TPTP Problem File: ITP164^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP164^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_743__3591834_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_743__3591834_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 445 ( 175 unt; 87 typ; 0 def)
% Number of atoms : 1137 ( 343 equ; 0 cnn)
% Maximal formula atoms : 6 ( 3 avg)
% Number of connectives : 2801 ( 20 ~; 1 |; 48 &;2266 @)
% ( 0 <=>; 466 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 269 ( 269 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 73 usr; 8 con; 0-3 aty)
% Number of variables : 1046 ( 110 ^; 930 !; 6 ?;1046 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:34:13.126
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Ounit_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(ty_n_t__Product____Type__Ounit,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (73)
thf(sy_c_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
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thf(sy_c_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
bot_bo1087887705t_unit: set_Product_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
order_1714329108nres_a: ( refine424419629nres_a > $o ) > refine424419629nres_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
top_to177290092t_unit: refine787176636t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
top_to231829469nres_a: refine424419629nres_a ).
thf(sy_c_Partial__Function_Oflat__ord_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
partia1658438072t_unit: refine787176636t_unit > refine787176636t_unit > refine787176636t_unit > $o ).
thf(sy_c_Partial__Function_Oflat__ord_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
partia906949161nres_a: refine424419629nres_a > refine424419629nres_a > refine424419629nres_a > $o ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
produc1076565719t_unit: product_unit > product_unit > produc971140967t_unit ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Ounit_001tf__a,type,
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thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Ounit,type,
produc1776699686t_unit: a > product_unit > produc1767851702t_unit ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_ORETURN_001t__Product____Type__Ounit,type,
refine1420258419t_unit: product_unit > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_ORETURN_001tf__a,type,
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thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
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thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001t__Product____Type__Ounit_001tf__a,type,
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thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001tf__a_001t__Product____Type__Ounit,type,
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thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oabs__fun_001tf__a_001tf__a,type,
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thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
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thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001t__Product____Type__Ounit_001tf__a,type,
refine119808503unit_a: refine787176636t_unit > ( product_unit > refine424419629nres_a ) > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001tf__a_001t__Product____Type__Ounit,type,
refine96995669t_unit: refine424419629nres_a > ( a > refine787176636t_unit ) > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001tf__a_001tf__a,type,
refine436832838nd_a_a: refine424419629nres_a > ( a > refine424419629nres_a ) > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
refine944483349t_unit: set_Pr451126599t_unit > refine787176636t_unit > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001t__Product____Type__Ounit_001tf__a,type,
refine2043866374unit_a: set_Pr1720557880unit_a > refine424419629nres_a > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__a_001t__Product____Type__Ounit,type,
refine2021053540t_unit: set_Pr1628433942t_unit > refine787176636t_unit > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oconc__fun_001tf__a_001tf__a,type,
refine1441824853un_a_a: set_Product_prod_a_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oinres_001t__Product____Type__Ounit,type,
refine558004794t_unit: refine787176636t_unit > product_unit > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oinres_001tf__a,type,
refine1001002027nres_a: refine424419629nres_a > a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onf__inres_001t__Product____Type__Ounit,type,
refine983493746t_unit: refine787176636t_unit > product_unit > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onf__inres_001tf__a,type,
refine1312857699nres_a: refine424419629nres_a > a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onofail_001t__Product____Type__Ounit,type,
refine579265252t_unit: refine787176636t_unit > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onofail_001tf__a,type,
refine412683989fail_a: refine424419629nres_a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001t__Product____Type__Ounit,type,
refine1777164439t_unit: set_Product_unit > refine787176636t_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Onres_ORES_001tf__a,type,
refine1198353288_RES_a: set_a > refine424419629nres_a ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
refine230495195t_unit: ( refine787176636t_unit > refine787176636t_unit ) > ( refine787176636t_unit > refine787176636t_unit ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
refine2089046860nres_a: ( refine787176636t_unit > refine424419629nres_a ) > ( refine424419629nres_a > refine787176636t_unit ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
refine327276970t_unit: ( refine424419629nres_a > refine787176636t_unit ) > ( refine787176636t_unit > refine424419629nres_a ) > $o ).
thf(sy_c_Refine__Misc_Ogalois__connection_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
refine2004812827nres_a: ( refine424419629nres_a > refine424419629nres_a ) > ( refine424419629nres_a > refine424419629nres_a ) > $o ).
thf(sy_c_Relation_ODomain_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
domain2090798924t_unit: set_Pr451126599t_unit > set_Product_unit ).
thf(sy_c_Relation_ODomain_001t__Product____Type__Ounit_001tf__a,type,
domain822362941unit_a: set_Pr1720557880unit_a > set_Product_unit ).
thf(sy_c_Relation_ODomain_001tf__a_001t__Product____Type__Ounit,type,
domain799550107t_unit: set_Pr1628433942t_unit > set_a ).
thf(sy_c_Relation_ODomain_001tf__a_001tf__a,type,
domain_a_a: set_Product_prod_a_a > set_a ).
thf(sy_c_Relation_Osingle__valued_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
single330234563t_unit: set_Pr451126599t_unit > $o ).
thf(sy_c_Relation_Osingle__valued_001t__Product____Type__Ounit_001tf__a,type,
single249782708unit_a: set_Pr1720557880unit_a > $o ).
thf(sy_c_Relation_Osingle__valued_001tf__a_001t__Product____Type__Ounit,type,
single226969874t_unit: set_Pr1628433942t_unit > $o ).
thf(sy_c_Relation_Osingle__valued_001tf__a_001tf__a,type,
single_valued_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
collec797068754t_unit: ( produc971140967t_unit > $o ) > set_Pr451126599t_unit ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mtf__a_J,type,
collec535904323unit_a: ( produc884009688unit_a > $o ) > set_Pr1720557880unit_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Ounit_J,type,
collec1419746337t_unit: ( produc1767851702t_unit > $o ) > set_Pr1628433942t_unit ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
insert_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
member1423014800t_unit: produc971140967t_unit > set_Pr451126599t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mtf__a_J,type,
member1211819009unit_a: produc884009688unit_a > set_Pr1720557880unit_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Ounit_J,type,
member2095661023t_unit: produc1767851702t_unit > set_Pr1628433942t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_f,type,
f: product_unit > refine424419629nres_a ).
% Relevant facts (352)
thf(fact_0_bind__SUCCEED,axiom,
! [F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ bot_bo658782032t_unit @ F )
= bot_bo658782032t_unit ) ).
% bind_SUCCEED
thf(fact_1_bind__SUCCEED,axiom,
! [F: a > refine787176636t_unit] :
( ( refine96995669t_unit @ bot_bo529555393nres_a @ F )
= bot_bo658782032t_unit ) ).
% bind_SUCCEED
thf(fact_2_bind__SUCCEED,axiom,
! [F: a > refine424419629nres_a] :
( ( refine436832838nd_a_a @ bot_bo529555393nres_a @ F )
= bot_bo529555393nres_a ) ).
% bind_SUCCEED
thf(fact_3_bind__SUCCEED,axiom,
! [F: product_unit > refine424419629nres_a] :
( ( refine119808503unit_a @ bot_bo658782032t_unit @ F )
= bot_bo529555393nres_a ) ).
% bind_SUCCEED
thf(fact_4_nres__monad1,axiom,
! [X: product_unit,F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ ( refine1420258419t_unit @ X ) @ F )
= ( F @ X ) ) ).
% nres_monad1
thf(fact_5_nres__monad1,axiom,
! [X: a,F: a > refine424419629nres_a] :
( ( refine436832838nd_a_a @ ( refine2063221604TURN_a @ X ) @ F )
= ( F @ X ) ) ).
% nres_monad1
thf(fact_6_nres__monad1,axiom,
! [X: a,F: a > refine787176636t_unit] :
( ( refine96995669t_unit @ ( refine2063221604TURN_a @ X ) @ F )
= ( F @ X ) ) ).
% nres_monad1
thf(fact_7_nres__monad1,axiom,
! [X: product_unit,F: product_unit > refine424419629nres_a] :
( ( refine119808503unit_a @ ( refine1420258419t_unit @ X ) @ F )
= ( F @ X ) ) ).
% nres_monad1
thf(fact_8_nres__monad2,axiom,
! [M: refine787176636t_unit] :
( ( refine681446406t_unit @ M @ refine1420258419t_unit )
= M ) ).
% nres_monad2
thf(fact_9_nres__monad2,axiom,
! [M: refine424419629nres_a] :
( ( refine436832838nd_a_a @ M @ refine2063221604TURN_a )
= M ) ).
% nres_monad2
thf(fact_10_abs__fun__strict,axiom,
! [R: set_Pr451126599t_unit] :
( ( refine838861686t_unit @ R @ bot_bo658782032t_unit )
= bot_bo658782032t_unit ) ).
% abs_fun_strict
thf(fact_11_abs__fun__strict,axiom,
! [R: set_Pr1720557880unit_a] :
( ( refine364464487unit_a @ R @ bot_bo658782032t_unit )
= bot_bo529555393nres_a ) ).
% abs_fun_strict
thf(fact_12_abs__fun__strict,axiom,
! [R: set_Pr1628433942t_unit] :
( ( refine341651653t_unit @ R @ bot_bo529555393nres_a )
= bot_bo658782032t_unit ) ).
% abs_fun_strict
thf(fact_13_abs__fun__strict,axiom,
! [R: set_Product_prod_a_a] :
( ( refine1136779702un_a_a @ R @ bot_bo529555393nres_a )
= bot_bo529555393nres_a ) ).
% abs_fun_strict
thf(fact_14_conc__fun__strict,axiom,
! [R: set_Pr451126599t_unit] :
( ( refine944483349t_unit @ R @ bot_bo658782032t_unit )
= bot_bo658782032t_unit ) ).
% conc_fun_strict
thf(fact_15_conc__fun__strict,axiom,
! [R: set_Pr1628433942t_unit] :
( ( refine2021053540t_unit @ R @ bot_bo658782032t_unit )
= bot_bo529555393nres_a ) ).
% conc_fun_strict
thf(fact_16_conc__fun__strict,axiom,
! [R: set_Pr1720557880unit_a] :
( ( refine2043866374unit_a @ R @ bot_bo529555393nres_a )
= bot_bo658782032t_unit ) ).
% conc_fun_strict
thf(fact_17_conc__fun__strict,axiom,
! [R: set_Product_prod_a_a] :
( ( refine1441824853un_a_a @ R @ bot_bo529555393nres_a )
= bot_bo529555393nres_a ) ).
% conc_fun_strict
thf(fact_18_nres__inequalities_I5_J,axiom,
! [X: product_unit] :
( bot_bo658782032t_unit
!= ( refine1420258419t_unit @ X ) ) ).
% nres_inequalities(5)
thf(fact_19_nres__inequalities_I5_J,axiom,
! [X: a] :
( bot_bo529555393nres_a
!= ( refine2063221604TURN_a @ X ) ) ).
% nres_inequalities(5)
thf(fact_20_inres__simps_I4_J,axiom,
( ( refine558004794t_unit @ bot_bo658782032t_unit )
= ( ^ [Uu: product_unit] : $false ) ) ).
% inres_simps(4)
thf(fact_21_inres__simps_I4_J,axiom,
( ( refine1001002027nres_a @ bot_bo529555393nres_a )
= ( ^ [Uu: a] : $false ) ) ).
% inres_simps(4)
thf(fact_22_nres__order__simps_I2_J,axiom,
! [M: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ M @ bot_bo658782032t_unit )
= ( M = bot_bo658782032t_unit ) ) ).
% nres_order_simps(2)
thf(fact_23_nres__order__simps_I2_J,axiom,
! [M: refine424419629nres_a] :
( ( ord_le519537037nres_a @ M @ bot_bo529555393nres_a )
= ( M = bot_bo529555393nres_a ) ) ).
% nres_order_simps(2)
thf(fact_24_nofail__simps_I4_J,axiom,
refine579265252t_unit @ bot_bo658782032t_unit ).
% nofail_simps(4)
thf(fact_25_nofail__simps_I4_J,axiom,
refine412683989fail_a @ bot_bo529555393nres_a ).
% nofail_simps(4)
thf(fact_26_bind__FAIL,axiom,
! [F: product_unit > refine424419629nres_a] :
( ( refine119808503unit_a @ top_to177290092t_unit @ F )
= top_to231829469nres_a ) ).
% bind_FAIL
thf(fact_27_bind__FAIL,axiom,
! [F: a > refine424419629nres_a] :
( ( refine436832838nd_a_a @ top_to231829469nres_a @ F )
= top_to231829469nres_a ) ).
% bind_FAIL
thf(fact_28_bind__FAIL,axiom,
! [F: a > refine787176636t_unit] :
( ( refine96995669t_unit @ top_to231829469nres_a @ F )
= top_to177290092t_unit ) ).
% bind_FAIL
thf(fact_29_bind__FAIL,axiom,
! [F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ top_to177290092t_unit @ F )
= top_to177290092t_unit ) ).
% bind_FAIL
thf(fact_30_nres__order__simps_I1_J,axiom,
! [M: refine424419629nres_a] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ M ) ).
% nres_order_simps(1)
thf(fact_31_nres__order__simps_I1_J,axiom,
! [M: refine787176636t_unit] : ( ord_le1051254044t_unit @ bot_bo658782032t_unit @ M ) ).
% nres_order_simps(1)
thf(fact_32_nres__more__simps_I6_J,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( refine1420258419t_unit @ X )
= ( refine1420258419t_unit @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(6)
thf(fact_33_nres__more__simps_I6_J,axiom,
! [X: a,Y: a] :
( ( ( refine2063221604TURN_a @ X )
= ( refine2063221604TURN_a @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(6)
thf(fact_34_nres__order__simps_I4_J,axiom,
! [M: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ M )
= ( M = top_to231829469nres_a ) ) ).
% nres_order_simps(4)
thf(fact_35_nres__order__simps_I4_J,axiom,
! [M: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ M )
= ( M = top_to177290092t_unit ) ) ).
% nres_order_simps(4)
thf(fact_36_nofail__simps_I1_J,axiom,
~ ( refine412683989fail_a @ top_to231829469nres_a ) ).
% nofail_simps(1)
thf(fact_37_nofail__simps_I1_J,axiom,
~ ( refine579265252t_unit @ top_to177290092t_unit ) ).
% nofail_simps(1)
thf(fact_38_inres__simps_I1_J,axiom,
( ( refine1001002027nres_a @ top_to231829469nres_a )
= ( ^ [Uu: a] : $true ) ) ).
% inres_simps(1)
thf(fact_39_inres__simps_I1_J,axiom,
( ( refine558004794t_unit @ top_to177290092t_unit )
= ( ^ [Uu: product_unit] : $true ) ) ).
% inres_simps(1)
thf(fact_40_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Product_prod_a_a,S: refine424419629nres_a] :
( ( top_to231829469nres_a
= ( refine1441824853un_a_a @ R @ S ) )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(2)
thf(fact_41_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr1720557880unit_a,S: refine424419629nres_a] :
( ( top_to177290092t_unit
= ( refine2043866374unit_a @ R @ S ) )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(2)
thf(fact_42_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr1628433942t_unit,S: refine787176636t_unit] :
( ( top_to231829469nres_a
= ( refine2021053540t_unit @ R @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(2)
thf(fact_43_conc__fun__fail__iff_I2_J,axiom,
! [R: set_Pr451126599t_unit,S: refine787176636t_unit] :
( ( top_to177290092t_unit
= ( refine944483349t_unit @ R @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(2)
thf(fact_44_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Product_prod_a_a,S: refine424419629nres_a] :
( ( ( refine1441824853un_a_a @ R @ S )
= top_to231829469nres_a )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(1)
thf(fact_45_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr1720557880unit_a,S: refine424419629nres_a] :
( ( ( refine2043866374unit_a @ R @ S )
= top_to177290092t_unit )
= ( S = top_to231829469nres_a ) ) ).
% conc_fun_fail_iff(1)
thf(fact_46_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr1628433942t_unit,S: refine787176636t_unit] :
( ( ( refine2021053540t_unit @ R @ S )
= top_to231829469nres_a )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(1)
thf(fact_47_conc__fun__fail__iff_I1_J,axiom,
! [R: set_Pr451126599t_unit,S: refine787176636t_unit] :
( ( ( refine944483349t_unit @ R @ S )
= top_to177290092t_unit )
= ( S = top_to177290092t_unit ) ) ).
% conc_fun_fail_iff(1)
thf(fact_48_conc__fun__FAIL,axiom,
! [R: set_Product_prod_a_a] :
( ( refine1441824853un_a_a @ R @ top_to231829469nres_a )
= top_to231829469nres_a ) ).
% conc_fun_FAIL
thf(fact_49_conc__fun__FAIL,axiom,
! [R: set_Pr1720557880unit_a] :
( ( refine2043866374unit_a @ R @ top_to231829469nres_a )
= top_to177290092t_unit ) ).
% conc_fun_FAIL
thf(fact_50_conc__fun__FAIL,axiom,
! [R: set_Pr1628433942t_unit] :
( ( refine2021053540t_unit @ R @ top_to177290092t_unit )
= top_to231829469nres_a ) ).
% conc_fun_FAIL
thf(fact_51_conc__fun__FAIL,axiom,
! [R: set_Pr451126599t_unit] :
( ( refine944483349t_unit @ R @ top_to177290092t_unit )
= top_to177290092t_unit ) ).
% conc_fun_FAIL
thf(fact_52_nres__order__simps_I20_J,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X ) @ ( refine1420258419t_unit @ Y ) )
= ( X = Y ) ) ).
% nres_order_simps(20)
thf(fact_53_nres__order__simps_I20_J,axiom,
! [X: a,Y: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine2063221604TURN_a @ Y ) )
= ( X = Y ) ) ).
% nres_order_simps(20)
thf(fact_54_abs__fun__simps_I1_J,axiom,
! [R: set_Product_prod_a_a] :
( ( refine1136779702un_a_a @ R @ top_to231829469nres_a )
= top_to231829469nres_a ) ).
% abs_fun_simps(1)
thf(fact_55_abs__fun__simps_I1_J,axiom,
! [R: set_Pr1628433942t_unit] :
( ( refine341651653t_unit @ R @ top_to231829469nres_a )
= top_to177290092t_unit ) ).
% abs_fun_simps(1)
thf(fact_56_abs__fun__simps_I1_J,axiom,
! [R: set_Pr1720557880unit_a] :
( ( refine364464487unit_a @ R @ top_to177290092t_unit )
= top_to231829469nres_a ) ).
% abs_fun_simps(1)
thf(fact_57_abs__fun__simps_I1_J,axiom,
! [R: set_Pr451126599t_unit] :
( ( refine838861686t_unit @ R @ top_to177290092t_unit )
= top_to177290092t_unit ) ).
% abs_fun_simps(1)
thf(fact_58_nofail__simps_I3_J,axiom,
! [X: product_unit] : ( refine579265252t_unit @ ( refine1420258419t_unit @ X ) ) ).
% nofail_simps(3)
thf(fact_59_nofail__simps_I3_J,axiom,
! [X: a] : ( refine412683989fail_a @ ( refine2063221604TURN_a @ X ) ) ).
% nofail_simps(3)
thf(fact_60_inres__simps_I3_J,axiom,
! [X: product_unit] :
( ( refine558004794t_unit @ ( refine1420258419t_unit @ X ) )
= ( ^ [Y2: product_unit,Z: product_unit] : Y2 = Z
@ X ) ) ).
% inres_simps(3)
thf(fact_61_inres__simps_I3_J,axiom,
! [X: a] :
( ( refine1001002027nres_a @ ( refine2063221604TURN_a @ X ) )
= ( ^ [Y2: a,Z: a] : Y2 = Z
@ X ) ) ).
% inres_simps(3)
thf(fact_62_intro__nofail_I2_J,axiom,
! [S: refine424419629nres_a] :
( ( top_to231829469nres_a != S )
= ( refine412683989fail_a @ S ) ) ).
% intro_nofail(2)
thf(fact_63_intro__nofail_I2_J,axiom,
! [S: refine787176636t_unit] :
( ( top_to177290092t_unit != S )
= ( refine579265252t_unit @ S ) ) ).
% intro_nofail(2)
thf(fact_64_nres__order__simps_I3_J,axiom,
! [M: refine424419629nres_a] : ( ord_le519537037nres_a @ M @ top_to231829469nres_a ) ).
% nres_order_simps(3)
thf(fact_65_nres__order__simps_I3_J,axiom,
! [M: refine787176636t_unit] : ( ord_le1051254044t_unit @ M @ top_to177290092t_unit ) ).
% nres_order_simps(3)
thf(fact_66_nres__inequalities_I3_J,axiom,
! [X: product_unit] :
( top_to177290092t_unit
!= ( refine1420258419t_unit @ X ) ) ).
% nres_inequalities(3)
thf(fact_67_nres__inequalities_I3_J,axiom,
! [X: a] :
( top_to231829469nres_a
!= ( refine2063221604TURN_a @ X ) ) ).
% nres_inequalities(3)
thf(fact_68_abs__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Product_prod_a_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ B ) @ C )
=> ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_69_abs__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,R: set_Pr1628433942t_unit,C: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R @ B ) @ C )
=> ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_70_abs__trans__additional_I1_J,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,R: set_Pr1720557880unit_a,C: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ord_le519537037nres_a @ ( refine364464487unit_a @ R @ B ) @ C )
=> ( ord_le519537037nres_a @ ( refine364464487unit_a @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_71_abs__trans__additional_I1_J,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,R: set_Pr451126599t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R @ B ) @ C )
=> ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R @ A ) @ C ) ) ) ).
% abs_trans_additional(1)
thf(fact_72_conc__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,R: set_Product_prod_a_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ ( refine1441824853un_a_a @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_73_conc__trans__additional_I1_J,axiom,
! [A: refine787176636t_unit,R: set_Pr1720557880unit_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A @ ( refine2043866374unit_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le1051254044t_unit @ A @ ( refine2043866374unit_a @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_74_conc__trans__additional_I1_J,axiom,
! [A: refine424419629nres_a,R: set_Pr1628433942t_unit,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A @ ( refine2021053540t_unit @ R @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ord_le519537037nres_a @ A @ ( refine2021053540t_unit @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_75_conc__trans__additional_I1_J,axiom,
! [A: refine787176636t_unit,R: set_Pr451126599t_unit,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ ( refine944483349t_unit @ R @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ord_le1051254044t_unit @ A @ ( refine944483349t_unit @ R @ C ) ) ) ) ).
% conc_trans_additional(1)
thf(fact_76_order__subst1,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_77_order__subst1,axiom,
! [A2: refine424419629nres_a,F: refine787176636t_unit > refine424419629nres_a,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_78_order__subst1,axiom,
! [A2: refine787176636t_unit,F: refine424419629nres_a > refine787176636t_unit,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_79_order__subst1,axiom,
! [A2: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_80_order__subst1,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C2: set_a] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_81_order__subst1,axiom,
! [A2: refine787176636t_unit,F: set_a > refine787176636t_unit,B2: set_a,C2: set_a] :
( ( ord_le1051254044t_unit @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_82_order__subst1,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_83_order__subst1,axiom,
! [A2: set_a,F: refine787176636t_unit > set_a,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_84_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_85_order__subst1,axiom,
! [A2: refine424419629nres_a,F: set_Pr451126599t_unit > refine424419629nres_a,B2: set_Pr451126599t_unit,C2: set_Pr451126599t_unit] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le2035129575t_unit @ B2 @ C2 )
=> ( ! [X2: set_Pr451126599t_unit,Y3: set_Pr451126599t_unit] :
( ( ord_le2035129575t_unit @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_86_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le1051254044t_unit @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,F: refine787176636t_unit > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ord_le1051254044t_unit @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C2: set_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,F: refine787176636t_unit > set_a,C2: set_a] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le1051254044t_unit @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_95_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_Pr451126599t_unit,C2: set_Pr451126599t_unit] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le2035129575t_unit @ ( F @ B2 ) @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le2035129575t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2035129575t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_96_ord__eq__le__subst,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_ord__eq__le__subst,axiom,
! [A2: refine787176636t_unit,F: refine424419629nres_a > refine787176636t_unit,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__eq__le__subst,axiom,
! [A2: refine424419629nres_a,F: refine787176636t_unit > refine424419629nres_a,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A2: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A2: set_a,F: refine787176636t_unit > set_a,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C2: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A2: refine787176636t_unit,F: set_a > refine787176636t_unit,B2: set_a,C2: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_105_ord__eq__le__subst,axiom,
! [A2: set_Pr451126599t_unit,F: refine424419629nres_a > set_Pr451126599t_unit,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le2035129575t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2035129575t_unit @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,F: refine787176636t_unit > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C2: set_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_111_ord__le__eq__subst,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,F: refine787176636t_unit > set_a,C2: set_a] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_112_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_115_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_Pr451126599t_unit,C2: set_Pr451126599t_unit] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X2 @ Y3 )
=> ( ord_le2035129575t_unit @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2035129575t_unit @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_116_eq__iff,axiom,
( ( ^ [Y2: set_Pr451126599t_unit,Z: set_Pr451126599t_unit] : Y2 = Z )
= ( ^ [X3: set_Pr451126599t_unit,Y4: set_Pr451126599t_unit] :
( ( ord_le2035129575t_unit @ X3 @ Y4 )
& ( ord_le2035129575t_unit @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_117_eq__iff,axiom,
( ( ^ [Y2: set_Pr1720557880unit_a,Z: set_Pr1720557880unit_a] : Y2 = Z )
= ( ^ [X3: set_Pr1720557880unit_a,Y4: set_Pr1720557880unit_a] :
( ( ord_le2070001880unit_a @ X3 @ Y4 )
& ( ord_le2070001880unit_a @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_118_eq__iff,axiom,
( ( ^ [Y2: set_Pr1628433942t_unit,Z: set_Pr1628433942t_unit] : Y2 = Z )
= ( ^ [X3: set_Pr1628433942t_unit,Y4: set_Pr1628433942t_unit] :
( ( ord_le1977877942t_unit @ X3 @ Y4 )
& ( ord_le1977877942t_unit @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_119_eq__iff,axiom,
( ( ^ [Y2: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y2 = Z )
= ( ^ [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X3 @ Y4 )
& ( ord_le1824328871od_a_a @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_120_eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : Y2 = Z )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_121_eq__iff,axiom,
( ( ^ [Y2: refine424419629nres_a,Z: refine424419629nres_a] : Y2 = Z )
= ( ^ [X3: refine424419629nres_a,Y4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y4 )
& ( ord_le519537037nres_a @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_122_eq__iff,axiom,
( ( ^ [Y2: refine787176636t_unit,Z: refine787176636t_unit] : Y2 = Z )
= ( ^ [X3: refine787176636t_unit,Y4: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y4 )
& ( ord_le1051254044t_unit @ Y4 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_123_antisym,axiom,
! [X: set_Pr451126599t_unit,Y: set_Pr451126599t_unit] :
( ( ord_le2035129575t_unit @ X @ Y )
=> ( ( ord_le2035129575t_unit @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_124_antisym,axiom,
! [X: set_Pr1720557880unit_a,Y: set_Pr1720557880unit_a] :
( ( ord_le2070001880unit_a @ X @ Y )
=> ( ( ord_le2070001880unit_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_125_antisym,axiom,
! [X: set_Pr1628433942t_unit,Y: set_Pr1628433942t_unit] :
( ( ord_le1977877942t_unit @ X @ Y )
=> ( ( ord_le1977877942t_unit @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_126_antisym,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ X @ Y )
=> ( ( ord_le1824328871od_a_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_127_antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_128_antisym,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X @ Y )
=> ( ( ord_le519537037nres_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_129_antisym,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X @ Y )
=> ( ( ord_le1051254044t_unit @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_130_eq__refl,axiom,
! [X: set_Pr451126599t_unit,Y: set_Pr451126599t_unit] :
( ( X = Y )
=> ( ord_le2035129575t_unit @ X @ Y ) ) ).
% eq_refl
thf(fact_131_eq__refl,axiom,
! [X: set_Pr1720557880unit_a,Y: set_Pr1720557880unit_a] :
( ( X = Y )
=> ( ord_le2070001880unit_a @ X @ Y ) ) ).
% eq_refl
thf(fact_132_eq__refl,axiom,
! [X: set_Pr1628433942t_unit,Y: set_Pr1628433942t_unit] :
( ( X = Y )
=> ( ord_le1977877942t_unit @ X @ Y ) ) ).
% eq_refl
thf(fact_133_eq__refl,axiom,
! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( X = Y )
=> ( ord_le1824328871od_a_a @ X @ Y ) ) ).
% eq_refl
thf(fact_134_eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% eq_refl
thf(fact_135_eq__refl,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a] :
( ( X = Y )
=> ( ord_le519537037nres_a @ X @ Y ) ) ).
% eq_refl
thf(fact_136_eq__refl,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit] :
( ( X = Y )
=> ( ord_le1051254044t_unit @ X @ Y ) ) ).
% eq_refl
thf(fact_137_order_Otrans,axiom,
! [A2: set_Pr451126599t_unit,B2: set_Pr451126599t_unit,C2: set_Pr451126599t_unit] :
( ( ord_le2035129575t_unit @ A2 @ B2 )
=> ( ( ord_le2035129575t_unit @ B2 @ C2 )
=> ( ord_le2035129575t_unit @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_138_order_Otrans,axiom,
! [A2: set_Pr1720557880unit_a,B2: set_Pr1720557880unit_a,C2: set_Pr1720557880unit_a] :
( ( ord_le2070001880unit_a @ A2 @ B2 )
=> ( ( ord_le2070001880unit_a @ B2 @ C2 )
=> ( ord_le2070001880unit_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_139_order_Otrans,axiom,
! [A2: set_Pr1628433942t_unit,B2: set_Pr1628433942t_unit,C2: set_Pr1628433942t_unit] :
( ( ord_le1977877942t_unit @ A2 @ B2 )
=> ( ( ord_le1977877942t_unit @ B2 @ C2 )
=> ( ord_le1977877942t_unit @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_140_order_Otrans,axiom,
! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C2: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A2 @ B2 )
=> ( ( ord_le1824328871od_a_a @ B2 @ C2 )
=> ( ord_le1824328871od_a_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_141_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_142_order_Otrans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ord_le519537037nres_a @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_143_order_Otrans,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ord_le1051254044t_unit @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_144_mem__Collect__eq,axiom,
! [A2: produc971140967t_unit,P: produc971140967t_unit > $o] :
( ( member1423014800t_unit @ A2 @ ( collec797068754t_unit @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A2: produc884009688unit_a,P: produc884009688unit_a > $o] :
( ( member1211819009unit_a @ A2 @ ( collec535904323unit_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A2: produc1767851702t_unit,P: produc1767851702t_unit > $o] :
( ( member2095661023t_unit @ A2 @ ( collec1419746337t_unit @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member449909584od_a_a @ A2 @ ( collec645855634od_a_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_148_mem__Collect__eq,axiom,
! [A2: product_unit,P: product_unit > $o] :
( ( member_Product_unit @ A2 @ ( collect_Product_unit @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_150_Collect__mem__eq,axiom,
! [A: set_Pr451126599t_unit] :
( ( collec797068754t_unit
@ ^ [X3: produc971140967t_unit] : ( member1423014800t_unit @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
! [A: set_Pr1720557880unit_a] :
( ( collec535904323unit_a
@ ^ [X3: produc884009688unit_a] : ( member1211819009unit_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_152_Collect__mem__eq,axiom,
! [A: set_Pr1628433942t_unit] :
( ( collec1419746337t_unit
@ ^ [X3: produc1767851702t_unit] : ( member2095661023t_unit @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_153_Collect__mem__eq,axiom,
! [A: set_Product_prod_a_a] :
( ( collec645855634od_a_a
@ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A: set_Product_unit] :
( ( collect_Product_unit
@ ^ [X3: product_unit] : ( member_Product_unit @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_156_Collect__cong,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ! [X2: product_unit] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_Product_unit @ P )
= ( collect_Product_unit @ Q ) ) ) ).
% Collect_cong
thf(fact_157_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_158_antisym__conv,axiom,
! [Y: set_Pr451126599t_unit,X: set_Pr451126599t_unit] :
( ( ord_le2035129575t_unit @ Y @ X )
=> ( ( ord_le2035129575t_unit @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_159_antisym__conv,axiom,
! [Y: set_Pr1720557880unit_a,X: set_Pr1720557880unit_a] :
( ( ord_le2070001880unit_a @ Y @ X )
=> ( ( ord_le2070001880unit_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_160_antisym__conv,axiom,
! [Y: set_Pr1628433942t_unit,X: set_Pr1628433942t_unit] :
( ( ord_le1977877942t_unit @ Y @ X )
=> ( ( ord_le1977877942t_unit @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_161_antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ Y @ X )
=> ( ( ord_le1824328871od_a_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_162_antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_163_antisym__conv,axiom,
! [Y: refine424419629nres_a,X: refine424419629nres_a] :
( ( ord_le519537037nres_a @ Y @ X )
=> ( ( ord_le519537037nres_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_164_antisym__conv,axiom,
! [Y: refine787176636t_unit,X: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ Y @ X )
=> ( ( ord_le1051254044t_unit @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_165_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr451126599t_unit,Z: set_Pr451126599t_unit] : Y2 = Z )
= ( ^ [A3: set_Pr451126599t_unit,B3: set_Pr451126599t_unit] :
( ( ord_le2035129575t_unit @ A3 @ B3 )
& ( ord_le2035129575t_unit @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_166_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr1720557880unit_a,Z: set_Pr1720557880unit_a] : Y2 = Z )
= ( ^ [A3: set_Pr1720557880unit_a,B3: set_Pr1720557880unit_a] :
( ( ord_le2070001880unit_a @ A3 @ B3 )
& ( ord_le2070001880unit_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_167_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Pr1628433942t_unit,Z: set_Pr1628433942t_unit] : Y2 = Z )
= ( ^ [A3: set_Pr1628433942t_unit,B3: set_Pr1628433942t_unit] :
( ( ord_le1977877942t_unit @ A3 @ B3 )
& ( ord_le1977877942t_unit @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_168_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_Product_prod_a_a,Z: set_Product_prod_a_a] : Y2 = Z )
= ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ord_le1824328871od_a_a @ A3 @ B3 )
& ( ord_le1824328871od_a_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_169_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : Y2 = Z )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_170_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: refine424419629nres_a,Z: refine424419629nres_a] : Y2 = Z )
= ( ^ [A3: refine424419629nres_a,B3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A3 @ B3 )
& ( ord_le519537037nres_a @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_171_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: refine787176636t_unit,Z: refine787176636t_unit] : Y2 = Z )
= ( ^ [A3: refine787176636t_unit,B3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A3 @ B3 )
& ( ord_le1051254044t_unit @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_172_ord__eq__le__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( A2 = B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C2 )
=> ( ord_le519537037nres_a @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_173_ord__eq__le__trans,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( A2 = B2 )
=> ( ( ord_le1051254044t_unit @ B2 @ C2 )
=> ( ord_le1051254044t_unit @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_174_ord__le__eq__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le519537037nres_a @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_175_ord__le__eq__trans,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le1051254044t_unit @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_176_order__class_Oorder_Oantisym,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_177_order__class_Oorder_Oantisym,axiom,
! [A2: refine787176636t_unit,B2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( ( ord_le1051254044t_unit @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_178_order__trans,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X @ Y )
=> ( ( ord_le519537037nres_a @ Y @ Z2 )
=> ( ord_le519537037nres_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_179_order__trans,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit,Z2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X @ Y )
=> ( ( ord_le1051254044t_unit @ Y @ Z2 )
=> ( ord_le1051254044t_unit @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_180_dual__order_Orefl,axiom,
! [A2: refine424419629nres_a] : ( ord_le519537037nres_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_181_dual__order_Orefl,axiom,
! [A2: refine787176636t_unit] : ( ord_le1051254044t_unit @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_182_dual__order_Otrans,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a,C2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( ( ord_le519537037nres_a @ C2 @ B2 )
=> ( ord_le519537037nres_a @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_183_dual__order_Otrans,axiom,
! [B2: refine787176636t_unit,A2: refine787176636t_unit,C2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ B2 @ A2 )
=> ( ( ord_le1051254044t_unit @ C2 @ B2 )
=> ( ord_le1051254044t_unit @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_184_top__greatest,axiom,
! [A2: refine424419629nres_a] : ( ord_le519537037nres_a @ A2 @ top_to231829469nres_a ) ).
% top_greatest
thf(fact_185_top__greatest,axiom,
! [A2: refine787176636t_unit] : ( ord_le1051254044t_unit @ A2 @ top_to177290092t_unit ) ).
% top_greatest
thf(fact_186_pwD1,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ S @ S2 )
=> ( ( refine412683989fail_a @ S2 )
=> ( refine412683989fail_a @ S ) ) ) ).
% pwD1
thf(fact_187_pwD1,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ S @ S2 )
=> ( ( refine579265252t_unit @ S2 )
=> ( refine579265252t_unit @ S ) ) ) ).
% pwD1
thf(fact_188_pwD2,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a,X: a] :
( ( ord_le519537037nres_a @ S @ S2 )
=> ( ( refine1001002027nres_a @ S @ X )
=> ( refine1001002027nres_a @ S2 @ X ) ) ) ).
% pwD2
thf(fact_189_pwD2,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit,X: product_unit] :
( ( ord_le1051254044t_unit @ S @ S2 )
=> ( ( refine558004794t_unit @ S @ X )
=> ( refine558004794t_unit @ S2 @ X ) ) ) ).
% pwD2
thf(fact_190_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: refine424419629nres_a,Z: refine424419629nres_a] : Y2 = Z )
= ( ^ [A3: refine424419629nres_a,B3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B3 @ A3 )
& ( ord_le519537037nres_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_191_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: refine787176636t_unit,Z: refine787176636t_unit] : Y2 = Z )
= ( ^ [A3: refine787176636t_unit,B3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ B3 @ A3 )
& ( ord_le1051254044t_unit @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_192_dual__order_Oantisym,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_193_dual__order_Oantisym,axiom,
! [B2: refine787176636t_unit,A2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ B2 @ A2 )
=> ( ( ord_le1051254044t_unit @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_194_pw__eqI,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S )
= ( refine412683989fail_a @ S2 ) )
=> ( ! [X2: a] :
( ( refine1001002027nres_a @ S @ X2 )
= ( refine1001002027nres_a @ S2 @ X2 ) )
=> ( S = S2 ) ) ) ).
% pw_eqI
thf(fact_195_pw__eqI,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S )
= ( refine579265252t_unit @ S2 ) )
=> ( ! [X2: product_unit] :
( ( refine558004794t_unit @ S @ X2 )
= ( refine558004794t_unit @ S2 @ X2 ) )
=> ( S = S2 ) ) ) ).
% pw_eqI
thf(fact_196_pw__leI,axiom,
! [S2: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S2 )
=> ( ( refine412683989fail_a @ S )
& ! [X2: a] :
( ( refine1001002027nres_a @ S @ X2 )
=> ( refine1001002027nres_a @ S2 @ X2 ) ) ) )
=> ( ord_le519537037nres_a @ S @ S2 ) ) ).
% pw_leI
thf(fact_197_pw__leI,axiom,
! [S2: refine787176636t_unit,S: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S2 )
=> ( ( refine579265252t_unit @ S )
& ! [X2: product_unit] :
( ( refine558004794t_unit @ S @ X2 )
=> ( refine558004794t_unit @ S2 @ X2 ) ) ) )
=> ( ord_le1051254044t_unit @ S @ S2 ) ) ).
% pw_leI
thf(fact_198_pw__leI_H,axiom,
! [S2: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S2 )
=> ( refine412683989fail_a @ S ) )
=> ( ! [X2: a] :
( ( refine412683989fail_a @ S2 )
=> ( ( refine1001002027nres_a @ S @ X2 )
=> ( refine1001002027nres_a @ S2 @ X2 ) ) )
=> ( ord_le519537037nres_a @ S @ S2 ) ) ) ).
% pw_leI'
thf(fact_199_pw__leI_H,axiom,
! [S2: refine787176636t_unit,S: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S2 )
=> ( refine579265252t_unit @ S ) )
=> ( ! [X2: product_unit] :
( ( refine579265252t_unit @ S2 )
=> ( ( refine558004794t_unit @ S @ X2 )
=> ( refine558004794t_unit @ S2 @ X2 ) ) )
=> ( ord_le1051254044t_unit @ S @ S2 ) ) ) ).
% pw_leI'
thf(fact_200_abs__trans,axiom,
! [R: set_Product_prod_a_a,C: refine424419629nres_a,B: refine424419629nres_a,R2: set_Product_prod_a_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ C ) @ B )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R2 @ B ) @ A )
=> ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R2 @ ( refine1136779702un_a_a @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_201_abs__trans,axiom,
! [R: set_Product_prod_a_a,C: refine424419629nres_a,B: refine424419629nres_a,R2: set_Pr1628433942t_unit,A: refine787176636t_unit] :
( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ C ) @ B )
=> ( ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R2 @ B ) @ A )
=> ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R2 @ ( refine1136779702un_a_a @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_202_abs__trans,axiom,
! [R: set_Pr1628433942t_unit,C: refine424419629nres_a,B: refine787176636t_unit,R2: set_Pr1720557880unit_a,A: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R @ C ) @ B )
=> ( ( ord_le519537037nres_a @ ( refine364464487unit_a @ R2 @ B ) @ A )
=> ( ord_le519537037nres_a @ ( refine364464487unit_a @ R2 @ ( refine341651653t_unit @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_203_abs__trans,axiom,
! [R: set_Pr1628433942t_unit,C: refine424419629nres_a,B: refine787176636t_unit,R2: set_Pr451126599t_unit,A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R @ C ) @ B )
=> ( ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R2 @ B ) @ A )
=> ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R2 @ ( refine341651653t_unit @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_204_abs__trans,axiom,
! [R: set_Pr1720557880unit_a,C: refine787176636t_unit,B: refine424419629nres_a,R2: set_Product_prod_a_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine364464487unit_a @ R @ C ) @ B )
=> ( ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R2 @ B ) @ A )
=> ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R2 @ ( refine364464487unit_a @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_205_abs__trans,axiom,
! [R: set_Pr1720557880unit_a,C: refine787176636t_unit,B: refine424419629nres_a,R2: set_Pr1628433942t_unit,A: refine787176636t_unit] :
( ( ord_le519537037nres_a @ ( refine364464487unit_a @ R @ C ) @ B )
=> ( ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R2 @ B ) @ A )
=> ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R2 @ ( refine364464487unit_a @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_206_abs__trans,axiom,
! [R: set_Pr451126599t_unit,C: refine787176636t_unit,B: refine787176636t_unit,R2: set_Pr1720557880unit_a,A: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R @ C ) @ B )
=> ( ( ord_le519537037nres_a @ ( refine364464487unit_a @ R2 @ B ) @ A )
=> ( ord_le519537037nres_a @ ( refine364464487unit_a @ R2 @ ( refine838861686t_unit @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_207_abs__trans,axiom,
! [R: set_Pr451126599t_unit,C: refine787176636t_unit,B: refine787176636t_unit,R2: set_Pr451126599t_unit,A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R @ C ) @ B )
=> ( ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R2 @ B ) @ A )
=> ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R2 @ ( refine838861686t_unit @ R @ C ) ) @ A ) ) ) ).
% abs_trans
thf(fact_208_inres__def,axiom,
( refine558004794t_unit
= ( ^ [S3: refine787176636t_unit,X3: product_unit] : ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X3 ) @ S3 ) ) ) ).
% inres_def
thf(fact_209_inres__def,axiom,
( refine1001002027nres_a
= ( ^ [S3: refine424419629nres_a,X3: a] : ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ S3 ) ) ) ).
% inres_def
thf(fact_210_pw__eq__iff,axiom,
( ( ^ [Y2: refine424419629nres_a,Z: refine424419629nres_a] : Y2 = Z )
= ( ^ [S3: refine424419629nres_a,S4: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S3 )
= ( refine412683989fail_a @ S4 ) )
& ! [X3: a] :
( ( refine1001002027nres_a @ S3 @ X3 )
= ( refine1001002027nres_a @ S4 @ X3 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_211_pw__eq__iff,axiom,
( ( ^ [Y2: refine787176636t_unit,Z: refine787176636t_unit] : Y2 = Z )
= ( ^ [S3: refine787176636t_unit,S4: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S3 )
= ( refine579265252t_unit @ S4 ) )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ S3 @ X3 )
= ( refine558004794t_unit @ S4 @ X3 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_212_pw__le__iff,axiom,
( ord_le519537037nres_a
= ( ^ [S3: refine424419629nres_a,S4: refine424419629nres_a] :
( ( refine412683989fail_a @ S4 )
=> ( ( refine412683989fail_a @ S3 )
& ! [X3: a] :
( ( refine1001002027nres_a @ S3 @ X3 )
=> ( refine1001002027nres_a @ S4 @ X3 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_213_pw__le__iff,axiom,
( ord_le1051254044t_unit
= ( ^ [S3: refine787176636t_unit,S4: refine787176636t_unit] :
( ( refine579265252t_unit @ S4 )
=> ( ( refine579265252t_unit @ S3 )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ S3 @ X3 )
=> ( refine558004794t_unit @ S4 @ X3 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_214_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Product_prod_a_a,B: refine424419629nres_a,R2: set_Product_prod_a_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ ( refine1441824853un_a_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_215_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Product_prod_a_a,B: refine424419629nres_a,R2: set_Pr1628433942t_unit,A: refine787176636t_unit] :
( ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine2021053540t_unit @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine1441824853un_a_a @ R @ ( refine2021053540t_unit @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_216_conc__trans,axiom,
! [C: refine787176636t_unit,R: set_Pr1720557880unit_a,B: refine424419629nres_a,R2: set_Product_prod_a_a,A: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ C @ ( refine2043866374unit_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine1441824853un_a_a @ R2 @ A ) )
=> ( ord_le1051254044t_unit @ C @ ( refine2043866374unit_a @ R @ ( refine1441824853un_a_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_217_conc__trans,axiom,
! [C: refine787176636t_unit,R: set_Pr1720557880unit_a,B: refine424419629nres_a,R2: set_Pr1628433942t_unit,A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ C @ ( refine2043866374unit_a @ R @ B ) )
=> ( ( ord_le519537037nres_a @ B @ ( refine2021053540t_unit @ R2 @ A ) )
=> ( ord_le1051254044t_unit @ C @ ( refine2043866374unit_a @ R @ ( refine2021053540t_unit @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_218_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Pr1628433942t_unit,B: refine787176636t_unit,R2: set_Pr1720557880unit_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ C @ ( refine2021053540t_unit @ R @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ ( refine2043866374unit_a @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine2021053540t_unit @ R @ ( refine2043866374unit_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_219_conc__trans,axiom,
! [C: refine424419629nres_a,R: set_Pr1628433942t_unit,B: refine787176636t_unit,R2: set_Pr451126599t_unit,A: refine787176636t_unit] :
( ( ord_le519537037nres_a @ C @ ( refine2021053540t_unit @ R @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ ( refine944483349t_unit @ R2 @ A ) )
=> ( ord_le519537037nres_a @ C @ ( refine2021053540t_unit @ R @ ( refine944483349t_unit @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_220_conc__trans,axiom,
! [C: refine787176636t_unit,R: set_Pr451126599t_unit,B: refine787176636t_unit,R2: set_Pr1720557880unit_a,A: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ C @ ( refine944483349t_unit @ R @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ ( refine2043866374unit_a @ R2 @ A ) )
=> ( ord_le1051254044t_unit @ C @ ( refine944483349t_unit @ R @ ( refine2043866374unit_a @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_221_conc__trans,axiom,
! [C: refine787176636t_unit,R: set_Pr451126599t_unit,B: refine787176636t_unit,R2: set_Pr451126599t_unit,A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ C @ ( refine944483349t_unit @ R @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ ( refine944483349t_unit @ R2 @ A ) )
=> ( ord_le1051254044t_unit @ C @ ( refine944483349t_unit @ R @ ( refine944483349t_unit @ R2 @ A ) ) ) ) ) ).
% conc_trans
thf(fact_222_le__nofailI,axiom,
! [M2: refine424419629nres_a,M: refine424419629nres_a] :
( ( ( refine412683989fail_a @ M2 )
=> ( ord_le519537037nres_a @ M @ M2 ) )
=> ( ord_le519537037nres_a @ M @ M2 ) ) ).
% le_nofailI
thf(fact_223_le__nofailI,axiom,
! [M2: refine787176636t_unit,M: refine787176636t_unit] :
( ( ( refine579265252t_unit @ M2 )
=> ( ord_le1051254044t_unit @ M @ M2 ) )
=> ( ord_le1051254044t_unit @ M @ M2 ) ) ).
% le_nofailI
thf(fact_224_top_Oextremum__unique,axiom,
! [A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A2 )
= ( A2 = top_to231829469nres_a ) ) ).
% top.extremum_unique
thf(fact_225_top_Oextremum__unique,axiom,
! [A2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ A2 )
= ( A2 = top_to177290092t_unit ) ) ).
% top.extremum_unique
thf(fact_226_pw__bind__leI,axiom,
! [S: refine424419629nres_a,M: refine424419629nres_a,F: a > refine424419629nres_a] :
( ( ( refine412683989fail_a @ S )
=> ( refine412683989fail_a @ M ) )
=> ( ! [X2: a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1001002027nres_a @ M @ X2 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ S ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_227_pw__bind__leI,axiom,
! [S: refine787176636t_unit,M: refine424419629nres_a,F: a > refine787176636t_unit] :
( ( ( refine579265252t_unit @ S )
=> ( refine412683989fail_a @ M ) )
=> ( ! [X2: a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1001002027nres_a @ M @ X2 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ S ) ) )
=> ( ord_le1051254044t_unit @ ( refine96995669t_unit @ M @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_228_pw__bind__leI,axiom,
! [S: refine424419629nres_a,M: refine787176636t_unit,F: product_unit > refine424419629nres_a] :
( ( ( refine412683989fail_a @ S )
=> ( refine579265252t_unit @ M ) )
=> ( ! [X2: product_unit] :
( ( refine579265252t_unit @ M )
=> ( ( refine558004794t_unit @ M @ X2 )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ S ) ) )
=> ( ord_le519537037nres_a @ ( refine119808503unit_a @ M @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_229_pw__bind__leI,axiom,
! [S: refine787176636t_unit,M: refine787176636t_unit,F: product_unit > refine787176636t_unit] :
( ( ( refine579265252t_unit @ S )
=> ( refine579265252t_unit @ M ) )
=> ( ! [X2: product_unit] :
( ( refine579265252t_unit @ M )
=> ( ( refine558004794t_unit @ M @ X2 )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ S ) ) )
=> ( ord_le1051254044t_unit @ ( refine681446406t_unit @ M @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_230_top_Oextremum__uniqueI,axiom,
! [A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A2 )
=> ( A2 = top_to231829469nres_a ) ) ).
% top.extremum_uniqueI
thf(fact_231_top_Oextremum__uniqueI,axiom,
! [A2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ top_to177290092t_unit @ A2 )
=> ( A2 = top_to177290092t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_232_pw__bind__le__iff,axiom,
! [M: refine424419629nres_a,F: a > refine424419629nres_a,S: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ S )
= ( ( ( refine412683989fail_a @ S )
=> ( refine412683989fail_a @ M ) )
& ! [X3: a] :
( ( ( refine412683989fail_a @ M )
& ( refine1001002027nres_a @ M @ X3 ) )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_233_pw__bind__le__iff,axiom,
! [M: refine424419629nres_a,F: a > refine787176636t_unit,S: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ ( refine96995669t_unit @ M @ F ) @ S )
= ( ( ( refine579265252t_unit @ S )
=> ( refine412683989fail_a @ M ) )
& ! [X3: a] :
( ( ( refine412683989fail_a @ M )
& ( refine1001002027nres_a @ M @ X3 ) )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_234_pw__bind__le__iff,axiom,
! [M: refine787176636t_unit,F: product_unit > refine424419629nres_a,S: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine119808503unit_a @ M @ F ) @ S )
= ( ( ( refine412683989fail_a @ S )
=> ( refine579265252t_unit @ M ) )
& ! [X3: product_unit] :
( ( ( refine579265252t_unit @ M )
& ( refine558004794t_unit @ M @ X3 ) )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_235_pw__bind__le__iff,axiom,
! [M: refine787176636t_unit,F: product_unit > refine787176636t_unit,S: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ ( refine681446406t_unit @ M @ F ) @ S )
= ( ( ( refine579265252t_unit @ S )
=> ( refine579265252t_unit @ M ) )
& ! [X3: product_unit] :
( ( ( refine579265252t_unit @ M )
& ( refine558004794t_unit @ M @ X3 ) )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_236_pw__conc__nofail,axiom,
! [R: set_Product_prod_a_a,S: refine424419629nres_a] :
( ( refine412683989fail_a @ ( refine1441824853un_a_a @ R @ S ) )
= ( refine412683989fail_a @ S ) ) ).
% pw_conc_nofail
thf(fact_237_pw__conc__nofail,axiom,
! [R: set_Pr1720557880unit_a,S: refine424419629nres_a] :
( ( refine579265252t_unit @ ( refine2043866374unit_a @ R @ S ) )
= ( refine412683989fail_a @ S ) ) ).
% pw_conc_nofail
thf(fact_238_pw__conc__nofail,axiom,
! [R: set_Pr1628433942t_unit,S: refine787176636t_unit] :
( ( refine412683989fail_a @ ( refine2021053540t_unit @ R @ S ) )
= ( refine579265252t_unit @ S ) ) ).
% pw_conc_nofail
thf(fact_239_pw__conc__nofail,axiom,
! [R: set_Pr451126599t_unit,S: refine787176636t_unit] :
( ( refine579265252t_unit @ ( refine944483349t_unit @ R @ S ) )
= ( refine579265252t_unit @ S ) ) ).
% pw_conc_nofail
thf(fact_240_not__nofail__inres,axiom,
! [S: refine424419629nres_a,X: a] :
( ~ ( refine412683989fail_a @ S )
=> ( refine1001002027nres_a @ S @ X ) ) ).
% not_nofail_inres
thf(fact_241_not__nofail__inres,axiom,
! [S: refine787176636t_unit,X: product_unit] :
( ~ ( refine579265252t_unit @ S )
=> ( refine558004794t_unit @ S @ X ) ) ).
% not_nofail_inres
thf(fact_242_bot_Oextremum__uniqueI,axiom,
! [A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ bot_bo529555393nres_a )
=> ( A2 = bot_bo529555393nres_a ) ) ).
% bot.extremum_uniqueI
thf(fact_243_bot_Oextremum__uniqueI,axiom,
! [A2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ bot_bo658782032t_unit )
=> ( A2 = bot_bo658782032t_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_244_bot_Oextremum__unique,axiom,
! [A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ bot_bo529555393nres_a )
= ( A2 = bot_bo529555393nres_a ) ) ).
% bot.extremum_unique
thf(fact_245_bot_Oextremum__unique,axiom,
! [A2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ bot_bo658782032t_unit )
= ( A2 = bot_bo658782032t_unit ) ) ).
% bot.extremum_unique
thf(fact_246_bot_Oextremum,axiom,
! [A2: refine424419629nres_a] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ A2 ) ).
% bot.extremum
thf(fact_247_bot_Oextremum,axiom,
! [A2: refine787176636t_unit] : ( ord_le1051254044t_unit @ bot_bo658782032t_unit @ A2 ) ).
% bot.extremum
thf(fact_248_pw__bind__nofail,axiom,
! [M: refine424419629nres_a,F: a > refine424419629nres_a] :
( ( refine412683989fail_a @ ( refine436832838nd_a_a @ M @ F ) )
= ( ( refine412683989fail_a @ M )
& ! [X3: a] :
( ( refine1001002027nres_a @ M @ X3 )
=> ( refine412683989fail_a @ ( F @ X3 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_249_pw__bind__nofail,axiom,
! [M: refine424419629nres_a,F: a > refine787176636t_unit] :
( ( refine579265252t_unit @ ( refine96995669t_unit @ M @ F ) )
= ( ( refine412683989fail_a @ M )
& ! [X3: a] :
( ( refine1001002027nres_a @ M @ X3 )
=> ( refine579265252t_unit @ ( F @ X3 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_250_pw__bind__nofail,axiom,
! [M: refine787176636t_unit,F: product_unit > refine424419629nres_a] :
( ( refine412683989fail_a @ ( refine119808503unit_a @ M @ F ) )
= ( ( refine579265252t_unit @ M )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ M @ X3 )
=> ( refine412683989fail_a @ ( F @ X3 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_251_pw__bind__nofail,axiom,
! [M: refine787176636t_unit,F: product_unit > refine787176636t_unit] :
( ( refine579265252t_unit @ ( refine681446406t_unit @ M @ F ) )
= ( ( refine579265252t_unit @ M )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ M @ X3 )
=> ( refine579265252t_unit @ ( F @ X3 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_252_bind__cong,axiom,
! [M3: refine787176636t_unit,M4: refine787176636t_unit,F: product_unit > refine424419629nres_a,F2: product_unit > refine424419629nres_a] :
( ( M3 = M4 )
=> ( ! [X2: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X2 ) @ M4 )
=> ( ( F @ X2 )
= ( F2 @ X2 ) ) )
=> ( ( refine119808503unit_a @ M3 @ F )
= ( refine119808503unit_a @ M4 @ F2 ) ) ) ) ).
% bind_cong
thf(fact_253_bind__cong,axiom,
! [M3: refine787176636t_unit,M4: refine787176636t_unit,F: product_unit > refine787176636t_unit,F2: product_unit > refine787176636t_unit] :
( ( M3 = M4 )
=> ( ! [X2: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X2 ) @ M4 )
=> ( ( F @ X2 )
= ( F2 @ X2 ) ) )
=> ( ( refine681446406t_unit @ M3 @ F )
= ( refine681446406t_unit @ M4 @ F2 ) ) ) ) ).
% bind_cong
thf(fact_254_bind__cong,axiom,
! [M3: refine424419629nres_a,M4: refine424419629nres_a,F: a > refine424419629nres_a,F2: a > refine424419629nres_a] :
( ( M3 = M4 )
=> ( ! [X2: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X2 ) @ M4 )
=> ( ( F @ X2 )
= ( F2 @ X2 ) ) )
=> ( ( refine436832838nd_a_a @ M3 @ F )
= ( refine436832838nd_a_a @ M4 @ F2 ) ) ) ) ).
% bind_cong
thf(fact_255_bind__cong,axiom,
! [M3: refine424419629nres_a,M4: refine424419629nres_a,F: a > refine787176636t_unit,F2: a > refine787176636t_unit] :
( ( M3 = M4 )
=> ( ! [X2: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X2 ) @ M4 )
=> ( ( F @ X2 )
= ( F2 @ X2 ) ) )
=> ( ( refine96995669t_unit @ M3 @ F )
= ( refine96995669t_unit @ M4 @ F2 ) ) ) ) ).
% bind_cong
thf(fact_256_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine787176636t_unit,M2: refine787176636t_unit,F: product_unit > refine424419629nres_a,F2: product_unit > refine424419629nres_a] :
( ( ord_le1051254044t_unit @ M @ M2 )
=> ( ! [X2: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X2 ) @ M )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ord_le519537037nres_a @ ( refine119808503unit_a @ M @ F ) @ ( refine119808503unit_a @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_257_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine787176636t_unit,M2: refine787176636t_unit,F: product_unit > refine787176636t_unit,F2: product_unit > refine787176636t_unit] :
( ( ord_le1051254044t_unit @ M @ M2 )
=> ( ! [X2: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X2 ) @ M )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ord_le1051254044t_unit @ ( refine681446406t_unit @ M @ F ) @ ( refine681446406t_unit @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_258_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine424419629nres_a,M2: refine424419629nres_a,F: a > refine424419629nres_a,F2: a > refine424419629nres_a] :
( ( ord_le519537037nres_a @ M @ M2 )
=> ( ! [X2: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X2 ) @ M )
=> ( ord_le519537037nres_a @ ( F @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M @ F ) @ ( refine436832838nd_a_a @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_259_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M: refine424419629nres_a,M2: refine424419629nres_a,F: a > refine787176636t_unit,F2: a > refine787176636t_unit] :
( ( ord_le519537037nres_a @ M @ M2 )
=> ( ! [X2: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X2 ) @ M )
=> ( ord_le1051254044t_unit @ ( F @ X2 ) @ ( F2 @ X2 ) ) )
=> ( ord_le1051254044t_unit @ ( refine96995669t_unit @ M @ F ) @ ( refine96995669t_unit @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_260_nres__inequalities_I2_J,axiom,
top_to177290092t_unit != bot_bo658782032t_unit ).
% nres_inequalities(2)
thf(fact_261_nres__inequalities_I2_J,axiom,
top_to231829469nres_a != bot_bo529555393nres_a ).
% nres_inequalities(2)
thf(fact_262_nres__inequalities_I4_J,axiom,
bot_bo658782032t_unit != top_to177290092t_unit ).
% nres_inequalities(4)
thf(fact_263_nres__inequalities_I4_J,axiom,
bot_bo529555393nres_a != top_to231829469nres_a ).
% nres_inequalities(4)
thf(fact_264_order__mono__setup_Orefl,axiom,
! [X: refine424419629nres_a] : ( ord_le519537037nres_a @ X @ X ) ).
% order_mono_setup.refl
thf(fact_265_order__mono__setup_Orefl,axiom,
! [X: refine787176636t_unit] : ( ord_le1051254044t_unit @ X @ X ) ).
% order_mono_setup.refl
thf(fact_266_nf__inres__def,axiom,
( refine1312857699nres_a
= ( ^ [M5: refine424419629nres_a,X3: a] :
( ( refine412683989fail_a @ M5 )
& ( refine1001002027nres_a @ M5 @ X3 ) ) ) ) ).
% nf_inres_def
thf(fact_267_nf__inres__def,axiom,
( refine983493746t_unit
= ( ^ [M5: refine787176636t_unit,X3: product_unit] :
( ( refine579265252t_unit @ M5 )
& ( refine558004794t_unit @ M5 @ X3 ) ) ) ) ).
% nf_inres_def
thf(fact_268_meta__le__everything__if__top,axiom,
! [M3: refine424419629nres_a,X: refine424419629nres_a] :
( ( M3 = top_to231829469nres_a )
=> ( ord_le519537037nres_a @ X @ M3 ) ) ).
% meta_le_everything_if_top
thf(fact_269_meta__le__everything__if__top,axiom,
! [M3: refine787176636t_unit,X: refine787176636t_unit] :
( ( M3 = top_to177290092t_unit )
=> ( ord_le1051254044t_unit @ X @ M3 ) ) ).
% meta_le_everything_if_top
thf(fact_270_pw__flat__le__iff,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( partia906949161nres_a @ bot_bo529555393nres_a @ S @ S2 )
= ( ? [X4: a] : ( refine1001002027nres_a @ S @ X4 )
=> ( ( ( refine412683989fail_a @ S )
= ( refine412683989fail_a @ S2 ) )
& ! [X3: a] :
( ( refine1001002027nres_a @ S @ X3 )
= ( refine1001002027nres_a @ S2 @ X3 ) ) ) ) ) ).
% pw_flat_le_iff
thf(fact_271_pw__flat__le__iff,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( partia1658438072t_unit @ bot_bo658782032t_unit @ S @ S2 )
= ( ? [X4: product_unit] : ( refine558004794t_unit @ S @ X4 )
=> ( ( ( refine579265252t_unit @ S )
= ( refine579265252t_unit @ S2 ) )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ S @ X3 )
= ( refine558004794t_unit @ S2 @ X3 ) ) ) ) ) ).
% pw_flat_le_iff
thf(fact_272_pw__abs__inres,axiom,
! [R: set_Product_prod_a_a,M: refine424419629nres_a,A2: a] :
( ( refine1001002027nres_a @ ( refine1136779702un_a_a @ R @ M ) @ A2 )
= ( ( refine412683989fail_a @ ( refine1136779702un_a_a @ R @ M ) )
=> ? [C3: a] :
( ( refine1001002027nres_a @ M @ C3 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ C3 @ A2 ) @ R ) ) ) ) ).
% pw_abs_inres
thf(fact_273_pw__abs__inres,axiom,
! [R: set_Pr1628433942t_unit,M: refine424419629nres_a,A2: product_unit] :
( ( refine558004794t_unit @ ( refine341651653t_unit @ R @ M ) @ A2 )
= ( ( refine579265252t_unit @ ( refine341651653t_unit @ R @ M ) )
=> ? [C3: a] :
( ( refine1001002027nres_a @ M @ C3 )
& ( member2095661023t_unit @ ( produc1776699686t_unit @ C3 @ A2 ) @ R ) ) ) ) ).
% pw_abs_inres
thf(fact_274_pw__abs__inres,axiom,
! [R: set_Pr1720557880unit_a,M: refine787176636t_unit,A2: a] :
( ( refine1001002027nres_a @ ( refine364464487unit_a @ R @ M ) @ A2 )
= ( ( refine412683989fail_a @ ( refine364464487unit_a @ R @ M ) )
=> ? [C3: product_unit] :
( ( refine558004794t_unit @ M @ C3 )
& ( member1211819009unit_a @ ( produc1799512520unit_a @ C3 @ A2 ) @ R ) ) ) ) ).
% pw_abs_inres
thf(fact_275_pw__abs__inres,axiom,
! [R: set_Pr451126599t_unit,M: refine787176636t_unit,A2: product_unit] :
( ( refine558004794t_unit @ ( refine838861686t_unit @ R @ M ) @ A2 )
= ( ( refine579265252t_unit @ ( refine838861686t_unit @ R @ M ) )
=> ? [C3: product_unit] :
( ( refine558004794t_unit @ M @ C3 )
& ( member1423014800t_unit @ ( produc1076565719t_unit @ C3 @ A2 ) @ R ) ) ) ) ).
% pw_abs_inres
thf(fact_276_pw__abs__nofail,axiom,
! [R: set_Product_prod_a_a,M: refine424419629nres_a] :
( ( refine412683989fail_a @ ( refine1136779702un_a_a @ R @ M ) )
= ( ( refine412683989fail_a @ M )
& ! [X3: a] :
( ( refine1001002027nres_a @ M @ X3 )
=> ( member_a @ X3 @ ( domain_a_a @ R ) ) ) ) ) ).
% pw_abs_nofail
thf(fact_277_pw__abs__nofail,axiom,
! [R: set_Pr1628433942t_unit,M: refine424419629nres_a] :
( ( refine579265252t_unit @ ( refine341651653t_unit @ R @ M ) )
= ( ( refine412683989fail_a @ M )
& ! [X3: a] :
( ( refine1001002027nres_a @ M @ X3 )
=> ( member_a @ X3 @ ( domain799550107t_unit @ R ) ) ) ) ) ).
% pw_abs_nofail
thf(fact_278_pw__abs__nofail,axiom,
! [R: set_Pr1720557880unit_a,M: refine787176636t_unit] :
( ( refine412683989fail_a @ ( refine364464487unit_a @ R @ M ) )
= ( ( refine579265252t_unit @ M )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ M @ X3 )
=> ( member_Product_unit @ X3 @ ( domain822362941unit_a @ R ) ) ) ) ) ).
% pw_abs_nofail
thf(fact_279_pw__abs__nofail,axiom,
! [R: set_Pr451126599t_unit,M: refine787176636t_unit] :
( ( refine579265252t_unit @ ( refine838861686t_unit @ R @ M ) )
= ( ( refine579265252t_unit @ M )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ M @ X3 )
=> ( member_Product_unit @ X3 @ ( domain2090798924t_unit @ R ) ) ) ) ) ).
% pw_abs_nofail
thf(fact_280_conc__abs__swap,axiom,
! [R: set_Product_prod_a_a,M4: refine424419629nres_a,M3: refine424419629nres_a] :
( ( single_valued_a_a @ R )
=> ( ( ord_le519537037nres_a @ M4 @ ( refine1441824853un_a_a @ R @ M3 ) )
= ( ord_le519537037nres_a @ ( refine1136779702un_a_a @ R @ M4 ) @ M3 ) ) ) ).
% conc_abs_swap
thf(fact_281_conc__abs__swap,axiom,
! [R: set_Pr1720557880unit_a,M4: refine787176636t_unit,M3: refine424419629nres_a] :
( ( single249782708unit_a @ R )
=> ( ( ord_le1051254044t_unit @ M4 @ ( refine2043866374unit_a @ R @ M3 ) )
= ( ord_le519537037nres_a @ ( refine364464487unit_a @ R @ M4 ) @ M3 ) ) ) ).
% conc_abs_swap
thf(fact_282_conc__abs__swap,axiom,
! [R: set_Pr1628433942t_unit,M4: refine424419629nres_a,M3: refine787176636t_unit] :
( ( single226969874t_unit @ R )
=> ( ( ord_le519537037nres_a @ M4 @ ( refine2021053540t_unit @ R @ M3 ) )
= ( ord_le1051254044t_unit @ ( refine341651653t_unit @ R @ M4 ) @ M3 ) ) ) ).
% conc_abs_swap
thf(fact_283_conc__abs__swap,axiom,
! [R: set_Pr451126599t_unit,M4: refine787176636t_unit,M3: refine787176636t_unit] :
( ( single330234563t_unit @ R )
=> ( ( ord_le1051254044t_unit @ M4 @ ( refine944483349t_unit @ R @ M3 ) )
= ( ord_le1051254044t_unit @ ( refine838861686t_unit @ R @ M4 ) @ M3 ) ) ) ).
% conc_abs_swap
thf(fact_284_pw__flat__ge__iff,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( partia906949161nres_a @ top_to231829469nres_a @ S @ S2 )
= ( ( refine412683989fail_a @ S )
=> ( ( refine412683989fail_a @ S2 )
& ! [X3: a] :
( ( refine1001002027nres_a @ S @ X3 )
= ( refine1001002027nres_a @ S2 @ X3 ) ) ) ) ) ).
% pw_flat_ge_iff
thf(fact_285_pw__flat__ge__iff,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ S @ S2 )
= ( ( refine579265252t_unit @ S )
=> ( ( refine579265252t_unit @ S2 )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ S @ X3 )
= ( refine558004794t_unit @ S2 @ X3 ) ) ) ) ) ).
% pw_flat_ge_iff
thf(fact_286_GreatestI2__order,axiom,
! [P: refine424419629nres_a > $o,X: refine424419629nres_a,Q: refine424419629nres_a > $o] :
( ( P @ X )
=> ( ! [Y3: refine424419629nres_a] :
( ( P @ Y3 )
=> ( ord_le519537037nres_a @ Y3 @ X ) )
=> ( ! [X2: refine424419629nres_a] :
( ( P @ X2 )
=> ( ! [Y5: refine424419629nres_a] :
( ( P @ Y5 )
=> ( ord_le519537037nres_a @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_1714329108nres_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_287_GreatestI2__order,axiom,
! [P: refine787176636t_unit > $o,X: refine787176636t_unit,Q: refine787176636t_unit > $o] :
( ( P @ X )
=> ( ! [Y3: refine787176636t_unit] :
( ( P @ Y3 )
=> ( ord_le1051254044t_unit @ Y3 @ X ) )
=> ( ! [X2: refine787176636t_unit] :
( ( P @ X2 )
=> ( ! [Y5: refine787176636t_unit] :
( ( P @ Y5 )
=> ( ord_le1051254044t_unit @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_453013155t_unit @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_288_Greatest__equality,axiom,
! [P: refine424419629nres_a > $o,X: refine424419629nres_a] :
( ( P @ X )
=> ( ! [Y3: refine424419629nres_a] :
( ( P @ Y3 )
=> ( ord_le519537037nres_a @ Y3 @ X ) )
=> ( ( order_1714329108nres_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_289_Greatest__equality,axiom,
! [P: refine787176636t_unit > $o,X: refine787176636t_unit] :
( ( P @ X )
=> ( ! [Y3: refine787176636t_unit] :
( ( P @ Y3 )
=> ( ord_le1051254044t_unit @ Y3 @ X ) )
=> ( ( order_453013155t_unit @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_290_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: refine424419629nres_a,M12: refine424419629nres_a,M22: refine424419629nres_a,M23: refine424419629nres_a] :
( ( B2
=> ( ord_le519537037nres_a @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le519537037nres_a @ M22 @ M23 ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B2 @ M1 @ M22 ) @ ( if_Ref1724547303nres_a @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_291_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: refine787176636t_unit,M12: refine787176636t_unit,M22: refine787176636t_unit,M23: refine787176636t_unit] :
( ( B2
=> ( ord_le1051254044t_unit @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le1051254044t_unit @ M22 @ M23 ) )
=> ( ord_le1051254044t_unit @ ( if_Ref1369692790t_unit @ B2 @ M1 @ M22 ) @ ( if_Ref1369692790t_unit @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_292_flat__ord__compat_I1_J,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a] :
( ( partia906949161nres_a @ bot_bo529555393nres_a @ X @ Y )
=> ( ord_le519537037nres_a @ X @ Y ) ) ).
% flat_ord_compat(1)
thf(fact_293_flat__ord__compat_I1_J,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit] :
( ( partia1658438072t_unit @ bot_bo658782032t_unit @ X @ Y )
=> ( ord_le1051254044t_unit @ X @ Y ) ) ).
% flat_ord_compat(1)
thf(fact_294_flat__ord__compat_I2_J,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a] :
( ( partia906949161nres_a @ top_to231829469nres_a @ X @ Y )
=> ( ord_le519537037nres_a @ Y @ X ) ) ).
% flat_ord_compat(2)
thf(fact_295_flat__ord__compat_I2_J,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ X @ Y )
=> ( ord_le1051254044t_unit @ Y @ X ) ) ).
% flat_ord_compat(2)
thf(fact_296_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I2_J,axiom,
! [M: refine787176636t_unit,M2: refine787176636t_unit,F: product_unit > refine424419629nres_a,F2: product_unit > refine424419629nres_a] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ M @ M2 )
=> ( ! [X2: product_unit] : ( partia906949161nres_a @ top_to231829469nres_a @ ( F @ X2 ) @ ( F2 @ X2 ) )
=> ( partia906949161nres_a @ top_to231829469nres_a @ ( refine119808503unit_a @ M @ F ) @ ( refine119808503unit_a @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(2)
thf(fact_297_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I2_J,axiom,
! [M: refine424419629nres_a,M2: refine424419629nres_a,F: a > refine424419629nres_a,F2: a > refine424419629nres_a] :
( ( partia906949161nres_a @ top_to231829469nres_a @ M @ M2 )
=> ( ! [X2: a] : ( partia906949161nres_a @ top_to231829469nres_a @ ( F @ X2 ) @ ( F2 @ X2 ) )
=> ( partia906949161nres_a @ top_to231829469nres_a @ ( refine436832838nd_a_a @ M @ F ) @ ( refine436832838nd_a_a @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(2)
thf(fact_298_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I2_J,axiom,
! [M: refine424419629nres_a,M2: refine424419629nres_a,F: a > refine787176636t_unit,F2: a > refine787176636t_unit] :
( ( partia906949161nres_a @ top_to231829469nres_a @ M @ M2 )
=> ( ! [X2: a] : ( partia1658438072t_unit @ top_to177290092t_unit @ ( F @ X2 ) @ ( F2 @ X2 ) )
=> ( partia1658438072t_unit @ top_to177290092t_unit @ ( refine96995669t_unit @ M @ F ) @ ( refine96995669t_unit @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(2)
thf(fact_299_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I2_J,axiom,
! [M: refine787176636t_unit,M2: refine787176636t_unit,F: product_unit > refine787176636t_unit,F2: product_unit > refine787176636t_unit] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ M @ M2 )
=> ( ! [X2: product_unit] : ( partia1658438072t_unit @ top_to177290092t_unit @ ( F @ X2 ) @ ( F2 @ X2 ) )
=> ( partia1658438072t_unit @ top_to177290092t_unit @ ( refine681446406t_unit @ M @ F ) @ ( refine681446406t_unit @ M2 @ F2 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(2)
thf(fact_300_conc__fun__R__mono,axiom,
! [R: set_Product_prod_a_a,R2: set_Product_prod_a_a,M: refine424419629nres_a] :
( ( ord_le1824328871od_a_a @ R @ R2 )
=> ( ord_le519537037nres_a @ ( refine1441824853un_a_a @ R @ M ) @ ( refine1441824853un_a_a @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_301_conc__fun__R__mono,axiom,
! [R: set_Pr1720557880unit_a,R2: set_Pr1720557880unit_a,M: refine424419629nres_a] :
( ( ord_le2070001880unit_a @ R @ R2 )
=> ( ord_le1051254044t_unit @ ( refine2043866374unit_a @ R @ M ) @ ( refine2043866374unit_a @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_302_conc__fun__R__mono,axiom,
! [R: set_Pr1628433942t_unit,R2: set_Pr1628433942t_unit,M: refine787176636t_unit] :
( ( ord_le1977877942t_unit @ R @ R2 )
=> ( ord_le519537037nres_a @ ( refine2021053540t_unit @ R @ M ) @ ( refine2021053540t_unit @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_303_conc__fun__R__mono,axiom,
! [R: set_Pr451126599t_unit,R2: set_Pr451126599t_unit,M: refine787176636t_unit] :
( ( ord_le2035129575t_unit @ R @ R2 )
=> ( ord_le1051254044t_unit @ ( refine944483349t_unit @ R @ M ) @ ( refine944483349t_unit @ R2 @ M ) ) ) ).
% conc_fun_R_mono
thf(fact_304_order__mono__setup_Omono__if,axiom,
! [T: refine424419629nres_a,T2: refine424419629nres_a,E: refine424419629nres_a,E2: refine424419629nres_a,B2: $o] :
( ( ord_le519537037nres_a @ T @ T2 )
=> ( ( ord_le519537037nres_a @ E @ E2 )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B2 @ T @ E ) @ ( if_Ref1724547303nres_a @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_305_order__mono__setup_Omono__if,axiom,
! [T: refine787176636t_unit,T2: refine787176636t_unit,E: refine787176636t_unit,E2: refine787176636t_unit,B2: $o] :
( ( ord_le1051254044t_unit @ T @ T2 )
=> ( ( ord_le1051254044t_unit @ E @ E2 )
=> ( ord_le1051254044t_unit @ ( if_Ref1369692790t_unit @ B2 @ T @ E ) @ ( if_Ref1369692790t_unit @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_306_ac__galois,axiom,
! [R: set_Product_prod_a_a] :
( ( single_valued_a_a @ R )
=> ( refine2004812827nres_a @ ( refine1136779702un_a_a @ R ) @ ( refine1441824853un_a_a @ R ) ) ) ).
% ac_galois
thf(fact_307_ac__galois,axiom,
! [R: set_Pr1720557880unit_a] :
( ( single249782708unit_a @ R )
=> ( refine2089046860nres_a @ ( refine364464487unit_a @ R ) @ ( refine2043866374unit_a @ R ) ) ) ).
% ac_galois
thf(fact_308_ac__galois,axiom,
! [R: set_Pr1628433942t_unit] :
( ( single226969874t_unit @ R )
=> ( refine327276970t_unit @ ( refine341651653t_unit @ R ) @ ( refine2021053540t_unit @ R ) ) ) ).
% ac_galois
thf(fact_309_ac__galois,axiom,
! [R: set_Pr451126599t_unit] :
( ( single330234563t_unit @ R )
=> ( refine230495195t_unit @ ( refine838861686t_unit @ R ) @ ( refine944483349t_unit @ R ) ) ) ).
% ac_galois
thf(fact_310_abs__fun__simps_I3_J,axiom,
! [X5: set_a,R: set_Product_prod_a_a] :
( ~ ( ord_less_eq_set_a @ X5 @ ( domain_a_a @ R ) )
=> ( ( refine1136779702un_a_a @ R @ ( refine1198353288_RES_a @ X5 ) )
= top_to231829469nres_a ) ) ).
% abs_fun_simps(3)
thf(fact_311_abs__fun__simps_I3_J,axiom,
! [X5: set_a,R: set_Pr1628433942t_unit] :
( ~ ( ord_less_eq_set_a @ X5 @ ( domain799550107t_unit @ R ) )
=> ( ( refine341651653t_unit @ R @ ( refine1198353288_RES_a @ X5 ) )
= top_to177290092t_unit ) ) ).
% abs_fun_simps(3)
thf(fact_312_abs__fun__simps_I3_J,axiom,
! [X5: set_Product_unit,R: set_Pr1720557880unit_a] :
( ~ ( ord_le1023748749t_unit @ X5 @ ( domain822362941unit_a @ R ) )
=> ( ( refine364464487unit_a @ R @ ( refine1777164439t_unit @ X5 ) )
= top_to231829469nres_a ) ) ).
% abs_fun_simps(3)
thf(fact_313_abs__fun__simps_I3_J,axiom,
! [X5: set_Product_unit,R: set_Pr451126599t_unit] :
( ~ ( ord_le1023748749t_unit @ X5 @ ( domain2090798924t_unit @ R ) )
=> ( ( refine838861686t_unit @ R @ ( refine1777164439t_unit @ X5 ) )
= top_to177290092t_unit ) ) ).
% abs_fun_simps(3)
thf(fact_314_inres__simps_I2_J,axiom,
! [X5: set_a] :
( ( refine1001002027nres_a @ ( refine1198353288_RES_a @ X5 ) )
= ( ^ [X3: a] : ( member_a @ X3 @ X5 ) ) ) ).
% inres_simps(2)
thf(fact_315_inres__simps_I2_J,axiom,
! [X5: set_Product_unit] :
( ( refine558004794t_unit @ ( refine1777164439t_unit @ X5 ) )
= ( ^ [X3: product_unit] : ( member_Product_unit @ X3 @ X5 ) ) ) ).
% inres_simps(2)
thf(fact_316_nres__more__simps_I2_J,axiom,
! [X5: set_Product_unit] :
( ( ( refine1777164439t_unit @ X5 )
= bot_bo658782032t_unit )
= ( X5 = bot_bo1087887705t_unit ) ) ).
% nres_more_simps(2)
thf(fact_317_nres__more__simps_I2_J,axiom,
! [X5: set_a] :
( ( ( refine1198353288_RES_a @ X5 )
= bot_bo529555393nres_a )
= ( X5 = bot_bot_set_a ) ) ).
% nres_more_simps(2)
thf(fact_318_nres__more__simps_I1_J,axiom,
! [X5: set_Product_unit] :
( ( bot_bo658782032t_unit
= ( refine1777164439t_unit @ X5 ) )
= ( X5 = bot_bo1087887705t_unit ) ) ).
% nres_more_simps(1)
thf(fact_319_nres__more__simps_I1_J,axiom,
! [X5: set_a] :
( ( bot_bo529555393nres_a
= ( refine1198353288_RES_a @ X5 ) )
= ( X5 = bot_bot_set_a ) ) ).
% nres_more_simps(1)
thf(fact_320_nres__order__simps_I21_J,axiom,
! [X: product_unit,Y6: set_Product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X ) @ ( refine1777164439t_unit @ Y6 ) )
= ( member_Product_unit @ X @ Y6 ) ) ).
% nres_order_simps(21)
thf(fact_321_nres__order__simps_I21_J,axiom,
! [X: a,Y6: set_a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X ) @ ( refine1198353288_RES_a @ Y6 ) )
= ( member_a @ X @ Y6 ) ) ).
% nres_order_simps(21)
thf(fact_322_galois__connection__def,axiom,
( refine2004812827nres_a
= ( ^ [Alpha: refine424419629nres_a > refine424419629nres_a,Gamma: refine424419629nres_a > refine424419629nres_a] :
! [C3: refine424419629nres_a,A3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ C3 @ ( Gamma @ A3 ) )
= ( ord_le519537037nres_a @ ( Alpha @ C3 ) @ A3 ) ) ) ) ).
% galois_connection_def
thf(fact_323_galois__connection__def,axiom,
( refine327276970t_unit
= ( ^ [Alpha: refine424419629nres_a > refine787176636t_unit,Gamma: refine787176636t_unit > refine424419629nres_a] :
! [C3: refine424419629nres_a,A3: refine787176636t_unit] :
( ( ord_le519537037nres_a @ C3 @ ( Gamma @ A3 ) )
= ( ord_le1051254044t_unit @ ( Alpha @ C3 ) @ A3 ) ) ) ) ).
% galois_connection_def
thf(fact_324_galois__connection__def,axiom,
( refine2089046860nres_a
= ( ^ [Alpha: refine787176636t_unit > refine424419629nres_a,Gamma: refine424419629nres_a > refine787176636t_unit] :
! [C3: refine787176636t_unit,A3: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ C3 @ ( Gamma @ A3 ) )
= ( ord_le519537037nres_a @ ( Alpha @ C3 ) @ A3 ) ) ) ) ).
% galois_connection_def
thf(fact_325_galois__connection__def,axiom,
( refine230495195t_unit
= ( ^ [Alpha: refine787176636t_unit > refine787176636t_unit,Gamma: refine787176636t_unit > refine787176636t_unit] :
! [C3: refine787176636t_unit,A3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ C3 @ ( Gamma @ A3 ) )
= ( ord_le1051254044t_unit @ ( Alpha @ C3 ) @ A3 ) ) ) ) ).
% galois_connection_def
thf(fact_326_galois__connection_Ointro,axiom,
! [Gamma2: refine424419629nres_a > refine424419629nres_a,Alpha2: refine424419629nres_a > refine424419629nres_a] :
( ! [C4: refine424419629nres_a,A4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le519537037nres_a @ ( Alpha2 @ C4 ) @ A4 ) )
=> ( refine2004812827nres_a @ Alpha2 @ Gamma2 ) ) ).
% galois_connection.intro
thf(fact_327_galois__connection_Ointro,axiom,
! [Gamma2: refine787176636t_unit > refine424419629nres_a,Alpha2: refine424419629nres_a > refine787176636t_unit] :
( ! [C4: refine424419629nres_a,A4: refine787176636t_unit] :
( ( ord_le519537037nres_a @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le1051254044t_unit @ ( Alpha2 @ C4 ) @ A4 ) )
=> ( refine327276970t_unit @ Alpha2 @ Gamma2 ) ) ).
% galois_connection.intro
thf(fact_328_galois__connection_Ointro,axiom,
! [Gamma2: refine424419629nres_a > refine787176636t_unit,Alpha2: refine787176636t_unit > refine424419629nres_a] :
( ! [C4: refine787176636t_unit,A4: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le519537037nres_a @ ( Alpha2 @ C4 ) @ A4 ) )
=> ( refine2089046860nres_a @ Alpha2 @ Gamma2 ) ) ).
% galois_connection.intro
thf(fact_329_galois__connection_Ointro,axiom,
! [Gamma2: refine787176636t_unit > refine787176636t_unit,Alpha2: refine787176636t_unit > refine787176636t_unit] :
( ! [C4: refine787176636t_unit,A4: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ C4 @ ( Gamma2 @ A4 ) )
= ( ord_le1051254044t_unit @ ( Alpha2 @ C4 ) @ A4 ) )
=> ( refine230495195t_unit @ Alpha2 @ Gamma2 ) ) ).
% galois_connection.intro
thf(fact_330_galois__connection_Ogalois,axiom,
! [Alpha2: refine424419629nres_a > refine424419629nres_a,Gamma2: refine424419629nres_a > refine424419629nres_a,C2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( refine2004812827nres_a @ Alpha2 @ Gamma2 )
=> ( ( ord_le519537037nres_a @ C2 @ ( Gamma2 @ A2 ) )
= ( ord_le519537037nres_a @ ( Alpha2 @ C2 ) @ A2 ) ) ) ).
% galois_connection.galois
thf(fact_331_galois__connection_Ogalois,axiom,
! [Alpha2: refine424419629nres_a > refine787176636t_unit,Gamma2: refine787176636t_unit > refine424419629nres_a,C2: refine424419629nres_a,A2: refine787176636t_unit] :
( ( refine327276970t_unit @ Alpha2 @ Gamma2 )
=> ( ( ord_le519537037nres_a @ C2 @ ( Gamma2 @ A2 ) )
= ( ord_le1051254044t_unit @ ( Alpha2 @ C2 ) @ A2 ) ) ) ).
% galois_connection.galois
thf(fact_332_galois__connection_Ogalois,axiom,
! [Alpha2: refine787176636t_unit > refine424419629nres_a,Gamma2: refine424419629nres_a > refine787176636t_unit,C2: refine787176636t_unit,A2: refine424419629nres_a] :
( ( refine2089046860nres_a @ Alpha2 @ Gamma2 )
=> ( ( ord_le1051254044t_unit @ C2 @ ( Gamma2 @ A2 ) )
= ( ord_le519537037nres_a @ ( Alpha2 @ C2 ) @ A2 ) ) ) ).
% galois_connection.galois
thf(fact_333_galois__connection_Ogalois,axiom,
! [Alpha2: refine787176636t_unit > refine787176636t_unit,Gamma2: refine787176636t_unit > refine787176636t_unit,C2: refine787176636t_unit,A2: refine787176636t_unit] :
( ( refine230495195t_unit @ Alpha2 @ Gamma2 )
=> ( ( ord_le1051254044t_unit @ C2 @ ( Gamma2 @ A2 ) )
= ( ord_le1051254044t_unit @ ( Alpha2 @ C2 ) @ A2 ) ) ) ).
% galois_connection.galois
thf(fact_334_nres__order__simps_I5_J,axiom,
! [X5: set_a,Y6: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ X5 ) @ ( refine1198353288_RES_a @ Y6 ) )
= ( ord_less_eq_set_a @ X5 @ Y6 ) ) ).
% nres_order_simps(5)
thf(fact_335_nres__order__simps_I5_J,axiom,
! [X5: set_Product_unit,Y6: set_Product_unit] :
( ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ X5 ) @ ( refine1777164439t_unit @ Y6 ) )
= ( ord_le1023748749t_unit @ X5 @ Y6 ) ) ).
% nres_order_simps(5)
thf(fact_336_less__eq__nres_Osimps_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ A2 ) @ ( refine1198353288_RES_a @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% less_eq_nres.simps(2)
thf(fact_337_less__eq__nres_Osimps_I2_J,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ A2 ) @ ( refine1777164439t_unit @ B2 ) )
= ( ord_le1023748749t_unit @ A2 @ B2 ) ) ).
% less_eq_nres.simps(2)
thf(fact_338_bot__nres__def,axiom,
( bot_bo658782032t_unit
= ( refine1777164439t_unit @ bot_bo1087887705t_unit ) ) ).
% bot_nres_def
thf(fact_339_bot__nres__def,axiom,
( bot_bo529555393nres_a
= ( refine1198353288_RES_a @ bot_bot_set_a ) ) ).
% bot_nres_def
thf(fact_340_SPEC__cons__rule,axiom,
! [M3: refine424419629nres_a,Phi: a > $o,Psi: a > $o] :
( ( ord_le519537037nres_a @ M3 @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
=> ( ! [X2: a] :
( ( Phi @ X2 )
=> ( Psi @ X2 ) )
=> ( ord_le519537037nres_a @ M3 @ ( refine1198353288_RES_a @ ( collect_a @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_341_SPEC__cons__rule,axiom,
! [M3: refine787176636t_unit,Phi: product_unit > $o,Psi: product_unit > $o] :
( ( ord_le1051254044t_unit @ M3 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) )
=> ( ! [X2: product_unit] :
( ( Phi @ X2 )
=> ( Psi @ X2 ) )
=> ( ord_le1051254044t_unit @ M3 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_342_SPEC__trans,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a,Postcond: a > $o] :
( ( ord_le519537037nres_a @ X @ Y )
=> ( ( ord_le519537037nres_a @ Y @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) )
=> ( ord_le519537037nres_a @ X @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_343_SPEC__trans,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit,Postcond: product_unit > $o] :
( ( ord_le1051254044t_unit @ X @ Y )
=> ( ( ord_le1051254044t_unit @ Y @ ( refine1777164439t_unit @ ( collect_Product_unit @ Postcond ) ) )
=> ( ord_le1051254044t_unit @ X @ ( refine1777164439t_unit @ ( collect_Product_unit @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_344_nres__order__simps_I22_J,axiom,
! [X5: set_Product_unit,Y: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ X5 ) @ ( refine1420258419t_unit @ Y ) )
= ( ord_le1023748749t_unit @ X5 @ ( insert_Product_unit @ Y @ bot_bo1087887705t_unit ) ) ) ).
% nres_order_simps(22)
thf(fact_345_nres__order__simps_I22_J,axiom,
! [X5: set_a,Y: a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ X5 ) @ ( refine2063221604TURN_a @ Y ) )
= ( ord_less_eq_set_a @ X5 @ ( insert_a @ Y @ bot_bot_set_a ) ) ) ).
% nres_order_simps(22)
thf(fact_346_nres__more__simps_I3_J,axiom,
! [X5: set_Product_unit,X: product_unit] :
( ( ( refine1777164439t_unit @ X5 )
= ( refine1420258419t_unit @ X ) )
= ( X5
= ( insert_Product_unit @ X @ bot_bo1087887705t_unit ) ) ) ).
% nres_more_simps(3)
thf(fact_347_nres__more__simps_I3_J,axiom,
! [X5: set_a,X: a] :
( ( ( refine1198353288_RES_a @ X5 )
= ( refine2063221604TURN_a @ X ) )
= ( X5
= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% nres_more_simps(3)
thf(fact_348_nres__more__simps_I5_J,axiom,
! [X: product_unit,X5: set_Product_unit] :
( ( ( refine1420258419t_unit @ X )
= ( refine1777164439t_unit @ X5 ) )
= ( ( insert_Product_unit @ X @ bot_bo1087887705t_unit )
= X5 ) ) ).
% nres_more_simps(5)
thf(fact_349_nres__more__simps_I5_J,axiom,
! [X: a,X5: set_a] :
( ( ( refine2063221604TURN_a @ X )
= ( refine1198353288_RES_a @ X5 ) )
= ( ( insert_a @ X @ bot_bot_set_a )
= X5 ) ) ).
% nres_more_simps(5)
thf(fact_350_RETURN__def,axiom,
( refine1420258419t_unit
= ( ^ [X3: product_unit] : ( refine1777164439t_unit @ ( insert_Product_unit @ X3 @ bot_bo1087887705t_unit ) ) ) ) ).
% RETURN_def
thf(fact_351_RETURN__def,axiom,
( refine2063221604TURN_a
= ( ^ [X3: a] : ( refine1198353288_RES_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% RETURN_def
% Helper facts (5)
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X: refine424419629nres_a,Y: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_T,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit] :
( ( if_Ref1369692790t_unit @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_T,axiom,
! [X: refine787176636t_unit,Y: refine787176636t_unit] :
( ( if_Ref1369692790t_unit @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( refine119808503unit_a @ bot_bo658782032t_unit @ f )
= bot_bo529555393nres_a ) ).
%------------------------------------------------------------------------------