%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP164^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:56:04 AM UTC 2026
% Result : Theorem 4.21s 2.06s
% Output : Refutation 4.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP164^2 : TPTP v9.2.1. Released v7.5.0.
% 0.13/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n020.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:28:42 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : ITP164^2 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.46 % Domain : Interactive Theorem Proving
% 0.31/0.46 % Problem : Sledgehammer Refine_Basic problem prob_743__3591834_1
% 0.31/0.46 % Version : Especial.
% 0.31/0.46 % English :
% 0.31/0.46
% 0.31/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.46 % Source : [Des21]
% 0.31/0.46 % Names : Refine_Basic/prob_743__3591834_1 [Des21]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.00 v7.5.0
% 0.31/0.46 % Syntax : Number of formulae : 340 ( 158 unt; 39 typ; 0 def)
% 0.31/0.46 % Number of atoms : 783 ( 261 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.31/0.46 % Number of connectives : 3660 ( 66 ~; 6 |; 40 &;3252 @)
% 0.31/0.46 % ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 28 ( 7 avg)
% 0.31/0.46 % Number of types : 3 ( 2 usr)
% 0.31/0.46 % Number of type conns : 136 ( 136 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 40 ( 37 usr; 2 con; 0-4 aty)
% 0.31/0.46 % Number of variables : 1124 ( 61 ^;1011 !; 12 ?;1124 :)
% 0.31/0.46 % ( 40 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.46 % SPC : TH1_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:19:33.107
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % Could-be-implicit typings (5)
% 0.31/0.46 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 0.31/0.46 refine1665802226e_nres: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Product__Type_Ounit,type,
% 0.31/0.46 product_unit: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.31/0.46 product_prod: $tType > $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Set_Oset,type,
% 0.31/0.46 set: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_tf_a,type,
% 0.31/0.46 a: $tType ).
% 0.31/0.46
% 0.31/0.46 % Explicit typings (34)
% 0.31/0.46 thf(sy_cl_HOL_Otype,type,
% 0.31/0.46 type:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.31/0.46 bot:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.31/0.46 ord:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.31/0.46 top:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.31/0.46 order:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.31/0.46 linorder:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.31/0.46 preorder:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.31/0.46 order_bot:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.31/0.46 order_top:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.31/0.46 comple187826305attice:
% 0.31/0.46 !>[A: $tType] : $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_If,type,
% 0.31/0.46 if:
% 0.31/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.31/0.46 bot_bot:
% 0.31/0.46 !>[A: $tType] : A ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.46 ord_less_eq:
% 0.31/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.31/0.46 order_Greatest:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.31/0.46 top_top:
% 0.31/0.46 !>[A: $tType] : A ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Partial__Function_Oflat__ord,type,
% 0.31/0.46 partial_flat_ord:
% 0.31/0.46 !>[A: $tType] : ( A > A > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.31/0.46 product_Pair:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 0.31/0.46 refine1687780735RETURN:
% 0.31/0.46 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
% 0.31/0.46 refine81118332bs_fun:
% 0.31/0.46 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 0.31/0.46 refine463715084e_bind:
% 0.31/0.46 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
% 0.31/0.46 refine1073749519nc_fun:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 0.31/0.46 refine1315500908_inres:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 0.31/0.46 refine406925620_inres:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 0.31/0.46 refine1102455758nofail:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 0.31/0.46 refine605929679le_RES:
% 0.31/0.46 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 0.31/0.46 refine1672542526he_RES:
% 0.31/0.46 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Refine__Misc_Ogalois__connection,type,
% 0.31/0.46 refine1150083786ection:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( A > B ) > ( B > A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Relation_ODomain,type,
% 0.31/0.46 domain:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Relation_Osingle__valued,type,
% 0.31/0.46 single_valued:
% 0.31/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Set_OCollect,type,
% 0.31/0.46 collect:
% 0.31/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Set_Oinsert,type,
% 0.31/0.46 insert:
% 0.31/0.46 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_Set_Othe__elem,type,
% 0.31/0.46 the_elem:
% 0.31/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_member,type,
% 0.31/0.46 member:
% 0.31/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_v_f,type,
% 0.31/0.46 f: product_unit > ( refine1665802226e_nres @ a ) ).
% 0.31/0.46
% 0.31/0.46 % Relevant facts (255)
% 0.31/0.46 thf(fact_0_bind__SUCCEED,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine463715084e_bind @ B @ A @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) @ F )
% 0.31/0.46 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bind_SUCCEED
% 0.31/0.46 thf(fact_1_bot__apply,axiom,
% 0.31/0.46 ! [C: $tType,D: $tType] :
% 0.31/0.46 ( ( bot @ C )
% 0.31/0.46 => ( ( bot_bot @ ( D > C ) )
% 0.31/0.46 = ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_apply
% 0.31/0.46 thf(fact_2_bot__fun__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( bot @ B )
% 0.31/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.31/0.46 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_fun_def
% 0.31/0.46 thf(fact_3_nres__monad1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,X2: B,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X2 ) @ F )
% 0.31/0.46 = ( F @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_monad1
% 0.31/0.46 thf(fact_4_nres__monad2,axiom,
% 0.31/0.46 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( refine463715084e_bind @ A @ A @ M @ ( refine1687780735RETURN @ A ) )
% 0.31/0.46 = M ) ).
% 0.31/0.46
% 0.31/0.46 % nres_monad2
% 0.31/0.46 thf(fact_5_abs__fun__strict,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.31/0.46 ( ( refine81118332bs_fun @ B @ A @ R @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
% 0.31/0.46 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_fun_strict
% 0.31/0.46 thf(fact_6_conc__fun__strict,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( refine1073749519nc_fun @ A @ B @ R @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
% 0.31/0.46 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_strict
% 0.31/0.46 thf(fact_7_nres__inequalities_I5_J,axiom,
% 0.31/0.46 ! [C: $tType,X2: C] :
% 0.31/0.46 ( ( bot_bot @ ( refine1665802226e_nres @ C ) )
% 0.31/0.46 != ( refine1687780735RETURN @ C @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_inequalities(5)
% 0.31/0.46 thf(fact_8_inres__simps_I4_J,axiom,
% 0.31/0.46 ! [D: $tType] :
% 0.31/0.46 ( ( refine1315500908_inres @ D @ ( bot_bot @ ( refine1665802226e_nres @ D ) ) )
% 0.31/0.46 = ( ^ [Uu: D] : $false ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_simps(4)
% 0.31/0.46 thf(fact_9_nres__order__simps_I2_J,axiom,
% 0.31/0.46 ! [B: $tType,M: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
% 0.31/0.46 = ( M
% 0.31/0.46 = ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(2)
% 0.31/0.46 thf(fact_10_nofail__simps_I4_J,axiom,
% 0.31/0.46 ! [D: $tType] : ( refine1102455758nofail @ D @ ( bot_bot @ ( refine1665802226e_nres @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_simps(4)
% 0.31/0.46 thf(fact_11_bind__FAIL,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bind_FAIL
% 0.31/0.46 thf(fact_12_nres__order__simps_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,M: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ M ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(1)
% 0.31/0.46 thf(fact_13_top__apply,axiom,
% 0.31/0.46 ! [C: $tType,D: $tType] :
% 0.31/0.46 ( ( top @ C )
% 0.31/0.46 => ( ( top_top @ ( D > C ) )
% 0.31/0.46 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_apply
% 0.31/0.46 thf(fact_14_nres__more__simps_I6_J,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y: A] :
% 0.31/0.46 ( ( ( refine1687780735RETURN @ A @ X2 )
% 0.31/0.46 = ( refine1687780735RETURN @ A @ Y ) )
% 0.31/0.46 = ( X2 = Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(6)
% 0.31/0.46 thf(fact_15_nres__order__simps_I4_J,axiom,
% 0.31/0.46 ! [D: $tType,M: refine1665802226e_nres @ D] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( top_top @ ( refine1665802226e_nres @ D ) ) @ M )
% 0.31/0.46 = ( M
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ D ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(4)
% 0.31/0.46 thf(fact_16_nofail__simps_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_simps(1)
% 0.31/0.46 thf(fact_17_inres__simps_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.31/0.46 = ( ^ [Uu: A] : $true ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_simps(1)
% 0.31/0.46 thf(fact_18_conc__fun__fail__iff_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 = ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 0.31/0.46 = ( S
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_fail_iff(2)
% 0.31/0.46 thf(fact_19_conc__fun__fail__iff_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.31/0.46 = ( S
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_fail_iff(1)
% 0.31/0.46 thf(fact_20_conc__fun__FAIL,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_FAIL
% 0.31/0.46 thf(fact_21_nres__order__simps_I20_J,axiom,
% 0.31/0.46 ! [W: $tType,X2: W,Y: W] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X2 ) @ ( refine1687780735RETURN @ W @ Y ) )
% 0.31/0.46 = ( X2 = Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(20)
% 0.31/0.46 thf(fact_22_abs__fun__simps_I1_J,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 0.31/0.46 ( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_fun_simps(1)
% 0.31/0.46 thf(fact_23_nofail__simps_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,X2: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_simps(3)
% 0.31/0.46 thf(fact_24_inres__simps_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,X2: C] :
% 0.31/0.46 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X2 ) )
% 0.31/0.46 = ( ^ [Y2: C,Z: C] : ( Y2 = Z )
% 0.31/0.46 @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_simps(3)
% 0.31/0.46 thf(fact_25_intro__nofail_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 != S )
% 0.31/0.46 = ( refine1102455758nofail @ A @ S ) ) ).
% 0.31/0.46
% 0.31/0.46 % intro_nofail(2)
% 0.31/0.46 thf(fact_26_nres__order__simps_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,M: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(3)
% 0.31/0.46 thf(fact_27_nres__inequalities_I3_J,axiom,
% 0.31/0.46 ! [C: $tType,X2: C] :
% 0.31/0.46 ( ( top_top @ ( refine1665802226e_nres @ C ) )
% 0.31/0.46 != ( refine1687780735RETURN @ C @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_inequalities(3)
% 0.31/0.46 thf(fact_28_abs__trans__additional_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans_additional(1)
% 0.31/0.46 thf(fact_29_conc__trans__additional_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans_additional(1)
% 0.31/0.46 thf(fact_30_le__funD,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funD
% 0.31/0.46 thf(fact_31_le__funE,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F: A > B,G: A > B,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funE
% 0.31/0.46 thf(fact_32_le__funI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ! [F: A > B,G: A > B] :
% 0.31/0.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_funI
% 0.31/0.46 thf(fact_33_le__fun__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ord @ B )
% 0.31/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.31/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.31/0.46 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_fun_def
% 0.31/0.46 thf(fact_34_order__subst1,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( order @ B )
% 0.31/0.46 & ( order @ A ) )
% 0.31/0.46 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.31/0.46 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.31/0.46 => ( ! [X3: B,Y3: B] :
% 0.31/0.46 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst1
% 0.31/0.46 thf(fact_35_order__subst2,axiom,
% 0.31/0.46 ! [A: $tType,C: $tType] :
% 0.31/0.46 ( ( ( order @ C )
% 0.31/0.46 & ( order @ A ) )
% 0.31/0.46 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 0.31/0.46 => ( ! [X3: A,Y3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_subst2
% 0.31/0.46 thf(fact_36_ord__eq__le__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B )
% 0.31/0.46 & ( ord @ A ) )
% 0.31/0.46 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 0.31/0.46 ( ( A3
% 0.31/0.46 = ( F @ B3 ) )
% 0.31/0.46 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 0.31/0.46 => ( ! [X3: B,Y3: B] :
% 0.31/0.46 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_subst
% 0.31/0.46 thf(fact_37_ord__le__eq__subst,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( ord @ B )
% 0.31/0.46 & ( ord @ A ) )
% 0.31/0.46 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ( F @ B3 )
% 0.31/0.46 = C3 )
% 0.31/0.46 => ( ! [X3: A,Y3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_subst
% 0.31/0.46 thf(fact_38_eq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [X: A,Y4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.46 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_iff
% 0.31/0.46 thf(fact_39_antisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.31/0.46 => ( X2 = Y ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym
% 0.31/0.46 thf(fact_40_linear,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linear
% 0.31/0.46 thf(fact_41_eq__refl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( X2 = Y )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % eq_refl
% 0.31/0.46 thf(fact_42_le__cases,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases
% 0.31/0.46 thf(fact_43_order_Otrans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order.trans
% 0.31/0.46 thf(fact_44_le__cases3,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A,Z2: A] :
% 0.31/0.46 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.31/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 0.31/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 0.31/0.46 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_cases3
% 0.31/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,P: A > $o] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.31/0.46 = ( P @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % mem_Collect_eq
% 0.31/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( collect @ A
% 0.31/0.46 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 0.31/0.46 = A2 ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mem_eq
% 0.31/0.46 thf(fact_47_Collect__cong,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 = ( Q @ X3 ) )
% 0.31/0.46 => ( ( collect @ A @ P )
% 0.31/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_cong
% 0.31/0.46 thf(fact_48_ext,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( F @ X3 )
% 0.31/0.46 = ( G @ X3 ) )
% 0.31/0.46 => ( F = G ) ) ).
% 0.31/0.46
% 0.31/0.46 % ext
% 0.31/0.46 thf(fact_49_antisym__conv,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [Y: A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 = ( X2 = Y ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % antisym_conv
% 0.31/0.46 thf(fact_50_order__class_Oorder_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.31/0.46 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.eq_iff
% 0.31/0.46 thf(fact_51_ord__eq__le__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( A3 = B3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_eq_le_trans
% 0.31/0.46 thf(fact_52_ord__le__eq__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( B3 = C3 )
% 0.31/0.46 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ord_le_eq_trans
% 0.31/0.46 thf(fact_53_order__class_Oorder_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A3: A,B3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_class.order.antisym
% 0.31/0.46 thf(fact_54_order__trans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A,Y: A,Z2: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.46 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_trans
% 0.31/0.46 thf(fact_55_dual__order_Orefl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.refl
% 0.31/0.46 thf(fact_56_linorder__wlog,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( linorder @ A )
% 0.31/0.46 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.31/0.46 ( ! [A5: A,B5: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.31/0.46 => ( P @ A5 @ B5 ) )
% 0.31/0.46 => ( ! [A5: A,B5: A] :
% 0.31/0.46 ( ( P @ B5 @ A5 )
% 0.31/0.46 => ( P @ A5 @ B5 ) )
% 0.31/0.46 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % linorder_wlog
% 0.31/0.46 thf(fact_57_dual__order_Otrans,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [B3: A,A3: A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 0.31/0.46 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.trans
% 0.31/0.46 thf(fact_58_top__greatest,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_greatest
% 0.31/0.46 thf(fact_59_pwD1,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 0.31/0.46 => ( ( refine1102455758nofail @ A @ S2 )
% 0.31/0.46 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pwD1
% 0.31/0.46 thf(fact_60_pwD2,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 0.31/0.46 => ( ( refine1315500908_inres @ A @ S @ X2 )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S2 @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pwD2
% 0.31/0.46 thf(fact_61_dual__order_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [A4: A,B4: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.31/0.46 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.eq_iff
% 0.31/0.46 thf(fact_62_dual__order_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [B3: A,A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.46 => ( A3 = B3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % dual_order.antisym
% 0.31/0.46 thf(fact_63_pw__eqI,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 = ( refine1102455758nofail @ A @ S2 ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.31/0.46 = ( refine1315500908_inres @ A @ S2 @ X3 ) )
% 0.31/0.46 => ( S = S2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_eqI
% 0.31/0.46 thf(fact_64_pw__leI,axiom,
% 0.31/0.46 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S2 )
% 0.31/0.46 => ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 & ! [X3: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S2 @ X3 ) ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_leI
% 0.31/0.46 thf(fact_65_pw__leI_H,axiom,
% 0.31/0.46 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S2 )
% 0.31/0.46 => ( refine1102455758nofail @ A @ S ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ S2 )
% 0.31/0.46 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S2 @ X3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_leI'
% 0.31/0.46 thf(fact_66_abs__trans,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C2: refine1665802226e_nres @ B,B2: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A2: refine1665802226e_nres @ C] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B2 ) @ A2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) ) @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_trans
% 0.31/0.46 thf(fact_67_inres__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1315500908_inres @ A )
% 0.31/0.46 = ( ^ [S3: refine1665802226e_nres @ A,X: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X ) @ S3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_def
% 0.31/0.46 thf(fact_68_pw__eq__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ^ [Y2: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S3 )
% 0.31/0.46 = ( refine1102455758nofail @ A @ S4 ) )
% 0.31/0.46 & ! [X: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S3 @ X )
% 0.31/0.46 = ( refine1315500908_inres @ A @ S4 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_eq_iff
% 0.31/0.46 thf(fact_69_pw__le__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ S4 )
% 0.31/0.46 => ( ( refine1102455758nofail @ A @ S3 )
% 0.31/0.46 & ! [X: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S3 @ X )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S4 @ X ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_le_iff
% 0.31/0.46 thf(fact_70_conc__trans,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A2: refine1665802226e_nres @ C] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_trans
% 0.31/0.46 thf(fact_71_le__nofailI,axiom,
% 0.31/0.46 ! [A: $tType,M2: refine1665802226e_nres @ A,M: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ M2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % le_nofailI
% 0.31/0.46 thf(fact_72_nofail__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1102455758nofail @ A )
% 0.31/0.46 = ( ^ [S3: refine1665802226e_nres @ A] :
% 0.31/0.46 ( S3
% 0.31/0.46 != ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_def
% 0.31/0.46 thf(fact_73_top_Oextremum__unique,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.31/0.46 = ( A3
% 0.31/0.46 = ( top_top @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top.extremum_unique
% 0.31/0.46 thf(fact_74_pw__bind__leI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 => ( refine1102455758nofail @ B @ M ) )
% 0.31/0.46 => ( ! [X3: B] :
% 0.31/0.46 ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 => ( ( refine1315500908_inres @ B @ M @ X3 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_bind_leI
% 0.31/0.46 thf(fact_75_top_Oextremum__uniqueI,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.31/0.46 => ( A3
% 0.31/0.46 = ( top_top @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top.extremum_uniqueI
% 0.31/0.46 thf(fact_76_not__nofail__iff,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ~ ( refine1102455758nofail @ A @ S ) )
% 0.31/0.46 = ( S
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_nofail_iff
% 0.31/0.46 thf(fact_77_pw__bind__le__iff,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S )
% 0.31/0.46 = ( ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 => ( refine1102455758nofail @ B @ M ) )
% 0.31/0.46 & ! [X: B] :
% 0.31/0.46 ( ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 & ( refine1315500908_inres @ B @ M @ X ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X ) @ S ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_bind_le_iff
% 0.31/0.46 thf(fact_78_pw__conc__nofail,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 0.31/0.46 = ( refine1102455758nofail @ B @ S ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_conc_nofail
% 0.31/0.46 thf(fact_79_not__nofail__inres,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,X2: A] :
% 0.31/0.46 ( ~ ( refine1102455758nofail @ A @ S )
% 0.31/0.46 => ( refine1315500908_inres @ A @ S @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % not_nofail_inres
% 0.31/0.46 thf(fact_80_bot_Oextremum__uniqueI,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A )
% 0.31/0.46 => ! [A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.31/0.46 => ( A3
% 0.31/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum_uniqueI
% 0.31/0.46 thf(fact_81_bot_Oextremum__unique,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A )
% 0.31/0.46 => ! [A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 0.31/0.46 = ( A3
% 0.31/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum_unique
% 0.31/0.46 thf(fact_82_bot_Oextremum,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_bot @ A )
% 0.31/0.46 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot.extremum
% 0.31/0.46 thf(fact_83_pw__bind__nofail,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M @ F ) )
% 0.31/0.46 = ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 & ! [X: B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ M @ X )
% 0.31/0.46 => ( refine1102455758nofail @ A @ ( F @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_bind_nofail
% 0.31/0.46 thf(fact_84_bind__cong,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 0.31/0.46 ( ( M3 = M4 )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M4 )
% 0.31/0.46 => ( ( F @ X3 )
% 0.31/0.46 = ( F3 @ X3 ) ) )
% 0.31/0.46 => ( ( refine463715084e_bind @ A @ B @ M3 @ F )
% 0.31/0.46 = ( refine463715084e_bind @ A @ B @ M4 @ F3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bind_cong
% 0.31/0.46 thf(fact_85_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M2 @ F3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(1)
% 0.31/0.46 thf(fact_86_nres__inequalities_I2_J,axiom,
% 0.31/0.46 ! [B: $tType] :
% 0.31/0.46 ( ( top_top @ ( refine1665802226e_nres @ B ) )
% 0.31/0.46 != ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_inequalities(2)
% 0.31/0.46 thf(fact_87_nres__inequalities_I4_J,axiom,
% 0.31/0.46 ! [D: $tType] :
% 0.31/0.46 ( ( bot_bot @ ( refine1665802226e_nres @ D ) )
% 0.31/0.46 != ( top_top @ ( refine1665802226e_nres @ D ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_inequalities(4)
% 0.31/0.46 thf(fact_88_order__mono__setup_Orefl,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_mono_setup.refl
% 0.31/0.46 thf(fact_89_nf__inres__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine406925620_inres @ A )
% 0.31/0.46 = ( ^ [M5: refine1665802226e_nres @ A,X: A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ M5 )
% 0.31/0.46 & ( refine1315500908_inres @ A @ M5 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nf_inres_def
% 0.31/0.46 thf(fact_90_meta__le__everything__if__top,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order_top @ A )
% 0.31/0.46 => ! [M3: A,X2: A] :
% 0.31/0.46 ( ( M3
% 0.31/0.46 = ( top_top @ A ) )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ M3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % meta_le_everything_if_top
% 0.31/0.46 thf(fact_91_pw__flat__le__iff,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ S @ S2 )
% 0.31/0.46 = ( ? [X4: A] : ( refine1315500908_inres @ A @ S @ X4 )
% 0.31/0.46 => ( ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 = ( refine1102455758nofail @ A @ S2 ) )
% 0.31/0.46 & ! [X: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S @ X )
% 0.31/0.46 = ( refine1315500908_inres @ A @ S2 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_flat_le_iff
% 0.31/0.46 thf(fact_92_pw__abs__inres,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B,A3: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) @ A3 )
% 0.31/0.46 = ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 0.31/0.46 => ? [C4: B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ M @ C4 )
% 0.31/0.46 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C4 @ A3 ) @ R ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_abs_inres
% 0.31/0.46 thf(fact_93_pw__abs__nofail,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 0.31/0.46 = ( ( refine1102455758nofail @ B @ M )
% 0.31/0.46 & ! [X: B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ M @ X )
% 0.31/0.46 => ( member @ B @ X @ ( domain @ B @ A @ R ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_abs_nofail
% 0.31/0.46 thf(fact_94_conc__abs__swap,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),M4: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( single_valued @ A @ B @ R )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ ( refine1073749519nc_fun @ A @ B @ R @ M3 ) )
% 0.31/0.46 = ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ M4 ) @ M3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_abs_swap
% 0.31/0.46 thf(fact_95_pw__flat__ge__iff,axiom,
% 0.31/0.46 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ S @ S2 )
% 0.31/0.46 = ( ( refine1102455758nofail @ A @ S )
% 0.31/0.46 => ( ( refine1102455758nofail @ A @ S2 )
% 0.31/0.46 & ! [X: A] :
% 0.31/0.46 ( ( refine1315500908_inres @ A @ S @ X )
% 0.31/0.46 = ( refine1315500908_inres @ A @ S2 @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % pw_flat_ge_iff
% 0.31/0.46 thf(fact_96_GreatestI2__order,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [P: A > $o,X2: A,Q: A > $o] :
% 0.31/0.46 ( ( P @ X2 )
% 0.31/0.46 => ( ! [Y3: A] :
% 0.31/0.46 ( ( P @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 => ( ! [Y5: A] :
% 0.31/0.46 ( ( P @ Y5 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 0.31/0.46 => ( Q @ X3 ) ) )
% 0.31/0.46 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % GreatestI2_order
% 0.31/0.46 thf(fact_97_Greatest__equality,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( order @ A )
% 0.31/0.46 => ! [P: A > $o,X2: A] :
% 0.31/0.46 ( ( P @ X2 )
% 0.31/0.46 => ( ! [Y3: A] :
% 0.31/0.46 ( ( P @ Y3 )
% 0.31/0.46 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 0.31/0.46 => ( ( order_Greatest @ A @ P )
% 0.31/0.46 = X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Greatest_equality
% 0.31/0.46 thf(fact_98_Refine__Misc_Oif__mono,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord @ A )
% 0.31/0.46 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 0.31/0.46 ( ( B3
% 0.31/0.46 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 0.31/0.46 => ( ( ~ B3
% 0.31/0.46 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Refine_Misc.if_mono
% 0.31/0.46 thf(fact_99_flat__ord_Odual__order_Orefl,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,A3: A] : ( partial_flat_ord @ A @ B3 @ A3 @ A3 ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.dual_order.refl
% 0.31/0.46 thf(fact_100_flat__le__mono__setup_Orefl,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,X2: A] : ( partial_flat_ord @ A @ B3 @ X2 @ X2 ) ).
% 0.31/0.46
% 0.31/0.46 % flat_le_mono_setup.refl
% 0.31/0.46 thf(fact_101_flat__ord_Odual__order_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A,B3: A,A3: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 0.31/0.46 => ( A3 = B3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.dual_order.antisym
% 0.31/0.46 thf(fact_102_flat__le__mono__setup_Omono__let,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,B3: A,F: B > A,F3: B > A,X2: B] :
% 0.31/0.46 ( ! [X3: B] : ( partial_flat_ord @ A @ B3 @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 0.31/0.46 => ( partial_flat_ord @ A @ B3 @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_le_mono_setup.mono_let
% 0.31/0.46 thf(fact_103_flat__ord_Odual__order_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A] :
% 0.31/0.46 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [A4: A,B4: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ B4 @ A4 )
% 0.31/0.46 & ( partial_flat_ord @ A @ Ba @ A4 @ B4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.dual_order.eq_iff
% 0.31/0.46 thf(fact_104_flat__le__mono__setup_Omono__if,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A,T: A,T2: A,E: A,E2: A,B3: $o] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ T @ T2 )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ Ba @ E @ E2 )
% 0.31/0.46 => ( partial_flat_ord @ A @ Ba @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_le_mono_setup.mono_if
% 0.31/0.46 thf(fact_105_flat__ord_Odual__order_Otrans,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A,B3: A,A3: A,C3: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ Ba @ C3 @ B3 )
% 0.31/0.46 => ( partial_flat_ord @ A @ Ba @ C3 @ A3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.dual_order.trans
% 0.31/0.46 thf(fact_106_flat__ord_Oord__le__eq__trans,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A,A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 0.31/0.46 => ( ( B3 = C3 )
% 0.31/0.46 => ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.ord_le_eq_trans
% 0.31/0.46 thf(fact_107_flat__ord_Oord__eq__le__trans,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A,Ba: A,C3: A] :
% 0.31/0.46 ( ( A3 = B3 )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ Ba @ B3 @ C3 )
% 0.31/0.46 => ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.ord_eq_le_trans
% 0.31/0.46 thf(fact_108_flat__ord_Oorder_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A,A3: A,B3: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
% 0.31/0.46 => ( A3 = B3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.order.antisym
% 0.31/0.46 thf(fact_109_flat__ord_Oorder_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A] :
% 0.31/0.46 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [A4: A,B4: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ A4 @ B4 )
% 0.31/0.46 & ( partial_flat_ord @ A @ Ba @ B4 @ A4 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.order.eq_iff
% 0.31/0.46 thf(fact_110_flat__ord_Oantisym__conv,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,Y: A,X2: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ B3 @ Y @ X2 )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
% 0.31/0.46 = ( X2 = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.antisym_conv
% 0.31/0.46 thf(fact_111_flat__ord_Oorder__trans,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,X2: A,Y: A,Z2: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ B3 @ Y @ Z2 )
% 0.31/0.46 => ( partial_flat_ord @ A @ B3 @ X2 @ Z2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.order_trans
% 0.31/0.46 thf(fact_112_flat__ord_Oorder_Otrans,axiom,
% 0.31/0.46 ! [A: $tType,Ba: A,A3: A,B3: A,C3: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ Ba @ B3 @ C3 )
% 0.31/0.46 => ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.order.trans
% 0.31/0.46 thf(fact_113_flat__ord_Oeq__refl,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y: A,B3: A] :
% 0.31/0.46 ( ( X2 = Y )
% 0.31/0.46 => ( partial_flat_ord @ A @ B3 @ X2 @ Y ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.eq_refl
% 0.31/0.46 thf(fact_114_flat__ord_Oantisym,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,X2: A,Y: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
% 0.31/0.46 => ( ( partial_flat_ord @ A @ B3 @ Y @ X2 )
% 0.31/0.46 => ( X2 = Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.antisym
% 0.31/0.46 thf(fact_115_flat__ord_Oeq__iff,axiom,
% 0.31/0.46 ! [A: $tType,B3: A] :
% 0.31/0.46 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [X: A,Y4: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ B3 @ X @ Y4 )
% 0.31/0.46 & ( partial_flat_ord @ A @ B3 @ Y4 @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord.eq_iff
% 0.31/0.46 thf(fact_116_nofail__antimono__fun,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
% 0.31/0.46 => ( ( refine1102455758nofail @ B @ ( G @ X2 ) )
% 0.31/0.46 => ( refine1102455758nofail @ B @ ( F @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_antimono_fun
% 0.31/0.46 thf(fact_117_flat__ord__compat_I1_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( comple187826305attice @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ ( bot_bot @ A ) @ X2 @ Y )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord_compat(1)
% 0.31/0.46 thf(fact_118_flat__ord__compat_I2_J,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( comple187826305attice @ A )
% 0.31/0.46 => ! [X2: A,Y: A] :
% 0.31/0.46 ( ( partial_flat_ord @ A @ ( top_top @ A ) @ X2 @ Y )
% 0.31/0.46 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % flat_ord_compat(2)
% 0.31/0.46 thf(fact_119_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 0.31/0.46 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ M @ M2 )
% 0.31/0.46 => ( ! [X3: A] : ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 0.31/0.46 => ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M2 @ F3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Refine_Basic_Mirabelle_tqojlsrkwy.bind_mono(2)
% 0.31/0.46 thf(fact_120_conc__fun__R__mono,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M: refine1665802226e_nres @ B] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % conc_fun_R_mono
% 0.31/0.46 thf(fact_121_order__mono__setup_Omono__let,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [F: B > A,F3: B > A,X2: B] :
% 0.31/0.46 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_mono_setup.mono_let
% 0.31/0.46 thf(fact_122_order__mono__setup_Omono__if,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( preorder @ A )
% 0.31/0.46 => ! [T: A,T2: A,E: A,E2: A,B3: $o] :
% 0.31/0.46 ( ( ord_less_eq @ A @ T @ T2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ E @ E2 )
% 0.31/0.46 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % order_mono_setup.mono_if
% 0.31/0.46 thf(fact_123_Domain__empty,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_empty
% 0.31/0.46 thf(fact_124_old_Oprod_Oinject,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
% 0.31/0.46 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.31/0.46 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.31/0.46 = ( ( A3 = A6 )
% 0.31/0.46 & ( B3 = B6 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.prod.inject
% 0.31/0.46 thf(fact_125_prod_Oinject,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.31/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.31/0.46 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.31/0.46 = ( ( X1 = Y1 )
% 0.31/0.46 & ( X22 = Y22 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod.inject
% 0.31/0.46 thf(fact_126_Domain_Oinducts,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X2: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
% 0.31/0.46 ( ( member @ A @ X2 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 => ( ! [A5: A,B5: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R3 )
% 0.31/0.46 => ( P @ A5 ) )
% 0.31/0.46 => ( P @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.inducts
% 0.31/0.46 thf(fact_127_Domain_ODomainI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 0.31/0.46 => ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.DomainI
% 0.31/0.46 thf(fact_128_bot__empty__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.31/0.46 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_empty_eq
% 0.31/0.46 thf(fact_129_surj__pair,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.31/0.46 ? [X3: A,Y3: B] :
% 0.31/0.46 ( P2
% 0.31/0.46 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % surj_pair
% 0.31/0.46 thf(fact_130_prod__cases,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.31/0.46 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.31/0.46 => ( P @ P2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_cases
% 0.31/0.46 thf(fact_131_Pair__inject,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
% 0.31/0.46 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.31/0.46 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.31/0.46 => ~ ( ( A3 = A6 )
% 0.31/0.46 => ( B3 != B6 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Pair_inject
% 0.31/0.46 thf(fact_132_prod__cases3,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/0.46 ~ ! [A5: A,B5: B,C5: C] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_cases3
% 0.31/0.46 thf(fact_133_prod__cases4,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.31/0.46 ~ ! [A5: A,B5: B,C5: C,D2: D] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_cases4
% 0.31/0.46 thf(fact_134_prod__cases5,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 0.31/0.46 ~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_cases5
% 0.31/0.46 thf(fact_135_prod__cases6,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) )] :
% 0.31/0.46 ~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F4 ) @ D2 @ ( product_Pair @ E3 @ F4 @ E4 @ F5 ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_cases6
% 0.31/0.46 thf(fact_136_prod__cases7,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 0.31/0.46 ~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4,G4: G3] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F4 @ G3 ) @ E4 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_cases7
% 0.31/0.46 thf(fact_137_prod__induct3,axiom,
% 0.31/0.46 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.31/0.46 ( ! [A5: A,B5: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) )
% 0.31/0.46 => ( P @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_induct3
% 0.31/0.46 thf(fact_138_prod__induct4,axiom,
% 0.31/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.31/0.46 ( ! [A5: A,B5: B,C5: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
% 0.31/0.46 => ( P @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_induct4
% 0.31/0.46 thf(fact_139_prod__induct5,axiom,
% 0.31/0.46 ! [E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 0.31/0.46 ( ! [A5: A,B5: B,C5: C,D2: D,E4: E3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) )
% 0.31/0.46 => ( P @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_induct5
% 0.31/0.46 thf(fact_140_prod__induct6,axiom,
% 0.31/0.46 ! [F4: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) )] :
% 0.31/0.46 ( ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F4 ) @ D2 @ ( product_Pair @ E3 @ F4 @ E4 @ F5 ) ) ) ) ) )
% 0.31/0.46 => ( P @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_induct6
% 0.31/0.46 thf(fact_141_prod__induct7,axiom,
% 0.31/0.46 ! [G3: $tType,F4: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 0.31/0.46 ( ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F4 @ G3 ) @ E4 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) )
% 0.31/0.46 => ( P @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % prod_induct7
% 0.31/0.46 thf(fact_142_old_Oprod_Oexhaust,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.31/0.46 ~ ! [A5: A,B5: B] :
% 0.31/0.46 ( Y
% 0.31/0.46 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.prod.exhaust
% 0.31/0.46 thf(fact_143_old_Oprod_Oinducts,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.31/0.46 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.31/0.46 => ( P @ Prod ) ) ).
% 0.31/0.46
% 0.31/0.46 % old.prod.inducts
% 0.31/0.46 thf(fact_144_subrelI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ! [X3: A,Y3: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 0.31/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S5 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subrelI
% 0.31/0.46 thf(fact_145_single__valued__subset,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 )
% 0.31/0.46 => ( ( single_valued @ A @ B @ S5 )
% 0.31/0.46 => ( single_valued @ A @ B @ R3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % single_valued_subset
% 0.31/0.46 thf(fact_146_single__valued__empty,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % single_valued_empty
% 0.31/0.46 thf(fact_147_Domain__empty__iff,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( ( domain @ A @ B @ R3 )
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = ( R3
% 0.31/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_empty_iff
% 0.31/0.46 thf(fact_148_Domain__mono,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_mono
% 0.31/0.46 thf(fact_149_single__valuedD,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),X2: A,Y: B,Z2: B] :
% 0.31/0.46 ( ( single_valued @ A @ B @ R3 )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R3 )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z2 ) @ R3 )
% 0.31/0.46 => ( Y = Z2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % single_valuedD
% 0.31/0.46 thf(fact_150_single__valuedI,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ! [X3: A,Y3: B,Z3: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 0.31/0.46 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z3 ) @ R3 )
% 0.31/0.46 => ( Y3 = Z3 ) ) )
% 0.31/0.46 => ( single_valued @ A @ B @ R3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % single_valuedI
% 0.31/0.46 thf(fact_151_single__valued__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( single_valued @ A @ B )
% 0.31/0.46 = ( ^ [R4: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ! [X: A,Y4: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R4 )
% 0.31/0.46 => ! [Z4: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z4 ) @ R4 )
% 0.31/0.46 => ( Y4 = Z4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % single_valued_def
% 0.31/0.46 thf(fact_152_DomainE,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 => ~ ! [B5: B] :
% 0.31/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % DomainE
% 0.31/0.46 thf(fact_153_Domain__iff,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 = ( ? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y4 ) @ R3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_iff
% 0.31/0.46 thf(fact_154_Domain_Ocases,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 => ~ ! [B5: B] :
% 0.31/0.46 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.cases
% 0.31/0.46 thf(fact_155_Domain_Osimps,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 0.31/0.46 = ( ? [A4: A,B4: B] :
% 0.31/0.46 ( ( A3 = A4 )
% 0.31/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain.simps
% 0.31/0.46 thf(fact_156_subset__empty,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = ( A2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_empty
% 0.31/0.46 thf(fact_157_empty__subsetI,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 0.31/0.46
% 0.31/0.46 % empty_subsetI
% 0.31/0.46 thf(fact_158_subsetI,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( member @ A @ X3 @ A2 )
% 0.31/0.46 => ( member @ A @ X3 @ B2 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetI
% 0.31/0.46 thf(fact_159_subset__antisym,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.31/0.46 => ( A2 = B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_antisym
% 0.31/0.46 thf(fact_160_empty__iff,axiom,
% 0.31/0.46 ! [A: $tType,C3: A] :
% 0.31/0.46 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_iff
% 0.31/0.46 thf(fact_161_all__not__in__conv,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( ! [X: A] :
% 0.31/0.46 ~ ( member @ A @ X @ A2 ) )
% 0.31/0.46 = ( A2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % all_not_in_conv
% 0.31/0.46 thf(fact_162_Collect__empty__eq,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o] :
% 0.31/0.46 ( ( ( collect @ A @ P )
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ~ ( P @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_empty_eq
% 0.31/0.46 thf(fact_163_empty__Collect__eq,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o] :
% 0.31/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.31/0.46 = ( collect @ A @ P ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ~ ( P @ X ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_Collect_eq
% 0.31/0.46 thf(fact_164_emptyE,axiom,
% 0.31/0.46 ! [A: $tType,A3: A] :
% 0.31/0.46 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % emptyE
% 0.31/0.46 thf(fact_165_equals0D,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,A3: A] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0D
% 0.31/0.46 thf(fact_166_equals0I,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ! [Y3: A] :
% 0.31/0.46 ~ ( member @ A @ Y3 @ A2 )
% 0.31/0.46 => ( A2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equals0I
% 0.31/0.46 thf(fact_167_ex__in__conv,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ( ? [X: A] : ( member @ A @ X @ A2 ) )
% 0.31/0.46 = ( A2
% 0.31/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ex_in_conv
% 0.31/0.46 thf(fact_168_bot__set__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.31/0.46 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_set_def
% 0.31/0.46 thf(fact_169_empty__not__UNIV,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.31/0.46 != ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % empty_not_UNIV
% 0.31/0.46 thf(fact_170_Collect__mono__iff,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.31/0.46 = ( ! [X: A] :
% 0.31/0.46 ( ( P @ X )
% 0.31/0.46 => ( Q @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono_iff
% 0.31/0.46 thf(fact_171_set__eq__subset,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 0.31/0.46 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.31/0.46 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % set_eq_subset
% 0.31/0.46 thf(fact_172_subset__trans,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_trans
% 0.31/0.46 thf(fact_173_Collect__mono,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ( ( P @ X3 )
% 0.31/0.46 => ( Q @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_mono
% 0.31/0.46 thf(fact_174_subset__refl,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.31/0.46
% 0.31/0.46 % subset_refl
% 0.31/0.46 thf(fact_175_subset__UNIV,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_UNIV
% 0.31/0.46 thf(fact_176_subset__iff,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.31/0.46 ! [T3: A] :
% 0.31/0.46 ( ( member @ A @ T3 @ A7 )
% 0.31/0.46 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_iff
% 0.31/0.46 thf(fact_177_equalityD2,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD2
% 0.31/0.46 thf(fact_178_equalityD1,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityD1
% 0.31/0.46 thf(fact_179_subset__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.31/0.46 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.31/0.46 ! [X: A] :
% 0.31/0.46 ( ( member @ A @ X @ A7 )
% 0.31/0.46 => ( member @ A @ X @ B7 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_eq
% 0.31/0.46 thf(fact_180_equalityE,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( A2 = B2 )
% 0.31/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % equalityE
% 0.31/0.46 thf(fact_181_subsetD,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( member @ A @ C3 @ A2 )
% 0.31/0.46 => ( member @ A @ C3 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subsetD
% 0.31/0.46 thf(fact_182_in__mono,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ( member @ A @ X2 @ A2 )
% 0.31/0.46 => ( member @ A @ X2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % in_mono
% 0.31/0.46 thf(fact_183_Collect__empty__eq__bot,axiom,
% 0.31/0.46 ! [A: $tType,P: A > $o] :
% 0.31/0.46 ( ( ( collect @ A @ P )
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = ( P
% 0.31/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Collect_empty_eq_bot
% 0.31/0.46 thf(fact_184_less__by__empty,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_by_empty
% 0.31/0.46 thf(fact_185_subset__emptyI,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ! [X3: A] :
% 0.31/0.46 ~ ( member @ A @ X3 @ A2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_emptyI
% 0.31/0.46 thf(fact_186_UNIV__I,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UNIV_I
% 0.31/0.46 thf(fact_187_UNIV__witness,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % UNIV_witness
% 0.31/0.46 thf(fact_188_top__set__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( top_top @ ( set @ A ) )
% 0.31/0.46 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_set_def
% 0.31/0.46 thf(fact_189_UNIV__eq__I,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A] :
% 0.31/0.46 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 0.31/0.46 => ( ( top_top @ ( set @ A ) )
% 0.31/0.46 = A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % UNIV_eq_I
% 0.31/0.46 thf(fact_190_top__empty__eq,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( top_top @ ( A > $o ) )
% 0.31/0.46 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % top_empty_eq
% 0.31/0.46 thf(fact_191_ssubst__Pair__rhs,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R3: A,S5: B,R: set @ ( product_prod @ A @ B ),S6: B] :
% 0.31/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S5 ) @ R )
% 0.31/0.46 => ( ( S6 = S5 )
% 0.31/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S6 ) @ R ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ssubst_Pair_rhs
% 0.31/0.46 thf(fact_192_iso__tuple__UNIV__I,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % iso_tuple_UNIV_I
% 0.31/0.46 thf(fact_193_ac__galois,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( single_valued @ A @ B @ R )
% 0.31/0.46 => ( refine1150083786ection @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R ) @ ( refine1073749519nc_fun @ A @ B @ R ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % ac_galois
% 0.31/0.46 thf(fact_194_abs__fun__simps_I3_J,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
% 0.31/0.46 ( ~ ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
% 0.31/0.46 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
% 0.31/0.46 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % abs_fun_simps(3)
% 0.31/0.46 thf(fact_195_nres_Oinject,axiom,
% 0.31/0.46 ! [A: $tType,X22: set @ A,Y22: set @ A] :
% 0.31/0.46 ( ( ( refine605929679le_RES @ A @ X22 )
% 0.31/0.46 = ( refine605929679le_RES @ A @ Y22 ) )
% 0.31/0.46 = ( X22 = Y22 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres.inject
% 0.31/0.46 thf(fact_196_nres__more__simps_I4_J,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A,Y6: set @ A] :
% 0.31/0.46 ( ( ( refine605929679le_RES @ A @ X5 )
% 0.31/0.46 = ( refine605929679le_RES @ A @ Y6 ) )
% 0.31/0.46 = ( X5 = Y6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(4)
% 0.31/0.46 thf(fact_197_top1I,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 0.31/0.46
% 0.31/0.46 % top1I
% 0.31/0.46 thf(fact_198_nofail__simps_I2_J,axiom,
% 0.31/0.46 ! [B: $tType,X5: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_simps(2)
% 0.31/0.46 thf(fact_199_inres__simps_I2_J,axiom,
% 0.31/0.46 ! [B: $tType,X5: set @ B] :
% 0.31/0.46 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X5 ) )
% 0.31/0.46 = ( ^ [X: B] : ( member @ B @ X @ X5 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % inres_simps(2)
% 0.31/0.46 thf(fact_200_nf__inres__RES,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A,X2: A] :
% 0.31/0.46 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X5 ) @ X2 )
% 0.31/0.46 = ( member @ A @ X2 @ X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nf_inres_RES
% 0.31/0.46 thf(fact_201_nf__inres__SPEC,axiom,
% 0.31/0.46 ! [A: $tType,Phi: A > $o,X2: A] :
% 0.31/0.46 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X2 )
% 0.31/0.46 = ( Phi @ X2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nf_inres_SPEC
% 0.31/0.46 thf(fact_202_nres__more__simps_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A] :
% 0.31/0.46 ( ( ( refine605929679le_RES @ A @ X5 )
% 0.31/0.46 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) )
% 0.31/0.46 = ( X5
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(2)
% 0.31/0.46 thf(fact_203_nres__more__simps_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A] :
% 0.31/0.46 ( ( ( bot_bot @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 = ( refine605929679le_RES @ A @ X5 ) )
% 0.31/0.46 = ( X5
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(1)
% 0.31/0.46 thf(fact_204_nres__order__simps_I21_J,axiom,
% 0.31/0.46 ! [X6: $tType,X2: X6,Y6: set @ X6] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ X6 ) @ ( refine1687780735RETURN @ X6 @ X2 ) @ ( refine605929679le_RES @ X6 @ Y6 ) )
% 0.31/0.46 = ( member @ X6 @ X2 @ Y6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(21)
% 0.31/0.46 thf(fact_205_nres__cases,axiom,
% 0.31/0.46 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( M
% 0.31/0.46 != ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 0.31/0.46 => ~ ! [X7: set @ A] :
% 0.31/0.46 ( M
% 0.31/0.46 != ( refine605929679le_RES @ A @ X7 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_cases
% 0.31/0.46 thf(fact_206_nres__inequalities_I1_J,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A] :
% 0.31/0.46 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 != ( refine605929679le_RES @ A @ X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_inequalities(1)
% 0.31/0.46 thf(fact_207_galois__connection__def,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ( comple187826305attice @ A )
% 0.31/0.46 & ( comple187826305attice @ B ) )
% 0.31/0.46 => ( ( refine1150083786ection @ A @ B )
% 0.31/0.46 = ( ^ [Alpha: A > B,Gamma: B > A] :
% 0.31/0.46 ! [C4: A,A4: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ C4 @ ( Gamma @ A4 ) )
% 0.31/0.46 = ( ord_less_eq @ B @ ( Alpha @ C4 ) @ A4 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % galois_connection_def
% 0.31/0.46 thf(fact_208_galois__connection_Ointro,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( comple187826305attice @ B )
% 0.31/0.46 & ( comple187826305attice @ A ) )
% 0.31/0.46 => ! [Gamma2: B > A,Alpha2: A > B] :
% 0.31/0.46 ( ! [C5: A,A5: B] :
% 0.31/0.46 ( ( ord_less_eq @ A @ C5 @ ( Gamma2 @ A5 ) )
% 0.31/0.46 = ( ord_less_eq @ B @ ( Alpha2 @ C5 ) @ A5 ) )
% 0.31/0.46 => ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % galois_connection.intro
% 0.31/0.46 thf(fact_209_galois__connection_Ogalois,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( comple187826305attice @ B )
% 0.31/0.46 & ( comple187826305attice @ A ) )
% 0.31/0.46 => ! [Alpha2: A > B,Gamma2: B > A,C3: A,A3: B] :
% 0.31/0.46 ( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
% 0.31/0.46 => ( ( ord_less_eq @ A @ C3 @ ( Gamma2 @ A3 ) )
% 0.31/0.46 = ( ord_less_eq @ B @ ( Alpha2 @ C3 ) @ A3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % galois_connection.galois
% 0.31/0.46 thf(fact_210_galois__connection_O_092_060alpha_062_092_060gamma_062__defl,axiom,
% 0.31/0.46 ! [A: $tType,B: $tType] :
% 0.31/0.46 ( ( ( comple187826305attice @ B )
% 0.31/0.46 & ( comple187826305attice @ A ) )
% 0.31/0.46 => ! [Alpha2: A > B,Gamma2: B > A,X2: B] :
% 0.31/0.46 ( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
% 0.31/0.46 => ( ord_less_eq @ B @ ( Alpha2 @ ( Gamma2 @ X2 ) ) @ X2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % galois_connection.\<alpha>\<gamma>_defl
% 0.31/0.46 thf(fact_211_galois__connection_O_092_060gamma_062_092_060alpha_062__infl,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType] :
% 0.31/0.46 ( ( ( comple187826305attice @ A )
% 0.31/0.46 & ( comple187826305attice @ B ) )
% 0.31/0.46 => ! [Alpha2: A > B,Gamma2: B > A,X2: A] :
% 0.31/0.46 ( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
% 0.31/0.46 => ( ord_less_eq @ A @ X2 @ ( Gamma2 @ ( Alpha2 @ X2 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % galois_connection.\<gamma>\<alpha>_infl
% 0.31/0.46 thf(fact_212_nres__order__simps_I5_J,axiom,
% 0.31/0.46 ! [E3: $tType,X5: set @ E3,Y6: set @ E3] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine605929679le_RES @ E3 @ X5 ) @ ( refine605929679le_RES @ E3 @ Y6 ) )
% 0.31/0.46 = ( ord_less_eq @ ( set @ E3 ) @ X5 @ Y6 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(5)
% 0.31/0.46 thf(fact_213_less__eq__nres_Osimps_I2_J,axiom,
% 0.31/0.46 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 0.31/0.46 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % less_eq_nres.simps(2)
% 0.31/0.46 thf(fact_214_bot__nres__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( bot_bot @ ( refine1665802226e_nres @ A ) )
% 0.31/0.46 = ( refine605929679le_RES @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % bot_nres_def
% 0.31/0.46 thf(fact_215_nofail__RES__conv,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1102455758nofail @ A )
% 0.31/0.46 = ( ^ [M5: refine1665802226e_nres @ A] :
% 0.31/0.46 ? [M6: set @ A] :
% 0.31/0.46 ( M5
% 0.31/0.46 = ( refine605929679le_RES @ A @ M6 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nofail_RES_conv
% 0.31/0.46 thf(fact_216_SPEC__cons__rule,axiom,
% 0.31/0.46 ! [A: $tType,M3: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 0.31/0.46 => ( ! [X3: A] :
% 0.31/0.46 ( ( Phi @ X3 )
% 0.31/0.46 => ( Psi @ X3 ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % SPEC_cons_rule
% 0.31/0.46 thf(fact_217_SPEC__trans,axiom,
% 0.31/0.46 ! [A: $tType,X2: refine1665802226e_nres @ A,Y: refine1665802226e_nres @ A,Postcond: A > $o] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ Y )
% 0.31/0.46 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
% 0.31/0.46 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % SPEC_trans
% 0.31/0.46 thf(fact_218_the__RES__inv,axiom,
% 0.31/0.46 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 0.31/0.46 ( ( refine1102455758nofail @ A @ M3 )
% 0.31/0.46 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M3 ) )
% 0.31/0.46 = M3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % the_RES_inv
% 0.31/0.46 thf(fact_219_nres__order__simps_I22_J,axiom,
% 0.31/0.46 ! [Y7: $tType,X5: set @ Y7,Y: Y7] :
% 0.31/0.46 ( ( ord_less_eq @ ( refine1665802226e_nres @ Y7 ) @ ( refine605929679le_RES @ Y7 @ X5 ) @ ( refine1687780735RETURN @ Y7 @ Y ) )
% 0.31/0.46 = ( ord_less_eq @ ( set @ Y7 ) @ X5 @ ( insert @ Y7 @ Y @ ( bot_bot @ ( set @ Y7 ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_order_simps(22)
% 0.31/0.46 thf(fact_220_insert__absorb2,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,A2: set @ A] :
% 0.31/0.46 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A2 ) )
% 0.31/0.46 = ( insert @ A @ X2 @ A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_absorb2
% 0.31/0.46 thf(fact_221_insert__iff,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 0.31/0.46 = ( ( A3 = B3 )
% 0.31/0.46 | ( member @ A @ A3 @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_iff
% 0.31/0.46 thf(fact_222_insertCI,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.31/0.46 ( ( ~ ( member @ A @ A3 @ B2 )
% 0.31/0.46 => ( A3 = B3 ) )
% 0.31/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insertCI
% 0.31/0.46 thf(fact_223_singletonI,axiom,
% 0.31/0.46 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % singletonI
% 0.31/0.46 thf(fact_224_insert__subset,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B2 )
% 0.31/0.46 = ( ( member @ A @ X2 @ B2 )
% 0.31/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_subset
% 0.31/0.46 thf(fact_225_singleton__insert__inj__eq,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,A3: A,A2: set @ A] :
% 0.31/0.46 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = ( insert @ A @ A3 @ A2 ) )
% 0.31/0.46 = ( ( A3 = B3 )
% 0.31/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % singleton_insert_inj_eq
% 0.31/0.46 thf(fact_226_singleton__insert__inj__eq_H,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 0.31/0.46 ( ( ( insert @ A @ A3 @ A2 )
% 0.31/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 = ( ( A3 = B3 )
% 0.31/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % singleton_insert_inj_eq'
% 0.31/0.46 thf(fact_227_Domain__insert,axiom,
% 0.31/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
% 0.31/0.46 ( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 ) )
% 0.31/0.46 = ( insert @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Domain_insert
% 0.31/0.46 thf(fact_228_nres__more__simps_I3_J,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A,X2: A] :
% 0.31/0.46 ( ( ( refine605929679le_RES @ A @ X5 )
% 0.31/0.46 = ( refine1687780735RETURN @ A @ X2 ) )
% 0.31/0.46 = ( X5
% 0.31/0.46 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(3)
% 0.31/0.46 thf(fact_229_nres__more__simps_I5_J,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,X5: set @ A] :
% 0.31/0.46 ( ( ( refine1687780735RETURN @ A @ X2 )
% 0.31/0.46 = ( refine605929679le_RES @ A @ X5 ) )
% 0.31/0.46 = ( ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = X5 ) ) ).
% 0.31/0.46
% 0.31/0.46 % nres_more_simps(5)
% 0.31/0.46 thf(fact_230_insert__UNIV,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] :
% 0.31/0.46 ( ( insert @ A @ X2 @ ( top_top @ ( set @ A ) ) )
% 0.31/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_UNIV
% 0.31/0.46 thf(fact_231_insert__mono,axiom,
% 0.31/0.46 ! [A: $tType,C2: set @ A,D3: set @ A,A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C2 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_mono
% 0.31/0.46 thf(fact_232_subset__insert,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ~ ( member @ A @ X2 @ A2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) )
% 0.31/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_insert
% 0.31/0.46 thf(fact_233_subset__insertI,axiom,
% 0.31/0.46 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_insertI
% 0.31/0.46 thf(fact_234_subset__insertI2,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_insertI2
% 0.31/0.46 thf(fact_235_singleton__inject,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A] :
% 0.31/0.46 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 => ( A3 = B3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % singleton_inject
% 0.31/0.46 thf(fact_236_insert__not__empty,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.31/0.46 ( ( insert @ A @ A3 @ A2 )
% 0.31/0.46 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_not_empty
% 0.31/0.46 thf(fact_237_doubleton__eq__iff,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A,C3: A,D4: A] :
% 0.31/0.46 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 = ( insert @ A @ C3 @ ( insert @ A @ D4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.31/0.46 = ( ( ( A3 = C3 )
% 0.31/0.46 & ( B3 = D4 ) )
% 0.31/0.46 | ( ( A3 = D4 )
% 0.31/0.46 & ( B3 = C3 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % doubleton_eq_iff
% 0.31/0.46 thf(fact_238_singleton__iff,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.31/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 = ( B3 = A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % singleton_iff
% 0.31/0.46 thf(fact_239_singletonD,axiom,
% 0.31/0.46 ! [A: $tType,B3: A,A3: A] :
% 0.31/0.46 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 => ( B3 = A3 ) ) ).
% 0.31/0.46
% 0.31/0.46 % singletonD
% 0.31/0.46 thf(fact_240_mk__disjoint__insert,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.31/0.46 ( ( member @ A @ A3 @ A2 )
% 0.31/0.46 => ? [B8: set @ A] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( insert @ A @ A3 @ B8 ) )
% 0.31/0.46 & ~ ( member @ A @ A3 @ B8 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % mk_disjoint_insert
% 0.31/0.46 thf(fact_241_insert__commute,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y: A,A2: set @ A] :
% 0.31/0.46 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A2 ) )
% 0.31/0.46 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_commute
% 0.31/0.46 thf(fact_242_insert__eq__iff,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
% 0.31/0.46 ( ~ ( member @ A @ A3 @ A2 )
% 0.31/0.46 => ( ~ ( member @ A @ B3 @ B2 )
% 0.31/0.46 => ( ( ( insert @ A @ A3 @ A2 )
% 0.31/0.46 = ( insert @ A @ B3 @ B2 ) )
% 0.31/0.46 = ( ( ( A3 = B3 )
% 0.31/0.46 => ( A2 = B2 ) )
% 0.31/0.46 & ( ( A3 != B3 )
% 0.31/0.46 => ? [C6: set @ A] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( insert @ A @ B3 @ C6 ) )
% 0.31/0.46 & ~ ( member @ A @ B3 @ C6 )
% 0.31/0.46 & ( B2
% 0.31/0.46 = ( insert @ A @ A3 @ C6 ) )
% 0.31/0.46 & ~ ( member @ A @ A3 @ C6 ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_eq_iff
% 0.31/0.46 thf(fact_243_insert__absorb,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,A2: set @ A] :
% 0.31/0.46 ( ( member @ A @ A3 @ A2 )
% 0.31/0.46 => ( ( insert @ A @ A3 @ A2 )
% 0.31/0.46 = A2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_absorb
% 0.31/0.46 thf(fact_244_insert__ident,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 0.31/0.46 ( ~ ( member @ A @ X2 @ A2 )
% 0.31/0.46 => ( ~ ( member @ A @ X2 @ B2 )
% 0.31/0.46 => ( ( ( insert @ A @ X2 @ A2 )
% 0.31/0.46 = ( insert @ A @ X2 @ B2 ) )
% 0.31/0.46 = ( A2 = B2 ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_ident
% 0.31/0.46 thf(fact_245_Set_Oset__insert,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,A2: set @ A] :
% 0.31/0.46 ( ( member @ A @ X2 @ A2 )
% 0.31/0.46 => ~ ! [B8: set @ A] :
% 0.31/0.46 ( ( A2
% 0.31/0.46 = ( insert @ A @ X2 @ B8 ) )
% 0.31/0.46 => ( member @ A @ X2 @ B8 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % Set.set_insert
% 0.31/0.46 thf(fact_246_insertI2,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 0.31/0.46 ( ( member @ A @ A3 @ B2 )
% 0.31/0.46 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insertI2
% 0.31/0.46 thf(fact_247_insertI1,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 0.31/0.46
% 0.31/0.46 % insertI1
% 0.31/0.46 thf(fact_248_insertE,axiom,
% 0.31/0.46 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 0.31/0.46 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 0.31/0.46 => ( ( A3 != B3 )
% 0.31/0.46 => ( member @ A @ A3 @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insertE
% 0.31/0.46 thf(fact_249_subset__singletonD,axiom,
% 0.31/0.46 ! [A: $tType,A2: set @ A,X2: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 => ( ( A2
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 | ( A2
% 0.31/0.46 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_singletonD
% 0.31/0.46 thf(fact_250_subset__singleton__iff,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A,A3: A] :
% 0.31/0.46 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 = ( ( X5
% 0.31/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.31/0.46 | ( X5
% 0.31/0.46 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % subset_singleton_iff
% 0.31/0.46 thf(fact_251_insert__subsetI,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,A2: set @ A,X5: set @ A] :
% 0.31/0.46 ( ( member @ A @ X2 @ A2 )
% 0.31/0.46 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
% 0.31/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X5 ) @ A2 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % insert_subsetI
% 0.31/0.46 thf(fact_252_RETURN__def,axiom,
% 0.31/0.46 ! [A: $tType] :
% 0.31/0.46 ( ( refine1687780735RETURN @ A )
% 0.31/0.46 = ( ^ [X: A] : ( refine605929679le_RES @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 % RETURN_def
% 0.31/0.46 thf(fact_253_the__RES_Osimps,axiom,
% 0.31/0.46 ! [A: $tType,X5: set @ A] :
% 0.31/0.46 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X5 ) )
% 0.31/0.46 = X5 ) ).
% 0.31/0.46
% 0.31/0.46 % the_RES.simps
% 0.31/0.46 thf(fact_254_the__elem__eq,axiom,
% 0.31/0.46 ! [A: $tType,X2: A] :
% 0.31/0.46 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.31/0.46 = X2 ) ).
% 0.31/0.46
% 0.31/0.46 % the_elem_eq
% 0.31/0.46
% 0.31/0.46 % Type constructors (42)
% 0.31/0.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( comple187826305attice @ A9 )
% 0.31/0.46 => ( comple187826305attice @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( order_top @ A9 )
% 0.31/0.46 => ( order_top @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( order_bot @ A9 )
% 0.31/0.46 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( preorder @ A9 )
% 0.31/0.46 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( order @ A9 )
% 0.31/0.46 => ( order @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( top @ A9 )
% 0.31/0.46 => ( top @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( ord @ A9 )
% 0.31/0.46 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.31/0.46 ! [A8: $tType,A9: $tType] :
% 0.31/0.46 ( ( bot @ A9 )
% 0.31/0.46 => ( bot @ ( A8 > A9 ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
% 0.31/0.46 ! [A8: $tType] : ( comple187826305attice @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 0.31/0.46 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 0.31/0.46 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.31/0.46 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.31/0.46 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
% 0.31/0.46 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.31/0.46 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
% 0.31/0.46 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_9,axiom,
% 0.31/0.46 comple187826305attice @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_10,axiom,
% 0.31/0.46 order_top @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 0.31/0.46 order_bot @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 0.31/0.46 preorder @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.31/0.46 linorder @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 0.31/0.46 order @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Otop_14,axiom,
% 0.31/0.46 top @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 0.31/0.46 ord @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 0.31/0.46 bot @ $o ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_17,axiom,
% 0.31/0.46 comple187826305attice @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_18,axiom,
% 0.31/0.46 order_top @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_19,axiom,
% 0.31/0.46 order_bot @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_20,axiom,
% 0.31/0.46 preorder @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_21,axiom,
% 0.31/0.46 linorder @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder_22,axiom,
% 0.31/0.46 order @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Otop_23,axiom,
% 0.31/0.46 top @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Oord_24,axiom,
% 0.31/0.46 ord @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Product__Type_Ounit___Orderings_Obot_25,axiom,
% 0.31/0.46 bot @ product_unit ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_26,axiom,
% 0.31/0.46 ! [A8: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_27,axiom,
% 0.31/0.46 ! [A8: $tType] : ( order_top @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__bot_28,axiom,
% 0.31/0.46 ! [A8: $tType] : ( order_bot @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_29,axiom,
% 0.31/0.46 ! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_30,axiom,
% 0.31/0.46 ! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_31,axiom,
% 0.31/0.46 ! [A8: $tType] : ( top @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_32,axiom,
% 0.31/0.46 ! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_33,axiom,
% 0.31/0.46 ! [A8: $tType] : ( bot @ ( refine1665802226e_nres @ A8 ) ) ).
% 0.31/0.46
% 0.31/0.46 % Helper facts (3)
% 0.31/0.46 thf(help_If_3_1_T,axiom,
% 0.31/0.46 ! [P: $o] :
% 0.31/0.46 ( ( P = $true )
% 0.31/0.46 | ( P = $false ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(help_If_2_1_T,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y: A] :
% 0.31/0.46 ( ( if @ A @ $false @ X2 @ Y )
% 0.31/0.46 = Y ) ).
% 0.31/0.46
% 0.31/0.46 thf(help_If_1_1_T,axiom,
% 0.31/0.46 ! [A: $tType,X2: A,Y: A] :
% 0.31/0.46 ( ( if @ A @ $true @ X2 @ Y )
% 0.31/0.46 = X2 ) ).
% 0.31/0.46
% 0.31/0.46 % Conjectures (1)
% 0.31/0.46 thf(conj_0,conjecture,
% 0.31/0.46 ( ( refine463715084e_bind @ product_unit @ a @ ( bot_bot @ ( refine1665802226e_nres @ product_unit ) ) @ f )
% 0.31/0.46 = ( bot_bot @ ( refine1665802226e_nres @ a ) ) ) ).
% 0.31/0.46
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 ------------------------
% 2.88/1.54 ---- Embedded in THF ---
% 2.88/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.88/1.54 %%% Generated on Tue Feb 24 08:28:43 EST 2026
% 2.88/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.88/1.54 %%% Logic specification used:
% 2.88/1.54 %%% thf(spec, logic, $$dhol).
% 2.88/1.54
% 2.88/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.kXsISafMJX/DTF2DTF15221.p
% 2.88/1.54 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres, type, refine1665802226e_nres: ($tType > $tType)).
% 2.88/1.54 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.88/1.54 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.88/1.54 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.88/1.54 thf(ty_tf_a, type, a: $tType).
% 2.88/1.54 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: ($o))).
% 2.88/1.54 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.88/1.54 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.88/1.54 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.88/1.54 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 2.88/1.54 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.88/1.54 thf(sy_c_Partial__Function_Oflat__ord, type, partial_flat_ord: (!> [A:$tType]: ((A > (A > (A > $o)))))).
% 2.88/1.54 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN, type, refine1687780735RETURN: (!> [A:$tType]: ((A > (refine1665802226e_nres @ A))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun, type, refine81118332bs_fun: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > ((refine1665802226e_nres @ B) > (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind, type, refine463715084e_bind: (!> [B:$tType,A:$tType]: (((refine1665802226e_nres @ B) > ((B > (refine1665802226e_nres @ A)) > (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun, type, refine1073749519nc_fun: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((refine1665802226e_nres @ B) > (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres, type, refine1315500908_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres, type, refine406925620_inres: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (A > $o))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail, type, refine1102455758nofail: (!> [A:$tType]: (((refine1665802226e_nres @ A) > $o)))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES, type, refine605929679le_RES: (!> [A:$tType]: (((set @ A) > (refine1665802226e_nres @ A))))).
% 2.88/1.54 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES, type, refine1672542526he_RES: (!> [A:$tType]: (((refine1665802226e_nres @ A) > (set @ A))))).
% 2.88/1.54 thf(sy_c_Refine__Misc_Ogalois__connection, type, refine1150083786ection: (!> [A:$tType,B:$tType]: (((A > B) > ((B > A) > $o))))).
% 2.88/1.54 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 2.88/1.54 thf(sy_c_Relation_Osingle__valued, type, single_valued: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > $o)))).
% 2.88/1.54 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.88/1.54 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.88/1.54 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.88/1.54 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.88/1.54 thf(sy_v_f, type, f: (product_unit > (refine1665802226e_nres @ a))).
% 2.88/1.54 thf(fact_0_bind__SUCCEED, axiom, (! [B:$tType,A:$tType,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ (bot_bot @ (refine1665802226e_nres @ B))) @ F) = (bot_bot @ (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(fact_1_bot__apply, axiom, (! [C:$tType,D:$tType]: (((bot @ C) => ((bot_bot @ (D > C)) = (^ [X:D]: ((bot_bot @ C)))))))).
% 2.88/1.54 thf(fact_2_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X:A]: ((bot_bot @ B)))))))).
% 2.88/1.54 thf(fact_3_nres__monad1, axiom, (! [A:$tType,B:$tType,X2:B,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ ((refine1687780735RETURN @ B) @ X2)) @ F) = (F @ X2))))).
% 2.88/1.54 thf(fact_4_nres__monad2, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A)]: ((((((refine463715084e_bind @ A) @ A) @ M) @ (refine1687780735RETURN @ A)) = M)))).
% 2.88/1.54 thf(fact_5_abs__fun__strict, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((refine81118332bs_fun @ B) @ A) @ R) @ (bot_bot @ (refine1665802226e_nres @ B))) = (bot_bot @ (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(fact_6_conc__fun__strict, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((((refine1073749519nc_fun @ A) @ B) @ R) @ (bot_bot @ (refine1665802226e_nres @ B))) = (bot_bot @ (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(fact_7_nres__inequalities_I5_J, axiom, (! [C:$tType,X2:C]: (((bot_bot @ (refine1665802226e_nres @ C)) != ((refine1687780735RETURN @ C) @ X2))))).
% 2.88/1.54 thf(fact_8_inres__simps_I4_J, axiom, (! [D:$tType]: ((((refine1315500908_inres @ D) @ (bot_bot @ (refine1665802226e_nres @ D))) = (^ [Uu:D]: ($false)))))).
% 2.88/1.54 thf(fact_9_nres__order__simps_I2_J, axiom, (! [B:$tType,M:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ B)) @ M) @ (bot_bot @ (refine1665802226e_nres @ B))) = (M = (bot_bot @ (refine1665802226e_nres @ B))))))).
% 2.88/1.54 thf(fact_10_nofail__simps_I4_J, axiom, (! [D:$tType]: (((refine1102455758nofail @ D) @ (bot_bot @ (refine1665802226e_nres @ D)))))).
% 2.88/1.54 thf(fact_11_bind__FAIL, axiom, (! [B:$tType,A:$tType,F:(B > (refine1665802226e_nres @ A))]: ((((((refine463715084e_bind @ B) @ A) @ (top_top @ (refine1665802226e_nres @ B))) @ F) = (top_top @ (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(fact_12_nres__order__simps_I1_J, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A)]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ (bot_bot @ (refine1665802226e_nres @ A))) @ M)))).
% 2.88/1.54 thf(fact_13_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X:D]: ((top_top @ C)))))))).
% 2.88/1.54 thf(fact_14_nres__more__simps_I6_J, axiom, (! [A:$tType,X2:A,Y:A]: (((((refine1687780735RETURN @ A) @ X2) = ((refine1687780735RETURN @ A) @ Y)) = (X2 = Y))))).
% 2.88/1.54 thf(fact_15_nres__order__simps_I4_J, axiom, (! [D:$tType,M:(refine1665802226e_nres @ D)]: (((((ord_less_eq @ (refine1665802226e_nres @ D)) @ (top_top @ (refine1665802226e_nres @ D))) @ M) = (M = (top_top @ (refine1665802226e_nres @ D))))))).
% 2.88/1.54 thf(fact_16_nofail__simps_I1_J, axiom, (! [A:$tType]: ((~ (((refine1102455758nofail @ A) @ (top_top @ (refine1665802226e_nres @ A)))))))).
% 2.88/1.54 thf(fact_17_inres__simps_I1_J, axiom, (! [A:$tType]: ((((refine1315500908_inres @ A) @ (top_top @ (refine1665802226e_nres @ A))) = (^ [Uu:A]: ($true)))))).
% 2.88/1.54 thf(fact_18_conc__fun__fail__iff_I2_J, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: ((((top_top @ (refine1665802226e_nres @ A)) = ((((refine1073749519nc_fun @ A) @ B) @ R) @ S)) = (S = (top_top @ (refine1665802226e_nres @ B))))))).
% 2.88/1.54 thf(fact_19_conc__fun__fail__iff_I1_J, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: (((((((refine1073749519nc_fun @ A) @ B) @ R) @ S) = (top_top @ (refine1665802226e_nres @ A))) = (S = (top_top @ (refine1665802226e_nres @ B))))))).
% 2.88/1.54 thf(fact_20_conc__fun__FAIL, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((((refine1073749519nc_fun @ A) @ B) @ R) @ (top_top @ (refine1665802226e_nres @ B))) = (top_top @ (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(fact_21_nres__order__simps_I20_J, axiom, (! [W:$tType,X2:W,Y:W]: (((((ord_less_eq @ (refine1665802226e_nres @ W)) @ ((refine1687780735RETURN @ W) @ X2)) @ ((refine1687780735RETURN @ W) @ Y)) = (X2 = Y))))).
% 2.88/1.54 thf(fact_22_abs__fun__simps_I1_J, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A))]: ((((((refine81118332bs_fun @ B) @ A) @ R) @ (top_top @ (refine1665802226e_nres @ B))) = (top_top @ (refine1665802226e_nres @ A)))))).
% 2.88/1.54 thf(fact_23_nofail__simps_I3_J, axiom, (! [C:$tType,X2:C]: (((refine1102455758nofail @ C) @ ((refine1687780735RETURN @ C) @ X2))))).
% 2.88/1.54 thf(fact_24_inres__simps_I3_J, axiom, (! [C:$tType,X2:C]: ((((refine1315500908_inres @ C) @ ((refine1687780735RETURN @ C) @ X2)) = ((^ [Y2:C,Z:C]: ((Y2 = Z))) @ X2))))).
% 2.88/1.54 thf(fact_25_intro__nofail_I2_J, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A)]: ((((top_top @ (refine1665802226e_nres @ A)) != S) = ((refine1102455758nofail @ A) @ S))))).
% 2.88/1.54 thf(fact_26_nres__order__simps_I3_J, axiom, (! [C:$tType,M:(refine1665802226e_nres @ C)]: ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ M) @ (top_top @ (refine1665802226e_nres @ C)))))).
% 2.88/1.54 thf(fact_27_nres__inequalities_I3_J, axiom, (! [C:$tType,X2:C]: (((top_top @ (refine1665802226e_nres @ C)) != ((refine1687780735RETURN @ C) @ X2))))).
% 2.88/1.54 thf(fact_28_abs__trans__additional_I1_J, axiom, (! [A:$tType,B:$tType,A2:(refine1665802226e_nres @ A),B2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ B2)) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ A2)) @ C2)))))).
% 2.88/1.54 thf(fact_29_conc__trans__additional_I1_J, axiom, (! [A:$tType,B:$tType,A2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),B2:(refine1665802226e_nres @ B),C2:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ B2) @ C2) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ A2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ C2))))))).
% 2.88/1.54 thf(fact_30_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.88/1.54 thf(fact_31_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2))))))))).
% 2.88/1.54 thf(fact_32_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.88/1.54 thf(fact_33_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G2 @ X))))))))))).
% 2.88/1.54 thf(fact_34_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 2.88/1.54 thf(fact_35_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C3:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C3) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C3)))))))))).
% 2.88/1.54 thf(fact_36_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C3:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C3) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C3))))))))))).
% 2.88/1.54 thf(fact_37_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C3:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C3) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C3)))))))))).
% 2.88/1.54 thf(fact_38_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X))))))))).
% 2.88/1.54 thf(fact_39_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.88/1.54 thf(fact_40_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.88/1.54 thf(fact_41_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.88/1.54 thf(fact_42_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.88/1.54 thf(fact_43_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.88/1.54 thf(fact_44_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z2:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z2)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z2)))) => (((((ord_less_eq @ A) @ X2) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y)))) => (((((ord_less_eq @ A) @ Z2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.88/1.54 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.88/1.54 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A2)))) = A2)))).
% 2.88/1.54 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.88/1.54 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.88/1.54 thf(fact_49_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.88/1.54 thf(fact_50_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 2.88/1.54 thf(fact_51_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.88/1.54 thf(fact_52_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C3) => (((ord_less_eq @ A) @ A3) @ C3))))))))).
% 2.88/1.54 thf(fact_53_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.88/1.54 thf(fact_54_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z2:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z2) => (((ord_less_eq @ A) @ X2) @ Z2))))))))).
% 2.88/1.54 thf(fact_55_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.88/1.54 thf(fact_56_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A3) @ B3))))))))).
% 2.88/1.54 thf(fact_57_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C3) @ B3) => (((ord_less_eq @ A) @ C3) @ A3))))))))).
% 2.88/1.54 thf(fact_58_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 2.88/1.54 thf(fact_59_pwD1, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2) => (((refine1102455758nofail @ A) @ S2) => ((refine1102455758nofail @ A) @ S)))))).
% 2.88/1.54 thf(fact_60_pwD2, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A),X2:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2) => ((((refine1315500908_inres @ A) @ S) @ X2) => (((refine1315500908_inres @ A) @ S2) @ X2)))))).
% 2.88/1.54 thf(fact_61_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 2.88/1.54 thf(fact_62_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.88/1.54 thf(fact_63_pw__eqI, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S) = ((refine1102455758nofail @ A) @ S2)) => ((! [X3:A]: (((((refine1315500908_inres @ A) @ S) @ X3) = (((refine1315500908_inres @ A) @ S2) @ X3)))) => (S = S2)))))).
% 2.88/1.54 thf(fact_64_pw__leI, axiom, (! [A:$tType,S2:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S2) => (((refine1102455758nofail @ A) @ S) & (! [X3:A]: (((((refine1315500908_inres @ A) @ S) @ X3) => (((refine1315500908_inres @ A) @ S2) @ X3)))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2))))).
% 2.88/1.54 thf(fact_65_pw__leI_H, axiom, (! [A:$tType,S2:(refine1665802226e_nres @ A),S:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S2) => ((refine1102455758nofail @ A) @ S)) => ((! [X3:A]: ((((refine1102455758nofail @ A) @ S2) => ((((refine1315500908_inres @ A) @ S) @ X3) => (((refine1315500908_inres @ A) @ S2) @ X3))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ S) @ S2)))))).
% 2.88/1.54 thf(fact_66_abs__trans, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ B) @ A)),C2:(refine1665802226e_nres @ B),B2:(refine1665802226e_nres @ A),R2:(set @ ((product_prod @ A) @ C)),A2:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ C2)) @ B2) => ((((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ A) @ C) @ R2) @ B2)) @ A2) => (((ord_less_eq @ (refine1665802226e_nres @ C)) @ ((((refine81118332bs_fun @ A) @ C) @ R2) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ C2))) @ A2)))))).
% 2.88/1.54 thf(fact_67_inres__def, axiom, (! [A:$tType]: (((refine1315500908_inres @ A) = (^ [S3:(refine1665802226e_nres @ A),X:A]: ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X)) @ S3))))))).
% 2.88/1.54 thf(fact_68_pw__eq__iff, axiom, (! [A:$tType]: (((^ [Y2:(refine1665802226e_nres @ A),Z:(refine1665802226e_nres @ A)]: ((Y2 = Z))) = (^ [S3:(refine1665802226e_nres @ A),S4:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ S3) = ((refine1102455758nofail @ A) @ S4)) & (! [X:A]: (((((refine1315500908_inres @ A) @ S3) @ X) = (((refine1315500908_inres @ A) @ S4) @ X))))))))))).
% 2.88/1.54 thf(fact_69_pw__le__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (refine1665802226e_nres @ A)) = (^ [S3:(refine1665802226e_nres @ A),S4:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ S4) => (((refine1102455758nofail @ A) @ S3) & (! [X:A]: (((((refine1315500908_inres @ A) @ S3) @ X) => (((refine1315500908_inres @ A) @ S4) @ X)))))))))))).
% 2.88/1.54 thf(fact_70_conc__trans, axiom, (! [A:$tType,B:$tType,C:$tType,C2:(refine1665802226e_nres @ A),R:(set @ ((product_prod @ A) @ B)),B2:(refine1665802226e_nres @ B),R2:(set @ ((product_prod @ B) @ C)),A2:(refine1665802226e_nres @ C)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ C2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ B2)) => ((((ord_less_eq @ (refine1665802226e_nres @ B)) @ B2) @ ((((refine1073749519nc_fun @ B) @ C) @ R2) @ A2)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ C2) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ ((((refine1073749519nc_fun @ B) @ C) @ R2) @ A2)))))))).
% 2.88/1.54 thf(fact_71_le__nofailI, axiom, (! [A:$tType,M2:(refine1665802226e_nres @ A),M:(refine1665802226e_nres @ A)]: (((((refine1102455758nofail @ A) @ M2) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ M2)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ M2))))).
% 2.88/1.54 thf(fact_72_nofail__def, axiom, (! [A:$tType]: (((refine1102455758nofail @ A) = (^ [S3:(refine1665802226e_nres @ A)]: ((S3 != (top_top @ (refine1665802226e_nres @ A))))))))).
% 2.88/1.54 thf(fact_73_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 2.88/1.54 thf(fact_74_pw__bind__leI, axiom, (! [B:$tType,A:$tType,S:(refine1665802226e_nres @ A),M:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A))]: (((((refine1102455758nofail @ A) @ S) => ((refine1102455758nofail @ B) @ M)) => ((! [X3:B]: ((((refine1102455758nofail @ B) @ M) => ((((refine1315500908_inres @ B) @ M) @ X3) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F @ X3)) @ S))))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ M) @ F)) @ S)))))).
% 2.88/1.54 thf(fact_75_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 2.88/1.54 thf(fact_76_not__nofail__iff, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A)]: (((~ (((refine1102455758nofail @ A) @ S))) = (S = (top_top @ (refine1665802226e_nres @ A))))))).
% 2.88/1.54 thf(fact_77_pw__bind__le__iff, axiom, (! [A:$tType,B:$tType,M:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A)),S:(refine1665802226e_nres @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine463715084e_bind @ B) @ A) @ M) @ F)) @ S) = ((((refine1102455758nofail @ A) @ S) => ((refine1102455758nofail @ B) @ M)) & (! [X:B]: (((((refine1102455758nofail @ B) @ M) & (((refine1315500908_inres @ B) @ M) @ X)) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ (F @ X)) @ S))))))))).
% 2.88/1.54 thf(fact_78_pw__conc__nofail, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(refine1665802226e_nres @ B)]: ((((refine1102455758nofail @ A) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ S)) = ((refine1102455758nofail @ B) @ S))))).
% 2.88/1.54 thf(fact_79_not__nofail__inres, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),X2:A]: (((~ (((refine1102455758nofail @ A) @ S))) => (((refine1315500908_inres @ A) @ S) @ X2))))).
% 2.88/1.54 thf(fact_80_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) => (A3 = (bot_bot @ A))))))))).
% 2.88/1.54 thf(fact_81_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (bot_bot @ A)) = (A3 = (bot_bot @ A))))))))).
% 2.88/1.54 thf(fact_82_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A3))))))).
% 2.88/1.54 thf(fact_83_pw__bind__nofail, axiom, (! [A:$tType,B:$tType,M:(refine1665802226e_nres @ B),F:(B > (refine1665802226e_nres @ A))]: ((((refine1102455758nofail @ A) @ ((((refine463715084e_bind @ B) @ A) @ M) @ F)) = (((refine1102455758nofail @ B) @ M) & (! [X:B]: (((((refine1315500908_inres @ B) @ M) @ X) => ((refine1102455758nofail @ A) @ (F @ X)))))))))).
% 2.88/1.54 thf(fact_84_bind__cong, axiom, (! [B:$tType,A:$tType,M3:(refine1665802226e_nres @ A),M4:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F3:(A > (refine1665802226e_nres @ B))]: (((M3 = M4) => ((! [X3:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X3)) @ M4) => ((F @ X3) = (F3 @ X3))))) => (((((refine463715084e_bind @ A) @ B) @ M3) @ F) = ((((refine463715084e_bind @ A) @ B) @ M4) @ F3))))))).
% 2.88/1.54 thf(fact_85_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J, axiom, (! [B:$tType,A:$tType,M:(refine1665802226e_nres @ A),M2:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F3:(A > (refine1665802226e_nres @ B))]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M) @ M2) => ((! [X3:A]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine1687780735RETURN @ A) @ X3)) @ M) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ (F @ X3)) @ (F3 @ X3))))) => (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine463715084e_bind @ A) @ B) @ M) @ F)) @ ((((refine463715084e_bind @ A) @ B) @ M2) @ F3))))))).
% 2.88/1.54 thf(fact_86_nres__inequalities_I2_J, axiom, (! [B:$tType]: (((top_top @ (refine1665802226e_nres @ B)) != (bot_bot @ (refine1665802226e_nres @ B)))))).
% 2.88/1.54 thf(fact_87_nres__inequalities_I4_J, axiom, (! [D:$tType]: (((bot_bot @ (refine1665802226e_nres @ D)) != (top_top @ (refine1665802226e_nres @ D)))))).
% 2.88/1.54 thf(fact_88_order__mono__setup_Orefl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.88/1.54 thf(fact_89_nf__inres__def, axiom, (! [A:$tType]: (((refine406925620_inres @ A) = (^ [M5:(refine1665802226e_nres @ A),X:A]: ((((refine1102455758nofail @ A) @ M5) & (((refine1315500908_inres @ A) @ M5) @ X)))))))).
% 2.88/1.54 thf(fact_90_meta__le__everything__if__top, axiom, (! [A:$tType]: (((order_top @ A) => (! [M3:A,X2:A]: (((M3 = (top_top @ A)) => (((ord_less_eq @ A) @ X2) @ M3)))))))).
% 2.88/1.54 thf(fact_91_pw__flat__le__iff, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A)]: ((((((partial_flat_ord @ (refine1665802226e_nres @ A)) @ (bot_bot @ (refine1665802226e_nres @ A))) @ S) @ S2) = ((? [X4:A]: ((((refine1315500908_inres @ A) @ S) @ X4))) => ((((refine1102455758nofail @ A) @ S) = ((refine1102455758nofail @ A) @ S2)) & (! [X:A]: (((((refine1315500908_inres @ A) @ S) @ X) = (((refine1315500908_inres @ A) @ S2) @ X)))))))))).
% 2.88/1.54 thf(fact_92_pw__abs__inres, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),M:(refine1665802226e_nres @ B),A3:A]: (((((refine1315500908_inres @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M)) @ A3) = (((refine1102455758nofail @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M)) => (? [C4:B]: (((((refine1315500908_inres @ B) @ M) @ C4) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ C4) @ A3)) @ R))))))))).
% 2.88/1.54 thf(fact_93_pw__abs__nofail, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),M:(refine1665802226e_nres @ B)]: ((((refine1102455758nofail @ A) @ ((((refine81118332bs_fun @ B) @ A) @ R) @ M)) = (((refine1102455758nofail @ B) @ M) & (! [X:B]: (((((refine1315500908_inres @ B) @ M) @ X) => (((member @ B) @ X) @ (((domain @ B) @ A) @ R)))))))))).
% 2.88/1.54 thf(fact_94_conc__abs__swap, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),M4:(refine1665802226e_nres @ A),M3:(refine1665802226e_nres @ B)]: (((((single_valued @ A) @ B) @ R) => ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M4) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ M3)) = (((ord_less_eq @ (refine1665802226e_nres @ B)) @ ((((refine81118332bs_fun @ A) @ B) @ R) @ M4)) @ M3)))))).
% 2.88/1.54 thf(fact_95_pw__flat__ge__iff, axiom, (! [A:$tType,S:(refine1665802226e_nres @ A),S2:(refine1665802226e_nres @ A)]: ((((((partial_flat_ord @ (refine1665802226e_nres @ A)) @ (top_top @ (refine1665802226e_nres @ A))) @ S) @ S2) = (((refine1102455758nofail @ A) @ S) => (((refine1102455758nofail @ A) @ S2) & (! [X:A]: (((((refine1315500908_inres @ A) @ S) @ X) = (((refine1315500908_inres @ A) @ S2) @ X)))))))))).
% 2.88/1.54 thf(fact_96_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A,Q:(A > $o)]: (((P @ X2) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ A) @ Y3) @ X2)))) => ((! [X3:A]: (((P @ X3) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ Y5) @ X3)))) => (Q @ X3))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 2.88/1.54 thf(fact_97_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A]: (((P @ X2) => ((! [Y3:A]: (((P @ Y3) => (((ord_less_eq @ A) @ Y3) @ X2)))) => (((order_Greatest @ A) @ P) = X2))))))))).
% 2.88/1.54 thf(fact_98_Refine__Misc_Oif__mono, axiom, (! [A:$tType]: (((ord @ A) => (! [B3:$o,M1:A,M12:A,M22:A,M23:A]: (((B3 => (((ord_less_eq @ A) @ M1) @ M12)) => (((~ (B3)) => (((ord_less_eq @ A) @ M22) @ M23)) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ M1) @ M22)) @ ((((if @ A) @ B3) @ M12) @ M23)))))))))).
% 2.88/1.54 thf(fact_99_flat__ord_Odual__order_Orefl, axiom, (! [A:$tType,B3:A,A3:A]: (((((partial_flat_ord @ A) @ B3) @ A3) @ A3)))).
% 2.88/1.54 thf(fact_100_flat__le__mono__setup_Orefl, axiom, (! [A:$tType,B3:A,X2:A]: (((((partial_flat_ord @ A) @ B3) @ X2) @ X2)))).
% 2.88/1.54 thf(fact_101_flat__ord_Odual__order_Oantisym, axiom, (! [A:$tType,Ba:A,B3:A,A3:A]: ((((((partial_flat_ord @ A) @ Ba) @ B3) @ A3) => (((((partial_flat_ord @ A) @ Ba) @ A3) @ B3) => (A3 = B3)))))).
% 2.88/1.54 thf(fact_102_flat__le__mono__setup_Omono__let, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),F3:(B > A),X2:B]: (((! [X3:B]: (((((partial_flat_ord @ A) @ B3) @ (F @ X3)) @ (F3 @ X3)))) => ((((partial_flat_ord @ A) @ B3) @ (F @ X2)) @ (F3 @ X2)))))).
% 2.88/1.54 thf(fact_103_flat__ord_Odual__order_Oeq__iff, axiom, (! [A:$tType,Ba:A]: (((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: ((((((partial_flat_ord @ A) @ Ba) @ B4) @ A4) & ((((partial_flat_ord @ A) @ Ba) @ A4) @ B4)))))))).
% 2.88/1.54 thf(fact_104_flat__le__mono__setup_Omono__if, axiom, (! [A:$tType,Ba:A,T:A,T2:A,E:A,E2:A,B3:$o]: ((((((partial_flat_ord @ A) @ Ba) @ T) @ T2) => (((((partial_flat_ord @ A) @ Ba) @ E) @ E2) => ((((partial_flat_ord @ A) @ Ba) @ ((((if @ A) @ B3) @ T) @ E)) @ ((((if @ A) @ B3) @ T2) @ E2))))))).
% 2.88/1.54 thf(fact_105_flat__ord_Odual__order_Otrans, axiom, (! [A:$tType,Ba:A,B3:A,A3:A,C3:A]: ((((((partial_flat_ord @ A) @ Ba) @ B3) @ A3) => (((((partial_flat_ord @ A) @ Ba) @ C3) @ B3) => ((((partial_flat_ord @ A) @ Ba) @ C3) @ A3)))))).
% 2.88/1.54 thf(fact_106_flat__ord_Oord__le__eq__trans, axiom, (! [A:$tType,Ba:A,A3:A,B3:A,C3:A]: ((((((partial_flat_ord @ A) @ Ba) @ A3) @ B3) => ((B3 = C3) => ((((partial_flat_ord @ A) @ Ba) @ A3) @ C3)))))).
% 2.88/1.54 thf(fact_107_flat__ord_Oord__eq__le__trans, axiom, (! [A:$tType,A3:A,B3:A,Ba:A,C3:A]: (((A3 = B3) => (((((partial_flat_ord @ A) @ Ba) @ B3) @ C3) => ((((partial_flat_ord @ A) @ Ba) @ A3) @ C3)))))).
% 2.88/1.54 thf(fact_108_flat__ord_Oorder_Oantisym, axiom, (! [A:$tType,Ba:A,A3:A,B3:A]: ((((((partial_flat_ord @ A) @ Ba) @ A3) @ B3) => (((((partial_flat_ord @ A) @ Ba) @ B3) @ A3) => (A3 = B3)))))).
% 2.88/1.54 thf(fact_109_flat__ord_Oorder_Oeq__iff, axiom, (! [A:$tType,Ba:A]: (((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [A4:A,B4:A]: ((((((partial_flat_ord @ A) @ Ba) @ A4) @ B4) & ((((partial_flat_ord @ A) @ Ba) @ B4) @ A4)))))))).
% 2.88/1.54 thf(fact_110_flat__ord_Oantisym__conv, axiom, (! [A:$tType,B3:A,Y:A,X2:A]: ((((((partial_flat_ord @ A) @ B3) @ Y) @ X2) => (((((partial_flat_ord @ A) @ B3) @ X2) @ Y) = (X2 = Y)))))).
% 2.88/1.54 thf(fact_111_flat__ord_Oorder__trans, axiom, (! [A:$tType,B3:A,X2:A,Y:A,Z2:A]: ((((((partial_flat_ord @ A) @ B3) @ X2) @ Y) => (((((partial_flat_ord @ A) @ B3) @ Y) @ Z2) => ((((partial_flat_ord @ A) @ B3) @ X2) @ Z2)))))).
% 2.88/1.54 thf(fact_112_flat__ord_Oorder_Otrans, axiom, (! [A:$tType,Ba:A,A3:A,B3:A,C3:A]: ((((((partial_flat_ord @ A) @ Ba) @ A3) @ B3) => (((((partial_flat_ord @ A) @ Ba) @ B3) @ C3) => ((((partial_flat_ord @ A) @ Ba) @ A3) @ C3)))))).
% 2.88/1.54 thf(fact_113_flat__ord_Oeq__refl, axiom, (! [A:$tType,X2:A,Y:A,B3:A]: (((X2 = Y) => ((((partial_flat_ord @ A) @ B3) @ X2) @ Y))))).
% 2.88/1.54 thf(fact_114_flat__ord_Oantisym, axiom, (! [A:$tType,B3:A,X2:A,Y:A]: ((((((partial_flat_ord @ A) @ B3) @ X2) @ Y) => (((((partial_flat_ord @ A) @ B3) @ Y) @ X2) => (X2 = Y)))))).
% 2.88/1.54 thf(fact_115_flat__ord_Oeq__iff, axiom, (! [A:$tType,B3:A]: (((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X:A,Y4:A]: ((((((partial_flat_ord @ A) @ B3) @ X) @ Y4) & ((((partial_flat_ord @ A) @ B3) @ Y4) @ X)))))))).
% 2.88/1.54 thf(fact_116_nofail__antimono__fun, axiom, (! [B:$tType,A:$tType,F:(A > (refine1665802226e_nres @ B)),G:(A > (refine1665802226e_nres @ B)),X2:A]: (((((ord_less_eq @ (A > (refine1665802226e_nres @ B))) @ F) @ G) => (((refine1102455758nofail @ B) @ (G @ X2)) => ((refine1102455758nofail @ B) @ (F @ X2))))))).
% 2.88/1.54 thf(fact_117_flat__ord__compat_I1_J, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [X2:A,Y:A]: ((((((partial_flat_ord @ A) @ (bot_bot @ A)) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.88/1.54 thf(fact_118_flat__ord__compat_I2_J, axiom, (! [A:$tType]: (((comple187826305attice @ A) => (! [X2:A,Y:A]: ((((((partial_flat_ord @ A) @ (top_top @ A)) @ X2) @ Y) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.88/1.54 thf(fact_119_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J, axiom, (! [B:$tType,A:$tType,M:(refine1665802226e_nres @ A),M2:(refine1665802226e_nres @ A),F:(A > (refine1665802226e_nres @ B)),F3:(A > (refine1665802226e_nres @ B))]: ((((((partial_flat_ord @ (refine1665802226e_nres @ A)) @ (top_top @ (refine1665802226e_nres @ A))) @ M) @ M2) => ((! [X3:A]: (((((partial_flat_ord @ (refine1665802226e_nres @ B)) @ (top_top @ (refine1665802226e_nres @ B))) @ (F @ X3)) @ (F3 @ X3)))) => ((((partial_flat_ord @ (refine1665802226e_nres @ B)) @ (top_top @ (refine1665802226e_nres @ B))) @ ((((refine463715084e_bind @ A) @ B) @ M) @ F)) @ ((((refine463715084e_bind @ A) @ B) @ M2) @ F3))))))).
% 2.88/1.54 thf(fact_120_conc__fun__R__mono, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),R2:(set @ ((product_prod @ A) @ B)),M:(refine1665802226e_nres @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ R2) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((((refine1073749519nc_fun @ A) @ B) @ R) @ M)) @ ((((refine1073749519nc_fun @ A) @ B) @ R2) @ M)))))).
% 2.88/1.54 thf(fact_121_order__mono__setup_Omono__let, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [F:(B > A),F3:(B > A),X2:B]: (((! [X3:B]: ((((ord_less_eq @ A) @ (F @ X3)) @ (F3 @ X3)))) => (((ord_less_eq @ A) @ (F @ X2)) @ (F3 @ X2))))))))).
% 2.88/1.54 thf(fact_122_order__mono__setup_Omono__if, axiom, (! [A:$tType]: (((preorder @ A) => (! [T:A,T2:A,E:A,E2:A,B3:$o]: (((((ord_less_eq @ A) @ T) @ T2) => ((((ord_less_eq @ A) @ E) @ E2) => (((ord_less_eq @ A) @ ((((if @ A) @ B3) @ T) @ E)) @ ((((if @ A) @ B3) @ T2) @ E2)))))))))).
% 2.88/1.54 thf(fact_123_Domain__empty, axiom, (! [B:$tType,A:$tType]: (((((domain @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (bot_bot @ (set @ A)))))).
% 2.88/1.54 thf(fact_124_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A6:A,B6:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A6) @ B6)) = ((A3 = A6) & (B3 = B6)))))).
% 2.88/1.54 thf(fact_125_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.88/1.54 thf(fact_126_Domain_Oinducts, axiom, (! [B:$tType,A:$tType,X2:A,R3:(set @ ((product_prod @ A) @ B)),P:(A > $o)]: (((((member @ A) @ X2) @ (((domain @ A) @ B) @ R3)) => ((! [A5:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R3) => (P @ A5)))) => (P @ X2)))))).
% 2.88/1.54 thf(fact_127_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R3) => (((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)))))).
% 2.88/1.54 thf(fact_128_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 2.88/1.54 thf(fact_129_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 2.88/1.54 thf(fact_130_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ P2))))).
% 2.88/1.54 thf(fact_131_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A6:A,B6:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A6) @ B6)) => (~ (((A3 = A6) => (B3 != B6)))))))).
% 2.88/1.54 thf(fact_132_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C5:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C5)))))))))).
% 2.88/1.54 thf(fact_133_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C5) @ D2))))))))))).
% 2.88/1.54 thf(fact_134_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D,E4:E3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E3)) @ C5) @ ((((product_Pair @ D) @ E3) @ D2) @ E4)))))))))))).
% 2.88/1.54 thf(fact_135_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E3:$tType,F4:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4)))))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D,E4:E3,F5:F4]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ F4)) @ D2) @ ((((product_Pair @ E3) @ F4) @ E4) @ F5))))))))))))).
% 2.88/1.54 thf(fact_136_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E3:$tType,F4:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3))))))]: ((~ ((! [A5:A,B5:B,C5:C,D2:D,E4:E3,F5:F4,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3)))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3))) @ D2) @ ((((product_Pair @ E3) @ ((product_prod @ F4) @ G3)) @ E4) @ ((((product_Pair @ F4) @ G3) @ F5) @ G4)))))))))))))).
% 2.88/1.54 thf(fact_137_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C5:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C5))))) => (P @ X2))))).
% 2.88/1.54 thf(fact_138_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C5:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C5) @ D2)))))) => (P @ X2))))).
% 2.88/1.54 thf(fact_139_prod__induct5, axiom, (! [E3:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3)))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))))]: (((! [A5:A,B5:B,C5:C,D2:D,E4:E3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E3)) @ C5) @ ((((product_Pair @ D) @ E3) @ D2) @ E4))))))) => (P @ X2))))).
% 2.88/1.54 thf(fact_140_prod__induct6, axiom, (! [F4:$tType,E3:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4)))))]: (((! [A5:A,B5:B,C5:C,D2:D,E4:E3,F5:F4]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F4))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ F4)) @ D2) @ ((((product_Pair @ E3) @ F4) @ E4) @ F5)))))))) => (P @ X2))))).
% 2.88/1.54 thf(fact_141_prod__induct7, axiom, (! [G3:$tType,F4:$tType,E3:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3)))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3))))))]: (((! [A5:A,B5:B,C5:C,D2:D,E4:E3,F5:F4,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3)))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ ((product_prod @ F4) @ G3))) @ D2) @ ((((product_Pair @ E3) @ ((product_prod @ F4) @ G3)) @ E4) @ ((((product_Pair @ F4) @ G3) @ F5) @ G4))))))))) => (P @ X2))))).
% 2.88/1.54 thf(fact_142_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 2.88/1.54 thf(fact_143_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ Prod))))).
% 2.88/1.54 thf(fact_144_subrelI, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S5:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S5)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S5))))).
% 2.88/1.54 thf(fact_145_single__valued__subset, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S5:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S5) => ((((single_valued @ A) @ B) @ S5) => (((single_valued @ A) @ B) @ R3)))))).
% 2.88/1.54 thf(fact_146_single__valued__empty, axiom, (! [B:$tType,A:$tType]: ((((single_valued @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.88/1.54 thf(fact_147_Domain__empty__iff, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: ((((((domain @ A) @ B) @ R3) = (bot_bot @ (set @ A))) = (R3 = (bot_bot @ (set @ ((product_prod @ A) @ B)))))))).
% 2.88/1.54 thf(fact_148_Domain__mono, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B)),S5:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R3) @ S5) => (((ord_less_eq @ (set @ A)) @ (((domain @ A) @ B) @ R3)) @ (((domain @ A) @ B) @ S5)))))).
% 2.88/1.54 thf(fact_149_single__valuedD, axiom, (! [A:$tType,B:$tType,R3:(set @ ((product_prod @ A) @ B)),X2:A,Y:B,Z2:B]: (((((single_valued @ A) @ B) @ R3) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R3) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Z2)) @ R3) => (Y = Z2))))))).
% 2.88/1.54 thf(fact_150_single__valuedI, axiom, (! [B:$tType,A:$tType,R3:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y3:B,Z3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Z3)) @ R3) => (Y3 = Z3))))) => (((single_valued @ A) @ B) @ R3))))).
% 2.88/1.54 thf(fact_151_single__valued__def, axiom, (! [B:$tType,A:$tType]: ((((single_valued @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ B))]: ((! [X:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ R4) => (! [Z4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Z4)) @ R4) => (Y4 = Z4))))))))))))).
% 2.88/1.54 thf(fact_152_DomainE, axiom, (! [B:$tType,A:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) => (~ ((! [B5:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B5)) @ R3))))))))))).
% 2.88/1.54 thf(fact_153_Domain__iff, axiom, (! [A:$tType,B:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) = (? [Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ Y4)) @ R3))))))).
% 2.88/1.54 thf(fact_154_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) => (~ ((! [B5:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B5)) @ R3))))))))))).
% 2.88/1.54 thf(fact_155_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A3:A,R3:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R3)) = (? [A4:A,B4:B]: (((A3 = A4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B4)) @ R3)))))))).
% 2.88/1.54 thf(fact_156_subset__empty, axiom, (! [A:$tType,A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_157_empty__subsetI, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A2)))).
% 2.88/1.54 thf(fact_158_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.88/1.54 thf(fact_159_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.88/1.54 thf(fact_160_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 2.88/1.54 thf(fact_161_all__not__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X:A]: ((~ ((((member @ A) @ X) @ A2))))) = (A2 = (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_162_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X:A]: ((~ ((P @ X))))))))).
% 2.88/1.54 thf(fact_163_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X:A]: ((~ ((P @ X))))))))).
% 2.88/1.54 thf(fact_164_emptyE, axiom, (! [A:$tType,A3:A]: ((~ ((((member @ A) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.88/1.54 thf(fact_165_equals0D, axiom, (! [A:$tType,A2:(set @ A),A3:A]: (((A2 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A3) @ A2))))))).
% 2.88/1.54 thf(fact_166_equals0I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A2))))) => (A2 = (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_167_ex__in__conv, axiom, (! [A:$tType,A2:(set @ A)]: (((? [X:A]: ((((member @ A) @ X) @ A2))) = (A2 != (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_168_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.88/1.54 thf(fact_169_empty__not__UNIV, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) != (top_top @ (set @ A)))))).
% 2.88/1.54 thf(fact_170_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P @ X) => (Q @ X)))))))).
% 2.88/1.54 thf(fact_171_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z:(set @ A)]: ((Y2 = Z))) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))).
% 2.88/1.54 thf(fact_172_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C2) => (((ord_less_eq @ (set @ A)) @ A2) @ C2)))))).
% 2.88/1.54 thf(fact_173_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.88/1.54 thf(fact_174_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.88/1.54 thf(fact_175_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.88/1.54 thf(fact_176_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A7) => (((member @ A) @ T3) @ B7)))))))))).
% 2.88/1.54 thf(fact_177_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.88/1.54 thf(fact_178_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.88/1.54 thf(fact_179_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A7) => (((member @ A) @ X) @ B7)))))))))).
% 2.88/1.54 thf(fact_180_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.88/1.54 thf(fact_181_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C3) @ A2) => (((member @ A) @ C3) @ B2)))))).
% 2.88/1.54 thf(fact_182_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X2) @ A2) => (((member @ A) @ X2) @ B2)))))).
% 2.88/1.54 thf(fact_183_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.88/1.54 thf(fact_184_less__by__empty, axiom, (! [A:$tType,A2:(set @ ((product_prod @ A) @ A)),B2:(set @ ((product_prod @ A) @ A))]: (((A2 = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A2) @ B2))))).
% 2.88/1.54 thf(fact_185_subset__emptyI, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((~ ((((member @ A) @ X3) @ A2))))) => (((ord_less_eq @ (set @ A)) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_186_UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.88/1.54 thf(fact_187_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 2.88/1.54 thf(fact_188_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.88/1.54 thf(fact_189_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.88/1.54 thf(fact_190_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A))))))))).
% 2.88/1.54 thf(fact_191_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R3:A,S5:B,R:(set @ ((product_prod @ A) @ B)),S6:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R3) @ S5)) @ R) => ((S6 = S5) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R3) @ S6)) @ R)))))).
% 2.88/1.54 thf(fact_192_iso__tuple__UNIV__I, axiom, (! [A:$tType,X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))).
% 2.88/1.54 thf(fact_193_ac__galois, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((single_valued @ A) @ B) @ R) => ((((refine1150083786ection @ (refine1665802226e_nres @ A)) @ (refine1665802226e_nres @ B)) @ (((refine81118332bs_fun @ A) @ B) @ R)) @ (((refine1073749519nc_fun @ A) @ B) @ R)))))).
% 2.88/1.54 thf(fact_194_abs__fun__simps_I3_J, axiom, (! [B:$tType,A:$tType,X5:(set @ B),R:(set @ ((product_prod @ B) @ A))]: (((~ ((((ord_less_eq @ (set @ B)) @ X5) @ (((domain @ B) @ A) @ R)))) => (((((refine81118332bs_fun @ B) @ A) @ R) @ ((refine605929679le_RES @ B) @ X5)) = (top_top @ (refine1665802226e_nres @ A))))))).
% 2.88/1.54 thf(fact_195_nres_Oinject, axiom, (! [A:$tType,X22:(set @ A),Y22:(set @ A)]: (((((refine605929679le_RES @ A) @ X22) = ((refine605929679le_RES @ A) @ Y22)) = (X22 = Y22))))).
% 2.88/1.54 thf(fact_196_nres__more__simps_I4_J, axiom, (! [A:$tType,X5:(set @ A),Y6:(set @ A)]: (((((refine605929679le_RES @ A) @ X5) = ((refine605929679le_RES @ A) @ Y6)) = (X5 = Y6))))).
% 2.88/1.54 thf(fact_197_top1I, axiom, (! [A:$tType,X2:A]: (((top_top @ (A > $o)) @ X2)))).
% 2.88/1.54 thf(fact_198_nofail__simps_I2_J, axiom, (! [B:$tType,X5:(set @ B)]: (((refine1102455758nofail @ B) @ ((refine605929679le_RES @ B) @ X5))))).
% 2.88/1.54 thf(fact_199_inres__simps_I2_J, axiom, (! [B:$tType,X5:(set @ B)]: ((((refine1315500908_inres @ B) @ ((refine605929679le_RES @ B) @ X5)) = (^ [X:B]: ((((member @ B) @ X) @ X5))))))).
% 2.88/1.54 thf(fact_200_nf__inres__RES, axiom, (! [A:$tType,X5:(set @ A),X2:A]: (((((refine406925620_inres @ A) @ ((refine605929679le_RES @ A) @ X5)) @ X2) = (((member @ A) @ X2) @ X5))))).
% 2.88/1.54 thf(fact_201_nf__inres__SPEC, axiom, (! [A:$tType,Phi:(A > $o),X2:A]: (((((refine406925620_inres @ A) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) @ X2) = (Phi @ X2))))).
% 2.88/1.54 thf(fact_202_nres__more__simps_I2_J, axiom, (! [A:$tType,X5:(set @ A)]: (((((refine605929679le_RES @ A) @ X5) = (bot_bot @ (refine1665802226e_nres @ A))) = (X5 = (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_203_nres__more__simps_I1_J, axiom, (! [A:$tType,X5:(set @ A)]: ((((bot_bot @ (refine1665802226e_nres @ A)) = ((refine605929679le_RES @ A) @ X5)) = (X5 = (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_204_nres__order__simps_I21_J, axiom, (! [X6:$tType,X2:X6,Y6:(set @ X6)]: (((((ord_less_eq @ (refine1665802226e_nres @ X6)) @ ((refine1687780735RETURN @ X6) @ X2)) @ ((refine605929679le_RES @ X6) @ Y6)) = (((member @ X6) @ X2) @ Y6))))).
% 2.88/1.54 thf(fact_205_nres__cases, axiom, (! [A:$tType,M:(refine1665802226e_nres @ A)]: (((M != (top_top @ (refine1665802226e_nres @ A))) => (~ ((! [X7:(set @ A)]: ((M != ((refine605929679le_RES @ A) @ X7)))))))))).
% 2.88/1.54 thf(fact_206_nres__inequalities_I1_J, axiom, (! [A:$tType,X5:(set @ A)]: (((top_top @ (refine1665802226e_nres @ A)) != ((refine605929679le_RES @ A) @ X5))))).
% 2.88/1.54 thf(fact_207_galois__connection__def, axiom, (! [B:$tType,A:$tType]: ((((comple187826305attice @ A) & (comple187826305attice @ B)) => (((refine1150083786ection @ A) @ B) = (^ [Alpha:(A > B),Gamma:(B > A)]: ((! [C4:A,A4:B]: (((((ord_less_eq @ A) @ C4) @ (Gamma @ A4)) = (((ord_less_eq @ B) @ (Alpha @ C4)) @ A4))))))))))).
% 2.88/1.54 thf(fact_208_galois__connection_Ointro, axiom, (! [A:$tType,B:$tType]: ((((comple187826305attice @ B) & (comple187826305attice @ A)) => (! [Gamma2:(B > A),Alpha2:(A > B)]: (((! [C5:A,A5:B]: (((((ord_less_eq @ A) @ C5) @ (Gamma2 @ A5)) = (((ord_less_eq @ B) @ (Alpha2 @ C5)) @ A5)))) => ((((refine1150083786ection @ A) @ B) @ Alpha2) @ Gamma2)))))))).
% 2.88/1.54 thf(fact_209_galois__connection_Ogalois, axiom, (! [A:$tType,B:$tType]: ((((comple187826305attice @ B) & (comple187826305attice @ A)) => (! [Alpha2:(A > B),Gamma2:(B > A),C3:A,A3:B]: ((((((refine1150083786ection @ A) @ B) @ Alpha2) @ Gamma2) => ((((ord_less_eq @ A) @ C3) @ (Gamma2 @ A3)) = (((ord_less_eq @ B) @ (Alpha2 @ C3)) @ A3))))))))).
% 2.88/1.54 thf(fact_210_galois__connection_O_092_060alpha_062_092_060gamma_062__defl, axiom, (! [A:$tType,B:$tType]: ((((comple187826305attice @ B) & (comple187826305attice @ A)) => (! [Alpha2:(A > B),Gamma2:(B > A),X2:B]: ((((((refine1150083786ection @ A) @ B) @ Alpha2) @ Gamma2) => (((ord_less_eq @ B) @ (Alpha2 @ (Gamma2 @ X2))) @ X2)))))))).
% 2.88/1.54 thf(fact_211_galois__connection_O_092_060gamma_062_092_060alpha_062__infl, axiom, (! [B:$tType,A:$tType]: ((((comple187826305attice @ A) & (comple187826305attice @ B)) => (! [Alpha2:(A > B),Gamma2:(B > A),X2:A]: ((((((refine1150083786ection @ A) @ B) @ Alpha2) @ Gamma2) => (((ord_less_eq @ A) @ X2) @ (Gamma2 @ (Alpha2 @ X2)))))))))).
% 2.88/1.54 thf(fact_212_nres__order__simps_I5_J, axiom, (! [E3:$tType,X5:(set @ E3),Y6:(set @ E3)]: (((((ord_less_eq @ (refine1665802226e_nres @ E3)) @ ((refine605929679le_RES @ E3) @ X5)) @ ((refine605929679le_RES @ E3) @ Y6)) = (((ord_less_eq @ (set @ E3)) @ X5) @ Y6))))).
% 2.88/1.54 thf(fact_213_less__eq__nres_Osimps_I2_J, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ ((refine605929679le_RES @ A) @ A3)) @ ((refine605929679le_RES @ A) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.88/1.54 thf(fact_214_bot__nres__def, axiom, (! [A:$tType]: (((bot_bot @ (refine1665802226e_nres @ A)) = ((refine605929679le_RES @ A) @ (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_215_nofail__RES__conv, axiom, (! [A:$tType]: (((refine1102455758nofail @ A) = (^ [M5:(refine1665802226e_nres @ A)]: ((? [M6:(set @ A)]: ((M5 = ((refine605929679le_RES @ A) @ M6)))))))))).
% 2.88/1.54 thf(fact_216_SPEC__cons__rule, axiom, (! [A:$tType,M3:(refine1665802226e_nres @ A),Phi:(A > $o),Psi:(A > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ M3) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Phi))) => ((! [X3:A]: (((Phi @ X3) => (Psi @ X3)))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ M3) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Psi)))))))).
% 2.88/1.54 thf(fact_217_SPEC__trans, axiom, (! [A:$tType,X2:(refine1665802226e_nres @ A),Y:(refine1665802226e_nres @ A),Postcond:(A > $o)]: (((((ord_less_eq @ (refine1665802226e_nres @ A)) @ X2) @ Y) => ((((ord_less_eq @ (refine1665802226e_nres @ A)) @ Y) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Postcond))) => (((ord_less_eq @ (refine1665802226e_nres @ A)) @ X2) @ ((refine605929679le_RES @ A) @ ((collect @ A) @ Postcond)))))))).
% 2.88/1.54 thf(fact_218_the__RES__inv, axiom, (! [A:$tType,M3:(refine1665802226e_nres @ A)]: ((((refine1102455758nofail @ A) @ M3) => (((refine605929679le_RES @ A) @ ((refine1672542526he_RES @ A) @ M3)) = M3))))).
% 2.88/1.54 thf(fact_219_nres__order__simps_I22_J, axiom, (! [Y7:$tType,X5:(set @ Y7),Y:Y7]: (((((ord_less_eq @ (refine1665802226e_nres @ Y7)) @ ((refine605929679le_RES @ Y7) @ X5)) @ ((refine1687780735RETURN @ Y7) @ Y)) = (((ord_less_eq @ (set @ Y7)) @ X5) @ (((insert @ Y7) @ Y) @ (bot_bot @ (set @ Y7)))))))).
% 2.88/1.54 thf(fact_220_insert__absorb2, axiom, (! [A:$tType,X2:A,A2:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ X2) @ A2)) = (((insert @ A) @ X2) @ A2))))).
% 2.88/1.54 thf(fact_221_insert__iff, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A2)) = ((A3 = B3) | (((member @ A) @ A3) @ A2)))))).
% 2.88/1.54 thf(fact_222_insertCI, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: ((((~ ((((member @ A) @ A3) @ B2))) => (A3 = B3)) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.88/1.54 thf(fact_223_singletonI, axiom, (! [A:$tType,A3:A]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.88/1.54 thf(fact_224_insert__subset, axiom, (! [A:$tType,X2:A,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X2) @ A2)) @ B2) = ((((member @ A) @ X2) @ B2) & (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.88/1.54 thf(fact_225_singleton__insert__inj__eq, axiom, (! [A:$tType,B3:A,A3:A,A2:(set @ A)]: ((((((insert @ A) @ B3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A3) @ A2)) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.88/1.54 thf(fact_226_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A3:A,A2:(set @ A),B3:A]: ((((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = ((A3 = B3) & (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A))))))))).
% 2.88/1.54 thf(fact_227_Domain__insert, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,R3:(set @ ((product_prod @ A) @ B))]: (((((domain @ A) @ B) @ (((insert @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R3)) = (((insert @ A) @ A3) @ (((domain @ A) @ B) @ R3)))))).
% 2.88/1.54 thf(fact_228_nres__more__simps_I3_J, axiom, (! [A:$tType,X5:(set @ A),X2:A]: (((((refine605929679le_RES @ A) @ X5) = ((refine1687780735RETURN @ A) @ X2)) = (X5 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))).
% 2.88/1.54 thf(fact_229_nres__more__simps_I5_J, axiom, (! [A:$tType,X2:A,X5:(set @ A)]: (((((refine1687780735RETURN @ A) @ X2) = ((refine605929679le_RES @ A) @ X5)) = ((((insert @ A) @ X2) @ (bot_bot @ (set @ A))) = X5))))).
% 2.88/1.54 thf(fact_230_insert__UNIV, axiom, (! [A:$tType,X2:A]: (((((insert @ A) @ X2) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 2.88/1.54 thf(fact_231_insert__mono, axiom, (! [A:$tType,C2:(set @ A),D3:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ C2) @ D3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A3) @ C2)) @ (((insert @ A) @ A3) @ D3)))))).
% 2.88/1.54 thf(fact_232_subset__insert, axiom, (! [A:$tType,X2:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X2) @ A2))) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X2) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.88/1.54 thf(fact_233_subset__insertI, axiom, (! [A:$tType,B2:(set @ A),A3:A]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((insert @ A) @ A3) @ B2))))).
% 2.88/1.54 thf(fact_234_subset__insertI2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),B3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B3) @ B2)))))).
% 2.88/1.54 thf(fact_235_singleton__inject, axiom, (! [A:$tType,A3:A,B3:A]: ((((((insert @ A) @ A3) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) => (A3 = B3))))).
% 2.88/1.54 thf(fact_236_insert__not__empty, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((insert @ A) @ A3) @ A2) != (bot_bot @ (set @ A)))))).
% 2.88/1.54 thf(fact_237_doubleton__eq__iff, axiom, (! [A:$tType,A3:A,B3:A,C3:A,D4:A]: ((((((insert @ A) @ A3) @ (((insert @ A) @ B3) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D4) @ (bot_bot @ (set @ A))))) = (((A3 = C3) & (B3 = D4)) | ((A3 = D4) & (B3 = C3))))))).
% 2.88/1.54 thf(fact_238_singleton__iff, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = (B3 = A3))))).
% 2.88/1.54 thf(fact_239_singletonD, axiom, (! [A:$tType,B3:A,A3:A]: (((((member @ A) @ B3) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) => (B3 = A3))))).
% 2.88/1.54 thf(fact_240_mk__disjoint__insert, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => (? [B8:(set @ A)]: (((A2 = (((insert @ A) @ A3) @ B8)) & (~ ((((member @ A) @ A3) @ B8)))))))))).
% 2.88/1.54 thf(fact_241_insert__commute, axiom, (! [A:$tType,X2:A,Y:A,A2:(set @ A)]: (((((insert @ A) @ X2) @ (((insert @ A) @ Y) @ A2)) = (((insert @ A) @ Y) @ (((insert @ A) @ X2) @ A2)))))).
% 2.88/1.54 thf(fact_242_insert__eq__iff, axiom, (! [A:$tType,A3:A,A2:(set @ A),B3:A,B2:(set @ A)]: (((~ ((((member @ A) @ A3) @ A2))) => ((~ ((((member @ A) @ B3) @ B2))) => (((((insert @ A) @ A3) @ A2) = (((insert @ A) @ B3) @ B2)) = (((A3 = B3) => (A2 = B2)) & ((A3 != B3) => (? [C6:(set @ A)]: (((A2 = (((insert @ A) @ B3) @ C6)) & ((~ ((((member @ A) @ B3) @ C6))) & ((B2 = (((insert @ A) @ A3) @ C6)) & (~ ((((member @ A) @ A3) @ C6)))))))))))))))).
% 2.88/1.54 thf(fact_243_insert__absorb, axiom, (! [A:$tType,A3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ A2) => ((((insert @ A) @ A3) @ A2) = A2))))).
% 2.88/1.54 thf(fact_244_insert__ident, axiom, (! [A:$tType,X2:A,A2:(set @ A),B2:(set @ A)]: (((~ ((((member @ A) @ X2) @ A2))) => ((~ ((((member @ A) @ X2) @ B2))) => (((((insert @ A) @ X2) @ A2) = (((insert @ A) @ X2) @ B2)) = (A2 = B2))))))).
% 2.88/1.54 thf(fact_245_Set_Oset__insert, axiom, (! [A:$tType,X2:A,A2:(set @ A)]: (((((member @ A) @ X2) @ A2) => (~ ((! [B8:(set @ A)]: (((A2 = (((insert @ A) @ X2) @ B8)) => (((member @ A) @ X2) @ B8)))))))))).
% 2.88/1.54 thf(fact_246_insertI2, axiom, (! [A:$tType,A3:A,B2:(set @ A),B3:A]: (((((member @ A) @ A3) @ B2) => (((member @ A) @ A3) @ (((insert @ A) @ B3) @ B2)))))).
% 2.88/1.54 thf(fact_247_insertI1, axiom, (! [A:$tType,A3:A,B2:(set @ A)]: ((((member @ A) @ A3) @ (((insert @ A) @ A3) @ B2))))).
% 2.88/1.54 thf(fact_248_insertE, axiom, (! [A:$tType,A3:A,B3:A,A2:(set @ A)]: (((((member @ A) @ A3) @ (((insert @ A) @ B3) @ A2)) => ((A3 != B3) => (((member @ A) @ A3) @ A2)))))).
% 2.88/1.54 thf(fact_249_subset__singletonD, axiom, (! [A:$tType,A2:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) => ((A2 = (bot_bot @ (set @ A))) | (A2 = (((insert @ A) @ X2) @ (bot_bot @ (set @ A))))))))).
% 2.88/1.54 thf(fact_250_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A3:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A3) @ (bot_bot @ (set @ A))))))))).
% 2.88/1.54 thf(fact_251_insert__subsetI, axiom, (! [A:$tType,X2:A,A2:(set @ A),X5:(set @ A)]: (((((member @ A) @ X2) @ A2) => ((((ord_less_eq @ (set @ A)) @ X5) @ A2) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X2) @ X5)) @ A2)))))).
% 2.88/1.54 thf(fact_252_RETURN__def, axiom, (! [A:$tType]: (((refine1687780735RETURN @ A) = (^ [X:A]: (((refine605929679le_RES @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))))))))).
% 2.88/1.54 thf(fact_253_the__RES_Osimps, axiom, (! [A:$tType,X5:(set @ A)]: ((((refine1672542526he_RES @ A) @ ((refine605929679le_RES @ A) @ X5)) = X5)))).
% 2.88/1.54 thf(fact_254_the__elem__eq, axiom, (! [A:$tType,X2:A]: ((((the_elem @ A) @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))) = X2)))).
% 2.88/1.54 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A8:$tType,A9:$tType]: (((comple187826305attice @ A9) => (comple187826305attice @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: (((order_top @ A9) => (order_top @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A8:$tType,A9:$tType]: (((order_bot @ A9) => (order_bot @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_fun___Orderings_Obot, axiom, (! [A8:$tType,A9:$tType]: (((bot @ A9) => (bot @ (A8 > A9)))))).
% 2.88/1.54 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1, axiom, (! [A8:$tType]: ((comple187826305attice @ (set @ A8))))).
% 2.88/1.54 thf(tcon_Set_Oset___Orderings_Oorder__top_2, axiom, (! [A8:$tType]: ((order_top @ (set @ A8))))).
% 2.88/1.54 thf(tcon_Set_Oset___Orderings_Oorder__bot_3, axiom, (! [A8:$tType]: ((order_bot @ (set @ A8))))).
% 2.88/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.88/1.54 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.88/1.54 thf(tcon_Set_Oset___Orderings_Otop_6, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 2.88/1.54 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.88/1.54 thf(tcon_Set_Oset___Orderings_Obot_8, axiom, (! [A8:$tType]: ((bot @ (set @ A8))))).
% 2.88/1.54 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_9, axiom, (comple187826305attice @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Oorder__top_10, axiom, (order_top @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11, axiom, (order_bot @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_12, axiom, (preorder @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_13, axiom, (order @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Otop_14, axiom, (top @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Oord_15, axiom, (ord @ $o)).
% 2.88/1.54 thf(tcon_HOL_Obool___Orderings_Obot_16, axiom, (bot @ $o)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_17, axiom, (comple187826305attice @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_18, axiom, (order_top @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_19, axiom, (order_bot @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_20, axiom, (preorder @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_21, axiom, (linorder @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Oorder_22, axiom, (order @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Otop_23, axiom, (top @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Oord_24, axiom, (ord @ product_unit)).
% 2.88/1.54 thf(tcon_Product__Type_Ounit___Orderings_Obot_25, axiom, (bot @ product_unit)).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_26, axiom, (! [A8:$tType]: ((comple187826305attice @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_27, axiom, (! [A8:$tType]: ((order_top @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__bot_28, axiom, (! [A8:$tType]: ((order_bot @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_29, axiom, (! [A8:$tType]: ((preorder @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_30, axiom, (! [A8:$tType]: ((order @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_31, axiom, (! [A8:$tType]: ((top @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_32, axiom, (! [A8:$tType]: ((ord @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_33, axiom, (! [A8:$tType]: ((bot @ (refine1665802226e_nres @ A8))))).
% 2.88/1.54 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.88/1.54 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $false) @ X2) @ Y) = Y)))).
% 2.88/1.54 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y:A]: ((((((if @ A) @ $true) @ X2) @ Y) = X2)))).
% 2.88/1.54 thf(conj_0, conjecture, (((((refine463715084e_bind @ product_unit) @ a) @ (bot_bot @ (refine1665802226e_nres @ product_unit))) @ f) = (bot_bot @ (refine1665802226e_nres @ a)))).
% 2.88/1.54 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.kXsISafMJX/DTF2DTF15221.p
% 2.88/1.54 ------------------------
% 2.88/1.57 ---- Cleaned THF ---
% 2.88/1.57 thf(ty_t_Refine__Basic__Mirabelle__tqojlsrkwy_Onres,type,
% 2.88/1.57 refine1665802226e_nres: $tType > $tType ).
% 2.88/1.57
% 2.88/1.57 thf(ty_t_Product__Type_Ounit,type,
% 2.88/1.57 product_unit: $tType ).
% 2.88/1.57
% 2.88/1.57 thf(ty_t_Product__Type_Oprod,type,
% 2.88/1.57 product_prod: $tType > $tType > $tType ).
% 2.88/1.57
% 2.88/1.57 thf(ty_t_Set_Oset,type,
% 2.88/1.57 set: $tType > $tType ).
% 2.88/1.57
% 2.88/1.57 thf(ty_tf_a,type,
% 2.88/1.57 a: $tType ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_HOL_Otype,type,
% 2.88/1.57 type:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Obot,type,
% 2.88/1.57 bot:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Oord,type,
% 2.88/1.57 ord:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Otop,type,
% 2.88/1.57 top:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Oorder,type,
% 2.88/1.57 order:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Olinorder,type,
% 2.88/1.57 linorder:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Opreorder,type,
% 2.88/1.57 preorder:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.88/1.57 order_bot:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Orderings_Oorder__top,type,
% 2.88/1.57 order_top:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 2.88/1.57 comple187826305attice:
% 2.88/1.57 !>[A: $tType] : $o ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_If,type,
% 2.88/1.57 if:
% 2.88/1.57 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.88/1.57 bot_bot:
% 2.88/1.57 !>[A: $tType] : A ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.88/1.57 ord_less_eq:
% 2.88/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 2.88/1.57 order_Greatest:
% 2.88/1.57 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.88/1.57 top_top:
% 2.88/1.57 !>[A: $tType] : A ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Partial__Function_Oflat__ord,type,
% 2.88/1.57 partial_flat_ord:
% 2.88/1.57 !>[A: $tType] : ( A > A > A > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Product__Type_OPair,type,
% 2.88/1.57 product_Pair:
% 2.88/1.57 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_ORETURN,type,
% 2.88/1.57 refine1687780735RETURN:
% 2.88/1.57 !>[A: $tType] : ( A > ( refine1665802226e_nres @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oabs__fun,type,
% 2.88/1.57 refine81118332bs_fun:
% 2.88/1.57 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Obind,type,
% 2.88/1.57 refine463715084e_bind:
% 2.88/1.57 !>[B: $tType,A: $tType] : ( ( refine1665802226e_nres @ B ) > ( B > ( refine1665802226e_nres @ A ) ) > ( refine1665802226e_nres @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oconc__fun,type,
% 2.88/1.57 refine1073749519nc_fun:
% 2.88/1.57 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( refine1665802226e_nres @ B ) > ( refine1665802226e_nres @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Oinres,type,
% 2.88/1.57 refine1315500908_inres:
% 2.88/1.57 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onf__inres,type,
% 2.88/1.57 refine406925620_inres:
% 2.88/1.57 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > A > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onofail,type,
% 2.88/1.57 refine1102455758nofail:
% 2.88/1.57 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Onres_ORES,type,
% 2.88/1.57 refine605929679le_RES:
% 2.88/1.57 !>[A: $tType] : ( ( set @ A ) > ( refine1665802226e_nres @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Basic__Mirabelle__tqojlsrkwy_Othe__RES,type,
% 2.88/1.57 refine1672542526he_RES:
% 2.88/1.57 !>[A: $tType] : ( ( refine1665802226e_nres @ A ) > ( set @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Refine__Misc_Ogalois__connection,type,
% 2.88/1.57 refine1150083786ection:
% 2.88/1.57 !>[A: $tType,B: $tType] : ( ( A > B ) > ( B > A ) > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Relation_ODomain,type,
% 2.88/1.57 domain:
% 2.88/1.57 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Relation_Osingle__valued,type,
% 2.88/1.57 single_valued:
% 2.88/1.57 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Set_OCollect,type,
% 2.88/1.57 collect:
% 2.88/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Set_Oinsert,type,
% 2.88/1.57 insert:
% 2.88/1.57 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_Set_Othe__elem,type,
% 2.88/1.57 the_elem:
% 2.88/1.57 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_c_member,type,
% 2.88/1.57 member:
% 2.88/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.88/1.57
% 2.88/1.57 thf(sy_v_f,type,
% 2.88/1.57 f: product_unit > ( refine1665802226e_nres @ a ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_0_bind__SUCCEED,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 2.88/1.57 ( ( refine463715084e_bind @ B @ A @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) @ F )
% 2.88/1.57 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_1_bot__apply,axiom,
% 2.88/1.57 ! [C: $tType,D: $tType] :
% 2.88/1.57 ( ( bot @ C )
% 2.88/1.57 => ( ( bot_bot @ ( D > C ) )
% 2.88/1.57 = ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_2_bot__fun__def,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( bot @ B )
% 2.88/1.57 => ( ( bot_bot @ ( A > B ) )
% 2.88/1.57 = ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_3_nres__monad1,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,X2: B,F: B > ( refine1665802226e_nres @ A )] :
% 2.88/1.57 ( ( refine463715084e_bind @ B @ A @ ( refine1687780735RETURN @ B @ X2 ) @ F )
% 2.88/1.57 = ( F @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_4_nres__monad2,axiom,
% 2.88/1.57 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( refine463715084e_bind @ A @ A @ M @ ( refine1687780735RETURN @ A ) )
% 2.88/1.57 = M ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_5_abs__fun__strict,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 2.88/1.57 ( ( refine81118332bs_fun @ B @ A @ R @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
% 2.88/1.57 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_6_conc__fun__strict,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( refine1073749519nc_fun @ A @ B @ R @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
% 2.88/1.57 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_7_nres__inequalities_I5_J,axiom,
% 2.88/1.57 ! [C: $tType,X2: C] :
% 2.88/1.57 ( ( bot_bot @ ( refine1665802226e_nres @ C ) )
% 2.88/1.57 != ( refine1687780735RETURN @ C @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_8_inres__simps_I4_J,axiom,
% 2.88/1.57 ! [D: $tType] :
% 2.88/1.57 ( ( refine1315500908_inres @ D @ ( bot_bot @ ( refine1665802226e_nres @ D ) ) )
% 2.88/1.57 = ( ^ [Uu: D] : $false ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_9_nres__order__simps_I2_J,axiom,
% 2.88/1.57 ! [B: $tType,M: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ M @ ( bot_bot @ ( refine1665802226e_nres @ B ) ) )
% 2.88/1.57 = ( M
% 2.88/1.57 = ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_10_nofail__simps_I4_J,axiom,
% 2.88/1.57 ! [D: $tType] : ( refine1102455758nofail @ D @ ( bot_bot @ ( refine1665802226e_nres @ D ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_11_bind__FAIL,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,F: B > ( refine1665802226e_nres @ A )] :
% 2.88/1.57 ( ( refine463715084e_bind @ B @ A @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ F )
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_12_nres__order__simps_I1_J,axiom,
% 2.88/1.57 ! [A: $tType,M: refine1665802226e_nres @ A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ M ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_13_top__apply,axiom,
% 2.88/1.57 ! [C: $tType,D: $tType] :
% 2.88/1.57 ( ( top @ C )
% 2.88/1.57 => ( ( top_top @ ( D > C ) )
% 2.88/1.57 = ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_14_nres__more__simps_I6_J,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,Y: A] :
% 2.88/1.57 ( ( ( refine1687780735RETURN @ A @ X2 )
% 2.88/1.57 = ( refine1687780735RETURN @ A @ Y ) )
% 2.88/1.57 = ( X2 = Y ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_15_nres__order__simps_I4_J,axiom,
% 2.88/1.57 ! [D: $tType,M: refine1665802226e_nres @ D] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ D ) @ ( top_top @ ( refine1665802226e_nres @ D ) ) @ M )
% 2.88/1.57 = ( M
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ D ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_16_nofail__simps_I1_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ~ ( refine1102455758nofail @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_17_inres__simps_I1_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 2.88/1.57 = ( ^ [Uu: A] : $true ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_18_conc__fun__fail__iff_I2_J,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 2.88/1.57 = ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 2.88/1.57 = ( S
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_19_conc__fun__fail__iff_I1_J,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( ( refine1073749519nc_fun @ A @ B @ R @ S )
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 2.88/1.57 = ( S
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ B ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_20_conc__fun__FAIL,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( refine1073749519nc_fun @ A @ B @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_21_nres__order__simps_I20_J,axiom,
% 2.88/1.57 ! [W: $tType,X2: W,Y: W] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ W ) @ ( refine1687780735RETURN @ W @ X2 ) @ ( refine1687780735RETURN @ W @ Y ) )
% 2.88/1.57 = ( X2 = Y ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_22_abs__fun__simps_I1_J,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
% 2.88/1.57 ( ( refine81118332bs_fun @ B @ A @ R @ ( top_top @ ( refine1665802226e_nres @ B ) ) )
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_23_nofail__simps_I3_J,axiom,
% 2.88/1.57 ! [C: $tType,X2: C] : ( refine1102455758nofail @ C @ ( refine1687780735RETURN @ C @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_24_inres__simps_I3_J,axiom,
% 2.88/1.57 ! [C: $tType,X2: C] :
% 2.88/1.57 ( ( refine1315500908_inres @ C @ ( refine1687780735RETURN @ C @ X2 ) )
% 2.88/1.57 = ( ^ [Y2: C,Z: C] : ( Y2 = Z )
% 2.88/1.57 @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_25_intro__nofail_I2_J,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 2.88/1.57 != S )
% 2.88/1.57 = ( refine1102455758nofail @ A @ S ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_26_nres__order__simps_I3_J,axiom,
% 2.88/1.57 ! [C: $tType,M: refine1665802226e_nres @ C] : ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ M @ ( top_top @ ( refine1665802226e_nres @ C ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_27_nres__inequalities_I3_J,axiom,
% 2.88/1.57 ! [C: $tType,X2: C] :
% 2.88/1.57 ( ( top_top @ ( refine1665802226e_nres @ C ) )
% 2.88/1.57 != ( refine1687780735RETURN @ C @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_28_abs__trans__additional_I1_J,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,B2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),C2: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ B2 )
% 2.88/1.57 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ B2 ) @ C2 )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ A2 ) @ C2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_29_conc__trans__additional_I1_J,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,A2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,C2: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 2.88/1.57 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ C2 )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ A2 @ ( refine1073749519nc_fun @ A @ B @ R @ C2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_30_le__funD,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( ord @ B )
% 2.88/1.57 => ! [F: A > B,G: A > B,X2: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.88/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_31_le__funE,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( ord @ B )
% 2.88/1.57 => ! [F: A > B,G: A > B,X2: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.88/1.57 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_32_le__funI,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( ord @ B )
% 2.88/1.57 => ! [F: A > B,G: A > B] :
% 2.88/1.57 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.88/1.57 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_33_le__fun__def,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( ord @ B )
% 2.88/1.57 => ( ( ord_less_eq @ ( A > B ) )
% 2.88/1.57 = ( ^ [F2: A > B,G2: A > B] :
% 2.88/1.57 ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_34_order__subst1,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( ( order @ B )
% 2.88/1.57 & ( order @ A ) )
% 2.88/1.57 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.88/1.57 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 2.88/1.57 => ( ! [X3: B,Y3: B] :
% 2.88/1.57 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 2.88/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.88/1.57 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_35_order__subst2,axiom,
% 2.88/1.57 ! [A: $tType,C: $tType] :
% 2.88/1.57 ( ( ( order @ C )
% 2.88/1.57 & ( order @ A ) )
% 2.88/1.57 => ! [A3: A,B3: A,F: A > C,C3: C] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.88/1.57 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
% 2.88/1.57 => ( ! [X3: A,Y3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.88/1.57 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.88/1.57 => ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_36_ord__eq__le__subst,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( ( ord @ B )
% 2.88/1.57 & ( ord @ A ) )
% 2.88/1.57 => ! [A3: A,F: B > A,B3: B,C3: B] :
% 2.88/1.57 ( ( A3
% 2.88/1.57 = ( F @ B3 ) )
% 2.88/1.57 => ( ( ord_less_eq @ B @ B3 @ C3 )
% 2.88/1.57 => ( ! [X3: B,Y3: B] :
% 2.88/1.57 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 2.88/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.88/1.57 => ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_37_ord__le__eq__subst,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( ( ord @ B )
% 2.88/1.57 & ( ord @ A ) )
% 2.88/1.57 => ! [A3: A,B3: A,F: A > B,C3: B] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.88/1.57 => ( ( ( F @ B3 )
% 2.88/1.57 = C3 )
% 2.88/1.57 => ( ! [X3: A,Y3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.88/1.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.88/1.57 => ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_38_eq__iff,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [X: A,Y4: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.88/1.57 & ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_39_antisym,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [X2: A,Y: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.57 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.88/1.57 => ( X2 = Y ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_40_linear,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( linorder @ A )
% 2.88/1.57 => ! [X2: A,Y: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.57 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_41_eq__refl,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( preorder @ A )
% 2.88/1.57 => ! [X2: A,Y: A] :
% 2.88/1.57 ( ( X2 = Y )
% 2.88/1.57 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_42_le__cases,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( linorder @ A )
% 2.88/1.57 => ! [X2: A,Y: A] :
% 2.88/1.57 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.57 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_43_order_Otrans,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [A3: A,B3: A,C3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 2.88/1.57 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_44_le__cases3,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( linorder @ A )
% 2.88/1.57 => ! [X2: A,Y: A,Z2: A] :
% 2.88/1.57 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.57 => ~ ( ord_less_eq @ A @ Y @ Z2 ) )
% 2.88/1.57 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.88/1.57 => ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.88/1.57 => ( ( ( ord_less_eq @ A @ X2 @ Z2 )
% 2.88/1.57 => ~ ( ord_less_eq @ A @ Z2 @ Y ) )
% 2.88/1.57 => ( ( ( ord_less_eq @ A @ Z2 @ Y )
% 2.88/1.57 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.88/1.57 => ( ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.88/1.57 => ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
% 2.88/1.57 => ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
% 2.88/1.57 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_45_mem__Collect__eq,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,P: A > $o] :
% 2.88/1.57 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.88/1.57 = ( P @ A3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_46_Collect__mem__eq,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] :
% 2.88/1.57 ( ( collect @ A
% 2.88/1.57 @ ^ [X: A] : ( member @ A @ X @ A2 ) )
% 2.88/1.57 = A2 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_47_Collect__cong,axiom,
% 2.88/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.88/1.57 ( ! [X3: A] :
% 2.88/1.57 ( ( P @ X3 )
% 2.88/1.57 = ( Q @ X3 ) )
% 2.88/1.57 => ( ( collect @ A @ P )
% 2.88/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_48_ext,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.88/1.57 ( ! [X3: A] :
% 2.88/1.57 ( ( F @ X3 )
% 2.88/1.57 = ( G @ X3 ) )
% 2.88/1.57 => ( F = G ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_49_antisym__conv,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [Y: A,X2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.57 = ( X2 = Y ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_50_order__class_Oorder_Oeq__iff,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [A4: A,B4: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.88/1.57 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_51_ord__eq__le__trans,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ord @ A )
% 2.88/1.57 => ! [A3: A,B3: A,C3: A] :
% 2.88/1.57 ( ( A3 = B3 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ B3 @ C3 )
% 2.88/1.57 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_52_ord__le__eq__trans,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ord @ A )
% 2.88/1.57 => ! [A3: A,B3: A,C3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.88/1.57 => ( ( B3 = C3 )
% 2.88/1.57 => ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_53_order__class_Oorder_Oantisym,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [A3: A,B3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.88/1.57 => ( A3 = B3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_54_order__trans,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( preorder @ A )
% 2.88/1.57 => ! [X2: A,Y: A,Z2: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.88/1.57 => ( ( ord_less_eq @ A @ Y @ Z2 )
% 2.88/1.57 => ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_55_dual__order_Orefl,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_56_linorder__wlog,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( linorder @ A )
% 2.88/1.57 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.88/1.57 ( ! [A5: A,B5: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.88/1.57 => ( P @ A5 @ B5 ) )
% 2.88/1.57 => ( ! [A5: A,B5: A] :
% 2.88/1.57 ( ( P @ B5 @ A5 )
% 2.88/1.57 => ( P @ A5 @ B5 ) )
% 2.88/1.57 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_57_dual__order_Otrans,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [B3: A,A3: A,C3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ C3 @ B3 )
% 2.88/1.57 => ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_58_top__greatest,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order_top @ A )
% 2.88/1.57 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_59_pwD1,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 2.88/1.57 => ( ( refine1102455758nofail @ A @ S2 )
% 2.88/1.57 => ( refine1102455758nofail @ A @ S ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_60_pwD2,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A,X2: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 )
% 2.88/1.57 => ( ( refine1315500908_inres @ A @ S @ X2 )
% 2.88/1.57 => ( refine1315500908_inres @ A @ S2 @ X2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_61_dual__order_Oeq__iff,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [A4: A,B4: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.88/1.57 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_62_dual__order_Oantisym,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [B3: A,A3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.88/1.57 => ( A3 = B3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_63_pw__eqI,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ( refine1102455758nofail @ A @ S )
% 2.88/1.57 = ( refine1102455758nofail @ A @ S2 ) )
% 2.88/1.57 => ( ! [X3: A] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ S @ X3 )
% 2.88/1.57 = ( refine1315500908_inres @ A @ S2 @ X3 ) )
% 2.88/1.57 => ( S = S2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_64_pw__leI,axiom,
% 2.88/1.57 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ( refine1102455758nofail @ A @ S2 )
% 2.88/1.57 => ( ( refine1102455758nofail @ A @ S )
% 2.88/1.57 & ! [X3: A] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ S @ X3 )
% 2.88/1.57 => ( refine1315500908_inres @ A @ S2 @ X3 ) ) ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_65_pw__leI_H,axiom,
% 2.88/1.57 ! [A: $tType,S2: refine1665802226e_nres @ A,S: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ( refine1102455758nofail @ A @ S2 )
% 2.88/1.57 => ( refine1102455758nofail @ A @ S ) )
% 2.88/1.57 => ( ! [X3: A] :
% 2.88/1.57 ( ( refine1102455758nofail @ A @ S2 )
% 2.88/1.57 => ( ( refine1315500908_inres @ A @ S @ X3 )
% 2.88/1.57 => ( refine1315500908_inres @ A @ S2 @ X3 ) ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ S @ S2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_66_abs__trans,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),C2: refine1665802226e_nres @ B,B2: refine1665802226e_nres @ A,R2: set @ ( product_prod @ A @ C ),A2: refine1665802226e_nres @ C] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) @ B2 )
% 2.88/1.57 => ( ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ B2 ) @ A2 )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ C ) @ ( refine81118332bs_fun @ A @ C @ R2 @ ( refine81118332bs_fun @ B @ A @ R @ C2 ) ) @ A2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_67_inres__def,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( refine1315500908_inres @ A )
% 2.88/1.57 = ( ^ [S3: refine1665802226e_nres @ A,X: A] : ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X ) @ S3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_68_pw__eq__iff,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ^ [Y2: refine1665802226e_nres @ A,Z: refine1665802226e_nres @ A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ( refine1102455758nofail @ A @ S3 )
% 2.88/1.57 = ( refine1102455758nofail @ A @ S4 ) )
% 2.88/1.57 & ! [X: A] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ S3 @ X )
% 2.88/1.57 = ( refine1315500908_inres @ A @ S4 @ X ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_69_pw__le__iff,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) )
% 2.88/1.57 = ( ^ [S3: refine1665802226e_nres @ A,S4: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( refine1102455758nofail @ A @ S4 )
% 2.88/1.57 => ( ( refine1102455758nofail @ A @ S3 )
% 2.88/1.57 & ! [X: A] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ S3 @ X )
% 2.88/1.57 => ( refine1315500908_inres @ A @ S4 @ X ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_70_conc__trans,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,C: $tType,C2: refine1665802226e_nres @ A,R: set @ ( product_prod @ A @ B ),B2: refine1665802226e_nres @ B,R2: set @ ( product_prod @ B @ C ),A2: refine1665802226e_nres @ C] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ B2 ) )
% 2.88/1.57 => ( ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ B2 @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ C2 @ ( refine1073749519nc_fun @ A @ B @ R @ ( refine1073749519nc_fun @ B @ C @ R2 @ A2 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_71_le__nofailI,axiom,
% 2.88/1.57 ! [A: $tType,M2: refine1665802226e_nres @ A,M: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ( refine1102455758nofail @ A @ M2 )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_72_nofail__def,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( refine1102455758nofail @ A )
% 2.88/1.57 = ( ^ [S3: refine1665802226e_nres @ A] :
% 2.88/1.57 ( S3
% 2.88/1.57 != ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_73_top_Oextremum__unique,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order_top @ A )
% 2.88/1.57 => ! [A3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.88/1.57 = ( A3
% 2.88/1.57 = ( top_top @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_74_pw__bind__leI,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,S: refine1665802226e_nres @ A,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 2.88/1.57 ( ( ( refine1102455758nofail @ A @ S )
% 2.88/1.57 => ( refine1102455758nofail @ B @ M ) )
% 2.88/1.57 => ( ! [X3: B] :
% 2.88/1.57 ( ( refine1102455758nofail @ B @ M )
% 2.88/1.57 => ( ( refine1315500908_inres @ B @ M @ X3 )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X3 ) @ S ) ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_75_top_Oextremum__uniqueI,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order_top @ A )
% 2.88/1.57 => ! [A3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.88/1.57 => ( A3
% 2.88/1.57 = ( top_top @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_76_not__nofail__iff,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ~ ( refine1102455758nofail @ A @ S ) )
% 2.88/1.57 = ( S
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_77_pw__bind__le__iff,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A ),S: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine463715084e_bind @ B @ A @ M @ F ) @ S )
% 2.88/1.57 = ( ( ( refine1102455758nofail @ A @ S )
% 2.88/1.57 => ( refine1102455758nofail @ B @ M ) )
% 2.88/1.57 & ! [X: B] :
% 2.88/1.57 ( ( ( refine1102455758nofail @ B @ M )
% 2.88/1.57 & ( refine1315500908_inres @ B @ M @ X ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( F @ X ) @ S ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_78_pw__conc__nofail,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( refine1102455758nofail @ A @ ( refine1073749519nc_fun @ A @ B @ R @ S ) )
% 2.88/1.57 = ( refine1102455758nofail @ B @ S ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_79_not__nofail__inres,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A,X2: A] :
% 2.88/1.57 ( ~ ( refine1102455758nofail @ A @ S )
% 2.88/1.57 => ( refine1315500908_inres @ A @ S @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_80_bot_Oextremum__uniqueI,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order_bot @ A )
% 2.88/1.57 => ! [A3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.88/1.57 => ( A3
% 2.88/1.57 = ( bot_bot @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_81_bot_Oextremum__unique,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order_bot @ A )
% 2.88/1.57 => ! [A3: A] :
% 2.88/1.57 ( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
% 2.88/1.57 = ( A3
% 2.88/1.57 = ( bot_bot @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_82_bot_Oextremum,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order_bot @ A )
% 2.88/1.57 => ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_83_pw__bind__nofail,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,M: refine1665802226e_nres @ B,F: B > ( refine1665802226e_nres @ A )] :
% 2.88/1.57 ( ( refine1102455758nofail @ A @ ( refine463715084e_bind @ B @ A @ M @ F ) )
% 2.88/1.57 = ( ( refine1102455758nofail @ B @ M )
% 2.88/1.57 & ! [X: B] :
% 2.88/1.57 ( ( refine1315500908_inres @ B @ M @ X )
% 2.88/1.57 => ( refine1102455758nofail @ A @ ( F @ X ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_84_bind__cong,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,M3: refine1665802226e_nres @ A,M4: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 2.88/1.57 ( ( M3 = M4 )
% 2.88/1.57 => ( ! [X3: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M4 )
% 2.88/1.57 => ( ( F @ X3 )
% 2.88/1.57 = ( F3 @ X3 ) ) )
% 2.88/1.57 => ( ( refine463715084e_bind @ A @ B @ M3 @ F )
% 2.88/1.57 = ( refine463715084e_bind @ A @ B @ M4 @ F3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_85_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I1_J,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M @ M2 )
% 2.88/1.57 => ( ! [X3: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1687780735RETURN @ A @ X3 ) @ M )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( F @ X3 ) @ ( F3 @ X3 ) ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M2 @ F3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_86_nres__inequalities_I2_J,axiom,
% 2.88/1.57 ! [B: $tType] :
% 2.88/1.57 ( ( top_top @ ( refine1665802226e_nres @ B ) )
% 2.88/1.57 != ( bot_bot @ ( refine1665802226e_nres @ B ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_87_nres__inequalities_I4_J,axiom,
% 2.88/1.57 ! [D: $tType] :
% 2.88/1.57 ( ( bot_bot @ ( refine1665802226e_nres @ D ) )
% 2.88/1.57 != ( top_top @ ( refine1665802226e_nres @ D ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_88_order__mono__setup_Orefl,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( preorder @ A )
% 2.88/1.57 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_89_nf__inres__def,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( refine406925620_inres @ A )
% 2.88/1.57 = ( ^ [M5: refine1665802226e_nres @ A,X: A] :
% 2.88/1.57 ( ( refine1102455758nofail @ A @ M5 )
% 2.88/1.57 & ( refine1315500908_inres @ A @ M5 @ X ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_90_meta__le__everything__if__top,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order_top @ A )
% 2.88/1.57 => ! [M3: A,X2: A] :
% 2.88/1.57 ( ( M3
% 2.88/1.57 = ( top_top @ A ) )
% 2.88/1.57 => ( ord_less_eq @ A @ X2 @ M3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_91_pw__flat__le__iff,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( bot_bot @ ( refine1665802226e_nres @ A ) ) @ S @ S2 )
% 2.88/1.57 = ( ? [X4: A] : ( refine1315500908_inres @ A @ S @ X4 )
% 2.88/1.57 => ( ( ( refine1102455758nofail @ A @ S )
% 2.88/1.57 = ( refine1102455758nofail @ A @ S2 ) )
% 2.88/1.57 & ! [X: A] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ S @ X )
% 2.88/1.57 = ( refine1315500908_inres @ A @ S2 @ X ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_92_pw__abs__inres,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B,A3: A] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) @ A3 )
% 2.88/1.57 = ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 2.88/1.57 => ? [C4: B] :
% 2.88/1.57 ( ( refine1315500908_inres @ B @ M @ C4 )
% 2.88/1.57 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ C4 @ A3 ) @ R ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_93_pw__abs__nofail,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),M: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( refine1102455758nofail @ A @ ( refine81118332bs_fun @ B @ A @ R @ M ) )
% 2.88/1.57 = ( ( refine1102455758nofail @ B @ M )
% 2.88/1.57 & ! [X: B] :
% 2.88/1.57 ( ( refine1315500908_inres @ B @ M @ X )
% 2.88/1.57 => ( member @ B @ X @ ( domain @ B @ A @ R ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_94_conc__abs__swap,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),M4: refine1665802226e_nres @ A,M3: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( single_valued @ A @ B @ R )
% 2.88/1.57 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M4 @ ( refine1073749519nc_fun @ A @ B @ R @ M3 ) )
% 2.88/1.57 = ( ord_less_eq @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R @ M4 ) @ M3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_95_pw__flat__ge__iff,axiom,
% 2.88/1.57 ! [A: $tType,S: refine1665802226e_nres @ A,S2: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ S @ S2 )
% 2.88/1.57 = ( ( refine1102455758nofail @ A @ S )
% 2.88/1.57 => ( ( refine1102455758nofail @ A @ S2 )
% 2.88/1.57 & ! [X: A] :
% 2.88/1.57 ( ( refine1315500908_inres @ A @ S @ X )
% 2.88/1.57 = ( refine1315500908_inres @ A @ S2 @ X ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_96_GreatestI2__order,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [P: A > $o,X2: A,Q: A > $o] :
% 2.88/1.57 ( ( P @ X2 )
% 2.88/1.57 => ( ! [Y3: A] :
% 2.88/1.57 ( ( P @ Y3 )
% 2.88/1.57 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 2.88/1.57 => ( ! [X3: A] :
% 2.88/1.57 ( ( P @ X3 )
% 2.88/1.57 => ( ! [Y5: A] :
% 2.88/1.57 ( ( P @ Y5 )
% 2.88/1.57 => ( ord_less_eq @ A @ Y5 @ X3 ) )
% 2.88/1.57 => ( Q @ X3 ) ) )
% 2.88/1.57 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_97_Greatest__equality,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( order @ A )
% 2.88/1.57 => ! [P: A > $o,X2: A] :
% 2.88/1.57 ( ( P @ X2 )
% 2.88/1.57 => ( ! [Y3: A] :
% 2.88/1.57 ( ( P @ Y3 )
% 2.88/1.57 => ( ord_less_eq @ A @ Y3 @ X2 ) )
% 2.88/1.57 => ( ( order_Greatest @ A @ P )
% 2.88/1.57 = X2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_98_Refine__Misc_Oif__mono,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ord @ A )
% 2.88/1.57 => ! [B3: $o,M1: A,M12: A,M22: A,M23: A] :
% 2.88/1.57 ( ( B3
% 2.88/1.57 => ( ord_less_eq @ A @ M1 @ M12 ) )
% 2.88/1.57 => ( ( ~ B3
% 2.88/1.57 => ( ord_less_eq @ A @ M22 @ M23 ) )
% 2.88/1.57 => ( ord_less_eq @ A @ ( if @ A @ B3 @ M1 @ M22 ) @ ( if @ A @ B3 @ M12 @ M23 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_99_flat__ord_Odual__order_Orefl,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,A3: A] : ( partial_flat_ord @ A @ B3 @ A3 @ A3 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_100_flat__le__mono__setup_Orefl,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,X2: A] : ( partial_flat_ord @ A @ B3 @ X2 @ X2 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_101_flat__ord_Odual__order_Oantisym,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A,B3: A,A3: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 2.88/1.57 => ( A3 = B3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_102_flat__le__mono__setup_Omono__let,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,B3: A,F: B > A,F3: B > A,X2: B] :
% 2.88/1.57 ( ! [X3: B] : ( partial_flat_ord @ A @ B3 @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 2.88/1.57 => ( partial_flat_ord @ A @ B3 @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_103_flat__ord_Odual__order_Oeq__iff,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A] :
% 2.88/1.57 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [A4: A,B4: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ B4 @ A4 )
% 2.88/1.57 & ( partial_flat_ord @ A @ Ba @ A4 @ B4 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_104_flat__le__mono__setup_Omono__if,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A,T: A,T2: A,E: A,E2: A,B3: $o] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ T @ T2 )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ Ba @ E @ E2 )
% 2.88/1.57 => ( partial_flat_ord @ A @ Ba @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_105_flat__ord_Odual__order_Otrans,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A,B3: A,A3: A,C3: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ Ba @ C3 @ B3 )
% 2.88/1.57 => ( partial_flat_ord @ A @ Ba @ C3 @ A3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_106_flat__ord_Oord__le__eq__trans,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A,A3: A,B3: A,C3: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 2.88/1.57 => ( ( B3 = C3 )
% 2.88/1.57 => ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_107_flat__ord_Oord__eq__le__trans,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B3: A,Ba: A,C3: A] :
% 2.88/1.57 ( ( A3 = B3 )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ Ba @ B3 @ C3 )
% 2.88/1.57 => ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_108_flat__ord_Oorder_Oantisym,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A,A3: A,B3: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ Ba @ B3 @ A3 )
% 2.88/1.57 => ( A3 = B3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_109_flat__ord_Oorder_Oeq__iff,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A] :
% 2.88/1.57 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [A4: A,B4: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ A4 @ B4 )
% 2.88/1.57 & ( partial_flat_ord @ A @ Ba @ B4 @ A4 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_110_flat__ord_Oantisym__conv,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,Y: A,X2: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ B3 @ Y @ X2 )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
% 2.88/1.57 = ( X2 = Y ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_111_flat__ord_Oorder__trans,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,X2: A,Y: A,Z2: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ B3 @ Y @ Z2 )
% 2.88/1.57 => ( partial_flat_ord @ A @ B3 @ X2 @ Z2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_112_flat__ord_Oorder_Otrans,axiom,
% 2.88/1.57 ! [A: $tType,Ba: A,A3: A,B3: A,C3: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ Ba @ A3 @ B3 )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ Ba @ B3 @ C3 )
% 2.88/1.57 => ( partial_flat_ord @ A @ Ba @ A3 @ C3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_113_flat__ord_Oeq__refl,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,Y: A,B3: A] :
% 2.88/1.57 ( ( X2 = Y )
% 2.88/1.57 => ( partial_flat_ord @ A @ B3 @ X2 @ Y ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_114_flat__ord_Oantisym,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,X2: A,Y: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ B3 @ X2 @ Y )
% 2.88/1.57 => ( ( partial_flat_ord @ A @ B3 @ Y @ X2 )
% 2.88/1.57 => ( X2 = Y ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_115_flat__ord_Oeq__iff,axiom,
% 2.88/1.57 ! [A: $tType,B3: A] :
% 2.88/1.57 ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [X: A,Y4: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ B3 @ X @ Y4 )
% 2.88/1.57 & ( partial_flat_ord @ A @ B3 @ Y4 @ X ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_116_nofail__antimono__fun,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,F: A > ( refine1665802226e_nres @ B ),G: A > ( refine1665802226e_nres @ B ),X2: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( A > ( refine1665802226e_nres @ B ) ) @ F @ G )
% 2.88/1.57 => ( ( refine1102455758nofail @ B @ ( G @ X2 ) )
% 2.88/1.57 => ( refine1102455758nofail @ B @ ( F @ X2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_117_flat__ord__compat_I1_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( comple187826305attice @ A )
% 2.88/1.57 => ! [X2: A,Y: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ ( bot_bot @ A ) @ X2 @ Y )
% 2.88/1.57 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_118_flat__ord__compat_I2_J,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( comple187826305attice @ A )
% 2.88/1.57 => ! [X2: A,Y: A] :
% 2.88/1.57 ( ( partial_flat_ord @ A @ ( top_top @ A ) @ X2 @ Y )
% 2.88/1.57 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_119_Refine__Basic__Mirabelle__tqojlsrkwy_Obind__mono_I2_J,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,M: refine1665802226e_nres @ A,M2: refine1665802226e_nres @ A,F: A > ( refine1665802226e_nres @ B ),F3: A > ( refine1665802226e_nres @ B )] :
% 2.88/1.57 ( ( partial_flat_ord @ ( refine1665802226e_nres @ A ) @ ( top_top @ ( refine1665802226e_nres @ A ) ) @ M @ M2 )
% 2.88/1.57 => ( ! [X3: A] : ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 2.88/1.57 => ( partial_flat_ord @ ( refine1665802226e_nres @ B ) @ ( top_top @ ( refine1665802226e_nres @ B ) ) @ ( refine463715084e_bind @ A @ B @ M @ F ) @ ( refine463715084e_bind @ A @ B @ M2 @ F3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_120_conc__fun__R__mono,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),M: refine1665802226e_nres @ B] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ R2 )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine1073749519nc_fun @ A @ B @ R @ M ) @ ( refine1073749519nc_fun @ A @ B @ R2 @ M ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_121_order__mono__setup_Omono__let,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( preorder @ A )
% 2.88/1.57 => ! [F: B > A,F3: B > A,X2: B] :
% 2.88/1.57 ( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F3 @ X3 ) )
% 2.88/1.57 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F3 @ X2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_122_order__mono__setup_Omono__if,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( preorder @ A )
% 2.88/1.57 => ! [T: A,T2: A,E: A,E2: A,B3: $o] :
% 2.88/1.57 ( ( ord_less_eq @ A @ T @ T2 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ E @ E2 )
% 2.88/1.57 => ( ord_less_eq @ A @ ( if @ A @ B3 @ T @ E ) @ ( if @ A @ B3 @ T2 @ E2 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_123_Domain__empty,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( domain @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_124_old_Oprod_Oinject,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
% 2.88/1.57 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.88/1.57 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 2.88/1.57 = ( ( A3 = A6 )
% 2.88/1.57 & ( B3 = B6 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_125_prod_Oinject,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.88/1.57 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.88/1.57 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.88/1.57 = ( ( X1 = Y1 )
% 2.88/1.57 & ( X22 = Y22 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_126_Domain_Oinducts,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,X2: A,R3: set @ ( product_prod @ A @ B ),P: A > $o] :
% 2.88/1.57 ( ( member @ A @ X2 @ ( domain @ A @ B @ R3 ) )
% 2.88/1.57 => ( ! [A5: A,B5: B] :
% 2.88/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R3 )
% 2.88/1.57 => ( P @ A5 ) )
% 2.88/1.57 => ( P @ X2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_127_Domain_ODomainI,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 2.88/1.57 => ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_128_bot__empty__eq,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( bot_bot @ ( A > $o ) )
% 2.88/1.57 = ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_129_surj__pair,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.88/1.57 ? [X3: A,Y3: B] :
% 2.88/1.57 ( P2
% 2.88/1.57 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_130_prod__cases,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.88/1.57 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.88/1.57 => ( P @ P2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_131_Pair__inject,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B6: B] :
% 2.88/1.57 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.88/1.57 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 2.88/1.57 => ~ ( ( A3 = A6 )
% 2.88/1.57 => ( B3 != B6 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_132_prod__cases3,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.88/1.57 ~ ! [A5: A,B5: B,C5: C] :
% 2.88/1.57 ( Y
% 2.88/1.57 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_133_prod__cases4,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.88/1.57 ~ ! [A5: A,B5: B,C5: C,D2: D] :
% 2.88/1.57 ( Y
% 2.88/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_134_prod__cases5,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 2.88/1.57 ~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3] :
% 2.88/1.57 ( Y
% 2.88/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_135_prod__cases6,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) )] :
% 2.88/1.57 ~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4] :
% 2.88/1.57 ( Y
% 2.88/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F4 ) @ D2 @ ( product_Pair @ E3 @ F4 @ E4 @ F5 ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_136_prod__cases7,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 2.88/1.57 ~ ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4,G4: G3] :
% 2.88/1.57 ( Y
% 2.88/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F4 @ G3 ) @ E4 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_137_prod__induct3,axiom,
% 2.88/1.57 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.88/1.57 ( ! [A5: A,B5: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) )
% 2.88/1.57 => ( P @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_138_prod__induct4,axiom,
% 2.88/1.57 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.88/1.57 ( ! [A5: A,B5: B,C5: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
% 2.88/1.57 => ( P @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_139_prod__induct5,axiom,
% 2.88/1.57 ! [E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 2.88/1.57 ( ! [A5: A,B5: B,C5: C,D2: D,E4: E3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) )
% 2.88/1.57 => ( P @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_140_prod__induct6,axiom,
% 2.88/1.57 ! [F4: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) )] :
% 2.88/1.57 ( ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F4 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F4 ) @ D2 @ ( product_Pair @ E3 @ F4 @ E4 @ F5 ) ) ) ) ) )
% 2.88/1.57 => ( P @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_141_prod__induct7,axiom,
% 2.88/1.57 ! [G3: $tType,F4: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
% 2.88/1.57 ( ! [A5: A,B5: B,C5: C,D2: D,E4: E3,F5: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F4 @ G3 ) @ E4 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) )
% 2.88/1.57 => ( P @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_142_old_Oprod_Oexhaust,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.88/1.57 ~ ! [A5: A,B5: B] :
% 2.88/1.57 ( Y
% 2.88/1.57 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_143_old_Oprod_Oinducts,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.88/1.57 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.88/1.57 => ( P @ Prod ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_144_subrelI,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ! [X3: A,Y3: B] :
% 2.88/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 2.88/1.57 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S5 ) )
% 2.88/1.57 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_145_single__valued__subset,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 )
% 2.88/1.57 => ( ( single_valued @ A @ B @ S5 )
% 2.88/1.57 => ( single_valued @ A @ B @ R3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_146_single__valued__empty,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_147_Domain__empty__iff,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( ( domain @ A @ B @ R3 )
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 = ( R3
% 2.88/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_148_Domain__mono,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R3 ) @ ( domain @ A @ B @ S5 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_149_single__valuedD,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),X2: A,Y: B,Z2: B] :
% 2.88/1.57 ( ( single_valued @ A @ B @ R3 )
% 2.88/1.57 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R3 )
% 2.88/1.57 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z2 ) @ R3 )
% 2.88/1.57 => ( Y = Z2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_150_single__valuedI,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ! [X3: A,Y3: B,Z3: B] :
% 2.88/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 2.88/1.57 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z3 ) @ R3 )
% 2.88/1.57 => ( Y3 = Z3 ) ) )
% 2.88/1.57 => ( single_valued @ A @ B @ R3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_151_single__valued__def,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( single_valued @ A @ B )
% 2.88/1.57 = ( ^ [R4: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ! [X: A,Y4: B] :
% 2.88/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R4 )
% 2.88/1.57 => ! [Z4: B] :
% 2.88/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z4 ) @ R4 )
% 2.88/1.57 => ( Y4 = Z4 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_152_DomainE,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 2.88/1.57 => ~ ! [B5: B] :
% 2.88/1.57 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_153_Domain__iff,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 2.88/1.57 = ( ? [Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y4 ) @ R3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_154_Domain_Ocases,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 2.88/1.57 => ~ ! [B5: B] :
% 2.88/1.57 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_155_Domain_Osimps,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,A3: A,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( member @ A @ A3 @ ( domain @ A @ B @ R3 ) )
% 2.88/1.57 = ( ? [A4: A,B4: B] :
% 2.88/1.57 ( ( A3 = A4 )
% 2.88/1.57 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ R3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_156_subset__empty,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 = ( A2
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_157_empty__subsetI,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_158_subsetI,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ! [X3: A] :
% 2.88/1.57 ( ( member @ A @ X3 @ A2 )
% 2.88/1.57 => ( member @ A @ X3 @ B2 ) )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_159_subset__antisym,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.88/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.88/1.57 => ( A2 = B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_160_empty__iff,axiom,
% 2.88/1.57 ! [A: $tType,C3: A] :
% 2.88/1.57 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_161_all__not__in__conv,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] :
% 2.88/1.57 ( ( ! [X: A] :
% 2.88/1.57 ~ ( member @ A @ X @ A2 ) )
% 2.88/1.57 = ( A2
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_162_Collect__empty__eq,axiom,
% 2.88/1.57 ! [A: $tType,P: A > $o] :
% 2.88/1.57 ( ( ( collect @ A @ P )
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 = ( ! [X: A] :
% 2.88/1.57 ~ ( P @ X ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_163_empty__Collect__eq,axiom,
% 2.88/1.57 ! [A: $tType,P: A > $o] :
% 2.88/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 2.88/1.57 = ( collect @ A @ P ) )
% 2.88/1.57 = ( ! [X: A] :
% 2.88/1.57 ~ ( P @ X ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_164_emptyE,axiom,
% 2.88/1.57 ! [A: $tType,A3: A] :
% 2.88/1.57 ~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_165_equals0D,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,A3: A] :
% 2.88/1.57 ( ( A2
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 => ~ ( member @ A @ A3 @ A2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_166_equals0I,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] :
% 2.88/1.57 ( ! [Y3: A] :
% 2.88/1.57 ~ ( member @ A @ Y3 @ A2 )
% 2.88/1.57 => ( A2
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_167_ex__in__conv,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] :
% 2.88/1.57 ( ( ? [X: A] : ( member @ A @ X @ A2 ) )
% 2.88/1.57 = ( A2
% 2.88/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_168_bot__set__def,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( bot_bot @ ( set @ A ) )
% 2.88/1.57 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_169_empty__not__UNIV,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( bot_bot @ ( set @ A ) )
% 2.88/1.57 != ( top_top @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_170_Collect__mono__iff,axiom,
% 2.88/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.88/1.57 = ( ! [X: A] :
% 2.88/1.57 ( ( P @ X )
% 2.88/1.57 => ( Q @ X ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_171_set__eq__subset,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ^ [Y2: set @ A,Z: set @ A] : ( Y2 = Z ) )
% 2.88/1.57 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.88/1.57 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_172_subset__trans,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C2: set @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.88/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C2 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ C2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_173_Collect__mono,axiom,
% 2.88/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.88/1.57 ( ! [X3: A] :
% 2.88/1.57 ( ( P @ X3 )
% 2.88/1.57 => ( Q @ X3 ) )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_174_subset__refl,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_175_subset__UNIV,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_176_subset__iff,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.88/1.57 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.88/1.57 ! [T3: A] :
% 2.88/1.57 ( ( member @ A @ T3 @ A7 )
% 2.88/1.57 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_177_equalityD2,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ( A2 = B2 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_178_equalityD1,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ( A2 = B2 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_179_subset__eq,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.88/1.57 = ( ^ [A7: set @ A,B7: set @ A] :
% 2.88/1.57 ! [X: A] :
% 2.88/1.57 ( ( member @ A @ X @ A7 )
% 2.88/1.57 => ( member @ A @ X @ B7 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_180_equalityE,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ( A2 = B2 )
% 2.88/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.88/1.57 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_181_subsetD,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,C3: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.88/1.57 => ( ( member @ A @ C3 @ A2 )
% 2.88/1.57 => ( member @ A @ C3 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_182_in__mono,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.88/1.57 => ( ( member @ A @ X2 @ A2 )
% 2.88/1.57 => ( member @ A @ X2 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_183_Collect__empty__eq__bot,axiom,
% 2.88/1.57 ! [A: $tType,P: A > $o] :
% 2.88/1.57 ( ( ( collect @ A @ P )
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 = ( P
% 2.88/1.57 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_184_less__by__empty,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
% 2.88/1.57 ( ( A2
% 2.88/1.57 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 2.88/1.57 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A2 @ B2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_185_subset__emptyI,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] :
% 2.88/1.57 ( ! [X3: A] :
% 2.88/1.57 ~ ( member @ A @ X3 @ A2 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_186_UNIV__I,axiom,
% 2.88/1.57 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_187_UNIV__witness,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_188_top__set__def,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( top_top @ ( set @ A ) )
% 2.88/1.57 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_189_UNIV__eq__I,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A] :
% 2.88/1.57 ( ! [X3: A] : ( member @ A @ X3 @ A2 )
% 2.88/1.57 => ( ( top_top @ ( set @ A ) )
% 2.88/1.57 = A2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_190_top__empty__eq,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( top_top @ ( A > $o ) )
% 2.88/1.57 = ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_191_ssubst__Pair__rhs,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R3: A,S5: B,R: set @ ( product_prod @ A @ B ),S6: B] :
% 2.88/1.57 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S5 ) @ R )
% 2.88/1.57 => ( ( S6 = S5 )
% 2.88/1.57 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S6 ) @ R ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_192_iso__tuple__UNIV__I,axiom,
% 2.88/1.57 ! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_193_ac__galois,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( single_valued @ A @ B @ R )
% 2.88/1.57 => ( refine1150083786ection @ ( refine1665802226e_nres @ A ) @ ( refine1665802226e_nres @ B ) @ ( refine81118332bs_fun @ A @ B @ R ) @ ( refine1073749519nc_fun @ A @ B @ R ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_194_abs__fun__simps_I3_J,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,X5: set @ B,R: set @ ( product_prod @ B @ A )] :
% 2.88/1.57 ( ~ ( ord_less_eq @ ( set @ B ) @ X5 @ ( domain @ B @ A @ R ) )
% 2.88/1.57 => ( ( refine81118332bs_fun @ B @ A @ R @ ( refine605929679le_RES @ B @ X5 ) )
% 2.88/1.57 = ( top_top @ ( refine1665802226e_nres @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_195_nres_Oinject,axiom,
% 2.88/1.57 ! [A: $tType,X22: set @ A,Y22: set @ A] :
% 2.88/1.57 ( ( ( refine605929679le_RES @ A @ X22 )
% 2.88/1.57 = ( refine605929679le_RES @ A @ Y22 ) )
% 2.88/1.57 = ( X22 = Y22 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_196_nres__more__simps_I4_J,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A,Y6: set @ A] :
% 2.88/1.57 ( ( ( refine605929679le_RES @ A @ X5 )
% 2.88/1.57 = ( refine605929679le_RES @ A @ Y6 ) )
% 2.88/1.57 = ( X5 = Y6 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_197_top1I,axiom,
% 2.88/1.57 ! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_198_nofail__simps_I2_J,axiom,
% 2.88/1.57 ! [B: $tType,X5: set @ B] : ( refine1102455758nofail @ B @ ( refine605929679le_RES @ B @ X5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_199_inres__simps_I2_J,axiom,
% 2.88/1.57 ! [B: $tType,X5: set @ B] :
% 2.88/1.57 ( ( refine1315500908_inres @ B @ ( refine605929679le_RES @ B @ X5 ) )
% 2.88/1.57 = ( ^ [X: B] : ( member @ B @ X @ X5 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_200_nf__inres__RES,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A,X2: A] :
% 2.88/1.57 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ X5 ) @ X2 )
% 2.88/1.57 = ( member @ A @ X2 @ X5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_201_nf__inres__SPEC,axiom,
% 2.88/1.57 ! [A: $tType,Phi: A > $o,X2: A] :
% 2.88/1.57 ( ( refine406925620_inres @ A @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) @ X2 )
% 2.88/1.57 = ( Phi @ X2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_202_nres__more__simps_I2_J,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A] :
% 2.88/1.57 ( ( ( refine605929679le_RES @ A @ X5 )
% 2.88/1.57 = ( bot_bot @ ( refine1665802226e_nres @ A ) ) )
% 2.88/1.57 = ( X5
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_203_nres__more__simps_I1_J,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A] :
% 2.88/1.57 ( ( ( bot_bot @ ( refine1665802226e_nres @ A ) )
% 2.88/1.57 = ( refine605929679le_RES @ A @ X5 ) )
% 2.88/1.57 = ( X5
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_204_nres__order__simps_I21_J,axiom,
% 2.88/1.57 ! [X6: $tType,X2: X6,Y6: set @ X6] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ X6 ) @ ( refine1687780735RETURN @ X6 @ X2 ) @ ( refine605929679le_RES @ X6 @ Y6 ) )
% 2.88/1.57 = ( member @ X6 @ X2 @ Y6 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_205_nres__cases,axiom,
% 2.88/1.57 ! [A: $tType,M: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( M
% 2.88/1.57 != ( top_top @ ( refine1665802226e_nres @ A ) ) )
% 2.88/1.57 => ~ ! [X7: set @ A] :
% 2.88/1.57 ( M
% 2.88/1.57 != ( refine605929679le_RES @ A @ X7 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_206_nres__inequalities_I1_J,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A] :
% 2.88/1.57 ( ( top_top @ ( refine1665802226e_nres @ A ) )
% 2.88/1.57 != ( refine605929679le_RES @ A @ X5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_207_galois__connection__def,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( ( comple187826305attice @ A )
% 2.88/1.57 & ( comple187826305attice @ B ) )
% 2.88/1.57 => ( ( refine1150083786ection @ A @ B )
% 2.88/1.57 = ( ^ [Alpha: A > B,Gamma: B > A] :
% 2.88/1.57 ! [C4: A,A4: B] :
% 2.88/1.57 ( ( ord_less_eq @ A @ C4 @ ( Gamma @ A4 ) )
% 2.88/1.57 = ( ord_less_eq @ B @ ( Alpha @ C4 ) @ A4 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_208_galois__connection_Ointro,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( ( comple187826305attice @ B )
% 2.88/1.57 & ( comple187826305attice @ A ) )
% 2.88/1.57 => ! [Gamma2: B > A,Alpha2: A > B] :
% 2.88/1.57 ( ! [C5: A,A5: B] :
% 2.88/1.57 ( ( ord_less_eq @ A @ C5 @ ( Gamma2 @ A5 ) )
% 2.88/1.57 = ( ord_less_eq @ B @ ( Alpha2 @ C5 ) @ A5 ) )
% 2.88/1.57 => ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_209_galois__connection_Ogalois,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( ( comple187826305attice @ B )
% 2.88/1.57 & ( comple187826305attice @ A ) )
% 2.88/1.57 => ! [Alpha2: A > B,Gamma2: B > A,C3: A,A3: B] :
% 2.88/1.57 ( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
% 2.88/1.57 => ( ( ord_less_eq @ A @ C3 @ ( Gamma2 @ A3 ) )
% 2.88/1.57 = ( ord_less_eq @ B @ ( Alpha2 @ C3 ) @ A3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_210_galois__connection_O_092_060alpha_062_092_060gamma_062__defl,axiom,
% 2.88/1.57 ! [A: $tType,B: $tType] :
% 2.88/1.57 ( ( ( comple187826305attice @ B )
% 2.88/1.57 & ( comple187826305attice @ A ) )
% 2.88/1.57 => ! [Alpha2: A > B,Gamma2: B > A,X2: B] :
% 2.88/1.57 ( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
% 2.88/1.57 => ( ord_less_eq @ B @ ( Alpha2 @ ( Gamma2 @ X2 ) ) @ X2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_211_galois__connection_O_092_060gamma_062_092_060alpha_062__infl,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType] :
% 2.88/1.57 ( ( ( comple187826305attice @ A )
% 2.88/1.57 & ( comple187826305attice @ B ) )
% 2.88/1.57 => ! [Alpha2: A > B,Gamma2: B > A,X2: A] :
% 2.88/1.57 ( ( refine1150083786ection @ A @ B @ Alpha2 @ Gamma2 )
% 2.88/1.57 => ( ord_less_eq @ A @ X2 @ ( Gamma2 @ ( Alpha2 @ X2 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_212_nres__order__simps_I5_J,axiom,
% 2.88/1.57 ! [E3: $tType,X5: set @ E3,Y6: set @ E3] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ E3 ) @ ( refine605929679le_RES @ E3 @ X5 ) @ ( refine605929679le_RES @ E3 @ Y6 ) )
% 2.88/1.57 = ( ord_less_eq @ ( set @ E3 ) @ X5 @ Y6 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_213_less__eq__nres_Osimps_I2_J,axiom,
% 2.88/1.57 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ ( refine605929679le_RES @ A @ A3 ) @ ( refine605929679le_RES @ A @ B3 ) )
% 2.88/1.57 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_214_bot__nres__def,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( bot_bot @ ( refine1665802226e_nres @ A ) )
% 2.88/1.57 = ( refine605929679le_RES @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_215_nofail__RES__conv,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( refine1102455758nofail @ A )
% 2.88/1.57 = ( ^ [M5: refine1665802226e_nres @ A] :
% 2.88/1.57 ? [M6: set @ A] :
% 2.88/1.57 ( M5
% 2.88/1.57 = ( refine605929679le_RES @ A @ M6 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_216_SPEC__cons__rule,axiom,
% 2.88/1.57 ! [A: $tType,M3: refine1665802226e_nres @ A,Phi: A > $o,Psi: A > $o] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Phi ) ) )
% 2.88/1.57 => ( ! [X3: A] :
% 2.88/1.57 ( ( Phi @ X3 )
% 2.88/1.57 => ( Psi @ X3 ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ M3 @ ( refine605929679le_RES @ A @ ( collect @ A @ Psi ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_217_SPEC__trans,axiom,
% 2.88/1.57 ! [A: $tType,X2: refine1665802226e_nres @ A,Y: refine1665802226e_nres @ A,Postcond: A > $o] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ Y )
% 2.88/1.57 => ( ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ Y @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) )
% 2.88/1.57 => ( ord_less_eq @ ( refine1665802226e_nres @ A ) @ X2 @ ( refine605929679le_RES @ A @ ( collect @ A @ Postcond ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_218_the__RES__inv,axiom,
% 2.88/1.57 ! [A: $tType,M3: refine1665802226e_nres @ A] :
% 2.88/1.57 ( ( refine1102455758nofail @ A @ M3 )
% 2.88/1.57 => ( ( refine605929679le_RES @ A @ ( refine1672542526he_RES @ A @ M3 ) )
% 2.88/1.57 = M3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_219_nres__order__simps_I22_J,axiom,
% 2.88/1.57 ! [Y7: $tType,X5: set @ Y7,Y: Y7] :
% 2.88/1.57 ( ( ord_less_eq @ ( refine1665802226e_nres @ Y7 ) @ ( refine605929679le_RES @ Y7 @ X5 ) @ ( refine1687780735RETURN @ Y7 @ Y ) )
% 2.88/1.57 = ( ord_less_eq @ ( set @ Y7 ) @ X5 @ ( insert @ Y7 @ Y @ ( bot_bot @ ( set @ Y7 ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_220_insert__absorb2,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,A2: set @ A] :
% 2.88/1.57 ( ( insert @ A @ X2 @ ( insert @ A @ X2 @ A2 ) )
% 2.88/1.57 = ( insert @ A @ X2 @ A2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_221_insert__iff,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.88/1.57 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 2.88/1.57 = ( ( A3 = B3 )
% 2.88/1.57 | ( member @ A @ A3 @ A2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_222_insertCI,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.88/1.57 ( ( ~ ( member @ A @ A3 @ B2 )
% 2.88/1.57 => ( A3 = B3 ) )
% 2.88/1.57 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_223_singletonI,axiom,
% 2.88/1.57 ! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_224_insert__subset,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ A2 ) @ B2 )
% 2.88/1.57 = ( ( member @ A @ X2 @ B2 )
% 2.88/1.57 & ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_225_singleton__insert__inj__eq,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,A3: A,A2: set @ A] :
% 2.88/1.57 ( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 = ( insert @ A @ A3 @ A2 ) )
% 2.88/1.57 = ( ( A3 = B3 )
% 2.88/1.57 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_226_singleton__insert__inj__eq_H,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,A2: set @ A,B3: A] :
% 2.88/1.57 ( ( ( insert @ A @ A3 @ A2 )
% 2.88/1.57 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 = ( ( A3 = B3 )
% 2.88/1.57 & ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_227_Domain__insert,axiom,
% 2.88/1.57 ! [B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B )] :
% 2.88/1.57 ( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 ) )
% 2.88/1.57 = ( insert @ A @ A3 @ ( domain @ A @ B @ R3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_228_nres__more__simps_I3_J,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A,X2: A] :
% 2.88/1.57 ( ( ( refine605929679le_RES @ A @ X5 )
% 2.88/1.57 = ( refine1687780735RETURN @ A @ X2 ) )
% 2.88/1.57 = ( X5
% 2.88/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_229_nres__more__simps_I5_J,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,X5: set @ A] :
% 2.88/1.57 ( ( ( refine1687780735RETURN @ A @ X2 )
% 2.88/1.57 = ( refine605929679le_RES @ A @ X5 ) )
% 2.88/1.57 = ( ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 = X5 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_230_insert__UNIV,axiom,
% 2.88/1.57 ! [A: $tType,X2: A] :
% 2.88/1.57 ( ( insert @ A @ X2 @ ( top_top @ ( set @ A ) ) )
% 2.88/1.57 = ( top_top @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_231_insert__mono,axiom,
% 2.88/1.57 ! [A: $tType,C2: set @ A,D3: set @ A,A3: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ C2 @ D3 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C2 ) @ ( insert @ A @ A3 @ D3 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_232_subset__insert,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ~ ( member @ A @ X2 @ A2 )
% 2.88/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ B2 ) )
% 2.88/1.57 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_233_subset__insertI,axiom,
% 2.88/1.57 ! [A: $tType,B2: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_234_subset__insertI2,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,B2: set @ A,B3: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_235_singleton__inject,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B3: A] :
% 2.88/1.57 ( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 = ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 => ( A3 = B3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_236_insert__not__empty,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,A2: set @ A] :
% 2.88/1.57 ( ( insert @ A @ A3 @ A2 )
% 2.88/1.57 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_237_doubleton__eq__iff,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B3: A,C3: A,D4: A] :
% 2.88/1.57 ( ( ( insert @ A @ A3 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 = ( insert @ A @ C3 @ ( insert @ A @ D4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.88/1.57 = ( ( ( A3 = C3 )
% 2.88/1.57 & ( B3 = D4 ) )
% 2.88/1.57 | ( ( A3 = D4 )
% 2.88/1.57 & ( B3 = C3 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_238_singleton__iff,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,A3: A] :
% 2.88/1.57 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 = ( B3 = A3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_239_singletonD,axiom,
% 2.88/1.57 ! [A: $tType,B3: A,A3: A] :
% 2.88/1.57 ( ( member @ A @ B3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 => ( B3 = A3 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_240_mk__disjoint__insert,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,A2: set @ A] :
% 2.88/1.57 ( ( member @ A @ A3 @ A2 )
% 2.88/1.57 => ? [B8: set @ A] :
% 2.88/1.57 ( ( A2
% 2.88/1.57 = ( insert @ A @ A3 @ B8 ) )
% 2.88/1.57 & ~ ( member @ A @ A3 @ B8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_241_insert__commute,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,Y: A,A2: set @ A] :
% 2.88/1.57 ( ( insert @ A @ X2 @ ( insert @ A @ Y @ A2 ) )
% 2.88/1.57 = ( insert @ A @ Y @ ( insert @ A @ X2 @ A2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_242_insert__eq__iff,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,A2: set @ A,B3: A,B2: set @ A] :
% 2.88/1.57 ( ~ ( member @ A @ A3 @ A2 )
% 2.88/1.57 => ( ~ ( member @ A @ B3 @ B2 )
% 2.88/1.57 => ( ( ( insert @ A @ A3 @ A2 )
% 2.88/1.57 = ( insert @ A @ B3 @ B2 ) )
% 2.88/1.57 = ( ( ( A3 = B3 )
% 2.88/1.57 => ( A2 = B2 ) )
% 2.88/1.57 & ( ( A3 != B3 )
% 2.88/1.57 => ? [C6: set @ A] :
% 2.88/1.57 ( ( A2
% 2.88/1.57 = ( insert @ A @ B3 @ C6 ) )
% 2.88/1.57 & ~ ( member @ A @ B3 @ C6 )
% 2.88/1.57 & ( B2
% 2.88/1.57 = ( insert @ A @ A3 @ C6 ) )
% 2.88/1.57 & ~ ( member @ A @ A3 @ C6 ) ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_243_insert__absorb,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,A2: set @ A] :
% 2.88/1.57 ( ( member @ A @ A3 @ A2 )
% 2.88/1.57 => ( ( insert @ A @ A3 @ A2 )
% 2.88/1.57 = A2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_244_insert__ident,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
% 2.88/1.57 ( ~ ( member @ A @ X2 @ A2 )
% 2.88/1.57 => ( ~ ( member @ A @ X2 @ B2 )
% 2.88/1.57 => ( ( ( insert @ A @ X2 @ A2 )
% 2.88/1.57 = ( insert @ A @ X2 @ B2 ) )
% 2.88/1.57 = ( A2 = B2 ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_245_Set_Oset__insert,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,A2: set @ A] :
% 2.88/1.57 ( ( member @ A @ X2 @ A2 )
% 2.88/1.57 => ~ ! [B8: set @ A] :
% 2.88/1.57 ( ( A2
% 2.88/1.57 = ( insert @ A @ X2 @ B8 ) )
% 2.88/1.57 => ( member @ A @ X2 @ B8 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_246_insertI2,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B2: set @ A,B3: A] :
% 2.88/1.57 ( ( member @ A @ A3 @ B2 )
% 2.88/1.57 => ( member @ A @ A3 @ ( insert @ A @ B3 @ B2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_247_insertI1,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B2: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B2 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_248_insertE,axiom,
% 2.88/1.57 ! [A: $tType,A3: A,B3: A,A2: set @ A] :
% 2.88/1.57 ( ( member @ A @ A3 @ ( insert @ A @ B3 @ A2 ) )
% 2.88/1.57 => ( ( A3 != B3 )
% 2.88/1.57 => ( member @ A @ A3 @ A2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_249_subset__singletonD,axiom,
% 2.88/1.57 ! [A: $tType,A2: set @ A,X2: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 => ( ( A2
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 | ( A2
% 2.88/1.57 = ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_250_subset__singleton__iff,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A,A3: A] :
% 2.88/1.57 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 = ( ( X5
% 2.88/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.88/1.57 | ( X5
% 2.88/1.57 = ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_251_insert__subsetI,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,A2: set @ A,X5: set @ A] :
% 2.88/1.57 ( ( member @ A @ X2 @ A2 )
% 2.88/1.57 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A2 )
% 2.88/1.57 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X5 ) @ A2 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_252_RETURN__def,axiom,
% 2.88/1.57 ! [A: $tType] :
% 2.88/1.57 ( ( refine1687780735RETURN @ A )
% 2.88/1.57 = ( ^ [X: A] : ( refine605929679le_RES @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_253_the__RES_Osimps,axiom,
% 2.88/1.57 ! [A: $tType,X5: set @ A] :
% 2.88/1.57 ( ( refine1672542526he_RES @ A @ ( refine605929679le_RES @ A @ X5 ) )
% 2.88/1.57 = X5 ) ).
% 2.88/1.57
% 2.88/1.57 thf(fact_254_the__elem__eq,axiom,
% 2.88/1.57 ! [A: $tType,X2: A] :
% 2.88/1.57 ( ( the_elem @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.88/1.57 = X2 ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( comple187826305attice @ A9 )
% 2.88/1.57 => ( comple187826305attice @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( order_top @ A9 )
% 2.88/1.57 => ( order_top @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( order_bot @ A9 )
% 2.88/1.57 => ( order_bot @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( preorder @ A9 )
% 2.88/1.57 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( order @ A9 )
% 2.88/1.57 => ( order @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Otop,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( top @ A9 )
% 2.88/1.57 => ( top @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( ord @ A9 )
% 2.88/1.57 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_fun___Orderings_Obot,axiom,
% 2.88/1.57 ! [A8: $tType,A9: $tType] :
% 2.88/1.57 ( ( bot @ A9 )
% 2.88/1.57 => ( bot @ ( A8 > A9 ) ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
% 2.88/1.57 ! [A8: $tType] : ( comple187826305attice @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 2.88/1.57 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 2.88/1.57 ! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.88/1.57 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.88/1.57 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Otop_6,axiom,
% 2.88/1.57 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 2.88/1.57 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
% 2.88/1.57 ! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_9,axiom,
% 2.88/1.57 comple187826305attice @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__top_10,axiom,
% 2.88/1.57 order_top @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 2.88/1.57 order_bot @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 2.88/1.57 preorder @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.88/1.57 linorder @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
% 2.88/1.57 order @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Otop_14,axiom,
% 2.88/1.57 top @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 2.88/1.57 ord @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 2.88/1.57 bot @ $o ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_17,axiom,
% 2.88/1.57 comple187826305attice @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_18,axiom,
% 2.88/1.57 order_top @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_19,axiom,
% 2.88/1.57 order_bot @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_20,axiom,
% 2.88/1.57 preorder @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_21,axiom,
% 2.88/1.57 linorder @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oorder_22,axiom,
% 2.88/1.57 order @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Otop_23,axiom,
% 2.88/1.57 top @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Oord_24,axiom,
% 2.88/1.57 ord @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Product__Type_Ounit___Orderings_Obot_25,axiom,
% 2.88/1.57 bot @ product_unit ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Complete__Lattices_Ocomplete__lattice_26,axiom,
% 2.88/1.57 ! [A8: $tType] : ( comple187826305attice @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__top_27,axiom,
% 2.88/1.57 ! [A8: $tType] : ( order_top @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder__bot_28,axiom,
% 2.88/1.57 ! [A8: $tType] : ( order_bot @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Opreorder_29,axiom,
% 2.88/1.57 ! [A8: $tType] : ( preorder @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oorder_30,axiom,
% 2.88/1.57 ! [A8: $tType] : ( order @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Otop_31,axiom,
% 2.88/1.57 ! [A8: $tType] : ( top @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Oord_32,axiom,
% 2.88/1.57 ! [A8: $tType] : ( ord @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(tcon_Refine__Basic__Mirabelle__tqojlsrkwy_Onres___Orderings_Obot_33,axiom,
% 2.88/1.57 ! [A8: $tType] : ( bot @ ( refine1665802226e_nres @ A8 ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(help_If_3_1_T,axiom,
% 2.88/1.57 ! [P: $o] :
% 2.88/1.57 ( ( P = $true )
% 2.88/1.57 | ( P = $false ) ) ).
% 2.88/1.57
% 2.88/1.57 thf(help_If_2_1_T,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,Y: A] :
% 2.88/1.57 ( ( if @ A @ $false @ X2 @ Y )
% 2.88/1.57 = Y ) ).
% 2.88/1.57
% 2.88/1.57 thf(help_If_1_1_T,axiom,
% 2.88/1.57 ! [A: $tType,X2: A,Y: A] :
% 2.88/1.57 ( ( if @ A @ $true @ X2 @ Y )
% 2.88/1.57 = X2 ) ).
% 2.88/1.57
% 2.88/1.57 thf(conj_0,conjecture,
% 2.88/1.57 ( ( refine463715084e_bind @ product_unit @ a @ ( bot_bot @ ( refine1665802226e_nres @ product_unit ) ) @ f )
% 2.88/1.57 = ( bot_bot @ ( refine1665802226e_nres @ a ) ) ) ).
% 2.88/1.57 ------------------------
% 2.88/1.59 % (15401)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/183Mi)
% 3.00/1.60 % (15403)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/27Mi)
% 3.00/1.60 % (15406)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/275Mi)
% 3.00/1.60 % (15407)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/18Mi)
% 3.00/1.60 % (15405)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/2Mi)
% 3.00/1.60 % (15405)Instruction limit reached!
% 3.00/1.60 % (15405)------------------------------
% 3.00/1.60 % (15405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.60 % (15405)Termination reason: Unknown
% 3.00/1.60 % (15405)Termination phase: shuffling
% 3.00/1.60
% 3.00/1.60 % (15405)Memory used [KB]: 1279
% 3.00/1.60 % (15405)Time elapsed: 0.003 s
% 3.00/1.60 % (15405)Instructions burned: 2 (million)
% 3.00/1.60 % (15405)------------------------------
% 3.00/1.60 % (15405)------------------------------
% 3.00/1.60 % (15402)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/4Mi)
% 3.00/1.61 % (15402)Instruction limit reached!
% 3.00/1.61 % (15402)------------------------------
% 3.00/1.61 % (15402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (15402)Termination reason: Unknown
% 3.00/1.61 % (15402)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (15402)Memory used [KB]: 1407
% 3.00/1.61 % (15402)Time elapsed: 0.004 s
% 3.00/1.61 % (15402)Instructions burned: 6 (million)
% 3.00/1.61 % (15402)------------------------------
% 3.00/1.61 % (15402)------------------------------
% 3.00/1.61 % (15404)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/2Mi)
% 3.00/1.61 % (15404)Instruction limit reached!
% 3.00/1.61 % (15404)------------------------------
% 3.00/1.61 % (15404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (15404)Termination reason: Unknown
% 3.00/1.61 % (15404)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (15404)Memory used [KB]: 1407
% 3.00/1.61 % (15404)Time elapsed: 0.004 s
% 3.00/1.61 % (15404)Instructions burned: 3 (million)
% 3.00/1.61 % (15404)------------------------------
% 3.00/1.61 % (15404)------------------------------
% 3.00/1.61 % (15407)Instruction limit reached!
% 3.00/1.61 % (15407)------------------------------
% 3.00/1.61 % (15407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (15407)Termination reason: Unknown
% 3.00/1.61 % (15407)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (15407)Memory used [KB]: 1791
% 3.00/1.61 % (15407)Time elapsed: 0.011 s
% 3.00/1.61 % (15407)Instructions burned: 19 (million)
% 3.00/1.61 % (15407)------------------------------
% 3.00/1.61 % (15407)------------------------------
% 3.00/1.61 % (15403)Instruction limit reached!
% 3.00/1.61 % (15403)------------------------------
% 3.00/1.61 % (15403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (15403)Termination reason: Unknown
% 3.00/1.61 % (15403)Termination phase: Property scanning
% 3.00/1.61
% 3.00/1.61 % (15403)Memory used [KB]: 1791
% 3.00/1.61 % (15403)Time elapsed: 0.015 s
% 3.00/1.61 % (15403)Instructions burned: 28 (million)
% 3.00/1.61 % (15403)------------------------------
% 3.00/1.61 % (15403)------------------------------
% 3.00/1.62 % (15413)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/3Mi)
% 3.00/1.62 % (15413)Instruction limit reached!
% 3.00/1.62 % (15413)------------------------------
% 3.00/1.62 % (15413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.62 % (15413)Termination reason: Unknown
% 3.00/1.62 % (15413)Termination phase: shuffling
% 3.00/1.62
% 3.00/1.62 % (15413)Memory used [KB]: 1407
% 3.00/1.62 % (15413)Time elapsed: 0.003 s
% 3.00/1.62 % (15413)Instructions burned: 3 (million)
% 3.00/1.62 % (15413)------------------------------
% 3.00/1.62 % (15413)------------------------------
% 3.00/1.62 % (15414)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/37Mi)
% 3.00/1.62 % (15406)First to succeed.
% 3.00/1.63 % (15419)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/1041Mi)
% 3.00/1.63 % (15416)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/15Mi)
% 3.00/1.63 % (15417)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/3Mi)
% 3.00/1.63 % (15417)Instruction limit reached!
% 3.00/1.63 % (15417)------------------------------
% 3.00/1.63 % (15417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (15417)Termination reason: Unknown
% 3.00/1.63 % (15417)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (15417)Memory used [KB]: 1407
% 3.00/1.63 % (15417)Time elapsed: 0.005 s
% 3.00/1.63 % (15417)Instructions burned: 4 (million)
% 3.00/1.63 % (15417)------------------------------
% 3.00/1.63 % (15417)------------------------------
% 3.00/1.63 % (15421)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/7Mi)
% 3.00/1.63 % (15416)Instruction limit reached!
% 3.00/1.63 % (15416)------------------------------
% 3.00/1.63 % (15416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (15416)Termination reason: Unknown
% 3.00/1.63 % (15416)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (15416)Memory used [KB]: 1663
% 3.00/1.63 % (15416)Time elapsed: 0.008 s
% 3.00/1.63 % (15416)Instructions burned: 17 (million)
% 3.00/1.63 % (15416)------------------------------
% 3.00/1.63 % (15416)------------------------------
% 3.00/1.63 % (15414)Instruction limit reached!
% 3.00/1.63 % (15414)------------------------------
% 3.00/1.63 % (15414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (15414)Termination reason: Unknown
% 3.00/1.63 % (15414)Termination phase: Property scanning
% 3.00/1.63
% 3.00/1.63 % (15414)Memory used [KB]: 2174
% 3.00/1.63 % (15414)Time elapsed: 0.013 s
% 3.00/1.63 % (15414)Instructions burned: 38 (million)
% 3.00/1.63 % (15414)------------------------------
% 3.00/1.63 % (15414)------------------------------
% 3.00/1.63 % (15421)Instruction limit reached!
% 3.00/1.63 % (15421)------------------------------
% 3.00/1.63 % (15421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (15421)Termination reason: Unknown
% 3.00/1.63 % (15421)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (15421)Memory used [KB]: 1535
% 3.00/1.63 % (15421)Time elapsed: 0.004 s
% 3.00/1.63 % (15421)Instructions burned: 8 (million)
% 3.00/1.63 % (15421)------------------------------
% 3.00/1.63 % (15421)------------------------------
% 3.00/1.64 % (15422)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/16Mi)
% 3.00/1.64 % (15423)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/3Mi)
% 3.00/1.64 % (15423)Instruction limit reached!
% 3.00/1.64 % (15423)------------------------------
% 3.00/1.64 % (15423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (15423)Termination reason: Unknown
% 3.00/1.64 % (15423)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (15423)Memory used [KB]: 1407
% 3.00/1.64 % (15423)Time elapsed: 0.002 s
% 3.00/1.64 % (15423)Instructions burned: 3 (million)
% 3.00/1.64 % (15423)------------------------------
% 3.00/1.64 % (15423)------------------------------
% 3.00/1.64 % (15424)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/3Mi)
% 3.00/1.64 % (15425)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/7Mi)
% 3.00/1.64 % (15424)Instruction limit reached!
% 3.00/1.64 % (15424)------------------------------
% 3.00/1.64 % (15424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (15424)Termination reason: Unknown
% 3.00/1.64 % (15424)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (15424)Memory used [KB]: 1407
% 3.00/1.64 % (15424)Time elapsed: 0.002 s
% 3.00/1.64 % (15424)Instructions burned: 3 (million)
% 3.00/1.64 % (15424)------------------------------
% 3.00/1.64 % (15424)------------------------------
% 3.00/1.65 % (15422)Instruction limit reached!
% 3.00/1.65 % (15422)------------------------------
% 3.00/1.65 % (15422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (15422)Termination reason: Unknown
% 3.00/1.65 % (15422)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (15422)Memory used [KB]: 1791
% 3.00/1.65 % (15422)Time elapsed: 0.008 s
% 3.00/1.65 % (15422)Instructions burned: 17 (million)
% 3.00/1.65 % (15422)------------------------------
% 3.00/1.65 % (15422)------------------------------
% 3.00/1.65 % (15425)Instruction limit reached!
% 3.00/1.65 % (15425)------------------------------
% 3.00/1.65 % (15425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (15425)Termination reason: Unknown
% 3.00/1.65 % (15425)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (15425)Memory used [KB]: 1535
% 3.00/1.65 % (15425)Time elapsed: 0.005 s
% 3.00/1.65 % (15425)Instructions burned: 9 (million)
% 3.00/1.65 % (15425)------------------------------
% 3.00/1.65 % (15425)------------------------------
% 3.00/1.65 % (15426)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/3Mi)
% 3.00/1.65 % (15426)Instruction limit reached!
% 3.00/1.65 % (15426)------------------------------
% 3.00/1.65 % (15426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (15426)Termination reason: Unknown
% 3.00/1.65 % (15426)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (15426)Memory used [KB]: 1407
% 3.00/1.65 % (15426)Time elapsed: 0.003 s
% 3.00/1.65 % (15426)Instructions burned: 4 (million)
% 3.00/1.65 % (15426)------------------------------
% 3.00/1.65 % (15426)------------------------------
% 3.00/1.66 % (15427)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/4Mi)
% 3.00/1.66 % (15427)Instruction limit reached!
% 3.00/1.66 % (15427)------------------------------
% 3.00/1.66 % (15427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (15427)Termination reason: Unknown
% 3.00/1.66 % (15427)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (15427)Memory used [KB]: 1407
% 3.00/1.66 % (15427)Time elapsed: 0.004 s
% 3.00/1.66 % (15427)Instructions burned: 6 (million)
% 3.00/1.66 % (15427)------------------------------
% 3.00/1.66 % (15427)------------------------------
% 3.00/1.66 % (15428)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/18Mi)
% 3.00/1.66 % (15429)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/710Mi)
% 3.00/1.66 % (15401)Instruction limit reached!
% 3.00/1.66 % (15401)------------------------------
% 3.00/1.66 % (15401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (15401)Termination reason: Unknown
% 3.00/1.66 % (15401)Termination phase: Saturation
% 3.00/1.66
% 3.00/1.66 % (15401)Memory used [KB]: 7291
% 3.00/1.66 % (15401)Time elapsed: 0.068 s
% 3.00/1.66 % (15401)Instructions burned: 183 (million)
% 3.00/1.66 % (15401)------------------------------
% 3.00/1.66 % (15401)------------------------------
% 3.00/1.66 % (15428)Instruction limit reached!
% 3.00/1.66 % (15428)------------------------------
% 3.00/1.66 % (15428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (15428)Termination reason: Unknown
% 3.00/1.66 % (15428)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (15428)Memory used [KB]: 1791
% 3.00/1.66 % (15428)Time elapsed: 0.008 s
% 3.00/1.66 % (15428)Instructions burned: 19 (million)
% 3.00/1.66 % (15428)------------------------------
% 3.00/1.66 % (15428)------------------------------
% 3.00/1.67 % (15430)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/6Mi)
% 3.00/1.67 % (15430)Instruction limit reached!
% 3.00/1.67 % (15430)------------------------------
% 3.00/1.67 % (15430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.67 % (15430)Termination reason: Unknown
% 3.00/1.67 % (15430)Termination phase: shuffling
% 3.00/1.67
% 3.00/1.67 % (15430)Memory used [KB]: 1407
% 3.00/1.67 % (15430)Time elapsed: 0.004 s
% 3.00/1.67 % (15430)Instructions burned: 6 (million)
% 3.00/1.67 % (15430)------------------------------
% 3.00/1.67 % (15430)------------------------------
% 3.00/1.67 % (15431)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/902Mi)
% 3.00/1.67 % (15432)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/21Mi)
% 3.00/1.68 % (15433)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/5Mi)
% 3.00/1.68 % (15433)Instruction limit reached!
% 3.00/1.68 % (15433)------------------------------
% 3.00/1.68 % (15433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (15433)Termination reason: Unknown
% 3.00/1.68 % (15433)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (15433)Memory used [KB]: 1407
% 3.00/1.68 % (15433)Time elapsed: 0.004 s
% 3.00/1.68 % (15433)Instructions burned: 7 (million)
% 3.00/1.68 % (15433)------------------------------
% 3.00/1.68 % (15433)------------------------------
% 3.00/1.68 % (15434)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_15221 for (2999ds/6Mi)
% 3.00/1.68 % (15432)Instruction limit reached!
% 3.00/1.68 % (15432)------------------------------
% 3.00/1.68 % (15432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (15432)Termination reason: Unknown
% 3.00/1.68 % (15432)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (15432)Memory used [KB]: 1918
% 3.00/1.68 % (15432)Time elapsed: 0.009 s
% 3.00/1.68 % (15432)Instructions burned: 23 (million)
% 3.00/1.68 % (15432)------------------------------
% 3.00/1.68 % (15432)------------------------------
% 3.00/1.68 % (15434)Instruction limit reached!
% 3.00/1.68 % (15434)------------------------------
% 3.00/1.68 % (15434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (15434)Termination reason: Unknown
% 3.00/1.68 % (15434)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (15434)Memory used [KB]: 1407
% 3.00/1.68 % (15434)Time elapsed: 0.003 s
% 3.00/1.68 % (15434)Instructions burned: 7 (million)
% 3.00/1.68 % (15434)------------------------------
% 3.00/1.68 % (15434)------------------------------
% 3.00/1.69 % (15435)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_15221 for (2998ds/377Mi)
% 3.00/1.69 % (15436)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_15221 for (2998ds/779Mi)
% 3.00/1.69 % (15437)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_15221 for (2998ds/19Mi)
% 3.00/1.70 % (15437)Instruction limit reached!
% 3.00/1.70 % (15437)------------------------------
% 3.00/1.70 % (15437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.70 % (15437)Termination reason: Unknown
% 3.00/1.70 % (15437)Termination phase: shuffling
% 3.00/1.70
% 3.00/1.70 % (15437)Memory used [KB]: 1791
% 3.00/1.70 % (15437)Time elapsed: 0.008 s
% 3.00/1.70 % (15437)Instructions burned: 20 (million)
% 3.00/1.70 % (15437)------------------------------
% 3.00/1.70 % (15437)------------------------------
% 3.00/1.71 % (15438)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_15221 for (2998ds/879Mi)
% 3.00/1.81 % (15435)Instruction limit reached!
% 3.00/1.81 % (15435)------------------------------
% 3.00/1.81 % (15435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.81 % (15435)Termination reason: Unknown
% 3.00/1.81 % (15435)Termination phase: Saturation
% 3.00/1.81
% 3.00/1.81 % (15435)Memory used [KB]: 8315
% 3.00/1.81 % (15435)Time elapsed: 0.127 s
% 3.00/1.81 % (15435)Instructions burned: 378 (million)
% 3.00/1.81 % (15435)------------------------------
% 3.00/1.81 % (15435)------------------------------
% 3.00/1.83 % (15439)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_15221 for (2997ds/17Mi)
% 3.00/1.84 % (15439)Instruction limit reached!
% 3.00/1.84 % (15439)------------------------------
% 3.00/1.84 % (15439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.84 % (15439)Termination reason: Unknown
% 3.00/1.84 % (15439)Termination phase: shuffling
% 3.00/1.84
% 3.00/1.84 % (15439)Memory used [KB]: 1791
% 3.00/1.84 % (15439)Time elapsed: 0.012 s
% 3.00/1.84 % (15439)Instructions burned: 18 (million)
% 3.00/1.84 % (15439)------------------------------
% 3.00/1.84 % (15439)------------------------------
% 3.00/1.86 % (15440)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_15221 for (2997ds/3Mi)
% 3.00/1.86 % (15440)Instruction limit reached!
% 3.00/1.86 % (15440)------------------------------
% 3.00/1.86 % (15440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.86 % (15440)Termination reason: Unknown
% 3.00/1.86 % (15440)Termination phase: shuffling
% 3.00/1.86
% 3.00/1.86 % (15440)Memory used [KB]: 1407
% 3.00/1.86 % (15440)Time elapsed: 0.003 s
% 3.00/1.86 % (15440)Instructions burned: 4 (million)
% 3.00/1.86 % (15440)------------------------------
% 3.00/1.86 % (15440)------------------------------
% 3.00/1.87 % (15441)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_15221 for (2997ds/30Mi)
% 3.00/1.88 % (15441)Instruction limit reached!
% 3.00/1.88 % (15441)------------------------------
% 3.00/1.88 % (15441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.88 % (15441)Termination reason: Unknown
% 3.00/1.88 % (15441)Termination phase: Property scanning
% 3.00/1.88
% 3.00/1.88 % (15441)Memory used [KB]: 1918
% 3.00/1.88 % (15441)Time elapsed: 0.012 s
% 3.00/1.88 % (15441)Instructions burned: 33 (million)
% 3.00/1.88 % (15441)------------------------------
% 3.00/1.88 % (15441)------------------------------
% 3.00/1.89 % (15442)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_15221 for (2996ds/127Mi)
% 3.00/1.93 % (15442)Instruction limit reached!
% 3.00/1.93 % (15442)------------------------------
% 3.00/1.93 % (15442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.93 % (15442)Termination reason: Unknown
% 3.00/1.93 % (15442)Termination phase: Saturation
% 3.00/1.93
% 3.00/1.93 % (15442)Memory used [KB]: 7036
% 3.00/1.93 % (15442)Time elapsed: 0.042 s
% 3.00/1.93 % (15442)Instructions burned: 127 (million)
% 3.00/1.93 % (15442)------------------------------
% 3.00/1.93 % (15442)------------------------------
% 3.00/1.94 % (15443)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_15221 for (2996ds/100Mi)
% 3.00/1.97 % (15443)Instruction limit reached!
% 3.00/1.97 % (15443)------------------------------
% 3.00/1.97 % (15443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.97 % (15443)Termination reason: Unknown
% 3.00/1.97 % (15443)Termination phase: Saturation
% 3.00/1.97
% 3.00/1.97 % (15443)Memory used [KB]: 6908
% 3.00/1.97 % (15443)Time elapsed: 0.031 s
% 3.00/1.97 % (15443)Instructions burned: 100 (million)
% 3.00/1.97 % (15443)------------------------------
% 3.00/1.97 % (15443)------------------------------
% 3.00/1.98 % (15444)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_15221 for (2996ds/3Mi)
% 3.00/1.98 % (15444)Instruction limit reached!
% 3.00/1.98 % (15444)------------------------------
% 3.00/1.98 % (15444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.98 % (15444)Termination reason: Unknown
% 3.00/1.98 % (15444)Termination phase: shuffling
% 3.00/1.98
% 3.00/1.98 % (15444)Memory used [KB]: 1407
% 3.00/1.98 % (15444)Time elapsed: 0.003 s
% 3.00/1.98 % (15444)Instructions burned: 4 (million)
% 3.00/1.98 % (15444)------------------------------
% 3.00/1.98 % (15444)------------------------------
% 3.00/1.99 % (15445)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_15221 for (2995ds/20Mi)
% 3.00/2.00 % (15445)Instruction limit reached!
% 3.00/2.00 % (15445)------------------------------
% 3.00/2.00 % (15445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/2.00 % (15445)Termination reason: Unknown
% 3.00/2.00 % (15445)Termination phase: Property scanning
% 3.00/2.00
% 3.00/2.00 % (15445)Memory used [KB]: 1791
% 3.00/2.00 % (15445)Time elapsed: 0.009 s
% 3.00/2.00 % (15445)Instructions burned: 20 (million)
% 3.00/2.00 % (15445)------------------------------
% 3.00/2.00 % (15445)------------------------------
% 3.00/2.01 % (15446)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_15221 for (2995ds/86Mi)
% 4.21/2.03 % (15446)Instruction limit reached!
% 4.21/2.03 % (15446)------------------------------
% 4.21/2.03 % (15446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.03 % (15446)Termination reason: Unknown
% 4.21/2.03 % (15446)Termination phase: Saturation
% 4.21/2.03
% 4.21/2.03 % (15446)Memory used [KB]: 6780
% 4.21/2.03 % (15446)Time elapsed: 0.029 s
% 4.21/2.03 % (15446)Instructions burned: 87 (million)
% 4.21/2.03 % (15446)------------------------------
% 4.21/2.03 % (15446)------------------------------
% 4.21/2.04 % (15447)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_15221 for (2995ds/107Mi)
% 4.21/2.05 % (15436)Also succeeded, but the first one will report.
% 4.21/2.06 % (15406)Refutation found. Thanks to Tanya!
% 4.21/2.06 % SZS status Theorem for DTF2THF_15221
% 4.21/2.06 % SZS output start Proof for DTF2THF_15221
% 4.21/2.06 thf(type_def_6, type, product_prod: ($tType * $tType) > $tType).
% 4.21/2.06 thf(type_def_7, type, refine1665802226e_nres: ($tType) > $tType).
% 4.21/2.06 thf(type_def_8, type, set: ($tType) > $tType).
% 4.21/2.06 thf(type_def_9, type, product_unit: $tType).
% 4.21/2.06 thf(type_def_10, type, a: $tType).
% 4.21/2.06 thf(func_def_0, type, refine1665802226e_nres: ($tType) > $tType).
% 4.21/2.06 thf(func_def_1, type, product_unit: $tType).
% 4.21/2.06 thf(func_def_2, type, product_prod: ($tType * $tType) > $tType).
% 4.21/2.06 thf(func_def_3, type, set: ($tType) > $tType).
% 4.21/2.06 thf(func_def_4, type, a: $tType).
% 4.21/2.06 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_6, type, bot: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_8, type, top: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_10, type, linorder: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_11, type, preorder: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_12, type, order_bot: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_13, type, order_top: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_14, type, comple187826305attice: !>[X0: $tType]:($o)).
% 4.21/2.06 thf(func_def_15, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 4.21/2.06 thf(func_def_16, type, bot_bot: !>[X0: $tType]:(X0)).
% 4.21/2.06 thf(func_def_17, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 4.21/2.06 thf(func_def_18, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 4.21/2.06 thf(func_def_19, type, top_top: !>[X0: $tType]:(X0)).
% 4.21/2.06 thf(func_def_20, type, partial_flat_ord: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 4.21/2.06 thf(func_def_21, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 4.21/2.06 thf(func_def_22, type, refine1687780735RETURN: !>[X0: $tType]:(X0 > refine1665802226e_nres @ X0)).
% 4.21/2.06 thf(func_def_23, type, refine81118332bs_fun: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > refine1665802226e_nres @ X0 > refine1665802226e_nres @ X1)).
% 4.21/2.06 thf(func_def_24, type, refine463715084e_bind: !>[X0: $tType, X1: $tType]:(refine1665802226e_nres @ X0 > (X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X1)).
% 4.21/2.06 thf(func_def_25, type, refine1073749519nc_fun: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > refine1665802226e_nres @ X1 > refine1665802226e_nres @ X0)).
% 4.21/2.06 thf(func_def_26, type, refine1315500908_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 4.21/2.06 thf(func_def_27, type, refine406925620_inres: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0 > $o)).
% 4.21/2.06 thf(func_def_28, type, refine1102455758nofail: !>[X0: $tType]:(refine1665802226e_nres @ X0 > $o)).
% 4.21/2.06 thf(func_def_29, type, refine605929679le_RES: !>[X0: $tType]:(set @ X0 > refine1665802226e_nres @ X0)).
% 4.21/2.06 thf(func_def_30, type, refine1672542526he_RES: !>[X0: $tType]:(refine1665802226e_nres @ X0 > set @ X0)).
% 4.21/2.06 thf(func_def_31, type, refine1150083786ection: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > $o)).
% 4.21/2.06 thf(func_def_32, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 4.21/2.06 thf(func_def_33, type, single_valued: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > $o)).
% 4.21/2.06 thf(func_def_34, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 4.21/2.06 thf(func_def_35, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 4.21/2.06 thf(func_def_36, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 4.21/2.06 thf(func_def_37, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 4.21/2.06 thf(func_def_38, type, f: product_unit > refine1665802226e_nres @ a).
% 4.21/2.06 thf(func_def_53, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 4.21/2.06 thf(func_def_54, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 4.21/2.06 thf(func_def_55, type, sK2: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 4.21/2.06 thf(func_def_56, type, sK3: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 4.21/2.06 thf(func_def_57, type, sK4: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 4.21/2.06 thf(func_def_58, type, sK5: !>[X0: $tType]:(refine1665802226e_nres @ X0 > X0)).
% 4.21/2.06 thf(func_def_59, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 4.21/2.06 thf(func_def_60, type, sK7: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 4.21/2.06 thf(func_def_61, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 4.21/2.06 thf(func_def_62, type, sK9: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 4.21/2.06 thf(func_def_63, type, sK10: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 4.21/2.06 thf(func_def_64, type, sK11: !>[X0: $tType, X1: $tType]:(refine1665802226e_nres @ X0 > (X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 4.21/2.06 thf(func_def_65, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > refine1665802226e_nres @ X1) > refine1665802226e_nres @ X0 > refine1665802226e_nres @ X1 > X0)).
% 4.21/2.06 thf(func_def_66, type, sK13: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 4.21/2.06 thf(func_def_67, type, sK14: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X0 > X1)).
% 4.21/2.06 thf(func_def_68, type, sK15: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 4.21/2.06 thf(func_def_69, type, sK16: !>[X0: $tType]:(X0)).
% 4.21/2.06 thf(func_def_70, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 4.21/2.06 thf(func_def_71, type, sK18: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 4.21/2.06 thf(func_def_72, type, sK19: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > refine1665802226e_nres @ X1 > X1)).
% 4.21/2.06 thf(func_def_73, type, sK20: !>[X0: $tType, X1: $tType]:((X1 > refine1665802226e_nres @ X0) > (X1 > refine1665802226e_nres @ X0) > X1)).
% 4.21/2.06 thf(func_def_74, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 4.21/2.06 thf(func_def_75, type, sK22: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 4.21/2.06 thf(func_def_76, type, sK23: !>[X0: $tType]:(set @ X0 > X0)).
% 4.21/2.06 thf(func_def_77, type, sK24: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 4.21/2.06 thf(func_def_78, type, sK25: !>[X0: $tType]:(refine1665802226e_nres @ X0 > refine1665802226e_nres @ X0 > X0)).
% 4.21/2.06 thf(func_def_80, type, ph27: !>[X0: $tType]:(X0)).
% 4.21/2.06 thf(f1262,plain,(
% 4.21/2.06 $false),
% 4.21/2.06 inference(subsumption_resolution,[status(thm)],[f1037,f1057])).
% 4.21/2.06 thf(f1057,plain,(
% 4.21/2.06 ( ! [X1 : $tType,X0 : $tType,X2 : X0 > refine1665802226e_nres @ X1] : ((bot_bot @ refine1665802226e_nres @ X1 = (refine463715084e_bind @ X0 @ X1 @ bot_bot @ refine1665802226e_nres @ X0 @ X2))) )),
% 4.21/2.06 inference(cnf_transformation,[status(thm)],[f815])).
% 4.21/2.06 thf(f815,plain,(
% 4.21/2.06 ! [X0 : $tType,X1 : $tType,X2 : X0 > refine1665802226e_nres @ X1] : (bot_bot @ refine1665802226e_nres @ X1 = (refine463715084e_bind @ X0 @ X1 @ bot_bot @ refine1665802226e_nres @ X0 @ X2))),
% 4.21/2.06 inference(rectify,[status(thm)],[f1])).
% 4.21/2.06 thf(f1,axiom,(
% 4.21/2.06 ! [X1 : $tType,X0 : $tType,X2 : X1 > refine1665802226e_nres @ X0] : ((refine463715084e_bind @ X1 @ X0 @ bot_bot @ refine1665802226e_nres @ X1 @ X2) = bot_bot @ refine1665802226e_nres @ X0)),
% 4.21/2.06 file('/export/starexec/sandbox2/tmp/tmp.kXsISafMJX/DTF2THF_15221.p',fact_0_bind__SUCCEED)).
% 4.21/2.06 thf(f1037,plain,(
% 4.21/2.06 ((refine463715084e_bind @ product_unit @ a @ bot_bot @ refine1665802226e_nres @ product_unit @ f) != bot_bot @ refine1665802226e_nres @ a)),
% 4.21/2.06 inference(cnf_transformation,[status(thm)],[f812])).
% 4.21/2.06 thf(f812,plain,(
% 4.21/2.06 ((refine463715084e_bind @ product_unit @ a @ bot_bot @ refine1665802226e_nres @ product_unit @ f) != bot_bot @ refine1665802226e_nres @ a)),
% 4.21/2.06 inference(flattening,[status(thm)],[f302])).
% 4.21/2.06 thf(f302,negated_conjecture,(
% 4.21/2.06 ~((refine463715084e_bind @ product_unit @ a @ bot_bot @ refine1665802226e_nres @ product_unit @ f) = bot_bot @ refine1665802226e_nres @ a)),
% 4.21/2.06 inference(negated_conjecture,[status(cth)],[f301])).
% 4.21/2.06 thf(f301,conjecture,(
% 4.21/2.06 ((refine463715084e_bind @ product_unit @ a @ bot_bot @ refine1665802226e_nres @ product_unit @ f) = bot_bot @ refine1665802226e_nres @ a)),
% 4.21/2.06 file('/export/starexec/sandbox2/tmp/tmp.kXsISafMJX/DTF2THF_15221.p',conj_0)).
% 4.21/2.06 % SZS output end Proof for DTF2THF_15221
% 4.21/2.06 % (15406)------------------------------
% 4.21/2.06 % (15406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.21/2.06 % (15406)Termination reason: Refutation
% 4.21/2.06
% 4.21/2.06 % (15406)Memory used [KB]: 6652
% 4.21/2.06 % (15406)Time elapsed: 0.028 s
% 4.21/2.06 % (15406)Instructions burned: 65 (million)
% 4.21/2.06 % (15406)------------------------------
% 4.21/2.06 % (15406)------------------------------
% 4.21/2.06 % (15396)Success in time 0.482 s
%------------------------------------------------------------------------------