TPTP Problem File: ITP165^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP165^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_892__3593680_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_892__3593680_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 355 ( 164 unt; 49 typ; 0 def)
% Number of atoms : 819 ( 279 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3130 ( 81 ~; 11 |; 33 &;2696 @)
% ( 0 <=>; 309 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 157 ( 157 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 5 con; 0-5 aty)
% Number of variables : 953 ( 53 ^; 849 !; 7 ?; 953 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:47.460
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
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_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (45)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
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_Partial__Function_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > A > A > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_OASSUME,type,
refine1946708059ASSUME: $o > ( refine1665802226e_nres @ product_unit ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
refine1687780735RETURN:
!>[A: $tType] : ( A > ( 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_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_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_Refine__Misc_Ostrict,type,
refine_strict:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set__Interval_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_S1,type,
s1: refine1665802226e_nres @ a ).
thf(sy_v_S2,type,
s2: refine1665802226e_nres @ a ).
thf(sy_v__092_060Phi_062,type,
phi: a > $o ).
thf(sy_v_b,type,
b: $o ).
% Relevant facts (254)
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_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_3_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_4_SPEC__trans,axiom,
! [A: $tType,X4: refine1665802226e_nres @ A,Y3: refine1665802226e_nres @ A,Postcond: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Y3 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_5_SPEC__cons__rule,axiom,
! [A: $tType,M: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
=> ( ! [X3: A] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_6_nf__inres__RES,axiom,
! [A: $tType,X: set @ A,X4: A] :
( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X ) @ X4 )
= ( member @ A @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_7_nf__inres__SPEC,axiom,
! [A: $tType,Phi: A > $o,X4: A] :
( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X4 )
= ( Phi @ X4 ) ) ).
% nf_inres_SPEC
thf(fact_8_order__mono__setup_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_mono_setup.refl
thf(fact_9_the__RES_Osimps,axiom,
! [A: $tType,X: set @ A] :
( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
= X ) ).
% the_RES.simps
thf(fact_10_nres__order__simps_I21_J,axiom,
! [X5: $tType,X4: X5,Y: set @ X5] :
( ( ord_less_eq @ ( refine1665802226e_nres @ X5 ) @ ( refine1687780735RETURN @ X5 @ X4 ) @ ( refine605929679le_RES @ X5 @ Y ) )
= ( member @ X5 @ X4 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_11_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_12_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funD
thf(fact_13_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funE
thf(fact_14_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_15_Refine__Misc_Oif__mono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B2: $o,M1: A,M12: A,M2: A,M22: A] :
( ( B2
=> ( ord_less_eq @ A @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_less_eq @ A @ M2 @ M22 ) )
=> ( ord_less_eq @ A @ ( if @ A @ B2 @ M1 @ M2 ) @ ( if @ A @ B2 @ M12 @ M22 ) ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_16_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X6: A] : ( ord_less_eq @ B @ ( F3 @ X6 ) @ ( G2 @ X6 ) ) ) ) ) ).
% le_fun_def
thf(fact_17_nres__more__simps_I6_J,axiom,
! [A: $tType,X4: A,Y3: A] :
( ( ( refine1687780735RETURN @ A @ X4 )
= ( refine1687780735RETURN @ A @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_more_simps(6)
thf(fact_18_nres__order__simps_I20_J,axiom,
! [W: $tType,X4: W,Y3: W] :
( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X4 ) @ ( refine1687780735RETURN @ W @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_order_simps(20)
thf(fact_19_less__eq__nres_Osimps_I2_J,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% less_eq_nres.simps(2)
thf(fact_20_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_21_RETURN__to__SPEC__rule,axiom,
! [A: $tType,M: refine1665802226e_nres @ A,V: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ( ^ [Y4: A,Z: A] : Y4 = Z
@ V ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine1687780735RETURN @ A @ V ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_22_RETURN__rule,axiom,
! [A: $tType,Phi: A > $o,X4: A] :
( ( Phi @ X4 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_23_order__mono__setup_Omono__let,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [F: B > A,F4: B > A,X4: B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F4 @ X3 ) )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F4 @ X4 ) ) ) ) ).
% order_mono_setup.mono_let
thf(fact_24_order__mono__setup_Omono__if,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [T: A,T2: A,E2: A,E3: A,B2: $o] :
( ( ord_less_eq @ A @ T @ T2 )
=> ( ( ord_less_eq @ A @ E2 @ E3 )
=> ( ord_less_eq @ A @ ( if @ A @ B2 @ T @ E2 ) @ ( if @ A @ B2 @ T2 @ E3 ) ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_25_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_26_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_27_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_28_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_29_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_30_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_31_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_32_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_33_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_34_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_35_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X4: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_36_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X4 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_37_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_38_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% le_cases
thf(fact_39_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 = Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% eq_refl
thf(fact_40_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% linear
thf(fact_41_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ) ).
% antisym
thf(fact_42_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X6: A,Y5: A] :
( ( ord_less_eq @ A @ X6 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X6 ) ) ) ) ) ).
% eq_iff
thf(fact_43_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y6: A] :
( ( ord_less_eq @ A @ X3 @ Y6 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X6: A] : ( member @ A @ X6 @ A5 ) )
= A5 ) ).
% 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_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y6: B] :
( ( ord_less_eq @ B @ X3 @ Y6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_49_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y6: A] :
( ( ord_less_eq @ A @ X3 @ Y6 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_50_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y6: B] :
( ( ord_less_eq @ B @ X3 @ Y6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_51_the__RES__inv,axiom,
! [A: $tType,M: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ M )
=> ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M ) )
= M ) ) ).
% the_RES_inv
thf(fact_52_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X4: A] :
( ( P @ X4 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X4 ) )
=> ( ( order_Greatest @ A @ P )
= X4 ) ) ) ) ).
% Greatest_equality
thf(fact_53_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X4: A,Q: A > $o] :
( ( P @ X4 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X4 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y7: A] :
( ( P @ Y7 )
=> ( ord_less_eq @ A @ Y7 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_54_transfer_Otransfer__Let,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [Alpha: C > A,F: B > C,F5: B > A,X4: B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( Alpha @ ( F @ X3 ) ) @ ( F5 @ X3 ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( F @ X4 ) ) @ ( F5 @ X4 ) ) ) ) ).
% transfer.transfer_Let
thf(fact_55_transfer_Otransfer__if,axiom,
! [C: $tType,A: $tType] :
( ( comple187826305attice @ A )
=> ! [B2: $o,Alpha: C > A,S1: C,S12: A,S2: C,S22: A] :
( ( B2
=> ( ord_less_eq @ A @ ( Alpha @ S1 ) @ S12 ) )
=> ( ( ~ B2
=> ( ord_less_eq @ A @ ( Alpha @ S2 ) @ S22 ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B2 @ S1 @ S2 ) ) @ ( if @ A @ B2 @ S12 @ S22 ) ) ) ) ) ).
% transfer.transfer_if
thf(fact_56_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X7: $o > A,Y8: $o > A] :
( ( ord_less_eq @ A @ ( X7 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq @ A @ ( X7 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_57_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_58_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ( C2 = D )
=> ( ord_less_eq @ A @ A2 @ D ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_59_bind__cong,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( M = M3 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M3 )
=> ( ( F @ X3 )
= ( F4 @ X3 ) ) )
=> ( ( refine463715084e_bind @ A @ B @ M @ F )
= ( refine463715084e_bind @ A @ B @ M3 @ F4 ) ) ) ) ).
% bind_cong
thf(fact_60_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
! [B: $tType,A: $tType,M4: refine1665802226e_nres @ A,M5: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ M5 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M4 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F4 @ X3 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M4 @ F ) @ ( refine463715084e_bind @ A @ B @ M5 @ F4 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
thf(fact_61_nofail__simps_I2_J,axiom,
! [B: $tType,X: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X ) ) ).
% nofail_simps(2)
thf(fact_62_nofail__simps_I3_J,axiom,
! [C: $tType,X4: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X4 ) ) ).
% nofail_simps(3)
thf(fact_63_nres__monad2,axiom,
! [A: $tType,M4: refine1665802226e_nres @ A] :
( ( refine463715084e_bind @ A @ A @ M4 @ ( refine1687780735RETURN @ A ) )
= M4 ) ).
% nres_monad2
thf(fact_64_nres__monad1,axiom,
! [A: $tType,B: $tType,X4: B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X4 ) @ F )
= ( F @ X4 ) ) ).
% nres_monad1
thf(fact_65_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X6: A] :
( ( member @ A @ X6 @ S )
=> ( P @ X6 ) ) ) ) ).
% subset_Collect_conv
thf(fact_66_nofail__antimono__fun,axiom,
! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X4: A] :
( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
=> ( ( refine1102455758nofail @ B @ ( G @ X4 ) )
=> ( refine1102455758nofail @ B @ ( F @ X4 ) ) ) ) ).
% nofail_antimono_fun
thf(fact_67_nofail__RES__conv,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [M6: refine1665802226e_nres @ A] :
? [M7: set @ A] :
( M6
= ( refine605929679le_RES @ A @ M7 ) ) ) ) ).
% nofail_RES_conv
thf(fact_68_pwD1,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 )
=> ( ( refine1102455758nofail @ A @ S3 )
=> ( refine1102455758nofail @ A @ S ) ) ) ).
% pwD1
thf(fact_69_le__nofailI,axiom,
! [A: $tType,M5: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ M5 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ M5 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ M5 ) ) ).
% le_nofailI
thf(fact_70_subsetI,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% subsetI
thf(fact_71_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( A5 = B5 ) ) ) ).
% subset_antisym
thf(fact_72_pw__bind__leI,axiom,
! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M4 ) )
=> ( ! [X3: B] :
( ( refine1102455758nofail @ B @ M4 )
=> ( ( refine1315500908_inres @ B @ M4 @ X3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M4 @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_73_pw__bind__le__iff,axiom,
! [A: $tType,B: $tType,M4: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M4 @ F ) @ S )
= ( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M4 ) )
& ! [X6: B] :
( ( ( refine1102455758nofail @ B @ M4 )
& ( refine1315500908_inres @ B @ M4 @ X6 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X6 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_74_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_75_in__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_76_subsetD,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_77_inres__simps_I2_J,axiom,
! [B: $tType,X: set @ B] :
( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X ) )
= ( ^ [X6: B] : ( member @ B @ X6 @ X ) ) ) ).
% inres_simps(2)
thf(fact_78_inres__simps_I3_J,axiom,
! [C: $tType,X4: C] :
( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X4 ) )
= ( ^ [Y4: C,Z: C] : Y4 = Z
@ X4 ) ) ).
% inres_simps(3)
thf(fact_79_pwD2,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A,X4: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 )
=> ( ( refine1315500908_inres @ A @ S @ X4 )
=> ( refine1315500908_inres @ A @ S3 @ X4 ) ) ) ).
% pwD2
thf(fact_80_pw__eqI,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S )
= ( refine1102455758nofail @ A @ S3 ) )
=> ( ! [X3: A] :
( ( refine1315500908_inres @ A @ S @ X3 )
= ( refine1315500908_inres @ A @ S3 @ X3 ) )
=> ( S = S3 ) ) ) ).
% pw_eqI
thf(fact_81_pw__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : Y4 = Z )
= ( ^ [S4: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S4 )
= ( refine1102455758nofail @ A @ S5 ) )
& ! [X6: A] :
( ( refine1315500908_inres @ A @ S4 @ X6 )
= ( refine1315500908_inres @ A @ S5 @ X6 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_82_not__nofail__inres,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,X4: A] :
( ~ ( refine1102455758nofail @ A @ S )
=> ( refine1315500908_inres @ A @ S @ X4 ) ) ).
% not_nofail_inres
thf(fact_83_pw__le__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
= ( ^ [S4: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ S5 )
=> ( ( refine1102455758nofail @ A @ S4 )
& ! [X6: A] :
( ( refine1315500908_inres @ A @ S4 @ X6 )
=> ( refine1315500908_inres @ A @ S5 @ X6 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_84_pw__leI_H,axiom,
! [A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S3 )
=> ( refine1102455758nofail @ A @ S ) )
=> ( ! [X3: A] :
( ( refine1102455758nofail @ A @ S3 )
=> ( ( refine1315500908_inres @ A @ S @ X3 )
=> ( refine1315500908_inres @ A @ S3 @ X3 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 ) ) ) ).
% pw_leI'
thf(fact_85_pw__leI,axiom,
! [A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S3 )
=> ( ( refine1102455758nofail @ A @ S )
& ! [X3: A] :
( ( refine1315500908_inres @ A @ S @ X3 )
=> ( refine1315500908_inres @ A @ S3 @ X3 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S3 ) ) ).
% pw_leI
thf(fact_86_pw__bind__nofail,axiom,
! [A: $tType,B: $tType,M4: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M4 @ F ) )
= ( ( refine1102455758nofail @ B @ M4 )
& ! [X6: B] :
( ( refine1315500908_inres @ B @ M4 @ X6 )
=> ( refine1102455758nofail @ A @ ( F @ X6 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_87_inres__def,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A )
= ( ^ [S4: refine1665802226e_nres @ A,X6: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X6 ) @ S4 ) ) ) ).
% inres_def
thf(fact_88_nf__inres__def,axiom,
! [A: $tType] :
( ( refine406925620_inres @ A )
= ( ^ [M6: refine1665802226e_nres @ A,X6: A] :
( ( refine1102455758nofail @ A @ M6 )
& ( refine1315500908_inres @ A @ M6 @ X6 ) ) ) ) ).
% nf_inres_def
thf(fact_89_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X6: A] :
( ( P @ X6 )
=> ( Q @ X6 ) ) ) ) ).
% Collect_mono_iff
thf(fact_90_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z: set @ A] : Y4 = 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_91_subset__trans,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% subset_trans
thf(fact_92_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_93_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_94_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_95_equalityD2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_96_equalityD1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_97_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X6: A] :
( ( member @ A @ X6 @ A6 )
=> ( member @ A @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_98_equalityE,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% equalityE
thf(fact_99_less__eq__nres_Oelims_I1_J,axiom,
! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: $o] :
( ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
= Y3 )
=> ( ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ~ Y3 )
=> ( ! [A4: set @ A] :
( ( X4
= ( refine605929679le_RES @ A @ A4 ) )
=> ! [B4: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B4 ) )
=> ( Y3
= ( ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) )
=> ~ ( ( X4
= ( refine1767639642_FAILi @ A ) )
=> ( ? [Uv: set @ A] :
( Xa
= ( refine605929679le_RES @ A @ Uv ) )
=> Y3 ) ) ) ) ) ).
% less_eq_nres.elims(1)
thf(fact_100_less__eq__nres_Oelims_I2_J,axiom,
! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
=> ( ( Xa
!= ( refine1767639642_FAILi @ A ) )
=> ~ ! [A4: set @ A] :
( ( X4
= ( refine605929679le_RES @ A @ A4 ) )
=> ! [B4: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B4 ) )
=> ~ ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) ) ).
% less_eq_nres.elims(2)
thf(fact_101_less__eq__nres_Oelims_I3_J,axiom,
! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A] :
( ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
=> ( ! [A4: set @ A] :
( ( X4
= ( refine605929679le_RES @ A @ A4 ) )
=> ! [B4: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) )
=> ~ ( ( X4
= ( refine1767639642_FAILi @ A ) )
=> ! [Uv: set @ A] :
( Xa
!= ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% less_eq_nres.elims(3)
thf(fact_102_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_103_nres__order__simps_I22_J,axiom,
! [Y9: $tType,X: set @ Y9,Y3: Y9] :
( ( ord_less_eq @ ( refine1665802226e_nres @ Y9 ) @ ( refine605929679le_RES @ Y9 @ X ) @ ( refine1687780735RETURN @ Y9 @ Y3 ) )
= ( ord_less_eq @ ( set @ Y9 ) @ X @ ( insert @ Y9 @ Y3 @ ( bot_bot @ ( set @ Y9 ) ) ) ) ) ).
% nres_order_simps(22)
thf(fact_104_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_105_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X6: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_106_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_107_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X6: A] :
~ ( member @ A @ X6 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_108_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X6: A] :
~ ( P @ X6 ) ) ) ).
% Collect_empty_eq
thf(fact_109_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X6: A] :
~ ( P @ X6 ) ) ) ).
% empty_Collect_eq
thf(fact_110_insertCI,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B5 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_111_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A5 ) ) ) ).
% insert_iff
thf(fact_112_insert__absorb2,axiom,
! [A: $tType,X4: A,A5: set @ A] :
( ( insert @ A @ X4 @ ( insert @ A @ X4 @ A5 ) )
= ( insert @ A @ X4 @ A5 ) ) ).
% insert_absorb2
thf(fact_113_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_114_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_115_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_116_insert__subset,axiom,
! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ A5 ) @ B5 )
= ( ( member @ A @ X4 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_117_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A5: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A5 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_118_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_119_nres__more__simps_I5_J,axiom,
! [A: $tType,X4: A,X: set @ A] :
( ( ( refine1687780735RETURN @ A @ X4 )
= ( refine605929679le_RES @ A @ X ) )
= ( ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
= X ) ) ).
% nres_more_simps(5)
thf(fact_120_nres__more__simps_I3_J,axiom,
! [A: $tType,X: set @ A,X4: A] :
( ( ( refine605929679le_RES @ A @ X )
= ( refine1687780735RETURN @ A @ X4 ) )
= ( X
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% nres_more_simps(3)
thf(fact_121_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_122_subset__singleton__iff,axiom,
! [A: $tType,X: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X
= ( bot_bot @ ( set @ A ) ) )
| ( X
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_123_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_124_insertE,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A5 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A5 ) ) ) ).
% insertE
thf(fact_125_equals0D,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A5 ) ) ).
% equals0D
thf(fact_126_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y6: A] :
~ ( member @ A @ Y6 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_127_insertI1,axiom,
! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% insertI1
thf(fact_128_insertI2,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( member @ A @ A2 @ B5 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_129_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X6: A] : ( member @ A @ X6 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_130_Set_Oset__insert,axiom,
! [A: $tType,X4: A,A5: set @ A] :
( ( member @ A @ X4 @ A5 )
=> ~ ! [B7: set @ A] :
( ( A5
= ( insert @ A @ X4 @ B7 ) )
=> ( member @ A @ X4 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_131_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_132_insert__ident,axiom,
! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X4 @ A5 )
=> ( ~ ( member @ A @ X4 @ B5 )
=> ( ( ( insert @ A @ X4 @ A5 )
= ( insert @ A @ X4 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_133_insert__absorb,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( insert @ A @ A2 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_134_insert__eq__iff,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A,B5: set @ A] :
( ~ ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ B2 @ B5 )
=> ( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ B5 ) )
= ( ( ( A2 = B2 )
=> ( A5 = B5 ) )
& ( ( A2 != B2 )
=> ? [C4: set @ A] :
( ( A5
= ( insert @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B5
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_135_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_136_insert__commute,axiom,
! [A: $tType,X4: A,Y3: A,A5: set @ A] :
( ( insert @ A @ X4 @ ( insert @ A @ Y3 @ A5 ) )
= ( insert @ A @ Y3 @ ( insert @ A @ X4 @ A5 ) ) ) ).
% insert_commute
thf(fact_137_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C2: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_138_insert__not__empty,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( insert @ A @ A2 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_139_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_140_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ? [B7: set @ A] :
( ( A5
= ( insert @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_141_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X6: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_142_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_143_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_144_memb__imp__not__empty,axiom,
! [A: $tType,X4: A,S: set @ A] :
( ( member @ A @ X4 @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_145_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_146_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_147_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_148_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_149_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_150_subset__insert,axiom,
! [A: $tType,X4: A,A5: set @ A,B5: set @ A] :
( ~ ( member @ A @ X4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X4 @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_151_insert__mono,axiom,
! [A: $tType,C3: set @ A,D3: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C3 ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_152_RETURN__def,axiom,
! [A: $tType] :
( ( refine1687780735RETURN @ A )
= ( ^ [X6: A] : ( refine605929679le_RES @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% RETURN_def
thf(fact_153_nres_Odistinct_I1_J,axiom,
! [A: $tType,X2: set @ A] :
( ( refine1767639642_FAILi @ A )
!= ( refine605929679le_RES @ A @ X2 ) ) ).
% nres.distinct(1)
thf(fact_154_nres_Oinduct,axiom,
! [A: $tType,P: ( refine1665802226e_nres @ A ) > $o,Nres: refine1665802226e_nres @ A] :
( ( P @ ( refine1767639642_FAILi @ A ) )
=> ( ! [X3: set @ A] : ( P @ ( refine605929679le_RES @ A @ X3 ) )
=> ( P @ Nres ) ) ) ).
% nres.induct
thf(fact_155_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_156_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 ) )
=> ( ! [V2: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V2 ) )
=> ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sup_nres.induct
thf(fact_157_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 ) )
=> ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_nres.induct
thf(fact_158_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 ) )
=> ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
=> ( ! [Uv: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_eq_nres.induct
thf(fact_159_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_160_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_161_the__elem__eq,axiom,
! [A: $tType,X4: A] :
( ( the_elem @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
= X4 ) ).
% the_elem_eq
thf(fact_162_is__singletonI,axiom,
! [A: $tType,X4: A] : ( is_singleton @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_163_insert__subsetI,axiom,
! [A: $tType,X4: A,A5: set @ A,X: set @ A] :
( ( member @ A @ X4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X4 @ X ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_164_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_165_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_166_nres__order__simps_I2_J,axiom,
! [B: $tType,M4: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M4 @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
= ( M4
= ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ) ).
% nres_order_simps(2)
thf(fact_167_nofail__simps_I4_J,axiom,
! [D2: $tType] : ( refine1102455758nofail @ D2 @ ( bot_bot @ ( refine1665802226e_nres @ D2 ) ) ) ).
% nofail_simps(4)
thf(fact_168_inres__simps_I4_J,axiom,
! [D2: $tType] :
( ( refine1315500908_inres @ D2 @ ( bot_bot @ ( refine1665802226e_nres @ D2 ) ) )
= ( ^ [Uu3: D2] : $false ) ) ).
% inres_simps(4)
thf(fact_169_bind__SUCCEED,axiom,
! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) @ F )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% bind_SUCCEED
thf(fact_170_nres__more__simps_I2_J,axiom,
! [A: $tType,X: set @ A] :
( ( ( refine605929679le_RES @ A @ X )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) )
= ( X
= ( bot_bot @ ( set @ A ) ) ) ) ).
% nres_more_simps(2)
thf(fact_171_nres__more__simps_I1_J,axiom,
! [A: $tType,X: set @ A] :
( ( ( bot_bot @ ( refine1665802226e_nres @ A ) )
= ( refine605929679le_RES @ A @ X ) )
= ( X
= ( bot_bot @ ( set @ A ) ) ) ) ).
% nres_more_simps(1)
thf(fact_172_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_173_nres__order__simps_I1_J,axiom,
! [A: $tType,M4: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ M4 ) ).
% nres_order_simps(1)
thf(fact_174_nres__inequalities_I5_J,axiom,
! [C: $tType,X4: C] :
( ( bot_bot @ ( refine1665802226e_nres @ C ) )
!= ( refine1687780735RETURN @ C @ X4 ) ) ).
% nres_inequalities(5)
thf(fact_175_bot__nres__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( refine1665802226e_nres @ A ) )
= ( refine605929679le_RES @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_nres_def
thf(fact_176_SUCCEED__rule,axiom,
! [A: $tType,Phi: A > $o] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ).
% SUCCEED_rule
thf(fact_177_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_178_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y6: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y6 @ A5 )
=> ( X3 = Y6 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_179_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X6: A] :
( A6
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_180_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X3: A] :
( A5
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_181_subset__Compl__singleton,axiom,
! [A: $tType,A5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_182_strictD__simp,axiom,
! [A: $tType] :
( ( bot @ A )
=> ! [F: A > A] :
( ( refine_strict @ A @ A @ F )
=> ( ( F @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ) ).
% strictD_simp
thf(fact_183_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_184_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_185_ComplI,axiom,
! [A: $tType,C2: A,A5: set @ A] :
( ~ ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% ComplI
thf(fact_186_Compl__iff,axiom,
! [A: $tType,C2: A,A5: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( ~ ( member @ A @ C2 @ A5 ) ) ) ).
% Compl_iff
thf(fact_187_Compl__eq__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A5 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( A5 = B5 ) ) ).
% Compl_eq_Compl_iff
thf(fact_188_Compl__anti__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% Compl_anti_mono
thf(fact_189_Compl__subset__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% Compl_subset_Compl_iff
thf(fact_190_ibind__strict_I1_J,axiom,
! [A: $tType,F: product_unit > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ product_unit @ A @ ( bot_bot @ ( refine1665802226e_nres @ product_unit ) ) @ F )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% ibind_strict(1)
thf(fact_191_ComplD,axiom,
! [A: $tType,C2: A,A5: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
=> ~ ( member @ A @ C2 @ A5 ) ) ).
% ComplD
thf(fact_192_double__complement,axiom,
! [A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= A5 ) ).
% double_complement
thf(fact_193_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_194_subset__Compl__self__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_195_strictD,axiom,
! [A: $tType,B: $tType] :
( ( ( bot @ B )
& ( bot @ A ) )
=> ! [F: A > B] :
( ( refine_strict @ A @ B @ F )
=> ( ( F @ ( bot_bot @ A ) )
= ( bot_bot @ B ) ) ) ) ).
% strictD
thf(fact_196_strictI,axiom,
! [A: $tType,B: $tType] :
( ( ( bot @ B )
& ( bot @ A ) )
=> ! [F: B > A] :
( ( ( F @ ( bot_bot @ B ) )
= ( bot_bot @ A ) )
=> ( refine_strict @ B @ A @ F ) ) ) ).
% strictI
thf(fact_197_strict__def,axiom,
! [B: $tType,A: $tType] :
( ( ( bot @ A )
& ( bot @ B ) )
=> ( ( refine_strict @ A @ B )
= ( ^ [F3: A > B] :
( ( F3 @ ( bot_bot @ A ) )
= ( bot_bot @ B ) ) ) ) ) ).
% strict_def
thf(fact_198_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_199_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X4 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% compl_le_compl_iff
thf(fact_200_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X6: A] : ( member @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_201_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_202_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_203_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_204_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X4 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X4 ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_205_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X4: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X4 ) )
=> ( ord_less_eq @ A @ X4 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_206_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X4 ) ) ) ) ).
% compl_mono
thf(fact_207_ASSUME__simps_I2_J,axiom,
( ( refine1946708059ASSUME @ $false )
= ( bot_bot @ ( refine1665802226e_nres @ product_unit ) ) ) ).
% ASSUME_simps(2)
thf(fact_208_pw__flat__le__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ S @ S3 )
= ( ? [X8: A] : ( refine1315500908_inres @ A @ S @ X8 )
=> ( ( ( refine1102455758nofail @ A @ S )
= ( refine1102455758nofail @ A @ S3 ) )
& ! [X6: A] :
( ( refine1315500908_inres @ A @ S @ X6 )
= ( refine1315500908_inres @ A @ S3 @ X6 ) ) ) ) ) ).
% pw_flat_le_iff
thf(fact_209_pw__ASSUME_I2_J,axiom,
! [Phi: $o,X4: product_unit] :
( ( refine1315500908_inres @ product_unit @ ( refine1946708059ASSUME @ Phi ) @ X4 )
= Phi ) ).
% pw_ASSUME(2)
thf(fact_210_pw__ASSUME_I1_J,axiom,
! [Phi: $o] : ( refine1102455758nofail @ product_unit @ ( refine1946708059ASSUME @ Phi ) ) ).
% pw_ASSUME(1)
thf(fact_211_flat__ord__compat_I1_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [X4: A,Y3: A] :
( ( partial_flat_ord @ A @ ( bot_bot @ A ) @ X4 @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% flat_ord_compat(1)
thf(fact_212_pw__flat__ge__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S3: refine1665802226e_nres @ A] :
( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ S @ S3 )
= ( ( refine1102455758nofail @ A @ S )
=> ( ( refine1102455758nofail @ A @ S3 )
& ! [X6: A] :
( ( refine1315500908_inres @ A @ S @ X6 )
= ( refine1315500908_inres @ A @ S3 @ X6 ) ) ) ) ) ).
% pw_flat_ge_iff
thf(fact_213_top__apply,axiom,
! [C: $tType,D2: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D2 > C ) )
= ( ^ [X6: D2] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_214_nres__order__simps_I4_J,axiom,
! [D2: $tType,M4: refine1665802226e_nres @ D2] :
( ( ord_less_eq @ ( refine1665802226e_nres @ D2 ) @ ( top_top @ ( refine1665802226e_nres @ D2 ) ) @ M4 )
= ( M4
= ( top_top @ ( refine1665802226e_nres @ D2 ) ) ) ) ).
% nres_order_simps(4)
thf(fact_215_nofail__simps_I1_J,axiom,
! [A: $tType] :
~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% nofail_simps(1)
thf(fact_216_nres__simp__internals_I2_J,axiom,
! [B: $tType] :
( ( refine1767639642_FAILi @ B )
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ).
% nres_simp_internals(2)
thf(fact_217_inres__simps_I1_J,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
= ( ^ [Uu3: A] : $true ) ) ).
% inres_simps(1)
thf(fact_218_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_219_compl__bot__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% compl_bot_eq
thf(fact_220_compl__top__eq,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% compl_top_eq
thf(fact_221_flat__ord__compat_I2_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [X4: A,Y3: A] :
( ( partial_flat_ord @ A @ ( top_top @ A ) @ X4 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% flat_ord_compat(2)
thf(fact_222_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_223_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_224_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_225_meta__le__everything__if__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [M: A,X4: A] :
( ( M
= ( top_top @ A ) )
=> ( ord_less_eq @ A @ X4 @ M ) ) ) ).
% meta_le_everything_if_top
thf(fact_226_nres__inequalities_I3_J,axiom,
! [C: $tType,X4: C] :
( ( top_top @ ( refine1665802226e_nres @ C ) )
!= ( refine1687780735RETURN @ C @ X4 ) ) ).
% nres_inequalities(3)
thf(fact_227_top__nres__def,axiom,
! [A: $tType] :
( ( top_top @ ( refine1665802226e_nres @ A ) )
= ( refine1767639642_FAILi @ A ) ) ).
% top_nres_def
thf(fact_228_not__nofail__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A] :
( ( ~ ( refine1102455758nofail @ A @ S ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% not_nofail_iff
thf(fact_229_nofail__def,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [S4: refine1665802226e_nres @ A] :
( S4
!= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% nofail_def
thf(fact_230_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_231_nres__order__simps_I3_J,axiom,
! [C: $tType,M4: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M4 @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% nres_order_simps(3)
thf(fact_232_nres__inequalities_I1_J,axiom,
! [A: $tType,X: set @ A] :
( ( top_top @ ( refine1665802226e_nres @ A ) )
!= ( refine605929679le_RES @ A @ X ) ) ).
% nres_inequalities(1)
thf(fact_233_nres__cases,axiom,
! [A: $tType,M4: refine1665802226e_nres @ A] :
( ( M4
!= ( top_top @ ( refine1665802226e_nres @ A ) ) )
=> ~ ! [X9: set @ A] :
( M4
!= ( refine605929679le_RES @ A @ X9 ) ) ) ).
% nres_cases
thf(fact_234_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_235_nres__inequalities_I4_J,axiom,
! [D2: $tType] :
( ( bot_bot @ ( refine1665802226e_nres @ D2 ) )
!= ( top_top @ ( refine1665802226e_nres @ D2 ) ) ) ).
% nres_inequalities(4)
thf(fact_236_nres__inequalities_I2_J,axiom,
! [B: $tType] :
( ( top_top @ ( refine1665802226e_nres @ B ) )
!= ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ).
% nres_inequalities(2)
thf(fact_237_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J,axiom,
! [B: $tType,A: $tType,M4: refine1665802226e_nres @ A,M5: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ M4 @ M5 )
=> ( ! [X3: A] : ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( F @ X3 ) @ ( F4 @ X3 ) )
=> ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( refine463715084e_bind @ A @ B @ M4 @ F ) @ ( refine463715084e_bind @ A @ B @ M5 @ F4 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(2)
thf(fact_238_flat__ord_OatLeastAtMost__singleton,axiom,
! [A: $tType,B2: A,A2: A] :
( ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ B2 ) @ A2 @ A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% flat_ord.atLeastAtMost_singleton
thf(fact_239_flat__ord_OatLeastAtMost__singleton__iff,axiom,
! [A: $tType,Ba: A,A2: A,B2: A,C2: A] :
( ( ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ Ba ) @ A2 @ B2 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( B2 = C2 ) ) ) ).
% flat_ord.atLeastAtMost_singleton_iff
thf(fact_240_flat__ord_OatLeastatMost__empty__iff,axiom,
! [A: $tType,Ba: A,A2: A,B2: A] :
( ( ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ Ba ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( partial_flat_ord @ A @ Ba @ A2 @ B2 ) ) ) ).
% flat_ord.atLeastatMost_empty_iff
thf(fact_241_flat__ord_OatLeastatMost__empty__iff2,axiom,
! [A: $tType,Ba: A,A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ Ba ) @ A2 @ B2 ) )
= ( ~ ( partial_flat_ord @ A @ Ba @ A2 @ B2 ) ) ) ).
% flat_ord.atLeastatMost_empty_iff2
thf(fact_242_flat__ord_OatLeastatMost__subset__iff,axiom,
! [A: $tType,Ba: A,A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ Ba ) @ A2 @ B2 ) @ ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ Ba ) @ C2 @ D ) )
= ( ~ ( partial_flat_ord @ A @ Ba @ A2 @ B2 )
| ( ( partial_flat_ord @ A @ Ba @ C2 @ A2 )
& ( partial_flat_ord @ A @ Ba @ B2 @ D ) ) ) ) ).
% flat_ord.atLeastatMost_subset_iff
thf(fact_243_insert__UNIV,axiom,
! [A: $tType,X4: A] :
( ( insert @ A @ X4 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_244_subset__UNIV,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_245_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_246_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_247_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_248_flat__ord_OatLeastAtMost__singleton_H,axiom,
! [A: $tType,A2: A,B2: A,Ba: A] :
( ( A2 = B2 )
=> ( ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ Ba ) @ A2 @ B2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% flat_ord.atLeastAtMost_singleton'
thf(fact_249_flat__ord_OIcc__subset__Iic__iff,axiom,
! [A: $tType,B2: A,L: A,H: A,H2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ B2 ) @ L @ H ) @ ( set_atMost @ A @ ( partial_flat_ord @ A @ B2 ) @ H2 ) )
= ( ~ ( partial_flat_ord @ A @ B2 @ L @ H )
| ( partial_flat_ord @ A @ B2 @ H @ H2 ) ) ) ).
% flat_ord.Icc_subset_Iic_iff
thf(fact_250_flat__ord_OIcc__subset__Ici__iff,axiom,
! [A: $tType,B2: A,L: A,H: A,L2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_atLeastAtMost @ A @ ( partial_flat_ord @ A @ B2 ) @ L @ H ) @ ( set_atLeast @ A @ ( partial_flat_ord @ A @ B2 ) @ L2 ) )
= ( ~ ( partial_flat_ord @ A @ B2 @ L @ H )
| ( partial_flat_ord @ A @ B2 @ L2 @ L ) ) ) ).
% flat_ord.Icc_subset_Ici_iff
thf(fact_251_UNIV__I,axiom,
! [A: $tType,X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_252_relprop__UNIV__orient,axiom,
! [A: $tType,R: set @ A] :
( ( R
= ( top_top @ ( set @ A ) ) )
=> ( ( top_top @ ( set @ A ) )
= R ) ) ).
% relprop_UNIV_orient
thf(fact_253_eq__UNIV__iff,axiom,
! [A: $tType,S: set @ A] :
( ( S
= ( top_top @ ( set @ A ) ) )
= ( ! [X6: A] : ( member @ A @ X6 @ S ) ) ) ).
% eq_UNIV_iff
% Type constructors (46)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 )
=> ( comple187826305attice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A7: $tType,A8: $tType] :
( ( boolean_algebra @ A8 )
=> ( boolean_algebra @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A8: $tType] :
( ( order_top @ A8 )
=> ( order_top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 )
=> ( order_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 )
=> ( top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 )
=> ( bot @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_2,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_3,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_4,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_10,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_11,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_12,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_15,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_16,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
bot @ $o ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_19,axiom,
comple187826305attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Oboolean__algebra_20,axiom,
boolean_algebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_21,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_22,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_23,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_24,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_25,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_26,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_27,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_28,axiom,
bot @ product_unit ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_29,axiom,
! [A7: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_30,axiom,
! [A7: $tType] : ( order_top @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__bot_31,axiom,
! [A7: $tType] : ( order_bot @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_32,axiom,
! [A7: $tType] : ( preorder @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_33,axiom,
! [A7: $tType] : ( order @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_34,axiom,
! [A7: $tType] : ( top @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_35,axiom,
! [A7: $tType] : ( ord @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_36,axiom,
! [A7: $tType] : ( bot @ ( refine1665802226e_nres @ A7 ) ) ).
% 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,X4: A,Y3: A] :
( ( if @ A @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X4: A,Y3: A] :
( ( if @ A @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
( b
=> ( ord_less_eq @ ( refine1665802226e_nres @ a ) @ s1 @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) ) ) ).
thf(conj_1,hypothesis,
( ~ b
=> ( ord_less_eq @ ( refine1665802226e_nres @ a ) @ s2 @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) ) ) ).
thf(conj_2,conjecture,
ord_less_eq @ ( refine1665802226e_nres @ a ) @ ( if @ ( refine1665802226e_nres @ a ) @ b @ s1 @ s2 ) @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) ).
%------------------------------------------------------------------------------