TPTP Problem File: ITP162^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP162^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_1909__3604894_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_1909__3604894_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 350 ( 128 unt; 38 typ; 0 def)
% Number of atoms : 789 ( 223 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3541 ( 46 ~; 3 |; 33 &;3068 @)
% ( 0 <=>; 391 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 190 ( 190 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 36 usr; 3 con; 0-5 aty)
% Number of variables : 1063 ( 97 ^; 931 !; 2 ?;1063 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:30.432
%------------------------------------------------------------------------------
% 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 (34)
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_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_Lattices_Olattice,type,
lattice:
!>[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_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde1561333602nf_top:
!>[A: $tType] : $o ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( 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_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_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_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a2: refine1665802226e_nres @ a ).
thf(sy_v_P,type,
p: a > $o ).
thf(sy_v_Q,type,
q: a > $o ).
% Relevant facts (251)
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_assms_I2_J,axiom,
ord_less_eq @ ( refine1665802226e_nres @ a ) @ a2 @ ( refine605929679le_RES @ a @ ( collect @ a @ q ) ) ).
% assms(2)
thf(fact_3_assms_I1_J,axiom,
ord_less_eq @ ( refine1665802226e_nres @ a ) @ a2 @ ( refine605929679le_RES @ a @ ( collect @ a @ p ) ) ).
% assms(1)
thf(fact_4__092_060open_062A_A_092_060le_062_Ainf_A_ISPEC_AP_J_A_ISPEC_AQ_J_092_060close_062,axiom,
ord_less_eq @ ( refine1665802226e_nres @ a ) @ a2 @ ( inf_inf @ ( refine1665802226e_nres @ a ) @ ( refine605929679le_RES @ a @ ( collect @ a @ p ) ) @ ( refine605929679le_RES @ a @ ( collect @ a @ q ) ) ) ).
% \<open>A \<le> inf (SPEC P) (SPEC Q)\<close>
thf(fact_5_if__rule,axiom,
! [A: $tType,B2: $o,S1: refine1665802226e_nres @ A,Phi: A > $o,S2: refine1665802226e_nres @ A] :
( ( B2
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S1 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
=> ( ( ~ B2
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ S1 @ S2 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% if_rule
thf(fact_6_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_7_SPEC__iff,axiom,
! [A: $tType,P: refine1665802226e_nres @ A,Q: A > $o,R: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ P
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [S3: A] :
( ( Q @ S3 )
=> ( R @ S3 ) ) ) ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ P
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [S3: A] :
( ~ ( Q @ S3 )
=> ~ ( R @ S3 ) ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ P
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [S3: A] :
( ( Q @ S3 )
= ( R @ S3 ) ) ) ) ) ) ) ).
% SPEC_iff
thf(fact_8_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_9_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_10_lhs__step__If,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B2: $o,T: A,M: A,E: A] :
( ( B2
=> ( ord_less_eq @ A @ T @ M ) )
=> ( ( ~ B2
=> ( ord_less_eq @ A @ E @ M ) )
=> ( ord_less_eq @ A @ ( if @ A @ B2 @ T @ E ) @ M ) ) ) ) ).
% lhs_step_If
thf(fact_11_use__spec__rule,axiom,
! [A: $tType,M: refine1665802226e_nres @ A,Psi: A > $o,Phi: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [S3: A] :
( ( Psi @ S3 )
=> ( Phi @ S3 ) ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% use_spec_rule
thf(fact_12_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_13_order__mono__setup_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_mono_setup.refl
thf(fact_14_the__RES_Osimps,axiom,
! [A: $tType,X: set @ A] :
( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
= X ) ).
% the_RES.simps
thf(fact_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_23_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,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_24_le__infD2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ).
% le_infD2
thf(fact_25_le__infD1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% le_infD1
thf(fact_26_inf__leI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ C2 ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf_leI
thf(fact_27_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_28_nres__order__simps_I5_J,axiom,
! [E2: $tType,X: set @ E2,Y: set @ E2] :
( ( ord_less_eq @ ( refine1665802226e_nres @ E2 ) @ ( refine605929679le_RES @ E2 @ X ) @ ( refine605929679le_RES @ E2 @ Y ) )
= ( ord_less_eq @ ( set @ E2 ) @ X @ Y ) ) ).
% nres_order_simps(5)
thf(fact_29_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_30_order__mono__setup_Omono__if,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [T: A,T2: A,E: A,E3: A,B2: $o] :
( ( ord_less_eq @ A @ T @ T2 )
=> ( ( ord_less_eq @ A @ E @ E3 )
=> ( ord_less_eq @ A @ ( if @ A @ B2 @ T @ E ) @ ( if @ A @ B2 @ T2 @ E3 ) ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_31_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_32_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_33_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_34_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_35_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_36_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_37_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_38_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_39_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_40_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_41_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_42_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_44_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_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_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_47_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_48_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_49_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 = Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% eq_refl
thf(fact_50_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_51_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_52_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_53_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,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_54_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,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_55_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,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_56_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_57_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ ( inf_inf @ A @ Y3 @ Z2 ) )
= ( ( ord_less_eq @ A @ X4 @ Y3 )
& ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_58_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X5: A] : ( inf_inf @ B @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% inf_apply
thf(fact_59_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_60_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ X4 @ X4 )
= X4 ) ) ).
% inf_idem
thf(fact_61_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_62_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ X4 @ Y3 ) )
= ( inf_inf @ A @ X4 @ Y3 ) ) ) ).
% inf_left_idem
thf(fact_63_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_64_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ Y3 )
= ( inf_inf @ A @ X4 @ Y3 ) ) ) ).
% inf_right_idem
thf(fact_65_inf__nres_Osimps_I3_J,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A2 ) @ ( refine605929679le_RES @ A @ B2 ) )
= ( refine605929679le_RES @ A @ ( inf_inf @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% inf_nres.simps(3)
thf(fact_66_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y3 @ Z2 ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X4 @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_67_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C2 ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_68_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X5: A,Y6: A] : ( inf_inf @ A @ Y6 @ X5 ) ) ) ) ).
% inf_commute
thf(fact_69_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A3: A,B3: A] : ( inf_inf @ A @ B3 @ A3 ) ) ) ) ).
% inf.commute
thf(fact_70_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ Z2 )
= ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y3 @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_71_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_72_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( inf_inf @ A @ K @ B2 ) )
=> ( ( inf_inf @ A @ A2 @ B5 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_73_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: A,K: A,A2: A,B2: A] :
( ( A5
= ( inf_inf @ A @ K @ A2 ) )
=> ( ( inf_inf @ A @ A5 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_74_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X5: A] : ( inf_inf @ B @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% inf_fun_def
thf(fact_75_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X5: A,Y6: A] : ( inf_inf @ A @ Y6 @ X5 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_76_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ Z2 )
= ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y3 @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_77_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y3 @ Z2 ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X4 @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_78_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y3: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ X4 @ Y3 ) )
= ( inf_inf @ A @ X4 @ Y3 ) ) ) ).
% inf_sup_aci(4)
thf(fact_79_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ Y3 ) ) ).
% inf_sup_ord(2)
thf(fact_80_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ X4 ) ) ).
% inf_sup_ord(1)
thf(fact_81_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ X4 ) ) ).
% inf_le1
thf(fact_82_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ Y3 ) ) ).
% inf_le2
thf(fact_83_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X4 @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X4 @ A2 )
=> ~ ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ).
% le_infE
thf(fact_84_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X4 @ A2 )
=> ( ( ord_less_eq @ A @ X4 @ B2 )
=> ( ord_less_eq @ A @ X4 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_85_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D ) ) ) ) ) ).
% inf_mono
thf(fact_86_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X4: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X4 ) ) ) ).
% le_infI1
thf(fact_87_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X4: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X4 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X4 ) ) ) ).
% le_infI2
thf(fact_88_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_89_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( inf_inf @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% inf.orderI
thf(fact_90_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F: A > A > A,X4: A,Y3: A] :
( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less_eq @ A @ X3 @ Z3 )
=> ( ord_less_eq @ A @ X3 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ) ).
% inf_unique
thf(fact_91_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X5: A,Y6: A] :
( ( inf_inf @ A @ X5 @ Y6 )
= X5 ) ) ) ) ).
% le_iff_inf
thf(fact_92_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb1
thf(fact_93_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_94_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( inf_inf @ A @ X4 @ Y3 )
= X4 ) ) ) ).
% inf_absorb1
thf(fact_95_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y3: A,X4: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ( inf_inf @ A @ X4 @ Y3 )
= Y3 ) ) ) ).
% inf_absorb2
thf(fact_96_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_97_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_98_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ X4 @ Z2 )
=> ( ord_less_eq @ A @ X4 @ ( inf_inf @ A @ Y3 @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_99_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( A3
= ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% inf.order_iff
thf(fact_100_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ).
% inf.cobounded1
thf(fact_101_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_102_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_103_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_104_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_105_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_106_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_107_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A5 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_108_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_109_Collect__subset,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( P @ X5 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_110_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X5: A] : ( member @ A @ X5 @ A6 )
@ ^ [X5: A] : ( member @ A @ X5 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_111_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X5: A] : ( member @ A @ X5 @ R )
@ ^ [X5: A] : ( member @ A @ X5 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_112_conj__subset__def,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5
@ ( collect @ A
@ ^ [X5: A] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_113_IntI,axiom,
! [A: $tType,C2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A5 )
=> ( ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_114_Int__iff,axiom,
! [A: $tType,C2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( ( member @ A @ C2 @ A5 )
& ( member @ A @ C2 @ B5 ) ) ) ).
% Int_iff
thf(fact_115_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_116_IntE,axiom,
! [A: $tType,C2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
=> ~ ( ( member @ A @ C2 @ A5 )
=> ~ ( member @ A @ C2 @ B5 ) ) ) ).
% IntE
thf(fact_117_IntD1,axiom,
! [A: $tType,C2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
=> ( member @ A @ C2 @ A5 ) ) ).
% IntD1
thf(fact_118_IntD2,axiom,
! [A: $tType,C2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
=> ( member @ A @ C2 @ B5 ) ) ).
% IntD2
thf(fact_119_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( member @ A @ X5 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_120_Int__assoc,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Int_assoc
thf(fact_121_Int__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_122_Int__Collect,axiom,
! [A: $tType,X4: A,A5: set @ A,P: A > $o] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_123_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_124_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X5: A] : ( member @ A @ X5 @ A6 )
@ ^ [X5: A] : ( member @ A @ X5 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_125_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X5: A] : ( member @ A @ X5 @ R )
@ ^ [X5: A] : ( member @ A @ X5 @ S ) )
= ( ^ [X5: A] : ( member @ A @ X5 @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_126_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X5: A] :
( ( P @ X5 )
& ( Q @ X5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_127_Int__left__absorb,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ).
% Int_left_absorb
thf(fact_128_Int__left__commute,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B5 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_129_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X4: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) ) ).
% predicate1D
thf(fact_130_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X4: A,Q: A > $o] :
( ( P @ X4 )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X4 ) ) ) ).
% rev_predicate1D
thf(fact_131_Int__Collect__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B5 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_132_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_133_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z: set @ A] : Y5 = 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_134_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_135_Int__greatest,axiom,
! [A: $tType,C3: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_136_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_137_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_138_Int__absorb2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= A5 ) ) ).
% Int_absorb2
thf(fact_139_Int__absorb1,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_140_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_141_equalityD2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ).
% equalityD2
thf(fact_142_equalityD1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( A5 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B5 ) ) ).
% equalityD1
thf(fact_143_Int__lower2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_144_Int__lower1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ A5 ) ).
% Int_lower1
thf(fact_145_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ A @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_146_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_147_Int__mono,axiom,
! [A: $tType,A5: set @ A,C3: set @ A,B5: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_148_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_149_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_150_inter__eq__subsetI,axiom,
! [A: $tType,S: set @ A,S4: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ S4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ S4 )
= ( inf_inf @ ( set @ A ) @ B5 @ S4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ S )
= ( inf_inf @ ( set @ A ) @ B5 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_151_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( P @ X5 ) ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ B5 )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_152_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ B5 )
& ( Q @ X5 ) ) )
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_153_Collect__restrict,axiom,
! [A: $tType,X: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ X )
& ( P @ X5 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_154_inf1I,axiom,
! [A: $tType,A5: A > $o,X4: A,B5: A > $o] :
( ( A5 @ X4 )
=> ( ( B5 @ X4 )
=> ( inf_inf @ ( A > $o ) @ A5 @ B5 @ X4 ) ) ) ).
% inf1I
thf(fact_155_inf1E,axiom,
! [A: $tType,A5: A > $o,B5: A > $o,X4: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B5 @ X4 )
=> ~ ( ( A5 @ X4 )
=> ~ ( B5 @ X4 ) ) ) ).
% inf1E
thf(fact_156_inf1D1,axiom,
! [A: $tType,A5: A > $o,B5: A > $o,X4: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B5 @ X4 )
=> ( A5 @ X4 ) ) ).
% inf1D1
thf(fact_157_inf1D2,axiom,
! [A: $tType,A5: A > $o,B5: A > $o,X4: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B5 @ X4 )
=> ( B5 @ X4 ) ) ).
% inf1D2
thf(fact_158_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_159_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( P @ X5 ) ) ) ) ).
% subset_Collect_conv
thf(fact_160_prop__restrict,axiom,
! [A: $tType,X4: A,Z4: set @ A,X: set @ A,P: A > $o] :
( ( member @ A @ X4 @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ X )
& ( P @ X5 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_161_Powp__mono,axiom,
! [A: $tType,A5: A > $o,B5: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A5 @ B5 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A5 ) @ ( powp @ A @ B5 ) ) ) ).
% Powp_mono
thf(fact_162_inf__nres_Oelims,axiom,
! [A: $tType,X4: refine1665802226e_nres @ A,Xa: refine1665802226e_nres @ A,Y3: refine1665802226e_nres @ A] :
( ( ( inf_inf @ ( refine1665802226e_nres @ A ) @ X4 @ Xa )
= Y3 )
=> ( ( ( Xa
= ( refine1767639642_FAILi @ A ) )
=> ( Y3 != X4 ) )
=> ( ( ( X4
= ( refine1767639642_FAILi @ A ) )
=> ! [V: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ V ) )
=> ( Y3
!= ( refine605929679le_RES @ A @ V ) ) ) )
=> ~ ! [A4: set @ A] :
( ( X4
= ( refine605929679le_RES @ A @ A4 ) )
=> ! [B4: set @ A] :
( ( Xa
= ( refine605929679le_RES @ A @ B4 ) )
=> ( Y3
!= ( refine605929679le_RES @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% inf_nres.elims
thf(fact_163_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X4: A,Q: A > $o] :
( ( P @ X4 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ 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_164_inf__nres_Osimps_I1_J,axiom,
! [A: $tType,X4: refine1665802226e_nres @ A] :
( ( inf_inf @ ( refine1665802226e_nres @ A ) @ X4 @ ( refine1767639642_FAILi @ A ) )
= X4 ) ).
% inf_nres.simps(1)
thf(fact_165_nres_Odistinct_I1_J,axiom,
! [A: $tType,X2: set @ A] :
( ( refine1767639642_FAILi @ A )
!= ( refine605929679le_RES @ A @ X2 ) ) ).
% nres.distinct(1)
thf(fact_166_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_167_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_168_sup__nres_Oinduct,axiom,
! [A: $tType,P: ( refine1665802226e_nres @ A ) > ( refine1665802226e_nres @ A ) > $o,A0: refine1665802226e_nres @ A,A1: refine1665802226e_nres @ A] :
( ! [Uu: refine1665802226e_nres @ A] : ( P @ Uu @ ( refine1767639642_FAILi @ A ) )
=> ( ! [V: set @ A] : ( P @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) )
=> ( ! [A4: set @ A,B4: set @ A] : ( P @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sup_nres.induct
thf(fact_169_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_170_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_171_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_172_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_173_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_174_inf__nres_Osimps_I2_J,axiom,
! [A: $tType,V2: set @ A] :
( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V2 ) )
= ( refine605929679le_RES @ A @ V2 ) ) ).
% inf_nres.simps(2)
thf(fact_175_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_176_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_177_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_178_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X4: A] :
( ( P @ X4 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X4 ) )
=> ( ( order_Greatest @ A @ P )
= X4 ) ) ) ) ).
% Greatest_equality
thf(fact_179_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_180_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_181_inf__RETURN__SPEC_I1_J,axiom,
! [A: $tType,X4: A,Phi: A > $o] :
( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
= ( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [Y6: A] :
( ( Y6 = X4 )
& ( Phi @ X4 ) ) ) ) ) ).
% inf_RETURN_SPEC(1)
thf(fact_182_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_183_nofail__simps_I2_J,axiom,
! [B: $tType,X: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X ) ) ).
% nofail_simps(2)
thf(fact_184_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_185_nofail__simps_I3_J,axiom,
! [C: $tType,X4: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X4 ) ) ).
% nofail_simps(3)
thf(fact_186_nres__order__simps_I21_J,axiom,
! [X6: $tType,X4: X6,Y: set @ X6] :
( ( ord_less_eq @ ( refine1665802226e_nres @ X6 ) @ ( refine1687780735RETURN @ X6 @ X4 ) @ ( refine605929679le_RES @ X6 @ Y ) )
= ( member @ X6 @ X4 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_187_inf__RETURN__SPEC_I2_J,axiom,
! [A: $tType,Phi: A > $o,X4: A] :
( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ ( refine1687780735RETURN @ A @ X4 ) )
= ( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [Y6: A] :
( ( Y6 = X4 )
& ( Phi @ X4 ) ) ) ) ) ).
% inf_RETURN_SPEC(2)
thf(fact_188_pwD1,axiom,
! [A: $tType,S: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S4 )
=> ( ( refine1102455758nofail @ A @ S4 )
=> ( refine1102455758nofail @ A @ S ) ) ) ).
% pwD1
thf(fact_189_le__nofailI,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ M3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ M3 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ M3 ) ) ).
% le_nofailI
thf(fact_190_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_191_nofail__RES__conv,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [M5: refine1665802226e_nres @ A] :
? [M6: set @ A] :
( M5
= ( refine605929679le_RES @ A @ M6 ) ) ) ) ).
% nofail_RES_conv
thf(fact_192_SPEC__eq__is__RETURN_I1_J,axiom,
! [A: $tType,X4: A] :
( ( refine605929679le_RES @ A
@ ( collect @ A
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ X4 ) ) )
= ( refine1687780735RETURN @ A @ X4 ) ) ).
% SPEC_eq_is_RETURN(1)
thf(fact_193_RETURN__SPEC__conv,axiom,
! [A: $tType] :
( ( refine1687780735RETURN @ A )
= ( ^ [R3: A] :
( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [X5: A] : X5 = R3 ) ) ) ) ).
% RETURN_SPEC_conv
thf(fact_194_SPEC__eq__is__RETURN_I2_J,axiom,
! [B: $tType,Y3: B] :
( ( refine605929679le_RES @ B
@ ( collect @ B
@ ^ [X5: B] : X5 = Y3 ) )
= ( refine1687780735RETURN @ B @ Y3 ) ) ).
% SPEC_eq_is_RETURN(2)
thf(fact_195_pw__inf__nofail,axiom,
! [A: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A] :
( ( refine1102455758nofail @ A @ ( inf_inf @ ( refine1665802226e_nres @ A ) @ A2 @ B2 ) )
= ( ( refine1102455758nofail @ A @ A2 )
| ( refine1102455758nofail @ A @ B2 ) ) ) ).
% pw_inf_nofail
thf(fact_196_le__RES__nofailI,axiom,
! [A: $tType,A2: refine1665802226e_nres @ A,X4: set @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine605929679le_RES @ A @ X4 ) )
=> ( refine1102455758nofail @ A @ A2 ) ) ).
% le_RES_nofailI
thf(fact_197_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_198_lhs__step__RES,axiom,
! [A: $tType,X: set @ A,M: refine1665802226e_nres @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ X ) @ M ) ) ).
% lhs_step_RES
thf(fact_199_RETURN__to__SPEC__rule,axiom,
! [A: $tType,M: refine1665802226e_nres @ A,V2: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ V2 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine1687780735RETURN @ A @ V2 ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_200_lhs__step__SPEC,axiom,
! [A: $tType,Phi: A > $o,M: refine1665802226e_nres @ A] :
( ! [X3: A] :
( ( Phi @ X3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ M ) ) ).
% lhs_step_SPEC
thf(fact_201_lhs__step__bind,axiom,
! [A: $tType,B: $tType,M7: refine1665802226e_nres @ B,M: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B )] :
( ( ( refine1102455758nofail @ B @ M7 )
=> ( refine1102455758nofail @ A @ M ) )
=> ( ! [X3: A] :
( ( refine406925620_inres @ A @ M @ X3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ M7 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ M7 ) ) ) ).
% lhs_step_bind
thf(fact_202_inf__RETURN__RES_I2_J,axiom,
! [A: $tType,X4: A,X: set @ A] :
( ( ( member @ A @ X4 @ X )
=> ( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ X ) @ ( refine1687780735RETURN @ A @ X4 ) )
= ( refine1687780735RETURN @ A @ X4 ) ) )
& ( ~ ( member @ A @ X4 @ X )
=> ( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ X ) @ ( refine1687780735RETURN @ A @ X4 ) )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% inf_RETURN_RES(2)
thf(fact_203_inf__RETURN__RES_I1_J,axiom,
! [A: $tType,X4: A,X: set @ A] :
( ( ( member @ A @ X4 @ X )
=> ( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine605929679le_RES @ A @ X ) )
= ( refine1687780735RETURN @ A @ X4 ) ) )
& ( ~ ( member @ A @ X4 @ X )
=> ( ( inf_inf @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine605929679le_RES @ A @ X ) )
= ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% inf_RETURN_RES(1)
thf(fact_204_bot__apply,axiom,
! [C: $tType,D3: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D3 > C ) )
= ( ^ [X5: D3] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_205_nres__monad3,axiom,
! [B: $tType,A: $tType,C: $tType,M4: refine1665802226e_nres @ C,F: C > ( refine1665802226e_nres @ B ),G: B > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ B @ A @ ( refine463715084e_bind @ C @ B @ M4 @ F ) @ G )
= ( refine463715084e_bind @ C @ A @ M4
@ ^ [X5: C] : ( refine463715084e_bind @ B @ A @ ( F @ X5 ) @ G ) ) ) ).
% nres_monad3
thf(fact_206_If__bind__distrib,axiom,
! [B: $tType,A: $tType,B2: $o,T: refine1665802226e_nres @ A,E: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B )] :
( ( B2
=> ( ( refine463715084e_bind @ A @ B @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ T @ E ) @ F )
= ( refine463715084e_bind @ A @ B @ T @ F ) ) )
& ( ~ B2
=> ( ( refine463715084e_bind @ A @ B @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ T @ E ) @ F )
= ( refine463715084e_bind @ A @ B @ E @ F ) ) ) ) ).
% If_bind_distrib
thf(fact_207_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ X4 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_208_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X4 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_209_nres__monad2,axiom,
! [A: $tType,M4: refine1665802226e_nres @ A] :
( ( refine463715084e_bind @ A @ A @ M4 @ ( refine1687780735RETURN @ A ) )
= M4 ) ).
% nres_monad2
thf(fact_210_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_211_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_212_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_213_nofail__simps_I4_J,axiom,
! [D3: $tType] : ( refine1102455758nofail @ D3 @ ( bot_bot @ ( refine1665802226e_nres @ D3 ) ) ) ).
% nofail_simps(4)
thf(fact_214_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
! [B: $tType,A: $tType,M4: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ M3 )
=> ( ! [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 @ M3 @ F4 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
thf(fact_215_bind__cong,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M7: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( M = M7 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M7 )
=> ( ( F @ X3 )
= ( F4 @ X3 ) ) )
=> ( ( refine463715084e_bind @ A @ B @ M @ F )
= ( refine463715084e_bind @ A @ B @ M7 @ F4 ) ) ) ) ).
% bind_cong
thf(fact_216_pw__RES__bind__choose_I1_J,axiom,
! [A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ X )
=> ( refine1102455758nofail @ A @ ( F @ X5 ) ) ) ) ) ).
% pw_RES_bind_choose(1)
thf(fact_217_let__to__bind__conv,axiom,
! [A: $tType,B: $tType] :
( ( ^ [S3: B,F3: B > ( refine1665802226e_nres @ A )] : ( F3 @ S3 ) )
= ( ^ [X5: B] : ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X5 ) ) ) ) ).
% let_to_bind_conv
thf(fact_218_nres__inequalities_I5_J,axiom,
! [C: $tType,X4: C] :
( ( bot_bot @ ( refine1665802226e_nres @ C ) )
!= ( refine1687780735RETURN @ C @ X4 ) ) ).
% nres_inequalities(5)
thf(fact_219_le__SPEC__bindI,axiom,
! [B: $tType,A: $tType,Phi: A > $o,X4: A,M: refine1665802226e_nres @ B,F: A > ( refine1665802226e_nres @ B )] :
( ( Phi @ X4 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( F @ X4 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine463715084e_bind @ A @ B @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ F ) ) ) ) ).
% le_SPEC_bindI
thf(fact_220_RES__bind__choose,axiom,
! [B: $tType,A: $tType,X4: A,X: set @ A,M: refine1665802226e_nres @ B,F: A > ( refine1665802226e_nres @ B )] :
( ( member @ A @ X4 @ X )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( F @ X4 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine463715084e_bind @ A @ B @ ( refine605929679le_RES @ A @ X ) @ F ) ) ) ) ).
% RES_bind_choose
thf(fact_221_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_222_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_223_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_224_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_225_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_226_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_227_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_228_bind__rule__complete,axiom,
! [A: $tType,B: $tType,M4: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),Phi: A > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M4 @ F ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
= ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M4
@ ( refine605929679le_RES @ B
@ ( collect @ B
@ ^ [X5: B] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ) ) ).
% bind_rule_complete
thf(fact_229_specify__left,axiom,
! [A: $tType,B: $tType,M: refine1665802226e_nres @ A,Phi: A > $o,F: A > ( refine1665802226e_nres @ B ),M4: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
=> ( ! [X3: A] :
( ( Phi @ X3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ M4 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ M4 ) ) ) ).
% specify_left
thf(fact_230_bind__rule,axiom,
! [A: $tType,B: $tType,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),Phi: B > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ^ [X5: A] : ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X5 ) @ ( refine605929679le_RES @ B @ ( collect @ B @ Phi ) ) ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M4 @ F ) @ ( refine605929679le_RES @ B @ ( collect @ B @ Phi ) ) ) ) ).
% bind_rule
thf(fact_231_bind__le__nofailI,axiom,
! [A: $tType,B: $tType,M: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),M7: refine1665802226e_nres @ B] :
( ( refine1102455758nofail @ A @ M )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ M7 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ M7 ) ) ) ).
% bind_le_nofailI
thf(fact_232_bind__le__shift,axiom,
! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),M7: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ M7 )
= ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M
@ ( if @ ( refine1665802226e_nres @ B ) @ ( refine1102455758nofail @ A @ M7 )
@ ( refine605929679le_RES @ B
@ ( collect @ B
@ ^ [X5: B] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ M7 ) ) )
@ ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ) ).
% bind_le_shift
thf(fact_233_bind__sim__select__rule,axiom,
! [A: $tType,B: $tType,C: $tType,M: refine1665802226e_nres @ B,F4: B > ( refine1665802226e_nres @ A ),Psi: A > $o,F: B > ( refine1665802226e_nres @ C ),Phi: C > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F4 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) )
=> ( ! [X3: B] :
( ( refine1102455758nofail @ B @ M )
=> ( ( refine1315500908_inres @ B @ M @ X3 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F4 @ X3 ) @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ X3 ) @ ( refine605929679le_RES @ C @ ( collect @ C @ Phi ) ) ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine463715084e_bind @ B @ C @ M @ F ) @ ( refine605929679le_RES @ C @ ( collect @ C @ Phi ) ) ) ) ) ).
% bind_sim_select_rule
thf(fact_234_top__apply,axiom,
! [C: $tType,D3: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D3 > C ) )
= ( ^ [X5: D3] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_235_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_236_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_237_inf__top__left,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X4 )
= X4 ) ) ).
% inf_top_left
thf(fact_238_inf__top__right,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ X4 @ ( top_top @ A ) )
= X4 ) ) ).
% inf_top_right
thf(fact_239_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [X4: A,Y3: A] :
( ( ( inf_inf @ A @ X4 @ Y3 )
= ( top_top @ A ) )
= ( ( X4
= ( top_top @ A ) )
& ( Y3
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_240_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [X4: A,Y3: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X4 @ Y3 ) )
= ( ( X4
= ( top_top @ A ) )
& ( Y3
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_241_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [A2: A,B2: A] :
( ( ( inf_inf @ A @ A2 @ B2 )
= ( top_top @ A ) )
= ( ( A2
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_242_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A2 )
= A2 ) ) ).
% inf_top.left_neutral
thf(fact_243_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [A2: A,B2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A2 @ B2 ) )
= ( ( A2
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_244_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1561333602nf_top @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ ( top_top @ A ) )
= A2 ) ) ).
% inf_top.right_neutral
thf(fact_245_inres__simps_I2_J,axiom,
! [B: $tType,X: set @ B] :
( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X ) )
= ( ^ [X5: B] : ( member @ B @ X5 @ X ) ) ) ).
% inres_simps(2)
thf(fact_246_nres__order__simps_I4_J,axiom,
! [D3: $tType,M4: refine1665802226e_nres @ D3] :
( ( ord_less_eq @ ( refine1665802226e_nres @ D3 ) @ ( top_top @ ( refine1665802226e_nres @ D3 ) ) @ M4 )
= ( M4
= ( top_top @ ( refine1665802226e_nres @ D3 ) ) ) ) ).
% nres_order_simps(4)
thf(fact_247_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_248_inres__simps_I3_J,axiom,
! [C: $tType,X4: C] :
( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X4 ) )
= ( ^ [Y5: C,Z: C] : Y5 = Z
@ X4 ) ) ).
% inres_simps(3)
thf(fact_249_nofail__simps_I1_J,axiom,
! [A: $tType] :
~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% nofail_simps(1)
thf(fact_250_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)
% Type constructors (57)
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice,axiom,
! [A7: $tType] : ( bounded_lattice @ ( refine1665802226e_nres @ A7 ) ) ).
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,
! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_4,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounded_lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounde1561333602nf_top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounded_lattice_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_inf @ A8 )
=> ( semilattice_inf @ ( 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___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 )
=> ( lattice @ ( 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___Lattices_Obounded__semilattice__inf__top_5,axiom,
! [A7: $tType] : ( bounde1561333602nf_top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_6,axiom,
! [A7: $tType] : ( bounded_lattice_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_7,axiom,
! [A7: $tType] : ( semilattice_inf @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_8,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_9,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_10,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_11,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_12,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_13,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_14,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_15,axiom,
bounde1561333602nf_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_16,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_17,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_18,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_19,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_20,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_21,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_22,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_23,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_24,axiom,
bot @ $o ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_25,axiom,
bounde1561333602nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_26,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_27,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_28,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_29,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_30,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_31,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_32,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_33,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_34,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_35,axiom,
bot @ product_unit ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__semilattice__inf__top_36,axiom,
! [A7: $tType] : ( bounde1561333602nf_top @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Obounded__lattice__bot_37,axiom,
! [A7: $tType] : ( bounded_lattice_bot @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Osemilattice__inf_38,axiom,
! [A7: $tType] : ( semilattice_inf @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__bot_39,axiom,
! [A7: $tType] : ( order_bot @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_40,axiom,
! [A7: $tType] : ( preorder @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Lattices_Olattice_41,axiom,
! [A7: $tType] : ( lattice @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_42,axiom,
! [A7: $tType] : ( order @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_43,axiom,
! [A7: $tType] : ( top @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_44,axiom,
! [A7: $tType] : ( ord @ ( refine1665802226e_nres @ A7 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_45,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 (1)
thf(conj_0,conjecture,
( ord_less_eq @ ( refine1665802226e_nres @ a ) @ a2
@ ( refine605929679le_RES @ a
@ ( collect @ a
@ ^ [V3: a] :
( ( p @ V3 )
& ( q @ V3 ) ) ) ) ) ).
%------------------------------------------------------------------------------