ) )
=> ? [X3: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ U2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U2 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_249_rtrancl__trancl__reflcl,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R6 ) @ ( id @ A ) ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_250_trancl__image__by__rtrancl,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),Vi: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ Vi ) @ Vi )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ Vi ) ) ).
% trancl_image_by_rtrancl
thf(fact_251_pw__bind__le__iff,axiom,
! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S )
= ( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M ) )
& ! [X: B] :
( ( ( refine1102455758nofail @ B @ M )
& ( refine1315500908_inres @ B @ M @ X ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_252_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X2: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X2 )
= ( top_top @ A ) ) ) ).
% sup_top_left
% Type constructors (42)
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice,axiom,
! [A9: $tType] : ( bounded_lattice @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_1,axiom,
bounded_lattice @ product_unit ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_2,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_3,axiom,
! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_4,axiom,
! [A9: $tType,A10: $tType] :
( ( bounded_lattice @ A10 )
=> ( bounded_lattice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A9: $tType,A10: $tType] :
( ( comple187826305attice @ A10 )
=> ( comple187826305attice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A9: $tType,A10: $tType] :
( ( bounded_lattice @ A10 )
=> ( bounded_lattice_top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A9: $tType,A10: $tType] :
( ( order_top @ A10 )
=> ( order_top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A10: $tType] :
( ( top @ A10 )
=> ( top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_5,axiom,
! [A9: $tType] : ( comple187826305attice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_6,axiom,
! [A9: $tType] : ( bounded_lattice_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_7,axiom,
! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_10,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_12,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_14,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
ord @ $o ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_19,axiom,
comple187826305attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_20,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_21,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_22,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_23,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_24,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_25,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_26,axiom,
ord @ product_unit ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_27,axiom,
! [A9: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice__top_28,axiom,
! [A9: $tType] : ( bounded_lattice_top @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_29,axiom,
! [A9: $tType] : ( order_top @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_30,axiom,
! [A9: $tType] : ( preorder @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_31,axiom,
! [A9: $tType] : ( order @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_32,axiom,
! [A9: $tType] : ( top @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_33,axiom,
! [A9: $tType] : ( ord @ ( refine1665802226e_nres @ A9 ) ) ).
% 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,X2: A,Y2: A] :
( ( if @ A @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq @ ( refine1665802226e_nres @ a ) @ x @ ( refine1073749519nc_fun @ a @ b @ r @ ( top_top @ ( refine1665802226e_nres @ b ) ) ) ).
%------------------------------------------------------------------------------