%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM165^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n025.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 08:41:49 AM UTC 2026
% Result : Theorem 7.26s 2.24s
% Output : Refutation 7.26s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : COM165^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n025.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
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 04:15:42 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : COM165^1 : TPTP v9.2.1. Released v7.0.0.
% 0.18/0.45 % Domain : Computing Theory
% 0.18/0.45 % Problem : Binary decision diagram 172
% 0.18/0.45 % Version : [Bla16] axioms : Especial.
% 0.18/0.45 % English :
% 0.18/0.45
% 0.18/0.45 % Refs : [OS08] Ortner & Schirmer (2008), BDD Normalisation
% 0.18/0.45 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.18/0.45 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.18/0.45 % Source : [Bla16]
% 0.18/0.45 % Names : bindag__172.p [Bla16]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.00 v7.2.0, 0.25 v7.1.0
% 0.18/0.45 % Syntax : Number of formulae : 350 ( 89 unt; 56 typ; 0 def)
% 0.18/0.45 % Number of atoms : 887 ( 197 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 13 ( 3 avg)
% 0.18/0.45 % Number of connectives : 3292 ( 110 ~; 28 |; 48 &;2628 @)
% 0.18/0.45 % ( 0 <=>; 478 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 21 ( 8 avg)
% 0.18/0.45 % Number of types : 4 ( 3 usr)
% 0.18/0.45 % Number of type conns : 240 ( 240 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 56 ( 53 usr; 9 con; 0-6 aty)
% 0.18/0.45 % Number of variables : 989 ( 55 ^; 858 !; 36 ?; 989 :)
% 0.18/0.45 % ( 40 !>; 0 ?*; 0 @-; 0 @+)
% 0.18/0.45 % SPC : TH1_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.18/0.45 % 2016-07-13 14:45:25.154
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 %----Could-be-implicit typings (5)
% 0.18/0.45 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 0.18/0.45 binDag_Mirabelle_dag: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Simpl__Heap_Oref,type,
% 0.18/0.45 simpl_ref: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Set_Oset,type,
% 0.18/0.45 set: $tType > $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_Nat_Onat,type,
% 0.18/0.45 nat: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_t_itself,type,
% 0.18/0.45 itself: $tType > $tType ).
% 0.18/0.45
% 0.18/0.45 %----Explicit typings (51)
% 0.18/0.45 thf(sy_cl_HOL_Otype,type,
% 0.18/0.45 type:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Obot,type,
% 0.18/0.45 bot:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Oord,type,
% 0.18/0.45 ord:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Oorder,type,
% 0.18/0.45 order:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Ono__bot,type,
% 0.18/0.45 no_bot:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Ono__top,type,
% 0.18/0.45 no_top:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.18/0.45 linorder:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Opreorder,type,
% 0.18/0.45 preorder:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.18/0.45 zero_neq_one:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.18/0.45 order_bot:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Owellorder,type,
% 0.18/0.45 wellorder:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.18/0.45 zero_less_one:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Odense__order,type,
% 0.18/0.45 dense_order:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.18/0.45 linordered_idom:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Fields_Olinordered__field,type,
% 0.18/0.45 linordered_field:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.18/0.45 dense_linorder:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.18/0.45 linordered_semidom:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.18/0.45 linord1659791738miring:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.18/0.45 canoni770627133id_add:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.18/0.45 condit1656338222tinuum:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.18/0.45 condit1037483654norder:
% 0.18/0.45 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 0.18/0.45 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 0.18/0.45 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 0.18/0.45 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 0.18/0.45 binDag1297733282se_dag:
% 0.18/0.45 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 0.18/0.45 binDag1442713106ec_dag:
% 0.18/0.45 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Osize__dag,type,
% 0.18/0.45 binDag1924123185ze_dag: binDag_Mirabelle_dag > nat ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 0.18/0.45 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 0.18/0.45 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.18/0.45 one_one:
% 0.18/0.45 !>[A: $tType] : A ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 0.18/0.45 sgn_sgn:
% 0.18/0.45 !>[A: $tType] : ( A > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.18/0.45 zero_zero:
% 0.18/0.45 !>[A: $tType] : A ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_If,type,
% 0.18/0.45 if:
% 0.18/0.45 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.18/0.45 size_size:
% 0.18/0.45 !>[A: $tType] : ( A > nat ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.18/0.45 bot_bot:
% 0.18/0.45 !>[A: $tType] : A ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.18/0.45 ord_less:
% 0.18/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.18/0.45 ord_less_eq:
% 0.18/0.45 !>[A: $tType] : ( A > A > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oordering,type,
% 0.18/0.45 ordering:
% 0.18/0.45 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Pure_Otype,type,
% 0.18/0.45 type2:
% 0.18/0.45 !>[A: $tType] : ( itself @ A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Relation_Oinv__imagep,type,
% 0.18/0.45 inv_imagep:
% 0.18/0.45 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 0.18/0.45 zero_neq_one_of_bool:
% 0.18/0.45 !>[A: $tType] : ( $o > A ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect,type,
% 0.18/0.45 collect:
% 0.18/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_Ois__empty,type,
% 0.18/0.45 is_empty:
% 0.18/0.45 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Zorn_Ochain__subset,type,
% 0.18/0.45 chain_subset:
% 0.18/0.45 !>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member,type,
% 0.18/0.45 member:
% 0.18/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_a____,type,
% 0.18/0.45 a: simpl_ref ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_l____,type,
% 0.18/0.45 l: binDag_Mirabelle_dag ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_r____,type,
% 0.18/0.45 r: binDag_Mirabelle_dag ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_thesis____,type,
% 0.18/0.45 thesis: $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_x,type,
% 0.18/0.45 x: binDag_Mirabelle_dag ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_y,type,
% 0.18/0.45 y: binDag_Mirabelle_dag ).
% 0.18/0.45
% 0.18/0.45 %----Relevant facts (256)
% 0.18/0.45 thf(fact_0_Node,axiom,
% 0.18/0.45 ( y
% 0.18/0.45 = ( binDag476092410e_Node @ l @ a @ r ) ) ).
% 0.18/0.45
% 0.18/0.45 % Node
% 0.18/0.45 thf(fact_1_dag_Oinject,axiom,
% 0.18/0.45 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 0.18/0.45 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 0.18/0.45 = ( ( X21 = Y21 )
% 0.18/0.45 & ( X22 = Y22 )
% 0.18/0.45 & ( X23 = Y23 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dag.inject
% 0.18/0.45 thf(fact_2_DAG_Osimps_I2_J,axiom,
% 0.18/0.45 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.18/0.45 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 0.18/0.45 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 0.18/0.45 & ( binDag_Mirabelle_DAG @ L )
% 0.18/0.45 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % DAG.simps(2)
% 0.18/0.45 thf(fact_3_x__less__y,axiom,
% 0.18/0.45 ord_less @ binDag_Mirabelle_dag @ x @ y ).
% 0.18/0.45
% 0.18/0.45 % x_less_y
% 0.18/0.45 thf(fact_4_set__of__Tip,axiom,
% 0.18/0.45 ( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
% 0.18/0.45 = ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % set_of_Tip
% 0.18/0.45 thf(fact_5_le__dag__set__of,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
% 0.18/0.45 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_dag_set_of
% 0.18/0.45 thf(fact_6_less__dag__set__of,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 0.18/0.45 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_dag_set_of
% 0.18/0.45 thf(fact_7_dag_Osimps_I7_J,axiom,
% 0.18/0.45 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.18/0.45 = ( F2 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X23 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dag.simps(7)
% 0.18/0.45 thf(fact_8_dag_Osimps_I5_J,axiom,
% 0.18/0.45 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.18/0.45 = ( F2 @ X21 @ X22 @ X23 ) ) ).
% 0.18/0.45
% 0.18/0.45 % dag.simps(5)
% 0.18/0.45 thf(fact_9_subdag__neq,axiom,
% 0.18/0.45 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag786255756subdag @ T @ S )
% 0.18/0.45 => ( T != S ) ) ).
% 0.18/0.45
% 0.18/0.45 % subdag_neq
% 0.18/0.45 thf(fact_10_subdag__trans,axiom,
% 0.18/0.45 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag786255756subdag @ T @ S )
% 0.18/0.45 => ( ( binDag786255756subdag @ S @ R )
% 0.18/0.45 => ( binDag786255756subdag @ T @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subdag_trans
% 0.18/0.45 thf(fact_11_subdag__not__sym,axiom,
% 0.18/0.45 ! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag786255756subdag @ S @ T )
% 0.18/0.45 => ~ ( binDag786255756subdag @ T @ S ) ) ).
% 0.18/0.45
% 0.18/0.45 % subdag_not_sym
% 0.18/0.45 thf(fact_12_DAG__y,axiom,
% 0.18/0.45 binDag_Mirabelle_DAG @ y ).
% 0.18/0.45
% 0.18/0.45 % DAG_y
% 0.18/0.45 thf(fact_13_DAG_Osimps_I1_J,axiom,
% 0.18/0.45 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 0.18/0.45
% 0.18/0.45 % DAG.simps(1)
% 0.18/0.45 thf(fact_14_dag_Osimps_I6_J,axiom,
% 0.18/0.45 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 0.18/0.45 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 0.18/0.45 = F1 ) ).
% 0.18/0.45
% 0.18/0.45 % dag.simps(6)
% 0.18/0.45 thf(fact_15_dag_Osimps_I4_J,axiom,
% 0.18/0.45 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 0.18/0.45 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 0.18/0.45 = F1 ) ).
% 0.18/0.45
% 0.18/0.45 % dag.simps(4)
% 0.18/0.45 thf(fact_16_dag_Odistinct_I1_J,axiom,
% 0.18/0.45 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.18/0.45 ( binDag_Mirabelle_Tip
% 0.18/0.45 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 0.18/0.45
% 0.18/0.45 % dag.distinct(1)
% 0.18/0.45 thf(fact_17_subdag_Osimps_I2_J,axiom,
% 0.18/0.45 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T )
% 0.18/0.45 = ( ( T = L )
% 0.18/0.45 | ( T = R )
% 0.18/0.45 | ( binDag786255756subdag @ L @ T )
% 0.18/0.45 | ( binDag786255756subdag @ R @ T ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subdag.simps(2)
% 0.18/0.45 thf(fact_18_subdag_Osimps_I1_J,axiom,
% 0.18/0.45 ! [T: binDag_Mirabelle_dag] :
% 0.18/0.45 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% 0.18/0.45
% 0.18/0.45 % subdag.simps(1)
% 0.18/0.45 thf(fact_19_DAG__less,axiom,
% 0.18/0.45 ! [Y: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag_Mirabelle_DAG @ Y )
% 0.18/0.45 => ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 0.18/0.45 => ( binDag_Mirabelle_DAG @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % DAG_less
% 0.18/0.45 thf(fact_20_dag_Oinduct,axiom,
% 0.18/0.45 ! [P: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( P @ binDag_Mirabelle_Tip )
% 0.18/0.45 => ( ! [X1: binDag_Mirabelle_dag,X2: simpl_ref,X3: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( P @ X1 )
% 0.18/0.45 => ( ( P @ X3 )
% 0.18/0.45 => ( P @ ( binDag476092410e_Node @ X1 @ X2 @ X3 ) ) ) )
% 0.18/0.45 => ( P @ Dag ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dag.induct
% 0.18/0.45 thf(fact_21_le__dag__def,axiom,
% 0.18/0.45 ( ( ord_less_eq @ binDag_Mirabelle_dag )
% 0.18/0.45 = ( ^ [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( S2 = T2 )
% 0.18/0.45 | ( ord_less @ binDag_Mirabelle_dag @ S2 @ T2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_dag_def
% 0.18/0.45 thf(fact_22_dag_Oexhaust,axiom,
% 0.18/0.45 ! [Y: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( Y != binDag_Mirabelle_Tip )
% 0.18/0.45 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 0.18/0.45 ( Y
% 0.18/0.45 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dag.exhaust
% 0.18/0.45 thf(fact_23_dag__less__le,axiom,
% 0.18/0.45 ( ( ord_less @ binDag_Mirabelle_dag )
% 0.18/0.45 = ( ^ [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y2 )
% 0.18/0.45 & ( X4 != Y2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % dag_less_le
% 0.18/0.45 thf(fact_24_le__dag__refl,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % le_dag_refl
% 0.18/0.45 thf(fact_25_le__dag__trans,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag,Z: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
% 0.18/0.45 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ Z )
% 0.18/0.45 => ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_dag_trans
% 0.18/0.45 thf(fact_26_less__dag__Tip,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag] :
% 0.18/0.45 ~ ( ord_less @ binDag_Mirabelle_dag @ X @ binDag_Mirabelle_Tip ) ).
% 0.18/0.45
% 0.18/0.45 % less_dag_Tip
% 0.18/0.45 thf(fact_27_less__dag__def,axiom,
% 0.18/0.45 ( ( ord_less @ binDag_Mirabelle_dag )
% 0.18/0.45 = ( ^ [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_dag_def
% 0.18/0.45 thf(fact_28_subdag__NodeD,axiom,
% 0.18/0.45 ! [T: binDag_Mirabelle_dag,Lt: binDag_Mirabelle_dag,A2: simpl_ref,Rt: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt @ A2 @ Rt ) )
% 0.18/0.45 => ( ( binDag786255756subdag @ T @ Lt )
% 0.18/0.45 & ( binDag786255756subdag @ T @ Rt ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subdag_NodeD
% 0.18/0.45 thf(fact_29_less__Node__dag,axiom,
% 0.18/0.45 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X )
% 0.18/0.45 => ( ( ord_less @ binDag_Mirabelle_dag @ L @ X )
% 0.18/0.45 & ( ord_less @ binDag_Mirabelle_dag @ R @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_Node_dag
% 0.18/0.45 thf(fact_30_less__dag__Node,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.18/0.45 = ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ L )
% 0.18/0.45 | ( ord_less_eq @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_dag_Node
% 0.18/0.45 thf(fact_31_le__dag__antisym,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
% 0.18/0.45 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ X )
% 0.18/0.45 => ( X = Y ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % le_dag_antisym
% 0.18/0.45 thf(fact_32_less__dag__Node_H,axiom,
% 0.18/0.45 ! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.18/0.45 ( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.18/0.45 = ( ( X = L )
% 0.18/0.45 | ( X = R )
% 0.18/0.45 | ( ord_less @ binDag_Mirabelle_dag @ X @ L )
% 0.18/0.45 | ( ord_less @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_dag_Node'
% 0.18/0.45 thf(fact_33_subset__empty,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A] :
% 0.18/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.18/0.45 = ( A3
% 0.18/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_empty
% 0.18/0.45 thf(fact_34_empty__subsetI,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.18/0.45
% 0.18/0.45 % empty_subsetI
% 0.18/0.45 thf(fact_35_subsetI,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.18/0.45 ( ! [X5: A] :
% 0.18/0.45 ( ( member @ A @ X5 @ A3 )
% 0.18/0.45 => ( member @ A @ X5 @ B2 ) )
% 0.18/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_36_subset__antisym,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.18/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.18/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 0.18/0.45 => ( A3 = B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_antisym
% 0.18/0.45 thf(fact_37_empty__iff,axiom,
% 0.18/0.45 ! [A: $tType,C: A] :
% 0.18/0.45 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % empty_iff
% 0.18/0.45 thf(fact_38_all__not__in__conv,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A] :
% 0.18/0.45 ( ( ! [X4: A] :
% 0.18/0.45 ~ ( member @ A @ X4 @ A3 ) )
% 0.18/0.45 = ( A3
% 0.18/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % all_not_in_conv
% 0.18/0.45 thf(fact_39_Collect__empty__eq,axiom,
% 0.18/0.45 ! [A: $tType,P: A > $o] :
% 0.18/0.45 ( ( ( collect @ A @ P )
% 0.18/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.18/0.45 = ( ! [X4: A] :
% 0.18/0.45 ~ ( P @ X4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_empty_eq
% 0.18/0.45 thf(fact_40_empty__Collect__eq,axiom,
% 0.18/0.45 ! [A: $tType,P: A > $o] :
% 0.18/0.45 ( ( ( bot_bot @ ( set @ A ) )
% 0.18/0.45 = ( collect @ A @ P ) )
% 0.18/0.45 = ( ! [X4: A] :
% 0.18/0.45 ~ ( P @ X4 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % empty_Collect_eq
% 0.18/0.45 thf(fact_41_bot__apply,axiom,
% 0.18/0.45 ! [C2: $tType,D: $tType] :
% 0.18/0.45 ( ( bot @ C2 @ ( type2 @ C2 ) )
% 0.18/0.45 => ( ( bot_bot @ ( D > C2 ) )
% 0.18/0.45 = ( ^ [X4: D] : ( bot_bot @ C2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % bot_apply
% 0.18/0.45 thf(fact_42_order__refl,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.18/0.45 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_43_subset__emptyI,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A] :
% 0.18/0.45 ( ! [X5: A] :
% 0.18/0.45 ~ ( member @ A @ X5 @ A3 )
% 0.18/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_emptyI
% 0.18/0.45 thf(fact_44_bot__set__def,axiom,
% 0.18/0.45 ! [A: $tType] :
% 0.18/0.45 ( ( bot_bot @ ( set @ A ) )
% 0.18/0.45 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % bot_set_def
% 0.18/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: $tType,A2: A,P: A > $o] :
% 0.18/0.45 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.18/0.45 = ( P @ A2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A: $tType,A3: set @ A] :
% 0.18/0.45 ( ( collect @ A
% 0.18/0.45 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.18/0.45 = A3 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_47_Collect__cong,axiom,
% 0.18/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.18/0.45 ( ! [X5: A] :
% 0.18/0.45 ( ( P @ X5 )
% 0.29/0.45 = ( Q @ X5 ) )
% 0.29/0.45 => ( ( collect @ A @ P )
% 0.29/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_cong
% 0.29/0.45 thf(fact_48_ext,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.29/0.45 ( ! [X5: A] :
% 0.29/0.45 ( ( F @ X5 )
% 0.29/0.45 = ( G @ X5 ) )
% 0.29/0.45 => ( F = G ) ) ).
% 0.29/0.45
% 0.29/0.45 % ext
% 0.29/0.45 thf(fact_49_dual__order_Oantisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.45 => ( A2 = B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.antisym
% 0.29/0.45 thf(fact_50_dual__order_Otrans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.29/0.45 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.trans
% 0.29/0.45 thf(fact_51_linorder__wlog,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > A > $o,A2: A,B3: A] :
% 0.29/0.45 ( ! [A4: A,B4: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.29/0.45 => ( P @ A4 @ B4 ) )
% 0.29/0.45 => ( ! [A4: A,B4: A] :
% 0.29/0.45 ( ( P @ B4 @ A4 )
% 0.29/0.45 => ( P @ A4 @ B4 ) )
% 0.29/0.45 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_wlog
% 0.29/0.45 thf(fact_52_dual__order_Orefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.refl
% 0.29/0.45 thf(fact_53_order__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.45 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_trans
% 0.29/0.45 thf(fact_54_order__class_Oorder_Oantisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.29/0.45 => ( A2 = B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.antisym
% 0.29/0.45 thf(fact_55_ord__le__eq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( B3 = C )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_trans
% 0.29/0.45 thf(fact_56_ord__eq__le__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A,C: A] :
% 0.29/0.45 ( ( A2 = B3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_trans
% 0.29/0.45 thf(fact_57_antisym__conv,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 = ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv
% 0.29/0.45 thf(fact_58_le__cases3,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.29/0.45 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.29/0.45 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.29/0.45 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases3
% 0.29/0.45 thf(fact_59_order_Otrans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.trans
% 0.29/0.45 thf(fact_60_le__cases,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases
% 0.29/0.45 thf(fact_61_eq__refl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X = Y )
% 0.29/0.45 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_refl
% 0.29/0.45 thf(fact_62_linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linear
% 0.29/0.45 thf(fact_63_antisym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.45 => ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym
% 0.29/0.45 thf(fact_64_eq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 0.29/0.45 = ( ^ [X4: A,Y2: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.45 & ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_65_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B3: A,F: A > B,C: B] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X5: A,Y4: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_66_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.29/0.45 => ( ! [X5: B,Y4: B] :
% 0.29/0.45 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_67_order__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C2: $tType] :
% 0.29/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X5: A,Y4: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_68_order__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.29/0.45 => ( ! [X5: B,Y4: B] :
% 0.29/0.45 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_69_le__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.45 = ( ^ [F3: A > B,G2: A > B] :
% 0.29/0.45 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_fun_def
% 0.29/0.45 thf(fact_70_le__funI,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ! [F: A > B,G: A > B] :
% 0.29/0.45 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.29/0.45 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funI
% 0.29/0.45 thf(fact_71_le__funE,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funE
% 0.29/0.45 thf(fact_72_le__funD,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.45 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_funD
% 0.29/0.45 thf(fact_73_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.29/0.45 => ( A2 != B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_implies_not_eq
% 0.29/0.45 thf(fact_74_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ( A2 != B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_implies_not_eq
% 0.29/0.45 thf(fact_75_not__less__iff__gr__or__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.45 = ( ( ord_less @ A @ Y @ X )
% 0.29/0.45 | ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_less_iff_gr_or_eq
% 0.29/0.45 thf(fact_76_dual__order_Ostrict__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.29/0.45 => ( ( ord_less @ A @ C @ B3 )
% 0.29/0.45 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_trans
% 0.29/0.45 thf(fact_77_less__imp__not__less,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_not_less
% 0.29/0.45 thf(fact_78_order_Ostrict__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( ord_less @ A @ B3 @ C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_trans
% 0.29/0.45 thf(fact_79_dual__order_Oirrefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A] :
% 0.29/0.45 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.irrefl
% 0.29/0.45 thf(fact_80_linorder__cases,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( X != Y )
% 0.29/0.45 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_cases
% 0.29/0.45 thf(fact_81_less__imp__triv,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,P: $o] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less @ A @ Y @ X )
% 0.29/0.45 => P ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_triv
% 0.29/0.45 thf(fact_82_less__imp__not__eq2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( Y != X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_not_eq2
% 0.29/0.45 thf(fact_83_antisym__conv3,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Y: A,X: A] :
% 0.29/0.45 ( ~ ( ord_less @ A @ Y @ X )
% 0.29/0.45 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.45 = ( X = Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv3
% 0.29/0.45 thf(fact_84_less__induct,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [P: A > $o,A2: A] :
% 0.29/0.45 ( ! [X5: A] :
% 0.29/0.45 ( ! [Y5: A] :
% 0.29/0.45 ( ( ord_less @ A @ Y5 @ X5 )
% 0.29/0.45 => ( P @ Y5 ) )
% 0.29/0.45 => ( P @ X5 ) )
% 0.29/0.45 => ( P @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_induct
% 0.29/0.45 thf(fact_85_less__not__sym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_not_sym
% 0.29/0.45 thf(fact_86_less__imp__not__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( X != Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_not_eq
% 0.29/0.45 thf(fact_87_dual__order_Oasym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.29/0.45 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.asym
% 0.29/0.45 thf(fact_88_ord__less__eq__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( B3 = C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_less_eq_trans
% 0.29/0.45 thf(fact_89_ord__eq__less__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A,C: A] :
% 0.29/0.45 ( ( A2 = B3 )
% 0.29/0.45 => ( ( ord_less @ A @ B3 @ C )
% 0.29/0.45 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_less_trans
% 0.29/0.45 thf(fact_90_less__irrefl,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_irrefl
% 0.29/0.45 thf(fact_91_less__linear,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 | ( X = Y )
% 0.29/0.45 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_linear
% 0.29/0.45 thf(fact_92_less__trans,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ( ord_less @ A @ Y @ Z )
% 0.29/0.45 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_trans
% 0.29/0.45 thf(fact_93_less__asym_H,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_asym'
% 0.29/0.45 thf(fact_94_less__asym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_asym
% 0.29/0.45 thf(fact_95_less__imp__neq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( X != Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_imp_neq
% 0.29/0.45 thf(fact_96_dense,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ? [Z3: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Z3 )
% 0.29/0.45 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense
% 0.29/0.45 thf(fact_97_order_Oasym,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.asym
% 0.29/0.45 thf(fact_98_neq__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 = ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % neq_iff
% 0.29/0.45 thf(fact_99_neqE,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A] :
% 0.29/0.45 ( ( X != Y )
% 0.29/0.45 => ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % neqE
% 0.29/0.45 thf(fact_100_gt__ex,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( no_top @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % gt_ex
% 0.29/0.45 thf(fact_101_lt__ex,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A] :
% 0.29/0.45 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % lt_ex
% 0.29/0.45 thf(fact_102_order__less__subst2,axiom,
% 0.29/0.45 ! [A: $tType,C2: $tType] :
% 0.29/0.45 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X5: A,Y4: A] :
% 0.29/0.45 ( ( ord_less @ A @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_subst2
% 0.29/0.45 thf(fact_103_order__less__subst1,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less @ B @ B3 @ C )
% 0.29/0.45 => ( ! [X5: B,Y4: B] :
% 0.29/0.45 ( ( ord_less @ B @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_less_subst1
% 0.29/0.45 thf(fact_104_ord__less__eq__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,B3: A,F: A > B,C: B] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X5: A,Y4: A] :
% 0.29/0.45 ( ( ord_less @ A @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_less_eq_subst
% 0.29/0.45 thf(fact_105_ord__eq__less__subst,axiom,
% 0.29/0.45 ! [A: $tType,B: $tType] :
% 0.29/0.45 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.45 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.45 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.29/0.45 ( ( A2
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less @ B @ B3 @ C )
% 0.29/0.45 => ( ! [X5: B,Y4: B] :
% 0.29/0.45 ( ( ord_less @ B @ X5 @ Y4 )
% 0.29/0.45 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.45 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_less_subst
% 0.29/0.45 thf(fact_106_bot__fun__def,axiom,
% 0.29/0.45 ! [B: $tType,A: $tType] :
% 0.29/0.45 ( ( bot @ B @ ( type2 @ B ) )
% 0.29/0.45 => ( ( bot_bot @ ( A > B ) )
% 0.29/0.45 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % bot_fun_def
% 0.29/0.45 thf(fact_107_ex__in__conv,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A] :
% 0.29/0.45 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.29/0.45 = ( A3
% 0.29/0.45 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ex_in_conv
% 0.29/0.45 thf(fact_108_equals0I,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A] :
% 0.29/0.45 ( ! [Y4: A] :
% 0.29/0.45 ~ ( member @ A @ Y4 @ A3 )
% 0.29/0.45 => ( A3
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equals0I
% 0.29/0.45 thf(fact_109_equals0D,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,A2: A] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.45 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equals0D
% 0.29/0.45 thf(fact_110_emptyE,axiom,
% 0.29/0.45 ! [A: $tType,A2: A] :
% 0.29/0.45 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % emptyE
% 0.29/0.45 thf(fact_111_Collect__mono__iff,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.29/0.45 = ( ! [X4: A] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 => ( Q @ X4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono_iff
% 0.29/0.45 thf(fact_112_contra__subsetD,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( ~ ( member @ A @ C @ B2 )
% 0.29/0.45 => ~ ( member @ A @ C @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % contra_subsetD
% 0.29/0.45 thf(fact_113_set__eq__subset,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ^ [Y3: set @ A,Z2: set @ A] : ( Y3 = Z2 ) )
% 0.29/0.45 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.45 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_eq_subset
% 0.29/0.45 thf(fact_114_subset__trans,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_trans
% 0.29/0.45 thf(fact_115_Collect__mono,axiom,
% 0.29/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.45 ( ! [X5: A] :
% 0.29/0.45 ( ( P @ X5 )
% 0.29/0.45 => ( Q @ X5 ) )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono
% 0.29/0.45 thf(fact_116_subset__refl,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.29/0.45
% 0.29/0.45 % subset_refl
% 0.29/0.45 thf(fact_117_rev__subsetD,axiom,
% 0.29/0.45 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 0.29/0.45 ( ( member @ A @ C @ A3 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % rev_subsetD
% 0.29/0.45 thf(fact_118_subset__iff,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.45 ! [T2: A] :
% 0.29/0.45 ( ( member @ A @ T2 @ A5 )
% 0.29/0.45 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_119_set__rev__mp,axiom,
% 0.29/0.45 ! [A: $tType,X: A,A3: set @ A,B2: set @ A] :
% 0.29/0.45 ( ( member @ A @ X @ A3 )
% 0.29/0.45 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( member @ A @ X @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_rev_mp
% 0.29/0.45 thf(fact_120_equalityD2,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.45 ( ( A3 = B2 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD2
% 0.29/0.45 thf(fact_121_equalityD1,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.45 ( ( A3 = B2 )
% 0.29/0.45 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD1
% 0.29/0.45 thf(fact_122_subset__eq,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.45 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.45 ! [X4: A] :
% 0.29/0.45 ( ( member @ A @ X4 @ A5 )
% 0.29/0.45 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_123_equalityE,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.45 ( ( A3 = B2 )
% 0.29/0.45 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityE
% 0.29/0.45 thf(fact_124_subsetCE,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( ( member @ A @ C @ A3 )
% 0.29/0.45 => ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetCE
% 0.29/0.45 thf(fact_125_subsetD,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( ( member @ A @ C @ A3 )
% 0.29/0.45 => ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_126_in__mono,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( ( member @ A @ X @ A3 )
% 0.29/0.45 => ( member @ A @ X @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % in_mono
% 0.29/0.45 thf(fact_127_set__mp,axiom,
% 0.29/0.45 ! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
% 0.29/0.45 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.45 => ( ( member @ A @ X @ A3 )
% 0.29/0.45 => ( member @ A @ X @ B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_mp
% 0.29/0.45 thf(fact_128_order_Onot__eq__order__implies__strict,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A] :
% 0.29/0.45 ( ( A2 != B3 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.45 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.not_eq_order_implies_strict
% 0.29/0.45 thf(fact_129_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.29/0.45 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_implies_order
% 0.29/0.45 thf(fact_130_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less @ A )
% 0.29/0.45 = ( ^ [B6: A,A6: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B6 @ A6 )
% 0.29/0.45 & ( A6 != B6 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_iff_order
% 0.29/0.45 thf(fact_131_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less_eq @ A )
% 0.29/0.45 = ( ^ [B6: A,A6: A] :
% 0.29/0.45 ( ( ord_less @ A @ B6 @ A6 )
% 0.29/0.45 | ( A6 = B6 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.order_iff_strict
% 0.29/0.45 thf(fact_132_order_Ostrict__implies__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [A2: A,B3: A] :
% 0.29/0.45 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.45 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_implies_order
% 0.29/0.45 thf(fact_133_dense__le__bounded,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [X: A,Y: A,Z: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ Y )
% 0.29/0.45 => ( ! [W: A] :
% 0.29/0.45 ( ( ord_less @ A @ X @ W )
% 0.29/0.45 => ( ( ord_less @ A @ W @ Y )
% 0.29/0.45 => ( ord_less_eq @ A @ W @ Z ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense_le_bounded
% 0.29/0.45 thf(fact_134_dense__ge__bounded,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [Z: A,X: A,Y: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ X )
% 0.29/0.45 => ( ! [W: A] :
% 0.29/0.45 ( ( ord_less @ A @ Z @ W )
% 0.29/0.45 => ( ( ord_less @ A @ W @ X )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.29/0.45 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dense_ge_bounded
% 0.29/0.45 thf(fact_135_dual__order_Ostrict__trans2,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less @ A @ B3 @ A2 )
% 0.29/0.45 => ( ( ord_less_eq @ A @ C @ B3 )
% 0.29/0.45 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_trans2
% 0.29/0.45 thf(fact_136_dual__order_Ostrict__trans1,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ! [B3: A,A2: A,C: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ B3 @ A2 )
% 0.29/0.45 => ( ( ord_less @ A @ C @ B3 )
% 0.29/0.45 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.strict_trans1
% 0.29/0.45 thf(fact_137_order_Ostrict__iff__order,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less @ A )
% 0.29/0.45 = ( ^ [A6: A,B6: A] :
% 0.29/0.45 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.29/0.45 & ( A6 != B6 ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.strict_iff_order
% 0.29/0.45 thf(fact_138_order_Oorder__iff__strict,axiom,
% 0.29/0.45 ! [A: $tType] :
% 0.29/0.45 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.45 => ( ( ord_less_eq @ A )
% 0.29/0.45 = ( ^ [A6: A,B6: A] :
% 0.29/0.46 ( ( ord_less @ A @ A6 @ B6 )
% 0.29/0.46 | ( A6 = B6 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.order_iff_strict
% 0.29/0.46 thf(fact_139_order_Ostrict__trans2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B3 @ C )
% 0.29/0.46 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans2
% 0.29/0.46 thf(fact_140_order_Ostrict__trans1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B3: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less @ A @ B3 @ C )
% 0.29/0.46 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans1
% 0.29/0.46 thf(fact_141_not__le__imp__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le_imp_less
% 0.29/0.46 thf(fact_142_less__le__not__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [X4: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 & ~ ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_not_le
% 0.29/0.46 thf(fact_143_le__imp__less__or__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 | ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_imp_less_or_eq
% 0.29/0.46 thf(fact_144_le__less__linear,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_linear
% 0.29/0.46 thf(fact_145_dense__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,Z: A] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X5 @ Z ) )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_le
% 0.29/0.46 thf(fact_146_dense__ge,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Z: A,Y: A] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z @ X5 )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X5 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_ge
% 0.29/0.46 thf(fact_147_less__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.46 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_trans
% 0.29/0.46 thf(fact_148_le__less__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less @ A @ Y @ Z )
% 0.29/0.46 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_trans
% 0.29/0.46 thf(fact_149_antisym__conv2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv2
% 0.29/0.46 thf(fact_150_antisym__conv1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv1
% 0.29/0.46 thf(fact_151_less__imp__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le
% 0.29/0.46 thf(fact_152_le__neq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.46 => ( ( A2 != B3 )
% 0.29/0.46 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_trans
% 0.29/0.46 thf(fact_153_not__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.29/0.46 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less
% 0.29/0.46 thf(fact_154_not__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.29/0.46 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le
% 0.29/0.46 thf(fact_155_order__less__le__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C2: $tType] :
% 0.29/0.46 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 0.29/0.46 => ( ! [X5: A,Y4: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Y4 )
% 0.29/0.46 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst2
% 0.29/0.46 thf(fact_156_order__less__le__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B3 @ C )
% 0.29/0.46 => ( ! [X5: B,Y4: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst1
% 0.29/0.46 thf(fact_157_order__le__less__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C2: $tType] :
% 0.29/0.46 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B3 )
% 0.29/0.46 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 0.29/0.46 => ( ! [X5: A,Y4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst2
% 0.29/0.46 thf(fact_158_order__le__less__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A2: A,F: B > A,B3: B,C: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 0.29/0.46 => ( ( ord_less @ B @ B3 @ C )
% 0.29/0.46 => ( ! [X5: B,Y4: B] :
% 0.29/0.46 ( ( ord_less @ B @ X5 @ Y4 )
% 0.29/0.46 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.29/0.46 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst1
% 0.29/0.46 thf(fact_159_less__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [X4: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 0.29/0.46 & ( X4 != Y2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le
% 0.29/0.46 thf(fact_160_le__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [X4: A,Y2: A] :
% 0.29/0.46 ( ( ord_less @ A @ X4 @ Y2 )
% 0.29/0.46 | ( X4 = Y2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less
% 0.29/0.46 thf(fact_161_leI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % leI
% 0.29/0.46 thf(fact_162_leD,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % leD
% 0.29/0.46 thf(fact_163_bot_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.29/0.46 => ( A2
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_uniqueI
% 0.29/0.46 thf(fact_164_bot_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_unique
% 0.29/0.46 thf(fact_165_bot_Oextremum,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum
% 0.29/0.46 thf(fact_166_bot_Onot__eq__extremum,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 != ( bot_bot @ A ) )
% 0.29/0.46 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.not_eq_extremum
% 0.29/0.46 thf(fact_167_bot_Oextremum__strict,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot.extremum_strict
% 0.29/0.46 thf(fact_168_complete__interval,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A,B3: A,P: A > $o] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B3 )
% 0.29/0.46 => ( ( P @ A2 )
% 0.29/0.46 => ( ~ ( P @ B3 )
% 0.29/0.46 => ? [C4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ C4 )
% 0.29/0.46 & ( ord_less_eq @ A @ C4 @ B3 )
% 0.29/0.46 & ! [X6: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ A2 @ X6 )
% 0.29/0.46 & ( ord_less @ A @ X6 @ C4 ) )
% 0.29/0.46 => ( P @ X6 ) )
% 0.29/0.46 & ! [D2: A] :
% 0.29/0.46 ( ! [X5: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 0.29/0.46 & ( ord_less @ A @ X5 @ D2 ) )
% 0.29/0.46 => ( P @ X5 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % complete_interval
% 0.29/0.46 thf(fact_169_pinf_I6_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(6)
% 0.29/0.46 thf(fact_170_pinf_I8_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(8)
% 0.29/0.46 thf(fact_171_minf_I6_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(6)
% 0.29/0.46 thf(fact_172_minf_I8_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(8)
% 0.29/0.46 thf(fact_173_Set_Ois__empty__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( is_empty @ A )
% 0.29/0.46 = ( ^ [A5: set @ A] :
% 0.29/0.46 ( A5
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.is_empty_def
% 0.29/0.46 thf(fact_174_Collect__empty__eq__bot,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( ( collect @ A @ P )
% 0.29/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.29/0.46 = ( P
% 0.29/0.46 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_empty_eq_bot
% 0.29/0.46 thf(fact_175_bot__empty__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.29/0.46 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_empty_eq
% 0.29/0.46 thf(fact_176_psubsetI,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( A3 != B2 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubsetI
% 0.29/0.46 thf(fact_177_less__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ( ( ord_less @ ( A > B ) )
% 0.29/0.46 = ( ^ [F3: A > B,G2: A > B] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.29/0.46 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_fun_def
% 0.29/0.46 thf(fact_178_not__psubset__empty,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A] :
% 0.29/0.46 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_psubset_empty
% 0.29/0.46 thf(fact_179_psubsetE,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubsetE
% 0.29/0.46 thf(fact_180_psubset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.46 & ( A5 != B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_eq
% 0.29/0.46 thf(fact_181_psubset__imp__subset,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_imp_subset
% 0.29/0.46 thf(fact_182_psubset__subset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_subset_trans
% 0.29/0.46 thf(fact_183_subset__not__subset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.46 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_not_subset_eq
% 0.29/0.46 thf(fact_184_subset__psubset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_psubset_trans
% 0.29/0.46 thf(fact_185_subset__iff__psubset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A5: set @ A,B5: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
% 0.29/0.46 | ( A5 = B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff_psubset_eq
% 0.29/0.46 thf(fact_186_minf_I11_J,axiom,
% 0.29/0.46 ! [C2: $tType,D: $tType] :
% 0.29/0.46 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.29/0.46 => ! [F4: D] :
% 0.29/0.46 ? [Z3: C2] :
% 0.29/0.46 ! [X6: C2] :
% 0.29/0.46 ( ( ord_less @ C2 @ X6 @ Z3 )
% 0.29/0.46 => ( F4 = F4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(11)
% 0.29/0.46 thf(fact_187_minf_I7_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(7)
% 0.29/0.46 thf(fact_188_minf_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ( ord_less @ A @ X6 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(5)
% 0.29/0.46 thf(fact_189_minf_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ( X6 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(4)
% 0.29/0.46 thf(fact_190_minf_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ( X6 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(3)
% 0.29/0.46 thf(fact_191_minf_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P2 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ( ( ( P @ X6 )
% 0.29/0.46 | ( Q @ X6 ) )
% 0.29/0.46 = ( ( P2 @ X6 )
% 0.29/0.46 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(2)
% 0.29/0.46 thf(fact_192_minf_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P2 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ X5 @ Z4 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ X6 @ Z3 )
% 0.29/0.46 => ( ( ( P @ X6 )
% 0.29/0.46 & ( Q @ X6 ) )
% 0.29/0.46 = ( ( P2 @ X6 )
% 0.29/0.46 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minf(1)
% 0.29/0.46 thf(fact_193_pinf_I11_J,axiom,
% 0.29/0.46 ! [C2: $tType,D: $tType] :
% 0.29/0.46 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.29/0.46 => ! [F4: D] :
% 0.29/0.46 ? [Z3: C2] :
% 0.29/0.46 ! [X6: C2] :
% 0.29/0.46 ( ( ord_less @ C2 @ Z3 @ X6 )
% 0.29/0.46 => ( F4 = F4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(11)
% 0.29/0.46 thf(fact_194_pinf_I7_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ( ord_less @ A @ T @ X6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(7)
% 0.29/0.46 thf(fact_195_pinf_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(5)
% 0.29/0.46 thf(fact_196_pinf_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ( X6 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(4)
% 0.29/0.46 thf(fact_197_pinf_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [T: A] :
% 0.29/0.46 ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ( X6 != T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(3)
% 0.29/0.46 thf(fact_198_pinf_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P2 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ( ( ( P @ X6 )
% 0.29/0.46 | ( Q @ X6 ) )
% 0.29/0.46 = ( ( P2 @ X6 )
% 0.29/0.46 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(2)
% 0.29/0.46 thf(fact_199_pinf_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.29/0.46 ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( P @ X5 )
% 0.29/0.46 = ( P2 @ X5 ) ) )
% 0.29/0.46 => ( ? [Z4: A] :
% 0.29/0.46 ! [X5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z4 @ X5 )
% 0.29/0.46 => ( ( Q @ X5 )
% 0.29/0.46 = ( Q2 @ X5 ) ) )
% 0.29/0.46 => ? [Z3: A] :
% 0.29/0.46 ! [X6: A] :
% 0.29/0.46 ( ( ord_less @ A @ Z3 @ X6 )
% 0.29/0.46 => ( ( ( P @ X6 )
% 0.29/0.46 & ( Q @ X6 ) )
% 0.29/0.46 = ( ( P2 @ X6 )
% 0.29/0.46 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % pinf(1)
% 0.29/0.46 thf(fact_200_ex__gt__or__lt,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ? [B4: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B4 )
% 0.29/0.46 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_gt_or_lt
% 0.29/0.46 thf(fact_201_dependent__wellorder__choice,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: ( A > B ) > A > B > $o] :
% 0.29/0.46 ( ! [R2: B,F5: A > B,G3: A > B,X5: A] :
% 0.29/0.46 ( ! [Y5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Y5 @ X5 )
% 0.29/0.46 => ( ( F5 @ Y5 )
% 0.29/0.46 = ( G3 @ Y5 ) ) )
% 0.29/0.46 => ( ( P @ F5 @ X5 @ R2 )
% 0.29/0.46 = ( P @ G3 @ X5 @ R2 ) ) )
% 0.29/0.46 => ( ! [X5: A,F5: A > B] :
% 0.29/0.46 ( ! [Y5: A] :
% 0.29/0.46 ( ( ord_less @ A @ Y5 @ X5 )
% 0.29/0.46 => ( P @ F5 @ Y5 @ ( F5 @ Y5 ) ) )
% 0.29/0.46 => ? [X12: B] : ( P @ F5 @ X5 @ X12 ) )
% 0.29/0.46 => ? [F5: A > B] :
% 0.29/0.46 ! [X6: A] : ( P @ F5 @ X6 @ ( F5 @ X6 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dependent_wellorder_choice
% 0.29/0.46 thf(fact_202_linordered__field__no__lb,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X6: A] :
% 0.29/0.46 ? [Y4: A] : ( ord_less @ A @ Y4 @ X6 ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_field_no_lb
% 0.29/0.46 thf(fact_203_linordered__field__no__ub,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X6: A] :
% 0.29/0.46 ? [X1: A] : ( ord_less @ A @ X6 @ X1 ) ) ).
% 0.29/0.46
% 0.29/0.46 % linordered_field_no_ub
% 0.29/0.46 thf(fact_204_psubsetD,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( member @ A @ C @ A3 )
% 0.29/0.46 => ( member @ A @ C @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubsetD
% 0.29/0.46 thf(fact_205_psubset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 0.29/0.46 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 0.29/0.46 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 0.29/0.46 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % psubset_trans
% 0.29/0.46 thf(fact_206_chain__subset__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( chain_subset @ A )
% 0.29/0.46 = ( ^ [C5: set @ ( set @ A )] :
% 0.29/0.46 ! [X4: set @ A] :
% 0.29/0.46 ( ( member @ ( set @ A ) @ X4 @ C5 )
% 0.29/0.46 => ! [Y2: set @ A] :
% 0.29/0.46 ( ( member @ ( set @ A ) @ Y2 @ C5 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y2 )
% 0.29/0.46 | ( ord_less_eq @ ( set @ A ) @ Y2 @ X4 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % chain_subset_def
% 0.29/0.46 thf(fact_207_linorder__neqE__linordered__idom,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X != Y )
% 0.29/0.46 => ( ~ ( ord_less @ A @ X @ Y )
% 0.29/0.46 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_linordered_idom
% 0.29/0.46 thf(fact_208_in__inv__imagep,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( inv_imagep @ A @ B )
% 0.29/0.46 = ( ^ [R3: A > A > $o,F3: B > A,X4: B,Y2: B] : ( R3 @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % in_inv_imagep
% 0.29/0.46 thf(fact_209_subdag__size,axiom,
% 0.29/0.46 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 0.29/0.46 ( ( binDag786255756subdag @ T @ S )
% 0.29/0.46 => ( ord_less @ nat @ ( size_size @ binDag_Mirabelle_dag @ S ) @ ( size_size @ binDag_Mirabelle_dag @ T ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subdag_size
% 0.29/0.46 thf(fact_210_dag_Osize_I3_J,axiom,
% 0.29/0.46 ( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
% 0.29/0.46 = ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % dag.size(3)
% 0.29/0.46 thf(fact_211_order_Oordering__axioms,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.ordering_axioms
% 0.29/0.46 thf(fact_212_ordering_Onot__eq__order__implies__strict,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( A2 != B3 )
% 0.29/0.46 => ( ( Less_eq @ A2 @ B3 )
% 0.29/0.46 => ( Less @ A2 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.not_eq_order_implies_strict
% 0.29/0.46 thf(fact_213_ordering_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less @ A2 @ B3 )
% 0.29/0.46 => ( A2 != B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.strict_implies_not_eq
% 0.29/0.46 thf(fact_214_ordering_Ostrict__implies__order,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less @ A2 @ B3 )
% 0.29/0.46 => ( Less_eq @ A2 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.strict_implies_order
% 0.29/0.46 thf(fact_215_ordering_Ostrict__iff__order,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less @ A2 @ B3 )
% 0.29/0.46 = ( ( Less_eq @ A2 @ B3 )
% 0.29/0.46 & ( A2 != B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.strict_iff_order
% 0.29/0.46 thf(fact_216_ordering_Oorder__iff__strict,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less_eq @ A2 @ B3 )
% 0.29/0.46 = ( ( Less @ A2 @ B3 )
% 0.29/0.46 | ( A2 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.order_iff_strict
% 0.29/0.46 thf(fact_217_ordering_Ostrict__trans2,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less @ A2 @ B3 )
% 0.29/0.46 => ( ( Less_eq @ B3 @ C )
% 0.29/0.46 => ( Less @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.strict_trans2
% 0.29/0.46 thf(fact_218_ordering_Ostrict__trans1,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less_eq @ A2 @ B3 )
% 0.29/0.46 => ( ( Less @ B3 @ C )
% 0.29/0.46 => ( Less @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.strict_trans1
% 0.29/0.46 thf(fact_219_ordering_Ostrict__trans,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less @ A2 @ B3 )
% 0.29/0.46 => ( ( Less @ B3 @ C )
% 0.29/0.46 => ( Less @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.strict_trans
% 0.29/0.46 thf(fact_220_ordering_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less_eq @ A2 @ B3 )
% 0.29/0.46 => ( ( Less_eq @ B3 @ A2 )
% 0.29/0.46 => ( A2 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.antisym
% 0.29/0.46 thf(fact_221_ordering_Oirrefl,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ~ ( Less @ A2 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.irrefl
% 0.29/0.46 thf(fact_222_ordering_Otrans,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less_eq @ A2 @ B3 )
% 0.29/0.46 => ( ( Less_eq @ B3 @ C )
% 0.29/0.46 => ( Less_eq @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.trans
% 0.29/0.46 thf(fact_223_ordering_Ointro,axiom,
% 0.29/0.46 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 0.29/0.46 ( ! [A4: A,B4: A] :
% 0.29/0.46 ( ( Less @ A4 @ B4 )
% 0.29/0.46 = ( ( Less_eq @ A4 @ B4 )
% 0.29/0.46 & ( A4 != B4 ) ) )
% 0.29/0.46 => ( ! [A4: A] : ( Less_eq @ A4 @ A4 )
% 0.29/0.46 => ( ! [A4: A,B4: A] :
% 0.29/0.46 ( ( Less_eq @ A4 @ B4 )
% 0.29/0.46 => ( ( Less_eq @ B4 @ A4 )
% 0.29/0.46 => ( A4 = B4 ) ) )
% 0.29/0.46 => ( ! [A4: A,B4: A,C4: A] :
% 0.29/0.46 ( ( Less_eq @ A4 @ B4 )
% 0.29/0.46 => ( ( Less_eq @ B4 @ C4 )
% 0.29/0.46 => ( Less_eq @ A4 @ C4 ) ) )
% 0.29/0.46 => ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.intro
% 0.29/0.46 thf(fact_224_ordering_Orefl,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( Less_eq @ A2 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.refl
% 0.29/0.46 thf(fact_225_ordering_Oasym,axiom,
% 0.29/0.46 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 0.29/0.46 ( ( ordering @ A @ Less_eq @ Less )
% 0.29/0.46 => ( ( Less @ A2 @ B3 )
% 0.29/0.46 => ~ ( Less @ B3 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering.asym
% 0.29/0.46 thf(fact_226_ordering__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordering @ A )
% 0.29/0.46 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.29/0.46 ( ! [A6: A,B6: A] :
% 0.29/0.46 ( ( Less2 @ A6 @ B6 )
% 0.29/0.46 = ( ( Less_eq2 @ A6 @ B6 )
% 0.29/0.46 & ( A6 != B6 ) ) )
% 0.29/0.46 & ! [A6: A] : ( Less_eq2 @ A6 @ A6 )
% 0.29/0.46 & ! [A6: A,B6: A] :
% 0.29/0.46 ( ( Less_eq2 @ A6 @ B6 )
% 0.29/0.46 => ( ( Less_eq2 @ B6 @ A6 )
% 0.29/0.46 => ( A6 = B6 ) ) )
% 0.29/0.46 & ! [A6: A,B6: A,C6: A] :
% 0.29/0.46 ( ( Less_eq2 @ A6 @ B6 )
% 0.29/0.46 => ( ( Less_eq2 @ B6 @ C6 )
% 0.29/0.46 => ( Less_eq2 @ A6 @ C6 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ordering_def
% 0.29/0.46 thf(fact_227_not__gr__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.29/0.46 = ( N
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_gr_zero
% 0.29/0.46 thf(fact_228_le__zero__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.29/0.46 = ( N
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_zero_eq
% 0.29/0.46 thf(fact_229_zero__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le
% 0.29/0.46 thf(fact_230_zero__less__iff__neq__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.29/0.46 = ( N
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_iff_neq_zero
% 0.29/0.46 thf(fact_231_gr__implies__not__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [M: A,N: A] :
% 0.29/0.46 ( ( ord_less @ A @ M @ N )
% 0.29/0.46 => ( N
% 0.29/0.46 != ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_implies_not_zero
% 0.29/0.46 thf(fact_232_not__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_zero
% 0.29/0.46 thf(fact_233_gr__zeroI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [N: A] :
% 0.29/0.46 ( ( N
% 0.29/0.46 != ( zero_zero @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_zeroI
% 0.29/0.46 thf(fact_234_dag_Osize__gen_I1_J,axiom,
% 0.29/0.46 ( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
% 0.29/0.46 = ( zero_zero @ nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % dag.size_gen(1)
% 0.29/0.46 thf(fact_235_less__numeral__extra_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_numeral_extra(3)
% 0.29/0.46 thf(fact_236_le__numeral__extra_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(3)
% 0.29/0.46 thf(fact_237_sgn__greater,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
% 0.29/0.46 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sgn_greater
% 0.29/0.46 thf(fact_238_sgn__less,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sgn_less
% 0.29/0.46 thf(fact_239_sgn__sgn,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
% 0.29/0.46 = ( sgn_sgn @ A @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sgn_sgn
% 0.29/0.46 thf(fact_240_sgn__0__0,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ( sgn_sgn @ A @ A2 )
% 0.29/0.46 = ( zero_zero @ A ) )
% 0.29/0.46 = ( A2
% 0.29/0.46 = ( zero_zero @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sgn_0_0
% 0.29/0.46 thf(fact_241_sgn__pos,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.29/0.46 => ( ( sgn_sgn @ A @ A2 )
% 0.29/0.46 = ( one_one @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % sgn_pos
% 0.29/0.46 thf(fact_242_of__bool__eq_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( zero_neq_one_of_bool @ A @ $false )
% 0.29/0.46 = ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_bool_eq(1)
% 0.29/0.46 thf(fact_243_of__bool__eq_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( zero_neq_one_of_bool @ A @ $true )
% 0.29/0.46 = ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_bool_eq(2)
% 0.29/0.46 thf(fact_244_not__one__less__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.29/0.46 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_one_less_zero
% 0.29/0.46 thf(fact_245_zero__less__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_less_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_less_one
% 0.29/0.46 thf(fact_246_zero__le__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_one
% 0.29/0.46 thf(fact_247_not__one__le__zero,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_one_le_zero
% 0.29/0.46 thf(fact_248_split__of__bool__asm,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > $o,P3: $o] :
% 0.29/0.46 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 0.29/0.46 = ( ~ ( ( P3
% 0.29/0.46 & ~ ( P @ ( one_one @ A ) ) )
% 0.29/0.46 | ( ~ P3
% 0.29/0.46 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % split_of_bool_asm
% 0.29/0.46 thf(fact_249_split__of__bool,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > $o,P3: $o] :
% 0.29/0.46 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 0.29/0.46 = ( ( P3
% 0.29/0.46 => ( P @ ( one_one @ A ) ) )
% 0.29/0.46 & ( ~ P3
% 0.29/0.46 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % split_of_bool
% 0.29/0.46 thf(fact_250_zero__neq__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( zero_zero @ A )
% 0.29/0.46 != ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_neq_one
% 0.29/0.46 thf(fact_251_of__bool__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( zero_neq_one_of_bool @ A )
% 0.29/0.46 = ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_bool_def
% 0.29/0.46 thf(fact_252_of__bool__eq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P3: $o,Q3: $o] :
% 0.29/0.46 ( ( ( zero_neq_one_of_bool @ A @ P3 )
% 0.29/0.46 = ( zero_neq_one_of_bool @ A @ Q3 ) )
% 0.29/0.46 = ( P3 = Q3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % of_bool_eq_iff
% 0.29/0.46 thf(fact_253_le__numeral__extra_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(4)
% 0.29/0.46 thf(fact_254_less__numeral__extra_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_numeral_extra(4)
% 0.29/0.46 thf(fact_255_le__numeral__extra_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(2)
% 0.29/0.46
% 0.29/0.46 %----Type constructors (33)
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.29/0.46 ! [A7: $tType,A8: $tType] :
% 0.29/0.46 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 0.29/0.46 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.29/0.46 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.46 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/0.46 linord1659791738miring @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.29/0.46 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
% 0.29/0.46 zero_less_one @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.29/0.46 wellorder @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 0.29/0.46 order_bot @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 0.29/0.46 zero_neq_one @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 0.29/0.46 preorder @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.29/0.46 linorder @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.29/0.46 no_top @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 0.29/0.46 order @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 0.29/0.46 ord @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 0.29/0.46 bot @ nat @ ( type2 @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 0.29/0.46 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.29/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 0.29/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 0.29/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
% 0.29/0.46 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 0.29/0.46 order_bot @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 0.29/0.46 preorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
% 0.29/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 0.29/0.46 order @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 0.29/0.46 ord @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 0.29/0.46 bot @ $o @ ( type2 @ $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_17,axiom,
% 0.29/0.46 preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_18,axiom,
% 0.29/0.46 order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.29/0.46
% 0.29/0.46 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_19,axiom,
% 0.29/0.46 ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 0.29/0.46
% 0.29/0.46 %----Helper facts (3)
% 0.29/0.46 thf(help_If_3_1_T,axiom,
% 0.29/0.46 ! [P: $o] :
% 0.29/0.46 ( ( P = $true )
% 0.29/0.46 | ( P = $false ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_2_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.29/0.46 = Y ) ).
% 0.29/0.46
% 0.29/0.46 thf(help_If_1_1_T,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A] :
% 0.29/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 %----Conjectures (2)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 ( ~ ( member @ simpl_ref @ a @ ( binDag1380252983set_of @ l ) )
% 0.29/0.46 => ( ~ ( member @ simpl_ref @ a @ ( binDag1380252983set_of @ r ) )
% 0.29/0.46 => thesis ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,conjecture,
% 0.29/0.46 thesis ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.57/1.46 ---- Embedded in THF ---
% 2.57/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.57/1.46 %%% Generated on Tue Feb 24 04:15:43 EST 2026
% 2.57/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.57/1.46 %%% Logic specification used:
% 2.57/1.46 %%% thf(spec, logic, $$dhol).
% 2.57/1.46
% 2.57/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.EA9G7loSYs/DTF2DTF18239.p
% 2.57/1.46 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag, type, binDag_Mirabelle_dag: $tType).
% 2.57/1.46 thf(ty_t_Simpl__Heap_Oref, type, simpl_ref: $tType).
% 2.57/1.46 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.57/1.46 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.57/1.46 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.57/1.46 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG, type, binDag_Mirabelle_DAG: (binDag_Mirabelle_dag > $o)).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode, type, binDag476092410e_Node: (binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > binDag_Mirabelle_dag)))).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag, type, binDag1297733282se_dag: (!> [A:$tType]: ((A > ((binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))) > (binDag_Mirabelle_dag > A)))))).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag, type, binDag1442713106ec_dag: (!> [A:$tType]: ((A > ((binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A))))) > (binDag_Mirabelle_dag > A)))))).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Osize__dag, type, binDag1924123185ze_dag: (binDag_Mirabelle_dag > nat)).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of, type, binDag1380252983set_of: (binDag_Mirabelle_dag > (set @ simpl_ref))).
% 2.57/1.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag, type, binDag786255756subdag: (binDag_Mirabelle_dag > (binDag_Mirabelle_dag > $o))).
% 2.57/1.46 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.57/1.46 thf(sy_c_Groups_Osgn__class_Osgn, type, sgn_sgn: (!> [A:$tType]: ((A > A)))).
% 2.57/1.46 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.57/1.46 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.57/1.46 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.57/1.46 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.57/1.46 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.57/1.46 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.57/1.46 thf(sy_c_Orderings_Oordering, type, ordering: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 2.57/1.46 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.57/1.46 thf(sy_c_Relation_Oinv__imagep, type, inv_imagep: (!> [B:$tType,A:$tType]: (((B > (B > $o)) > ((A > B) > (A > (A > $o))))))).
% 2.57/1.46 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool, type, zero_neq_one_of_bool: (!> [A:$tType]: (($o > A)))).
% 2.57/1.46 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.57/1.46 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.57/1.46 thf(sy_c_Zorn_Ochain__subset, type, chain_subset: (!> [A:$tType]: (((set @ (set @ A)) > $o)))).
% 2.57/1.46 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.57/1.46 thf(sy_v_a____, type, a: simpl_ref).
% 2.57/1.46 thf(sy_v_l____, type, l: binDag_Mirabelle_dag).
% 2.57/1.46 thf(sy_v_r____, type, r: binDag_Mirabelle_dag).
% 2.57/1.46 thf(sy_v_thesis____, type, thesis: $o).
% 2.57/1.46 thf(sy_v_x, type, x: binDag_Mirabelle_dag).
% 2.57/1.46 thf(sy_v_y, type, y: binDag_Mirabelle_dag).
% 2.57/1.46 thf(fact_0_Node, axiom, (y = (((binDag476092410e_Node @ l) @ a) @ r))).
% 2.57/1.46 thf(fact_1_dag_Oinject, axiom, (! [X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag,Y21:binDag_Mirabelle_dag,Y22:simpl_ref,Y23:binDag_Mirabelle_dag]: ((((((binDag476092410e_Node @ X21) @ X22) @ X23) = (((binDag476092410e_Node @ Y21) @ Y22) @ Y23)) = ((X21 = Y21) & ((X22 = Y22) & (X23 = Y23))))))).
% 2.57/1.46 thf(fact_2_DAG_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((binDag_Mirabelle_DAG @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ L)))) & ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ R)))) & ((binDag_Mirabelle_DAG @ L) & (binDag_Mirabelle_DAG @ R)))))))).
% 2.57/1.46 thf(fact_3_x__less__y, axiom, (((ord_less @ binDag_Mirabelle_dag) @ x) @ y)).
% 2.57/1.46 thf(fact_4_set__of__Tip, axiom, ((binDag1380252983set_of @ binDag_Mirabelle_Tip) = (bot_bot @ (set @ simpl_ref)))).
% 2.57/1.46 thf(fact_5_le__dag__set__of, axiom, (! [X:binDag_Mirabelle_dag,Y:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ Y) => (((ord_less_eq @ (set @ simpl_ref)) @ (binDag1380252983set_of @ X)) @ (binDag1380252983set_of @ Y)))))).
% 2.57/1.46 thf(fact_6_less__dag__set__of, axiom, (! [X:binDag_Mirabelle_dag,Y:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X) @ Y) => (((ord_less_eq @ (set @ simpl_ref)) @ (binDag1380252983set_of @ X)) @ (binDag1380252983set_of @ Y)))))).
% 2.57/1.46 thf(fact_7_dag_Osimps_I7_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A))))),X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((((F2 @ X21) @ X22) @ X23) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ X21)) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ X23)))))).
% 2.57/1.46 thf(fact_8_dag_Osimps_I5_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))),X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((((((binDag1297733282se_dag @ A) @ F1) @ F2) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((F2 @ X21) @ X22) @ X23))))).
% 2.57/1.46 thf(fact_9_subdag__neq, axiom, (! [T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ S) => (T != S))))).
% 2.57/1.46 thf(fact_10_subdag__trans, axiom, (! [T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag,R:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ S) => (((binDag786255756subdag @ S) @ R) => ((binDag786255756subdag @ T) @ R)))))).
% 2.57/1.46 thf(fact_11_subdag__not__sym, axiom, (! [S:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ S) @ T) => (~ (((binDag786255756subdag @ T) @ S))))))).
% 2.57/1.46 thf(fact_12_DAG__y, axiom, (binDag_Mirabelle_DAG @ y)).
% 2.57/1.46 thf(fact_13_DAG_Osimps_I1_J, axiom, (binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip)).
% 2.57/1.46 thf(fact_14_dag_Osimps_I6_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A)))))]: ((((((binDag1442713106ec_dag @ A) @ F1) @ F2) @ binDag_Mirabelle_Tip) = F1)))).
% 2.57/1.46 thf(fact_15_dag_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F2:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A)))]: ((((((binDag1297733282se_dag @ A) @ F1) @ F2) @ binDag_Mirabelle_Tip) = F1)))).
% 2.57/1.46 thf(fact_16_dag_Odistinct_I1_J, axiom, (! [X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((binDag_Mirabelle_Tip != (((binDag476092410e_Node @ X21) @ X22) @ X23))))).
% 2.57/1.46 thf(fact_17_subdag_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ (((binDag476092410e_Node @ L) @ A2) @ R)) @ T) = ((T = L) | ((T = R) | (((binDag786255756subdag @ L) @ T) | ((binDag786255756subdag @ R) @ T)))))))).
% 2.57/1.46 thf(fact_18_subdag_Osimps_I1_J, axiom, (! [T:binDag_Mirabelle_dag]: ((~ (((binDag786255756subdag @ binDag_Mirabelle_Tip) @ T)))))).
% 2.57/1.46 thf(fact_19_DAG__less, axiom, (! [Y:binDag_Mirabelle_dag,X:binDag_Mirabelle_dag]: (((binDag_Mirabelle_DAG @ Y) => ((((ord_less @ binDag_Mirabelle_dag) @ X) @ Y) => (binDag_Mirabelle_DAG @ X)))))).
% 2.57/1.46 thf(fact_20_dag_Oinduct, axiom, (! [P:(binDag_Mirabelle_dag > $o),Dag:binDag_Mirabelle_dag]: (((P @ binDag_Mirabelle_Tip) => ((! [X1:binDag_Mirabelle_dag,X2:simpl_ref,X3:binDag_Mirabelle_dag]: (((P @ X1) => ((P @ X3) => (P @ (((binDag476092410e_Node @ X1) @ X2) @ X3)))))) => (P @ Dag)))))).
% 2.57/1.46 thf(fact_21_le__dag__def, axiom, ((ord_less_eq @ binDag_Mirabelle_dag) = (^ [S2:binDag_Mirabelle_dag,T2:binDag_Mirabelle_dag]: (((S2 = T2) | (((ord_less @ binDag_Mirabelle_dag) @ S2) @ T2)))))).
% 2.57/1.46 thf(fact_22_dag_Oexhaust, axiom, (! [Y:binDag_Mirabelle_dag]: (((Y != binDag_Mirabelle_Tip) => (~ ((! [X212:binDag_Mirabelle_dag,X222:simpl_ref,X232:binDag_Mirabelle_dag]: ((Y != (((binDag476092410e_Node @ X212) @ X222) @ X232)))))))))).
% 2.57/1.46 thf(fact_23_dag__less__le, axiom, ((ord_less @ binDag_Mirabelle_dag) = (^ [X4:binDag_Mirabelle_dag,Y2:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X4) @ Y2) & (X4 != Y2)))))).
% 2.57/1.46 thf(fact_24_le__dag__refl, axiom, (! [X:binDag_Mirabelle_dag]: ((((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ X)))).
% 2.57/1.46 thf(fact_25_le__dag__trans, axiom, (! [X:binDag_Mirabelle_dag,Y:binDag_Mirabelle_dag,Z:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ Y) => ((((ord_less_eq @ binDag_Mirabelle_dag) @ Y) @ Z) => (((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ Z)))))).
% 2.57/1.46 thf(fact_26_less__dag__Tip, axiom, (! [X:binDag_Mirabelle_dag]: ((~ ((((ord_less @ binDag_Mirabelle_dag) @ X) @ binDag_Mirabelle_Tip)))))).
% 2.57/1.46 thf(fact_27_less__dag__def, axiom, ((ord_less @ binDag_Mirabelle_dag) = (^ [S2:binDag_Mirabelle_dag,T2:binDag_Mirabelle_dag]: (((binDag786255756subdag @ T2) @ S2))))).
% 2.57/1.46 thf(fact_28_subdag__NodeD, axiom, (! [T:binDag_Mirabelle_dag,Lt:binDag_Mirabelle_dag,A2:simpl_ref,Rt:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ (((binDag476092410e_Node @ Lt) @ A2) @ Rt)) => (((binDag786255756subdag @ T) @ Lt) & ((binDag786255756subdag @ T) @ Rt)))))).
% 2.57/1.46 thf(fact_29_less__Node__dag, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag,X:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ (((binDag476092410e_Node @ L) @ A2) @ R)) @ X) => ((((ord_less @ binDag_Mirabelle_dag) @ L) @ X) & (((ord_less @ binDag_Mirabelle_dag) @ R) @ X)))))).
% 2.57/1.46 thf(fact_30_less__dag__Node, axiom, (! [X:binDag_Mirabelle_dag,L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X) @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ L) | (((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ R)))))).
% 2.57/1.46 thf(fact_31_le__dag__antisym, axiom, (! [X:binDag_Mirabelle_dag,Y:binDag_Mirabelle_dag]: (((((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ Y) => ((((ord_less_eq @ binDag_Mirabelle_dag) @ Y) @ X) => (X = Y)))))).
% 2.57/1.46 thf(fact_32_less__dag__Node_H, axiom, (! [X:binDag_Mirabelle_dag,L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((((ord_less @ binDag_Mirabelle_dag) @ X) @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((X = L) | ((X = R) | ((((ord_less @ binDag_Mirabelle_dag) @ X) @ L) | (((ord_less @ binDag_Mirabelle_dag) @ X) @ R)))))))).
% 2.57/1.46 thf(fact_33_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.57/1.46 thf(fact_34_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 2.57/1.46 thf(fact_35_subsetI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((! [X5:A]: (((((member @ A) @ X5) @ A3) => (((member @ A) @ X5) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.57/1.46 thf(fact_36_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A3) => (A3 = B2)))))).
% 2.57/1.46 thf(fact_37_empty__iff, axiom, (! [A:$tType,C:A]: ((~ ((((member @ A) @ C) @ (bot_bot @ (set @ A)))))))).
% 2.57/1.46 thf(fact_38_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.57/1.46 thf(fact_39_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 2.57/1.46 thf(fact_40_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 2.57/1.46 thf(fact_41_bot__apply, axiom, (! [C2:$tType,D:$tType]: ((((bot @ C2) @ (type2 @ C2)) => ((bot_bot @ (D > C2)) = (^ [X4:D]: ((bot_bot @ C2)))))))).
% 2.57/1.46 thf(fact_42_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.57/1.46 thf(fact_43_subset__emptyI, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X5:A]: ((~ ((((member @ A) @ X5) @ A3))))) => (((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.57/1.46 thf(fact_44_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.57/1.46 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.57/1.46 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 2.57/1.46 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.57/1.46 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X5:A]: (((F @ X5) = (G @ X5)))) => (F = G))))).
% 2.57/1.46 thf(fact_49_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ A2) @ B3) => (A2 = B3))))))))).
% 2.57/1.46 thf(fact_50_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.57/1.46 thf(fact_51_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B3:A]: (((! [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) => ((P @ A4) @ B4)))) => ((! [A4:A,B4:A]: ((((P @ B4) @ A4) => ((P @ A4) @ B4)))) => ((P @ A2) @ B3))))))))).
% 2.57/1.46 thf(fact_52_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.57/1.46 thf(fact_53_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.57/1.46 thf(fact_54_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ A2) => (A2 = B3))))))))).
% 2.57/1.46 thf(fact_55_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((B3 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_56_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((A2 = B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_57_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.57/1.46 thf(fact_58_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.57/1.46 thf(fact_59_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_60_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.57/1.46 thf(fact_61_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.57/1.46 thf(fact_62_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.57/1.46 thf(fact_63_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.57/1.46 thf(fact_64_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y3:A,Z2:A]: ((Y3 = Z2))) = (^ [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X4))))))))).
% 2.57/1.46 thf(fact_65_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B3) => (((F @ B3) = C) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ B) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.57/1.46 thf(fact_66_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((A2 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.57/1.46 thf(fact_67_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less_eq @ C2) @ (F @ B3)) @ C) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ C2) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.57/1.46 thf(fact_68_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.57/1.46 thf(fact_69_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G2 @ X4))))))))))).
% 2.57/1.46 thf(fact_70_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X5:A]: ((((ord_less_eq @ B) @ (F @ X5)) @ (G @ X5)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.57/1.46 thf(fact_71_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.57/1.46 thf(fact_72_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.57/1.46 thf(fact_73_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (A2 != B3)))))))).
% 2.57/1.46 thf(fact_74_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (A2 != B3)))))))).
% 2.57/1.46 thf(fact_75_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 2.57/1.46 thf(fact_76_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.57/1.46 thf(fact_77_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.57/1.46 thf(fact_78_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_79_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.57/1.46 thf(fact_80_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.57/1.46 thf(fact_81_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 2.57/1.46 thf(fact_82_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 2.57/1.46 thf(fact_83_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.57/1.46 thf(fact_84_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => (P @ Y5)))) => (P @ X5)))) => (P @ A2)))))))).
% 2.57/1.46 thf(fact_85_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.57/1.46 thf(fact_86_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.57/1.46 thf(fact_87_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (~ ((((ord_less @ A) @ A2) @ B3)))))))))).
% 2.57/1.46 thf(fact_88_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((B3 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_89_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((A2 = B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_90_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.57/1.46 thf(fact_91_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.57/1.46 thf(fact_92_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.57/1.46 thf(fact_93_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.57/1.46 thf(fact_94_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.57/1.46 thf(fact_95_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.57/1.46 thf(fact_96_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))).
% 2.57/1.46 thf(fact_97_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (~ ((((ord_less @ A) @ B3) @ A2)))))))))).
% 2.57/1.46 thf(fact_98_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.57/1.46 thf(fact_99_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.57/1.46 thf(fact_100_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.57/1.46 thf(fact_101_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X))))))))).
% 2.57/1.46 thf(fact_102_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less @ C2) @ (F @ B3)) @ C) => ((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => (((ord_less @ C2) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.57/1.46 thf(fact_103_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X5:B,Y4:B]: (((((ord_less @ B) @ X5) @ Y4) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.57/1.46 thf(fact_104_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B3) => (((F @ B3) = C) => ((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => (((ord_less @ B) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 2.57/1.46 thf(fact_105_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((A2 = (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X5:B,Y4:B]: (((((ord_less @ B) @ X5) @ Y4) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.57/1.46 thf(fact_106_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X4:A]: ((bot_bot @ B)))))))).
% 2.57/1.46 thf(fact_107_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.57/1.46 thf(fact_108_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y4:A]: ((~ ((((member @ A) @ Y4) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.57/1.46 thf(fact_109_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.57/1.46 thf(fact_110_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.57/1.46 thf(fact_111_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 2.57/1.46 thf(fact_112_contra__subsetD, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((~ ((((member @ A) @ C) @ B2))) => (~ ((((member @ A) @ C) @ A3)))))))).
% 2.57/1.46 thf(fact_113_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y3:(set @ A),Z2:(set @ A)]: ((Y3 = Z2))) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))).
% 2.57/1.46 thf(fact_114_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less_eq @ (set @ A)) @ A3) @ C3)))))).
% 2.57/1.46 thf(fact_115_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) => (Q @ X5)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.57/1.46 thf(fact_116_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.57/1.46 thf(fact_117_rev__subsetD, axiom, (! [A:$tType,C:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ C) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((member @ A) @ C) @ B2)))))).
% 2.57/1.46 thf(fact_118_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A5) => (((member @ A) @ T2) @ B5)))))))))).
% 2.57/1.46 thf(fact_119_set__rev__mp, axiom, (! [A:$tType,X:A,A3:(set @ A),B2:(set @ A)]: (((((member @ A) @ X) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((member @ A) @ X) @ B2)))))).
% 2.57/1.46 thf(fact_120_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A3))))).
% 2.57/1.46 thf(fact_121_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.57/1.46 thf(fact_122_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A5) => (((member @ A) @ X4) @ B5)))))))))).
% 2.57/1.46 thf(fact_123_equalityE, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((A3 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A3)))))))))).
% 2.57/1.46 thf(fact_124_subsetCE, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B2)))))).
% 2.57/1.46 thf(fact_125_subsetD, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B2)))))).
% 2.57/1.46 thf(fact_126_in__mono, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B2)))))).
% 2.57/1.46 thf(fact_127_set__mp, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ X) @ A3) => (((member @ A) @ X) @ B2)))))).
% 2.57/1.46 thf(fact_128_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((A2 != B3) => ((((ord_less_eq @ A) @ A2) @ B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.57/1.46 thf(fact_129_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A]: (((((ord_less @ A) @ B3) @ A2) => (((ord_less_eq @ A) @ B3) @ A2)))))))).
% 2.57/1.46 thf(fact_130_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B6:A,A6:A]: (((((ord_less_eq @ A) @ B6) @ A6) & (A6 != B6))))))))).
% 2.57/1.46 thf(fact_131_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B6:A,A6:A]: (((((ord_less @ A) @ B6) @ A6) | (A6 = B6))))))))).
% 2.57/1.46 thf(fact_132_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less @ A) @ A2) @ B3) => (((ord_less_eq @ A) @ A2) @ B3)))))))).
% 2.57/1.46 thf(fact_133_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.57/1.46 thf(fact_134_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,X:A,Y:A]: (((((ord_less @ A) @ Z) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.57/1.46 thf(fact_135_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((ord_less @ A) @ B3) @ A2) => ((((ord_less_eq @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.57/1.46 thf(fact_136_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B3:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B3) @ A2) => ((((ord_less @ A) @ C) @ B3) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.57/1.46 thf(fact_137_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) & (A6 != B6))))))))).
% 2.57/1.46 thf(fact_138_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A6:A,B6:A]: (((((ord_less @ A) @ A6) @ B6) | (A6 = B6))))))))).
% 2.57/1.46 thf(fact_139_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less_eq @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_140_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less @ A) @ B3) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.57/1.46 thf(fact_141_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 2.57/1.46 thf(fact_142_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) & (~ ((((ord_less_eq @ A) @ Y2) @ X4))))))))))).
% 2.57/1.46 thf(fact_143_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 2.57/1.46 thf(fact_144_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 2.57/1.46 thf(fact_145_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y:A,Z:A]: (((! [X5:A]: (((((ord_less @ A) @ X5) @ Y) => (((ord_less_eq @ A) @ X5) @ Z)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.57/1.46 thf(fact_146_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,Y:A]: (((! [X5:A]: (((((ord_less @ A) @ Z) @ X5) => (((ord_less_eq @ A) @ Y) @ X5)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.57/1.46 thf(fact_147_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.57/1.46 thf(fact_148_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.57/1.46 thf(fact_149_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.57/1.46 thf(fact_150_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.57/1.46 thf(fact_151_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.57/1.46 thf(fact_152_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B3:A]: (((((ord_less_eq @ A) @ A2) @ B3) => ((A2 != B3) => (((ord_less @ A) @ A2) @ B3))))))))).
% 2.57/1.46 thf(fact_153_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.57/1.46 thf(fact_154_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 2.57/1.46 thf(fact_155_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B3) => ((((ord_less_eq @ C2) @ (F @ B3)) @ C) => ((! [X5:A,Y4:A]: (((((ord_less @ A) @ X5) @ Y4) => (((ord_less @ C2) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.57/1.46 thf(fact_156_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C) => ((! [X5:B,Y4:B]: (((((ord_less_eq @ B) @ X5) @ Y4) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.57/1.46 thf(fact_157_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B3:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B3) => ((((ord_less @ C2) @ (F @ B3)) @ C) => ((! [X5:A,Y4:A]: (((((ord_less_eq @ A) @ X5) @ Y4) => (((ord_less_eq @ C2) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.57/1.46 thf(fact_158_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B3:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B3)) => ((((ord_less @ B) @ B3) @ C) => ((! [X5:B,Y4:B]: (((((ord_less @ B) @ X5) @ Y4) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.57/1.46 thf(fact_159_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X4:A,Y2:A]: (((((ord_less_eq @ A) @ X4) @ Y2) & (X4 != Y2))))))))).
% 2.57/1.46 thf(fact_160_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X4:A,Y2:A]: (((((ord_less @ A) @ X4) @ Y2) | (X4 = Y2))))))))).
% 2.57/1.46 thf(fact_161_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.57/1.46 thf(fact_162_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 2.57/1.46 thf(fact_163_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 2.57/1.46 thf(fact_164_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 2.57/1.46 thf(fact_165_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 2.57/1.46 thf(fact_166_bot_Onot__eq__extremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))).
% 2.57/1.46 thf(fact_167_bot_Oextremum__strict, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))).
% 2.57/1.46 thf(fact_168_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B3:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B3) => ((P @ A2) => ((~ ((P @ B3))) => (? [C4:A]: (((((ord_less_eq @ A) @ A2) @ C4) & ((((ord_less_eq @ A) @ C4) @ B3) & ((! [X6:A]: ((((((ord_less_eq @ A) @ A2) @ X6) & (((ord_less @ A) @ X6) @ C4)) => (P @ X6)))) & (! [D2:A]: (((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less @ A) @ X5) @ D2)) => (P @ X5)))) => (((ord_less_eq @ A) @ D2) @ C4)))))))))))))))))).
% 2.57/1.46 thf(fact_169_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (~ ((((ord_less_eq @ A) @ X6) @ T)))))))))))))).
% 2.57/1.46 thf(fact_170_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((ord_less_eq @ A) @ T) @ X6)))))))))))).
% 2.57/1.46 thf(fact_171_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((ord_less_eq @ A) @ X6) @ T)))))))))))).
% 2.57/1.46 thf(fact_172_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X6)))))))))))))).
% 2.57/1.46 thf(fact_173_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A5:(set @ A)]: ((A5 = (bot_bot @ (set @ A))))))))).
% 2.57/1.46 thf(fact_174_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.57/1.46 thf(fact_175_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (bot_bot @ (set @ A))))))))).
% 2.57/1.46 thf(fact_176_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((A3 != B2) => (((ord_less @ (set @ A)) @ A3) @ B2)))))).
% 2.57/1.46 thf(fact_177_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F3))))))))))).
% 2.57/1.46 thf(fact_178_not__psubset__empty, axiom, (! [A:$tType,A3:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A3) @ (bot_bot @ (set @ A)))))))).
% 2.57/1.46 thf(fact_179_psubsetE, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A3)))))))).
% 2.57/1.46 thf(fact_180_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (A5 != B5)))))))).
% 2.57/1.46 thf(fact_181_psubset__imp__subset, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => (((ord_less_eq @ (set @ A)) @ A3) @ B2))))).
% 2.57/1.46 thf(fact_182_psubset__subset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 2.57/1.46 thf(fact_183_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A5) @ B5) & (~ ((((ord_less_eq @ (set @ A)) @ B5) @ A5)))))))))).
% 2.57/1.46 thf(fact_184_subset__psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 2.57/1.46 thf(fact_185_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A5:(set @ A),B5:(set @ A)]: (((((ord_less @ (set @ A)) @ A5) @ B5) | (A5 = B5)))))))).
% 2.57/1.46 thf(fact_186_minf_I11_J, axiom, (! [C2:$tType,D:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F4:D]: ((? [Z3:C2]: ((! [X6:C2]: (((((ord_less @ C2) @ X6) @ Z3) => (F4 = F4)))))))))))).
% 2.57/1.46 thf(fact_187_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (~ ((((ord_less @ A) @ T) @ X6)))))))))))))).
% 2.57/1.46 thf(fact_188_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((ord_less @ A) @ X6) @ T)))))))))))).
% 2.57/1.46 thf(fact_189_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (X6 != T)))))))))))).
% 2.57/1.46 thf(fact_190_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (X6 != T)))))))))))).
% 2.57/1.46 thf(fact_191_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((P @ X6) | (Q @ X6)) = ((P2 @ X6) | (Q2 @ X6)))))))))))))))).
% 2.57/1.46 thf(fact_192_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ X6) @ Z3) => (((P @ X6) & (Q @ X6)) = ((P2 @ X6) & (Q2 @ X6)))))))))))))))).
% 2.57/1.46 thf(fact_193_pinf_I11_J, axiom, (! [C2:$tType,D:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F4:D]: ((? [Z3:C2]: ((! [X6:C2]: (((((ord_less @ C2) @ Z3) @ X6) => (F4 = F4)))))))))))).
% 2.57/1.46 thf(fact_194_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((ord_less @ A) @ T) @ X6)))))))))))).
% 2.57/1.46 thf(fact_195_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (~ ((((ord_less @ A) @ X6) @ T)))))))))))))).
% 2.57/1.46 thf(fact_196_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (X6 != T)))))))))))).
% 2.57/1.46 thf(fact_197_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (X6 != T)))))))))))).
% 2.57/1.46 thf(fact_198_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((P @ X6) | (Q @ X6)) = ((P2 @ X6) | (Q2 @ X6)))))))))))))))).
% 2.57/1.46 thf(fact_199_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z3:A]: ((! [X6:A]: (((((ord_less @ A) @ Z3) @ X6) => (((P @ X6) & (Q @ X6)) = ((P2 @ X6) & (Q2 @ X6)))))))))))))))).
% 2.57/1.46 thf(fact_200_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B4:A]: (((((ord_less @ A) @ A2) @ B4) | (((ord_less @ A) @ B4) @ A2)))))))))).
% 2.57/1.46 thf(fact_201_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:((A > B) > (A > (B > $o)))]: (((! [R2:B,F5:(A > B),G3:(A > B),X5:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => ((F5 @ Y5) = (G3 @ Y5))))) => ((((P @ F5) @ X5) @ R2) = (((P @ G3) @ X5) @ R2))))) => ((! [X5:A,F5:(A > B)]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => (((P @ F5) @ Y5) @ (F5 @ Y5))))) => (? [X12:B]: ((((P @ F5) @ X5) @ X12)))))) => (? [F5:(A > B)]: ((! [X6:A]: ((((P @ F5) @ X6) @ (F5 @ X6)))))))))))))).
% 2.57/1.46 thf(fact_202_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X6:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X6))))))))).
% 2.57/1.46 thf(fact_203_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X6:A]: ((? [X1:A]: ((((ord_less @ A) @ X6) @ X1))))))))).
% 2.57/1.46 thf(fact_204_psubsetD, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A3) @ B2) => ((((member @ A) @ C) @ A3) => (((member @ A) @ C) @ B2)))))).
% 2.57/1.46 thf(fact_205_psubset__trans, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less @ (set @ A)) @ A3) @ B2) => ((((ord_less @ (set @ A)) @ B2) @ C3) => (((ord_less @ (set @ A)) @ A3) @ C3)))))).
% 2.57/1.46 thf(fact_206_chain__subset__def, axiom, (! [A:$tType]: (((chain_subset @ A) = (^ [C5:(set @ (set @ A))]: ((! [X4:(set @ A)]: (((((member @ (set @ A)) @ X4) @ C5) => (! [Y2:(set @ A)]: (((((member @ (set @ A)) @ Y2) @ C5) => ((((ord_less_eq @ (set @ A)) @ X4) @ Y2) | (((ord_less_eq @ (set @ A)) @ Y2) @ X4)))))))))))))).
% 2.57/1.46 thf(fact_207_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.57/1.46 thf(fact_208_in__inv__imagep, axiom, (! [B:$tType,A:$tType]: ((((inv_imagep @ A) @ B) = (^ [R3:(A > (A > $o)),F3:(B > A),X4:B,Y2:B]: (((R3 @ (F3 @ X4)) @ (F3 @ Y2)))))))).
% 2.57/1.46 thf(fact_209_subdag__size, axiom, (! [T:binDag_Mirabelle_dag,S:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T) @ S) => (((ord_less @ nat) @ ((size_size @ binDag_Mirabelle_dag) @ S)) @ ((size_size @ binDag_Mirabelle_dag) @ T)))))).
% 2.57/1.46 thf(fact_210_dag_Osize_I3_J, axiom, (((size_size @ binDag_Mirabelle_dag) @ binDag_Mirabelle_Tip) = (zero_zero @ nat))).
% 2.57/1.46 thf(fact_211_order_Oordering__axioms, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (((ordering @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 2.57/1.46 thf(fact_212_ordering_Onot__eq__order__implies__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A]: (((((ordering @ A) @ Less_eq) @ Less) => ((A2 != B3) => (((Less_eq @ A2) @ B3) => ((Less @ A2) @ B3))))))).
% 2.57/1.46 thf(fact_213_ordering_Ostrict__implies__not__eq, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B3) => (A2 != B3)))))).
% 2.57/1.46 thf(fact_214_ordering_Ostrict__implies__order, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B3) => ((Less_eq @ A2) @ B3)))))).
% 2.57/1.46 thf(fact_215_ordering_Ostrict__iff__order, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B3) = (((Less_eq @ A2) @ B3) & (A2 != B3))))))).
% 2.57/1.46 thf(fact_216_ordering_Oorder__iff__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B3) = (((Less @ A2) @ B3) | (A2 = B3))))))).
% 2.57/1.46 thf(fact_217_ordering_Ostrict__trans2, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B3) => (((Less_eq @ B3) @ C) => ((Less @ A2) @ C))))))).
% 2.57/1.46 thf(fact_218_ordering_Ostrict__trans1, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B3) => (((Less @ B3) @ C) => ((Less @ A2) @ C))))))).
% 2.57/1.46 thf(fact_219_ordering_Ostrict__trans, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B3) => (((Less @ B3) @ C) => ((Less @ A2) @ C))))))).
% 2.57/1.46 thf(fact_220_ordering_Oantisym, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B3) => (((Less_eq @ B3) @ A2) => (A2 = B3))))))).
% 2.57/1.46 thf(fact_221_ordering_Oirrefl, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (~ (((Less @ A2) @ A2))))))).
% 2.57/1.46 thf(fact_222_ordering_Otrans, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B3) => (((Less_eq @ B3) @ C) => ((Less_eq @ A2) @ C))))))).
% 2.57/1.46 thf(fact_223_ordering_Ointro, axiom, (! [A:$tType,Less:(A > (A > $o)),Less_eq:(A > (A > $o))]: (((! [A4:A,B4:A]: ((((Less @ A4) @ B4) = (((Less_eq @ A4) @ B4) & (A4 != B4))))) => ((! [A4:A]: (((Less_eq @ A4) @ A4))) => ((! [A4:A,B4:A]: ((((Less_eq @ A4) @ B4) => (((Less_eq @ B4) @ A4) => (A4 = B4))))) => ((! [A4:A,B4:A,C4:A]: ((((Less_eq @ A4) @ B4) => (((Less_eq @ B4) @ C4) => ((Less_eq @ A4) @ C4))))) => (((ordering @ A) @ Less_eq) @ Less)))))))).
% 2.57/1.46 thf(fact_224_ordering_Orefl, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A]: (((((ordering @ A) @ Less_eq) @ Less) => ((Less_eq @ A2) @ A2))))).
% 2.57/1.46 thf(fact_225_ordering_Oasym, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B3:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B3) => (~ (((Less @ B3) @ A2)))))))).
% 2.57/1.46 thf(fact_226_ordering__def, axiom, (! [A:$tType]: (((ordering @ A) = (^ [Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: (((! [A6:A,B6:A]: ((((Less2 @ A6) @ B6) = (((Less_eq2 @ A6) @ B6) & (A6 != B6))))) & ((! [A6:A]: (((Less_eq2 @ A6) @ A6))) & ((! [A6:A,B6:A]: ((((Less_eq2 @ A6) @ B6) => (((Less_eq2 @ B6) @ A6) => (A6 = B6))))) & (! [A6:A,B6:A,C6:A]: ((((Less_eq2 @ A6) @ B6) => (((Less_eq2 @ B6) @ C6) => ((Less_eq2 @ A6) @ C6)))))))))))))).
% 2.57/1.46 thf(fact_227_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.57/1.46 thf(fact_228_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.57/1.46 thf(fact_229_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.57/1.46 thf(fact_230_zero__less__iff__neq__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.57/1.46 thf(fact_231_gr__implies__not__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.57/1.46 thf(fact_232_not__less__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.57/1.46 thf(fact_233_gr__zeroI, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.57/1.46 thf(fact_234_dag_Osize__gen_I1_J, axiom, ((binDag1924123185ze_dag @ binDag_Mirabelle_Tip) = (zero_zero @ nat))).
% 2.57/1.46 thf(fact_235_less__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.57/1.46 thf(fact_236_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.57/1.46 thf(fact_237_sgn__greater, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((sgn_sgn @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.57/1.46 thf(fact_238_sgn__less, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((sgn_sgn @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.57/1.46 thf(fact_239_sgn__sgn, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: ((((sgn_sgn @ A) @ ((sgn_sgn @ A) @ A2)) = ((sgn_sgn @ A) @ A2)))))))).
% 2.57/1.46 thf(fact_240_sgn__0__0, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.57/1.46 thf(fact_241_sgn__pos, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((sgn_sgn @ A) @ A2) = (one_one @ A))))))))).
% 2.57/1.46 thf(fact_242_of__bool__eq_I1_J, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (((zero_neq_one_of_bool @ A) @ $false) = (zero_zero @ A)))))).
% 2.57/1.46 thf(fact_243_of__bool__eq_I2_J, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (((zero_neq_one_of_bool @ A) @ $true) = (one_one @ A)))))).
% 2.57/1.46 thf(fact_244_not__one__less__zero, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.57/1.46 thf(fact_245_zero__less__one, axiom, (! [A:$tType]: ((((zero_less_one @ A) @ (type2 @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.57/1.46 thf(fact_246_zero__le__one, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.57/1.46 thf(fact_247_not__one__le__zero, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.57/1.46 thf(fact_248_split__of__bool__asm, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (! [P:(A > $o),P3:$o]: (((P @ ((zero_neq_one_of_bool @ A) @ P3)) = (~ (((P3 & (~ ((P @ (one_one @ A))))) | ((~ (P3)) & (~ ((P @ (zero_zero @ A))))))))))))))).
% 2.57/1.46 thf(fact_249_split__of__bool, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (! [P:(A > $o),P3:$o]: (((P @ ((zero_neq_one_of_bool @ A) @ P3)) = ((P3 => (P @ (one_one @ A))) & ((~ (P3)) => (P @ (zero_zero @ A))))))))))).
% 2.57/1.46 thf(fact_250_zero__neq__one, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => ((zero_zero @ A) != (one_one @ A)))))).
% 2.57/1.46 thf(fact_251_of__bool__def, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => ((zero_neq_one_of_bool @ A) = (^ [P4:$o]: (((((if @ A) @ P4) @ (one_one @ A)) @ (zero_zero @ A))))))))).
% 2.57/1.46 thf(fact_252_of__bool__eq__iff, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (! [P3:$o,Q3:$o]: (((((zero_neq_one_of_bool @ A) @ P3) = ((zero_neq_one_of_bool @ A) @ Q3)) = (P3 = Q3)))))))).
% 2.57/1.46 thf(fact_253_le__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.57/1.46 thf(fact_254_less__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 2.57/1.46 thf(fact_255_le__numeral__extra_I2_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.57/1.46 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: ((((order_bot @ A8) @ (type2 @ A8)) => ((order_bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.57/1.46 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.57/1.46 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.57/1.46 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.57/1.46 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: ((((bot @ A8) @ (type2 @ A8)) => ((bot @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.57/1.46 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, ((linord1659791738miring @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Rings_Ozero__less__one, axiom, ((zero_less_one @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1, axiom, ((order_bot @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Rings_Ozero__neq__one, axiom, ((zero_neq_one @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Opreorder_2, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Oorder_3, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Oord_4, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Nat_Onat___Orderings_Obot_5, axiom, ((bot @ nat) @ (type2 @ nat))).
% 2.57/1.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_6, axiom, (! [A7:$tType]: (((order_bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.57/1.46 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.57/1.46 thf(tcon_Set_Oset___Orderings_Oorder_8, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.57/1.46 thf(tcon_Set_Oset___Orderings_Oord_9, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.57/1.46 thf(tcon_Set_Oset___Orderings_Obot_10, axiom, (! [A7:$tType]: (((bot @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.57/1.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 2.57/1.46 thf(tcon_HOL_Obool___Orderings_Opreorder_12, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.57/1.46 thf(tcon_HOL_Obool___Orderings_Olinorder_13, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.57/1.46 thf(tcon_HOL_Obool___Orderings_Oorder_14, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.57/1.46 thf(tcon_HOL_Obool___Orderings_Oord_15, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.57/1.46 thf(tcon_HOL_Obool___Orderings_Obot_16, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.57/1.46 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_17, axiom, ((preorder @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.57/1.46 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_18, axiom, ((order @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.57/1.46 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_19, axiom, ((ord @ binDag_Mirabelle_dag) @ (type2 @ binDag_Mirabelle_dag))).
% 2.57/1.46 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.57/1.46 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.57/1.46 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.57/1.46 thf(conj_0, hypothesis, ((~ ((((member @ simpl_ref) @ a) @ (binDag1380252983set_of @ l)))) => ((~ ((((member @ simpl_ref) @ a) @ (binDag1380252983set_of @ r)))) => thesis))).
% 2.57/1.46 thf(conj_1, conjecture, thesis).
% 2.57/1.46 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.EA9G7loSYs/DTF2DTF18239.p
% 2.57/1.46 ------------------------
% 2.81/1.50 ---- Cleaned THF ---
% 2.81/1.50 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 2.81/1.50 binDag_Mirabelle_dag: $tType ).
% 2.81/1.50
% 2.81/1.50 thf(ty_t_Simpl__Heap_Oref,type,
% 2.81/1.50 simpl_ref: $tType ).
% 2.81/1.50
% 2.81/1.50 thf(ty_t_Set_Oset,type,
% 2.81/1.50 set: $tType > $tType ).
% 2.81/1.50
% 2.81/1.50 thf(ty_t_Nat_Onat,type,
% 2.81/1.50 nat: $tType ).
% 2.81/1.50
% 2.81/1.50 thf(ty_t_itself,type,
% 2.81/1.50 itself: $tType > $tType ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_HOL_Otype,type,
% 2.81/1.50 type:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Obot,type,
% 2.81/1.50 bot:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Oord,type,
% 2.81/1.50 ord:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Oorder,type,
% 2.81/1.50 order:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Ono__bot,type,
% 2.81/1.50 no_bot:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Ono__top,type,
% 2.81/1.50 no_top:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Olinorder,type,
% 2.81/1.50 linorder:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Opreorder,type,
% 2.81/1.50 preorder:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Rings_Ozero__neq__one,type,
% 2.81/1.50 zero_neq_one:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.81/1.50 order_bot:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Owellorder,type,
% 2.81/1.50 wellorder:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Rings_Ozero__less__one,type,
% 2.81/1.50 zero_less_one:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Odense__order,type,
% 2.81/1.50 dense_order:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.81/1.50 linordered_idom:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Fields_Olinordered__field,type,
% 2.81/1.50 linordered_field:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.81/1.50 dense_linorder:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.81/1.50 linordered_semidom:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.81/1.50 linord1659791738miring:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.81/1.50 canoni770627133id_add:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.81/1.50 condit1656338222tinuum:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.81/1.50 condit1037483654norder:
% 2.81/1.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 2.81/1.50 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 2.81/1.50 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 2.81/1.50 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 2.81/1.50 binDag1297733282se_dag:
% 2.81/1.50 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 2.81/1.50 binDag1442713106ec_dag:
% 2.81/1.50 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Osize__dag,type,
% 2.81/1.50 binDag1924123185ze_dag: binDag_Mirabelle_dag > nat ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 2.81/1.50 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 2.81/1.50 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.81/1.50 one_one:
% 2.81/1.50 !>[A: $tType] : A ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 2.81/1.50 sgn_sgn:
% 2.81/1.50 !>[A: $tType] : ( A > A ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.81/1.50 zero_zero:
% 2.81/1.50 !>[A: $tType] : A ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_If,type,
% 2.81/1.50 if:
% 2.81/1.50 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.81/1.50 size_size:
% 2.81/1.50 !>[A: $tType] : ( A > nat ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.81/1.50 bot_bot:
% 2.81/1.50 !>[A: $tType] : A ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.81/1.50 ord_less:
% 2.81/1.50 !>[A: $tType] : ( A > A > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.81/1.50 ord_less_eq:
% 2.81/1.50 !>[A: $tType] : ( A > A > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Orderings_Oordering,type,
% 2.81/1.50 ordering:
% 2.81/1.50 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Pure_Otype,type,
% 2.81/1.50 type2:
% 2.81/1.50 !>[A: $tType] : ( itself @ A ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Relation_Oinv__imagep,type,
% 2.81/1.50 inv_imagep:
% 2.81/1.50 !>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 2.81/1.50 zero_neq_one_of_bool:
% 2.81/1.50 !>[A: $tType] : ( $o > A ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Set_OCollect,type,
% 2.81/1.50 collect:
% 2.81/1.50 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Set_Ois__empty,type,
% 2.81/1.50 is_empty:
% 2.81/1.50 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_Zorn_Ochain__subset,type,
% 2.81/1.50 chain_subset:
% 2.81/1.50 !>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_c_member,type,
% 2.81/1.50 member:
% 2.81/1.50 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.81/1.50
% 2.81/1.50 thf(sy_v_a____,type,
% 2.81/1.50 a: simpl_ref ).
% 2.81/1.50
% 2.81/1.50 thf(sy_v_l____,type,
% 2.81/1.50 l: binDag_Mirabelle_dag ).
% 2.81/1.50
% 2.81/1.50 thf(sy_v_r____,type,
% 2.81/1.50 r: binDag_Mirabelle_dag ).
% 2.81/1.50
% 2.81/1.50 thf(sy_v_thesis____,type,
% 2.81/1.50 thesis: $o ).
% 2.81/1.50
% 2.81/1.50 thf(sy_v_x,type,
% 2.81/1.50 x: binDag_Mirabelle_dag ).
% 2.81/1.50
% 2.81/1.50 thf(sy_v_y,type,
% 2.81/1.50 y: binDag_Mirabelle_dag ).
% 2.81/1.50
% 2.81/1.50 thf(fact_0_Node,axiom,
% 2.81/1.50 ( y
% 2.81/1.50 = ( binDag476092410e_Node @ l @ a @ r ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_1_dag_Oinject,axiom,
% 2.81/1.50 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 2.81/1.50 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 2.81/1.50 = ( ( X21 = Y21 )
% 2.81/1.50 & ( X22 = Y22 )
% 2.81/1.50 & ( X23 = Y23 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_2_DAG_Osimps_I2_J,axiom,
% 2.81/1.50 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.81/1.50 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 2.81/1.50 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 2.81/1.50 & ( binDag_Mirabelle_DAG @ L )
% 2.81/1.50 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_3_x__less__y,axiom,
% 2.81/1.50 ord_less @ binDag_Mirabelle_dag @ x @ y ).
% 2.81/1.50
% 2.81/1.50 thf(fact_4_set__of__Tip,axiom,
% 2.81/1.50 ( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
% 2.81/1.50 = ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_5_le__dag__set__of,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
% 2.81/1.50 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_6_less__dag__set__of,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 2.81/1.50 => ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_7_dag_Osimps_I7_J,axiom,
% 2.81/1.50 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.81/1.50 = ( F2 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X23 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_8_dag_Osimps_I5_J,axiom,
% 2.81/1.50 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.81/1.50 = ( F2 @ X21 @ X22 @ X23 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_9_subdag__neq,axiom,
% 2.81/1.50 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag786255756subdag @ T @ S )
% 2.81/1.50 => ( T != S ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_10_subdag__trans,axiom,
% 2.81/1.50 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag786255756subdag @ T @ S )
% 2.81/1.50 => ( ( binDag786255756subdag @ S @ R )
% 2.81/1.50 => ( binDag786255756subdag @ T @ R ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_11_subdag__not__sym,axiom,
% 2.81/1.50 ! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag786255756subdag @ S @ T )
% 2.81/1.50 => ~ ( binDag786255756subdag @ T @ S ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_12_DAG__y,axiom,
% 2.81/1.50 binDag_Mirabelle_DAG @ y ).
% 2.81/1.50
% 2.81/1.50 thf(fact_13_DAG_Osimps_I1_J,axiom,
% 2.81/1.50 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 2.81/1.50
% 2.81/1.50 thf(fact_14_dag_Osimps_I6_J,axiom,
% 2.81/1.50 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 2.81/1.50 ( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 2.81/1.50 = F1 ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_15_dag_Osimps_I4_J,axiom,
% 2.81/1.50 ! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 2.81/1.50 ( ( binDag1297733282se_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
% 2.81/1.50 = F1 ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_16_dag_Odistinct_I1_J,axiom,
% 2.81/1.50 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.81/1.50 ( binDag_Mirabelle_Tip
% 2.81/1.50 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_17_subdag_Osimps_I2_J,axiom,
% 2.81/1.50 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T )
% 2.81/1.50 = ( ( T = L )
% 2.81/1.50 | ( T = R )
% 2.81/1.50 | ( binDag786255756subdag @ L @ T )
% 2.81/1.50 | ( binDag786255756subdag @ R @ T ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_18_subdag_Osimps_I1_J,axiom,
% 2.81/1.50 ! [T: binDag_Mirabelle_dag] :
% 2.81/1.50 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_19_DAG__less,axiom,
% 2.81/1.50 ! [Y: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag_Mirabelle_DAG @ Y )
% 2.81/1.50 => ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
% 2.81/1.50 => ( binDag_Mirabelle_DAG @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_20_dag_Oinduct,axiom,
% 2.81/1.50 ! [P: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( P @ binDag_Mirabelle_Tip )
% 2.81/1.50 => ( ! [X1: binDag_Mirabelle_dag,X2: simpl_ref,X3: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( P @ X1 )
% 2.81/1.50 => ( ( P @ X3 )
% 2.81/1.50 => ( P @ ( binDag476092410e_Node @ X1 @ X2 @ X3 ) ) ) )
% 2.81/1.50 => ( P @ Dag ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_21_le__dag__def,axiom,
% 2.81/1.50 ( ( ord_less_eq @ binDag_Mirabelle_dag )
% 2.81/1.50 = ( ^ [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( S2 = T2 )
% 2.81/1.50 | ( ord_less @ binDag_Mirabelle_dag @ S2 @ T2 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_22_dag_Oexhaust,axiom,
% 2.81/1.50 ! [Y: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( Y != binDag_Mirabelle_Tip )
% 2.81/1.50 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 2.81/1.50 ( Y
% 2.81/1.50 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_23_dag__less__le,axiom,
% 2.81/1.50 ( ( ord_less @ binDag_Mirabelle_dag )
% 2.81/1.50 = ( ^ [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y2 )
% 2.81/1.50 & ( X4 != Y2 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_24_le__dag__refl,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X @ X ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_25_le__dag__trans,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag,Z: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
% 2.81/1.50 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ Z )
% 2.81/1.50 => ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Z ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_26_less__dag__Tip,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag] :
% 2.81/1.50 ~ ( ord_less @ binDag_Mirabelle_dag @ X @ binDag_Mirabelle_Tip ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_27_less__dag__def,axiom,
% 2.81/1.50 ( ( ord_less @ binDag_Mirabelle_dag )
% 2.81/1.50 = ( ^ [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_28_subdag__NodeD,axiom,
% 2.81/1.50 ! [T: binDag_Mirabelle_dag,Lt: binDag_Mirabelle_dag,A2: simpl_ref,Rt: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt @ A2 @ Rt ) )
% 2.81/1.50 => ( ( binDag786255756subdag @ T @ Lt )
% 2.81/1.50 & ( binDag786255756subdag @ T @ Rt ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_29_less__Node__dag,axiom,
% 2.81/1.50 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X )
% 2.81/1.50 => ( ( ord_less @ binDag_Mirabelle_dag @ L @ X )
% 2.81/1.50 & ( ord_less @ binDag_Mirabelle_dag @ R @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_30_less__dag__Node,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.81/1.50 = ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ L )
% 2.81/1.50 | ( ord_less_eq @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_31_le__dag__antisym,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
% 2.81/1.50 => ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ X )
% 2.81/1.50 => ( X = Y ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_32_less__dag__Node_H,axiom,
% 2.81/1.50 ! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.81/1.50 ( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.81/1.50 = ( ( X = L )
% 2.81/1.50 | ( X = R )
% 2.81/1.50 | ( ord_less @ binDag_Mirabelle_dag @ X @ L )
% 2.81/1.50 | ( ord_less @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_33_subset__empty,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.81/1.50 = ( A3
% 2.81/1.50 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_34_empty__subsetI,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_35_subsetI,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.50 ( ! [X5: A] :
% 2.81/1.50 ( ( member @ A @ X5 @ A3 )
% 2.81/1.50 => ( member @ A @ X5 @ B2 ) )
% 2.81/1.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_36_subset__antisym,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
% 2.81/1.50 => ( A3 = B2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_37_empty__iff,axiom,
% 2.81/1.50 ! [A: $tType,C: A] :
% 2.81/1.50 ~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_38_all__not__in__conv,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] :
% 2.81/1.50 ( ( ! [X4: A] :
% 2.81/1.50 ~ ( member @ A @ X4 @ A3 ) )
% 2.81/1.50 = ( A3
% 2.81/1.50 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_39_Collect__empty__eq,axiom,
% 2.81/1.50 ! [A: $tType,P: A > $o] :
% 2.81/1.50 ( ( ( collect @ A @ P )
% 2.81/1.50 = ( bot_bot @ ( set @ A ) ) )
% 2.81/1.50 = ( ! [X4: A] :
% 2.81/1.50 ~ ( P @ X4 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_40_empty__Collect__eq,axiom,
% 2.81/1.50 ! [A: $tType,P: A > $o] :
% 2.81/1.50 ( ( ( bot_bot @ ( set @ A ) )
% 2.81/1.50 = ( collect @ A @ P ) )
% 2.81/1.50 = ( ! [X4: A] :
% 2.81/1.50 ~ ( P @ X4 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_41_bot__apply,axiom,
% 2.81/1.50 ! [C2: $tType,D: $tType] :
% 2.81/1.50 ( ( bot @ C2 @ ( type2 @ C2 ) )
% 2.81/1.50 => ( ( bot_bot @ ( D > C2 ) )
% 2.81/1.50 = ( ^ [X4: D] : ( bot_bot @ C2 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_42_order__refl,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_43_subset__emptyI,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] :
% 2.81/1.50 ( ! [X5: A] :
% 2.81/1.50 ~ ( member @ A @ X5 @ A3 )
% 2.81/1.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_44_bot__set__def,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( bot_bot @ ( set @ A ) )
% 2.81/1.50 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_45_mem__Collect__eq,axiom,
% 2.81/1.50 ! [A: $tType,A2: A,P: A > $o] :
% 2.81/1.50 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.81/1.50 = ( P @ A2 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_46_Collect__mem__eq,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] :
% 2.81/1.50 ( ( collect @ A
% 2.81/1.50 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.81/1.50 = A3 ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_47_Collect__cong,axiom,
% 2.81/1.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.81/1.50 ( ! [X5: A] :
% 2.81/1.50 ( ( P @ X5 )
% 2.81/1.50 = ( Q @ X5 ) )
% 2.81/1.50 => ( ( collect @ A @ P )
% 2.81/1.50 = ( collect @ A @ Q ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_48_ext,axiom,
% 2.81/1.50 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.81/1.50 ( ! [X5: A] :
% 2.81/1.50 ( ( F @ X5 )
% 2.81/1.50 = ( G @ X5 ) )
% 2.81/1.50 => ( F = G ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_49_dual__order_Oantisym,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [B3: A,A2: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.81/1.50 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.50 => ( A2 = B3 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_50_dual__order_Otrans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [B3: A,A2: A,C: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.81/1.50 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.81/1.50 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_51_linorder__wlog,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [P: A > A > $o,A2: A,B3: A] :
% 2.81/1.50 ( ! [A4: A,B4: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.81/1.50 => ( P @ A4 @ B4 ) )
% 2.81/1.50 => ( ! [A4: A,B4: A] :
% 2.81/1.50 ( ( P @ B4 @ A4 )
% 2.81/1.50 => ( P @ A4 @ B4 ) )
% 2.81/1.50 => ( P @ A2 @ B3 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_52_dual__order_Orefl,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_53_order__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A,Z: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.50 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.81/1.50 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_54_order__class_Oorder_Oantisym,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.81/1.50 => ( A2 = B3 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_55_ord__le__eq__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( ord @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A,C: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( B3 = C )
% 2.81/1.50 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_56_ord__eq__le__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( ord @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A,C: A] :
% 2.81/1.50 ( ( A2 = B3 )
% 2.81/1.50 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.81/1.50 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_57_antisym__conv,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [Y: A,X: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ Y @ X )
% 2.81/1.50 => ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.50 = ( X = Y ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_58_le__cases3,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A,Z: A] :
% 2.81/1.50 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.50 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.81/1.50 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.81/1.50 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.81/1.50 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.81/1.50 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.81/1.50 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.81/1.50 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.81/1.50 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.81/1.50 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.81/1.50 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.81/1.50 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_59_order_Otrans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A,C: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.81/1.50 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_60_le__cases,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.81/1.50 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_61_eq__refl,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( X = Y )
% 2.81/1.50 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_62_linear,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.50 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_63_antisym,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.50 => ( ( ord_less_eq @ A @ Y @ X )
% 2.81/1.50 => ( X = Y ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_64_eq__iff,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ( ( ^ [Y3: A,Z2: A] : ( Y3 = Z2 ) )
% 2.81/1.50 = ( ^ [X4: A,Y2: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.81/1.50 & ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_65_ord__le__eq__subst,axiom,
% 2.81/1.50 ! [A: $tType,B: $tType] :
% 2.81/1.50 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 & ( ord @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,B3: A,F: A > B,C: B] :
% 2.81/1.50 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( ( F @ B3 )
% 2.81/1.50 = C )
% 2.81/1.50 => ( ! [X5: A,Y4: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_66_ord__eq__le__subst,axiom,
% 2.81/1.50 ! [A: $tType,B: $tType] :
% 2.81/1.50 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 & ( ord @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.81/1.50 ( ( A2
% 2.81/1.50 = ( F @ B3 ) )
% 2.81/1.50 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.81/1.50 => ( ! [X5: B,Y4: B] :
% 2.81/1.50 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_67_order__subst2,axiom,
% 2.81/1.50 ! [A: $tType,C2: $tType] :
% 2.81/1.50 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.81/1.50 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 2.81/1.50 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 2.81/1.50 => ( ! [X5: A,Y4: A] :
% 2.81/1.50 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_68_order__subst1,axiom,
% 2.81/1.50 ! [A: $tType,B: $tType] :
% 2.81/1.50 ( ( ( order @ B @ ( type2 @ B ) )
% 2.81/1.50 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.81/1.50 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 2.81/1.50 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.81/1.50 => ( ! [X5: B,Y4: B] :
% 2.81/1.50 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_69_le__fun__def,axiom,
% 2.81/1.50 ! [B: $tType,A: $tType] :
% 2.81/1.50 ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 => ( ( ord_less_eq @ ( A > B ) )
% 2.81/1.50 = ( ^ [F3: A > B,G2: A > B] :
% 2.81/1.50 ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_70_le__funI,axiom,
% 2.81/1.50 ! [B: $tType,A: $tType] :
% 2.81/1.50 ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 => ! [F: A > B,G: A > B] :
% 2.81/1.50 ( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
% 2.81/1.50 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_71_le__funE,axiom,
% 2.81/1.50 ! [B: $tType,A: $tType] :
% 2.81/1.50 ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 => ! [F: A > B,G: A > B,X: A] :
% 2.81/1.50 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.81/1.50 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_72_le__funD,axiom,
% 2.81/1.50 ! [B: $tType,A: $tType] :
% 2.81/1.50 ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 => ! [F: A > B,G: A > B,X: A] :
% 2.81/1.50 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.81/1.50 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_73_dual__order_Ostrict__implies__not__eq,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [B3: A,A2: A] :
% 2.81/1.50 ( ( ord_less @ A @ B3 @ A2 )
% 2.81/1.50 => ( A2 != B3 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_74_order_Ostrict__implies__not__eq,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.50 => ( A2 != B3 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_75_not__less__iff__gr__or__eq,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.81/1.50 = ( ( ord_less @ A @ Y @ X )
% 2.81/1.50 | ( X = Y ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_76_dual__order_Ostrict__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [B3: A,A2: A,C: A] :
% 2.81/1.50 ( ( ord_less @ A @ B3 @ A2 )
% 2.81/1.50 => ( ( ord_less @ A @ C @ B3 )
% 2.81/1.50 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_77_less__imp__not__less,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_78_order_Ostrict__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A,C: A] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( ord_less @ A @ B3 @ C )
% 2.81/1.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_79_dual__order_Oirrefl,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A] :
% 2.81/1.50 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_80_linorder__cases,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ~ ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ( ( X != Y )
% 2.81/1.50 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_81_less__imp__triv,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A,P: $o] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ( ( ord_less @ A @ Y @ X )
% 2.81/1.50 => P ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_82_less__imp__not__eq2,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ( Y != X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_83_antisym__conv3,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [Y: A,X: A] :
% 2.81/1.50 ( ~ ( ord_less @ A @ Y @ X )
% 2.81/1.50 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.81/1.50 = ( X = Y ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_84_less__induct,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [P: A > $o,A2: A] :
% 2.81/1.50 ( ! [X5: A] :
% 2.81/1.50 ( ! [Y5: A] :
% 2.81/1.50 ( ( ord_less @ A @ Y5 @ X5 )
% 2.81/1.50 => ( P @ Y5 ) )
% 2.81/1.50 => ( P @ X5 ) )
% 2.81/1.50 => ( P @ A2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_85_less__not__sym,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_86_less__imp__not__eq,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ( X != Y ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_87_dual__order_Oasym,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [B3: A,A2: A] :
% 2.81/1.50 ( ( ord_less @ A @ B3 @ A2 )
% 2.81/1.50 => ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_88_ord__less__eq__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( ord @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A,C: A] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( B3 = C )
% 2.81/1.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_89_ord__eq__less__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( ord @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A,C: A] :
% 2.81/1.50 ( ( A2 = B3 )
% 2.81/1.50 => ( ( ord_less @ A @ B3 @ C )
% 2.81/1.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_90_less__irrefl,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A] :
% 2.81/1.50 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_91_less__linear,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 | ( X = Y )
% 2.81/1.50 | ( ord_less @ A @ Y @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_92_less__trans,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A,Z: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ( ( ord_less @ A @ Y @ Z )
% 2.81/1.50 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_93_less__asym_H,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.50 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_94_less__asym,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_95_less__imp__neq,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ( X != Y ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_96_dense,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ? [Z3: A] :
% 2.81/1.50 ( ( ord_less @ A @ X @ Z3 )
% 2.81/1.50 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_97_order_Oasym,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [A2: A,B3: A] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.50 => ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_98_neq__iff,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( X != Y )
% 2.81/1.50 = ( ( ord_less @ A @ X @ Y )
% 2.81/1.50 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_99_neqE,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A,Y: A] :
% 2.81/1.50 ( ( X != Y )
% 2.81/1.50 => ( ~ ( ord_less @ A @ X @ Y )
% 2.81/1.50 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_100_gt__ex,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( no_top @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A] :
% 2.81/1.50 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_101_lt__ex,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.81/1.50 => ! [X: A] :
% 2.81/1.50 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_102_order__less__subst2,axiom,
% 2.81/1.50 ! [A: $tType,C2: $tType] :
% 2.81/1.50 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.81/1.50 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 2.81/1.50 => ( ! [X5: A,Y4: A] :
% 2.81/1.50 ( ( ord_less @ A @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_103_order__less__subst1,axiom,
% 2.81/1.50 ! [A: $tType,B: $tType] :
% 2.81/1.50 ( ( ( order @ B @ ( type2 @ B ) )
% 2.81/1.50 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 2.81/1.50 => ( ( ord_less @ B @ B3 @ C )
% 2.81/1.50 => ( ! [X5: B,Y4: B] :
% 2.81/1.50 ( ( ord_less @ B @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_104_ord__less__eq__subst,axiom,
% 2.81/1.50 ! [A: $tType,B: $tType] :
% 2.81/1.50 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 & ( ord @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,B3: A,F: A > B,C: B] :
% 2.81/1.50 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.50 => ( ( ( F @ B3 )
% 2.81/1.50 = C )
% 2.81/1.50 => ( ! [X5: A,Y4: A] :
% 2.81/1.50 ( ( ord_less @ A @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_105_ord__eq__less__subst,axiom,
% 2.81/1.50 ! [A: $tType,B: $tType] :
% 2.81/1.50 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.50 & ( ord @ A @ ( type2 @ A ) ) )
% 2.81/1.50 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.81/1.50 ( ( A2
% 2.81/1.50 = ( F @ B3 ) )
% 2.81/1.50 => ( ( ord_less @ B @ B3 @ C )
% 2.81/1.50 => ( ! [X5: B,Y4: B] :
% 2.81/1.50 ( ( ord_less @ B @ X5 @ Y4 )
% 2.81/1.50 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.50 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_106_bot__fun__def,axiom,
% 2.81/1.50 ! [B: $tType,A: $tType] :
% 2.81/1.50 ( ( bot @ B @ ( type2 @ B ) )
% 2.81/1.50 => ( ( bot_bot @ ( A > B ) )
% 2.81/1.50 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_107_ex__in__conv,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] :
% 2.81/1.50 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.81/1.50 = ( A3
% 2.81/1.50 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_108_equals0I,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] :
% 2.81/1.50 ( ! [Y4: A] :
% 2.81/1.50 ~ ( member @ A @ Y4 @ A3 )
% 2.81/1.50 => ( A3
% 2.81/1.50 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_109_equals0D,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,A2: A] :
% 2.81/1.50 ( ( A3
% 2.81/1.50 = ( bot_bot @ ( set @ A ) ) )
% 2.81/1.50 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_110_emptyE,axiom,
% 2.81/1.50 ! [A: $tType,A2: A] :
% 2.81/1.50 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_111_Collect__mono__iff,axiom,
% 2.81/1.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.81/1.50 = ( ! [X4: A] :
% 2.81/1.50 ( ( P @ X4 )
% 2.81/1.50 => ( Q @ X4 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_112_contra__subsetD,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ( ~ ( member @ A @ C @ B2 )
% 2.81/1.50 => ~ ( member @ A @ C @ A3 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_113_set__eq__subset,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( ^ [Y3: set @ A,Z2: set @ A] : ( Y3 = Z2 ) )
% 2.81/1.50 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.81/1.50 & ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_114_subset__trans,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.81/1.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_115_Collect__mono,axiom,
% 2.81/1.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.81/1.50 ( ! [X5: A] :
% 2.81/1.50 ( ( P @ X5 )
% 2.81/1.50 => ( Q @ X5 ) )
% 2.81/1.50 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_116_subset__refl,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_117_rev__subsetD,axiom,
% 2.81/1.50 ! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
% 2.81/1.50 ( ( member @ A @ C @ A3 )
% 2.81/1.50 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ( member @ A @ C @ B2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_118_subset__iff,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) )
% 2.81/1.50 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.81/1.50 ! [T2: A] :
% 2.81/1.50 ( ( member @ A @ T2 @ A5 )
% 2.81/1.50 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_119_set__rev__mp,axiom,
% 2.81/1.50 ! [A: $tType,X: A,A3: set @ A,B2: set @ A] :
% 2.81/1.50 ( ( member @ A @ X @ A3 )
% 2.81/1.50 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ( member @ A @ X @ B2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_120_equalityD2,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.50 ( ( A3 = B2 )
% 2.81/1.50 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_121_equalityD1,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.50 ( ( A3 = B2 )
% 2.81/1.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_122_subset__eq,axiom,
% 2.81/1.50 ! [A: $tType] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) )
% 2.81/1.50 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.81/1.50 ! [X4: A] :
% 2.81/1.50 ( ( member @ A @ X4 @ A5 )
% 2.81/1.50 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_123_equalityE,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.50 ( ( A3 = B2 )
% 2.81/1.50 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_124_subsetCE,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ( ( member @ A @ C @ A3 )
% 2.81/1.50 => ( member @ A @ C @ B2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_125_subsetD,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 2.81/1.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.50 => ( ( member @ A @ C @ A3 )
% 2.81/1.50 => ( member @ A @ C @ B2 ) ) ) ).
% 2.81/1.50
% 2.81/1.50 thf(fact_126_in__mono,axiom,
% 2.81/1.50 ! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
% 2.81/1.51 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ( member @ A @ X @ A3 )
% 2.81/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_127_set__mp,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
% 2.81/1.51 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ( member @ A @ X @ A3 )
% 2.81/1.51 => ( member @ A @ X @ B2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_128_order_Onot__eq__order__implies__strict,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A,B3: A] :
% 2.81/1.51 ( ( A2 != B3 )
% 2.81/1.51 => ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.51 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_129_dual__order_Ostrict__implies__order,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [B3: A,A2: A] :
% 2.81/1.51 ( ( ord_less @ A @ B3 @ A2 )
% 2.81/1.51 => ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_130_dual__order_Ostrict__iff__order,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( ord_less @ A )
% 2.81/1.51 = ( ^ [B6: A,A6: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ B6 @ A6 )
% 2.81/1.51 & ( A6 != B6 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_131_dual__order_Oorder__iff__strict,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( ord_less_eq @ A )
% 2.81/1.51 = ( ^ [B6: A,A6: A] :
% 2.81/1.51 ( ( ord_less @ A @ B6 @ A6 )
% 2.81/1.51 | ( A6 = B6 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_132_order_Ostrict__implies__order,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A,B3: A] :
% 2.81/1.51 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.51 => ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_133_dense__le__bounded,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A,Z: A] :
% 2.81/1.51 ( ( ord_less @ A @ X @ Y )
% 2.81/1.51 => ( ! [W: A] :
% 2.81/1.51 ( ( ord_less @ A @ X @ W )
% 2.81/1.51 => ( ( ord_less @ A @ W @ Y )
% 2.81/1.51 => ( ord_less_eq @ A @ W @ Z ) ) )
% 2.81/1.51 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_134_dense__ge__bounded,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [Z: A,X: A,Y: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z @ X )
% 2.81/1.51 => ( ! [W: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z @ W )
% 2.81/1.51 => ( ( ord_less @ A @ W @ X )
% 2.81/1.51 => ( ord_less_eq @ A @ Y @ W ) ) )
% 2.81/1.51 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_135_dual__order_Ostrict__trans2,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [B3: A,A2: A,C: A] :
% 2.81/1.51 ( ( ord_less @ A @ B3 @ A2 )
% 2.81/1.51 => ( ( ord_less_eq @ A @ C @ B3 )
% 2.81/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_136_dual__order_Ostrict__trans1,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [B3: A,A2: A,C: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ B3 @ A2 )
% 2.81/1.51 => ( ( ord_less @ A @ C @ B3 )
% 2.81/1.51 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_137_order_Ostrict__iff__order,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( ord_less @ A )
% 2.81/1.51 = ( ^ [A6: A,B6: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.81/1.51 & ( A6 != B6 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_138_order_Oorder__iff__strict,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( ord_less_eq @ A )
% 2.81/1.51 = ( ^ [A6: A,B6: A] :
% 2.81/1.51 ( ( ord_less @ A @ A6 @ B6 )
% 2.81/1.51 | ( A6 = B6 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_139_order_Ostrict__trans2,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A,B3: A,C: A] :
% 2.81/1.51 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.51 => ( ( ord_less_eq @ A @ B3 @ C )
% 2.81/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_140_order_Ostrict__trans1,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A,B3: A,C: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.51 => ( ( ord_less @ A @ B3 @ C )
% 2.81/1.51 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_141_not__le__imp__less,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [Y: A,X: A] :
% 2.81/1.51 ( ~ ( ord_less_eq @ A @ Y @ X )
% 2.81/1.51 => ( ord_less @ A @ X @ Y ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_142_less__le__not__le,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( ord_less @ A )
% 2.81/1.51 = ( ^ [X4: A,Y2: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.81/1.51 & ~ ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_143_le__imp__less__or__eq,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.51 => ( ( ord_less @ A @ X @ Y )
% 2.81/1.51 | ( X = Y ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_144_le__less__linear,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.51 | ( ord_less @ A @ Y @ X ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_145_dense__le,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [Y: A,Z: A] :
% 2.81/1.51 ( ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ X5 @ Y )
% 2.81/1.51 => ( ord_less_eq @ A @ X5 @ Z ) )
% 2.81/1.51 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_146_dense__ge,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [Z: A,Y: A] :
% 2.81/1.51 ( ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z @ X5 )
% 2.81/1.51 => ( ord_less_eq @ A @ Y @ X5 ) )
% 2.81/1.51 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_147_less__le__trans,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A,Z: A] :
% 2.81/1.51 ( ( ord_less @ A @ X @ Y )
% 2.81/1.51 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.81/1.51 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_148_le__less__trans,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A,Z: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.51 => ( ( ord_less @ A @ Y @ Z )
% 2.81/1.51 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_149_antisym__conv2,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.51 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.81/1.51 = ( X = Y ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_150_antisym__conv1,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ~ ( ord_less @ A @ X @ Y )
% 2.81/1.51 => ( ( ord_less_eq @ A @ X @ Y )
% 2.81/1.51 = ( X = Y ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_151_less__imp__le,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( preorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ( ord_less @ A @ X @ Y )
% 2.81/1.51 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_152_le__neq__trans,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A,B3: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.51 => ( ( A2 != B3 )
% 2.81/1.51 => ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_153_not__less,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 2.81/1.51 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_154_not__le,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 2.81/1.51 = ( ord_less @ A @ Y @ X ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_155_order__less__le__subst2,axiom,
% 2.81/1.51 ! [A: $tType,C2: $tType] :
% 2.81/1.51 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.81/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.51 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 2.81/1.51 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.51 => ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
% 2.81/1.51 => ( ! [X5: A,Y4: A] :
% 2.81/1.51 ( ( ord_less @ A @ X5 @ Y4 )
% 2.81/1.51 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_156_order__less__le__subst1,axiom,
% 2.81/1.51 ! [A: $tType,B: $tType] :
% 2.81/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.81/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.51 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.81/1.51 ( ( ord_less @ A @ A2 @ ( F @ B3 ) )
% 2.81/1.51 => ( ( ord_less_eq @ B @ B3 @ C )
% 2.81/1.51 => ( ! [X5: B,Y4: B] :
% 2.81/1.51 ( ( ord_less_eq @ B @ X5 @ Y4 )
% 2.81/1.51 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_157_order__le__less__subst2,axiom,
% 2.81/1.51 ! [A: $tType,C2: $tType] :
% 2.81/1.51 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.81/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.51 => ! [A2: A,B3: A,F: A > C2,C: C2] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A2 @ B3 )
% 2.81/1.51 => ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
% 2.81/1.51 => ( ! [X5: A,Y4: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ X5 @ Y4 )
% 2.81/1.51 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.51 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_158_order__le__less__subst1,axiom,
% 2.81/1.51 ! [A: $tType,B: $tType] :
% 2.81/1.51 ( ( ( order @ B @ ( type2 @ B ) )
% 2.81/1.51 & ( order @ A @ ( type2 @ A ) ) )
% 2.81/1.51 => ! [A2: A,F: B > A,B3: B,C: B] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
% 2.81/1.51 => ( ( ord_less @ B @ B3 @ C )
% 2.81/1.51 => ( ! [X5: B,Y4: B] :
% 2.81/1.51 ( ( ord_less @ B @ X5 @ Y4 )
% 2.81/1.51 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 2.81/1.51 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_159_less__le,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( ord_less @ A )
% 2.81/1.51 = ( ^ [X4: A,Y2: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ X4 @ Y2 )
% 2.81/1.51 & ( X4 != Y2 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_160_le__less,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( ord_less_eq @ A )
% 2.81/1.51 = ( ^ [X4: A,Y2: A] :
% 2.81/1.51 ( ( ord_less @ A @ X4 @ Y2 )
% 2.81/1.51 | ( X4 = Y2 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_161_leI,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ~ ( ord_less @ A @ X @ Y )
% 2.81/1.51 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_162_leD,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [Y: A,X: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ Y @ X )
% 2.81/1.51 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_163_bot_Oextremum__uniqueI,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.81/1.51 => ( A2
% 2.81/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_164_bot_Oextremum__unique,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.81/1.51 = ( A2
% 2.81/1.51 = ( bot_bot @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_165_bot_Oextremum,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_166_bot_Onot__eq__extremum,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( A2
% 2.81/1.51 != ( bot_bot @ A ) )
% 2.81/1.51 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_167_bot_Oextremum__strict,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_168_complete__interval,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A,B3: A,P: A > $o] :
% 2.81/1.51 ( ( ord_less @ A @ A2 @ B3 )
% 2.81/1.51 => ( ( P @ A2 )
% 2.81/1.51 => ( ~ ( P @ B3 )
% 2.81/1.51 => ? [C4: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ A2 @ C4 )
% 2.81/1.51 & ( ord_less_eq @ A @ C4 @ B3 )
% 2.81/1.51 & ! [X6: A] :
% 2.81/1.51 ( ( ( ord_less_eq @ A @ A2 @ X6 )
% 2.81/1.51 & ( ord_less @ A @ X6 @ C4 ) )
% 2.81/1.51 => ( P @ X6 ) )
% 2.81/1.51 & ! [D2: A] :
% 2.81/1.51 ( ! [X5: A] :
% 2.81/1.51 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 2.81/1.51 & ( ord_less @ A @ X5 @ D2 ) )
% 2.81/1.51 => ( P @ X5 ) )
% 2.81/1.51 => ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_169_pinf_I6_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_170_pinf_I8_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_171_minf_I6_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_172_minf_I8_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_173_Set_Ois__empty__def,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( is_empty @ A )
% 2.81/1.51 = ( ^ [A5: set @ A] :
% 2.81/1.51 ( A5
% 2.81/1.51 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_174_Collect__empty__eq__bot,axiom,
% 2.81/1.51 ! [A: $tType,P: A > $o] :
% 2.81/1.51 ( ( ( collect @ A @ P )
% 2.81/1.51 = ( bot_bot @ ( set @ A ) ) )
% 2.81/1.51 = ( P
% 2.81/1.51 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_175_bot__empty__eq,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( bot_bot @ ( A > $o ) )
% 2.81/1.51 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_176_psubsetI,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.51 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ( A3 != B2 )
% 2.81/1.51 => ( ord_less @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_177_less__fun__def,axiom,
% 2.81/1.51 ! [B: $tType,A: $tType] :
% 2.81/1.51 ( ( ord @ B @ ( type2 @ B ) )
% 2.81/1.51 => ( ( ord_less @ ( A > B ) )
% 2.81/1.51 = ( ^ [F3: A > B,G2: A > B] :
% 2.81/1.51 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 2.81/1.51 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_178_not__psubset__empty,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A] :
% 2.81/1.51 ~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_179_psubsetE,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_180_psubset__eq,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) )
% 2.81/1.51 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.81/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.81/1.51 & ( A5 != B5 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_181_psubset__imp__subset,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_182_psubset__subset__trans,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.81/1.51 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_183_subset__not__subset__eq,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) )
% 2.81/1.51 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.81/1.51 ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
% 2.81/1.51 & ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_184_subset__psubset__trans,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.81/1.51 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.81/1.51 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_185_subset__iff__psubset__eq,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( ord_less_eq @ ( set @ A ) )
% 2.81/1.51 = ( ^ [A5: set @ A,B5: set @ A] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
% 2.81/1.51 | ( A5 = B5 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_186_minf_I11_J,axiom,
% 2.81/1.51 ! [C2: $tType,D: $tType] :
% 2.81/1.51 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.81/1.51 => ! [F4: D] :
% 2.81/1.51 ? [Z3: C2] :
% 2.81/1.51 ! [X6: C2] :
% 2.81/1.51 ( ( ord_less @ C2 @ X6 @ Z3 )
% 2.81/1.51 => ( F4 = F4 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_187_minf_I7_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_188_minf_I5_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ( ord_less @ A @ X6 @ T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_189_minf_I4_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ( X6 != T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_190_minf_I3_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ( X6 != T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_191_minf_I2_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.81/1.51 ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.81/1.51 => ( ( P @ X5 )
% 2.81/1.51 = ( P2 @ X5 ) ) )
% 2.81/1.51 => ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.81/1.51 => ( ( Q @ X5 )
% 2.81/1.51 = ( Q2 @ X5 ) ) )
% 2.81/1.51 => ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ( ( ( P @ X6 )
% 2.81/1.51 | ( Q @ X6 ) )
% 2.81/1.51 = ( ( P2 @ X6 )
% 2.81/1.51 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_192_minf_I1_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.81/1.51 ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.81/1.51 => ( ( P @ X5 )
% 2.81/1.51 = ( P2 @ X5 ) ) )
% 2.81/1.51 => ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ X5 @ Z4 )
% 2.81/1.51 => ( ( Q @ X5 )
% 2.81/1.51 = ( Q2 @ X5 ) ) )
% 2.81/1.51 => ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ X6 @ Z3 )
% 2.81/1.51 => ( ( ( P @ X6 )
% 2.81/1.51 & ( Q @ X6 ) )
% 2.81/1.51 = ( ( P2 @ X6 )
% 2.81/1.51 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_193_pinf_I11_J,axiom,
% 2.81/1.51 ! [C2: $tType,D: $tType] :
% 2.81/1.51 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.81/1.51 => ! [F4: D] :
% 2.81/1.51 ? [Z3: C2] :
% 2.81/1.51 ! [X6: C2] :
% 2.81/1.51 ( ( ord_less @ C2 @ Z3 @ X6 )
% 2.81/1.51 => ( F4 = F4 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_194_pinf_I7_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ( ord_less @ A @ T @ X6 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_195_pinf_I5_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_196_pinf_I4_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ( X6 != T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_197_pinf_I3_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [T: A] :
% 2.81/1.51 ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ( X6 != T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_198_pinf_I2_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.81/1.51 ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.81/1.51 => ( ( P @ X5 )
% 2.81/1.51 = ( P2 @ X5 ) ) )
% 2.81/1.51 => ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.81/1.51 => ( ( Q @ X5 )
% 2.81/1.51 = ( Q2 @ X5 ) ) )
% 2.81/1.51 => ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ( ( ( P @ X6 )
% 2.81/1.51 | ( Q @ X6 ) )
% 2.81/1.51 = ( ( P2 @ X6 )
% 2.81/1.51 | ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_199_pinf_I1_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.81/1.51 ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.81/1.51 => ( ( P @ X5 )
% 2.81/1.51 = ( P2 @ X5 ) ) )
% 2.81/1.51 => ( ? [Z4: A] :
% 2.81/1.51 ! [X5: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z4 @ X5 )
% 2.81/1.51 => ( ( Q @ X5 )
% 2.81/1.51 = ( Q2 @ X5 ) ) )
% 2.81/1.51 => ? [Z3: A] :
% 2.81/1.51 ! [X6: A] :
% 2.81/1.51 ( ( ord_less @ A @ Z3 @ X6 )
% 2.81/1.51 => ( ( ( P @ X6 )
% 2.81/1.51 & ( Q @ X6 ) )
% 2.81/1.51 = ( ( P2 @ X6 )
% 2.81/1.51 & ( Q2 @ X6 ) ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_200_ex__gt__or__lt,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ? [B4: A] :
% 2.81/1.51 ( ( ord_less @ A @ A2 @ B4 )
% 2.81/1.51 | ( ord_less @ A @ B4 @ A2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_201_dependent__wellorder__choice,axiom,
% 2.81/1.51 ! [B: $tType,A: $tType] :
% 2.81/1.51 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P: ( A > B ) > A > B > $o] :
% 2.81/1.51 ( ! [R2: B,F5: A > B,G3: A > B,X5: A] :
% 2.81/1.51 ( ! [Y5: A] :
% 2.81/1.51 ( ( ord_less @ A @ Y5 @ X5 )
% 2.81/1.51 => ( ( F5 @ Y5 )
% 2.81/1.51 = ( G3 @ Y5 ) ) )
% 2.81/1.51 => ( ( P @ F5 @ X5 @ R2 )
% 2.81/1.51 = ( P @ G3 @ X5 @ R2 ) ) )
% 2.81/1.51 => ( ! [X5: A,F5: A > B] :
% 2.81/1.51 ( ! [Y5: A] :
% 2.81/1.51 ( ( ord_less @ A @ Y5 @ X5 )
% 2.81/1.51 => ( P @ F5 @ Y5 @ ( F5 @ Y5 ) ) )
% 2.81/1.51 => ? [X12: B] : ( P @ F5 @ X5 @ X12 ) )
% 2.81/1.51 => ? [F5: A > B] :
% 2.81/1.51 ! [X6: A] : ( P @ F5 @ X6 @ ( F5 @ X6 ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_202_linordered__field__no__lb,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X6: A] :
% 2.81/1.51 ? [Y4: A] : ( ord_less @ A @ Y4 @ X6 ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_203_linordered__field__no__ub,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X6: A] :
% 2.81/1.51 ? [X1: A] : ( ord_less @ A @ X6 @ X1 ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_204_psubsetD,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ( member @ A @ C @ A3 )
% 2.81/1.51 => ( member @ A @ C @ B2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_205_psubset__trans,axiom,
% 2.81/1.51 ! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
% 2.81/1.51 ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
% 2.81/1.51 => ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
% 2.81/1.51 => ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_206_chain__subset__def,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( chain_subset @ A )
% 2.81/1.51 = ( ^ [C5: set @ ( set @ A )] :
% 2.81/1.51 ! [X4: set @ A] :
% 2.81/1.51 ( ( member @ ( set @ A ) @ X4 @ C5 )
% 2.81/1.51 => ! [Y2: set @ A] :
% 2.81/1.51 ( ( member @ ( set @ A ) @ Y2 @ C5 )
% 2.81/1.51 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y2 )
% 2.81/1.51 | ( ord_less_eq @ ( set @ A ) @ Y2 @ X4 ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_207_linorder__neqE__linordered__idom,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A,Y: A] :
% 2.81/1.51 ( ( X != Y )
% 2.81/1.51 => ( ~ ( ord_less @ A @ X @ Y )
% 2.81/1.51 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_208_in__inv__imagep,axiom,
% 2.81/1.51 ! [B: $tType,A: $tType] :
% 2.81/1.51 ( ( inv_imagep @ A @ B )
% 2.81/1.51 = ( ^ [R3: A > A > $o,F3: B > A,X4: B,Y2: B] : ( R3 @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_209_subdag__size,axiom,
% 2.81/1.51 ! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
% 2.81/1.51 ( ( binDag786255756subdag @ T @ S )
% 2.81/1.51 => ( ord_less @ nat @ ( size_size @ binDag_Mirabelle_dag @ S ) @ ( size_size @ binDag_Mirabelle_dag @ T ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_210_dag_Osize_I3_J,axiom,
% 2.81/1.51 ( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
% 2.81/1.51 = ( zero_zero @ nat ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_211_order_Oordering__axioms,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( order @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_212_ordering_Onot__eq__order__implies__strict,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( A2 != B3 )
% 2.81/1.51 => ( ( Less_eq @ A2 @ B3 )
% 2.81/1.51 => ( Less @ A2 @ B3 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_213_ordering_Ostrict__implies__not__eq,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less @ A2 @ B3 )
% 2.81/1.51 => ( A2 != B3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_214_ordering_Ostrict__implies__order,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less @ A2 @ B3 )
% 2.81/1.51 => ( Less_eq @ A2 @ B3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_215_ordering_Ostrict__iff__order,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less @ A2 @ B3 )
% 2.81/1.51 = ( ( Less_eq @ A2 @ B3 )
% 2.81/1.51 & ( A2 != B3 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_216_ordering_Oorder__iff__strict,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less_eq @ A2 @ B3 )
% 2.81/1.51 = ( ( Less @ A2 @ B3 )
% 2.81/1.51 | ( A2 = B3 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_217_ordering_Ostrict__trans2,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less @ A2 @ B3 )
% 2.81/1.51 => ( ( Less_eq @ B3 @ C )
% 2.81/1.51 => ( Less @ A2 @ C ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_218_ordering_Ostrict__trans1,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less_eq @ A2 @ B3 )
% 2.81/1.51 => ( ( Less @ B3 @ C )
% 2.81/1.51 => ( Less @ A2 @ C ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_219_ordering_Ostrict__trans,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less @ A2 @ B3 )
% 2.81/1.51 => ( ( Less @ B3 @ C )
% 2.81/1.51 => ( Less @ A2 @ C ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_220_ordering_Oantisym,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less_eq @ A2 @ B3 )
% 2.81/1.51 => ( ( Less_eq @ B3 @ A2 )
% 2.81/1.51 => ( A2 = B3 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_221_ordering_Oirrefl,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ~ ( Less @ A2 @ A2 ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_222_ordering_Otrans,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less_eq @ A2 @ B3 )
% 2.81/1.51 => ( ( Less_eq @ B3 @ C )
% 2.81/1.51 => ( Less_eq @ A2 @ C ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_223_ordering_Ointro,axiom,
% 2.81/1.51 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 2.81/1.51 ( ! [A4: A,B4: A] :
% 2.81/1.51 ( ( Less @ A4 @ B4 )
% 2.81/1.51 = ( ( Less_eq @ A4 @ B4 )
% 2.81/1.51 & ( A4 != B4 ) ) )
% 2.81/1.51 => ( ! [A4: A] : ( Less_eq @ A4 @ A4 )
% 2.81/1.51 => ( ! [A4: A,B4: A] :
% 2.81/1.51 ( ( Less_eq @ A4 @ B4 )
% 2.81/1.51 => ( ( Less_eq @ B4 @ A4 )
% 2.81/1.51 => ( A4 = B4 ) ) )
% 2.81/1.51 => ( ! [A4: A,B4: A,C4: A] :
% 2.81/1.51 ( ( Less_eq @ A4 @ B4 )
% 2.81/1.51 => ( ( Less_eq @ B4 @ C4 )
% 2.81/1.51 => ( Less_eq @ A4 @ C4 ) ) )
% 2.81/1.51 => ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_224_ordering_Orefl,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( Less_eq @ A2 @ A2 ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_225_ordering_Oasym,axiom,
% 2.81/1.51 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
% 2.81/1.51 ( ( ordering @ A @ Less_eq @ Less )
% 2.81/1.51 => ( ( Less @ A2 @ B3 )
% 2.81/1.51 => ~ ( Less @ B3 @ A2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_226_ordering__def,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( ordering @ A )
% 2.81/1.51 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 2.81/1.51 ( ! [A6: A,B6: A] :
% 2.81/1.51 ( ( Less2 @ A6 @ B6 )
% 2.81/1.51 = ( ( Less_eq2 @ A6 @ B6 )
% 2.81/1.51 & ( A6 != B6 ) ) )
% 2.81/1.51 & ! [A6: A] : ( Less_eq2 @ A6 @ A6 )
% 2.81/1.51 & ! [A6: A,B6: A] :
% 2.81/1.51 ( ( Less_eq2 @ A6 @ B6 )
% 2.81/1.51 => ( ( Less_eq2 @ B6 @ A6 )
% 2.81/1.51 => ( A6 = B6 ) ) )
% 2.81/1.51 & ! [A6: A,B6: A,C6: A] :
% 2.81/1.51 ( ( Less_eq2 @ A6 @ B6 )
% 2.81/1.51 => ( ( Less_eq2 @ B6 @ C6 )
% 2.81/1.51 => ( Less_eq2 @ A6 @ C6 ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_227_not__gr__zero,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [N: A] :
% 2.81/1.51 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.81/1.51 = ( N
% 2.81/1.51 = ( zero_zero @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_228_le__zero__eq,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [N: A] :
% 2.81/1.51 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.81/1.51 = ( N
% 2.81/1.51 = ( zero_zero @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_229_zero__le,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_230_zero__less__iff__neq__zero,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [N: A] :
% 2.81/1.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.81/1.51 = ( N
% 2.81/1.51 != ( zero_zero @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_231_gr__implies__not__zero,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [M: A,N: A] :
% 2.81/1.51 ( ( ord_less @ A @ M @ N )
% 2.81/1.51 => ( N
% 2.81/1.51 != ( zero_zero @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_232_not__less__zero,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [N: A] :
% 2.81/1.51 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_233_gr__zeroI,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [N: A] :
% 2.81/1.51 ( ( N
% 2.81/1.51 != ( zero_zero @ A ) )
% 2.81/1.51 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_234_dag_Osize__gen_I1_J,axiom,
% 2.81/1.51 ( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
% 2.81/1.51 = ( zero_zero @ nat ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_235_less__numeral__extra_I3_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_236_le__numeral__extra_I3_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_237_sgn__greater,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
% 2.81/1.51 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_238_sgn__less,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.81/1.51 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_239_sgn__sgn,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
% 2.81/1.51 = ( sgn_sgn @ A @ A2 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_240_sgn__0__0,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( ( sgn_sgn @ A @ A2 )
% 2.81/1.51 = ( zero_zero @ A ) )
% 2.81/1.51 = ( A2
% 2.81/1.51 = ( zero_zero @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_241_sgn__pos,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [A2: A] :
% 2.81/1.51 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.81/1.51 => ( ( sgn_sgn @ A @ A2 )
% 2.81/1.51 = ( one_one @ A ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_242_of__bool__eq_I1_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( zero_neq_one_of_bool @ A @ $false )
% 2.81/1.51 = ( zero_zero @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_243_of__bool__eq_I2_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( zero_neq_one_of_bool @ A @ $true )
% 2.81/1.51 = ( one_one @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_244_not__one__less__zero,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.81/1.51 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_245_zero__less__one,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_less_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_246_zero__le__one,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_247_not__one__le__zero,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.81/1.51 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_248_split__of__bool__asm,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P: A > $o,P3: $o] :
% 2.81/1.51 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 2.81/1.51 = ( ~ ( ( P3
% 2.81/1.51 & ~ ( P @ ( one_one @ A ) ) )
% 2.81/1.51 | ( ~ P3
% 2.81/1.51 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_249_split__of__bool,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P: A > $o,P3: $o] :
% 2.81/1.51 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 2.81/1.51 = ( ( P3
% 2.81/1.51 => ( P @ ( one_one @ A ) ) )
% 2.81/1.51 & ( ~ P3
% 2.81/1.51 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_250_zero__neq__one,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( zero_zero @ A )
% 2.81/1.51 != ( one_one @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_251_of__bool__def,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ( zero_neq_one_of_bool @ A )
% 2.81/1.51 = ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_252_of__bool__eq__iff,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.81/1.51 => ! [P3: $o,Q3: $o] :
% 2.81/1.51 ( ( ( zero_neq_one_of_bool @ A @ P3 )
% 2.81/1.51 = ( zero_neq_one_of_bool @ A @ Q3 ) )
% 2.81/1.51 = ( P3 = Q3 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_253_le__numeral__extra_I4_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_254_less__numeral__extra_I4_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(fact_255_le__numeral__extra_I2_J,axiom,
% 2.81/1.51 ! [A: $tType] :
% 2.81/1.51 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.81/1.51 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.81/1.51 ! [A7: $tType,A8: $tType] :
% 2.81/1.51 ( ( order_bot @ A8 @ ( type2 @ A8 ) )
% 2.81/1.51 => ( order_bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.81/1.51 ! [A7: $tType,A8: $tType] :
% 2.81/1.51 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.81/1.51 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.81/1.51 ! [A7: $tType,A8: $tType] :
% 2.81/1.51 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.81/1.51 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_fun___Orderings_Oord,axiom,
% 2.81/1.51 ! [A7: $tType,A8: $tType] :
% 2.81/1.51 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.81/1.51 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_fun___Orderings_Obot,axiom,
% 2.81/1.51 ! [A7: $tType,A8: $tType] :
% 2.81/1.51 ( ( bot @ A8 @ ( type2 @ A8 ) )
% 2.81/1.51 => ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.81/1.51 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.81/1.51 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.81/1.51 linord1659791738miring @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 2.81/1.51 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
% 2.81/1.51 zero_less_one @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.81/1.51 wellorder @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
% 2.81/1.51 order_bot @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 2.81/1.51 zero_neq_one @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
% 2.81/1.51 preorder @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.81/1.51 linorder @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.81/1.51 no_top @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
% 2.81/1.51 order @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
% 2.81/1.51 ord @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
% 2.81/1.51 bot @ nat @ ( type2 @ nat ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
% 2.81/1.51 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 2.81/1.51 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
% 2.81/1.51 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
% 2.81/1.51 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
% 2.81/1.51 ! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
% 2.81/1.51 order_bot @ $o @ ( type2 @ $o ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
% 2.81/1.51 preorder @ $o @ ( type2 @ $o ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
% 2.81/1.51 linorder @ $o @ ( type2 @ $o ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 2.81/1.51 order @ $o @ ( type2 @ $o ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 2.81/1.51 ord @ $o @ ( type2 @ $o ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
% 2.81/1.51 bot @ $o @ ( type2 @ $o ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_17,axiom,
% 2.81/1.51 preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_18,axiom,
% 2.81/1.51 order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.81/1.51
% 2.81/1.51 thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_19,axiom,
% 2.81/1.51 ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
% 2.81/1.51
% 2.81/1.51 thf(help_If_3_1_T,axiom,
% 2.81/1.51 ! [P: $o] :
% 2.81/1.51 ( ( P = $true )
% 2.81/1.51 | ( P = $false ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(help_If_2_1_T,axiom,
% 2.81/1.51 ! [A: $tType,X: A,Y: A] :
% 2.81/1.51 ( ( if @ A @ $false @ X @ Y )
% 2.81/1.51 = Y ) ).
% 2.81/1.51
% 2.81/1.51 thf(help_If_1_1_T,axiom,
% 2.81/1.51 ! [A: $tType,X: A,Y: A] :
% 2.81/1.51 ( ( if @ A @ $true @ X @ Y )
% 2.81/1.51 = X ) ).
% 2.81/1.51
% 2.81/1.51 thf(conj_0,hypothesis,
% 2.81/1.51 ( ~ ( member @ simpl_ref @ a @ ( binDag1380252983set_of @ l ) )
% 2.81/1.51 => ( ~ ( member @ simpl_ref @ a @ ( binDag1380252983set_of @ r ) )
% 2.81/1.51 => thesis ) ) ).
% 2.81/1.51
% 2.81/1.51 thf(conj_1,conjecture,
% 2.81/1.51 thesis ).
% 2.81/1.51 ------------------------
% 2.81/1.53 % (18459)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_18239 for (2999ds/4Mi)
% 2.81/1.53 % (18463)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_18239 for (2999ds/275Mi)
% 2.81/1.53 % (18464)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/18Mi)
% 2.81/1.53 % (18459)Instruction limit reached!
% 2.81/1.53 % (18459)------------------------------
% 2.81/1.53 % (18459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.53 % (18459)Termination reason: Unknown
% 2.81/1.53 % (18459)Termination phase: shuffling
% 2.81/1.53
% 2.81/1.53 % (18459)Memory used [KB]: 1407
% 2.81/1.53 % (18459)Time elapsed: 0.003 s
% 2.81/1.53 % (18459)Instructions burned: 5 (million)
% 2.81/1.53 % (18459)------------------------------
% 2.81/1.53 % (18459)------------------------------
% 2.81/1.53 % (18458)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/183Mi)
% 2.81/1.53 % (18464)Instruction limit reached!
% 2.81/1.53 % (18464)------------------------------
% 2.81/1.53 % (18464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.53 % (18464)Termination reason: Unknown
% 2.81/1.53 % (18464)Termination phase: Property scanning
% 2.81/1.53
% 2.81/1.53 % (18464)Memory used [KB]: 1791
% 2.81/1.53 % (18464)Time elapsed: 0.008 s
% 2.81/1.53 % (18464)Instructions burned: 19 (million)
% 2.81/1.53 % (18464)------------------------------
% 2.81/1.53 % (18464)------------------------------
% 2.81/1.53 % (18460)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_18239 for (2999ds/27Mi)
% 2.81/1.53 % (18461)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/2Mi)
% 2.81/1.53 % (18462)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_18239 for (2999ds/2Mi)
% 2.81/1.54 % (18461)Instruction limit reached!
% 2.81/1.54 % (18461)------------------------------
% 2.81/1.54 % (18461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.54 % (18461)Termination reason: Unknown
% 2.81/1.54 % (18461)Termination phase: shuffling
% 2.81/1.54
% 2.81/1.54 % (18461)Memory used [KB]: 1279
% 2.81/1.54 % (18461)Time elapsed: 0.004 s
% 2.81/1.54 % (18461)Instructions burned: 3 (million)
% 2.81/1.54 % (18461)------------------------------
% 2.81/1.54 % (18461)------------------------------
% 2.81/1.54 % (18462)Instruction limit reached!
% 2.81/1.54 % (18462)------------------------------
% 2.81/1.54 % (18462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.54 % (18462)Termination reason: Unknown
% 2.81/1.54 % (18462)Termination phase: shuffling
% 2.81/1.54
% 2.81/1.54 % (18462)Memory used [KB]: 1407
% 2.81/1.54 % (18462)Time elapsed: 0.004 s
% 2.81/1.54 % (18462)Instructions burned: 3 (million)
% 2.81/1.54 % (18462)------------------------------
% 2.81/1.54 % (18462)------------------------------
% 2.81/1.54 % (18465)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/3Mi)
% 2.81/1.54 % (18465)Instruction limit reached!
% 2.81/1.54 % (18465)------------------------------
% 2.81/1.54 % (18465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.54 % (18465)Termination reason: Unknown
% 2.81/1.54 % (18465)Termination phase: shuffling
% 2.81/1.54
% 2.81/1.54 % (18465)Memory used [KB]: 1407
% 2.81/1.54 % (18465)Time elapsed: 0.003 s
% 2.81/1.54 % (18465)Instructions burned: 4 (million)
% 2.81/1.54 % (18465)------------------------------
% 2.81/1.54 % (18465)------------------------------
% 2.81/1.54 % (18466)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_18239 for (2999ds/37Mi)
% 2.81/1.55 % (18463)Refutation not found, incomplete strategy
% 2.81/1.55 % (18463)------------------------------
% 2.81/1.55 % (18463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.55 % (18463)Termination reason: Refutation not found, incomplete strategy
% 2.81/1.55
% 2.81/1.55
% 2.81/1.55 % (18463)Memory used [KB]: 6524
% 2.81/1.55 % (18463)Time elapsed: 0.020 s
% 2.81/1.55 % (18463)Instructions burned: 54 (million)
% 2.81/1.55 % (18463)------------------------------
% 2.81/1.55 % (18463)------------------------------
% 2.81/1.55 % (18460)Instruction limit reached!
% 2.81/1.55 % (18460)------------------------------
% 2.81/1.55 % (18460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.55 % (18460)Termination reason: Unknown
% 2.81/1.55 % (18460)Termination phase: Property scanning
% 2.81/1.55
% 2.81/1.55 % (18460)Memory used [KB]: 1791
% 2.81/1.55 % (18460)Time elapsed: 0.018 s
% 2.81/1.55 % (18460)Instructions burned: 27 (million)
% 2.81/1.55 % (18460)------------------------------
% 2.81/1.55 % (18460)------------------------------
% 2.81/1.55 % (18469)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_18239 for (2999ds/1041Mi)
% 2.81/1.55 % (18468)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_18239 for (2999ds/3Mi)
% 2.81/1.55 % (18467)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_18239 for (2999ds/15Mi)
% 2.81/1.55 % (18466)Instruction limit reached!
% 2.81/1.55 % (18466)------------------------------
% 2.81/1.55 % (18466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.55 % (18466)Termination reason: Unknown
% 2.81/1.55 % (18466)Termination phase: Property scanning
% 2.81/1.55
% 2.81/1.55 % (18466)Memory used [KB]: 2046
% 2.81/1.55 % (18466)Time elapsed: 0.013 s
% 2.81/1.55 % (18466)Instructions burned: 37 (million)
% 2.81/1.55 % (18466)------------------------------
% 2.81/1.55 % (18466)------------------------------
% 2.81/1.55 % (18468)Instruction limit reached!
% 2.81/1.55 % (18468)------------------------------
% 2.81/1.55 % (18468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.55 % (18468)Termination reason: Unknown
% 2.81/1.55 % (18468)Termination phase: shuffling
% 2.81/1.55
% 2.81/1.55 % (18468)Memory used [KB]: 1407
% 2.81/1.55 % (18468)Time elapsed: 0.004 s
% 2.81/1.55 % (18468)Instructions burned: 4 (million)
% 2.81/1.55 % (18468)------------------------------
% 2.81/1.55 % (18468)------------------------------
% 2.81/1.56 % (18470)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/7Mi)
% 2.81/1.56 % (18470)Instruction limit reached!
% 2.81/1.56 % (18470)------------------------------
% 2.81/1.56 % (18470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.56 % (18470)Termination reason: Unknown
% 2.81/1.56 % (18470)Termination phase: shuffling
% 2.81/1.56
% 2.81/1.56 % (18470)Memory used [KB]: 1535
% 2.81/1.56 % (18470)Time elapsed: 0.004 s
% 2.81/1.56 % (18470)Instructions burned: 9 (million)
% 2.81/1.56 % (18470)------------------------------
% 2.81/1.56 % (18470)------------------------------
% 2.81/1.56 % (18471)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/16Mi)
% 2.81/1.56 % (18467)Instruction limit reached!
% 2.81/1.56 % (18467)------------------------------
% 2.81/1.56 % (18467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.56 % (18467)Termination reason: Unknown
% 2.81/1.56 % (18467)Termination phase: shuffling
% 2.81/1.56
% 2.81/1.56 % (18467)Memory used [KB]: 1663
% 2.81/1.56 % (18467)Time elapsed: 0.012 s
% 2.81/1.56 % (18467)Instructions burned: 16 (million)
% 2.81/1.56 % (18467)------------------------------
% 2.81/1.56 % (18467)------------------------------
% 2.81/1.56 % (18472)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_18239 for (2999ds/3Mi)
% 2.81/1.57 % (18471)Instruction limit reached!
% 2.81/1.57 % (18471)------------------------------
% 2.81/1.57 % (18471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.57 % (18471)Termination reason: Unknown
% 2.81/1.57 % (18471)Termination phase: shuffling
% 2.81/1.57
% 2.81/1.57 % (18471)Memory used [KB]: 1663
% 2.81/1.57 % (18471)Time elapsed: 0.007 s
% 2.81/1.57 % (18471)Instructions burned: 17 (million)
% 2.81/1.57 % (18471)------------------------------
% 2.81/1.57 % (18471)------------------------------
% 2.81/1.57 % (18472)Instruction limit reached!
% 2.81/1.57 % (18472)------------------------------
% 2.81/1.57 % (18472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.57 % (18472)Termination reason: Unknown
% 2.81/1.57 % (18472)Termination phase: shuffling
% 2.81/1.57
% 2.81/1.57 % (18472)Memory used [KB]: 1279
% 2.81/1.57 % (18472)Time elapsed: 0.003 s
% 2.81/1.57 % (18472)Instructions burned: 4 (million)
% 2.81/1.57 % (18472)------------------------------
% 2.81/1.57 % (18472)------------------------------
% 2.81/1.57 % (18473)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_18239 for (2999ds/3Mi)
% 2.81/1.57 % (18474)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/7Mi)
% 2.81/1.57 % (18473)Instruction limit reached!
% 2.81/1.57 % (18473)------------------------------
% 2.81/1.57 % (18473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.57 % (18473)Termination reason: Unknown
% 2.81/1.57 % (18473)Termination phase: shuffling
% 2.81/1.57
% 2.81/1.57 % (18473)Memory used [KB]: 1407
% 2.81/1.57 % (18473)Time elapsed: 0.003 s
% 2.81/1.57 % (18473)Instructions burned: 3 (million)
% 2.81/1.57 % (18473)------------------------------
% 2.81/1.57 % (18473)------------------------------
% 2.81/1.57 % (18474)Instruction limit reached!
% 2.81/1.57 % (18474)------------------------------
% 2.81/1.57 % (18474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.57 % (18474)Termination reason: Unknown
% 2.81/1.57 % (18474)Termination phase: shuffling
% 2.81/1.57
% 2.81/1.57 % (18474)Memory used [KB]: 1535
% 2.81/1.57 % (18474)Time elapsed: 0.004 s
% 2.81/1.57 % (18474)Instructions burned: 9 (million)
% 2.81/1.57 % (18474)------------------------------
% 2.81/1.57 % (18474)------------------------------
% 2.81/1.58 % (18476)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/4Mi)
% 2.81/1.58 % (18477)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/18Mi)
% 2.81/1.58 % (18476)Instruction limit reached!
% 2.81/1.58 % (18476)------------------------------
% 2.81/1.58 % (18476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.58 % (18476)Termination reason: Unknown
% 2.81/1.58 % (18476)Termination phase: shuffling
% 2.81/1.58
% 2.81/1.58 % (18476)Memory used [KB]: 1407
% 2.81/1.58 % (18476)Time elapsed: 0.003 s
% 2.81/1.58 % (18476)Instructions burned: 4 (million)
% 2.81/1.58 % (18476)------------------------------
% 2.81/1.58 % (18476)------------------------------
% 2.81/1.58 % (18478)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/710Mi)
% 2.81/1.58 % (18479)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/6Mi)
% 2.81/1.58 % (18475)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/3Mi)
% 2.81/1.58 % (18479)Instruction limit reached!
% 2.81/1.58 % (18479)------------------------------
% 2.81/1.58 % (18479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.58 % (18479)Termination reason: Unknown
% 2.81/1.58 % (18479)Termination phase: shuffling
% 2.81/1.58
% 2.81/1.58 % (18479)Memory used [KB]: 1407
% 2.81/1.58 % (18479)Time elapsed: 0.003 s
% 2.81/1.58 % (18479)Instructions burned: 6 (million)
% 2.81/1.58 % (18479)------------------------------
% 2.81/1.58 % (18479)------------------------------
% 2.81/1.58 % (18475)Instruction limit reached!
% 2.81/1.58 % (18475)------------------------------
% 2.81/1.58 % (18475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.58 % (18475)Termination reason: Unknown
% 2.81/1.58 % (18475)Termination phase: shuffling
% 2.81/1.58
% 2.81/1.58 % (18475)Memory used [KB]: 1407
% 2.81/1.58 % (18475)Time elapsed: 0.004 s
% 2.81/1.58 % (18475)Instructions burned: 4 (million)
% 2.81/1.58 % (18475)------------------------------
% 2.81/1.58 % (18475)------------------------------
% 2.81/1.58 % (18477)Instruction limit reached!
% 2.81/1.58 % (18477)------------------------------
% 2.81/1.58 % (18477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.58 % (18477)Termination reason: Unknown
% 2.81/1.58 % (18477)Termination phase: shuffling
% 2.81/1.58
% 2.81/1.58 % (18477)Memory used [KB]: 1663
% 2.81/1.58 % (18477)Time elapsed: 0.008 s
% 2.81/1.58 % (18477)Instructions burned: 18 (million)
% 2.81/1.58 % (18477)------------------------------
% 2.81/1.58 % (18477)------------------------------
% 2.81/1.59 % (18480)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_18239 for (2999ds/902Mi)
% 2.81/1.59 % (18481)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_18239 for (2999ds/21Mi)
% 2.81/1.59 % (18483)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_18239 for (2999ds/6Mi)
% 2.81/1.59 % (18483)Instruction limit reached!
% 2.81/1.59 % (18483)------------------------------
% 2.81/1.59 % (18483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.59 % (18483)Termination reason: Unknown
% 2.81/1.59 % (18483)Termination phase: shuffling
% 2.81/1.59
% 2.81/1.59 % (18483)Memory used [KB]: 1407
% 2.81/1.59 % (18483)Time elapsed: 0.004 s
% 2.81/1.59 % (18483)Instructions burned: 7 (million)
% 2.81/1.59 % (18483)------------------------------
% 2.81/1.59 % (18483)------------------------------
% 2.85/1.60 % (18481)Instruction limit reached!
% 2.85/1.60 % (18481)------------------------------
% 2.85/1.60 % (18481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.60 % (18481)Termination reason: Unknown
% 2.85/1.60 % (18481)Termination phase: Property scanning
% 2.85/1.60
% 2.85/1.60 % (18481)Memory used [KB]: 1791
% 2.85/1.60 % (18481)Time elapsed: 0.008 s
% 2.85/1.60 % (18481)Instructions burned: 21 (million)
% 2.85/1.60 % (18481)------------------------------
% 2.85/1.60 % (18481)------------------------------
% 2.85/1.60 % (18482)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/5Mi)
% 2.85/1.60 % (18458)Instruction limit reached!
% 2.85/1.60 % (18458)------------------------------
% 2.85/1.60 % (18458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.60 % (18458)Termination reason: Unknown
% 2.85/1.60 % (18458)Termination phase: Saturation
% 2.85/1.60
% 2.85/1.60 % (18458)Memory used [KB]: 7036
% 2.85/1.60 % (18458)Time elapsed: 0.071 s
% 2.85/1.60 % (18458)Instructions burned: 183 (million)
% 2.85/1.60 % (18458)------------------------------
% 2.85/1.60 % (18458)------------------------------
% 2.85/1.60 % (18482)Instruction limit reached!
% 2.85/1.60 % (18482)------------------------------
% 2.85/1.60 % (18482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.60 % (18482)Termination reason: Unknown
% 2.85/1.60 % (18482)Termination phase: shuffling
% 2.85/1.60
% 2.85/1.60 % (18482)Memory used [KB]: 1407
% 2.85/1.60 % (18482)Time elapsed: 0.005 s
% 2.85/1.60 % (18482)Instructions burned: 6 (million)
% 2.85/1.60 % (18482)------------------------------
% 2.85/1.60 % (18482)------------------------------
% 2.85/1.60 % (18484)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/377Mi)
% 2.85/1.61 % (18485)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_18239 for (2999ds/779Mi)
% 2.85/1.61 % (18486)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_18239 for (2999ds/19Mi)
% 2.85/1.62 % (18486)Instruction limit reached!
% 2.85/1.62 % (18486)------------------------------
% 2.85/1.62 % (18486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.62 % (18486)Termination reason: Unknown
% 2.85/1.62 % (18486)Termination phase: shuffling
% 2.85/1.62
% 2.85/1.62 % (18486)Memory used [KB]: 1663
% 2.85/1.62 % (18486)Time elapsed: 0.008 s
% 2.85/1.62 % (18486)Instructions burned: 21 (million)
% 2.85/1.62 % (18486)------------------------------
% 2.85/1.62 % (18486)------------------------------
% 2.85/1.62 % (18487)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_18239 for (2999ds/879Mi)
% 2.85/1.63 % (18488)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_18239 for (2998ds/17Mi)
% 2.85/1.63 % (18488)Instruction limit reached!
% 2.85/1.63 % (18488)------------------------------
% 2.85/1.63 % (18488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.63 % (18488)Termination reason: Unknown
% 2.85/1.63 % (18488)Termination phase: shuffling
% 2.85/1.63
% 2.85/1.63 % (18488)Memory used [KB]: 1663
% 2.85/1.63 % (18488)Time elapsed: 0.008 s
% 2.85/1.63 % (18488)Instructions burned: 17 (million)
% 2.85/1.63 % (18488)------------------------------
% 2.85/1.63 % (18488)------------------------------
% 2.85/1.64 % (18489)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18239 for (2998ds/3Mi)
% 2.85/1.65 % (18489)Instruction limit reached!
% 2.85/1.65 % (18489)------------------------------
% 2.85/1.65 % (18489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.65 % (18489)Termination reason: Unknown
% 2.85/1.65 % (18489)Termination phase: shuffling
% 2.85/1.65
% 2.85/1.65 % (18489)Memory used [KB]: 1407
% 2.85/1.65 % (18489)Time elapsed: 0.003 s
% 2.85/1.65 % (18489)Instructions burned: 3 (million)
% 2.85/1.65 % (18489)------------------------------
% 2.85/1.65 % (18489)------------------------------
% 2.85/1.66 % (18490)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_18239 for (2998ds/30Mi)
% 2.85/1.67 % (18490)Instruction limit reached!
% 2.85/1.67 % (18490)------------------------------
% 2.85/1.67 % (18490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.67 % (18490)Termination reason: Unknown
% 2.85/1.67 % (18490)Termination phase: Preprocessing 1
% 2.85/1.67
% 2.85/1.67 % (18490)Memory used [KB]: 1791
% 2.85/1.67 % (18490)Time elapsed: 0.012 s
% 2.85/1.67 % (18490)Instructions burned: 31 (million)
% 2.85/1.67 % (18490)------------------------------
% 2.85/1.67 % (18490)------------------------------
% 2.85/1.68 % (18491)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_18239 for (2998ds/127Mi)
% 2.85/1.71 % (18484)Instruction limit reached!
% 2.85/1.71 % (18484)------------------------------
% 2.85/1.71 % (18484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.71 % (18484)Termination reason: Unknown
% 2.85/1.71 % (18484)Termination phase: Saturation
% 2.85/1.71
% 2.85/1.71 % (18484)Memory used [KB]: 8059
% 2.85/1.71 % (18484)Time elapsed: 0.110 s
% 2.85/1.71 % (18484)Instructions burned: 377 (million)
% 2.85/1.71 % (18484)------------------------------
% 2.85/1.71 % (18484)------------------------------
% 2.85/1.72 % (18491)Instruction limit reached!
% 2.85/1.72 % (18491)------------------------------
% 2.85/1.72 % (18491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.72 % (18491)Termination reason: Unknown
% 2.85/1.72 % (18491)Termination phase: Saturation
% 2.85/1.72
% 2.85/1.72 % (18491)Memory used [KB]: 6908
% 2.85/1.72 % (18491)Time elapsed: 0.041 s
% 2.85/1.72 % (18491)Instructions burned: 127 (million)
% 2.85/1.72 % (18491)------------------------------
% 2.85/1.72 % (18491)------------------------------
% 2.85/1.72 % (18492)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_18239 for (2998ds/100Mi)
% 2.85/1.73 % (18493)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_18239 for (2998ds/3Mi)
% 2.85/1.73 % (18493)Instruction limit reached!
% 2.85/1.73 % (18493)------------------------------
% 2.85/1.73 % (18493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.73 % (18493)Termination reason: Unknown
% 2.85/1.73 % (18493)Termination phase: shuffling
% 2.85/1.73
% 2.85/1.73 % (18493)Memory used [KB]: 1407
% 2.85/1.73 % (18493)Time elapsed: 0.003 s
% 2.85/1.73 % (18493)Instructions burned: 4 (million)
% 2.85/1.73 % (18493)------------------------------
% 2.85/1.73 % (18493)------------------------------
% 2.85/1.74 % (18494)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_18239 for (2997ds/20Mi)
% 2.85/1.74 % (18494)Instruction limit reached!
% 2.85/1.74 % (18494)------------------------------
% 2.85/1.74 % (18494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.74 % (18494)Termination reason: Unknown
% 2.85/1.74 % (18494)Termination phase: Property scanning
% 2.85/1.74
% 2.85/1.74 % (18494)Memory used [KB]: 1791
% 2.85/1.74 % (18494)Time elapsed: 0.009 s
% 2.85/1.74 % (18494)Instructions burned: 23 (million)
% 2.85/1.74 % (18494)------------------------------
% 2.85/1.74 % (18494)------------------------------
% 2.85/1.75 % (18492)Instruction limit reached!
% 2.85/1.75 % (18492)------------------------------
% 2.85/1.75 % (18492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.75 % (18492)Termination reason: Unknown
% 2.85/1.75 % (18492)Termination phase: Property scanning
% 2.85/1.75
% 2.85/1.75 % (18492)Memory used [KB]: 2174
% 2.85/1.75 % (18492)Time elapsed: 0.028 s
% 2.85/1.75 % (18492)Instructions burned: 102 (million)
% 2.85/1.75 % (18492)------------------------------
% 2.85/1.75 % (18492)------------------------------
% 2.85/1.75 % (18495)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_18239 for (2997ds/86Mi)
% 2.85/1.76 % (18496)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_18239 for (2997ds/107Mi)
% 2.85/1.78 % (18495)Instruction limit reached!
% 2.85/1.78 % (18495)------------------------------
% 2.85/1.78 % (18495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.78 % (18495)Termination reason: Unknown
% 2.85/1.78 % (18495)Termination phase: Saturation
% 2.85/1.78
% 2.85/1.78 % (18495)Memory used [KB]: 6780
% 2.85/1.78 % (18495)Time elapsed: 0.027 s
% 2.85/1.78 % (18495)Instructions burned: 88 (million)
% 2.85/1.78 % (18495)------------------------------
% 2.85/1.78 % (18495)------------------------------
% 2.85/1.79 % (18497)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_18239 for (2997ds/39Mi)
% 2.85/1.79 % (18496)Instruction limit reached!
% 2.85/1.79 % (18496)------------------------------
% 2.85/1.79 % (18496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.79 % (18496)Termination reason: Unknown
% 2.85/1.79 % (18496)Termination phase: Saturation
% 2.85/1.79
% 2.85/1.79 % (18496)Memory used [KB]: 6780
% 2.85/1.79 % (18496)Time elapsed: 0.034 s
% 2.85/1.79 % (18496)Instructions burned: 109 (million)
% 2.85/1.79 % (18496)------------------------------
% 2.85/1.79 % (18496)------------------------------
% 2.85/1.79 % (18478)Instruction limit reached!
% 2.85/1.79 % (18478)------------------------------
% 2.85/1.79 % (18478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.79 % (18478)Termination reason: Unknown
% 2.85/1.79 % (18478)Termination phase: Saturation
% 2.85/1.79
% 2.85/1.79 % (18478)Memory used [KB]: 7931
% 2.85/1.79 % (18478)Time elapsed: 0.218 s
% 2.85/1.79 % (18478)Instructions burned: 711 (million)
% 2.85/1.79 % (18478)------------------------------
% 2.85/1.79 % (18478)------------------------------
% 2.85/1.80 % (18497)Instruction limit reached!
% 2.85/1.80 % (18497)------------------------------
% 2.85/1.80 % (18497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.80 % (18497)Termination reason: Unknown
% 2.85/1.80 % (18497)Termination phase: Property scanning
% 2.85/1.80
% 2.85/1.80 % (18497)Memory used [KB]: 2046
% 2.85/1.80 % (18497)Time elapsed: 0.012 s
% 2.85/1.80 % (18497)Instructions burned: 41 (million)
% 2.85/1.80 % (18497)------------------------------
% 2.85/1.80 % (18497)------------------------------
% 2.85/1.80 % (18498)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_18239 for (2997ds/448Mi)
% 2.85/1.80 % (18499)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_18239 for (2997ds/46Mi)
% 2.85/1.81 % (18500)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_18239 for (2997ds/98Mi)
% 2.85/1.82 % (18499)Instruction limit reached!
% 2.85/1.82 % (18499)------------------------------
% 2.85/1.82 % (18499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.82 % (18499)Termination reason: Unknown
% 2.85/1.82 % (18499)Termination phase: Preprocessing 3
% 2.85/1.82
% 2.85/1.82 % (18499)Memory used [KB]: 2046
% 2.85/1.82 % (18499)Time elapsed: 0.016 s
% 2.85/1.82 % (18499)Instructions burned: 46 (million)
% 2.85/1.82 % (18499)------------------------------
% 2.85/1.82 % (18499)------------------------------
% 2.85/1.83 % (18501)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/507Mi)
% 2.85/1.84 % (18500)Instruction limit reached!
% 2.85/1.84 % (18500)------------------------------
% 2.85/1.84 % (18500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.84 % (18500)Termination reason: Unknown
% 2.85/1.84 % (18500)Termination phase: Property scanning
% 2.85/1.84
% 2.85/1.84 % (18500)Memory used [KB]: 2174
% 2.85/1.84 % (18500)Time elapsed: 0.027 s
% 2.85/1.84 % (18500)Instructions burned: 98 (million)
% 2.85/1.84 % (18500)------------------------------
% 2.85/1.84 % (18500)------------------------------
% 2.85/1.85 % (18502)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/149Mi)
% 2.85/1.85 % (18485)Instruction limit reached!
% 2.85/1.85 % (18485)------------------------------
% 2.85/1.85 % (18485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.85 % (18485)Termination reason: Unknown
% 2.85/1.85 % (18485)Termination phase: Saturation
% 2.85/1.85
% 2.85/1.85 % (18485)Memory used [KB]: 8315
% 2.85/1.85 % (18485)Time elapsed: 0.241 s
% 2.85/1.85 % (18485)Instructions burned: 781 (million)
% 2.85/1.85 % (18485)------------------------------
% 2.85/1.85 % (18485)------------------------------
% 2.85/1.86 % (18503)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/39Mi)
% 2.85/1.87 % (18480)Instruction limit reached!
% 2.85/1.87 % (18480)------------------------------
% 2.85/1.87 % (18480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.87 % (18480)Termination reason: Unknown
% 2.85/1.87 % (18480)Termination phase: Saturation
% 2.85/1.87
% 2.85/1.87 % (18480)Memory used [KB]: 9083
% 2.85/1.87 % (18480)Time elapsed: 0.283 s
% 2.85/1.87 % (18480)Instructions burned: 903 (million)
% 2.85/1.87 % (18480)------------------------------
% 2.85/1.87 % (18480)------------------------------
% 2.85/1.87 % (18503)Instruction limit reached!
% 2.85/1.87 % (18503)------------------------------
% 2.85/1.87 % (18503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.87 % (18503)Termination reason: Unknown
% 2.85/1.87 % (18503)Termination phase: Preprocessing 3
% 2.85/1.87
% 2.85/1.87 % (18503)Memory used [KB]: 2046
% 2.85/1.87 % (18503)Time elapsed: 0.014 s
% 2.85/1.87 % (18503)Instructions burned: 40 (million)
% 2.85/1.87 % (18503)------------------------------
% 2.85/1.87 % (18503)------------------------------
% 2.85/1.88 % (18504)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/6Mi)
% 2.85/1.88 % (18505)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/1087Mi)
% 2.85/1.88 % (18504)Instruction limit reached!
% 2.85/1.88 % (18504)------------------------------
% 2.85/1.88 % (18504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.88 % (18504)Termination reason: Unknown
% 2.85/1.88 % (18504)Termination phase: shuffling
% 2.85/1.88
% 2.85/1.88 % (18504)Memory used [KB]: 1407
% 2.85/1.88 % (18504)Time elapsed: 0.003 s
% 2.85/1.88 % (18504)Instructions burned: 7 (million)
% 2.85/1.88 % (18504)------------------------------
% 2.85/1.88 % (18504)------------------------------
% 2.85/1.89 % (18502)Instruction limit reached!
% 2.85/1.89 % (18502)------------------------------
% 2.85/1.89 % (18502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.89 % (18502)Termination reason: Unknown
% 2.85/1.89 % (18502)Termination phase: Saturation
% 2.85/1.89
% 2.85/1.89 % (18502)Memory used [KB]: 6908
% 2.85/1.89 % (18502)Time elapsed: 0.043 s
% 2.85/1.89 % (18502)Instructions burned: 149 (million)
% 2.85/1.89 % (18502)------------------------------
% 2.85/1.89 % (18502)------------------------------
% 2.85/1.89 % (18506)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/47Mi)
% 2.85/1.89 % (18469)Instruction limit reached!
% 2.85/1.89 % (18469)------------------------------
% 2.85/1.89 % (18469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.89 % (18469)Termination reason: Unknown
% 2.85/1.89 % (18469)Termination phase: Saturation
% 2.85/1.89
% 2.85/1.89 % (18469)Memory used [KB]: 9338
% 2.85/1.89 % (18469)Time elapsed: 0.344 s
% 2.85/1.89 % (18469)Instructions burned: 1042 (million)
% 2.85/1.89 % (18469)------------------------------
% 2.85/1.89 % (18469)------------------------------
% 2.85/1.90 % (18507)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/44Mi)
% 2.85/1.90 % (18508)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/432Mi)
% 2.85/1.90 % (18506)Instruction limit reached!
% 2.85/1.90 % (18506)------------------------------
% 2.85/1.90 % (18506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.90 % (18506)Termination reason: Unknown
% 2.85/1.90 % (18506)Termination phase: Property scanning
% 2.85/1.90
% 2.85/1.90 % (18506)Memory used [KB]: 2174
% 2.85/1.90 % (18506)Time elapsed: 0.016 s
% 2.85/1.90 % (18506)Instructions burned: 50 (million)
% 2.85/1.90 % (18506)------------------------------
% 2.85/1.90 % (18506)------------------------------
% 2.85/1.91 % (18507)Instruction limit reached!
% 2.85/1.91 % (18507)------------------------------
% 2.85/1.91 % (18507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.91 % (18507)Termination reason: Unknown
% 2.85/1.91 % (18507)Termination phase: Property scanning
% 2.85/1.91
% 2.85/1.91 % (18507)Memory used [KB]: 2046
% 2.85/1.91 % (18507)Time elapsed: 0.013 s
% 2.85/1.91 % (18507)Instructions burned: 45 (million)
% 2.85/1.91 % (18507)------------------------------
% 2.85/1.91 % (18507)------------------------------
% 2.85/1.91 % (18509)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/62Mi)
% 2.85/1.92 % (18510)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_18239 for (2996ds/31Mi)
% 2.85/1.93 % (18510)Instruction limit reached!
% 2.85/1.93 % (18510)------------------------------
% 2.85/1.93 % (18510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.93 % (18510)Termination reason: Unknown
% 2.85/1.93 % (18510)Termination phase: Preprocessing 1
% 2.85/1.93
% 2.85/1.93 % (18510)Memory used [KB]: 1791
% 2.85/1.93 % (18510)Time elapsed: 0.011 s
% 2.85/1.93 % (18510)Instructions burned: 32 (million)
% 2.85/1.93 % (18510)------------------------------
% 2.85/1.93 % (18510)------------------------------
% 2.85/1.93 % (18509)Instruction limit reached!
% 2.85/1.93 % (18509)------------------------------
% 2.85/1.93 % (18509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.93 % (18509)Termination reason: Unknown
% 2.85/1.93 % (18509)Termination phase: Preprocessing 1
% 2.85/1.93
% 2.85/1.93 % (18509)Memory used [KB]: 2046
% 2.85/1.93 % (18509)Time elapsed: 0.018 s
% 2.85/1.93 % (18509)Instructions burned: 62 (million)
% 2.85/1.93 % (18509)------------------------------
% 2.85/1.93 % (18509)------------------------------
% 2.85/1.94 % (18487)Instruction limit reached!
% 2.85/1.94 % (18487)------------------------------
% 2.85/1.94 % (18487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.94 % (18487)Termination reason: Unknown
% 2.85/1.94 % (18487)Termination phase: Saturation
% 2.85/1.94
% 2.85/1.94 % (18487)Memory used [KB]: 8059
% 2.85/1.94 % (18487)Time elapsed: 0.318 s
% 2.85/1.94 % (18487)Instructions burned: 880 (million)
% 2.85/1.94 % (18487)------------------------------
% 2.85/1.94 % (18487)------------------------------
% 2.85/1.94 % (18498)Instruction limit reached!
% 2.85/1.94 % (18498)------------------------------
% 2.85/1.94 % (18498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.94 % (18498)Termination reason: Unknown
% 2.85/1.94 % (18498)Termination phase: Saturation
% 2.85/1.94
% 2.85/1.94 % (18498)Memory used [KB]: 7803
% 2.85/1.94 % (18498)Time elapsed: 0.135 s
% 2.85/1.94 % (18498)Instructions burned: 450 (million)
% 2.85/1.94 % (18498)------------------------------
% 2.85/1.94 % (18498)------------------------------
% 2.85/1.94 % (18511)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/96Mi)
% 2.85/1.94 % (18512)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/453Mi)
% 2.85/1.95 % (18514)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/83Mi)
% 2.85/1.95 % (18513)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/71Mi)
% 2.85/1.97 % (18511)Instruction limit reached!
% 2.85/1.97 % (18511)------------------------------
% 2.85/1.97 % (18511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.97 % (18511)Termination reason: Unknown
% 2.85/1.97 % (18511)Termination phase: Saturation
% 2.85/1.97
% 2.85/1.97 % (18511)Memory used [KB]: 6780
% 2.85/1.97 % (18511)Time elapsed: 0.030 s
% 2.85/1.97 % (18511)Instructions burned: 99 (million)
% 2.85/1.97 % (18511)------------------------------
% 2.85/1.97 % (18511)------------------------------
% 2.85/1.97 % (18513)Instruction limit reached!
% 2.85/1.97 % (18513)------------------------------
% 2.85/1.97 % (18513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.97 % (18513)Termination reason: Unknown
% 2.85/1.97 % (18513)Termination phase: Property scanning
% 2.85/1.97
% 2.85/1.97 % (18513)Memory used [KB]: 2174
% 2.85/1.97 % (18513)Time elapsed: 0.025 s
% 2.85/1.97 % (18513)Instructions burned: 74 (million)
% 2.85/1.97 % (18513)------------------------------
% 2.85/1.97 % (18513)------------------------------
% 2.85/1.97 % (18514)Instruction limit reached!
% 2.85/1.97 % (18514)------------------------------
% 2.85/1.97 % (18514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.97 % (18514)Termination reason: Unknown
% 2.85/1.97 % (18514)Termination phase: Saturation
% 2.85/1.97
% 2.85/1.97 % (18514)Memory used [KB]: 6652
% 2.85/1.97 % (18514)Time elapsed: 0.027 s
% 2.85/1.97 % (18514)Instructions burned: 86 (million)
% 2.85/1.97 % (18514)------------------------------
% 2.85/1.97 % (18514)------------------------------
% 2.85/1.98 % (18515)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/11Mi)
% 2.85/1.98 % (18515)Instruction limit reached!
% 2.85/1.98 % (18515)------------------------------
% 2.85/1.98 % (18515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.98 % (18515)Termination reason: Unknown
% 2.85/1.98 % (18515)Termination phase: shuffling
% 2.85/1.98
% 2.85/1.98 % (18515)Memory used [KB]: 1535
% 2.85/1.98 % (18515)Time elapsed: 0.004 s
% 2.85/1.98 % (18515)Instructions burned: 11 (million)
% 2.85/1.98 % (18515)------------------------------
% 2.85/1.98 % (18515)------------------------------
% 2.85/1.98 % (18516)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/41Mi)
% 2.85/1.98 % (18517)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/6Mi)
% 2.85/1.98 % (18517)Instruction limit reached!
% 2.85/1.98 % (18517)------------------------------
% 2.85/1.98 % (18517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.98 % (18517)Termination reason: Unknown
% 2.85/1.98 % (18517)Termination phase: shuffling
% 2.85/1.98
% 2.85/1.98 % (18517)Memory used [KB]: 1407
% 2.85/1.98 % (18517)Time elapsed: 0.003 s
% 2.85/1.98 % (18517)Instructions burned: 6 (million)
% 2.85/1.98 % (18517)------------------------------
% 2.85/1.98 % (18517)------------------------------
% 2.85/1.99 % (18518)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/150Mi)
% 2.85/1.99 % (18501)Instruction limit reached!
% 2.85/1.99 % (18501)------------------------------
% 2.85/1.99 % (18501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.99 % (18501)Termination reason: Unknown
% 2.85/1.99 % (18501)Termination phase: Saturation
% 2.85/1.99
% 2.85/1.99 % (18501)Memory used [KB]: 7931
% 2.85/1.99 % (18501)Time elapsed: 0.161 s
% 2.85/1.99 % (18501)Instructions burned: 508 (million)
% 2.85/1.99 % (18501)------------------------------
% 2.85/1.99 % (18501)------------------------------
% 2.85/1.99 % (18516)Instruction limit reached!
% 2.85/1.99 % (18516)------------------------------
% 2.85/1.99 % (18516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.99 % (18516)Termination reason: Unknown
% 2.85/1.99 % (18516)Termination phase: Property scanning
% 2.85/1.99
% 2.85/1.99 % (18516)Memory used [KB]: 2046
% 2.85/1.99 % (18516)Time elapsed: 0.014 s
% 2.85/1.99 % (18516)Instructions burned: 42 (million)
% 2.85/1.99 % (18516)------------------------------
% 2.85/1.99 % (18516)------------------------------
% 2.85/1.99 % (18519)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/343Mi)
% 2.85/2.00 % (18520)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/109Mi)
% 2.85/2.00 % (18521)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_18239 for (2995ds/253Mi)
% 6.27/2.03 % (18518)Instruction limit reached!
% 6.27/2.03 % (18518)------------------------------
% 6.27/2.03 % (18518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.27/2.03 % (18518)Termination reason: Unknown
% 6.27/2.03 % (18518)Termination phase: Saturation
% 6.27/2.03
% 6.27/2.03 % (18518)Memory used [KB]: 6908
% 6.27/2.03 % (18518)Time elapsed: 0.042 s
% 6.27/2.03 % (18518)Instructions burned: 150 (million)
% 6.27/2.03 % (18518)------------------------------
% 6.27/2.03 % (18518)------------------------------
% 6.27/2.03 % (18520)Instruction limit reached!
% 6.27/2.03 % (18520)------------------------------
% 6.27/2.03 % (18520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.27/2.03 % (18520)Termination reason: Unknown
% 6.27/2.03 % (18520)Termination phase: Saturation
% 6.27/2.03
% 6.27/2.03 % (18520)Memory used [KB]: 6780
% 6.27/2.03 % (18520)Time elapsed: 0.033 s
% 6.27/2.03 % (18520)Instructions burned: 111 (million)
% 6.27/2.03 % (18520)------------------------------
% 6.27/2.03 % (18520)------------------------------
% 6.27/2.04 % (18508)Instruction limit reached!
% 6.27/2.04 % (18508)------------------------------
% 6.27/2.04 % (18508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.27/2.04 % (18508)Termination reason: Unknown
% 6.27/2.04 % (18508)Termination phase: Saturation
% 6.27/2.04
% 6.27/2.04 % (18508)Memory used [KB]: 3454
% 6.27/2.04 % (18508)Time elapsed: 0.135 s
% 6.27/2.04 % (18508)Instructions burned: 433 (million)
% 6.27/2.04 % (18508)------------------------------
% 6.27/2.04 % (18508)------------------------------
% 6.27/2.04 % (18522)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_18239 for (2994ds/140Mi)
% 6.27/2.04 % (18523)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_18239 for (2994ds/614Mi)
% 6.27/2.05 % (18524)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_18239 for (2994ds/202Mi)
% 6.27/2.06 % (18524)Refutation not found, incomplete strategy
% 6.27/2.06 % (18524)------------------------------
% 6.27/2.06 % (18524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.27/2.06 % (18524)Termination reason: Refutation not found, incomplete strategy
% 6.27/2.06
% 6.27/2.06
% 6.27/2.06 % (18524)Memory used [KB]: 6396
% 6.27/2.06 % (18524)Time elapsed: 0.016 s
% 6.27/2.06 % (18524)Instructions burned: 46 (million)
% 6.27/2.06 % (18524)------------------------------
% 6.27/2.06 % (18524)------------------------------
% 6.27/2.07 % (18525)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_18239 for (2994ds/339Mi)
% 6.27/2.07 % (18512)Instruction limit reached!
% 6.27/2.07 % (18512)------------------------------
% 6.27/2.07 % (18512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.27/2.07 % (18512)Termination reason: Unknown
% 6.27/2.07 % (18512)Termination phase: Saturation
% 6.27/2.07
% 6.27/2.07 % (18512)Memory used [KB]: 7675
% 6.27/2.07 % (18512)Time elapsed: 0.135 s
% 6.27/2.07 % (18512)Instructions burned: 454 (million)
% 6.27/2.07 % (18512)------------------------------
% 6.27/2.07 % (18512)------------------------------
% 6.27/2.08 % (18521)Instruction limit reached!
% 6.27/2.08 % (18521)------------------------------
% 6.27/2.08 % (18521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.27/2.08 % (18521)Termination reason: Unknown
% 6.27/2.08 % (18521)Termination phase: Saturation
% 6.27/2.08
% 6.27/2.08 % (18521)Memory used [KB]: 7036
% 6.27/2.08 % (18521)Time elapsed: 0.082 s
% 6.27/2.08 % (18521)Instructions burned: 253 (million)
% 6.27/2.08 % (18521)------------------------------
% 6.27/2.08 % (18521)------------------------------
% 6.27/2.08 % (18522)Instruction limit reached!
% 6.27/2.08 % (18522)------------------------------
% 6.27/2.08 % (18522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.27/2.08 % (18522)Termination reason: Unknown
% 6.27/2.08 % (18522)Termination phase: Saturation
% 6.27/2.08
% 6.27/2.08 % (18522)Memory used [KB]: 7036
% 6.27/2.08 % (18522)Time elapsed: 0.069 s
% 6.27/2.08 % (18522)Instructions burned: 140 (million)
% 6.27/2.08 % (18522)------------------------------
% 6.27/2.08 % (18522)------------------------------
% 6.27/2.08 % (18526)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=283:si=on:rtra=on_0 on DTF2THF_18239 for (2994ds/283Mi)
% 6.27/2.09 % (18527)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_18239 for (2994ds/134Mi)
% 6.27/2.09 % (18528)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_18239 for (2994ds/298Mi)
% 6.78/2.13 % (18527)Instruction limit reached!
% 6.78/2.13 % (18527)------------------------------
% 6.78/2.13 % (18527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.78/2.13 % (18527)Termination reason: Unknown
% 6.78/2.13 % (18527)Termination phase: Saturation
% 6.78/2.13
% 6.78/2.13 % (18527)Memory used [KB]: 6908
% 6.78/2.13 % (18527)Time elapsed: 0.041 s
% 6.78/2.13 % (18527)Instructions burned: 135 (million)
% 6.78/2.13 % (18527)------------------------------
% 6.78/2.13 % (18527)------------------------------
% 6.78/2.14 % (18529)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_18239 for (2993ds/192Mi)
% 6.78/2.15 % (18519)Instruction limit reached!
% 6.78/2.15 % (18519)------------------------------
% 6.78/2.15 % (18519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.78/2.15 % (18519)Termination reason: Unknown
% 6.78/2.15 % (18519)Termination phase: Saturation
% 6.78/2.15
% 6.78/2.15 % (18519)Memory used [KB]: 7164
% 6.78/2.15 % (18519)Time elapsed: 0.157 s
% 6.78/2.15 % (18519)Instructions burned: 343 (million)
% 6.78/2.15 % (18519)------------------------------
% 6.78/2.15 % (18519)------------------------------
% 6.78/2.16 % (18530)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_18239 for (2993ds/258Mi)
% 6.78/2.17 % (18526)Instruction limit reached!
% 6.78/2.17 % (18526)------------------------------
% 6.78/2.17 % (18526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.78/2.17 % (18526)Termination reason: Unknown
% 6.78/2.17 % (18526)Termination phase: Saturation
% 6.78/2.17
% 6.78/2.17 % (18526)Memory used [KB]: 7291
% 6.78/2.17 % (18526)Time elapsed: 0.089 s
% 6.78/2.17 % (18526)Instructions burned: 283 (million)
% 6.78/2.17 % (18526)------------------------------
% 6.78/2.17 % (18526)------------------------------
% 6.78/2.17 % (18525)Instruction limit reached!
% 6.78/2.17 % (18525)------------------------------
% 6.78/2.17 % (18525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.78/2.17 % (18525)Termination reason: Unknown
% 6.78/2.17 % (18525)Termination phase: Saturation
% 6.78/2.17
% 6.78/2.17 % (18525)Memory used [KB]: 7803
% 6.78/2.17 % (18525)Time elapsed: 0.105 s
% 6.78/2.17 % (18525)Instructions burned: 340 (million)
% 6.78/2.17 % (18525)------------------------------
% 6.78/2.17 % (18525)------------------------------
% 6.78/2.18 % (18528)Instruction limit reached!
% 6.78/2.18 % (18528)------------------------------
% 6.78/2.18 % (18528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.78/2.18 % (18528)Termination reason: Unknown
% 6.78/2.18 % (18528)Termination phase: Saturation
% 6.78/2.18
% 6.78/2.18 % (18528)Memory used [KB]: 7675
% 6.78/2.18 % (18528)Time elapsed: 0.086 s
% 6.78/2.18 % (18528)Instructions burned: 301 (million)
% 6.78/2.18 % (18528)------------------------------
% 6.78/2.18 % (18528)------------------------------
% 6.78/2.18 % (18531)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_18239 for (2993ds/90Mi)
% 6.78/2.18 % (18532)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_18239 for (2993ds/231Mi)
% 6.78/2.19 % (18533)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_18239 for (2993ds/458Mi)
% 7.26/2.20 % (18529)Instruction limit reached!
% 7.26/2.20 % (18529)------------------------------
% 7.26/2.20 % (18529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.26/2.20 % (18529)Termination reason: Unknown
% 7.26/2.20 % (18529)Termination phase: Saturation
% 7.26/2.20
% 7.26/2.20 % (18529)Memory used [KB]: 7803
% 7.26/2.20 % (18529)Time elapsed: 0.059 s
% 7.26/2.20 % (18529)Instructions burned: 195 (million)
% 7.26/2.20 % (18529)------------------------------
% 7.26/2.20 % (18529)------------------------------
% 7.26/2.21 % (18531)Instruction limit reached!
% 7.26/2.21 % (18531)------------------------------
% 7.26/2.21 % (18531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.26/2.21 % (18531)Termination reason: Unknown
% 7.26/2.21 % (18531)Termination phase: Saturation
% 7.26/2.21
% 7.26/2.21 % (18531)Memory used [KB]: 6780
% 7.26/2.21 % (18531)Time elapsed: 0.027 s
% 7.26/2.21 % (18531)Instructions burned: 90 (million)
% 7.26/2.21 % (18531)------------------------------
% 7.26/2.21 % (18531)------------------------------
% 7.26/2.21 % (18534)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_18239 for (2993ds/31Mi)
% 7.26/2.22 % (18523)Instruction limit reached!
% 7.26/2.22 % (18523)------------------------------
% 7.26/2.22 % (18523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.26/2.22 % (18523)Termination reason: Unknown
% 7.26/2.22 % (18523)Termination phase: Saturation
% 7.26/2.22
% 7.26/2.22 % (18523)Memory used [KB]: 8827
% 7.26/2.22 % (18523)Time elapsed: 0.202 s
% 7.26/2.22 % (18523)Instructions burned: 614 (million)
% 7.26/2.22 % (18523)------------------------------
% 7.26/2.22 % (18523)------------------------------
% 7.26/2.22 % (18535)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_18239 for (2993ds/297Mi)
% 7.26/2.22 % (18505)Instruction limit reached!
% 7.26/2.22 % (18505)------------------------------
% 7.26/2.22 % (18505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.26/2.22 % (18505)Termination reason: Unknown
% 7.26/2.22 % (18505)Termination phase: Saturation
% 7.26/2.22
% 7.26/2.22 % (18505)Memory used [KB]: 8059
% 7.26/2.22 % (18505)Time elapsed: 0.340 s
% 7.26/2.22 % (18505)Instructions burned: 1089 (million)
% 7.26/2.22 % (18505)------------------------------
% 7.26/2.22 % (18505)------------------------------
% 7.26/2.22 % (18534)Instruction limit reached!
% 7.26/2.22 % (18534)------------------------------
% 7.26/2.22 % (18534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.26/2.22 % (18534)Termination reason: Unknown
% 7.26/2.22 % (18534)Termination phase: Preprocessing 1
% 7.26/2.22
% 7.26/2.22 % (18534)Memory used [KB]: 1791
% 7.26/2.22 % (18534)Time elapsed: 0.011 s
% 7.26/2.22 % (18534)Instructions burned: 32 (million)
% 7.26/2.22 % (18534)------------------------------
% 7.26/2.22 % (18534)------------------------------
% 7.26/2.23 % (18536)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_18239 for (2992ds/987Mi)
% 7.26/2.23 % (18537)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_18239 for (2992ds/182Mi)
% 7.26/2.23 % (18538)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_18239 for (2992ds/182Mi)
% 7.26/2.24 % (18530)First to succeed.
% 7.26/2.24 % (18530)Refutation found. Thanks to Tanya!
% 7.26/2.24 % SZS status Theorem for DTF2THF_18239
% 7.26/2.24 % SZS output start Proof for DTF2THF_18239
% 7.26/2.24 thf(type_def_5, type, itself: ($tType) > $tType).
% 7.26/2.24 thf(type_def_7, type, binDag_Mirabelle_dag: $tType).
% 7.26/2.24 thf(type_def_8, type, simpl_ref: $tType).
% 7.26/2.24 thf(type_def_9, type, nat: $tType).
% 7.26/2.24 thf(type_def_10, type, set: ($tType) > $tType).
% 7.26/2.24 thf(func_def_0, type, binDag_Mirabelle_dag: $tType).
% 7.26/2.24 thf(func_def_1, type, simpl_ref: $tType).
% 7.26/2.24 thf(func_def_2, type, set: ($tType) > $tType).
% 7.26/2.24 thf(func_def_3, type, nat: $tType).
% 7.26/2.24 thf(func_def_4, type, itself: ($tType) > $tType).
% 7.26/2.24 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_6, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_9, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_10, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_13, type, zero_neq_one: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_14, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_15, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_16, type, zero_less_one: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_17, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_18, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_19, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_20, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_21, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_22, type, linord1659791738miring: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_23, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_24, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_25, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 7.26/2.24 thf(func_def_26, type, binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o).
% 7.26/2.24 thf(func_def_27, type, binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_28, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_29, type, binDag1297733282se_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0) > binDag_Mirabelle_dag > X0)).
% 7.26/2.24 thf(func_def_30, type, binDag1442713106ec_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0 > X0 > X0) > binDag_Mirabelle_dag > X0)).
% 7.26/2.24 thf(func_def_31, type, binDag1924123185ze_dag: binDag_Mirabelle_dag > nat).
% 7.26/2.24 thf(func_def_32, type, binDag1380252983set_of: binDag_Mirabelle_dag > set @ simpl_ref).
% 7.26/2.24 thf(func_def_33, type, binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o).
% 7.26/2.24 thf(func_def_34, type, one_one: !>[X0: $tType]:(X0)).
% 7.26/2.24 thf(func_def_35, type, sgn_sgn: !>[X0: $tType]:(X0 > X0)).
% 7.26/2.24 thf(func_def_36, type, zero_zero: !>[X0: $tType]:(X0)).
% 7.26/2.24 thf(func_def_37, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 7.26/2.24 thf(func_def_38, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 7.26/2.24 thf(func_def_39, type, bot_bot: !>[X0: $tType]:(X0)).
% 7.26/2.24 thf(func_def_40, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 7.26/2.24 thf(func_def_41, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 7.26/2.24 thf(func_def_42, type, ordering: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 7.26/2.24 thf(func_def_43, type, type2: !>[X0: $tType]:(itself @ X0)).
% 7.26/2.24 thf(func_def_44, type, inv_imagep: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X1 > X1 > $o)).
% 7.26/2.24 thf(func_def_45, type, zero_neq_one_of_bool: !>[X0: $tType]:($o > X0)).
% 7.26/2.24 thf(func_def_46, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 7.26/2.24 thf(func_def_47, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 7.26/2.24 thf(func_def_48, type, chain_subset: !>[X0: $tType]:(set @ set @ X0 > $o)).
% 7.26/2.24 thf(func_def_49, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 7.26/2.24 thf(func_def_50, type, a: simpl_ref).
% 7.26/2.24 thf(func_def_51, type, l: binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_52, type, r: binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_54, type, x: binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_55, type, y: binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_72, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_73, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_74, type, sK2: !>[X0: $tType]:(set @ X0 > X0)).
% 7.26/2.24 thf(func_def_75, type, sK3: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_76, type, sK4: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_77, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_78, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_79, type, sK7: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_80, type, sK8: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_81, type, sK9: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_82, type, sK10: binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_83, type, sK11: binDag_Mirabelle_dag > simpl_ref).
% 7.26/2.24 thf(func_def_84, type, sK12: binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_85, type, sK13: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 7.26/2.24 thf(func_def_86, type, sK14: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_87, type, sK15: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_88, type, sK16: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_89, type, sK17: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_90, type, sK18: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_91, type, sK19: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_92, type, sK20: !>[X0: $tType]:(set @ X0 > X0)).
% 7.26/2.24 thf(func_def_93, type, sK21: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_94, type, sK22: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_95, type, sK23: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_96, type, sK24: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_97, type, sK25: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_98, type, sK26: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_99, type, sK27: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 7.26/2.24 thf(func_def_100, type, sK28: (binDag_Mirabelle_dag > $o) > binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_101, type, sK29: (binDag_Mirabelle_dag > $o) > simpl_ref).
% 7.26/2.24 thf(func_def_102, type, sK30: (binDag_Mirabelle_dag > $o) > binDag_Mirabelle_dag).
% 7.26/2.24 thf(func_def_103, type, sK31: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_104, type, sK32: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_105, type, sK33: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_106, type, sK34: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_107, type, sK35: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_108, type, sK36: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 7.26/2.24 thf(func_def_109, type, sK37: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_110, type, sK38: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_111, type, sK39: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_112, type, sK40: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_113, type, sK41: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_114, type, sK42: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_115, type, sK43: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_116, type, sK44: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_117, type, sK45: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_118, type, sK46: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_119, type, sK47: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 7.26/2.24 thf(func_def_121, type, ph49: !>[X0: $tType]:(X0)).
% 7.26/2.24 thf(f1882,plain,(
% 7.26/2.24 $false),
% 7.26/2.24 inference(avatar_sat_refutation,[status(thm)],[f1344,f1823,f1876])).
% 7.26/2.24 thf(f1876,plain,(
% 7.26/2.24 ~spl48_1),
% 7.26/2.24 inference(avatar_contradiction_clause,[status(thm)],[f1875])).
% 7.26/2.24 thf(f1875,plain,(
% 7.26/2.24 $false | ~spl48_1),
% 7.26/2.24 inference(trivial_inequality_removal,[status(thm)],[f1863])).
% 7.26/2.24 thf(f1863,plain,(
% 7.26/2.24 ($false = $true) | ~spl48_1),
% 7.26/2.24 inference(superposition,[status(thm)],[f1837,f1300])).
% 7.26/2.24 thf(f1300,plain,(
% 7.26/2.24 ($true = (binDag_Mirabelle_DAG @ (binDag476092410e_Node @ l @ a @ r)))),
% 7.26/2.24 inference(definition_unfolding,[status(thm)],[f1253,f1162])).
% 7.26/2.24 thf(f1162,plain,(
% 7.26/2.24 (y = (binDag476092410e_Node @ l @ a @ r))),
% 7.26/2.24 inference(cnf_transformation,[status(thm)],[f1])).
% 7.26/2.24 thf(f1,axiom,(
% 7.26/2.24 (y = (binDag476092410e_Node @ l @ a @ r))),
% 7.26/2.24 file('/export/starexec/sandbox2/tmp/tmp.EA9G7loSYs/DTF2THF_18239.p',fact_0_Node)).
% 7.26/2.24 thf(f1253,plain,(
% 7.26/2.24 ((binDag_Mirabelle_DAG @ y) = $true)),
% 7.26/2.24 inference(cnf_transformation,[status(thm)],[f402])).
% 7.26/2.24 thf(f402,plain,(
% 7.26/2.24 ((binDag_Mirabelle_DAG @ y) = $true)),
% 7.26/2.24 inference(fool_elimination,[status(thm)],[f401])).
% 7.26/2.24 thf(f401,plain,(
% 7.26/2.24 (binDag_Mirabelle_DAG @ y)),
% 7.26/2.24 inference(rectify,[status(thm)],[f13])).
% 7.26/2.24 thf(f13,axiom,(
% 7.26/2.24 (binDag_Mirabelle_DAG @ y)),
% 7.26/2.24 file('/export/starexec/sandbox2/tmp/tmp.EA9G7loSYs/DTF2THF_18239.p',fact_12_DAG__y)).
% 7.26/2.24 thf(f1837,plain,(
% 7.26/2.24 ( ! [X0 : binDag_Mirabelle_dag] : (($false = (binDag_Mirabelle_DAG @ (binDag476092410e_Node @ l @ a @ X0)))) ) | ~spl48_1),
% 7.26/2.24 inference(boolean_simplification,[status(thm)],[f1836])).
% 7.26/2.24 thf(f1836,plain,(
% 7.26/2.24 ( ! [X0 : binDag_Mirabelle_dag] : ((($false & (~ (member @ simpl_ref @ a @ (binDag1380252983set_of @ X0)))) = (binDag_Mirabelle_DAG @ (binDag476092410e_Node @ l @ a @ X0)))) ) | ~spl48_1),
% 7.26/2.24 inference(boolean_simplification,[status(thm)],[f1835])).
% 7.26/2.24 thf(f1835,plain,(
% 7.26/2.24 ( ! [X0 : binDag_Mirabelle_dag] : ((((((binDag_Mirabelle_DAG @ l) & (binDag_Mirabelle_DAG @ X0)) & $false) & (~ (member @ simpl_ref @ a @ (binDag1380252983set_of @ X0)))) = (binDag_Mirabelle_DAG @ (binDag476092410e_Node @ l @ a @ X0)))) ) | ~spl48_1),
% 7.26/2.24 inference(boolean_simplification,[status(thm)],[f1824])).
% 7.26/2.24 thf(f1824,plain,(
% 7.26/2.24 ( ! [X0 : binDag_Mirabelle_dag] : (((binDag_Mirabelle_DAG @ (binDag476092410e_Node @ l @ a @ X0)) = ((((binDag_Mirabelle_DAG @ l) & (binDag_Mirabelle_DAG @ X0)) & (~ $true)) & (~ (member @ simpl_ref @ a @ (binDag1380252983set_of @ X0)))))) ) | ~spl48_1),
% 7.26/2.24 inference(superposition,[status(thm)],[f1226,f1339])).
% 7.26/2.24 thf(f1339,plain,(
% 7.26/2.24 ((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) = $true) | ~spl48_1),
% 7.26/2.24 inference(avatar_component_clause,[status(thm)],[f1337])).
% 7.26/2.24 thf(f1337,definition,(
% 7.26/2.24 spl48_1 <=> ((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) = $true)),
% 7.26/2.24 introduced(definition,[new_symbols(naming,[spl48_1])],[avatar_definition])).
% 7.26/2.24 thf(f1226,plain,(
% 7.26/2.24 ( ! [X2 : binDag_Mirabelle_dag,X0 : binDag_Mirabelle_dag,X1 : simpl_ref] : ((((((binDag_Mirabelle_DAG @ X0) & (binDag_Mirabelle_DAG @ X2)) & (~ (member @ simpl_ref @ X1 @ (binDag1380252983set_of @ X0)))) & (~ (member @ simpl_ref @ X1 @ (binDag1380252983set_of @ X2)))) = (binDag_Mirabelle_DAG @ (binDag476092410e_Node @ X0 @ X1 @ X2)))) )),
% 7.26/2.24 inference(cnf_transformation,[status(thm)],[f702])).
% 7.26/2.24 thf(f702,plain,(
% 7.26/2.24 ! [X0 : binDag_Mirabelle_dag,X1 : simpl_ref,X2 : binDag_Mirabelle_dag] : (((((binDag_Mirabelle_DAG @ X0) & (binDag_Mirabelle_DAG @ X2)) & (~ (member @ simpl_ref @ X1 @ (binDag1380252983set_of @ X0)))) & (~ (member @ simpl_ref @ X1 @ (binDag1380252983set_of @ X2)))) = (binDag_Mirabelle_DAG @ (binDag476092410e_Node @ X0 @ X1 @ X2)))),
% 7.26/2.24 inference(fool_elimination,[status(thm)],[f701])).
% 7.26/2.24 thf(f701,plain,(
% 7.26/2.24 ! [X0 : binDag_Mirabelle_dag,X1 : simpl_ref,X2 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_DAG @ (binDag476092410e_Node @ X0 @ X1 @ X2)) = ~(member @ simpl_ref @ X1 @ (binDag1380252983set_of @ X2)) & ~(member @ simpl_ref @ X1 @ (binDag1380252983set_of @ X0)) & (binDag_Mirabelle_DAG @ X2) & (binDag_Mirabelle_DAG @ X0))),
% 7.26/2.24 inference(rectify,[status(thm)],[f3])).
% 7.26/2.24 thf(f3,axiom,(
% 7.26/2.24 ! [X8 : binDag_Mirabelle_dag,X9 : simpl_ref,X10 : binDag_Mirabelle_dag] : ((binDag_Mirabelle_DAG @ (binDag476092410e_Node @ X8 @ X9 @ X10)) = ~(member @ simpl_ref @ X9 @ (binDag1380252983set_of @ X10)) & ~(member @ simpl_ref @ X9 @ (binDag1380252983set_of @ X8)) & (binDag_Mirabelle_DAG @ X10) & (binDag_Mirabelle_DAG @ X8))),
% 7.26/2.24 file('/export/starexec/sandbox2/tmp/tmp.EA9G7loSYs/DTF2THF_18239.p',fact_2_DAG_Osimps_I2_J)).
% 7.26/2.24 thf(f1823,plain,(
% 7.26/2.24 ~spl48_2),
% 7.26/2.24 inference(avatar_contradiction_clause,[status(thm)],[f1822])).
% 7.26/2.24 thf(f1822,plain,(
% 7.26/2.24 $false | ~spl48_2),
% 7.26/2.24 inference(trivial_inequality_removal,[status(thm)],[f1807])).
% 7.26/2.24 thf(f1807,plain,(
% 7.26/2.24 ($false = $true) | ~spl48_2),
% 7.26/2.24 inference(superposition,[status(thm)],[f1300,f1781])).
% 7.26/2.24 thf(f1781,plain,(
% 7.26/2.24 ( ! [X0 : binDag_Mirabelle_dag] : (((binDag_Mirabelle_DAG @ (binDag476092410e_Node @ X0 @ a @ r)) = $false)) ) | ~spl48_2),
% 7.26/2.24 inference(boolean_simplification,[status(thm)],[f1780])).
% 7.26/2.24 thf(f1780,plain,(
% 7.26/2.24 ( ! [X0 : binDag_Mirabelle_dag] : ((((((binDag_Mirabelle_DAG @ X0) & (binDag_Mirabelle_DAG @ r)) & (~ (member @ simpl_ref @ a @ (binDag1380252983set_of @ X0)))) & $false) = (binDag_Mirabelle_DAG @ (binDag476092410e_Node @ X0 @ a @ r)))) ) | ~spl48_2),
% 7.26/2.24 inference(boolean_simplification,[status(thm)],[f1762])).
% 7.26/2.24 thf(f1762,plain,(
% 7.26/2.24 ( ! [X0 : binDag_Mirabelle_dag] : (((binDag_Mirabelle_DAG @ (binDag476092410e_Node @ X0 @ a @ r)) = ((((binDag_Mirabelle_DAG @ X0) & (binDag_Mirabelle_DAG @ r)) & (~ (member @ simpl_ref @ a @ (binDag1380252983set_of @ X0)))) & (~ $true)))) ) | ~spl48_2),
% 7.26/2.24 inference(superposition,[status(thm)],[f1226,f1343])).
% 7.26/2.24 thf(f1343,plain,(
% 7.26/2.24 ((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) = $true) | ~spl48_2),
% 7.26/2.24 inference(avatar_component_clause,[status(thm)],[f1341])).
% 7.26/2.24 thf(f1341,definition,(
% 7.26/2.24 spl48_2 <=> ((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) = $true)),
% 7.26/2.24 introduced(definition,[new_symbols(naming,[spl48_2])],[avatar_definition])).
% 7.26/2.24 thf(f1344,plain,(
% 7.26/2.24 spl48_1 | spl48_2),
% 7.26/2.24 inference(avatar_split_clause,[status(thm)],[f1335,f1341,f1337])).
% 7.26/2.24 thf(f1335,plain,(
% 7.26/2.24 ((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) = $true) | ((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) = $true)),
% 7.26/2.24 inference(subsumption_resolution,[status(thm)],[f1187,f1197])).
% 7.26/2.24 thf(f1197,plain,(
% 7.26/2.24 (thesis != $true)),
% 7.26/2.24 inference(cnf_transformation,[status(thm)],[f858])).
% 7.26/2.24 thf(f858,plain,(
% 7.26/2.24 (thesis != $true)),
% 7.26/2.24 inference(flattening,[status(thm)],[f516])).
% 7.26/2.24 thf(f516,plain,(
% 7.26/2.24 ~(thesis = $true)),
% 7.26/2.24 inference(fool_elimination,[status(thm)],[f515])).
% 7.26/2.24 thf(f515,plain,(
% 7.26/2.24 ~thesis),
% 7.26/2.24 inference(rectify,[status(thm)],[f295])).
% 7.26/2.24 thf(f295,negated_conjecture,(
% 7.26/2.24 ~thesis),
% 7.26/2.24 inference(negated_conjecture,[status(cth)],[f294])).
% 7.26/2.24 thf(f294,conjecture,(
% 7.26/2.24 thesis),
% 7.26/2.24 file('/export/starexec/sandbox2/tmp/tmp.EA9G7loSYs/DTF2THF_18239.p',conj_1)).
% 7.26/2.24 thf(f1187,plain,(
% 7.26/2.24 (thesis = $true) | ((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) = $true) | ((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) = $true)),
% 7.26/2.24 inference(cnf_transformation,[status(thm)],[f885])).
% 7.26/2.24 thf(f885,plain,(
% 7.26/2.24 (thesis = $true) | ((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) = $true) | ((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) = $true)),
% 7.26/2.24 inference(flattening,[status(thm)],[f884])).
% 7.26/2.24 thf(f884,plain,(
% 7.26/2.24 ((thesis = $true) | ((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) = $true)) | ((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) = $true)),
% 7.26/2.24 inference(ennf_transformation,[status(thm)],[f857])).
% 7.26/2.24 thf(f857,plain,(
% 7.26/2.24 ((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) != $true) => (((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) != $true) => (thesis = $true))),
% 7.26/2.24 inference(flattening,[status(thm)],[f472])).
% 7.26/2.24 thf(f472,plain,(
% 7.26/2.24 ~((member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) = $true) => (~((member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) = $true) => (thesis = $true))),
% 7.26/2.24 inference(fool_elimination,[status(thm)],[f471])).
% 7.26/2.24 thf(f471,plain,(
% 7.26/2.24 ~(member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) => (~(member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) => thesis)),
% 7.26/2.24 inference(rectify,[status(thm)],[f293])).
% 7.26/2.24 thf(f293,axiom,(
% 7.26/2.24 ~(member @ simpl_ref @ a @ (binDag1380252983set_of @ l)) => (~(member @ simpl_ref @ a @ (binDag1380252983set_of @ r)) => thesis)),
% 7.26/2.24 file('/export/starexec/sandbox2/tmp/tmp.EA9G7loSYs/DTF2THF_18239.p',conj_0)).
% 7.26/2.24 % SZS output end Proof for DTF2THF_18239
% 7.26/2.24 % (18530)------------------------------
% 7.26/2.24 % (18530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.26/2.24 % (18530)Termination reason: Refutation
% 7.26/2.24
% 7.26/2.24 % (18530)Memory used [KB]: 7164
% 7.26/2.24 % (18530)Time elapsed: 0.084 s
% 7.26/2.24 % (18530)Instructions burned: 255 (million)
% 7.26/2.24 % (18530)------------------------------
% 7.26/2.24 % (18530)------------------------------
% 7.26/2.24 % (18457)Success in time 0.728 s
%------------------------------------------------------------------------------