TPTP Problem File: ITP163^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP163^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Refine_Basic problem prob_374__3586996_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_374__3586996_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 304 ( 78 unt; 34 typ; 0 def)
% Number of atoms : 936 ( 248 equ; 0 cnn)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 2423 ( 117 ~; 11 |; 24 &;1741 @)
% ( 0 <=>; 530 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 127 ( 127 >; 0 *; 0 +; 0 <<)
% Number of symbols : 31 ( 28 usr; 4 con; 0-3 aty)
% Number of variables : 888 ( 86 ^; 794 !; 8 ?; 888 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:33:43.925
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J_J,type,
set_Pr384614482nres_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
produc1609974940nres_a: $tType ).
thf(ty_n_t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
refine424419629nres_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (28)
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
bNF_Ca459084161nres_a: set_Pr384614482nres_a > ( refine424419629nres_a > refine424419629nres_a ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
bNF_Ca953716171_set_a: set_Pr384614482nres_a > ( refine424419629nres_a > set_a ) > $o ).
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_Oord__class_Oless_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
ord_le1480771457nres_a: refine424419629nres_a > refine424419629nres_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_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_Product__Type_OPair_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J,type,
produc1136923598nres_a: refine424419629nres_a > refine424419629nres_a > produc1609974940nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_ORETURN_001tf__a,type,
refine2063221604TURN_a: a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oinres_001tf__a,type,
refine1001002027nres_a: refine424419629nres_a > a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oless__eq__nres__rel_001tf__a,type,
refine1048213316_rel_a: produc1609974940nres_a > produc1609974940nres_a > $o ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Oless__nres__rel_001tf__a,type,
refine260105272_rel_a: produc1609974940nres_a > produc1609974940nres_a > $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_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_001tf__a,type,
refine1198353288_RES_a: set_a > refine424419629nres_a ).
thf(sy_c_Refine__Basic__Mirabelle__kwjuvthmas_Othe__RES_001tf__a,type,
refine1822134885_RES_a: refine424419629nres_a > set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
accp_P624619091nres_a: ( produc1609974940nres_a > produc1609974940nres_a > $o ) > produc1609974940nres_a > $o ).
thf(sy_c_Zorn_Ochain__subset_001tf__a,type,
chain_subset_a: set_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_Mt__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_J,type,
member1468928819nres_a: produc1609974940nres_a > set_Pr384614482nres_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v__092_060Phi_062,type,
phi: a > $o ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (264)
thf(fact_0_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_1_nres_Oinject,axiom,
! [X2: set_a,Y2: set_a] :
( ( ( refine1198353288_RES_a @ X2 )
= ( refine1198353288_RES_a @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nres.inject
thf(fact_2_nf__inres__RES,axiom,
! [X: set_a,X3: a] :
( ( refine1312857699nres_a @ ( refine1198353288_RES_a @ X ) @ X3 )
= ( member_a @ X3 @ X ) ) ).
% nf_inres_RES
thf(fact_3_the__RES_Osimps,axiom,
! [X: set_a] :
( ( refine1822134885_RES_a @ ( refine1198353288_RES_a @ X ) )
= X ) ).
% the_RES.simps
thf(fact_4_inres__simps_I2_J,axiom,
! [X: set_a] :
( ( refine1001002027nres_a @ ( refine1198353288_RES_a @ X ) )
= ( ^ [X4: a] : ( member_a @ X4 @ X ) ) ) ).
% inres_simps(2)
thf(fact_5_nofail__simps_I2_J,axiom,
! [X: set_a] : ( refine412683989fail_a @ ( refine1198353288_RES_a @ X ) ) ).
% nofail_simps(2)
thf(fact_6_nofail__RES__conv,axiom,
( refine412683989fail_a
= ( ^ [M: refine424419629nres_a] :
? [M2: set_a] :
( M
= ( refine1198353288_RES_a @ M2 ) ) ) ) ).
% nofail_RES_conv
thf(fact_7_nres_Odistinct_I1_J,axiom,
! [X2: set_a] :
( refine464223677AILi_a
!= ( refine1198353288_RES_a @ X2 ) ) ).
% nres.distinct(1)
thf(fact_8_nres_Oinduct,axiom,
! [P: refine424419629nres_a > $o,Nres: refine424419629nres_a] :
( ( P @ refine464223677AILi_a )
=> ( ! [X5: set_a] : ( P @ ( refine1198353288_RES_a @ X5 ) )
=> ( P @ Nres ) ) ) ).
% nres.induct
thf(fact_9_nres_Oexhaust,axiom,
! [Y3: refine424419629nres_a] :
( ( Y3 != refine464223677AILi_a )
=> ~ ! [X22: set_a] :
( Y3
!= ( refine1198353288_RES_a @ X22 ) ) ) ).
% nres.exhaust
thf(fact_10_sup__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [Uu: refine424419629nres_a] : ( P @ Uu @ refine464223677AILi_a )
=> ( ! [V: set_a] : ( P @ refine464223677AILi_a @ ( refine1198353288_RES_a @ V ) )
=> ( ! [A: set_a,B: set_a] : ( P @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% sup_nres.induct
thf(fact_11_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 )
=> ( ! [A: set_a,B: set_a] : ( P @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_nres.induct
thf(fact_12_less__eq__nres_Oinduct,axiom,
! [P: refine424419629nres_a > refine424419629nres_a > $o,A0: refine424419629nres_a,A1: refine424419629nres_a] :
( ! [Uu: refine424419629nres_a] : ( P @ Uu @ refine464223677AILi_a )
=> ( ! [A: set_a,B: set_a] : ( P @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) )
=> ( ! [Uv: set_a] : ( P @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% less_eq_nres.induct
thf(fact_13_the__RES__inv,axiom,
! [M3: refine424419629nres_a] :
( ( refine412683989fail_a @ M3 )
=> ( ( refine1198353288_RES_a @ ( refine1822134885_RES_a @ M3 ) )
= M3 ) ) ).
% the_RES_inv
thf(fact_14_pw__eqI,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S )
= ( refine412683989fail_a @ S2 ) )
=> ( ! [X5: a] :
( ( refine1001002027nres_a @ S @ X5 )
= ( refine1001002027nres_a @ S2 @ X5 ) )
=> ( S = S2 ) ) ) ).
% pw_eqI
thf(fact_15_pw__eq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [S3: refine424419629nres_a,S4: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S3 )
= ( refine412683989fail_a @ S4 ) )
& ! [X4: a] :
( ( refine1001002027nres_a @ S3 @ X4 )
= ( refine1001002027nres_a @ S4 @ X4 ) ) ) ) ) ).
% pw_eq_iff
thf(fact_16_not__nofail__inres,axiom,
! [S: refine424419629nres_a,X3: a] :
( ~ ( refine412683989fail_a @ S )
=> ( refine1001002027nres_a @ S @ X3 ) ) ).
% not_nofail_inres
thf(fact_17_nf__inres__def,axiom,
( refine1312857699nres_a
= ( ^ [M: refine424419629nres_a,X4: a] :
( ( refine412683989fail_a @ M )
& ( refine1001002027nres_a @ M @ X4 ) ) ) ) ).
% nf_inres_def
thf(fact_18_inres__simps_I3_J,axiom,
! [X3: a] :
( ( refine1001002027nres_a @ ( refine2063221604TURN_a @ X3 ) )
= ( ^ [Y4: a,Z: a] : Y4 = Z
@ X3 ) ) ).
% inres_simps(3)
thf(fact_19_nofail__simps_I3_J,axiom,
! [X3: a] : ( refine412683989fail_a @ ( refine2063221604TURN_a @ X3 ) ) ).
% nofail_simps(3)
thf(fact_20_pw__leI,axiom,
! [S2: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S2 )
=> ( ( refine412683989fail_a @ S )
& ! [X5: a] :
( ( refine1001002027nres_a @ S @ X5 )
=> ( refine1001002027nres_a @ S2 @ X5 ) ) ) )
=> ( ord_le519537037nres_a @ S @ S2 ) ) ).
% pw_leI
thf(fact_21_pw__leI_H,axiom,
! [S2: refine424419629nres_a,S: refine424419629nres_a] :
( ( ( refine412683989fail_a @ S2 )
=> ( refine412683989fail_a @ S ) )
=> ( ! [X5: a] :
( ( refine412683989fail_a @ S2 )
=> ( ( refine1001002027nres_a @ S @ X5 )
=> ( refine1001002027nres_a @ S2 @ X5 ) ) )
=> ( ord_le519537037nres_a @ S @ S2 ) ) ) ).
% pw_leI'
thf(fact_22_pw__le__iff,axiom,
( ord_le519537037nres_a
= ( ^ [S3: refine424419629nres_a,S4: refine424419629nres_a] :
( ( refine412683989fail_a @ S4 )
=> ( ( refine412683989fail_a @ S3 )
& ! [X4: a] :
( ( refine1001002027nres_a @ S3 @ X4 )
=> ( refine1001002027nres_a @ S4 @ X4 ) ) ) ) ) ) ).
% pw_le_iff
thf(fact_23_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_24_less__nres_Osimps_I2_J,axiom,
! [Uv2: set_a] : ( ord_le1480771457nres_a @ ( refine1198353288_RES_a @ Uv2 ) @ refine464223677AILi_a ) ).
% less_nres.simps(2)
thf(fact_25_less__eq__nres_Ocases,axiom,
! [X3: produc1609974940nres_a] :
( ! [Uu: refine424419629nres_a] :
( X3
!= ( produc1136923598nres_a @ Uu @ refine464223677AILi_a ) )
=> ( ! [A: set_a,B: set_a] :
( X3
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) )
=> ~ ! [Uv: set_a] :
( X3
!= ( produc1136923598nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv ) ) ) ) ) ).
% less_eq_nres.cases
thf(fact_26_less__nres_Ocases,axiom,
! [X3: produc1609974940nres_a] :
( ! [Uu: refine424419629nres_a] :
( X3
!= ( produc1136923598nres_a @ refine464223677AILi_a @ Uu ) )
=> ( ! [Uv: set_a] :
( X3
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ Uv ) @ refine464223677AILi_a ) )
=> ~ ! [A: set_a,B: set_a] :
( X3
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) ) ) ) ).
% less_nres.cases
thf(fact_27_sup__nres_Ocases,axiom,
! [X3: produc1609974940nres_a] :
( ! [Uu: refine424419629nres_a] :
( X3
!= ( produc1136923598nres_a @ Uu @ refine464223677AILi_a ) )
=> ( ! [V: set_a] :
( X3
!= ( produc1136923598nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ V ) ) )
=> ~ ! [A: set_a,B: set_a] :
( X3
!= ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) ) ) ) ).
% sup_nres.cases
thf(fact_28_nres__more__simps_I6_J,axiom,
! [X3: a,Y3: a] :
( ( ( refine2063221604TURN_a @ X3 )
= ( refine2063221604TURN_a @ Y3 ) )
= ( X3 = Y3 ) ) ).
% nres_more_simps(6)
thf(fact_29_nres__order__simps_I20_J,axiom,
! [X3: a,Y3: a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ ( refine2063221604TURN_a @ Y3 ) )
= ( X3 = Y3 ) ) ).
% nres_order_simps(20)
thf(fact_30_nres__order__simps_I21_J,axiom,
! [X3: a,Y: set_a] :
( ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X3 ) @ ( refine1198353288_RES_a @ Y ) )
= ( member_a @ X3 @ Y ) ) ).
% nres_order_simps(21)
thf(fact_31_inres__def,axiom,
( refine1001002027nres_a
= ( ^ [S3: refine424419629nres_a,X4: a] : ( ord_le519537037nres_a @ ( refine2063221604TURN_a @ X4 ) @ S3 ) ) ) ).
% inres_def
thf(fact_32_less__nres_Osimps_I1_J,axiom,
! [Uu2: refine424419629nres_a] :
~ ( ord_le1480771457nres_a @ refine464223677AILi_a @ Uu2 ) ).
% less_nres.simps(1)
thf(fact_33_less__eq__nres_Osimps_I1_J,axiom,
! [Uu2: refine424419629nres_a] : ( ord_le519537037nres_a @ Uu2 @ refine464223677AILi_a ) ).
% less_eq_nres.simps(1)
thf(fact_34_le__nofailI,axiom,
! [M4: refine424419629nres_a,M5: refine424419629nres_a] :
( ( ( refine412683989fail_a @ M4 )
=> ( ord_le519537037nres_a @ M5 @ M4 ) )
=> ( ord_le519537037nres_a @ M5 @ M4 ) ) ).
% le_nofailI
thf(fact_35_pwD1,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ S @ S2 )
=> ( ( refine412683989fail_a @ S2 )
=> ( refine412683989fail_a @ S ) ) ) ).
% pwD1
thf(fact_36_pwD2,axiom,
! [S: refine424419629nres_a,S2: refine424419629nres_a,X3: a] :
( ( ord_le519537037nres_a @ S @ S2 )
=> ( ( refine1001002027nres_a @ S @ X3 )
=> ( refine1001002027nres_a @ S2 @ X3 ) ) ) ).
% pwD2
thf(fact_37_old_Oprod_Oinject,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,A3: refine424419629nres_a,B3: refine424419629nres_a] :
( ( ( produc1136923598nres_a @ A2 @ B2 )
= ( produc1136923598nres_a @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_38_prod_Oinject,axiom,
! [X1: refine424419629nres_a,X2: refine424419629nres_a,Y1: refine424419629nres_a,Y2: refine424419629nres_a] :
( ( ( produc1136923598nres_a @ X1 @ X2 )
= ( produc1136923598nres_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_39_order__mono__setup_Orefl,axiom,
! [X3: refine424419629nres_a] : ( ord_le519537037nres_a @ X3 @ X3 ) ).
% order_mono_setup.refl
thf(fact_40_order__mono__setup_Orefl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_mono_setup.refl
thf(fact_41_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A4: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_43_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_44_order_Onot__eq__order__implies__strict,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( A2 != B2 )
=> ( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ord_le1480771457nres_a @ A2 @ B2 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_45_order_Onot__eq__order__implies__strict,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_46_dual__order_Ostrict__implies__order,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ B2 @ A2 )
=> ( ord_le519537037nres_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_47_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_48_dual__order_Ostrict__iff__order,axiom,
( ord_le1480771457nres_a
= ( ^ [B4: refine424419629nres_a,A5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_49_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B4: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_50_dual__order_Oorder__iff__strict,axiom,
( ord_le519537037nres_a
= ( ^ [B4: refine424419629nres_a,A5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_51_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A5: set_a] :
( ( ord_less_set_a @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_52_order_Ostrict__implies__order,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ord_le519537037nres_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_53_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_54_less__eq__nres_Osimps_I2_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_le519537037nres_a @ ( refine1198353288_RES_a @ A2 ) @ ( refine1198353288_RES_a @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% less_eq_nres.simps(2)
thf(fact_55_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_56_less__nres_Osimps_I3_J,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_le1480771457nres_a @ ( refine1198353288_RES_a @ A2 ) @ ( refine1198353288_RES_a @ B2 ) )
= ( ord_less_set_a @ A2 @ B2 ) ) ).
% less_nres.simps(3)
thf(fact_57_order__subst1,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_58_order__subst1,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C: set_a] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_59_order__subst1,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_60_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_61_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_62_order__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_63_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_64_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_65_ord__eq__le__subst,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_66_ord__eq__le__subst,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_ord__eq__le__subst,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_68_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_69_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_ord__le__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le519537037nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_eq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [X4: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y6 )
& ( ord_le519537037nres_a @ Y6 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_74_eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
& ( ord_less_eq_set_a @ Y6 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_75_antisym,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_76_antisym,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_77_eq__refl,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( X3 = Y3 )
=> ( ord_le519537037nres_a @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_78_eq__refl,axiom,
! [X3: set_a,Y3: set_a] :
( ( X3 = Y3 )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_79_order_Otrans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ord_le519537037nres_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_80_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_81_antisym__conv,axiom,
! [Y3: refine424419629nres_a,X3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ Y3 @ X3 )
=> ( ( ord_le519537037nres_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_82_antisym__conv,axiom,
! [Y3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_83_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [A5: refine424419629nres_a,B4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A5 @ B4 )
& ( ord_le519537037nres_a @ B4 @ A5 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_84_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_85_ord__eq__le__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( A2 = B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ord_le519537037nres_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_86_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_87_ord__le__eq__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le519537037nres_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_88_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_89_order__class_Oorder_Oantisym,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_90_order__class_Oorder_Oantisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_91_order__trans,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ Z2 )
=> ( ord_le519537037nres_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_92_order__trans,axiom,
! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_93_dual__order_Orefl,axiom,
! [A2: refine424419629nres_a] : ( ord_le519537037nres_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_94_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_95_dual__order_Otrans,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( ( ord_le519537037nres_a @ C @ B2 )
=> ( ord_le519537037nres_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_96_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_97_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: refine424419629nres_a,Z: refine424419629nres_a] : Y4 = Z )
= ( ^ [A5: refine424419629nres_a,B4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B4 @ A5 )
& ( ord_le519537037nres_a @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_98_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_99_dual__order_Oantisym,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_100_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_101_order__mono__setup_Omono__if,axiom,
! [T: refine424419629nres_a,T2: refine424419629nres_a,E: refine424419629nres_a,E2: refine424419629nres_a,B2: $o] :
( ( ord_le519537037nres_a @ T @ T2 )
=> ( ( ord_le519537037nres_a @ E @ E2 )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B2 @ T @ E ) @ ( if_Ref1724547303nres_a @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_102_order__mono__setup_Omono__if,axiom,
! [T: set_a,T2: set_a,E: set_a,E2: set_a,B2: $o] :
( ( ord_less_eq_set_a @ T @ T2 )
=> ( ( ord_less_eq_set_a @ E @ E2 )
=> ( ord_less_eq_set_a @ ( if_set_a @ B2 @ T @ E ) @ ( if_set_a @ B2 @ T2 @ E2 ) ) ) ) ).
% order_mono_setup.mono_if
thf(fact_103_surj__pair,axiom,
! [P2: produc1609974940nres_a] :
? [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( P2
= ( produc1136923598nres_a @ X5 @ Y5 ) ) ).
% surj_pair
thf(fact_104_prod__cases,axiom,
! [P: produc1609974940nres_a > $o,P2: produc1609974940nres_a] :
( ! [A: refine424419629nres_a,B: refine424419629nres_a] : ( P @ ( produc1136923598nres_a @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_105_Pair__inject,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,A3: refine424419629nres_a,B3: refine424419629nres_a] :
( ( ( produc1136923598nres_a @ A2 @ B2 )
= ( produc1136923598nres_a @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_106_old_Oprod_Oexhaust,axiom,
! [Y3: produc1609974940nres_a] :
~ ! [A: refine424419629nres_a,B: refine424419629nres_a] :
( Y3
!= ( produc1136923598nres_a @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_107_old_Oprod_Oinducts,axiom,
! [P: produc1609974940nres_a > $o,Prod: produc1609974940nres_a] :
( ! [A: refine424419629nres_a,B: refine424419629nres_a] : ( P @ ( produc1136923598nres_a @ A @ B ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_108_ord__eq__less__subst,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_109_ord__eq__less__subst,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_110_ord__eq__less__subst,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_111_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_112_ord__less__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_113_ord__less__eq__subst,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_114_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_115_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_116_order__less__subst1,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_117_order__less__subst1,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C: set_a] :
( ( ord_le1480771457nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_118_order__less__subst1,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_119_order__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_120_order__less__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ord_le1480771457nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_121_order__less__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_122_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_le1480771457nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_123_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_124_order_Oasym,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ~ ( ord_le1480771457nres_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_125_order_Oasym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_126_less__imp__neq,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_127_less__imp__neq,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_128_less__asym,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ~ ( ord_le1480771457nres_a @ Y3 @ X3 ) ) ).
% less_asym
thf(fact_129_less__asym,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X3 ) ) ).
% less_asym
thf(fact_130_less__asym_H,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ~ ( ord_le1480771457nres_a @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_131_less__asym_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_132_less__trans,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( ( ord_le1480771457nres_a @ Y3 @ Z2 )
=> ( ord_le1480771457nres_a @ X3 @ Z2 ) ) ) ).
% less_trans
thf(fact_133_less__trans,axiom,
! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X3 @ Z2 ) ) ) ).
% less_trans
thf(fact_134_less__irrefl,axiom,
! [X3: refine424419629nres_a] :
~ ( ord_le1480771457nres_a @ X3 @ X3 ) ).
% less_irrefl
thf(fact_135_less__irrefl,axiom,
! [X3: set_a] :
~ ( ord_less_set_a @ X3 @ X3 ) ).
% less_irrefl
thf(fact_136_ord__eq__less__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( A2 = B2 )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ord_le1480771457nres_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_137_ord__eq__less__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_138_ord__less__eq__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le1480771457nres_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_139_ord__less__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_140_dual__order_Oasym,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ B2 @ A2 )
=> ~ ( ord_le1480771457nres_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_141_dual__order_Oasym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_142_less__imp__not__eq,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_not_eq
thf(fact_143_less__imp__not__eq,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_not_eq
thf(fact_144_less__not__sym,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ~ ( ord_le1480771457nres_a @ Y3 @ X3 ) ) ).
% less_not_sym
thf(fact_145_less__not__sym,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X3 ) ) ).
% less_not_sym
thf(fact_146_less__imp__not__eq2,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% less_imp_not_eq2
thf(fact_147_less__imp__not__eq2,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% less_imp_not_eq2
thf(fact_148_less__imp__triv,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a,P: $o] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( ( ord_le1480771457nres_a @ Y3 @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_149_less__imp__triv,axiom,
! [X3: set_a,Y3: set_a,P: $o] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_150_dual__order_Oirrefl,axiom,
! [A2: refine424419629nres_a] :
~ ( ord_le1480771457nres_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_151_dual__order_Oirrefl,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_152_order_Ostrict__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ord_le1480771457nres_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_153_order_Ostrict__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_154_less__imp__not__less,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ~ ( ord_le1480771457nres_a @ Y3 @ X3 ) ) ).
% less_imp_not_less
thf(fact_155_less__imp__not__less,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ~ ( ord_less_set_a @ Y3 @ X3 ) ) ).
% less_imp_not_less
thf(fact_156_dual__order_Ostrict__trans,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ B2 @ A2 )
=> ( ( ord_le1480771457nres_a @ C @ B2 )
=> ( ord_le1480771457nres_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_157_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_158_order_Ostrict__implies__not__eq,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_159_order_Ostrict__implies__not__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_160_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_161_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_162_less__eq__nres_Oelims_I3_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ~ ( ord_le519537037nres_a @ X3 @ Xa )
=> ( ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) )
=> ~ ( ( X3 = refine464223677AILi_a )
=> ! [Uv: set_a] :
( Xa
!= ( refine1198353288_RES_a @ Uv ) ) ) ) ) ).
% less_eq_nres.elims(3)
thf(fact_163_less__eq__nres_Oelims_I2_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Xa )
=> ( ( Xa != refine464223677AILi_a )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ~ ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ).
% less_eq_nres.elims(2)
thf(fact_164_less__eq__nres_Oelims_I1_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a,Y3: $o] :
( ( ( ord_le519537037nres_a @ X3 @ Xa )
= Y3 )
=> ( ( ( Xa = refine464223677AILi_a )
=> ~ Y3 )
=> ( ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( Y3
= ( ~ ( ord_less_eq_set_a @ A @ B ) ) ) ) )
=> ~ ( ( X3 = refine464223677AILi_a )
=> ( ? [Uv: set_a] :
( Xa
= ( refine1198353288_RES_a @ Uv ) )
=> Y3 ) ) ) ) ) ).
% less_eq_nres.elims(1)
thf(fact_165_less__nres_Oelims_I3_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ~ ( ord_le1480771457nres_a @ X3 @ Xa )
=> ( ( X3 != refine464223677AILi_a )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ord_less_set_a @ A @ B ) ) ) ) ) ).
% less_nres.elims(3)
thf(fact_166_less__nres_Oelims_I2_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Xa )
=> ( ( ? [Uv: set_a] :
( X3
= ( refine1198353288_RES_a @ Uv ) )
=> ( Xa != refine464223677AILi_a ) )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ~ ( ord_less_set_a @ A @ B ) ) ) ) ) ).
% less_nres.elims(2)
thf(fact_167_less__nres_Oelims_I1_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a,Y3: $o] :
( ( ( ord_le1480771457nres_a @ X3 @ Xa )
= Y3 )
=> ( ( ( X3 = refine464223677AILi_a )
=> Y3 )
=> ( ( ? [Uv: set_a] :
( X3
= ( refine1198353288_RES_a @ Uv ) )
=> ( ( Xa = refine464223677AILi_a )
=> ~ Y3 ) )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( Y3
= ( ~ ( ord_less_set_a @ A @ B ) ) ) ) ) ) ) ) ).
% less_nres.elims(1)
thf(fact_168_leD,axiom,
! [Y3: refine424419629nres_a,X3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ Y3 @ X3 )
=> ~ ( ord_le1480771457nres_a @ X3 @ Y3 ) ) ).
% leD
thf(fact_169_leD,axiom,
! [Y3: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ~ ( ord_less_set_a @ X3 @ Y3 ) ) ).
% leD
thf(fact_170_le__less,axiom,
( ord_le519537037nres_a
= ( ^ [X4: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% le_less
thf(fact_171_le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_set_a @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% le_less
thf(fact_172_less__le,axiom,
( ord_le1480771457nres_a
= ( ^ [X4: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% less_le
thf(fact_173_less__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% less_le
thf(fact_174_order__le__less__subst1,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_175_order__le__less__subst1,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C: set_a] :
( ( ord_le519537037nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_176_order__le__less__subst1,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_177_order__le__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_178_order__le__less__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le1480771457nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_179_order__le__less__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_180_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_le1480771457nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_181_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_182_order__less__le__subst1,axiom,
! [A2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_183_order__less__le__subst1,axiom,
! [A2: set_a,F: refine424419629nres_a > set_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_184_order__less__le__subst1,axiom,
! [A2: refine424419629nres_a,F: set_a > refine424419629nres_a,B2: set_a,C: set_a] :
( ( ord_le1480771457nres_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_le519537037nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_185_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_186_order__less__le__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_187_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_le1480771457nres_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_le1480771457nres_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_188_order__less__le__subst2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,F: refine424419629nres_a > set_a,C: set_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: refine424419629nres_a,Y5: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_189_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y5: set_a] :
( ( ord_less_set_a @ X5 @ Y5 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y5 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_190_le__neq__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le1480771457nres_a @ A2 @ B2 ) ) ) ).
% le_neq_trans
thf(fact_191_le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% le_neq_trans
thf(fact_192_antisym__conv1,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ~ ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( ( ord_le519537037nres_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_193_antisym__conv1,axiom,
! [X3: set_a,Y3: set_a] :
( ~ ( ord_less_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_194_antisym__conv2,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y3 )
=> ( ( ~ ( ord_le1480771457nres_a @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_195_antisym__conv2,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ~ ( ord_less_set_a @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_196_less__imp__le,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( ord_le519537037nres_a @ X3 @ Y3 ) ) ).
% less_imp_le
thf(fact_197_less__imp__le,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ X3 @ Y3 ) ) ).
% less_imp_le
thf(fact_198_le__less__trans,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y3 )
=> ( ( ord_le1480771457nres_a @ Y3 @ Z2 )
=> ( ord_le1480771457nres_a @ X3 @ Z2 ) ) ) ).
% le_less_trans
thf(fact_199_le__less__trans,axiom,
! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X3 @ Z2 ) ) ) ).
% le_less_trans
thf(fact_200_less__le__trans,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a,Z2: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Y3 )
=> ( ( ord_le519537037nres_a @ Y3 @ Z2 )
=> ( ord_le1480771457nres_a @ X3 @ Z2 ) ) ) ).
% less_le_trans
thf(fact_201_less__le__trans,axiom,
! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_set_a @ X3 @ Z2 ) ) ) ).
% less_le_trans
thf(fact_202_le__imp__less__or__eq,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Y3 )
=> ( ( ord_le1480771457nres_a @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% le_imp_less_or_eq
thf(fact_203_le__imp__less__or__eq,axiom,
! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_set_a @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% le_imp_less_or_eq
thf(fact_204_less__le__not__le,axiom,
( ord_le1480771457nres_a
= ( ^ [X4: refine424419629nres_a,Y6: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X4 @ Y6 )
& ~ ( ord_le519537037nres_a @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_205_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y6: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y6 )
& ~ ( ord_less_eq_set_a @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_206_order_Ostrict__trans1,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A2 @ B2 )
=> ( ( ord_le1480771457nres_a @ B2 @ C )
=> ( ord_le1480771457nres_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_207_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_208_order_Ostrict__trans2,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A2 @ B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ord_le1480771457nres_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_209_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_210_order_Oorder__iff__strict,axiom,
( ord_le519537037nres_a
= ( ^ [A5: refine424419629nres_a,B4: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_211_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_set_a @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_212_order_Ostrict__iff__order,axiom,
( ord_le1480771457nres_a
= ( ^ [A5: refine424419629nres_a,B4: refine424419629nres_a] :
( ( ord_le519537037nres_a @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_213_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_214_dual__order_Ostrict__trans1,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le519537037nres_a @ B2 @ A2 )
=> ( ( ord_le1480771457nres_a @ C @ B2 )
=> ( ord_le1480771457nres_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_215_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_216_dual__order_Ostrict__trans2,axiom,
! [B2: refine424419629nres_a,A2: refine424419629nres_a,C: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ B2 @ A2 )
=> ( ( ord_le519537037nres_a @ C @ B2 )
=> ( ord_le1480771457nres_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_217_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_218_relChain__def,axiom,
( bNF_Ca459084161nres_a
= ( ^ [R: set_Pr384614482nres_a,As: refine424419629nres_a > refine424419629nres_a] :
! [I: refine424419629nres_a,J: refine424419629nres_a] :
( ( member1468928819nres_a @ ( produc1136923598nres_a @ I @ J ) @ R )
=> ( ord_le519537037nres_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_219_relChain__def,axiom,
( bNF_Ca953716171_set_a
= ( ^ [R: set_Pr384614482nres_a,As: refine424419629nres_a > set_a] :
! [I: refine424419629nres_a,J: refine424419629nres_a] :
( ( member1468928819nres_a @ ( produc1136923598nres_a @ I @ J ) @ R )
=> ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).
% relChain_def
thf(fact_220_verit__comp__simplify1_I1_J,axiom,
! [A2: refine424419629nres_a] :
~ ( ord_le1480771457nres_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_221_verit__comp__simplify1_I1_J,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_222_psubsetI,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( A4 != B5 )
=> ( ord_less_set_a @ A4 @ B5 ) ) ) ).
% psubsetI
thf(fact_223_subset__antisym,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_224_subsetI,axiom,
! [A4: set_a,B5: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( member_a @ X5 @ B5 ) )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ).
% subsetI
thf(fact_225_in__mono,axiom,
! [A4: set_a,B5: set_a,X3: a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_226_subsetD,axiom,
! [A4: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_227_equalityE,axiom,
! [A4: set_a,B5: set_a] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ~ ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_228_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( member_a @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_229_equalityD1,axiom,
! [A4: set_a,B5: set_a] :
( ( A4 = B5 )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_230_equalityD2,axiom,
! [A4: set_a,B5: set_a] :
( ( A4 = B5 )
=> ( ord_less_eq_set_a @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_231_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A6 )
=> ( member_a @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_232_subset__refl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% subset_refl
thf(fact_233_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_234_subset__trans,axiom,
! [A4: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_eq_set_a @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_235_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : Y4 = Z )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_236_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_237_psubsetD,axiom,
! [A4: set_a,B5: set_a,C: a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ( member_a @ C @ A4 )
=> ( member_a @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_238_psubset__trans,axiom,
! [A4: set_a,B5: set_a,C2: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ( ord_less_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A4 @ C2 ) ) ) ).
% psubset_trans
thf(fact_239_psubsetE,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ).
% psubsetE
thf(fact_240_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_241_psubset__imp__subset,axiom,
! [A4: set_a,B5: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ord_less_eq_set_a @ A4 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_242_psubset__subset__trans,axiom,
! [A4: set_a,B5: set_a,C2: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
=> ( ( ord_less_eq_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_243_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_244_subset__psubset__trans,axiom,
! [A4: set_a,B5: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
=> ( ( ord_less_set_a @ B5 @ C2 )
=> ( ord_less_set_a @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_245_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_set_a @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_246_less__nres_Opelims_I1_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a,Y3: $o] :
( ( ( ord_le1480771457nres_a @ X3 @ Xa )
= Y3 )
=> ( ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ X3 @ Xa ) )
=> ( ( ( X3 = refine464223677AILi_a )
=> ( ~ Y3
=> ~ ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ refine464223677AILi_a @ Xa ) ) ) )
=> ( ! [Uv: set_a] :
( ( X3
= ( refine1198353288_RES_a @ Uv ) )
=> ( ( Xa = refine464223677AILi_a )
=> ( Y3
=> ~ ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ Uv ) @ refine464223677AILi_a ) ) ) ) )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ( Y3
= ( ord_less_set_a @ A @ B ) )
=> ~ ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) ) ) ) ) ) ) ) ) ).
% less_nres.pelims(1)
thf(fact_247_less__nres_Opelims_I2_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ( ord_le1480771457nres_a @ X3 @ Xa )
=> ( ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ X3 @ Xa ) )
=> ( ! [Uv: set_a] :
( ( X3
= ( refine1198353288_RES_a @ Uv ) )
=> ( ( Xa = refine464223677AILi_a )
=> ~ ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ Uv ) @ refine464223677AILi_a ) ) ) )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) )
=> ~ ( ord_less_set_a @ A @ B ) ) ) ) ) ) ) ).
% less_nres.pelims(2)
thf(fact_248_less__nres_Opelims_I3_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ~ ( ord_le1480771457nres_a @ X3 @ Xa )
=> ( ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ X3 @ Xa ) )
=> ( ( ( X3 = refine464223677AILi_a )
=> ~ ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ refine464223677AILi_a @ Xa ) ) )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ( accp_P624619091nres_a @ refine260105272_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) )
=> ( ord_less_set_a @ A @ B ) ) ) ) ) ) ) ).
% less_nres.pelims(3)
thf(fact_249_less__eq__nres_Opelims_I3_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ~ ( ord_le519537037nres_a @ X3 @ Xa )
=> ( ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ X3 @ Xa ) )
=> ( ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ) )
=> ~ ( ( X3 = refine464223677AILi_a )
=> ! [Uv: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ Uv ) )
=> ~ ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv ) ) ) ) ) ) ) ) ).
% less_eq_nres.pelims(3)
thf(fact_250_less__eq__nres_Opelims_I2_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a] :
( ( ord_le519537037nres_a @ X3 @ Xa )
=> ( ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ X3 @ Xa ) )
=> ( ( ( Xa = refine464223677AILi_a )
=> ~ ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ X3 @ refine464223677AILi_a ) ) )
=> ~ ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) )
=> ~ ( ord_less_eq_set_a @ A @ B ) ) ) ) ) ) ) ).
% less_eq_nres.pelims(2)
thf(fact_251_less__eq__nres_Opelims_I1_J,axiom,
! [X3: refine424419629nres_a,Xa: refine424419629nres_a,Y3: $o] :
( ( ( ord_le519537037nres_a @ X3 @ Xa )
= Y3 )
=> ( ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ X3 @ Xa ) )
=> ( ( ( Xa = refine464223677AILi_a )
=> ( Y3
=> ~ ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ X3 @ refine464223677AILi_a ) ) ) )
=> ( ! [A: set_a] :
( ( X3
= ( refine1198353288_RES_a @ A ) )
=> ! [B: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ B ) )
=> ( ( Y3
= ( ord_less_eq_set_a @ A @ B ) )
=> ~ ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ ( refine1198353288_RES_a @ A ) @ ( refine1198353288_RES_a @ B ) ) ) ) ) )
=> ~ ( ( X3 = refine464223677AILi_a )
=> ! [Uv: set_a] :
( ( Xa
= ( refine1198353288_RES_a @ Uv ) )
=> ( ~ Y3
=> ~ ( accp_P624619091nres_a @ refine1048213316_rel_a @ ( produc1136923598nres_a @ refine464223677AILi_a @ ( refine1198353288_RES_a @ Uv ) ) ) ) ) ) ) ) ) ) ).
% less_eq_nres.pelims(1)
thf(fact_252_GreatestI2__order,axiom,
! [P: refine424419629nres_a > $o,X3: refine424419629nres_a,Q: refine424419629nres_a > $o] :
( ( P @ X3 )
=> ( ! [Y5: refine424419629nres_a] :
( ( P @ Y5 )
=> ( ord_le519537037nres_a @ Y5 @ X3 ) )
=> ( ! [X5: refine424419629nres_a] :
( ( P @ X5 )
=> ( ! [Y7: refine424419629nres_a] :
( ( P @ Y7 )
=> ( ord_le519537037nres_a @ Y7 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_1714329108nres_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_253_GreatestI2__order,axiom,
! [P: set_a > $o,X3: set_a,Q: set_a > $o] :
( ( P @ X3 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X3 ) )
=> ( ! [X5: set_a] :
( ( P @ X5 )
=> ( ! [Y7: set_a] :
( ( P @ Y7 )
=> ( ord_less_eq_set_a @ Y7 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_254_Greatest__equality,axiom,
! [P: refine424419629nres_a > $o,X3: refine424419629nres_a] :
( ( P @ X3 )
=> ( ! [Y5: refine424419629nres_a] :
( ( P @ Y5 )
=> ( ord_le519537037nres_a @ Y5 @ X3 ) )
=> ( ( order_1714329108nres_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_255_Greatest__equality,axiom,
! [P: set_a > $o,X3: set_a] :
( ( P @ X3 )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X3 ) )
=> ( ( order_Greatest_set_a @ P )
= X3 ) ) ) ).
% Greatest_equality
thf(fact_256_subset__Collect__conv,axiom,
! [S: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ S @ ( collect_a @ P ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ S )
=> ( P @ X4 ) ) ) ) ).
% subset_Collect_conv
thf(fact_257_relprop__triggers_I6_J,axiom,
! [R2: set_a,R3: set_a] :
( ( ord_less_eq_set_a @ R2 @ R3 )
=> ( ord_less_eq_set_a @ R2 @ R3 ) ) ).
% relprop_triggers(6)
thf(fact_258_ord__eq__le__eq__trans,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,C: refine424419629nres_a,D: refine424419629nres_a] :
( ( A2 = B2 )
=> ( ( ord_le519537037nres_a @ B2 @ C )
=> ( ( C = D )
=> ( ord_le519537037nres_a @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_259_ord__eq__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a,D: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ( C = D )
=> ( ord_less_eq_set_a @ A2 @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_260_bex2I,axiom,
! [A2: refine424419629nres_a,B2: refine424419629nres_a,S: set_Pr384614482nres_a,P: refine424419629nres_a > refine424419629nres_a > $o] :
( ( member1468928819nres_a @ ( produc1136923598nres_a @ A2 @ B2 ) @ S )
=> ( ( ( member1468928819nres_a @ ( produc1136923598nres_a @ A2 @ B2 ) @ S )
=> ( P @ A2 @ B2 ) )
=> ? [A: refine424419629nres_a,B: refine424419629nres_a] :
( ( member1468928819nres_a @ ( produc1136923598nres_a @ A @ B ) @ S )
& ( P @ A @ B ) ) ) ) ).
% bex2I
thf(fact_261_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: refine424419629nres_a,M12: refine424419629nres_a,M22: refine424419629nres_a,M23: refine424419629nres_a] :
( ( B2
=> ( ord_le519537037nres_a @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_le519537037nres_a @ M22 @ M23 ) )
=> ( ord_le519537037nres_a @ ( if_Ref1724547303nres_a @ B2 @ M1 @ M22 ) @ ( if_Ref1724547303nres_a @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_262_Refine__Misc_Oif__mono,axiom,
! [B2: $o,M1: set_a,M12: set_a,M22: set_a,M23: set_a] :
( ( B2
=> ( ord_less_eq_set_a @ M1 @ M12 ) )
=> ( ( ~ B2
=> ( ord_less_eq_set_a @ M22 @ M23 ) )
=> ( ord_less_eq_set_a @ ( if_set_a @ B2 @ M1 @ M22 ) @ ( if_set_a @ B2 @ M12 @ M23 ) ) ) ) ).
% Refine_Misc.if_mono
thf(fact_263_chain__subset__def,axiom,
( chain_subset_a
= ( ^ [C3: set_set_a] :
! [X4: set_a] :
( ( member_set_a @ X4 @ C3 )
=> ! [Y6: set_a] :
( ( member_set_a @ Y6 @ C3 )
=> ( ( ord_less_eq_set_a @ X4 @ Y6 )
| ( ord_less_eq_set_a @ Y6 @ X4 ) ) ) ) ) ) ).
% chain_subset_def
% Helper facts (5)
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X3: set_a,Y3: set_a] :
( ( if_set_a @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X3: set_a,Y3: set_a] :
( ( if_set_a @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_3_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Refine____Basic____Mirabelle____kwjuvthmas__Onres_Itf__a_J_T,axiom,
! [X3: refine424419629nres_a,Y3: refine424419629nres_a] :
( ( if_Ref1724547303nres_a @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( refine1312857699nres_a @ ( refine1198353288_RES_a @ ( collect_a @ phi ) ) @ x )
= ( phi @ x ) ) ).
%------------------------------------------------------------------------------