TPTP Problem File: ITP161^2.p
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%------------------------------------------------------------------------------
% File : ITP161^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_1708__3602512_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_1708__3602512_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 342 ( 129 unt; 51 typ; 0 def)
% Number of atoms : 783 ( 206 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4514 ( 83 ~; 4 |; 34 &;4008 @)
% ( 0 <=>; 385 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 225 ( 225 >; 0 *; 0 +; 0 <<)
% Number of symbols : 51 ( 48 usr; 4 con; 0-6 aty)
% Number of variables : 1297 ( 58 ^;1159 !; 20 ?;1297 :)
% ( 60 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:09.061
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
refine1665802226e_nres: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Orderings_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_Oantimono,type,
antimono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_OASSERT,type,
refine1814851989ASSERT: $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_Oabs__fun,type,
refine81118332bs_fun:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
refine463715084e_bind:
!>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
refine1073749519nc_fun:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
refine1315500908_inres:
!>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Olift__assn,type,
refine1580981607t_assn:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( B > $o ) > 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_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v__092_060Phi_062,type,
phi: a > $o ).
thf(sy_v_f,type,
f: a > ( refine1665802226e_nres @ b ) ).
thf(sy_v_m,type,
m: refine1665802226e_nres @ b ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (250)
thf(fact_0_assms_I1_J,axiom,
phi @ x ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
ord_less_eq @ ( refine1665802226e_nres @ b ) @ m @ ( f @ x ) ).
% assms(2)
thf(fact_2_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_3_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_4_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_5_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_6_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_7_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_8_lhs__step__If,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [B2: $o,T2: A,M: A,E: A] :
( ( B2
=> ( ord_less_eq @ A @ T2 @ M ) )
=> ( ( ~ B2
=> ( ord_less_eq @ A @ E @ M ) )
=> ( ord_less_eq @ A @ ( if @ A @ B2 @ T2 @ E ) @ M ) ) ) ) ).
% lhs_step_If
thf(fact_9_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_10_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_11_order__mono__setup_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_mono_setup.refl
thf(fact_12_the__RES_Osimps,axiom,
! [A: $tType,X: set @ A] :
( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X ) )
= X ) ).
% the_RES.simps
thf(fact_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_order__mono__setup_Omono__if,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [T2: A,T3: A,E: A,E3: A,B2: $o] :
( ( ord_less_eq @ A @ T2 @ T3 )
=> ( ( ord_less_eq @ A @ E @ E3 )
=> ( ord_less_eq @ A @ ( if @ A @ B2 @ T2 @ E ) @ ( if @ A @ B2 @ T3 @ E3 ) ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_26_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_27_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_28_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_29_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_30_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 = Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% eq_refl
thf(fact_41_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_42_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_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_45_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_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_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_48_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_49_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_50_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_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 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ 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 )
=> ( ! [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_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,S12: C,S1: A,S22: C,S2: A] :
( ( B2
=> ( ord_less_eq @ A @ ( Alpha @ S12 ) @ S1 ) )
=> ( ( ~ B2
=> ( ord_less_eq @ A @ ( Alpha @ S22 ) @ S2 ) )
=> ( ord_less_eq @ A @ ( Alpha @ ( if @ C @ B2 @ S12 @ S22 ) ) @ ( if @ A @ B2 @ S1 @ S2 ) ) ) ) ) ).
% transfer.transfer_if
thf(fact_56_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X6: $o > A,Y8: $o > A] :
( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq @ A @ ( X6 @ $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_intro__spec__refine__iff,axiom,
! [C: $tType,A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A ),R: set @ ( product_prod @ A @ C ),M3: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) @ ( refine1073749519nc_fun @ A @ C @ R @ M3 ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ X )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ ( refine1073749519nc_fun @ A @ C @ R @ M3 ) ) ) ) ) ).
% intro_spec_refine_iff
thf(fact_60_intro__bind__refine__id,axiom,
! [A: $tType,B: $tType,C: $tType,M: refine1665802226e_nres @ A,M4: A,F: A > ( refine1665802226e_nres @ B ),R: set @ ( product_prod @ B @ C ),M5: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M
@ ( refine605929679le_RES @ A
@ ( collect @ A
@ ( ^ [Y5: A,Z: A] : Y5 = Z
@ M4 ) ) ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ M4 ) @ ( refine1073749519nc_fun @ B @ C @ R @ M5 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine1073749519nc_fun @ B @ C @ R @ M5 ) ) ) ) ).
% intro_bind_refine_id
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_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_63_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_64_pw__conc__nofail,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
= ( refine1102455758nofail @ B @ S ) ) ).
% pw_conc_nofail
thf(fact_65_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_66_le__nofailI,axiom,
! [A: $tType,M6: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ M6 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 ) ) ).
% le_nofailI
thf(fact_67_nofail__RES__conv,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [M7: refine1665802226e_nres @ A] :
? [M8: set @ A] :
( M7
= ( refine605929679le_RES @ A @ M8 ) ) ) ) ).
% nofail_RES_conv
thf(fact_68_if__refine,axiom,
! [A: $tType,B: $tType,B2: $o,B5: $o,S1: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),S13: refine1665802226e_nres @ B,S2: refine1665802226e_nres @ A,S23: refine1665802226e_nres @ B] :
( ( B2 = B5 )
=> ( ( B2
=> ( B5
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S1 @ ( refine1073749519nc_fun @ A @ B @ R @ S13 ) ) ) )
=> ( ( ~ B2
=> ( ~ B5
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S2 @ ( refine1073749519nc_fun @ A @ B @ R @ S23 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( if @ ( refine1665802226e_nres @ A ) @ B2 @ S1 @ S2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( if @ ( refine1665802226e_nres @ B ) @ B5 @ S13 @ S23 ) ) ) ) ) ) ).
% if_refine
thf(fact_69_conc__trans,axiom,
! [A: $tType,B: $tType,C: $tType,C3: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B6: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A5: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C3 @ ( refine1073749519nc_fun @ A @ B @ R @ B6 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B6 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A5 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C3 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A5 ) ) ) ) ) ).
% conc_trans
thf(fact_70_intro__prgR,axiom,
! [A: $tType,B: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),A2: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ A2 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ A2 ) ) ) ).
% intro_prgR
thf(fact_71_ref__two__step,axiom,
! [A: $tType,B: $tType,A5: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B6: refine1665802226e_nres @ B,C3: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ ( refine1073749519nc_fun @ A @ B @ R @ B6 ) )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B6 @ C3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ ( refine1073749519nc_fun @ A @ B @ R @ C3 ) ) ) ) ).
% ref_two_step
thf(fact_72_conc__fun__R__mono,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M3: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M3 ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M3 ) ) ) ).
% conc_fun_R_mono
thf(fact_73_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_74_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_75_rhs__step__bind__RES,axiom,
! [B: $tType,C: $tType,A: $tType,X7: A,X8: set @ A,M: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ C ),F4: A > ( refine1665802226e_nres @ C )] :
( ( member @ A @ X7 @ X8 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( F4 @ X7 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( refine463715084e_bind @ A @ C @ ( refine605929679le_RES @ A @ X8 ) @ F4 ) ) ) ) ) ).
% rhs_step_bind_RES
thf(fact_76_rhs__step__bind__SPEC,axiom,
! [B: $tType,C: $tType,A: $tType,Phi: A > $o,X7: A,M: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ C ),F4: A > ( refine1665802226e_nres @ C )] :
( ( Phi @ X7 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( F4 @ X7 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( refine1073749519nc_fun @ B @ C @ R @ ( refine463715084e_bind @ A @ C @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ F4 ) ) ) ) ) ).
% rhs_step_bind_SPEC
thf(fact_77_subsetI,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% subsetI
thf(fact_78_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( A5 = B6 ) ) ) ).
% subset_antisym
thf(fact_79_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_80_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_81_in__mono,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B6 ) ) ) ).
% in_mono
thf(fact_82_subsetD,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B6 ) ) ) ).
% subsetD
thf(fact_83_equalityE,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( A5 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% equalityE
thf(fact_84_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_85_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_86_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_87_pw__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y5: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : Y5 = Z )
= ( ^ [S4: refine1665802226e_nres @ A,S5: refine1665802226e_nres @ A] :
( ( ( refine1102455758nofail @ A @ S4 )
= ( refine1102455758nofail @ A @ S5 ) )
& ! [X5: A] :
( ( refine1315500908_inres @ A @ S4 @ X5 )
= ( refine1315500908_inres @ A @ S5 @ X5 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_88_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_89_pw__RES__bind__choose_I2_J,axiom,
! [A: $tType,B: $tType,X: set @ B,F: B > ( refine1665802226e_nres @ A ),Y3: A] :
( ( refine1315500908_inres @ A @ ( refine463715084e_bind @ B @ A @ ( refine605929679le_RES @ B @ X ) @ F ) @ Y3 )
= ( ? [X5: B] :
( ( member @ B @ X5 @ X )
& ( refine1315500908_inres @ A @ ( F @ X5 ) @ Y3 ) ) ) ) ).
% pw_RES_bind_choose(2)
thf(fact_90_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_91_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_92_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 )
& ! [X5: A] :
( ( refine1315500908_inres @ A @ S4 @ X5 )
=> ( refine1315500908_inres @ A @ S5 @ X5 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_93_pw__bind__nofail,axiom,
! [A: $tType,B: $tType,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M3 @ F ) )
= ( ( refine1102455758nofail @ B @ M3 )
& ! [X5: B] :
( ( refine1315500908_inres @ B @ M3 @ X5 )
=> ( refine1102455758nofail @ A @ ( F @ X5 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_94_nf__inres__def,axiom,
! [A: $tType] :
( ( refine406925620_inres @ A )
= ( ^ [M7: refine1665802226e_nres @ A,X5: A] :
( ( refine1102455758nofail @ A @ M7 )
& ( refine1315500908_inres @ A @ M7 @ X5 ) ) ) ) ).
% nf_inres_def
thf(fact_95_pw__bind__le__iff,axiom,
! [A: $tType,B: $tType,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M3 @ F ) @ S )
= ( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M3 ) )
& ! [X5: B] :
( ( ( refine1102455758nofail @ B @ M3 )
& ( refine1315500908_inres @ B @ M3 @ X5 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X5 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_96_pw__bind__leI,axiom,
! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
( ( ( refine1102455758nofail @ A @ S )
=> ( refine1102455758nofail @ B @ M3 ) )
=> ( ! [X3: B] :
( ( refine1102455758nofail @ B @ M3 )
=> ( ( refine1315500908_inres @ B @ M3 @ X3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M3 @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_97_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_98_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z: set @ A] : Y5 = Z )
= ( ^ [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_99_subset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% subset_trans
thf(fact_100_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_101_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_102_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [T4: A] :
( ( member @ A @ T4 @ A6 )
=> ( member @ A @ T4 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_103_equalityD2,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( A5 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ).
% equalityD2
thf(fact_104_equalityD1,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( A5 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% equalityD1
thf(fact_105_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B7: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ A @ X5 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_106_pw__ref__iff,axiom,
! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),S3: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ ( refine1073749519nc_fun @ A @ B @ R @ S3 ) )
= ( ( refine1102455758nofail @ B @ S3 )
=> ( ( refine1102455758nofail @ A @ S )
& ! [X5: A] :
( ( refine1315500908_inres @ A @ S @ X5 )
=> ? [S6: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ S6 ) @ R )
& ( refine1315500908_inres @ B @ S3 @ S6 ) ) ) ) ) ) ).
% pw_ref_iff
thf(fact_107_pw__ref__I,axiom,
! [B: $tType,A: $tType,S3: refine1665802226e_nres @ A,S: refine1665802226e_nres @ B,R: set @ ( product_prod @ B @ A )] :
( ( ( refine1102455758nofail @ A @ S3 )
=> ( ( refine1102455758nofail @ B @ S )
& ! [X3: B] :
( ( refine1315500908_inres @ B @ S @ X3 )
=> ? [S7: A] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ S7 ) @ R )
& ( refine1315500908_inres @ A @ S3 @ S7 ) ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ S @ ( refine1073749519nc_fun @ B @ A @ R @ S3 ) ) ) ).
% pw_ref_I
thf(fact_108_rhs__step__bind,axiom,
! [A: $tType,C: $tType,D2: $tType,B: $tType,M: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),M4: refine1665802226e_nres @ B,X4: A,Lhs: refine1665802226e_nres @ C,S: set @ ( product_prod @ C @ D2 ),F4: B > ( refine1665802226e_nres @ D2 )] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ ( refine1073749519nc_fun @ A @ B @ R @ M4 ) )
=> ( ( refine1315500908_inres @ A @ M @ X4 )
=> ( ! [X9: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ X9 ) @ R )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ Lhs @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ X9 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ Lhs @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( refine463715084e_bind @ B @ D2 @ M4 @ F4 ) ) ) ) ) ) ).
% rhs_step_bind
thf(fact_109_RES__refine,axiom,
! [A: $tType,B: $tType,S: set @ A,S3: set @ B,R: set @ ( product_prod @ A @ B )] :
( ! [S8: A] :
( ( member @ A @ S8 @ S )
=> ? [X10: B] :
( ( member @ B @ X10 @ S3 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S8 @ X10 ) @ R ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ S ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ S3 ) ) ) ) ).
% RES_refine
thf(fact_110_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_111_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_112_less__eq__nres_Ocases,axiom,
! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
( ! [Uu: refine1665802226e_nres @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
=> ( ! [A4: set @ A,B4: set @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) )
=> ~ ! [Uv: set @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ Uv ) ) ) ) ) ).
% less_eq_nres.cases
thf(fact_113_less__nres_Ocases,axiom,
! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
( ! [Uu: refine1665802226e_nres @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ Uu ) )
=> ( ! [Uv: set @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ Uv ) @ ( refine1767639642_FAILi @ A ) ) )
=> ~ ! [A4: set @ A,B4: set @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) ) ) ) ).
% less_nres.cases
thf(fact_114_sup__nres_Ocases,axiom,
! [A: $tType,X4: product_prod @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A )] :
( ! [Uu: refine1665802226e_nres @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ Uu @ ( refine1767639642_FAILi @ A ) ) )
=> ( ! [V: set @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine1767639642_FAILi @ A ) @ ( refine605929679le_RES @ A @ V ) ) )
=> ~ ! [A4: set @ A,B4: set @ A] :
( X4
!= ( product_Pair @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A4 ) @ ( refine605929679le_RES @ A @ B4 ) ) ) ) ) ).
% sup_nres.cases
thf(fact_115_introR,axiom,
! [B: $tType,A: $tType,A2: A,A7: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ A7 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ A7 ) @ R ) ) ).
% introR
thf(fact_116_bex2I,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S )
=> ( P @ A2 @ B2 ) )
=> ? [A4: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ S )
& ( P @ A4 @ B4 ) ) ) ) ).
% bex2I
thf(fact_117_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X4: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F6: A > B,A4: A,B4: A] :
( X4
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F6 @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% pairself.cases
thf(fact_118_pairself_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( A > B ) > ( product_prod @ A @ A ) > $o,A0: A > B,A1: product_prod @ A @ A] :
( ! [F6: A > B,A4: A,B4: A] : ( P @ F6 @ ( product_Pair @ A @ A @ A4 @ B4 ) )
=> ( P @ A0 @ A1 ) ) ).
% pairself.induct
thf(fact_119_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_120_nres_Odistinct_I1_J,axiom,
! [A: $tType,X2: set @ A] :
( ( refine1767639642_FAILi @ A )
!= ( refine605929679le_RES @ A @ X2 ) ) ).
% nres.distinct(1)
thf(fact_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_Let__refine_H,axiom,
! [A: $tType,C: $tType,D2: $tType,B: $tType,M: A,M4: B,R: set @ ( product_prod @ A @ B ),F: A > ( refine1665802226e_nres @ C ),S: set @ ( product_prod @ C @ D2 ),F4: B > ( refine1665802226e_nres @ D2 )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ M @ M4 ) @ R )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ M @ M4 ) @ R )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ M ) @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ M4 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( F @ M ) @ ( refine1073749519nc_fun @ C @ D2 @ S @ ( F4 @ M4 ) ) ) ) ) ).
% Let_refine'
thf(fact_129_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_130_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B5 ) )
= ( ( A2 = A7 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_131_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_132_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S9: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ S9 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S9 ) ) ).
% subrelI
thf(fact_133_lift__assn__def,axiom,
! [B: $tType,A: $tType] :
( ( refine1580981607t_assn @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),Phi3: B > $o,S10: A] :
? [S6: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S10 @ S6 ) @ R4 )
& ( Phi3 @ S6 ) ) ) ) ).
% lift_assn_def
thf(fact_134_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% surj_pair
thf(fact_135_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_136_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B5: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B5 ) )
=> ~ ( ( A2 = A7 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_137_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C4: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) ) ).
% prod_cases3
thf(fact_138_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
~ ! [A4: A,B4: B,C4: C,D3: D2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) ) ).
% prod_cases4
thf(fact_139_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E4 ) ) ) ) ) ).
% prod_cases5
thf(fact_140_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F7: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) )] :
~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F7 ) @ D3 @ ( product_Pair @ E2 @ F7 @ E4 @ F6 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_141_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D2: $tType,E2: $tType,F7: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) )] :
~ ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7,G4: G3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F7 @ G3 ) @ E4 @ ( product_Pair @ F7 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_142_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C4 ) ) )
=> ( P @ X4 ) ) ).
% prod_induct3
thf(fact_143_prod__induct4,axiom,
! [D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) )] :
( ! [A4: A,B4: B,C4: C,D3: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D2 ) @ B4 @ ( product_Pair @ C @ D2 @ C4 @ D3 ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct4
thf(fact_144_prod__induct5,axiom,
! [E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) )] :
( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ E2 ) @ C4 @ ( product_Pair @ D2 @ E2 @ D3 @ E4 ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct5
thf(fact_145_prod__induct6,axiom,
! [F7: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) )] :
( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ F7 ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ F7 ) @ D3 @ ( product_Pair @ E2 @ F7 @ E4 @ F6 ) ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct6
thf(fact_146_prod__induct7,axiom,
! [G3: $tType,F7: $tType,E2: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) )] :
( ! [A4: A,B4: B,C4: C,D3: D2,E4: E2,F6: F7,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) ) @ C4 @ ( product_Pair @ D2 @ ( product_prod @ E2 @ ( product_prod @ F7 @ G3 ) ) @ D3 @ ( product_Pair @ E2 @ ( product_prod @ F7 @ G3 ) @ E4 @ ( product_Pair @ F7 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct7
thf(fact_147_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y3
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_148_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_149_lift__assnI,axiom,
! [B: $tType,A: $tType,S9: A,S11: B,R: set @ ( product_prod @ A @ B ),Phi: B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ S9 @ S11 ) @ R )
=> ( ( Phi @ S11 )
=> ( refine1580981607t_assn @ A @ B @ R @ Phi @ S9 ) ) ) ).
% lift_assnI
thf(fact_150_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_151_return__refine__prop__return,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,X4: B,R: set @ ( product_prod @ B @ A )] :
( ( refine1102455758nofail @ A @ M )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X4 ) @ ( refine1073749519nc_fun @ B @ A @ R @ M ) )
=> ~ ! [X9: A] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ X9 ) @ R )
=> ~ ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X9 ) @ M ) ) ) ) ).
% return_refine_prop_return
thf(fact_152_ret__le__down__conv,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,C2: B,R: set @ ( product_prod @ B @ A )] :
( ( refine1102455758nofail @ A @ M )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ C2 ) @ ( refine1073749519nc_fun @ B @ A @ R @ M ) )
= ( ? [A3: A] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C2 @ A3 ) @ R )
& ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ A3 ) @ M ) ) ) ) ) ).
% ret_le_down_conv
thf(fact_153_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_154_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_155_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_156_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_157_nres__monad2,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A] :
( ( refine463715084e_bind @ A @ A @ M3 @ ( refine1687780735RETURN @ A ) )
= M3 ) ).
% nres_monad2
thf(fact_158_nofail__simps_I3_J,axiom,
! [C: $tType,X4: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X4 ) ) ).
% nofail_simps(3)
thf(fact_159_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_160_nres__order__simps_I21_J,axiom,
! [X11: $tType,X4: X11,Y: set @ X11] :
( ( ord_less_eq @ ( refine1665802226e_nres @ X11 ) @ ( refine1687780735RETURN @ X11 @ X4 ) @ ( refine605929679le_RES @ X11 @ Y ) )
= ( member @ X11 @ X4 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_161_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_162_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_163_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_164_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_165_if__RETURN__refine,axiom,
! [A: $tType,B: $tType,B2: $o,B5: $o,S1: A,R: set @ ( product_prod @ A @ B ),S13: refine1665802226e_nres @ B,S2: A,S23: refine1665802226e_nres @ B] :
( ( B2 = B5 )
=> ( ( B2
=> ( B5
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ S1 ) @ ( refine1073749519nc_fun @ A @ B @ R @ S13 ) ) ) )
=> ( ( ~ B2
=> ( ~ B5
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ S2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ S23 ) ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ ( if @ A @ B2 @ S1 @ S2 ) ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( if @ ( refine1665802226e_nres @ B ) @ B5 @ S13 @ S23 ) ) ) ) ) ) ).
% if_RETURN_refine
thf(fact_166_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ M6 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F4 @ X3 ) ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M6 @ F4 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
thf(fact_167_bind__cong,axiom,
! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( M = M4 )
=> ( ! [X3: A] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M4 )
=> ( ( F @ X3 )
= ( F4 @ X3 ) ) )
=> ( ( refine463715084e_bind @ A @ B @ M @ F )
= ( refine463715084e_bind @ A @ B @ M4 @ F4 ) ) ) ) ).
% bind_cong
thf(fact_168_inres__def,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A )
= ( ^ [S4: refine1665802226e_nres @ A,X5: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X5 ) @ S4 ) ) ) ).
% inres_def
thf(fact_169_RETURN__refine,axiom,
! [A: $tType,B: $tType,X4: A,X7: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ X7 ) @ R )
=> ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X4 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ X7 ) ) ) ) ).
% RETURN_refine
thf(fact_170_RETURN__ref__RETURND,axiom,
! [B: $tType,A: $tType,C2: A,R: set @ ( product_prod @ A @ B ),A2: B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ C2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1687780735RETURN @ B @ A2 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A2 ) @ R ) ) ).
% RETURN_ref_RETURND
thf(fact_171_RETURN__ref__SPECD,axiom,
! [A: $tType,B: $tType,C2: A,R: set @ ( product_prod @ A @ B ),Phi: B > $o] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ C2 ) @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine605929679le_RES @ B @ ( collect @ B @ Phi ) ) ) )
=> ~ ! [A4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ C2 @ A4 ) @ R )
=> ~ ( Phi @ A4 ) ) ) ).
% RETURN_ref_SPECD
thf(fact_172_RETURN__SPEC__refine,axiom,
! [B: $tType,A: $tType,X4: B,R: set @ ( product_prod @ B @ A ),Phi: A > $o] :
( ? [X12: A] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ X12 ) @ R )
& ( Phi @ X12 ) )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine1687780735RETURN @ B @ X4 ) @ ( refine1073749519nc_fun @ B @ A @ R @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) ) ) ) ).
% RETURN_SPEC_refine
thf(fact_173_pw__abs__inres,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M3: refine1665802226e_nres @ B,A2: A] :
( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) @ A2 )
= ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) )
=> ? [C5: B] :
( ( refine1315500908_inres @ B @ M3 @ C5 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C5 @ A2 ) @ R ) ) ) ) ).
% pw_abs_inres
thf(fact_174_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
! [I2: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R5 )
=> ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_175_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% uncurry_apply
thf(fact_176_abs__trans__additional_I1_J,axiom,
! [A: $tType,B: $tType,A5: refine1665802226e_nres @ A,B6: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C3: refine1665802226e_nres @ B] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B6 ) @ C3 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A5 ) @ C3 ) ) ) ).
% abs_trans_additional(1)
thf(fact_177_abs__trans,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C3: refine1665802226e_nres @ B,B6: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A5: refine1665802226e_nres @ C] :
( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C3 ) @ B6 )
=> ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B6 ) @ A5 )
=> ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C3 ) ) @ A5 ) ) ) ).
% abs_trans
thf(fact_178_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% uncurry_curry_id
thf(fact_179_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
= F ) ).
% curry_uncurry_id
thf(fact_180_pw__abs__nofail,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M3: refine1665802226e_nres @ B] :
( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M3 ) )
= ( ( refine1102455758nofail @ B @ M3 )
& ! [X5: B] :
( ( refine1315500908_inres @ B @ M3 @ X5 )
=> ( member @ B @ X5 @ ( domain @ B @ A @ R ) ) ) ) ) ).
% pw_abs_nofail
thf(fact_181_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_182_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F3: ( product_prod @ B @ C ) > A,A3: B,B3: C] : ( F3 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% curry_conv
thf(fact_183_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_184_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_185_DomainE,axiom,
! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R3 ) ) ).
% DomainE
thf(fact_186_Domain__iff,axiom,
! [A: $tType,B: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
= ( ? [Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ Y6 ) @ R3 ) ) ) ).
% Domain_iff
thf(fact_187_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B4 ) @ R3 ) ) ).
% Domain.cases
thf(fact_188_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A2: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) )
= ( ? [A3: A,B3: B] :
( ( A2 = A3 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 ) ) ) ) ).
% Domain.simps
thf(fact_189_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
=> ( member @ A @ A2 @ ( domain @ A @ B @ R3 ) ) ) ).
% Domain.DomainI
thf(fact_190_Domain_Oinducts,axiom,
! [B: $tType,A: $tType,X4: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
( ( member @ A @ X4 @ ( domain @ A @ B @ R3 ) )
=> ( ! [A4: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 )
=> ( P @ A4 ) )
=> ( P @ X4 ) ) ) ).
% Domain.inducts
thf(fact_191_Domain__mono,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S9: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S9 )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S9 ) ) ) ).
% Domain_mono
thf(fact_192_for__in__RI,axiom,
! [B: $tType,A: $tType,X4: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X4 @ ( domain @ A @ B @ R ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ ( fun_of_rel @ A @ B @ R @ X4 ) ) @ R ) ) ).
% for_in_RI
thf(fact_193_abs__fun__simps_I3_J,axiom,
! [B: $tType,A: $tType,X: set @ B,R: set @ ( product_prod @ B @ A )] :
( ~ ( ord_less_eq @ ( set @ B ) @ X @ ( domain @ B @ A @ R ) )
=> ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% abs_fun_simps(3)
thf(fact_194_abs__fun__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: set @ B,R: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ B ) @ X @ ( domain @ B @ A @ R ) )
=> ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X ) )
= ( refine605929679le_RES @ A @ ( image @ B @ A @ R @ X ) ) ) ) ).
% abs_fun_simps(2)
thf(fact_195_top__apply,axiom,
! [C: $tType,D2: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D2 > C ) )
= ( ^ [X5: D2] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_196_ImageI,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,R3: set @ ( product_prod @ A @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% ImageI
thf(fact_197_nres__order__simps_I4_J,axiom,
! [D2: $tType,M3: refine1665802226e_nres @ D2] :
( ( ord_less_eq @ ( refine1665802226e_nres @ D2 ) @ ( top_top @ ( refine1665802226e_nres @ D2 ) ) @ M3 )
= ( M3
= ( top_top @ ( refine1665802226e_nres @ D2 ) ) ) ) ).
% nres_order_simps(4)
thf(fact_198_conc__fun__FAIL,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% conc_fun_FAIL
thf(fact_199_conc__fun__fail__iff_I1_J,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
= ( top_top @ ( refine1665802226e_nres @ A ) ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% conc_fun_fail_iff(1)
thf(fact_200_conc__fun__fail__iff_I2_J,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
( ( ( top_top @ ( refine1665802226e_nres @ A ) )
= ( refine1073749519nc_fun @ A @ B @ R @ S ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% conc_fun_fail_iff(2)
thf(fact_201_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_202_nofail__simps_I1_J,axiom,
! [A: $tType] :
~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% nofail_simps(1)
thf(fact_203_inres__simps_I1_J,axiom,
! [A: $tType] :
( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
= ( ^ [Uu3: A] : $true ) ) ).
% inres_simps(1)
thf(fact_204_nres__simp__internals_I2_J,axiom,
! [B: $tType] :
( ( refine1767639642_FAILi @ B )
= ( top_top @ ( refine1665802226e_nres @ B ) ) ) ).
% nres_simp_internals(2)
thf(fact_205_abs__fun__simps_I1_J,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% abs_fun_simps(1)
thf(fact_206_FAIL__refine,axiom,
! [A: $tType,B: $tType,X: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X @ ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% FAIL_refine
thf(fact_207_rev__ImageI,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R3 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_208_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ B2 ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_209_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
=> ~ ! [X3: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R3 )
=> ~ ( member @ B @ X3 @ A5 ) ) ) ).
% ImageE
thf(fact_210_nres__inequalities_I3_J,axiom,
! [C: $tType,X4: C] :
( ( top_top @ ( refine1665802226e_nres @ C ) )
!= ( refine1687780735RETURN @ C @ X4 ) ) ).
% nres_inequalities(3)
thf(fact_211_top__nres__def,axiom,
! [A: $tType] :
( ( top_top @ ( refine1665802226e_nres @ A ) )
= ( refine1767639642_FAILi @ A ) ) ).
% top_nres_def
thf(fact_212_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_213_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_214_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_215_nres__order__simps_I3_J,axiom,
! [C: $tType,M3: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M3 @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% nres_order_simps(3)
thf(fact_216_nres__inequalities_I1_J,axiom,
! [A: $tType,X: set @ A] :
( ( top_top @ ( refine1665802226e_nres @ A ) )
!= ( refine605929679le_RES @ A @ X ) ) ).
% nres_inequalities(1)
thf(fact_217_nres__cases,axiom,
! [A: $tType,M3: refine1665802226e_nres @ A] :
( ( M3
!= ( top_top @ ( refine1665802226e_nres @ A ) ) )
=> ~ ! [X13: set @ A] :
( M3
!= ( refine605929679le_RES @ A @ X13 ) ) ) ).
% nres_cases
thf(fact_218_not__nofail__iff,axiom,
! [A: $tType,S: refine1665802226e_nres @ A] :
( ( ~ ( refine1102455758nofail @ A @ S ) )
= ( S
= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% not_nofail_iff
thf(fact_219_nofail__def,axiom,
! [A: $tType] :
( ( refine1102455758nofail @ A )
= ( ^ [S4: refine1665802226e_nres @ A] :
( S4
!= ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% nofail_def
thf(fact_220_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_221_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_222_Image__mono,axiom,
! [B: $tType,A: $tType,R6: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),A8: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R6 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R6 @ A8 ) @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% Image_mono
thf(fact_223_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_224_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 )
& ! [X5: A] :
( ( refine1315500908_inres @ A @ S @ X5 )
= ( refine1315500908_inres @ A @ S3 @ X5 ) ) ) ) ) ).
% pw_flat_ge_iff
thf(fact_225_subset__UNIV,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_226_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J,axiom,
! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M6: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F4: A > ( refine1665802226e_nres @ B )] :
( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ M3 @ M6 )
=> ( ! [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 @ M3 @ F ) @ ( refine463715084e_bind @ A @ B @ M6 @ F4 ) ) ) ) ).
% Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(2)
thf(fact_227_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_228_imonad1,axiom,
! [A: $tType,U: product_unit,F: product_unit > ( refine1665802226e_nres @ A )] :
( ( refine463715084e_bind @ product_unit @ A @ ( refine1687780735RETURN @ product_unit @ U ) @ F )
= ( F @ U ) ) ).
% imonad1
thf(fact_229_flat__ord_OantimonoD,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B,X4: A,Y3: A] :
( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
=> ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) ) ) ) ).
% flat_ord.antimonoD
thf(fact_230_flat__ord_OantimonoE,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B,X4: A,Y3: A] :
( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
=> ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X4 ) ) ) ) ) ).
% flat_ord.antimonoE
thf(fact_231_UNIV__I,axiom,
! [A: $tType,X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_232_relprop__UNIV__orient,axiom,
! [A: $tType,R: set @ A] :
( ( R
= ( top_top @ ( set @ A ) ) )
=> ( ( top_top @ ( set @ A ) )
= R ) ) ).
% relprop_UNIV_orient
thf(fact_233_eq__UNIV__iff,axiom,
! [A: $tType,S: set @ A] :
( ( S
= ( top_top @ ( set @ A ) ) )
= ( ! [X5: A] : ( member @ A @ X5 @ S ) ) ) ).
% eq_UNIV_iff
thf(fact_234_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_235_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_236_UNIV__eq__I,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A5 )
=> ( ( top_top @ ( set @ A ) )
= A5 ) ) ).
% UNIV_eq_I
thf(fact_237_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_238_order_Oantimono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( antimono @ A @ B )
= ( antimono @ A @ B ) ) ) ).
% order.antimono.cong
thf(fact_239_flat__ord_Oantimono__def,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B] :
( ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
= ( ! [X5: A,Y6: A] :
( ( partial_flat_ord @ A @ B2 @ X5 @ Y6 )
=> ( ord_less_eq @ B @ ( F @ Y6 ) @ ( F @ X5 ) ) ) ) ) ) ).
% flat_ord.antimono_def
thf(fact_240_flat__ord_OantimonoI,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B] :
( ! [X3: A,Y4: A] :
( ( partial_flat_ord @ A @ B2 @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( antimono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F ) ) ) ).
% flat_ord.antimonoI
thf(fact_241_flat__ord_Omono__def,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B] :
( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
= ( ! [X5: A,Y6: A] :
( ( partial_flat_ord @ A @ B2 @ X5 @ Y6 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y6 ) ) ) ) ) ) ).
% flat_ord.mono_def
thf(fact_242_flat__ord_OmonoI,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B] :
( ! [X3: A,Y4: A] :
( ( partial_flat_ord @ A @ B2 @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F ) ) ) ).
% flat_ord.monoI
thf(fact_243_top1I,axiom,
! [A: $tType,X4: A] : ( top_top @ ( A > $o ) @ X4 ) ).
% top1I
thf(fact_244_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_245_flat__ord_OmonoD,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B,X4: A,Y3: A] :
( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
=> ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) ) ) ) ).
% flat_ord.monoD
thf(fact_246_flat__ord_OmonoE,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [B2: A,F: A > B,X4: A,Y3: A] :
( ( mono @ A @ B @ ( partial_flat_ord @ A @ B2 ) @ F )
=> ( ( partial_flat_ord @ A @ B2 @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) ) ) ) ).
% flat_ord.monoE
thf(fact_247_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 )
= ( ? [X14: A] : ( refine1315500908_inres @ A @ S @ X14 )
=> ( ( ( refine1102455758nofail @ A @ S )
= ( refine1102455758nofail @ A @ S3 ) )
& ! [X5: A] :
( ( refine1315500908_inres @ A @ S @ X5 )
= ( refine1315500908_inres @ A @ S3 @ X5 ) ) ) ) ) ).
% pw_flat_le_iff
thf(fact_248_ASSERT__simps_I2_J,axiom,
( ( refine1814851989ASSERT @ $false )
= ( top_top @ ( refine1665802226e_nres @ product_unit ) ) ) ).
% ASSERT_simps(2)
thf(fact_249_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X5: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
% Type constructors (37)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A9: $tType,A10: $tType] :
( ( comple187826305attice @ A10 )
=> ( comple187826305attice @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A9: $tType,A10: $tType] :
( ( order_top @ A10 )
=> ( order_top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A10: $tType] :
( ( top @ A10 )
=> ( top @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A10: $tType] :
( ( bot @ A10 )
=> ( bot @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
! [A9: $tType] : ( comple187826305attice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_5,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_7,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_12,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
bot @ $o ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_15,axiom,
comple187826305attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_16,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_17,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_18,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_19,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_20,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_21,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_22,axiom,
bot @ product_unit ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_23,axiom,
! [A9: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_24,axiom,
! [A9: $tType] : ( order_top @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_25,axiom,
! [A9: $tType] : ( preorder @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_26,axiom,
! [A9: $tType] : ( order @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_27,axiom,
! [A9: $tType] : ( top @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_28,axiom,
! [A9: $tType] : ( ord @ ( refine1665802226e_nres @ A9 ) ) ).
thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_29,axiom,
! [A9: $tType] : ( bot @ ( refine1665802226e_nres @ A9 ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,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 @ b ) @ m @ ( refine463715084e_bind @ a @ b @ ( refine605929679le_RES @ a @ ( collect @ a @ phi ) ) @ f ) ).
%------------------------------------------------------------------------------