TPTP Problem File: ITP165^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP165^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_892__3593680_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Refine_Basic/prob_892__3593680_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 424 ( 168 unt; 62 typ; 0 def)
% Number of atoms : 1137 ( 342 equ; 0 cnn)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 2725 ( 62 ~; 2 |; 45 &;2081 @)
% ( 0 <=>; 535 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 328 ( 328 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 56 usr; 13 con; 0-4 aty)
% Number of variables : 1034 ( 110 ^; 917 !; 7 ?;1034 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:34:27.762
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
refine787176636t_unit: $tType ).
thf(ty_n_t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
refine424419629nres_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (56)
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
if_o_R1705037948nres_a: $o > ( $o > refine424419629nres_a ) > ( $o > refine424419629nres_a ) > $o > refine424419629nres_a ).
thf(sy_c_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
if_Ref1369692790t_unit: $o > refine787176636t_unit > refine787176636t_unit > refine787176636t_unit ).
thf(sy_c_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
if_Ref1724547303nres_a: $o > refine424419629nres_a > refine424419629nres_a > refine424419629nres_a ).
thf(sy_c_If_001t__Set__Oset_Itf__a_J,type,
if_set_a: $o > set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
bot_bo658782032t_unit: refine787176636t_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
bot_bo529555393nres_a: refine424419629nres_a ).
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,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_J,type,
ord_le1960259811nres_a: ( $o > $o > refine424419629nres_a ) > ( $o > $o > refine424419629nres_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_J,type,
ord_le1633311077t_unit: ( $o > refine787176636t_unit ) > ( $o > refine787176636t_unit ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
ord_le516598614nres_a: ( $o > refine424419629nres_a ) > ( $o > refine424419629nres_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_Itf__a_J_J,type,
ord_less_eq_o_set_a: ( $o > set_a ) > ( $o > set_a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
ord_le1051254044t_unit: refine787176636t_unit > refine787176636t_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
ord_le519537037nres_a: refine424419629nres_a > refine424419629nres_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_le1023748749t_unit: set_Product_unit > set_Product_unit > $o ).
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_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
order_245837199nres_a: ( ( $o > refine424419629nres_a ) > $o ) > $o > refine424419629nres_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
order_453013155t_unit: ( refine787176636t_unit > $o ) > refine787176636t_unit ).
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_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_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_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_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_Refine__Basic__Mirabelle__kwjuvthmas_OASSUME,type,
refine23983189ASSUME: $o > refine787176636t_unit ).
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,
refine2063221604TURN_a: a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Obind_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
refine681446406t_unit: refine787176636t_unit > ( product_unit > refine787176636t_unit ) > refine787176636t_unit ).
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_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_OFAILi_001tf__a,type,
refine464223677AILi_a: refine424419629nres_a ).
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__Basic__Mirabelle__kwjuvthmas_Othe__RES_001t__Product____Type__Ounit,type,
refine1208325748t_unit: refine787176636t_unit > set_Product_unit ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Othe__RES_001tf__a,type,
refine1822134885_RES_a: refine424419629nres_a > set_a ).
thf(sy_c_Refine__Misc_Ostrict_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J,type,
refine459374679t_unit: ( refine787176636t_unit > refine787176636t_unit ) > $o ).
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_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_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_S1,type,
s1: refine424419629nres_a ).
thf(sy_v_S2,type,
s2: refine424419629nres_a ).
thf(sy_v__092_060Phi_062,type,
phi: a > $o ).
thf(sy_v_b,type,
b: $o ).
% Relevant facts (350)
thf(fact_0_nres__more__simps_I4_J,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( refine1777164439t_unit @ X )
= ( refine1777164439t_unit @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(4)
thf(fact_1_nres__more__simps_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( ( refine1198353288_RES_a @ X )
= ( refine1198353288_RES_a @ Y ) )
= ( X = Y ) ) ).
% nres_more_simps(4)
thf(fact_2_nres_Oinject,axiom,
! [X2: set_Product_unit,Y2: set_Product_unit] :
( ( ( refine1777164439t_unit @ X2 )
= ( refine1777164439t_unit @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_3_nres_Oinject,axiom,
! [X2: set_a,Y2: set_a] :
( ( ( refine1198353288_RES_a @ X2 )
= ( refine1198353288_RES_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_4_RES__rule,axiom,
! [S: set_Product_unit,Phi: product_unit > $o] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ S ) @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) ) ) ).
% RES_rule
thf(fact_5_RES__rule,axiom,
! [S: set_a,Phi: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ( Phi @ X3 ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ S ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ).
% RES_rule
thf(fact_6_SPEC__rule,axiom,
! [Phi: product_unit > $o,Phi2: product_unit > $o] :
( ! [X3: product_unit] :
( ( Phi @ X3 )
=> ( Phi2 @ X3 ) )
=> ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_7_SPEC__rule,axiom,
! [Phi: a > $o,Phi2: a > $o] :
( ! [X3: a] :
( ( Phi @ X3 )
=> ( Phi2 @ X3 ) )
=> ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi2 ) ) ) ) ).
% SPEC_rule
thf(fact_8_SPEC__trans,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit,Postcond: product_unit > $o] :
( ( ord_le1051254044t_unit @ X4 @ Y3 )
=> ( ( ord_le1051254044t_unit @ Y3 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Postcond ) ) )
=> ( ord_le1051254044t_unit @ X4 @ ( refine1777164439t_unit @ ( collect_Product_unit @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_9_SPEC__trans,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Postcond: a > $o] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) )
=> ( ord_le519537037nres_a @ X4 @ ( refine1198353288_RES_a @ ( collect_a @ Postcond ) ) ) ) ) ).
% SPEC_trans
thf(fact_10_SPEC__cons__rule,axiom,
! [M: refine787176636t_unit,Phi: product_unit > $o,Psi: product_unit > $o] :
( ( ord_le1051254044t_unit @ M @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) )
=> ( ! [X3: product_unit] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_le1051254044t_unit @ M @ ( refine1777164439t_unit @ ( collect_Product_unit @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_11_SPEC__cons__rule,axiom,
! [M: refine424419629nres_a,Phi: a > $o,Psi: a > $o] :
( ( ord_le519537037nres_a @ M @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) )
=> ( ! [X3: a] :
( ( Phi @ X3 )
=> ( Psi @ X3 ) )
=> ( ord_le519537037nres_a @ M @ ( refine1198353288_RES_a @ ( collect_a @ Psi ) ) ) ) ) ).
% SPEC_cons_rule
thf(fact_12_nf__inres__RES,axiom,
! [X: set_Product_unit,X4: product_unit] :
( ( refine983493746t_unit @ ( refine1777164439t_unit @ X ) @ X4 )
= ( member_Product_unit @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_13_nf__inres__RES,axiom,
! [X: set_a,X4: a] :
( ( refine1312857699nres_a @ ( refine1198353288_RES_a @ X ) @ X4 )
= ( member_a @ X4 @ X ) ) ).
% nf_inres_RES
thf(fact_14_nf__inres__SPEC,axiom,
! [Phi: product_unit > $o,X4: product_unit] :
( ( refine983493746t_unit @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) @ X4 )
= ( Phi @ X4 ) ) ).
% nf_inres_SPEC
thf(fact_15_nf__inres__SPEC,axiom,
! [Phi: a > $o,X4: a] :
( ( refine1312857699nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) @ X4 )
= ( Phi @ X4 ) ) ).
% nf_inres_SPEC
thf(fact_16_order__mono__setup_Orefl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_17_order__mono__setup_Orefl,axiom,
! [X4: $o > refine424419629nres_a] : ( ord_le516598614nres_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_18_order__mono__setup_Orefl,axiom,
! [X4: refine787176636t_unit] : ( ord_le1051254044t_unit @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_19_order__mono__setup_Orefl,axiom,
! [X4: refine424419629nres_a] : ( ord_le519537037nres_a @ X4 @ X4 ) ).
% order_mono_setup.refl
thf(fact_20_the__RES_Osimps,axiom,
! [X: set_Product_unit] :
( ( refine1208325748t_unit @ ( refine1777164439t_unit @ X ) )
= X ) ).
% the_RES.simps
thf(fact_21_the__RES_Osimps,axiom,
! [X: set_a] :
( ( refine1822134885_RES_a @ ( refine1198353288_RES_a @ X ) )
= X ) ).
% the_RES.simps
thf(fact_22_nres__order__simps_I21_J,axiom,
! [X4: product_unit,Y: set_Product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X4 ) @ ( refine1777164439t_unit @ Y ) )
= ( member_Product_unit @ X4 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_23_nres__order__simps_I21_J,axiom,
! [X4: a,Y: set_a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1198353288_RES_a @ Y ) )
= ( member_a @ X4 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_24_le__funD,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a,X4: $o] :
( ( ord_le516598614nres_a @ F @ G )
=> ( ord_le519537037nres_a @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funD
thf(fact_25_le__funE,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a,X4: $o] :
( ( ord_le516598614nres_a @ F @ G )
=> ( ord_le519537037nres_a @ ( F @ X4 ) @ ( G @ X4 ) ) ) ).
% le_funE
thf(fact_26_le__funI,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a] :
( ! [X3: $o] : ( ord_le519537037nres_a @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le516598614nres_a @ F @ G ) ) ).
% le_funI
thf(fact_27_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: set_a,M12: set_a,M2: set_a,M22: set_a] :
( ( B
=> ( ord_less_eq_set_a @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_less_eq_set_a @ M2 @ M22 ) )
=> ( ord_less_eq_set_a @ ( if_set_a @ B @ M1 @ M2 ) @ ( if_set_a @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_28_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: $o > refine424419629nres_a,M12: $o > refine424419629nres_a,M2: $o > refine424419629nres_a,M22: $o > refine424419629nres_a] :
( ( B
=> ( ord_le516598614nres_a @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_le516598614nres_a @ M2 @ M22 ) )
=> ( ord_le516598614nres_a @ ( if_o_R1705037948nres_a @ B @ M1 @ M2 ) @ ( if_o_R1705037948nres_a @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_29_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: refine787176636t_unit,M12: refine787176636t_unit,M2: refine787176636t_unit,M22: refine787176636t_unit] :
( ( B
=> ( ord_le1051254044t_unit @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_le1051254044t_unit @ M2 @ M22 ) )
=> ( ord_le1051254044t_unit @ ( if_Ref1369692790t_unit @ B @ M1 @ M2 ) @ ( if_Ref1369692790t_unit @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_30_Refine__Misc_Oif__mono,axiom,
! [B: $o,M1: refine424419629nres_a,M12: refine424419629nres_a,M2: refine424419629nres_a,M22: refine424419629nres_a] :
( ( B
=> ( ord_le519537037nres_a @ M1 @ M12 ) )
=> ( ( ~ B
=> ( ord_le519537037nres_a @ M2 @ M22 ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B @ M1 @ M2 ) @ ( if_Ref1724547303nres_a @ B @ M12 @ M22 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_31_le__fun__def,axiom,
( ord_le516598614nres_a
= ( ^ [F2: $o > refine424419629nres_a,G2: $o > refine424419629nres_a] :
! [X5: $o] : ( ord_le519537037nres_a @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ).
% le_fun_def
thf(fact_32_nres__more__simps_I6_J,axiom,
! [X4: a,Y3: a] :
( ( ( refine2063221604TURN_a @ X4 )
= ( refine2063221604TURN_a @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_more_simps(6)
thf(fact_33_nres__more__simps_I6_J,axiom,
! [X4: product_unit,Y3: product_unit] :
( ( ( refine1420258419t_unit @ X4 )
= ( refine1420258419t_unit @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_more_simps(6)
thf(fact_34_nres__order__simps_I20_J,axiom,
! [X4: product_unit,Y3: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X4 ) @ ( refine1420258419t_unit @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_order_simps(20)
thf(fact_35_nres__order__simps_I20_J,axiom,
! [X4: a,Y3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine2063221604TURN_a @ Y3 ) )
= ( X4 = Y3 ) ) ).
% nres_order_simps(20)
thf(fact_36_less__eq__nres_Osimps_I2_J,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ A ) @ ( refine1777164439t_unit @ B ) )
= ( ord_le1023748749t_unit @ A @ B ) ) ).
% less_eq_nres.simps(2)
thf(fact_37_less__eq__nres_Osimps_I2_J,axiom,
! [A: set_a,B: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ).
% less_eq_nres.simps(2)
thf(fact_38_nres__order__simps_I5_J,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ord_le1051254044t_unit @ ( refine1777164439t_unit @ X ) @ ( refine1777164439t_unit @ Y ) )
= ( ord_le1023748749t_unit @ X @ Y ) ) ).
% nres_order_simps(5)
thf(fact_39_nres__order__simps_I5_J,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ X ) @ ( refine1198353288_RES_a @ Y ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% nres_order_simps(5)
thf(fact_40_RETURN__to__SPEC__rule,axiom,
! [M: refine787176636t_unit,V: product_unit] :
( ( ord_le1051254044t_unit @ M
@ ( refine1777164439t_unit
@ ( collect_Product_unit
@ ( ^ [Y4: product_unit,Z: product_unit] : Y4 = Z
@ V ) ) ) )
=> ( ord_le1051254044t_unit @ M @ ( refine1420258419t_unit @ V ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_41_RETURN__to__SPEC__rule,axiom,
! [M: refine424419629nres_a,V: a] :
( ( ord_le519537037nres_a @ M
@ ( refine1198353288_RES_a
@ ( collect_a
@ ( ^ [Y4: a,Z: a] : Y4 = Z
@ V ) ) ) )
=> ( ord_le519537037nres_a @ M @ ( refine2063221604TURN_a @ V ) ) ) ).
% RETURN_to_SPEC_rule
thf(fact_42_RETURN__rule,axiom,
! [Phi: product_unit > $o,X4: product_unit] :
( ( Phi @ X4 )
=> ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X4 ) @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_43_RETURN__rule,axiom,
! [Phi: a > $o,X4: a] :
( ( Phi @ X4 )
=> ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ) ).
% RETURN_rule
thf(fact_44_order__mono__setup_Omono__if,axiom,
! [T: set_a,T2: set_a,E: set_a,E2: set_a,B: $o] :
( ( ord_less_eq_set_a @ T @ T2 )
=> ( ( ord_less_eq_set_a @ E @ E2 )
=> ( ord_less_eq_set_a @ ( if_set_a @ B @ T @ E ) @ ( if_set_a @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_45_order__mono__setup_Omono__if,axiom,
! [T: $o > refine424419629nres_a,T2: $o > refine424419629nres_a,E: $o > refine424419629nres_a,E2: $o > refine424419629nres_a,B: $o] :
( ( ord_le516598614nres_a @ T @ T2 )
=> ( ( ord_le516598614nres_a @ E @ E2 )
=> ( ord_le516598614nres_a @ ( if_o_R1705037948nres_a @ B @ T @ E ) @ ( if_o_R1705037948nres_a @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_46_order__mono__setup_Omono__if,axiom,
! [T: refine787176636t_unit,T2: refine787176636t_unit,E: refine787176636t_unit,E2: refine787176636t_unit,B: $o] :
( ( ord_le1051254044t_unit @ T @ T2 )
=> ( ( ord_le1051254044t_unit @ E @ E2 )
=> ( ord_le1051254044t_unit @ ( if_Ref1369692790t_unit @ B @ T @ E ) @ ( if_Ref1369692790t_unit @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_47_order__mono__setup_Omono__if,axiom,
! [T: refine424419629nres_a,T2: refine424419629nres_a,E: refine424419629nres_a,E2: refine424419629nres_a,B: $o] :
( ( ord_le519537037nres_a @ T @ T2 )
=> ( ( ord_le519537037nres_a @ E @ E2 )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B @ T @ E ) @ ( if_Ref1724547303nres_a @ B @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_48_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_49_dual__order_Oantisym,axiom,
! [B: $o > refine424419629nres_a,A: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ B @ A )
=> ( ( ord_le516598614nres_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_50_dual__order_Oantisym,axiom,
! [B: refine787176636t_unit,A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ B @ A )
=> ( ( ord_le1051254044t_unit @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_51_dual__order_Oantisym,axiom,
! [B: refine424419629nres_a,A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B @ A )
=> ( ( ord_le519537037nres_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_52_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_53_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > refine424419629nres_a,Z: $o > refine424419629nres_a] : Y4 = Z )
= ( ^ [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ B2 @ A2 )
& ( ord_le516598614nres_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_54_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: refine787176636t_unit,Z: refine787176636t_unit] : Y4 = Z )
= ( ^ [A2: refine787176636t_unit,B2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ B2 @ A2 )
& ( ord_le1051254044t_unit @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_55_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
& ( ord_le519537037nres_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_56_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_57_dual__order_Otrans,axiom,
! [B: $o > refine424419629nres_a,A: $o > refine424419629nres_a,C: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ B @ A )
=> ( ( ord_le516598614nres_a @ C @ B )
=> ( ord_le516598614nres_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_58_dual__order_Otrans,axiom,
! [B: refine787176636t_unit,A: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ B @ A )
=> ( ( ord_le1051254044t_unit @ C @ B )
=> ( ord_le1051254044t_unit @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_59_dual__order_Otrans,axiom,
! [B: refine424419629nres_a,A: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B @ A )
=> ( ( ord_le519537037nres_a @ C @ B )
=> ( ord_le519537037nres_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_60_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_61_dual__order_Orefl,axiom,
! [A: $o > refine424419629nres_a] : ( ord_le516598614nres_a @ A @ A ) ).
% dual_order.refl
thf(fact_62_dual__order_Orefl,axiom,
! [A: refine787176636t_unit] : ( ord_le1051254044t_unit @ A @ A ) ).
% dual_order.refl
thf(fact_63_dual__order_Orefl,axiom,
! [A: refine424419629nres_a] : ( ord_le519537037nres_a @ A @ A ) ).
% dual_order.refl
thf(fact_64_order__trans,axiom,
! [X4: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_65_order__trans,axiom,
! [X4: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a,Z2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X4 @ Y3 )
=> ( ( ord_le516598614nres_a @ Y3 @ Z2 )
=> ( ord_le516598614nres_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_66_order__trans,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit,Z2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X4 @ Y3 )
=> ( ( ord_le1051254044t_unit @ Y3 @ Z2 )
=> ( ord_le1051254044t_unit @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_67_order__trans,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ Z2 )
=> ( ord_le519537037nres_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_68_order__class_Oorder_Oantisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_69_order__class_Oorder_Oantisym,axiom,
! [A: $o > refine424419629nres_a,B: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A @ B )
=> ( ( ord_le516598614nres_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_70_order__class_Oorder_Oantisym,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ord_le1051254044t_unit @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_71_order__class_Oorder_Oantisym,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_72_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_73_ord__le__eq__trans,axiom,
! [A: $o > refine424419629nres_a,B: $o > refine424419629nres_a,C: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A @ B )
=> ( ( B = C )
=> ( ord_le516598614nres_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_74_ord__le__eq__trans,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( B = C )
=> ( ord_le1051254044t_unit @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_75_ord__le__eq__trans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( B = C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_76_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_77_ord__eq__le__trans,axiom,
! [A: $o > refine424419629nres_a,B: $o > refine424419629nres_a,C: $o > refine424419629nres_a] :
( ( A = B )
=> ( ( ord_le516598614nres_a @ B @ C )
=> ( ord_le516598614nres_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_78_ord__eq__le__trans,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( A = B )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ord_le1051254044t_unit @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_79_ord__eq__le__trans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A = B )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_80_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_81_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: $o > refine424419629nres_a,Z: $o > refine424419629nres_a] : Y4 = Z )
= ( ^ [A2: $o > refine424419629nres_a,B2: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A2 @ B2 )
& ( ord_le516598614nres_a @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_82_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: refine787176636t_unit,Z: refine787176636t_unit] : Y4 = Z )
= ( ^ [A2: refine787176636t_unit,B2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A2 @ B2 )
& ( ord_le1051254044t_unit @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_83_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
& ( ord_le519537037nres_a @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_84_antisym__conv,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_85_antisym__conv,axiom,
! [Y3: $o > refine424419629nres_a,X4: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ Y3 @ X4 )
=> ( ( ord_le516598614nres_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_86_antisym__conv,axiom,
! [Y3: refine787176636t_unit,X4: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ Y3 @ X4 )
=> ( ( ord_le1051254044t_unit @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_87_antisym__conv,axiom,
! [Y3: refine424419629nres_a,X4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ Y3 @ X4 )
=> ( ( ord_le519537037nres_a @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_88_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_89_order_Otrans,axiom,
! [A: $o > refine424419629nres_a,B: $o > refine424419629nres_a,C: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ A @ B )
=> ( ( ord_le516598614nres_a @ B @ C )
=> ( ord_le516598614nres_a @ A @ C ) ) ) ).
% order.trans
thf(fact_90_order_Otrans,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ord_le1051254044t_unit @ A @ C ) ) ) ).
% order.trans
thf(fact_91_order_Otrans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ord_le519537037nres_a @ A @ C ) ) ) ).
% order.trans
thf(fact_92_eq__refl,axiom,
! [X4: set_a,Y3: set_a] :
( ( X4 = Y3 )
=> ( ord_less_eq_set_a @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_93_eq__refl,axiom,
! [X4: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a] :
( ( X4 = Y3 )
=> ( ord_le516598614nres_a @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_94_eq__refl,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( X4 = Y3 )
=> ( ord_le1051254044t_unit @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_95_eq__refl,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( X4 = Y3 )
=> ( ord_le519537037nres_a @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_96_antisym,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_97_antisym,axiom,
! [X4: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X4 @ Y3 )
=> ( ( ord_le516598614nres_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_98_antisym,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X4 @ Y3 )
=> ( ( ord_le1051254044t_unit @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_99_antisym,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_100_eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
& ( ord_less_eq_set_a @ Y5 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_101_eq__iff,axiom,
( ( ^ [Y4: $o > refine424419629nres_a,Z: $o > refine424419629nres_a] : Y4 = Z )
= ( ^ [X5: $o > refine424419629nres_a,Y5: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X5 @ Y5 )
& ( ord_le516598614nres_a @ Y5 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_102_eq__iff,axiom,
( ( ^ [Y4: refine787176636t_unit,Z: refine787176636t_unit] : Y4 = Z )
= ( ^ [X5: refine787176636t_unit,Y5: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X5 @ Y5 )
& ( ord_le1051254044t_unit @ Y5 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_103_eq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
& ( ord_le519537037nres_a @ Y5 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_104_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_109_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__le__eq__subst,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,F: refine787176636t_unit > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_111_ord__le__eq__subst,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,F: refine787176636t_unit > set_a,C: set_a] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_112_ord__le__eq__subst,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_ord__le__eq__subst,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > $o > refine424419629nres_a,C: $o > refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le516598614nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le516598614nres_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_mem__Collect__eq,axiom,
! [A: product_unit,P: product_unit > $o] :
( ( member_Product_unit @ A @ ( collect_Product_unit @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_115_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A3: set_Product_unit] :
( ( collect_Product_unit
@ ^ [X5: product_unit] : ( member_Product_unit @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_118_Collect__cong,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ! [X3: product_unit] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_Product_unit @ P )
= ( collect_Product_unit @ Q ) ) ) ).
% Collect_cong
thf(fact_119_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_120_ord__eq__le__subst,axiom,
! [A: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_121_ord__eq__le__subst,axiom,
! [A: set_a,F: refine424419629nres_a > set_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_122_ord__eq__le__subst,axiom,
! [A: refine787176636t_unit,F: refine424419629nres_a > refine787176636t_unit,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_123_ord__eq__le__subst,axiom,
! [A: refine424419629nres_a,F: set_a > refine424419629nres_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_124_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_125_ord__eq__le__subst,axiom,
! [A: refine787176636t_unit,F: set_a > refine787176636t_unit,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_126_ord__eq__le__subst,axiom,
! [A: refine424419629nres_a,F: refine787176636t_unit > refine424419629nres_a,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_127_ord__eq__le__subst,axiom,
! [A: set_a,F: refine787176636t_unit > set_a,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_128_ord__eq__le__subst,axiom,
! [A: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_129_ord__eq__le__subst,axiom,
! [A: $o > refine424419629nres_a,F: refine424419629nres_a > $o > refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le516598614nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le516598614nres_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_130_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le519537037nres_a @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_131_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_132_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le1051254044t_unit @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le519537037nres_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_134_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_135_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le1051254044t_unit @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_136_order__subst2,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,F: refine787176636t_unit > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ord_le519537037nres_a @ ( F @ B ) @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_137_order__subst2,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,F: refine787176636t_unit > set_a,C: set_a] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ B )
=> ( ( ord_le1051254044t_unit @ ( F @ B ) @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,F: refine424419629nres_a > $o > refine424419629nres_a,C: $o > refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ B )
=> ( ( ord_le516598614nres_a @ ( F @ B ) @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le516598614nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le516598614nres_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_140_order__subst1,axiom,
! [A: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A: refine424419629nres_a,F: set_a > refine424419629nres_a,B: set_a,C: set_a] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: refine424419629nres_a,F: refine787176636t_unit > refine424419629nres_a,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: set_a,F: refine424419629nres_a > set_a,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A: set_a,F: refine787176636t_unit > set_a,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: refine787176636t_unit,F: refine424419629nres_a > refine787176636t_unit,B: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1051254044t_unit @ A @ ( F @ B ) )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ! [X3: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: refine787176636t_unit,F: set_a > refine787176636t_unit,B: set_a,C: set_a] :
( ( ord_le1051254044t_unit @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: refine787176636t_unit,F: refine787176636t_unit > refine787176636t_unit,B: refine787176636t_unit,C: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ ( F @ B ) )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ! [X3: refine787176636t_unit,Y6: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ X3 @ Y6 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le1051254044t_unit @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst1,axiom,
! [A: refine424419629nres_a,F: ( $o > refine424419629nres_a ) > refine424419629nres_a,B: $o > refine424419629nres_a,C: $o > refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ ( F @ B ) )
=> ( ( ord_le516598614nres_a @ B @ C )
=> ( ! [X3: $o > refine424419629nres_a,Y6: $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ X3 @ Y6 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F @ Y6 ) ) )
=> ( ord_le519537037nres_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_150_the__RES__inv,axiom,
! [M: refine787176636t_unit] :
( ( refine579265252t_unit @ M )
=> ( ( refine1777164439t_unit @ ( refine1208325748t_unit @ M ) )
= M ) ) ).
% the_RES_inv
thf(fact_151_the__RES__inv,axiom,
! [M: refine424419629nres_a] :
( ( refine412683989fail_a @ M )
=> ( ( refine1198353288_RES_a @ ( refine1822134885_RES_a @ M ) )
= M ) ) ).
% the_RES_inv
thf(fact_152_Greatest__equality,axiom,
! [P: set_a > $o,X4: set_a] :
( ( P @ X4 )
=> ( ! [Y6: set_a] :
( ( P @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X4 ) )
=> ( ( order_Greatest_set_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_153_Greatest__equality,axiom,
! [P: ( $o > refine424419629nres_a ) > $o,X4: $o > refine424419629nres_a] :
( ( P @ X4 )
=> ( ! [Y6: $o > refine424419629nres_a] :
( ( P @ Y6 )
=> ( ord_le516598614nres_a @ Y6 @ X4 ) )
=> ( ( order_245837199nres_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_154_Greatest__equality,axiom,
! [P: refine787176636t_unit > $o,X4: refine787176636t_unit] :
( ( P @ X4 )
=> ( ! [Y6: refine787176636t_unit] :
( ( P @ Y6 )
=> ( ord_le1051254044t_unit @ Y6 @ X4 ) )
=> ( ( order_453013155t_unit @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_155_Greatest__equality,axiom,
! [P: refine424419629nres_a > $o,X4: refine424419629nres_a] :
( ( P @ X4 )
=> ( ! [Y6: refine424419629nres_a] :
( ( P @ Y6 )
=> ( ord_le519537037nres_a @ Y6 @ X4 ) )
=> ( ( order_1714329108nres_a @ P )
= X4 ) ) ) ).
% Greatest_equality
thf(fact_156_GreatestI2__order,axiom,
! [P: set_a > $o,X4: set_a,Q: set_a > $o] :
( ( P @ X4 )
=> ( ! [Y6: set_a] :
( ( P @ Y6 )
=> ( ord_less_eq_set_a @ Y6 @ X4 ) )
=> ( ! [X3: set_a] :
( ( P @ X3 )
=> ( ! [Y7: set_a] :
( ( P @ Y7 )
=> ( ord_less_eq_set_a @ Y7 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_157_GreatestI2__order,axiom,
! [P: ( $o > refine424419629nres_a ) > $o,X4: $o > refine424419629nres_a,Q: ( $o > refine424419629nres_a ) > $o] :
( ( P @ X4 )
=> ( ! [Y6: $o > refine424419629nres_a] :
( ( P @ Y6 )
=> ( ord_le516598614nres_a @ Y6 @ X4 ) )
=> ( ! [X3: $o > refine424419629nres_a] :
( ( P @ X3 )
=> ( ! [Y7: $o > refine424419629nres_a] :
( ( P @ Y7 )
=> ( ord_le516598614nres_a @ Y7 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_245837199nres_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_158_GreatestI2__order,axiom,
! [P: refine787176636t_unit > $o,X4: refine787176636t_unit,Q: refine787176636t_unit > $o] :
( ( P @ X4 )
=> ( ! [Y6: refine787176636t_unit] :
( ( P @ Y6 )
=> ( ord_le1051254044t_unit @ Y6 @ X4 ) )
=> ( ! [X3: refine787176636t_unit] :
( ( P @ X3 )
=> ( ! [Y7: refine787176636t_unit] :
( ( P @ Y7 )
=> ( ord_le1051254044t_unit @ Y7 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_453013155t_unit @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_159_GreatestI2__order,axiom,
! [P: refine424419629nres_a > $o,X4: refine424419629nres_a,Q: refine424419629nres_a > $o] :
( ( P @ X4 )
=> ( ! [Y6: refine424419629nres_a] :
( ( P @ Y6 )
=> ( ord_le519537037nres_a @ Y6 @ X4 ) )
=> ( ! [X3: refine424419629nres_a] :
( ( P @ X3 )
=> ( ! [Y7: refine424419629nres_a] :
( ( P @ Y7 )
=> ( ord_le519537037nres_a @ Y7 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_1714329108nres_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_160_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_set_a
= ( ^ [X6: $o > set_a,Y8: $o > set_a] :
( ( ord_less_eq_set_a @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq_set_a @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_161_le__rel__bool__arg__iff,axiom,
( ord_le1960259811nres_a
= ( ^ [X6: $o > $o > refine424419629nres_a,Y8: $o > $o > refine424419629nres_a] :
( ( ord_le516598614nres_a @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le516598614nres_a @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_162_le__rel__bool__arg__iff,axiom,
( ord_le1633311077t_unit
= ( ^ [X6: $o > refine787176636t_unit,Y8: $o > refine787176636t_unit] :
( ( ord_le1051254044t_unit @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le1051254044t_unit @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_163_le__rel__bool__arg__iff,axiom,
( ord_le516598614nres_a
= ( ^ [X6: $o > refine424419629nres_a,Y8: $o > refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le519537037nres_a @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_164_ord__eq__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( C = D )
=> ( ord_less_eq_set_a @ A @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_165_ord__eq__le__eq__trans,axiom,
! [A: $o > refine424419629nres_a,B: $o > refine424419629nres_a,C: $o > refine424419629nres_a,D: $o > refine424419629nres_a] :
( ( A = B )
=> ( ( ord_le516598614nres_a @ B @ C )
=> ( ( C = D )
=> ( ord_le516598614nres_a @ A @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_166_ord__eq__le__eq__trans,axiom,
! [A: refine787176636t_unit,B: refine787176636t_unit,C: refine787176636t_unit,D: refine787176636t_unit] :
( ( A = B )
=> ( ( ord_le1051254044t_unit @ B @ C )
=> ( ( C = D )
=> ( ord_le1051254044t_unit @ A @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_167_ord__eq__le__eq__trans,axiom,
! [A: refine424419629nres_a,B: refine424419629nres_a,C: refine424419629nres_a,D: refine424419629nres_a] :
( ( A = B )
=> ( ( ord_le519537037nres_a @ B @ C )
=> ( ( C = D )
=> ( ord_le519537037nres_a @ A @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_168_bind__cong,axiom,
! [M: refine424419629nres_a,M3: refine424419629nres_a,F: a > refine424419629nres_a,F3: a > refine424419629nres_a] :
( ( M = M3 )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M3 )
=> ( ( F @ X3 )
= ( F3 @ X3 ) ) )
=> ( ( refine436832838nd_a_a @ M @ F )
= ( refine436832838nd_a_a @ M3 @ F3 ) ) ) ) ).
% bind_cong
thf(fact_169_bind__cong,axiom,
! [M: refine787176636t_unit,M3: refine787176636t_unit,F: product_unit > refine787176636t_unit,F3: product_unit > refine787176636t_unit] :
( ( M = M3 )
=> ( ! [X3: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X3 ) @ M3 )
=> ( ( F @ X3 )
= ( F3 @ X3 ) ) )
=> ( ( refine681446406t_unit @ M @ F )
= ( refine681446406t_unit @ M3 @ F3 ) ) ) ) ).
% bind_cong
thf(fact_170_bind__cong,axiom,
! [M: refine787176636t_unit,M3: refine787176636t_unit,F: product_unit > refine424419629nres_a,F3: product_unit > refine424419629nres_a] :
( ( M = M3 )
=> ( ! [X3: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X3 ) @ M3 )
=> ( ( F @ X3 )
= ( F3 @ X3 ) ) )
=> ( ( refine119808503unit_a @ M @ F )
= ( refine119808503unit_a @ M3 @ F3 ) ) ) ) ).
% bind_cong
thf(fact_171_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine424419629nres_a,M5: refine424419629nres_a,F: a > refine787176636t_unit,F3: a > refine787176636t_unit] :
( ( ord_le519537037nres_a @ M4 @ M5 )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M4 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_le1051254044t_unit @ ( refine96995669t_unit @ M4 @ F ) @ ( refine96995669t_unit @ M5 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_172_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine787176636t_unit,M5: refine787176636t_unit,F: product_unit > refine787176636t_unit,F3: product_unit > refine787176636t_unit] :
( ( ord_le1051254044t_unit @ M4 @ M5 )
=> ( ! [X3: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X3 ) @ M4 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_le1051254044t_unit @ ( refine681446406t_unit @ M4 @ F ) @ ( refine681446406t_unit @ M5 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_173_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine787176636t_unit,M5: refine787176636t_unit,F: product_unit > refine424419629nres_a,F3: product_unit > refine424419629nres_a] :
( ( ord_le1051254044t_unit @ M4 @ M5 )
=> ( ! [X3: product_unit] :
( ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X3 ) @ M4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_le519537037nres_a @ ( refine119808503unit_a @ M4 @ F ) @ ( refine119808503unit_a @ M5 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_174_Refine__Basic__Mirabelle__kwjuvthmas_Obind__mono_I1_J,axiom,
! [M4: refine424419629nres_a,M5: refine424419629nres_a,F: a > refine424419629nres_a,F3: a > refine424419629nres_a] :
( ( ord_le519537037nres_a @ M4 @ M5 )
=> ( ! [X3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ M4 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M4 @ F ) @ ( refine436832838nd_a_a @ M5 @ F3 ) ) ) ) ).
% Refine_Basic_Mirabelle_kwjuvthmas.bind_mono(1)
thf(fact_175_nofail__simps_I2_J,axiom,
! [X: set_Product_unit] : ( refine579265252t_unit @ ( refine1777164439t_unit @ X ) ) ).
% nofail_simps(2)
thf(fact_176_nofail__simps_I2_J,axiom,
! [X: set_a] : ( refine412683989fail_a @ ( refine1198353288_RES_a @ X ) ) ).
% nofail_simps(2)
thf(fact_177_nofail__simps_I3_J,axiom,
! [X4: a] : ( refine412683989fail_a @ ( refine2063221604TURN_a @ X4 ) ) ).
% nofail_simps(3)
thf(fact_178_nofail__simps_I3_J,axiom,
! [X4: product_unit] : ( refine579265252t_unit @ ( refine1420258419t_unit @ X4 ) ) ).
% nofail_simps(3)
thf(fact_179_nres__monad2,axiom,
! [M4: refine424419629nres_a] :
( ( refine436832838nd_a_a @ M4 @ refine2063221604TURN_a )
= M4 ) ).
% nres_monad2
thf(fact_180_nres__monad2,axiom,
! [M4: refine787176636t_unit] :
( ( refine681446406t_unit @ M4 @ refine1420258419t_unit )
= M4 ) ).
% nres_monad2
thf(fact_181_nres__monad1,axiom,
! [X4: a,F: a > refine424419629nres_a] :
( ( refine436832838nd_a_a @ ( refine2063221604TURN_a @ X4 ) @ F )
= ( F @ X4 ) ) ).
% nres_monad1
thf(fact_182_nres__monad1,axiom,
! [X4: product_unit,F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ ( refine1420258419t_unit @ X4 ) @ F )
= ( F @ X4 ) ) ).
% nres_monad1
thf(fact_183_nres__monad1,axiom,
! [X4: product_unit,F: product_unit > refine424419629nres_a] :
( ( refine119808503unit_a @ ( refine1420258419t_unit @ X4 ) @ F )
= ( F @ X4 ) ) ).
% nres_monad1
thf(fact_184_subset__Collect__conv,axiom,
! [S: set_Product_unit,P: product_unit > $o] :
( ( ord_le1023748749t_unit @ S @ ( collect_Product_unit @ P ) )
= ( ! [X5: product_unit] :
( ( member_Product_unit @ X5 @ S )
=> ( P @ X5 ) ) ) ) ).
% subset_Collect_conv
thf(fact_185_subset__Collect__conv,axiom,
! [S: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ S @ ( collect_a @ P ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ S )
=> ( P @ X5 ) ) ) ) ).
% subset_Collect_conv
thf(fact_186_nofail__antimono__fun,axiom,
! [F: $o > refine424419629nres_a,G: $o > refine424419629nres_a,X4: $o] :
( ( ord_le516598614nres_a @ F @ G )
=> ( ( refine412683989fail_a @ ( G @ X4 ) )
=> ( refine412683989fail_a @ ( F @ X4 ) ) ) ) ).
% nofail_antimono_fun
thf(fact_187_nofail__RES__conv,axiom,
( refine579265252t_unit
= ( ^ [M6: refine787176636t_unit] :
? [M7: set_Product_unit] :
( M6
= ( refine1777164439t_unit @ M7 ) ) ) ) ).
% nofail_RES_conv
thf(fact_188_nofail__RES__conv,axiom,
( refine412683989fail_a
= ( ^ [M6: refine424419629nres_a] :
? [M7: set_a] :
( M6
= ( refine1198353288_RES_a @ M7 ) ) ) ) ).
% nofail_RES_conv
thf(fact_189_pwD1,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ S @ S2 )
=> ( ( refine412683989fail_a @ S2 )
=> ( refine412683989fail_a @ S ) ) ) ).
% pwD1
thf(fact_190_pwD1,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ S @ S2 )
=> ( ( refine579265252t_unit @ S2 )
=> ( refine579265252t_unit @ S ) ) ) ).
% pwD1
thf(fact_191_le__nofailI,axiom,
! [M5: refine424419629nres_a,M4: refine424419629nres_a] :
( ( ( refine412683989fail_a @ M5 )
=> ( ord_le519537037nres_a @ M4 @ M5 ) )
=> ( ord_le519537037nres_a @ M4 @ M5 ) ) ).
% le_nofailI
thf(fact_192_le__nofailI,axiom,
! [M5: refine787176636t_unit,M4: refine787176636t_unit] :
( ( ( refine579265252t_unit @ M5 )
=> ( ord_le1051254044t_unit @ M4 @ M5 ) )
=> ( ord_le1051254044t_unit @ M4 @ M5 ) ) ).
% le_nofailI
thf(fact_193_subsetI,axiom,
! [A3: set_Product_unit,B3: set_Product_unit] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A3 )
=> ( member_Product_unit @ X3 @ B3 ) )
=> ( ord_le1023748749t_unit @ A3 @ B3 ) ) ).
% subsetI
thf(fact_194_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_195_subset__antisym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_196_pw__bind__leI,axiom,
! [S: refine787176636t_unit,M4: refine424419629nres_a,F: a > refine787176636t_unit] :
( ( ( refine579265252t_unit @ S )
=> ( refine412683989fail_a @ M4 ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ M4 )
=> ( ( refine1001002027nres_a @ M4 @ X3 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ S ) ) )
=> ( ord_le1051254044t_unit @ ( refine96995669t_unit @ M4 @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_197_pw__bind__leI,axiom,
! [S: refine424419629nres_a,M4: refine424419629nres_a,F: a > refine424419629nres_a] :
( ( ( refine412683989fail_a @ S )
=> ( refine412683989fail_a @ M4 ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ M4 )
=> ( ( refine1001002027nres_a @ M4 @ X3 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ S ) ) )
=> ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M4 @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_198_pw__bind__leI,axiom,
! [S: refine424419629nres_a,M4: refine787176636t_unit,F: product_unit > refine424419629nres_a] :
( ( ( refine412683989fail_a @ S )
=> ( refine579265252t_unit @ M4 ) )
=> ( ! [X3: product_unit] :
( ( refine579265252t_unit @ M4 )
=> ( ( refine558004794t_unit @ M4 @ X3 )
=> ( ord_le519537037nres_a @ ( F @ X3 ) @ S ) ) )
=> ( ord_le519537037nres_a @ ( refine119808503unit_a @ M4 @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_199_pw__bind__leI,axiom,
! [S: refine787176636t_unit,M4: refine787176636t_unit,F: product_unit > refine787176636t_unit] :
( ( ( refine579265252t_unit @ S )
=> ( refine579265252t_unit @ M4 ) )
=> ( ! [X3: product_unit] :
( ( refine579265252t_unit @ M4 )
=> ( ( refine558004794t_unit @ M4 @ X3 )
=> ( ord_le1051254044t_unit @ ( F @ X3 ) @ S ) ) )
=> ( ord_le1051254044t_unit @ ( refine681446406t_unit @ M4 @ F ) @ S ) ) ) ).
% pw_bind_leI
thf(fact_200_pw__bind__le__iff,axiom,
! [M4: refine424419629nres_a,F: a > refine787176636t_unit,S: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ ( refine96995669t_unit @ M4 @ F ) @ S )
= ( ( ( refine579265252t_unit @ S )
=> ( refine412683989fail_a @ M4 ) )
& ! [X5: a] :
( ( ( refine412683989fail_a @ M4 )
& ( refine1001002027nres_a @ M4 @ X5 ) )
=> ( ord_le1051254044t_unit @ ( F @ X5 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_201_pw__bind__le__iff,axiom,
! [M4: refine424419629nres_a,F: a > refine424419629nres_a,S: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine436832838nd_a_a @ M4 @ F ) @ S )
= ( ( ( refine412683989fail_a @ S )
=> ( refine412683989fail_a @ M4 ) )
& ! [X5: a] :
( ( ( refine412683989fail_a @ M4 )
& ( refine1001002027nres_a @ M4 @ X5 ) )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_202_pw__bind__le__iff,axiom,
! [M4: refine787176636t_unit,F: product_unit > refine424419629nres_a,S: refine424419629nres_a] :
( ( ord_le519537037nres_a @ ( refine119808503unit_a @ M4 @ F ) @ S )
= ( ( ( refine412683989fail_a @ S )
=> ( refine579265252t_unit @ M4 ) )
& ! [X5: product_unit] :
( ( ( refine579265252t_unit @ M4 )
& ( refine558004794t_unit @ M4 @ X5 ) )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_203_pw__bind__le__iff,axiom,
! [M4: refine787176636t_unit,F: product_unit > refine787176636t_unit,S: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ ( refine681446406t_unit @ M4 @ F ) @ S )
= ( ( ( refine579265252t_unit @ S )
=> ( refine579265252t_unit @ M4 ) )
& ! [X5: product_unit] :
( ( ( refine579265252t_unit @ M4 )
& ( refine558004794t_unit @ M4 @ X5 ) )
=> ( ord_le1051254044t_unit @ ( F @ X5 ) @ S ) ) ) ) ).
% pw_bind_le_iff
thf(fact_204_relprop__triggers_I6_J,axiom,
! [R: set_a,R2: set_a] :
( ( ord_less_eq_set_a @ R @ R2 )
=> ( ord_less_eq_set_a @ R @ R2 ) ) ).
% relprop_triggers(6)
thf(fact_205_in__mono,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,X4: product_unit] :
( ( ord_le1023748749t_unit @ A3 @ B3 )
=> ( ( member_Product_unit @ X4 @ A3 )
=> ( member_Product_unit @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_206_in__mono,axiom,
! [A3: set_a,B3: set_a,X4: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B3 ) ) ) ).
% in_mono
thf(fact_207_subsetD,axiom,
! [A3: set_Product_unit,B3: set_Product_unit,C: product_unit] :
( ( ord_le1023748749t_unit @ A3 @ B3 )
=> ( ( member_Product_unit @ C @ A3 )
=> ( member_Product_unit @ C @ B3 ) ) ) ).
% subsetD
thf(fact_208_subsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_209_inres__simps_I2_J,axiom,
! [X: set_Product_unit] :
( ( refine558004794t_unit @ ( refine1777164439t_unit @ X ) )
= ( ^ [X5: product_unit] : ( member_Product_unit @ X5 @ X ) ) ) ).
% inres_simps(2)
thf(fact_210_inres__simps_I2_J,axiom,
! [X: set_a] :
( ( refine1001002027nres_a @ ( refine1198353288_RES_a @ X ) )
= ( ^ [X5: a] : ( member_a @ X5 @ X ) ) ) ).
% inres_simps(2)
thf(fact_211_inres__simps_I3_J,axiom,
! [X4: a] :
( ( refine1001002027nres_a @ ( refine2063221604TURN_a @ X4 ) )
= ( ^ [Y4: a,Z: a] : Y4 = Z
@ X4 ) ) ).
% inres_simps(3)
thf(fact_212_inres__simps_I3_J,axiom,
! [X4: product_unit] :
( ( refine558004794t_unit @ ( refine1420258419t_unit @ X4 ) )
= ( ^ [Y4: product_unit,Z: product_unit] : Y4 = Z
@ X4 ) ) ).
% inres_simps(3)
thf(fact_213_pwD2,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a,X4: a] :
( ( ord_le519537037nres_a @ S @ S2 )
=> ( ( refine1001002027nres_a @ S @ X4 )
=> ( refine1001002027nres_a @ S2 @ X4 ) ) ) ).
% pwD2
thf(fact_214_pwD2,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit,X4: product_unit] :
( ( ord_le1051254044t_unit @ S @ S2 )
=> ( ( refine558004794t_unit @ S @ X4 )
=> ( refine558004794t_unit @ S2 @ X4 ) ) ) ).
% pwD2
thf(fact_215_pw__eqI,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S )
= ( refine412683989fail_a @ S2 ) )
=> ( ! [X3: a] :
( ( refine1001002027nres_a @ S @ X3 )
= ( refine1001002027nres_a @ S2 @ X3 ) )
=> ( S = S2 ) ) ) ).
% pw_eqI
thf(fact_216_pw__eqI,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S )
= ( refine579265252t_unit @ S2 ) )
=> ( ! [X3: product_unit] :
( ( refine558004794t_unit @ S @ X3 )
= ( refine558004794t_unit @ S2 @ X3 ) )
=> ( S = S2 ) ) ) ).
% pw_eqI
thf(fact_217_pw__eq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [S3: refine424419629nres_a,S4: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S3 )
= ( refine412683989fail_a @ S4 ) )
& ! [X5: a] :
( ( refine1001002027nres_a @ S3 @ X5 )
= ( refine1001002027nres_a @ S4 @ X5 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_218_pw__eq__iff,axiom,
( ( ^ [Y4: refine787176636t_unit,Z: refine787176636t_unit] : Y4 = Z )
= ( ^ [S3: refine787176636t_unit,S4: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S3 )
= ( refine579265252t_unit @ S4 ) )
& ! [X5: product_unit] :
( ( refine558004794t_unit @ S3 @ X5 )
= ( refine558004794t_unit @ S4 @ X5 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_219_not__nofail__inres,axiom,
! [S: refine424419629nres_a,X4: a] :
( ~ ( refine412683989fail_a @ S )
=> ( refine1001002027nres_a @ S @ X4 ) ) ).
% not_nofail_inres
thf(fact_220_not__nofail__inres,axiom,
! [S: refine787176636t_unit,X4: product_unit] :
( ~ ( refine579265252t_unit @ S )
=> ( refine558004794t_unit @ S @ X4 ) ) ).
% not_nofail_inres
thf(fact_221_pw__le__iff,axiom,
( ord_le519537037nres_a
= ( ^ [S3: refine424419629nres_a,S4: refine424419629nres_a] :
( ( refine412683989fail_a @ S4 )
=> ( ( refine412683989fail_a @ S3 )
& ! [X5: a] :
( ( refine1001002027nres_a @ S3 @ X5 )
=> ( refine1001002027nres_a @ S4 @ X5 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_222_pw__le__iff,axiom,
( ord_le1051254044t_unit
= ( ^ [S3: refine787176636t_unit,S4: refine787176636t_unit] :
( ( refine579265252t_unit @ S4 )
=> ( ( refine579265252t_unit @ S3 )
& ! [X5: product_unit] :
( ( refine558004794t_unit @ S3 @ X5 )
=> ( refine558004794t_unit @ S4 @ X5 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_223_pw__leI_H,axiom,
! [S2: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S2 )
=> ( refine412683989fail_a @ S ) )
=> ( ! [X3: a] :
( ( refine412683989fail_a @ S2 )
=> ( ( refine1001002027nres_a @ S @ X3 )
=> ( refine1001002027nres_a @ S2 @ X3 ) ) )
=> ( ord_le519537037nres_a @ S @ S2 ) ) ) ).
% pw_leI'
thf(fact_224_pw__leI_H,axiom,
! [S2: refine787176636t_unit,S: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S2 )
=> ( refine579265252t_unit @ S ) )
=> ( ! [X3: product_unit] :
( ( refine579265252t_unit @ S2 )
=> ( ( refine558004794t_unit @ S @ X3 )
=> ( refine558004794t_unit @ S2 @ X3 ) ) )
=> ( ord_le1051254044t_unit @ S @ S2 ) ) ) ).
% pw_leI'
thf(fact_225_pw__leI,axiom,
! [S2: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S2 )
=> ( ( refine412683989fail_a @ S )
& ! [X3: a] :
( ( refine1001002027nres_a @ S @ X3 )
=> ( refine1001002027nres_a @ S2 @ X3 ) ) ) )
=> ( ord_le519537037nres_a @ S @ S2 ) ) ).
% pw_leI
thf(fact_226_pw__leI,axiom,
! [S2: refine787176636t_unit,S: refine787176636t_unit] :
( ( ( refine579265252t_unit @ S2 )
=> ( ( refine579265252t_unit @ S )
& ! [X3: product_unit] :
( ( refine558004794t_unit @ S @ X3 )
=> ( refine558004794t_unit @ S2 @ X3 ) ) ) )
=> ( ord_le1051254044t_unit @ S @ S2 ) ) ).
% pw_leI
thf(fact_227_pw__bind__nofail,axiom,
! [M4: refine424419629nres_a,F: a > refine787176636t_unit] :
( ( refine579265252t_unit @ ( refine96995669t_unit @ M4 @ F ) )
= ( ( refine412683989fail_a @ M4 )
& ! [X5: a] :
( ( refine1001002027nres_a @ M4 @ X5 )
=> ( refine579265252t_unit @ ( F @ X5 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_228_pw__bind__nofail,axiom,
! [M4: refine787176636t_unit,F: product_unit > refine424419629nres_a] :
( ( refine412683989fail_a @ ( refine119808503unit_a @ M4 @ F ) )
= ( ( refine579265252t_unit @ M4 )
& ! [X5: product_unit] :
( ( refine558004794t_unit @ M4 @ X5 )
=> ( refine412683989fail_a @ ( F @ X5 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_229_pw__bind__nofail,axiom,
! [M4: refine424419629nres_a,F: a > refine424419629nres_a] :
( ( refine412683989fail_a @ ( refine436832838nd_a_a @ M4 @ F ) )
= ( ( refine412683989fail_a @ M4 )
& ! [X5: a] :
( ( refine1001002027nres_a @ M4 @ X5 )
=> ( refine412683989fail_a @ ( F @ X5 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_230_pw__bind__nofail,axiom,
! [M4: refine787176636t_unit,F: product_unit > refine787176636t_unit] :
( ( refine579265252t_unit @ ( refine681446406t_unit @ M4 @ F ) )
= ( ( refine579265252t_unit @ M4 )
& ! [X5: product_unit] :
( ( refine558004794t_unit @ M4 @ X5 )
=> ( refine579265252t_unit @ ( F @ X5 ) ) ) ) ) ).
% pw_bind_nofail
thf(fact_231_inres__def,axiom,
( refine1001002027nres_a
= ( ^ [S3: refine424419629nres_a,X5: a] : ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X5 ) @ S3 ) ) ) ).
% inres_def
thf(fact_232_inres__def,axiom,
( refine558004794t_unit
= ( ^ [S3: refine787176636t_unit,X5: product_unit] : ( ord_le1051254044t_unit @ ( refine1420258419t_unit @ X5 ) @ S3 ) ) ) ).
% inres_def
thf(fact_233_nf__inres__def,axiom,
( refine983493746t_unit
= ( ^ [M6: refine787176636t_unit,X5: product_unit] :
( ( refine579265252t_unit @ M6 )
& ( refine558004794t_unit @ M6 @ X5 ) ) ) ) ).
% nf_inres_def
thf(fact_234_nf__inres__def,axiom,
( refine1312857699nres_a
= ( ^ [M6: refine424419629nres_a,X5: a] :
( ( refine412683989fail_a @ M6 )
& ( refine1001002027nres_a @ M6 @ X5 ) ) ) ) ).
% nf_inres_def
thf(fact_235_Collect__mono__iff,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ( ord_le1023748749t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q ) )
= ( ! [X5: product_unit] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_236_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_237_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_238_subset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_239_Collect__mono,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ! [X3: product_unit] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1023748749t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q ) ) ) ).
% Collect_mono
thf(fact_240_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_241_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_242_subset__iff,axiom,
( ord_le1023748749t_unit
= ( ^ [A4: set_Product_unit,B4: set_Product_unit] :
! [T3: product_unit] :
( ( member_Product_unit @ T3 @ A4 )
=> ( member_Product_unit @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_243_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A4 )
=> ( member_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_244_equalityD2,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_245_equalityD1,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_246_subset__eq,axiom,
( ord_le1023748749t_unit
= ( ^ [A4: set_Product_unit,B4: set_Product_unit] :
! [X5: product_unit] :
( ( member_Product_unit @ X5 @ A4 )
=> ( member_Product_unit @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_247_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( member_a @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_248_less__eq__nres_Oelims_I1_J,axiom,
! [X4: refine424419629nres_a,Xa: refine424419629nres_a,Y3: $o] :
( ( ( ord_le519537037nres_a @ X4 @ Xa )
= Y3 )
=> ( ( ( Xa = refine464223677AILi_a )
=> ~ Y3 )
=> ( ! [A5: set_a] :
( ( X4
= ( refine1198353288_RES_a @ A5 ) )
=> ! [B5: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B5 ) )
=> ( Y3
= ( ~ ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) )
=> ~ ( ( X4 = refine464223677AILi_a )
=> ( ? [Uv: set_a] :
( Xa
= ( refine1198353288_RES_a @ Uv ) )
=> Y3 ) ) ) ) ) ).
% less_eq_nres.elims(1)
thf(fact_249_less__eq__nres_Oelims_I2_J,axiom,
! [X4: refine424419629nres_a,Xa: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Xa )
=> ( ( Xa != refine464223677AILi_a )
=> ~ ! [A5: set_a] :
( ( X4
= ( refine1198353288_RES_a @ A5 ) )
=> ! [B5: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B5 ) )
=> ~ ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ) ).
% less_eq_nres.elims(2)
thf(fact_250_less__eq__nres_Oelims_I3_J,axiom,
! [X4: refine424419629nres_a,Xa: refine424419629nres_a] :
( ~ ( ord_le519537037nres_a @ X4 @ Xa )
=> ( ! [A5: set_a] :
( ( X4
= ( refine1198353288_RES_a @ A5 ) )
=> ! [B5: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B5 ) )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) )
=> ~ ( ( X4 = refine464223677AILi_a )
=> ! [Uv: set_a] :
( Xa
!= ( refine1198353288_RES_a @ Uv ) ) ) ) ) ).
% less_eq_nres.elims(3)
thf(fact_251_nres__order__simps_I22_J,axiom,
! [X: set_a,Y3: a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ X ) @ ( refine2063221604TURN_a @ Y3 ) )
= ( ord_less_eq_set_a @ X @ ( insert_a @ Y3 @ bot_bot_set_a ) ) ) ).
% nres_order_simps(22)
thf(fact_252_less__eq__nres_Osimps_I3_J,axiom,
! [Uv2: set_a] :
~ ( ord_le519537037nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv2 ) ) ).
% less_eq_nres.simps(3)
thf(fact_253_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_254_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X5: a] :
~ ( member_a @ X5 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_255_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X5: a] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_256_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X5: a] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_257_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_258_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_259_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_260_insert__subset,axiom,
! [X4: a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A3 ) @ B3 )
= ( ( member_a @ X4 @ B3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_261_nres__more__simps_I5_J,axiom,
! [X4: a,X: set_a] :
( ( ( refine2063221604TURN_a @ X4 )
= ( refine1198353288_RES_a @ X ) )
= ( ( insert_a @ X4 @ bot_bot_set_a )
= X ) ) ).
% nres_more_simps(5)
thf(fact_262_nres__more__simps_I3_J,axiom,
! [X: set_a,X4: a] :
( ( ( refine1198353288_RES_a @ X )
= ( refine2063221604TURN_a @ X4 ) )
= ( X
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% nres_more_simps(3)
thf(fact_263_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_264_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_265_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_266_equals0I,axiom,
! [A3: set_a] :
( ! [Y6: a] :
~ ( member_a @ Y6 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_267_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_268_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_269_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X5: a] : ( member_a @ X5 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_270_Set_Oset__insert,axiom,
! [X4: a,A3: set_a] :
( ( member_a @ X4 @ A3 )
=> ~ ! [B6: set_a] :
( ( A3
= ( insert_a @ X4 @ B6 ) )
=> ( member_a @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_271_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_272_insert__ident,axiom,
! [X4: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X4 @ A3 )
=> ( ~ ( member_a @ X4 @ B3 )
=> ( ( ( insert_a @ X4 @ A3 )
= ( insert_a @ X4 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_273_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_274_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B3
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_275_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_276_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B6: set_a] :
( ( A3
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_277_set__notEmptyE,axiom,
! [S: set_a] :
( ( S != bot_bot_set_a )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_278_memb__imp__not__empty,axiom,
! [X4: a,S: set_a] :
( ( member_a @ X4 @ S )
=> ( S != bot_bot_set_a ) ) ).
% memb_imp_not_empty
thf(fact_279_bot_Oextremum,axiom,
! [A: refine787176636t_unit] : ( ord_le1051254044t_unit @ bot_bo658782032t_unit @ A ) ).
% bot.extremum
thf(fact_280_bot_Oextremum,axiom,
! [A: refine424419629nres_a] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ A ) ).
% bot.extremum
thf(fact_281_bot_Oextremum__unique,axiom,
! [A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ bot_bo658782032t_unit )
= ( A = bot_bo658782032t_unit ) ) ).
% bot.extremum_unique
thf(fact_282_bot_Oextremum__unique,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ bot_bo529555393nres_a )
= ( A = bot_bo529555393nres_a ) ) ).
% bot.extremum_unique
thf(fact_283_bot_Oextremum__uniqueI,axiom,
! [A: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ A @ bot_bo658782032t_unit )
=> ( A = bot_bo658782032t_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_284_bot_Oextremum__uniqueI,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A @ bot_bo529555393nres_a )
=> ( A = bot_bo529555393nres_a ) ) ).
% bot.extremum_uniqueI
thf(fact_285_subset__insert,axiom,
! [X4: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X4 @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X4 @ B3 ) )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_286_RETURN__def,axiom,
( refine2063221604TURN_a
= ( ^ [X5: a] : ( refine1198353288_RES_a @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ).
% RETURN_def
thf(fact_287_nres_Odistinct_I1_J,axiom,
! [X2: set_a] :
( refine464223677AILi_a
!= ( refine1198353288_RES_a @ X2 ) ) ).
% nres.distinct(1)
thf(fact_288_nres_Oinduct,axiom,
! [P: refine424419629nres_a > $o,Nres: refine424419629nres_a] :
( ( P @ refine464223677AILi_a )
=> ( ! [X3: set_a] : ( P @ ( refine1198353288_RES_a @ X3 ) )
=> ( P @ Nres ) ) ) ).
% nres.induct
thf(fact_289_nres_Oexhaust,axiom,
! [Y3: refine424419629nres_a] :
( ( Y3 != refine464223677AILi_a )
=> ~ ! [X22: set_a] :
( Y3
!= ( refine1198353288_RES_a @ X22 ) ) ) ).
% nres.exhaust
thf(fact_290_sup__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [Uu: refine424419629nres_a] : ( P @ Uu @ refine464223677AILi_a )
=> ( ! [V2: set_a] : ( P @ refine464223677AILi_a @ ( refine1198353288_RES_a @ V2 ) )
=> ( ! [A5: set_a,B5: set_a] : ( P @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sup_nres.induct
thf(fact_291_less__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [X_1: refine424419629nres_a] : ( P @ refine464223677AILi_a @ X_1 )
=> ( ! [Uv: set_a] : ( P @ ( refine1198353288_RES_a @ Uv ) @ refine464223677AILi_a )
=> ( ! [A5: set_a,B5: set_a] : ( P @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_nres.induct
thf(fact_292_less__eq__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [Uu: refine424419629nres_a] : ( P @ Uu @ refine464223677AILi_a )
=> ( ! [A5: set_a,B5: set_a] : ( P @ ( refine1198353288_RES_a @ A5 ) @ ( refine1198353288_RES_a @ B5 ) )
=> ( ! [Uv: set_a] : ( P @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_eq_nres.induct
thf(fact_293_less__eq__nres_Osimps_I1_J,axiom,
! [Uu2: refine424419629nres_a] : ( ord_le519537037nres_a @ Uu2 @ refine464223677AILi_a ) ).
% less_eq_nres.simps(1)
thf(fact_294_insert__subsetI,axiom,
! [X4: a,A3: set_a,X: set_a] :
( ( member_a @ X4 @ A3 )
=> ( ( ord_less_eq_set_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X4 @ X ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_295_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_296_nres__order__simps_I2_J,axiom,
! [M4: refine787176636t_unit] :
( ( ord_le1051254044t_unit @ M4 @ bot_bo658782032t_unit )
= ( M4 = bot_bo658782032t_unit ) ) ).
% nres_order_simps(2)
thf(fact_297_nres__order__simps_I2_J,axiom,
! [M4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ M4 @ bot_bo529555393nres_a )
= ( M4 = bot_bo529555393nres_a ) ) ).
% nres_order_simps(2)
thf(fact_298_nofail__simps_I4_J,axiom,
refine579265252t_unit @ bot_bo658782032t_unit ).
% nofail_simps(4)
thf(fact_299_inres__simps_I4_J,axiom,
( ( refine558004794t_unit @ bot_bo658782032t_unit )
= ( ^ [Uu3: product_unit] : $false ) ) ).
% inres_simps(4)
thf(fact_300_bind__SUCCEED,axiom,
! [F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ bot_bo658782032t_unit @ F )
= bot_bo658782032t_unit ) ).
% bind_SUCCEED
thf(fact_301_nres__more__simps_I2_J,axiom,
! [X: set_Product_unit] :
( ( ( refine1777164439t_unit @ X )
= bot_bo658782032t_unit )
= ( X = bot_bo1087887705t_unit ) ) ).
% nres_more_simps(2)
thf(fact_302_nres__more__simps_I2_J,axiom,
! [X: set_a] :
( ( ( refine1198353288_RES_a @ X )
= bot_bo529555393nres_a )
= ( X = bot_bot_set_a ) ) ).
% nres_more_simps(2)
thf(fact_303_nres__more__simps_I1_J,axiom,
! [X: set_Product_unit] :
( ( bot_bo658782032t_unit
= ( refine1777164439t_unit @ X ) )
= ( X = bot_bo1087887705t_unit ) ) ).
% nres_more_simps(1)
thf(fact_304_nres__more__simps_I1_J,axiom,
! [X: set_a] :
( ( bot_bo529555393nres_a
= ( refine1198353288_RES_a @ X ) )
= ( X = bot_bot_set_a ) ) ).
% nres_more_simps(1)
thf(fact_305_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_306_nres__order__simps_I1_J,axiom,
! [M4: refine787176636t_unit] : ( ord_le1051254044t_unit @ bot_bo658782032t_unit @ M4 ) ).
% nres_order_simps(1)
thf(fact_307_nres__order__simps_I1_J,axiom,
! [M4: refine424419629nres_a] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ M4 ) ).
% nres_order_simps(1)
thf(fact_308_nres__inequalities_I5_J,axiom,
! [X4: product_unit] :
( bot_bo658782032t_unit
!= ( refine1420258419t_unit @ X4 ) ) ).
% nres_inequalities(5)
thf(fact_309_bot__nres__def,axiom,
( bot_bo658782032t_unit
= ( refine1777164439t_unit @ bot_bo1087887705t_unit ) ) ).
% bot_nres_def
thf(fact_310_bot__nres__def,axiom,
( bot_bo529555393nres_a
= ( refine1198353288_RES_a @ bot_bot_set_a ) ) ).
% bot_nres_def
thf(fact_311_SUCCEED__rule,axiom,
! [Phi: product_unit > $o] : ( ord_le1051254044t_unit @ bot_bo658782032t_unit @ ( refine1777164439t_unit @ ( collect_Product_unit @ Phi ) ) ) ).
% SUCCEED_rule
thf(fact_312_SUCCEED__rule,axiom,
! [Phi: a > $o] : ( ord_le519537037nres_a @ bot_bo529555393nres_a @ ( refine1198353288_RES_a @ ( collect_a @ Phi ) ) ) ).
% SUCCEED_rule
thf(fact_313_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X3: a,Y6: a] :
( ( member_a @ X3 @ A3 )
=> ( ( member_a @ Y6 @ A3 )
=> ( X3 = Y6 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_314_subset__Compl__singleton,axiom,
! [A3: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_315_strictD__simp,axiom,
! [F: refine787176636t_unit > refine787176636t_unit] :
( ( refine459374679t_unit @ F )
=> ( ( F @ bot_bo658782032t_unit )
= bot_bo658782032t_unit ) ) ).
% strictD_simp
thf(fact_316_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_317_ComplI,axiom,
! [C: a,A3: set_a] :
( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) ) ) ).
% ComplI
thf(fact_318_Compl__iff,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
= ( ~ ( member_a @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_319_ibind__strict_I1_J,axiom,
! [F: product_unit > refine787176636t_unit] :
( ( refine681446406t_unit @ bot_bo658782032t_unit @ F )
= bot_bo658782032t_unit ) ).
% ibind_strict(1)
thf(fact_320_ComplD,axiom,
! [C: a,A3: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A3 ) )
=> ~ ( member_a @ C @ A3 ) ) ).
% ComplD
thf(fact_321_strictD,axiom,
! [F: refine787176636t_unit > refine787176636t_unit] :
( ( refine459374679t_unit @ F )
=> ( ( F @ bot_bo658782032t_unit )
= bot_bo658782032t_unit ) ) ).
% strictD
thf(fact_322_strictI,axiom,
! [F: refine787176636t_unit > refine787176636t_unit] :
( ( ( F @ bot_bo658782032t_unit )
= bot_bo658782032t_unit )
=> ( refine459374679t_unit @ F ) ) ).
% strictI
thf(fact_323_strict__def,axiom,
( refine459374679t_unit
= ( ^ [F2: refine787176636t_unit > refine787176636t_unit] :
( ( F2 @ bot_bo658782032t_unit )
= bot_bo658782032t_unit ) ) ) ).
% strict_def
thf(fact_324_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X5: a] : ( member_a @ X5 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_325_ASSUME__simps_I2_J,axiom,
( ( refine23983189ASSUME @ $false )
= bot_bo658782032t_unit ) ).
% ASSUME_simps(2)
thf(fact_326_pw__flat__le__iff,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( partia1658438072t_unit @ bot_bo658782032t_unit @ S @ S2 )
= ( ? [X7: product_unit] : ( refine558004794t_unit @ S @ X7 )
=> ( ( ( refine579265252t_unit @ S )
= ( refine579265252t_unit @ S2 ) )
& ! [X5: product_unit] :
( ( refine558004794t_unit @ S @ X5 )
= ( refine558004794t_unit @ S2 @ X5 ) ) ) ) ) ).
% pw_flat_le_iff
thf(fact_327_pw__ASSUME_I2_J,axiom,
! [Phi: $o,X4: product_unit] :
( ( refine558004794t_unit @ ( refine23983189ASSUME @ Phi ) @ X4 )
= Phi ) ).
% pw_ASSUME(2)
thf(fact_328_pw__ASSUME_I1_J,axiom,
! [Phi: $o] : ( refine579265252t_unit @ ( refine23983189ASSUME @ Phi ) ) ).
% pw_ASSUME(1)
thf(fact_329_flat__ord__compat_I1_J,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( partia1658438072t_unit @ bot_bo658782032t_unit @ X4 @ Y3 )
=> ( ord_le1051254044t_unit @ X4 @ Y3 ) ) ).
% flat_ord_compat(1)
thf(fact_330_flat__ord__compat_I1_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( partia906949161nres_a @ bot_bo529555393nres_a @ X4 @ Y3 )
=> ( ord_le519537037nres_a @ X4 @ Y3 ) ) ).
% flat_ord_compat(1)
thf(fact_331_pw__flat__ge__iff,axiom,
! [S: refine787176636t_unit,S2: refine787176636t_unit] :
( ( partia1658438072t_unit @ top_to177290092t_unit @ S @ S2 )
= ( ( refine579265252t_unit @ S )
=> ( ( refine579265252t_unit @ S2 )
& ! [X5: product_unit] :
( ( refine558004794t_unit @ S @ X5 )
= ( refine558004794t_unit @ S2 @ X5 ) ) ) ) ) ).
% pw_flat_ge_iff
thf(fact_332_nres__order__simps_I4_J,axiom,
! [M4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ M4 )
= ( M4 = top_to231829469nres_a ) ) ).
% nres_order_simps(4)
thf(fact_333_nofail__simps_I1_J,axiom,
~ ( refine579265252t_unit @ top_to177290092t_unit ) ).
% nofail_simps(1)
thf(fact_334_inres__simps_I1_J,axiom,
( ( refine558004794t_unit @ top_to177290092t_unit )
= ( ^ [Uu3: product_unit] : $true ) ) ).
% inres_simps(1)
thf(fact_335_flat__ord__compat_I2_J,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( partia906949161nres_a @ top_to231829469nres_a @ X4 @ Y3 )
=> ( ord_le519537037nres_a @ Y3 @ X4 ) ) ).
% flat_ord_compat(2)
thf(fact_336_top__greatest,axiom,
! [A: refine424419629nres_a] : ( ord_le519537037nres_a @ A @ top_to231829469nres_a ) ).
% top_greatest
thf(fact_337_top_Oextremum__unique,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A )
= ( A = top_to231829469nres_a ) ) ).
% top.extremum_unique
thf(fact_338_top_Oextremum__uniqueI,axiom,
! [A: refine424419629nres_a] :
( ( ord_le519537037nres_a @ top_to231829469nres_a @ A )
=> ( A = top_to231829469nres_a ) ) ).
% top.extremum_uniqueI
thf(fact_339_meta__le__everything__if__top,axiom,
! [M: refine424419629nres_a,X4: refine424419629nres_a] :
( ( M = top_to231829469nres_a )
=> ( ord_le519537037nres_a @ X4 @ M ) ) ).
% meta_le_everything_if_top
thf(fact_340_not__nofail__iff,axiom,
! [S: refine787176636t_unit] :
( ( ~ ( refine579265252t_unit @ S ) )
= ( S = top_to177290092t_unit ) ) ).
% not_nofail_iff
thf(fact_341_nofail__def,axiom,
( refine579265252t_unit
= ( ^ [S3: refine787176636t_unit] : S3 != top_to177290092t_unit ) ) ).
% nofail_def
thf(fact_342_intro__nofail_I2_J,axiom,
! [S: refine787176636t_unit] :
( ( top_to177290092t_unit != S )
= ( refine579265252t_unit @ S ) ) ).
% intro_nofail(2)
thf(fact_343_nres__order__simps_I3_J,axiom,
! [M4: refine424419629nres_a] : ( ord_le519537037nres_a @ M4 @ top_to231829469nres_a ) ).
% nres_order_simps(3)
thf(fact_344_nres__inequalities_I1_J,axiom,
! [X: set_a] :
( top_to231829469nres_a
!= ( refine1198353288_RES_a @ X ) ) ).
% nres_inequalities(1)
thf(fact_345_nres__cases,axiom,
! [M4: refine424419629nres_a] :
( ( M4 != top_to231829469nres_a )
=> ~ ! [X8: set_a] :
( M4
!= ( refine1198353288_RES_a @ X8 ) ) ) ).
% nres_cases
thf(fact_346_nres__inequalities_I4_J,axiom,
bot_bo658782032t_unit != top_to177290092t_unit ).
% nres_inequalities(4)
thf(fact_347_nres__inequalities_I2_J,axiom,
top_to177290092t_unit != bot_bo658782032t_unit ).
% nres_inequalities(2)
thf(fact_348_UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_I
thf(fact_349_eq__UNIV__iff,axiom,
! [S: set_a] :
( ( S = top_top_set_a )
= ( ! [X5: a] : ( member_a @ X5 @ S ) ) ) ).
% eq_UNIV_iff
% Helper facts (9)
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y3: set_a] :
( ( if_set_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y3: set_a] :
( ( if_set_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X4: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_T,axiom,
! [X4: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a] :
( ( if_o_R1705037948nres_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001_062_I_Eo_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_T,axiom,
! [X4: $o > refine424419629nres_a,Y3: $o > refine424419629nres_a] :
( ( if_o_R1705037948nres_a @ $true @ X4 @ Y3 )
= X4 ) ).
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,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( if_Ref1369692790t_unit @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_It__Product____Type__Ounit_J_T,axiom,
! [X4: refine787176636t_unit,Y3: refine787176636t_unit] :
( ( if_Ref1369692790t_unit @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
( b
=> ( ord_le519537037nres_a @ s1 @ ( refine1198353288_RES_a @ ( collect_a @ phi ) ) ) ) ).
thf(conj_1,hypothesis,
( ~ b
=> ( ord_le519537037nres_a @ s2 @ ( refine1198353288_RES_a @ ( collect_a @ phi ) ) ) ) ).
thf(conj_2,conjecture,
ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ b @ s1 @ s2 ) @ ( refine1198353288_RES_a @ ( collect_a @ phi ) ) ).
%------------------------------------------------------------------------------