%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM163^1 : 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 : n014.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 08:41:48 AM UTC 2026
% Result : Theorem 2.95s 1.67s
% Output : Refutation 2.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM163^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n014.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 04:15:42 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : COM163^1 : TPTP v9.2.1. Released v7.0.0.
% 0.31/0.46 % Domain : Computing Theory
% 0.31/0.46 % Problem : Binary decision diagram 95
% 0.31/0.46 % Version : [Bla16] axioms : Especial.
% 0.31/0.46 % English :
% 0.31/0.46
% 0.31/0.46 % Refs : [OS08] Ortner & Schirmer (2008), BDD Normalisation
% 0.31/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.31/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.31/0.46 % Source : [Bla16]
% 0.31/0.46 % Names : bindag__95.p [Bla16]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.1.0
% 0.31/0.46 % Syntax : Number of formulae : 340 ( 55 unt; 57 typ; 0 def)
% 0.31/0.46 % Number of atoms : 945 ( 226 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 13 ( 3 avg)
% 0.31/0.46 % Number of connectives : 3514 ( 96 ~; 25 |; 58 &;2805 @)
% 0.31/0.46 % ( 0 <=>; 530 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.31/0.46 % Number of types : 4 ( 3 usr)
% 0.31/0.46 % Number of type conns : 246 ( 246 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 57 ( 54 usr; 6 con; 0-4 aty)
% 0.31/0.46 % Number of variables : 987 ( 32 ^; 872 !; 38 ?; 987 :)
% 0.31/0.46 % ( 45 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.46 % SPC : TH1_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.31/0.46 % 2016-07-13 14:45:00.755
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 %----Could-be-implicit typings (5)
% 0.31/0.46 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 0.31/0.46 binDag_Mirabelle_dag: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Simpl__Heap_Oref,type,
% 0.31/0.46 simpl_ref: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Set_Oset,type,
% 0.31/0.46 set: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_Nat_Onat,type,
% 0.31/0.46 nat: $tType ).
% 0.31/0.46
% 0.31/0.46 thf(ty_t_itself,type,
% 0.31/0.46 itself: $tType > $tType ).
% 0.31/0.46
% 0.31/0.46 %----Explicit typings (52)
% 0.31/0.46 thf(sy_cl_HOL_Otype,type,
% 0.31/0.46 type:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.31/0.46 field:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.31/0.46 ord:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.31/0.46 order:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Ono__bot,type,
% 0.31/0.46 no_bot:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Ono__top,type,
% 0.31/0.46 no_top:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.31/0.46 linorder:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.31/0.46 preorder:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.31/0.46 zero_neq_one:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.31/0.46 wellorder:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.31/0.46 zero_less_one:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.31/0.46 division_ring:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.31/0.46 semidom_divide:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Odense__order,type,
% 0.31/0.46 dense_order:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.31/0.46 linordered_idom:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.31/0.46 linordered_field:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.31/0.46 dense_linorder:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.31/0.46 linordered_semidom:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.31/0.46 ordered_ab_group_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 0.31/0.46 linord219039673up_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.31/0.46 linord1659791738miring:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.31/0.46 canoni770627133id_add:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.31/0.46 condit1656338222tinuum:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.31/0.46 condit1037483654norder:
% 0.31/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 0.31/0.46 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 0.31/0.46 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 0.31/0.46 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 0.31/0.46
% 0.31/0.46 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 0.31/0.46 binDag1297733282se_dag:
% 0.31/0.46 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 0.31/0.47 binDag1442713106ec_dag:
% 0.31/0.47 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Osize__dag,type,
% 0.31/0.47 binDag1924123185ze_dag: binDag_Mirabelle_dag > nat ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 0.31/0.47 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 0.31/0.47 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.31/0.47 one_one:
% 0.31/0.47 !>[A: $tType] : A ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 0.31/0.47 sgn_sgn:
% 0.31/0.47 !>[A: $tType] : ( A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.31/0.47 uminus_uminus:
% 0.31/0.47 !>[A: $tType] : ( A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.31/0.47 zero_zero:
% 0.31/0.47 !>[A: $tType] : A ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_If,type,
% 0.31/0.47 if:
% 0.31/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.31/0.47 size_size:
% 0.31/0.47 !>[A: $tType] : ( A > nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oord__class_OLeast,type,
% 0.31/0.47 ord_Least:
% 0.31/0.47 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.31/0.47 ord_less:
% 0.31/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.47 ord_less_eq:
% 0.31/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.31/0.47 order_antimono:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.31/0.47 order_strict_mono:
% 0.31/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oordering,type,
% 0.31/0.47 ordering:
% 0.31/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Pure_Otype,type,
% 0.31/0.47 type2:
% 0.31/0.47 !>[A: $tType] : ( itself @ A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.31/0.47 divide_divide:
% 0.31/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 0.31/0.47 zero_neq_one_of_bool:
% 0.31/0.47 !>[A: $tType] : ( $o > A ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Set_OCollect,type,
% 0.31/0.47 collect:
% 0.31/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_member,type,
% 0.31/0.47 member:
% 0.31/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_x,type,
% 0.31/0.47 x: binDag_Mirabelle_dag ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_y,type,
% 0.31/0.47 y: binDag_Mirabelle_dag ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_z,type,
% 0.31/0.47 z: binDag_Mirabelle_dag ).
% 0.31/0.47
% 0.31/0.47 %----Relevant facts (256)
% 0.31/0.47 thf(fact_0_True,axiom,
% 0.31/0.47 x = y ).
% 0.31/0.47
% 0.31/0.47 % True
% 0.31/0.47 thf(fact_1_y__z,axiom,
% 0.31/0.47 ord_less_eq @ binDag_Mirabelle_dag @ y @ z ).
% 0.31/0.47
% 0.31/0.47 % y_z
% 0.31/0.47 thf(fact_2_x__y,axiom,
% 0.31/0.47 ord_less_eq @ binDag_Mirabelle_dag @ x @ y ).
% 0.31/0.47
% 0.31/0.47 % x_y
% 0.31/0.47 thf(fact_3_le__dag__refl,axiom,
% 0.31/0.47 ! [X: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X @ X ) ).
% 0.31/0.47
% 0.31/0.47 % le_dag_refl
% 0.31/0.47 thf(fact_4_order__refl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_refl
% 0.31/0.47 thf(fact_5_le__funD,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.31/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_funD
% 0.31/0.47 thf(fact_6_le__funE,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.31/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_funE
% 0.31/0.47 thf(fact_7_le__funI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 => ! [F: A > B,G: A > B] :
% 0.31/0.47 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.31/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_funI
% 0.31/0.47 thf(fact_8_le__fun__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.31/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.31/0.47 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_fun_def
% 0.31/0.47 thf(fact_9_order__subst1,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.31/0.47 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.31/0.47 => ( ! [X2: B,Y: B] :
% 0.31/0.47 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst1
% 0.31/0.47 thf(fact_10_order__subst2,axiom,
% 0.31/0.47 ! [A: $tType,C2: $tType] :
% 0.31/0.47 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.31/0.47 => ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_subst2
% 0.31/0.47 thf(fact_11_ord__eq__le__subst,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( F @ B2 ) )
% 0.31/0.47 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.31/0.47 => ( ! [X2: B,Y: B] :
% 0.31/0.47 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_subst
% 0.31/0.47 thf(fact_12_ord__le__eq__subst,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ( F @ B2 )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_subst
% 0.31/0.47 thf(fact_13_eq__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.31/0.47 = ( ^ [X3: A,Y3: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.31/0.47 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_iff
% 0.31/0.47 thf(fact_14_antisym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Y4 @ X )
% 0.31/0.47 => ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym
% 0.31/0.47 thf(fact_15_linear,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linear
% 0.31/0.47 thf(fact_16_dual__order_Oantisym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.antisym
% 0.31/0.47 thf(fact_17_dual__order_Otrans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.31/0.47 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.trans
% 0.31/0.47 thf(fact_18_linorder__wlog,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ! [A3: A,B3: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.31/0.47 => ( P @ A3 @ B3 ) )
% 0.31/0.47 => ( ! [A3: A,B3: A] :
% 0.31/0.47 ( ( P @ B3 @ A3 )
% 0.31/0.47 => ( P @ A3 @ B3 ) )
% 0.31/0.47 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_wlog
% 0.31/0.47 thf(fact_19_dual__order_Orefl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.refl
% 0.31/0.47 thf(fact_20_order__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.31/0.47 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_trans
% 0.31/0.47 thf(fact_21_order__class_Oorder_Oantisym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_class.order.antisym
% 0.31/0.47 thf(fact_22_ord__le__eq__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( B2 = C )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_le_eq_trans
% 0.31/0.47 thf(fact_23_ord__eq__le__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( A2 = B2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_le_trans
% 0.31/0.47 thf(fact_24_antisym__conv,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [Y4: A,X: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 = ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv
% 0.31/0.47 thf(fact_25_le__cases3,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 0.31/0.47 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.31/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases3
% 0.31/0.47 thf(fact_26_order_Otrans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.trans
% 0.31/0.47 thf(fact_27_le__cases,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_cases
% 0.31/0.47 thf(fact_28_eq__refl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( X = Y4 )
% 0.31/0.47 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_refl
% 0.31/0.47 thf(fact_29_antimono__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ( ( order_antimono @ A @ B )
% 0.31/0.47 = ( ^ [F2: A > B] :
% 0.31/0.47 ! [X3: A,Y3: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antimono_def
% 0.31/0.47 thf(fact_30_antimonoI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B] :
% 0.31/0.47 ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) ) )
% 0.31/0.47 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antimonoI
% 0.31/0.47 thf(fact_31_antimonoE,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B,X: A,Y4: A] :
% 0.31/0.47 ( ( order_antimono @ A @ B @ F )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antimonoE
% 0.31/0.47 thf(fact_32_antimonoD,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B,X: A,Y4: A] :
% 0.31/0.47 ( ( order_antimono @ A @ B @ F )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antimonoD
% 0.31/0.47 thf(fact_33_strict__mono__less__eq,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B,X: A,Y4: A] :
% 0.31/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.31/0.47 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % strict_mono_less_eq
% 0.31/0.47 thf(fact_34_le__dag__def,axiom,
% 0.31/0.47 ( ( ord_less_eq @ binDag_Mirabelle_dag )
% 0.31/0.47 = ( ^ [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( S = T )
% 0.31/0.47 | ( ord_less @ binDag_Mirabelle_dag @ S @ T ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_dag_def
% 0.31/0.47 thf(fact_35_LeastI2__wellorder__ex,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,Q: A > $o] :
% 0.31/0.47 ( ? [X1: A] : ( P @ X1 )
% 0.31/0.47 => ( ! [A3: A] :
% 0.31/0.47 ( ( P @ A3 )
% 0.31/0.47 => ( ! [B4: A] :
% 0.31/0.47 ( ( P @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 0.31/0.47 => ( Q @ A3 ) ) )
% 0.31/0.47 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LeastI2_wellorder_ex
% 0.31/0.47 thf(fact_36_LeastI2__wellorder,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,A2: A,Q: A > $o] :
% 0.31/0.47 ( ( P @ A2 )
% 0.31/0.47 => ( ! [A3: A] :
% 0.31/0.47 ( ( P @ A3 )
% 0.31/0.47 => ( ! [B4: A] :
% 0.31/0.47 ( ( P @ B4 )
% 0.31/0.47 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 0.31/0.47 => ( Q @ A3 ) ) )
% 0.31/0.47 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LeastI2_wellorder
% 0.31/0.47 thf(fact_37_Least__equality,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,X: A] :
% 0.31/0.47 ( ( P @ X )
% 0.31/0.47 => ( ! [Y: A] :
% 0.31/0.47 ( ( P @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ X @ Y ) )
% 0.31/0.47 => ( ( ord_Least @ A @ P )
% 0.31/0.47 = X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Least_equality
% 0.31/0.47 thf(fact_38_LeastI2__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.31/0.47 ( ( P @ X )
% 0.31/0.47 => ( ! [Y: A] :
% 0.31/0.47 ( ( P @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ X @ Y ) )
% 0.31/0.47 => ( ! [X2: A] :
% 0.31/0.47 ( ( P @ X2 )
% 0.31/0.47 => ( ! [Y5: A] :
% 0.31/0.47 ( ( P @ Y5 )
% 0.31/0.47 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 0.31/0.47 => ( Q @ X2 ) ) )
% 0.31/0.47 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LeastI2_order
% 0.31/0.47 thf(fact_39_ord__eq__less__subst,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 = ( F @ B2 ) )
% 0.31/0.47 => ( ( ord_less @ B @ B2 @ C )
% 0.31/0.47 => ( ! [X2: B,Y: B] :
% 0.31/0.47 ( ( ord_less @ B @ X2 @ Y )
% 0.31/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_subst
% 0.31/0.47 thf(fact_40_ord__less__eq__subst,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ( F @ B2 )
% 0.31/0.47 = C )
% 0.31/0.47 => ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_subst
% 0.31/0.47 thf(fact_41_order__less__subst1,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.31/0.47 => ( ( ord_less @ B @ B2 @ C )
% 0.31/0.47 => ( ! [X2: B,Y: B] :
% 0.31/0.47 ( ( ord_less @ B @ X2 @ Y )
% 0.31/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst1
% 0.31/0.47 thf(fact_42_order__less__subst2,axiom,
% 0.31/0.47 ! [A: $tType,C2: $tType] :
% 0.31/0.47 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.31/0.47 => ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_subst2
% 0.31/0.47 thf(fact_43_lt__ex,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % lt_ex
% 0.31/0.47 thf(fact_44_gt__ex,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( no_top @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% 0.31/0.47
% 0.31/0.47 % gt_ex
% 0.31/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.31/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.31/0.47 = ( P @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % mem_Collect_eq
% 0.31/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/0.47 ! [A: $tType,A4: set @ A] :
% 0.31/0.47 ( ( collect @ A
% 0.31/0.47 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.31/0.47 = A4 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mem_eq
% 0.31/0.47 thf(fact_47_Collect__cong,axiom,
% 0.31/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( P @ X2 )
% 0.31/0.47 = ( Q @ X2 ) )
% 0.31/0.47 => ( ( collect @ A @ P )
% 0.31/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_cong
% 0.31/0.47 thf(fact_48_ext,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( F @ X2 )
% 0.31/0.47 = ( G @ X2 ) )
% 0.31/0.47 => ( F = G ) ) ).
% 0.31/0.47
% 0.31/0.47 % ext
% 0.31/0.47 thf(fact_49_neqE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( X != Y4 )
% 0.31/0.47 => ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neqE
% 0.31/0.47 thf(fact_50_neq__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( X != Y4 )
% 0.31/0.47 = ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neq_iff
% 0.31/0.47 thf(fact_51_order_Oasym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.asym
% 0.31/0.47 thf(fact_52_dense,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ? [Z3: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Z3 )
% 0.31/0.47 & ( ord_less @ A @ Z3 @ Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense
% 0.31/0.47 thf(fact_53_LeastI2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,A2: A,Q: A > $o] :
% 0.31/0.47 ( ( P @ A2 )
% 0.31/0.47 => ( ! [X2: A] :
% 0.31/0.47 ( ( P @ X2 )
% 0.31/0.47 => ( Q @ X2 ) )
% 0.31/0.47 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LeastI2
% 0.31/0.47 thf(fact_54_less__imp__neq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( X != Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_neq
% 0.31/0.47 thf(fact_55_strict__monoD,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B,X: A,Y4: A] :
% 0.31/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.31/0.47 => ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % strict_monoD
% 0.31/0.47 thf(fact_56_strict__monoI,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B] :
% 0.31/0.47 ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % strict_monoI
% 0.31/0.47 thf(fact_57_less__asym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_asym
% 0.31/0.47 thf(fact_58_less__asym_H,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_asym'
% 0.31/0.47 thf(fact_59_less__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less @ A @ Y4 @ Z2 )
% 0.31/0.47 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_trans
% 0.31/0.47 thf(fact_60_LeastI__ex,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o] :
% 0.31/0.47 ( ? [X1: A] : ( P @ X1 )
% 0.31/0.47 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LeastI_ex
% 0.31/0.47 thf(fact_61_less__linear,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 | ( X = Y4 )
% 0.31/0.47 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_linear
% 0.31/0.47 thf(fact_62_less__irrefl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_irrefl
% 0.31/0.47 thf(fact_63_LeastI2__ex,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,Q: A > $o] :
% 0.31/0.47 ( ? [X1: A] : ( P @ X1 )
% 0.31/0.47 => ( ! [X2: A] :
% 0.31/0.47 ( ( P @ X2 )
% 0.31/0.47 => ( Q @ X2 ) )
% 0.31/0.47 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % LeastI2_ex
% 0.31/0.47 thf(fact_64_ord__eq__less__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( A2 = B2 )
% 0.31/0.47 => ( ( ord_less @ A @ B2 @ C )
% 0.31/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_eq_less_trans
% 0.31/0.47 thf(fact_65_ord__less__eq__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( B2 = C )
% 0.31/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ord_less_eq_trans
% 0.31/0.47 thf(fact_66_dual__order_Oasym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.31/0.47 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.asym
% 0.31/0.47 thf(fact_67_less__imp__not__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( X != Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_eq
% 0.31/0.47 thf(fact_68_strict__mono__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ( ( order_strict_mono @ A @ B )
% 0.31/0.47 = ( ^ [F2: A > B] :
% 0.31/0.47 ! [X3: A,Y3: A] :
% 0.31/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.31/0.47 => ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % strict_mono_def
% 0.31/0.47 thf(fact_69_less__not__sym,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_not_sym
% 0.31/0.47 thf(fact_70_less__induct,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,A2: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ! [Y5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Y5 @ X2 )
% 0.31/0.47 => ( P @ Y5 ) )
% 0.31/0.47 => ( P @ X2 ) )
% 0.31/0.47 => ( P @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_induct
% 0.31/0.47 thf(fact_71_antisym__conv3,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [Y4: A,X: A] :
% 0.31/0.47 ( ~ ( ord_less @ A @ Y4 @ X )
% 0.31/0.47 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.47 = ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv3
% 0.31/0.47 thf(fact_72_less__imp__not__eq2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( Y4 != X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_eq2
% 0.31/0.47 thf(fact_73_less__imp__triv,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A,P: $o] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less @ A @ Y4 @ X )
% 0.31/0.47 => P ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_triv
% 0.31/0.47 thf(fact_74_linorder__cases,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ( X != Y4 )
% 0.31/0.47 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_cases
% 0.31/0.47 thf(fact_75_dual__order_Oirrefl,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.irrefl
% 0.31/0.47 thf(fact_76_order_Ostrict__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less @ A @ B2 @ C )
% 0.31/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_trans
% 0.31/0.47 thf(fact_77_strict__mono__less,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B,X: A,Y4: A] :
% 0.31/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.31/0.47 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 0.31/0.47 = ( ord_less @ A @ X @ Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % strict_mono_less
% 0.31/0.47 thf(fact_78_less__imp__not__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_not_less
% 0.31/0.47 thf(fact_79_dual__order_Ostrict__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less @ A @ C @ B2 )
% 0.31/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_trans
% 0.31/0.47 thf(fact_80_not__less__iff__gr__or__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.47 = ( ( ord_less @ A @ Y4 @ X )
% 0.31/0.47 | ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less_iff_gr_or_eq
% 0.31/0.47 thf(fact_81_order_Ostrict__implies__not__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( A2 != B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_implies_not_eq
% 0.31/0.47 thf(fact_82_dual__order_Ostrict__implies__not__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.31/0.47 => ( A2 != B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_implies_not_eq
% 0.31/0.47 thf(fact_83_strict__mono__eq,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.31/0.47 => ! [F: A > B,X: A,Y4: A] :
% 0.31/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.31/0.47 => ( ( ( F @ X )
% 0.31/0.47 = ( F @ Y4 ) )
% 0.31/0.47 = ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % strict_mono_eq
% 0.31/0.47 thf(fact_84_leD,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [Y4: A,X: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.31/0.47 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % leD
% 0.31/0.47 thf(fact_85_leI,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % leI
% 0.31/0.47 thf(fact_86_le__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [X3: A,Y3: A] :
% 0.31/0.47 ( ( ord_less @ A @ X3 @ Y3 )
% 0.31/0.47 | ( X3 = Y3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less
% 0.31/0.47 thf(fact_87_less__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [X3: A,Y3: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.31/0.47 & ( X3 != Y3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le
% 0.31/0.47 thf(fact_88_order__le__less__subst1,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.31/0.47 => ( ( ord_less @ B @ B2 @ C )
% 0.31/0.47 => ( ! [X2: B,Y: B] :
% 0.31/0.47 ( ( ord_less @ B @ X2 @ Y )
% 0.31/0.47 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst1
% 0.31/0.47 thf(fact_89_order__le__less__subst2,axiom,
% 0.31/0.47 ! [A: $tType,C2: $tType] :
% 0.31/0.47 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.31/0.47 => ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_le_less_subst2
% 0.31/0.47 thf(fact_90_order__less__le__subst1,axiom,
% 0.31/0.47 ! [A: $tType,B: $tType] :
% 0.31/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.31/0.47 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.31/0.47 => ( ! [X2: B,Y: B] :
% 0.31/0.47 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_le_subst1
% 0.31/0.47 thf(fact_91_order__less__le__subst2,axiom,
% 0.31/0.47 ! [A: $tType,C2: $tType] :
% 0.31/0.47 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.31/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.31/0.47 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.31/0.47 => ( ! [X2: A,Y: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Y )
% 0.31/0.47 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.31/0.47 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order_less_le_subst2
% 0.31/0.47 thf(fact_92_not__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 0.31/0.47 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_le
% 0.31/0.47 thf(fact_93_not__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less
% 0.31/0.47 thf(fact_94_le__neq__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( A2 != B2 )
% 0.31/0.47 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_neq_trans
% 0.31/0.47 thf(fact_95_less__imp__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_le
% 0.31/0.47 thf(fact_96_antisym__conv1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 = ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv1
% 0.31/0.47 thf(fact_97_antisym__conv2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.31/0.47 = ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % antisym_conv2
% 0.31/0.47 thf(fact_98_le__less__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less @ A @ Y4 @ Z2 )
% 0.31/0.47 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less_trans
% 0.31/0.47 thf(fact_99_less__le__trans,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.31/0.47 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le_trans
% 0.31/0.47 thf(fact_100_dense__ge,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [Z2: A,Y4: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z2 @ X2 )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_ge
% 0.31/0.47 thf(fact_101_dense__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [Y4: A,Z2: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Y4 )
% 0.31/0.47 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_le
% 0.31/0.47 thf(fact_102_le__less__linear,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_less_linear
% 0.31/0.47 thf(fact_103_le__imp__less__or__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.31/0.47 => ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 | ( X = Y4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_imp_less_or_eq
% 0.31/0.47 thf(fact_104_less__le__not__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [X3: A,Y3: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.31/0.47 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_le_not_le
% 0.31/0.47 thf(fact_105_not__le__imp__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [Y4: A,X: A] :
% 0.31/0.47 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 0.31/0.47 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_le_imp_less
% 0.31/0.47 thf(fact_106_order_Ostrict__trans1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less @ A @ B2 @ C )
% 0.31/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_trans1
% 0.31/0.47 thf(fact_107_order_Ostrict__trans2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.31/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_trans2
% 0.31/0.47 thf(fact_108_order_Oorder__iff__strict,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [A5: A,B5: A] :
% 0.31/0.47 ( ( ord_less @ A @ A5 @ B5 )
% 0.31/0.47 | ( A5 = B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.order_iff_strict
% 0.31/0.47 thf(fact_109_order_Ostrict__iff__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [A5: A,B5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.31/0.47 & ( A5 != B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_iff_order
% 0.31/0.47 thf(fact_110_dual__order_Ostrict__trans1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A,C: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less @ A @ C @ B2 )
% 0.31/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_trans1
% 0.31/0.47 thf(fact_111_dual__order_Ostrict__trans2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A,C: A] :
% 0.31/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.31/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_trans2
% 0.31/0.47 thf(fact_112_dense__ge__bounded,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [Z2: A,X: A,Y4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z2 @ X )
% 0.31/0.47 => ( ! [W: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z2 @ W )
% 0.31/0.47 => ( ( ord_less @ A @ W @ X )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_ge_bounded
% 0.31/0.47 thf(fact_113_dense__le__bounded,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A,Z2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ! [W: A] :
% 0.31/0.47 ( ( ord_less @ A @ X @ W )
% 0.31/0.47 => ( ( ord_less @ A @ W @ Y4 )
% 0.31/0.47 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.31/0.47 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dense_le_bounded
% 0.31/0.47 thf(fact_114_order_Ostrict__implies__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.strict_implies_order
% 0.31/0.47 thf(fact_115_dual__order_Oorder__iff__strict,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A )
% 0.31/0.47 = ( ^ [B5: A,A5: A] :
% 0.31/0.47 ( ( ord_less @ A @ B5 @ A5 )
% 0.31/0.47 | ( A5 = B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.order_iff_strict
% 0.31/0.47 thf(fact_116_dual__order_Ostrict__iff__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( ord_less @ A )
% 0.31/0.47 = ( ^ [B5: A,A5: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.31/0.47 & ( A5 != B5 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_iff_order
% 0.31/0.47 thf(fact_117_dual__order_Ostrict__implies__order,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.31/0.47 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dual_order.strict_implies_order
% 0.31/0.47 thf(fact_118_order_Onot__eq__order__implies__strict,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( A2 != B2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.31/0.47 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.not_eq_order_implies_strict
% 0.31/0.47 thf(fact_119_minf_I8_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(8)
% 0.31/0.47 thf(fact_120_minf_I6_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(6)
% 0.31/0.47 thf(fact_121_pinf_I8_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(8)
% 0.31/0.47 thf(fact_122_pinf_I6_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(6)
% 0.31/0.47 thf(fact_123_complete__interval,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A,P: A > $o] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.31/0.47 => ( ( P @ A2 )
% 0.31/0.47 => ( ~ ( P @ B2 )
% 0.31/0.47 => ? [C3: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.31/0.47 & ( ord_less_eq @ A @ C3 @ B2 )
% 0.31/0.47 & ! [X4: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 0.31/0.47 & ( ord_less @ A @ X4 @ C3 ) )
% 0.31/0.47 => ( P @ X4 ) )
% 0.31/0.47 & ! [D: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.31/0.47 & ( ord_less @ A @ X2 @ D ) )
% 0.31/0.47 => ( P @ X2 ) )
% 0.31/0.47 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % complete_interval
% 0.31/0.47 thf(fact_124_less__dag__def,axiom,
% 0.31/0.47 ( ( ord_less @ binDag_Mirabelle_dag )
% 0.31/0.47 = ( ^ [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T @ S ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_dag_def
% 0.31/0.47 thf(fact_125_ex__gt__or__lt,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ? [B3: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ B3 )
% 0.31/0.47 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ex_gt_or_lt
% 0.31/0.47 thf(fact_126_linorder__neqE__linordered__idom,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A,Y4: A] :
% 0.31/0.47 ( ( X != Y4 )
% 0.31/0.47 => ( ~ ( ord_less @ A @ X @ Y4 )
% 0.31/0.47 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_neqE_linordered_idom
% 0.31/0.47 thf(fact_127_subdag__neq,axiom,
% 0.31/0.47 ! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag786255756subdag @ T2 @ S2 )
% 0.31/0.47 => ( T2 != S2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subdag_neq
% 0.31/0.47 thf(fact_128_subdag__trans,axiom,
% 0.31/0.47 ! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag786255756subdag @ T2 @ S2 )
% 0.31/0.47 => ( ( binDag786255756subdag @ S2 @ R )
% 0.31/0.47 => ( binDag786255756subdag @ T2 @ R ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subdag_trans
% 0.31/0.47 thf(fact_129_subdag__not__sym,axiom,
% 0.31/0.47 ! [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag786255756subdag @ S2 @ T2 )
% 0.31/0.47 => ~ ( binDag786255756subdag @ T2 @ S2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subdag_not_sym
% 0.31/0.47 thf(fact_130_less__fun__def,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.31/0.47 => ( ( ord_less @ ( A > B ) )
% 0.31/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.31/0.47 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.31/0.47 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_fun_def
% 0.31/0.47 thf(fact_131_minf_I11_J,axiom,
% 0.31/0.47 ! [C2: $tType,D2: $tType] :
% 0.31/0.47 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.31/0.47 => ! [F3: D2] :
% 0.31/0.47 ? [Z3: C2] :
% 0.31/0.47 ! [X4: C2] :
% 0.31/0.47 ( ( ord_less @ C2 @ X4 @ Z3 )
% 0.31/0.47 => ( F3 = F3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(11)
% 0.31/0.47 thf(fact_132_minf_I7_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(7)
% 0.31/0.47 thf(fact_133_minf_I5_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(5)
% 0.31/0.47 thf(fact_134_minf_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ( X4 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(4)
% 0.31/0.47 thf(fact_135_minf_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ( X4 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(3)
% 0.31/0.47 thf(fact_136_minf_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.31/0.47 => ( ( P @ X2 )
% 0.31/0.47 = ( P2 @ X2 ) ) )
% 0.31/0.47 => ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.31/0.47 => ( ( Q @ X2 )
% 0.31/0.47 = ( Q2 @ X2 ) ) )
% 0.31/0.47 => ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ( ( ( P @ X4 )
% 0.31/0.47 | ( Q @ X4 ) )
% 0.31/0.47 = ( ( P2 @ X4 )
% 0.31/0.47 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(2)
% 0.31/0.47 thf(fact_137_minf_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.31/0.47 => ( ( P @ X2 )
% 0.31/0.47 = ( P2 @ X2 ) ) )
% 0.31/0.47 => ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ X2 @ Z4 )
% 0.31/0.47 => ( ( Q @ X2 )
% 0.31/0.47 = ( Q2 @ X2 ) ) )
% 0.31/0.47 => ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ X4 @ Z3 )
% 0.31/0.47 => ( ( ( P @ X4 )
% 0.31/0.47 & ( Q @ X4 ) )
% 0.31/0.47 = ( ( P2 @ X4 )
% 0.31/0.47 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % minf(1)
% 0.31/0.47 thf(fact_138_pinf_I11_J,axiom,
% 0.31/0.47 ! [C2: $tType,D2: $tType] :
% 0.31/0.47 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.31/0.47 => ! [F3: D2] :
% 0.31/0.47 ? [Z3: C2] :
% 0.31/0.47 ! [X4: C2] :
% 0.31/0.47 ( ( ord_less @ C2 @ Z3 @ X4 )
% 0.31/0.47 => ( F3 = F3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(11)
% 0.31/0.47 thf(fact_139_pinf_I7_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(7)
% 0.31/0.47 thf(fact_140_pinf_I5_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(5)
% 0.31/0.47 thf(fact_141_pinf_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ( X4 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(4)
% 0.31/0.47 thf(fact_142_pinf_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [T2: A] :
% 0.31/0.47 ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ( X4 != T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(3)
% 0.31/0.47 thf(fact_143_pinf_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.31/0.47 => ( ( P @ X2 )
% 0.31/0.47 = ( P2 @ X2 ) ) )
% 0.31/0.47 => ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.31/0.47 => ( ( Q @ X2 )
% 0.31/0.47 = ( Q2 @ X2 ) ) )
% 0.31/0.47 => ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ( ( ( P @ X4 )
% 0.31/0.47 | ( Q @ X4 ) )
% 0.31/0.47 = ( ( P2 @ X4 )
% 0.31/0.47 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(2)
% 0.31/0.47 thf(fact_144_pinf_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.31/0.47 ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.31/0.47 => ( ( P @ X2 )
% 0.31/0.47 = ( P2 @ X2 ) ) )
% 0.31/0.47 => ( ? [Z4: A] :
% 0.31/0.47 ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z4 @ X2 )
% 0.31/0.47 => ( ( Q @ X2 )
% 0.31/0.47 = ( Q2 @ X2 ) ) )
% 0.31/0.47 => ? [Z3: A] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( ord_less @ A @ Z3 @ X4 )
% 0.31/0.47 => ( ( ( P @ X4 )
% 0.31/0.47 & ( Q @ X4 ) )
% 0.31/0.47 = ( ( P2 @ X4 )
% 0.31/0.47 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % pinf(1)
% 0.31/0.47 thf(fact_145_dependent__wellorder__choice,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: ( A > B ) > A > B > $o] :
% 0.31/0.47 ( ! [R2: B,F4: A > B,G3: A > B,X2: A] :
% 0.31/0.47 ( ! [Y5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Y5 @ X2 )
% 0.31/0.47 => ( ( F4 @ Y5 )
% 0.31/0.47 = ( G3 @ Y5 ) ) )
% 0.31/0.47 => ( ( P @ F4 @ X2 @ R2 )
% 0.31/0.47 = ( P @ G3 @ X2 @ R2 ) ) )
% 0.31/0.47 => ( ! [X2: A,F4: A > B] :
% 0.31/0.47 ( ! [Y5: A] :
% 0.31/0.47 ( ( ord_less @ A @ Y5 @ X2 )
% 0.31/0.47 => ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
% 0.31/0.47 => ? [X1: B] : ( P @ F4 @ X2 @ X1 ) )
% 0.31/0.47 => ? [F4: A > B] :
% 0.31/0.47 ! [X4: A] : ( P @ F4 @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dependent_wellorder_choice
% 0.31/0.47 thf(fact_146_linordered__field__no__lb,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X4: A] :
% 0.31/0.47 ? [Y: A] : ( ord_less @ A @ Y @ X4 ) ) ).
% 0.31/0.47
% 0.31/0.47 % linordered_field_no_lb
% 0.31/0.47 thf(fact_147_linordered__field__no__ub,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X4: A] :
% 0.31/0.47 ? [X12: A] : ( ord_less @ A @ X4 @ X12 ) ) ).
% 0.31/0.47
% 0.31/0.47 % linordered_field_no_ub
% 0.31/0.47 thf(fact_148_subdag_Osimps_I1_J,axiom,
% 0.31/0.47 ! [T2: binDag_Mirabelle_dag] :
% 0.31/0.47 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T2 ) ).
% 0.31/0.47
% 0.31/0.47 % subdag.simps(1)
% 0.31/0.47 thf(fact_149_DAG_Osimps_I1_J,axiom,
% 0.31/0.47 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 0.31/0.47
% 0.31/0.47 % DAG.simps(1)
% 0.31/0.47 thf(fact_150_dag_Osimps_I4_J,axiom,
% 0.31/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 0.31/0.47 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 0.31/0.47 = F1 ) ).
% 0.31/0.47
% 0.31/0.47 % dag.simps(4)
% 0.31/0.47 thf(fact_151_dag_Osimps_I6_J,axiom,
% 0.31/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 0.31/0.47 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 0.31/0.47 = F1 ) ).
% 0.31/0.47
% 0.31/0.47 % dag.simps(6)
% 0.31/0.47 thf(fact_152_subdag__size,axiom,
% 0.31/0.47 ! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag786255756subdag @ T2 @ S2 )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ binDag_Mirabelle_dag @ S2 ) @ ( size_size @ binDag_Mirabelle_dag @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subdag_size
% 0.31/0.47 thf(fact_153_dag_Osimps_I7_J,axiom,
% 0.31/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.31/0.47 = ( F22 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X23 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.simps(7)
% 0.31/0.47 thf(fact_154_dag_Osimps_I5_J,axiom,
% 0.31/0.47 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 0.31/0.47 = ( F22 @ X21 @ X22 @ X23 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.simps(5)
% 0.31/0.47 thf(fact_155_dag_Oinject,axiom,
% 0.31/0.47 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 0.31/0.47 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 0.31/0.47 = ( ( X21 = Y21 )
% 0.31/0.47 & ( X22 = Y22 )
% 0.31/0.47 & ( X23 = Y23 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.inject
% 0.31/0.47 thf(fact_156_subdag__NodeD,axiom,
% 0.31/0.47 ! [T2: binDag_Mirabelle_dag,Lt: binDag_Mirabelle_dag,A2: simpl_ref,Rt: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag786255756subdag @ T2 @ ( binDag476092410e_Node @ Lt @ A2 @ Rt ) )
% 0.31/0.47 => ( ( binDag786255756subdag @ T2 @ Lt )
% 0.31/0.47 & ( binDag786255756subdag @ T2 @ Rt ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subdag_NodeD
% 0.31/0.47 thf(fact_157_subdag_Osimps_I2_J,axiom,
% 0.31/0.47 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T2 )
% 0.31/0.47 = ( ( T2 = L )
% 0.31/0.47 | ( T2 = R )
% 0.31/0.47 | ( binDag786255756subdag @ L @ T2 )
% 0.31/0.47 | ( binDag786255756subdag @ R @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subdag.simps(2)
% 0.31/0.47 thf(fact_158_dag_Oexhaust,axiom,
% 0.31/0.47 ! [Y4: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( Y4 != binDag_Mirabelle_Tip )
% 0.31/0.47 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 0.31/0.47 ( Y4
% 0.31/0.47 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.exhaust
% 0.31/0.47 thf(fact_159_dag_Oinduct,axiom,
% 0.31/0.47 ! [P: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( P @ binDag_Mirabelle_Tip )
% 0.31/0.47 => ( ! [X12: binDag_Mirabelle_dag,X24: simpl_ref,X32: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( P @ X12 )
% 0.31/0.47 => ( ( P @ X32 )
% 0.31/0.47 => ( P @ ( binDag476092410e_Node @ X12 @ X24 @ X32 ) ) ) )
% 0.31/0.47 => ( P @ Dag ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.induct
% 0.31/0.47 thf(fact_160_dag_Odistinct_I1_J,axiom,
% 0.31/0.47 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 0.31/0.47 ( binDag_Mirabelle_Tip
% 0.31/0.47 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.distinct(1)
% 0.31/0.47 thf(fact_161_DAG_Osimps_I2_J,axiom,
% 0.31/0.47 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 0.31/0.47 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 0.31/0.47 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 0.31/0.47 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 0.31/0.47 & ( binDag_Mirabelle_DAG @ L )
% 0.31/0.47 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % DAG.simps(2)
% 0.31/0.47 thf(fact_162_dag_Osize_I3_J,axiom,
% 0.31/0.47 ( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
% 0.31/0.47 = ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.size(3)
% 0.31/0.47 thf(fact_163_order_Oordering__axioms,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % order.ordering_axioms
% 0.31/0.47 thf(fact_164_ordering_Onot__eq__order__implies__strict,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( A2 != B2 )
% 0.31/0.47 => ( ( Less_eq @ A2 @ B2 )
% 0.31/0.47 => ( Less @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.not_eq_order_implies_strict
% 0.31/0.47 thf(fact_165_ordering_Ostrict__implies__not__eq,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less @ A2 @ B2 )
% 0.31/0.47 => ( A2 != B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.strict_implies_not_eq
% 0.31/0.47 thf(fact_166_ordering_Ostrict__implies__order,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less @ A2 @ B2 )
% 0.31/0.47 => ( Less_eq @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.strict_implies_order
% 0.31/0.47 thf(fact_167_ordering_Ostrict__iff__order,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less @ A2 @ B2 )
% 0.31/0.47 = ( ( Less_eq @ A2 @ B2 )
% 0.31/0.47 & ( A2 != B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.strict_iff_order
% 0.31/0.47 thf(fact_168_ordering_Oorder__iff__strict,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less_eq @ A2 @ B2 )
% 0.31/0.47 = ( ( Less @ A2 @ B2 )
% 0.31/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.order_iff_strict
% 0.31/0.47 thf(fact_169_ordering_Ostrict__trans2,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less @ A2 @ B2 )
% 0.31/0.47 => ( ( Less_eq @ B2 @ C )
% 0.31/0.47 => ( Less @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.strict_trans2
% 0.31/0.47 thf(fact_170_ordering_Ostrict__trans1,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less_eq @ A2 @ B2 )
% 0.31/0.47 => ( ( Less @ B2 @ C )
% 0.31/0.47 => ( Less @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.strict_trans1
% 0.31/0.47 thf(fact_171_ordering_Ostrict__trans,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less @ A2 @ B2 )
% 0.31/0.47 => ( ( Less @ B2 @ C )
% 0.31/0.47 => ( Less @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.strict_trans
% 0.31/0.47 thf(fact_172_ordering_Oantisym,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less_eq @ A2 @ B2 )
% 0.31/0.47 => ( ( Less_eq @ B2 @ A2 )
% 0.31/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.antisym
% 0.31/0.47 thf(fact_173_ordering_Oirrefl,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ~ ( Less @ A2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.irrefl
% 0.31/0.47 thf(fact_174_ordering_Otrans,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less_eq @ A2 @ B2 )
% 0.31/0.47 => ( ( Less_eq @ B2 @ C )
% 0.31/0.47 => ( Less_eq @ A2 @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.trans
% 0.31/0.47 thf(fact_175_ordering_Ointro,axiom,
% 0.31/0.47 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 0.31/0.47 ( ! [A3: A,B3: A] :
% 0.31/0.47 ( ( Less @ A3 @ B3 )
% 0.31/0.47 = ( ( Less_eq @ A3 @ B3 )
% 0.31/0.47 & ( A3 != B3 ) ) )
% 0.31/0.47 => ( ! [A3: A] : ( Less_eq @ A3 @ A3 )
% 0.31/0.47 => ( ! [A3: A,B3: A] :
% 0.31/0.47 ( ( Less_eq @ A3 @ B3 )
% 0.31/0.47 => ( ( Less_eq @ B3 @ A3 )
% 0.31/0.47 => ( A3 = B3 ) ) )
% 0.31/0.47 => ( ! [A3: A,B3: A,C3: A] :
% 0.31/0.47 ( ( Less_eq @ A3 @ B3 )
% 0.31/0.47 => ( ( Less_eq @ B3 @ C3 )
% 0.31/0.47 => ( Less_eq @ A3 @ C3 ) ) )
% 0.31/0.47 => ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.intro
% 0.31/0.47 thf(fact_176_ordering_Orefl,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( Less_eq @ A2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.refl
% 0.31/0.47 thf(fact_177_ordering_Oasym,axiom,
% 0.31/0.47 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 0.31/0.47 ( ( ordering @ A @ Less_eq @ Less )
% 0.31/0.47 => ( ( Less @ A2 @ B2 )
% 0.31/0.47 => ~ ( Less @ B2 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering.asym
% 0.31/0.47 thf(fact_178_ordering__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordering @ A )
% 0.31/0.47 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 0.31/0.47 ( ! [A5: A,B5: A] :
% 0.31/0.47 ( ( Less2 @ A5 @ B5 )
% 0.31/0.47 = ( ( Less_eq2 @ A5 @ B5 )
% 0.31/0.47 & ( A5 != B5 ) ) )
% 0.31/0.47 & ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
% 0.31/0.47 & ! [A5: A,B5: A] :
% 0.31/0.47 ( ( Less_eq2 @ A5 @ B5 )
% 0.31/0.47 => ( ( Less_eq2 @ B5 @ A5 )
% 0.31/0.47 => ( A5 = B5 ) ) )
% 0.31/0.47 & ! [A5: A,B5: A,C4: A] :
% 0.31/0.47 ( ( Less_eq2 @ A5 @ B5 )
% 0.31/0.47 => ( ( Less_eq2 @ B5 @ C4 )
% 0.31/0.47 => ( Less_eq2 @ A5 @ C4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ordering_def
% 0.31/0.47 thf(fact_179_not__gr__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_gr_zero
% 0.31/0.47 thf(fact_180_le__zero__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_zero_eq
% 0.31/0.47 thf(fact_181_dag_Osize__gen_I1_J,axiom,
% 0.31/0.47 ( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
% 0.31/0.47 = ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % dag.size_gen(1)
% 0.31/0.47 thf(fact_182_zero__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_le
% 0.31/0.47 thf(fact_183_zero__less__iff__neq__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.31/0.47 = ( N
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_less_iff_neq_zero
% 0.31/0.47 thf(fact_184_gr__implies__not__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [M: A,N: A] :
% 0.31/0.47 ( ( ord_less @ A @ M @ N )
% 0.31/0.47 => ( N
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr_implies_not_zero
% 0.31/0.47 thf(fact_185_not__less__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less_zero
% 0.31/0.47 thf(fact_186_gr__zeroI,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( N
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr_zeroI
% 0.31/0.47 thf(fact_187_less__numeral__extra_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(3)
% 0.31/0.47 thf(fact_188_le__numeral__extra_I3_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(3)
% 0.31/0.47 thf(fact_189_sgn__greater,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sgn_greater
% 0.31/0.47 thf(fact_190_sgn__sgn,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
% 0.31/0.47 = ( sgn_sgn @ A @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sgn_sgn
% 0.31/0.47 thf(fact_191_sgn__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.31/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sgn_less
% 0.31/0.47 thf(fact_192_sgn__0__0,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ( sgn_sgn @ A @ A2 )
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sgn_0_0
% 0.31/0.47 thf(fact_193_sgn__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.47 => ( ( sgn_sgn @ A @ A2 )
% 0.31/0.47 = ( one_one @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sgn_pos
% 0.31/0.47 thf(fact_194_of__bool__eq_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( zero_neq_one_of_bool @ A @ $false )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % of_bool_eq(1)
% 0.31/0.47 thf(fact_195_of__bool__eq_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( zero_neq_one_of_bool @ A @ $true )
% 0.31/0.47 = ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % of_bool_eq(2)
% 0.31/0.47 thf(fact_196_less__numeral__extra_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(2)
% 0.31/0.47 thf(fact_197_less__numeral__extra_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(1)
% 0.31/0.47 thf(fact_198_le__numeral__extra_I1_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(1)
% 0.31/0.47 thf(fact_199_le__numeral__extra_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(2)
% 0.31/0.47 thf(fact_200_less__numeral__extra_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_numeral_extra(4)
% 0.31/0.47 thf(fact_201_le__numeral__extra_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_numeral_extra(4)
% 0.31/0.47 thf(fact_202_of__bool__eq__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P3: $o,Q3: $o] :
% 0.31/0.47 ( ( ( zero_neq_one_of_bool @ A @ P3 )
% 0.31/0.47 = ( zero_neq_one_of_bool @ A @ Q3 ) )
% 0.31/0.47 = ( P3 = Q3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % of_bool_eq_iff
% 0.31/0.47 thf(fact_203_zero__less__one,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_less_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_less_one
% 0.31/0.47 thf(fact_204_not__one__less__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_one_less_zero
% 0.31/0.47 thf(fact_205_zero__le__one,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_le_one
% 0.31/0.47 thf(fact_206_not__one__le__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_one_le_zero
% 0.31/0.47 thf(fact_207_of__bool__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( zero_neq_one_of_bool @ A )
% 0.31/0.47 = ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % of_bool_def
% 0.31/0.47 thf(fact_208_zero__neq__one,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ( zero_zero @ A )
% 0.31/0.47 != ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_neq_one
% 0.31/0.47 thf(fact_209_split__of__bool,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P3: $o] :
% 0.31/0.47 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 0.31/0.47 = ( ( P3
% 0.31/0.47 => ( P @ ( one_one @ A ) ) )
% 0.31/0.47 & ( ~ P3
% 0.31/0.47 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % split_of_bool
% 0.31/0.47 thf(fact_210_split__of__bool__asm,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [P: A > $o,P3: $o] :
% 0.31/0.47 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 0.31/0.47 = ( ~ ( ( P3
% 0.31/0.47 & ~ ( P @ ( one_one @ A ) ) )
% 0.31/0.47 | ( ~ P3
% 0.31/0.47 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % split_of_bool_asm
% 0.31/0.47 thf(fact_211_sgn__1__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ( sgn_sgn @ A @ A2 )
% 0.31/0.47 = ( one_one @ A ) )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sgn_1_pos
% 0.31/0.47 thf(fact_212_sgn__neg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( sgn_sgn @ A @ A2 )
% 0.31/0.47 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % sgn_neg
% 0.31/0.47 thf(fact_213_divide__le__eq__1__neg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_le_eq_1_neg
% 0.31/0.47 thf(fact_214_neg__le__iff__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_le_iff_le
% 0.31/0.47 thf(fact_215_neg__less__iff__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.47 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_less_iff_less
% 0.31/0.47 thf(fact_216_division__ring__divide__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % division_ring_divide_zero
% 0.31/0.47 thf(fact_217_divide__cancel__right,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,C: A,B2: A] :
% 0.31/0.47 ( ( ( divide_divide @ A @ A2 @ C )
% 0.31/0.47 = ( divide_divide @ A @ B2 @ C ) )
% 0.31/0.47 = ( ( C
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_cancel_right
% 0.31/0.47 thf(fact_218_divide__cancel__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [C: A,A2: A,B2: A] :
% 0.31/0.47 ( ( ( divide_divide @ A @ C @ A2 )
% 0.31/0.47 = ( divide_divide @ A @ C @ B2 ) )
% 0.31/0.47 = ( ( C
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_cancel_left
% 0.31/0.47 thf(fact_219_divide__eq__0__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ( divide_divide @ A @ A2 @ B2 )
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 = ( ( A2
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 | ( B2
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_eq_0_iff
% 0.31/0.47 thf(fact_220_divide__zero__left,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_zero_left
% 0.31/0.47 thf(fact_221_divide__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 = ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_zero
% 0.31/0.47 thf(fact_222_divide__1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 0.31/0.47 = A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_1
% 0.31/0.47 thf(fact_223_neg__0__le__iff__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_0_le_iff_le
% 0.31/0.47 thf(fact_224_neg__le__0__iff__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.31/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_le_0_iff_le
% 0.31/0.47 thf(fact_225_less__eq__neg__nonpos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_eq_neg_nonpos
% 0.31/0.47 thf(fact_226_neg__less__eq__nonneg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.31/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_less_eq_nonneg
% 0.31/0.47 thf(fact_227_neg__less__0__iff__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_less_0_iff_less
% 0.31/0.47 thf(fact_228_neg__0__less__iff__less,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_0_less_iff_less
% 0.31/0.47 thf(fact_229_neg__less__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % neg_less_pos
% 0.31/0.47 thf(fact_230_less__neg__neg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 0.31/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_neg_neg
% 0.31/0.47 thf(fact_231_zero__eq__1__divide__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ( zero_zero @ A )
% 0.31/0.47 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_eq_1_divide_iff
% 0.31/0.47 thf(fact_232_one__divide__eq__0__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % one_divide_eq_0_iff
% 0.31/0.47 thf(fact_233_eq__divide__eq__1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ( one_one @ A )
% 0.31/0.47 = ( divide_divide @ A @ B2 @ A2 ) )
% 0.31/0.47 = ( ( A2
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 & ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_divide_eq_1
% 0.31/0.47 thf(fact_234_divide__eq__eq__1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [B2: A,A2: A] :
% 0.31/0.47 ( ( ( divide_divide @ A @ B2 @ A2 )
% 0.31/0.47 = ( one_one @ A ) )
% 0.31/0.47 = ( ( A2
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 & ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_eq_eq_1
% 0.31/0.47 thf(fact_235_divide__self__if,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ( A2
% 0.31/0.47 = ( zero_zero @ A ) )
% 0.31/0.47 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.31/0.47 = ( zero_zero @ A ) ) )
% 0.31/0.47 & ( ( A2
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.31/0.47 = ( one_one @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_self_if
% 0.31/0.47 thf(fact_236_divide__self,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( division_ring @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.31/0.47 = ( one_one @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_self
% 0.31/0.47 thf(fact_237_one__eq__divide__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ( one_one @ A )
% 0.31/0.47 = ( divide_divide @ A @ A2 @ B2 ) )
% 0.31/0.47 = ( ( B2
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 & ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % one_eq_divide_iff
% 0.31/0.47 thf(fact_238_divide__eq__1__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ( divide_divide @ A @ A2 @ B2 )
% 0.31/0.47 = ( one_one @ A ) )
% 0.31/0.47 = ( ( B2
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 & ( A2 = B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_eq_1_iff
% 0.31/0.47 thf(fact_239_div__self,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 => ( ( divide_divide @ A @ A2 @ A2 )
% 0.31/0.47 = ( one_one @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % div_self
% 0.31/0.47 thf(fact_240_divide__minus1,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [X: A] :
% 0.31/0.47 ( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.31/0.47 = ( uminus_uminus @ A @ X ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_minus1
% 0.31/0.47 thf(fact_241_divide__le__0__1__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_le_0_1_iff
% 0.31/0.47 thf(fact_242_zero__le__divide__1__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_le_divide_1_iff
% 0.31/0.47 thf(fact_243_divide__less__0__1__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 0.31/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_less_0_1_iff
% 0.31/0.47 thf(fact_244_divide__less__eq__1__neg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 0.31/0.47 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_less_eq_1_neg
% 0.31/0.47 thf(fact_245_divide__less__eq__1__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.47 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 0.31/0.47 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_less_eq_1_pos
% 0.31/0.47 thf(fact_246_less__divide__eq__1__neg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 0.31/0.47 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_divide_eq_1_neg
% 0.31/0.47 thf(fact_247_less__divide__eq__1__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.47 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 0.31/0.47 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_divide_eq_1_pos
% 0.31/0.47 thf(fact_248_zero__less__divide__1__iff,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 0.31/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_less_divide_1_iff
% 0.31/0.47 thf(fact_249_le__divide__eq__1__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_divide_eq_1_pos
% 0.31/0.47 thf(fact_250_le__divide__eq__1__neg,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.31/0.47 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 0.31/0.47 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_divide_eq_1_neg
% 0.31/0.47 thf(fact_251_divide__le__eq__1__pos,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.31/0.47 => ! [A2: A,B2: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 0.31/0.47 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % divide_le_eq_1_pos
% 0.31/0.47 thf(fact_252_less__minus__one__simps_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_minus_one_simps(4)
% 0.31/0.47 thf(fact_253_less__minus__one__simps_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_minus_one_simps(2)
% 0.31/0.47 thf(fact_254_le__minus__one__simps_I2_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_minus_one_simps(2)
% 0.31/0.47 thf(fact_255_le__minus__one__simps_I4_J,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.31/0.47 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_minus_one_simps(4)
% 0.31/0.47
% 0.31/0.47 %----Type constructors (23)
% 0.31/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.31/0.47 ! [A6: $tType,A7: $tType] :
% 0.31/0.47 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.31/0.47 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.47 ! [A6: $tType,A7: $tType] :
% 0.31/0.47 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.31/0.47 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.31/0.47 ! [A6: $tType,A7: $tType] :
% 0.31/0.47 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.31/0.47 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.31/0.47 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.31/0.47 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.31/0.47 linord1659791738miring @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.31/0.47 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Osemidom__divide,axiom,
% 0.31/0.47 semidom_divide @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
% 0.31/0.47 zero_less_one @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.31/0.47 wellorder @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 0.31/0.47 zero_neq_one @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.31/0.47 preorder @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.31/0.47 linorder @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.31/0.47 no_top @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.31/0.47 order @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.31/0.47 ord @ nat @ ( type2 @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.31/0.47 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.31/0.47 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.31/0.47 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.31/0.47 preorder @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.31/0.47 linorder @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.31/0.47 order @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.31/0.47 ord @ $o @ ( type2 @ $o ) ).
% 0.31/0.47
% 0.31/0.47 %----Helper facts (3)
% 0.31/0.47 thf(help_If_3_1_T,axiom,
% 0.31/0.47 ! [P: $o] :
% 0.31/0.47 ( ( P = $true )
% 0.31/0.47 | ( P = $false ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_2_1_T,axiom,
% 0.31/0.47 ! [A: $tType,X: A,Y4: A] :
% 0.31/0.47 ( ( if @ A @ $false @ X @ Y4 )
% 0.31/0.47 = Y4 ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_1_1_T,axiom,
% 0.31/0.47 ! [A: $tType,X: A,Y4: A] :
% 0.31/0.47 ( ( if @ A @ $true @ X @ Y4 )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 %----Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 ord_less_eq @ binDag_Mirabelle_dag @ x @ z ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.67/1.59 ---- Embedded in THF ---
% 2.67/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.67/1.59 %%% Generated on Tue Feb 24 04:15:43 EST 2026
% 2.67/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.67/1.59 %%% Logic specification used:
% 2.67/1.59 %%% thf(spec, logic, $$dhol).
% 2.67/1.59
% 2.67/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.EGMR65SasR/DTF2DTF19573.p
% 2.67/1.59 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag, type, binDag_Mirabelle_dag: $tType).
% 2.67/1.59 thf(ty_t_Simpl__Heap_Oref, type, simpl_ref: $tType).
% 2.67/1.59 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.67/1.59 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.67/1.59 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.67/1.59 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord219039673up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.67/1.59 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG, type, binDag_Mirabelle_DAG: (binDag_Mirabelle_dag > $o)).
% 2.67/1.59 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode, type, binDag476092410e_Node: (binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > binDag_Mirabelle_dag)))).
% 2.67/1.59 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 2.67/1.59 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag, type, binDag1297733282se_dag: (!> [A:$tType]: ((A > ((binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))) > (binDag_Mirabelle_dag > A)))))).
% 2.67/1.59 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)))))).
% 2.67/1.59 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Osize__dag, type, binDag1924123185ze_dag: (binDag_Mirabelle_dag > nat)).
% 2.67/1.59 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of, type, binDag1380252983set_of: (binDag_Mirabelle_dag > (set @ simpl_ref))).
% 2.67/1.59 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag, type, binDag786255756subdag: (binDag_Mirabelle_dag > (binDag_Mirabelle_dag > $o))).
% 2.67/1.59 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.67/1.59 thf(sy_c_Groups_Osgn__class_Osgn, type, sgn_sgn: (!> [A:$tType]: ((A > A)))).
% 2.67/1.59 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.67/1.59 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.67/1.59 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.67/1.59 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.67/1.59 thf(sy_c_Orderings_Oord__class_OLeast, type, ord_Least: (!> [A:$tType]: (((A > $o) > A)))).
% 2.67/1.59 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.67/1.59 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.67/1.59 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.67/1.59 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.67/1.59 thf(sy_c_Orderings_Oordering, type, ordering: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 2.67/1.59 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.67/1.59 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 2.67/1.59 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool, type, zero_neq_one_of_bool: (!> [A:$tType]: (($o > A)))).
% 2.67/1.59 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.67/1.59 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.67/1.59 thf(sy_v_x, type, x: binDag_Mirabelle_dag).
% 2.67/1.59 thf(sy_v_y, type, y: binDag_Mirabelle_dag).
% 2.67/1.59 thf(sy_v_z, type, z: binDag_Mirabelle_dag).
% 2.67/1.59 thf(fact_0_True, axiom, (x = y)).
% 2.67/1.59 thf(fact_1_y__z, axiom, (((ord_less_eq @ binDag_Mirabelle_dag) @ y) @ z)).
% 2.67/1.59 thf(fact_2_x__y, axiom, (((ord_less_eq @ binDag_Mirabelle_dag) @ x) @ y)).
% 2.67/1.59 thf(fact_3_le__dag__refl, axiom, (! [X:binDag_Mirabelle_dag]: ((((ord_less_eq @ binDag_Mirabelle_dag) @ X) @ X)))).
% 2.67/1.59 thf(fact_4_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.67/1.59 thf(fact_5_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))))))))).
% 2.67/1.59 thf(fact_6_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))))))))).
% 2.67/1.59 thf(fact_7_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.67/1.59 thf(fact_8_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 2.67/1.59 thf(fact_9_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less_eq @ B) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.59 thf(fact_10_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.67/1.59 thf(fact_11_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less_eq @ B) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.59 thf(fact_12_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.67/1.59 thf(fact_13_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 2.67/1.59 thf(fact_14_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X) => (X = Y4))))))))).
% 2.67/1.59 thf(fact_15_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.67/1.59 thf(fact_16_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.67/1.59 thf(fact_17_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.67/1.59 thf(fact_18_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((! [A3:A,B3:A]: ((((P @ B3) @ A3) => ((P @ A3) @ B3)))) => ((P @ A2) @ B2))))))))).
% 2.67/1.59 thf(fact_19_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.67/1.59 thf(fact_20_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.67/1.59 thf(fact_21_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.67/1.59 thf(fact_22_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_23_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_24_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 2.67/1.59 thf(fact_25_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ Z2)))) => (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y4)))) => (((((ord_less_eq @ A) @ Z2) @ Y4) => (~ ((((ord_less_eq @ A) @ Y4) @ X)))) => (((((ord_less_eq @ A) @ Y4) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y4))))))))))))))))).
% 2.67/1.59 thf(fact_26_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_27_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.67/1.59 thf(fact_28_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X = Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 2.67/1.59 thf(fact_29_antimono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_antimono @ A) @ B) = (^ [F2:(A > B)]: ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F2 @ Y3)) @ (F2 @ X3)))))))))))).
% 2.67/1.59 thf(fact_30_antimonoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X2))))) => (((order_antimono @ A) @ B) @ F)))))))).
% 2.67/1.59 thf(fact_31_antimonoE, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X)))))))))).
% 2.67/1.59 thf(fact_32_antimonoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X)))))))))).
% 2.67/1.59 thf(fact_33_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X)) @ (F @ Y4)) = (((ord_less_eq @ A) @ X) @ Y4))))))))).
% 2.67/1.59 thf(fact_34_le__dag__def, axiom, ((ord_less_eq @ binDag_Mirabelle_dag) = (^ [S:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: (((S = T) | (((ord_less @ binDag_Mirabelle_dag) @ S) @ T)))))).
% 2.67/1.59 thf(fact_35_LeastI2__wellorder__ex, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),Q:(A > $o)]: (((? [X1:A]: ((P @ X1))) => ((! [A3:A]: (((P @ A3) => ((! [B4:A]: (((P @ B4) => (((ord_less_eq @ A) @ A3) @ B4)))) => (Q @ A3))))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 2.67/1.59 thf(fact_36_LeastI2__wellorder, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [A3:A]: (((P @ A3) => ((! [B4:A]: (((P @ B4) => (((ord_less_eq @ A) @ A3) @ B4)))) => (Q @ A3))))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 2.67/1.59 thf(fact_37_Least__equality, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ A) @ X) @ Y)))) => (((ord_Least @ A) @ P) = X))))))))).
% 2.67/1.59 thf(fact_38_LeastI2__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ A) @ X) @ Y)))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ X2) @ Y5)))) => (Q @ X2))))) => (Q @ ((ord_Least @ A) @ P))))))))))).
% 2.67/1.59 thf(fact_39_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less @ B) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.59 thf(fact_40_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 2.67/1.59 thf(fact_41_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less @ B) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.59 thf(fact_42_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.67/1.59 thf(fact_43_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y:A]: ((((ord_less @ A) @ Y) @ X))))))))).
% 2.67/1.59 thf(fact_44_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X12:A]: ((((ord_less @ A) @ X) @ X12))))))))).
% 2.67/1.59 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.67/1.59 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 2.67/1.59 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.67/1.59 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.67/1.59 thf(fact_49_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less @ A) @ Y4) @ X))))))))).
% 2.67/1.59 thf(fact_50_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) = ((((ord_less @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 2.67/1.59 thf(fact_51_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.67/1.59 thf(fact_52_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y4))))))))))).
% 2.67/1.59 thf(fact_53_LeastI2, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 2.67/1.59 thf(fact_54_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 2.67/1.59 thf(fact_55_strict__monoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ A) @ X) @ Y4) => (((ord_less @ B) @ (F @ X)) @ (F @ Y4)))))))))).
% 2.67/1.59 thf(fact_56_strict__monoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y))))) => (((order_strict_mono @ A) @ B) @ F)))))))).
% 2.67/1.59 thf(fact_57_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 2.67/1.59 thf(fact_58_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.67/1.59 thf(fact_59_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.67/1.59 thf(fact_60_LeastI__ex, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o)]: (((? [X1:A]: ((P @ X1))) => (P @ ((ord_Least @ A) @ P))))))))).
% 2.67/1.59 thf(fact_61_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) | ((X = Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 2.67/1.59 thf(fact_62_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.67/1.59 thf(fact_63_LeastI2__ex, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),Q:(A > $o)]: (((? [X1:A]: ((P @ X1))) => ((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 2.67/1.59 thf(fact_64_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_65_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_66_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 2.67/1.59 thf(fact_67_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 2.67/1.59 thf(fact_68_strict__mono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_strict_mono @ A) @ B) = (^ [F2:(A > B)]: ((! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less @ B) @ (F2 @ X3)) @ (F2 @ Y3)))))))))))).
% 2.67/1.59 thf(fact_69_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 2.67/1.59 thf(fact_70_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => (P @ Y5)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.67/1.59 thf(fact_71_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((~ ((((ord_less @ A) @ Y4) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 2.67/1.59 thf(fact_72_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (Y4 != X)))))))).
% 2.67/1.59 thf(fact_73_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,P:$o]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ X) => P)))))))).
% 2.67/1.59 thf(fact_74_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((X != Y4) => (((ord_less @ A) @ Y4) @ X))))))))).
% 2.67/1.59 thf(fact_75_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.67/1.59 thf(fact_76_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_77_strict__mono__less, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X)) @ (F @ Y4)) = (((ord_less @ A) @ X) @ Y4))))))))).
% 2.67/1.59 thf(fact_78_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 2.67/1.59 thf(fact_79_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.67/1.59 thf(fact_80_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = ((((ord_less @ A) @ Y4) @ X) | (X = Y4))))))))).
% 2.67/1.59 thf(fact_81_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.67/1.59 thf(fact_82_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.67/1.59 thf(fact_83_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X) = (F @ Y4)) = (X = Y4))))))))).
% 2.67/1.59 thf(fact_84_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less @ A) @ X) @ Y4)))))))))).
% 2.67/1.59 thf(fact_85_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.67/1.59 thf(fact_86_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) | (X3 = Y3))))))))).
% 2.67/1.59 thf(fact_87_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (X3 != Y3))))))))).
% 2.67/1.59 thf(fact_88_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less @ B) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.59 thf(fact_89_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.67/1.59 thf(fact_90_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less_eq @ B) @ X2) @ Y) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.67/1.59 thf(fact_91_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.67/1.59 thf(fact_92_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) = (((ord_less @ A) @ Y4) @ X)))))))).
% 2.67/1.59 thf(fact_93_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 2.67/1.59 thf(fact_94_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.67/1.59 thf(fact_95_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 2.67/1.59 thf(fact_96_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 2.67/1.59 thf(fact_97_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 2.67/1.59 thf(fact_98_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.67/1.59 thf(fact_99_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 2.67/1.59 thf(fact_100_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,Y4:A]: (((! [X2:A]: (((((ord_less @ A) @ Z2) @ X2) => (((ord_less_eq @ A) @ Y4) @ X2)))) => (((ord_less_eq @ A) @ Y4) @ Z2)))))))).
% 2.67/1.59 thf(fact_101_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y4:A,Z2:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less_eq @ A) @ X2) @ Z2)))) => (((ord_less_eq @ A) @ Y4) @ Z2)))))))).
% 2.67/1.59 thf(fact_102_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X)))))))).
% 2.67/1.59 thf(fact_103_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less @ A) @ X) @ Y4) | (X = Y4))))))))).
% 2.67/1.59 thf(fact_104_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X3))))))))))).
% 2.67/1.59 thf(fact_105_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y4) @ X))) => (((ord_less @ A) @ X) @ Y4)))))))).
% 2.67/1.59 thf(fact_106_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_107_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.67/1.59 thf(fact_108_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) | (A5 = B5))))))))).
% 2.67/1.59 thf(fact_109_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (A5 != B5))))))))).
% 2.67/1.59 thf(fact_110_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.67/1.59 thf(fact_111_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.67/1.59 thf(fact_112_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z2:A,X:A,Y4:A]: (((((ord_less @ A) @ Z2) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z2) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y4) @ W))))) => (((ord_less_eq @ A) @ Y4) @ Z2))))))))).
% 2.67/1.59 thf(fact_113_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y4) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y4) @ Z2))))))))).
% 2.67/1.59 thf(fact_114_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.67/1.59 thf(fact_115_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: (((((ord_less @ A) @ B5) @ A5) | (A5 = B5))))))))).
% 2.67/1.59 thf(fact_116_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B5:A,A5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (A5 != B5))))))))).
% 2.67/1.59 thf(fact_117_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.67/1.59 thf(fact_118_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.67/1.59 thf(fact_119_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (~ ((((ord_less_eq @ A) @ T2) @ X4)))))))))))))).
% 2.67/1.59 thf(fact_120_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((ord_less_eq @ A) @ X4) @ T2)))))))))))).
% 2.67/1.59 thf(fact_121_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((ord_less_eq @ A) @ T2) @ X4)))))))))))).
% 2.67/1.59 thf(fact_122_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ T2)))))))))))))).
% 2.67/1.59 thf(fact_123_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X4:A]: ((((((ord_less_eq @ A) @ A2) @ X4) & (((ord_less @ A) @ X4) @ C3)) => (P @ X4)))) & (! [D:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ D)) => (P @ X2)))) => (((ord_less_eq @ A) @ D) @ C3)))))))))))))))))).
% 2.67/1.59 thf(fact_124_less__dag__def, axiom, ((ord_less @ binDag_Mirabelle_dag) = (^ [S:binDag_Mirabelle_dag,T:binDag_Mirabelle_dag]: (((binDag786255756subdag @ T) @ S))))).
% 2.67/1.59 thf(fact_125_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B3:A]: (((((ord_less @ A) @ A2) @ B3) | (((ord_less @ A) @ B3) @ A2)))))))))).
% 2.67/1.59 thf(fact_126_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less @ A) @ Y4) @ X))))))))).
% 2.67/1.59 thf(fact_127_subdag__neq, axiom, (! [T2:binDag_Mirabelle_dag,S2:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T2) @ S2) => (T2 != S2))))).
% 2.67/1.59 thf(fact_128_subdag__trans, axiom, (! [T2:binDag_Mirabelle_dag,S2:binDag_Mirabelle_dag,R:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T2) @ S2) => (((binDag786255756subdag @ S2) @ R) => ((binDag786255756subdag @ T2) @ R)))))).
% 2.67/1.59 thf(fact_129_subdag__not__sym, axiom, (! [S2:binDag_Mirabelle_dag,T2:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ S2) @ T2) => (~ (((binDag786255756subdag @ T2) @ S2))))))).
% 2.67/1.59 thf(fact_130_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 2.67/1.59 thf(fact_131_minf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ X4) @ Z3) => (F3 = F3)))))))))))).
% 2.67/1.59 thf(fact_132_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (~ ((((ord_less @ A) @ T2) @ X4)))))))))))))).
% 2.67/1.59 thf(fact_133_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((ord_less @ A) @ X4) @ T2)))))))))))).
% 2.67/1.59 thf(fact_134_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (X4 != T2)))))))))))).
% 2.67/1.59 thf(fact_135_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (X4 != T2)))))))))))).
% 2.67/1.59 thf(fact_136_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 2.67/1.59 thf(fact_137_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z3) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 2.67/1.59 thf(fact_138_pinf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ Z3) @ X4) => (F3 = F3)))))))))))).
% 2.67/1.59 thf(fact_139_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((ord_less @ A) @ T2) @ X4)))))))))))).
% 2.67/1.59 thf(fact_140_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (~ ((((ord_less @ A) @ X4) @ T2)))))))))))))).
% 2.67/1.59 thf(fact_141_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (X4 != T2)))))))))))).
% 2.67/1.59 thf(fact_142_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (X4 != T2)))))))))))).
% 2.67/1.59 thf(fact_143_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 2.67/1.59 thf(fact_144_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X4:A]: (((((ord_less @ A) @ Z3) @ X4) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 2.67/1.59 thf(fact_145_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:((A > B) > (A > (B > $o)))]: (((! [R2:B,F4:(A > B),G3:(A > B),X2:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => ((F4 @ Y5) = (G3 @ Y5))))) => ((((P @ F4) @ X2) @ R2) = (((P @ G3) @ X2) @ R2))))) => ((! [X2:A,F4:(A > B)]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => (((P @ F4) @ Y5) @ (F4 @ Y5))))) => (? [X1:B]: ((((P @ F4) @ X2) @ X1)))))) => (? [F4:(A > B)]: ((! [X4:A]: ((((P @ F4) @ X4) @ (F4 @ X4)))))))))))))).
% 2.67/1.59 thf(fact_146_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X4:A]: ((? [Y:A]: ((((ord_less @ A) @ Y) @ X4))))))))).
% 2.67/1.59 thf(fact_147_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X4:A]: ((? [X12:A]: ((((ord_less @ A) @ X4) @ X12))))))))).
% 2.67/1.59 thf(fact_148_subdag_Osimps_I1_J, axiom, (! [T2:binDag_Mirabelle_dag]: ((~ (((binDag786255756subdag @ binDag_Mirabelle_Tip) @ T2)))))).
% 2.67/1.59 thf(fact_149_DAG_Osimps_I1_J, axiom, (binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip)).
% 2.67/1.59 thf(fact_150_dag_Osimps_I4_J, axiom, (! [A:$tType,F1:A,F22:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A)))]: ((((((binDag1297733282se_dag @ A) @ F1) @ F22) @ binDag_Mirabelle_Tip) = F1)))).
% 2.67/1.59 thf(fact_151_dag_Osimps_I6_J, axiom, (! [A:$tType,F1:A,F22:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > (A > (A > A)))))]: ((((((binDag1442713106ec_dag @ A) @ F1) @ F22) @ binDag_Mirabelle_Tip) = F1)))).
% 2.67/1.59 thf(fact_152_subdag__size, axiom, (! [T2:binDag_Mirabelle_dag,S2:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T2) @ S2) => (((ord_less @ nat) @ ((size_size @ binDag_Mirabelle_dag) @ S2)) @ ((size_size @ binDag_Mirabelle_dag) @ T2)))))).
% 2.67/1.59 thf(fact_153_dag_Osimps_I7_J, axiom, (! [A:$tType,F1:A,F22:(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) @ F22) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((((F22 @ X21) @ X22) @ X23) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F22) @ X21)) @ ((((binDag1442713106ec_dag @ A) @ F1) @ F22) @ X23)))))).
% 2.67/1.59 thf(fact_154_dag_Osimps_I5_J, axiom, (! [A:$tType,F1:A,F22:(binDag_Mirabelle_dag > (simpl_ref > (binDag_Mirabelle_dag > A))),X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((((((binDag1297733282se_dag @ A) @ F1) @ F22) @ (((binDag476092410e_Node @ X21) @ X22) @ X23)) = (((F22 @ X21) @ X22) @ X23))))).
% 2.67/1.59 thf(fact_155_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))))))).
% 2.67/1.59 thf(fact_156_subdag__NodeD, axiom, (! [T2:binDag_Mirabelle_dag,Lt:binDag_Mirabelle_dag,A2:simpl_ref,Rt:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ T2) @ (((binDag476092410e_Node @ Lt) @ A2) @ Rt)) => (((binDag786255756subdag @ T2) @ Lt) & ((binDag786255756subdag @ T2) @ Rt)))))).
% 2.67/1.59 thf(fact_157_subdag_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag,T2:binDag_Mirabelle_dag]: ((((binDag786255756subdag @ (((binDag476092410e_Node @ L) @ A2) @ R)) @ T2) = ((T2 = L) | ((T2 = R) | (((binDag786255756subdag @ L) @ T2) | ((binDag786255756subdag @ R) @ T2)))))))).
% 2.67/1.59 thf(fact_158_dag_Oexhaust, axiom, (! [Y4:binDag_Mirabelle_dag]: (((Y4 != binDag_Mirabelle_Tip) => (~ ((! [X212:binDag_Mirabelle_dag,X222:simpl_ref,X232:binDag_Mirabelle_dag]: ((Y4 != (((binDag476092410e_Node @ X212) @ X222) @ X232)))))))))).
% 2.67/1.59 thf(fact_159_dag_Oinduct, axiom, (! [P:(binDag_Mirabelle_dag > $o),Dag:binDag_Mirabelle_dag]: (((P @ binDag_Mirabelle_Tip) => ((! [X12:binDag_Mirabelle_dag,X24:simpl_ref,X32:binDag_Mirabelle_dag]: (((P @ X12) => ((P @ X32) => (P @ (((binDag476092410e_Node @ X12) @ X24) @ X32)))))) => (P @ Dag)))))).
% 2.67/1.59 thf(fact_160_dag_Odistinct_I1_J, axiom, (! [X21:binDag_Mirabelle_dag,X22:simpl_ref,X23:binDag_Mirabelle_dag]: ((binDag_Mirabelle_Tip != (((binDag476092410e_Node @ X21) @ X22) @ X23))))).
% 2.67/1.59 thf(fact_161_DAG_Osimps_I2_J, axiom, (! [L:binDag_Mirabelle_dag,A2:simpl_ref,R:binDag_Mirabelle_dag]: (((binDag_Mirabelle_DAG @ (((binDag476092410e_Node @ L) @ A2) @ R)) = ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ L)))) & ((~ ((((member @ simpl_ref) @ A2) @ (binDag1380252983set_of @ R)))) & ((binDag_Mirabelle_DAG @ L) & (binDag_Mirabelle_DAG @ R)))))))).
% 2.67/1.59 thf(fact_162_dag_Osize_I3_J, axiom, (((size_size @ binDag_Mirabelle_dag) @ binDag_Mirabelle_Tip) = (zero_zero @ nat))).
% 2.67/1.59 thf(fact_163_order_Oordering__axioms, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (((ordering @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 2.67/1.59 thf(fact_164_ordering_Onot__eq__order__implies__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => ((A2 != B2) => (((Less_eq @ A2) @ B2) => ((Less @ A2) @ B2))))))).
% 2.67/1.59 thf(fact_165_ordering_Ostrict__implies__not__eq, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B2) => (A2 != B2)))))).
% 2.67/1.59 thf(fact_166_ordering_Ostrict__implies__order, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B2) => ((Less_eq @ A2) @ B2)))))).
% 2.67/1.59 thf(fact_167_ordering_Ostrict__iff__order, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B2) = (((Less_eq @ A2) @ B2) & (A2 != B2))))))).
% 2.67/1.59 thf(fact_168_ordering_Oorder__iff__strict, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) = (((Less @ A2) @ B2) | (A2 = B2))))))).
% 2.67/1.59 thf(fact_169_ordering_Ostrict__trans2, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B2) => (((Less_eq @ B2) @ C) => ((Less @ A2) @ C))))))).
% 2.67/1.59 thf(fact_170_ordering_Ostrict__trans1, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) => (((Less @ B2) @ C) => ((Less @ A2) @ C))))))).
% 2.67/1.59 thf(fact_171_ordering_Ostrict__trans, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B2) => (((Less @ B2) @ C) => ((Less @ A2) @ C))))))).
% 2.67/1.59 thf(fact_172_ordering_Oantisym, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) => (((Less_eq @ B2) @ A2) => (A2 = B2))))))).
% 2.67/1.59 thf(fact_173_ordering_Oirrefl, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (~ (((Less @ A2) @ A2))))))).
% 2.67/1.59 thf(fact_174_ordering_Otrans, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less_eq @ A2) @ B2) => (((Less_eq @ B2) @ C) => ((Less_eq @ A2) @ C))))))).
% 2.67/1.59 thf(fact_175_ordering_Ointro, axiom, (! [A:$tType,Less:(A > (A > $o)),Less_eq:(A > (A > $o))]: (((! [A3:A,B3:A]: ((((Less @ A3) @ B3) = (((Less_eq @ A3) @ B3) & (A3 != B3))))) => ((! [A3:A]: (((Less_eq @ A3) @ A3))) => ((! [A3:A,B3:A]: ((((Less_eq @ A3) @ B3) => (((Less_eq @ B3) @ A3) => (A3 = B3))))) => ((! [A3:A,B3:A,C3:A]: ((((Less_eq @ A3) @ B3) => (((Less_eq @ B3) @ C3) => ((Less_eq @ A3) @ C3))))) => (((ordering @ A) @ Less_eq) @ Less)))))))).
% 2.67/1.59 thf(fact_176_ordering_Orefl, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A]: (((((ordering @ A) @ Less_eq) @ Less) => ((Less_eq @ A2) @ A2))))).
% 2.67/1.59 thf(fact_177_ordering_Oasym, axiom, (! [A:$tType,Less_eq:(A > (A > $o)),Less:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq) @ Less) => (((Less @ A2) @ B2) => (~ (((Less @ B2) @ A2)))))))).
% 2.67/1.59 thf(fact_178_ordering__def, axiom, (! [A:$tType]: (((ordering @ A) = (^ [Less_eq2:(A > (A > $o)),Less2:(A > (A > $o))]: (((! [A5:A,B5:A]: ((((Less2 @ A5) @ B5) = (((Less_eq2 @ A5) @ B5) & (A5 != B5))))) & ((! [A5:A]: (((Less_eq2 @ A5) @ A5))) & ((! [A5:A,B5:A]: ((((Less_eq2 @ A5) @ B5) => (((Less_eq2 @ B5) @ A5) => (A5 = B5))))) & (! [A5:A,B5:A,C4:A]: ((((Less_eq2 @ A5) @ B5) => (((Less_eq2 @ B5) @ C4) => ((Less_eq2 @ A5) @ C4)))))))))))))).
% 2.67/1.59 thf(fact_179_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_180_le__zero__eq, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_181_dag_Osize__gen_I1_J, axiom, ((binDag1924123185ze_dag @ binDag_Mirabelle_Tip) = (zero_zero @ nat))).
% 2.67/1.59 thf(fact_182_zero__le, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 2.67/1.59 thf(fact_183_zero__less__iff__neq__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_184_gr__implies__not__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_185_not__less__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.67/1.59 thf(fact_186_gr__zeroI, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type2 @ A)) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.67/1.59 thf(fact_187_less__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_188_le__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))).
% 2.67/1.59 thf(fact_189_sgn__greater, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((sgn_sgn @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_190_sgn__sgn, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: ((((sgn_sgn @ A) @ ((sgn_sgn @ A) @ A2)) = ((sgn_sgn @ A) @ A2)))))))).
% 2.67/1.59 thf(fact_191_sgn__less, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((sgn_sgn @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_192_sgn__0__0, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_193_sgn__pos, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((sgn_sgn @ A) @ A2) = (one_one @ A))))))))).
% 2.67/1.59 thf(fact_194_of__bool__eq_I1_J, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (((zero_neq_one_of_bool @ A) @ $false) = (zero_zero @ A)))))).
% 2.67/1.59 thf(fact_195_of__bool__eq_I2_J, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => (((zero_neq_one_of_bool @ A) @ $true) = (one_one @ A)))))).
% 2.67/1.59 thf(fact_196_less__numeral__extra_I2_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_197_less__numeral__extra_I1_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.67/1.59 thf(fact_198_le__numeral__extra_I1_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.67/1.59 thf(fact_199_le__numeral__extra_I2_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_200_less__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 2.67/1.59 thf(fact_201_le__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 2.67/1.59 thf(fact_202_of__bool__eq__iff, axiom, (! [A:$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)))))))).
% 2.67/1.59 thf(fact_203_zero__less__one, axiom, (! [A:$tType]: ((((zero_less_one @ A) @ (type2 @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.67/1.59 thf(fact_204_not__one__less__zero, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_205_zero__le__one, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 2.67/1.59 thf(fact_206_not__one__le__zero, axiom, (! [A:$tType]: ((((linord1659791738miring @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_207_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))))))))).
% 2.67/1.59 thf(fact_208_zero__neq__one, axiom, (! [A:$tType]: ((((zero_neq_one @ A) @ (type2 @ A)) => ((zero_zero @ A) != (one_one @ A)))))).
% 2.67/1.59 thf(fact_209_split__of__bool, axiom, (! [A:$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))))))))))).
% 2.67/1.59 thf(fact_210_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))))))))))))))).
% 2.67/1.59 thf(fact_211_sgn__1__pos, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((sgn_sgn @ A) @ A2) = (one_one @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_212_sgn__neg, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((sgn_sgn @ A) @ A2) = ((uminus_uminus @ A) @ (one_one @ A)))))))))).
% 2.67/1.59 thf(fact_213_divide__le__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ A2) @ B2))))))))).
% 2.67/1.59 thf(fact_214_neg__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.67/1.59 thf(fact_215_neg__less__iff__less, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.67/1.59 thf(fact_216_division__ring__divide__zero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_217_divide__cancel__right, axiom, (! [A:$tType]: ((((field @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: ((((((divide_divide @ A) @ A2) @ C) = (((divide_divide @ A) @ B2) @ C)) = ((C = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.67/1.59 thf(fact_218_divide__cancel__left, axiom, (! [A:$tType]: ((((field @ A) @ (type2 @ A)) => (! [C:A,A2:A,B2:A]: ((((((divide_divide @ A) @ C) @ A2) = (((divide_divide @ A) @ C) @ B2)) = ((C = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.67/1.59 thf(fact_219_divide__eq__0__iff, axiom, (! [A:$tType]: ((((field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 2.67/1.59 thf(fact_220_divide__zero__left, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type2 @ A)) => (! [A2:A]: (((((divide_divide @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_221_divide__zero, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type2 @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.67/1.59 thf(fact_222_divide__1, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type2 @ A)) => (! [A2:A]: (((((divide_divide @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 2.67/1.59 thf(fact_223_neg__0__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_224_neg__le__0__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_225_less__eq__neg__nonpos, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_226_neg__less__eq__nonneg, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_227_neg__less__0__iff__less, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_228_neg__0__less__iff__less, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_229_neg__less__pos, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A2)) @ A2) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_230_less__neg__neg, axiom, (! [A:$tType]: ((((linord219039673up_add @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ A2) @ ((uminus_uminus @ A) @ A2)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_231_zero__eq__1__divide__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: ((((zero_zero @ A) = (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (A2 = (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_232_one__divide__eq__0__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_233_eq__divide__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: ((((one_one @ A) = (((divide_divide @ A) @ B2) @ A2)) = ((A2 != (zero_zero @ A)) & (A2 = B2))))))))).
% 2.67/1.59 thf(fact_234_divide__eq__eq__1, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: ((((((divide_divide @ A) @ B2) @ A2) = (one_one @ A)) = ((A2 != (zero_zero @ A)) & (A2 = B2))))))))).
% 2.67/1.59 thf(fact_235_divide__self__if, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: ((((A2 = (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (zero_zero @ A))) & ((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A)))))))))).
% 2.67/1.59 thf(fact_236_divide__self, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A))))))))).
% 2.67/1.59 thf(fact_237_one__eq__divide__iff, axiom, (! [A:$tType]: ((((field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((one_one @ A) = (((divide_divide @ A) @ A2) @ B2)) = ((B2 != (zero_zero @ A)) & (A2 = B2))))))))).
% 2.67/1.59 thf(fact_238_divide__eq__1__iff, axiom, (! [A:$tType]: ((((field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((((divide_divide @ A) @ A2) @ B2) = (one_one @ A)) = ((B2 != (zero_zero @ A)) & (A2 = B2))))))))).
% 2.67/1.59 thf(fact_239_div__self, axiom, (! [A:$tType]: ((((semidom_divide @ A) @ (type2 @ A)) => (! [A2:A]: (((A2 != (zero_zero @ A)) => ((((divide_divide @ A) @ A2) @ A2) = (one_one @ A))))))))).
% 2.67/1.59 thf(fact_240_divide__minus1, axiom, (! [A:$tType]: ((((field @ A) @ (type2 @ A)) => (! [X:A]: (((((divide_divide @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ X)))))))).
% 2.67/1.59 thf(fact_241_divide__le__0__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_242_zero__le__divide__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_243_divide__less__0__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.67/1.59 thf(fact_244_divide__less__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ A2) @ B2))))))))).
% 2.67/1.59 thf(fact_245_divide__less__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less @ A) @ B2) @ A2))))))))).
% 2.67/1.59 thf(fact_246_less__divide__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less @ A) @ B2) @ A2))))))))).
% 2.67/1.59 thf(fact_247_less__divide__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less @ A) @ A2) @ B2))))))))).
% 2.67/1.59 thf(fact_248_zero__less__divide__1__iff, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.67/1.59 thf(fact_249_le__divide__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ A2) @ B2))))))))).
% 2.67/1.59 thf(fact_250_le__divide__eq__1__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A2)) = (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 2.67/1.59 thf(fact_251_divide__le__eq__1__pos, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A2)) @ (one_one @ A)) = (((ord_less_eq @ A) @ B2) @ A2))))))))).
% 2.67/1.59 thf(fact_252_less__minus__one__simps_I4_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 2.67/1.59 thf(fact_253_less__minus__one__simps_I2_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))).
% 2.67/1.59 thf(fact_254_le__minus__one__simps_I2_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))).
% 2.67/1.59 thf(fact_255_le__minus__one__simps_I4_J, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 2.67/1.59 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.67/1.59 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.67/1.59 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 2.67/1.59 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, ((linord1659791738miring @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Rings_Osemidom__divide, axiom, ((semidom_divide @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Rings_Ozero__less__one, axiom, ((zero_less_one @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Rings_Ozero__neq__one, axiom, ((zero_neq_one @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.67/1.59 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A6:$tType]: (((preorder @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.67/1.59 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.67/1.59 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.67/1.59 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.67/1.59 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.67/1.59 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y4:A]: ((((((if @ A) @ $false) @ X) @ Y4) = Y4)))).
% 2.67/1.59 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y4:A]: ((((((if @ A) @ $true) @ X) @ Y4) = X)))).
% 2.67/1.59 thf(conj_0, conjecture, (((ord_less_eq @ binDag_Mirabelle_dag) @ x) @ z)).
% 2.67/1.59 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.EGMR65SasR/DTF2DTF19573.p
% 2.67/1.59 ------------------------
% 2.95/1.61 ---- Cleaned THF ---
% 2.95/1.61 thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
% 2.95/1.61 binDag_Mirabelle_dag: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Simpl__Heap_Oref,type,
% 2.95/1.61 simpl_ref: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Set_Oset,type,
% 2.95/1.61 set: $tType > $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_Nat_Onat,type,
% 2.95/1.61 nat: $tType ).
% 2.95/1.61
% 2.95/1.61 thf(ty_t_itself,type,
% 2.95/1.61 itself: $tType > $tType ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_HOL_Otype,type,
% 2.95/1.61 type:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Fields_Ofield,type,
% 2.95/1.61 field:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Oord,type,
% 2.95/1.61 ord:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Oorder,type,
% 2.95/1.61 order:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Ono__bot,type,
% 2.95/1.61 no_bot:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Ono__top,type,
% 2.95/1.61 no_top:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Olinorder,type,
% 2.95/1.61 linorder:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Opreorder,type,
% 2.95/1.61 preorder:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Ozero__neq__one,type,
% 2.95/1.61 zero_neq_one:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Owellorder,type,
% 2.95/1.61 wellorder:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Ozero__less__one,type,
% 2.95/1.61 zero_less_one:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Fields_Odivision__ring,type,
% 2.95/1.61 division_ring:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Osemidom__divide,type,
% 2.95/1.61 semidom_divide:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Odense__order,type,
% 2.95/1.61 dense_order:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.95/1.61 linordered_idom:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Fields_Olinordered__field,type,
% 2.95/1.61 linordered_field:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Orderings_Odense__linorder,type,
% 2.95/1.61 dense_linorder:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.95/1.61 linordered_semidom:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.95/1.61 ordered_ab_group_add:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 2.95/1.61 linord219039673up_add:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.95/1.61 linord1659791738miring:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.95/1.61 canoni770627133id_add:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 2.95/1.61 condit1656338222tinuum:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 2.95/1.61 condit1037483654norder:
% 2.95/1.61 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
% 2.95/1.61 binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
% 2.95/1.61 binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
% 2.95/1.61 binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
% 2.95/1.61 binDag1297733282se_dag:
% 2.95/1.61 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
% 2.95/1.61 binDag1442713106ec_dag:
% 2.95/1.61 !>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Osize__dag,type,
% 2.95/1.61 binDag1924123185ze_dag: binDag_Mirabelle_dag > nat ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
% 2.95/1.61 binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
% 2.95/1.61 binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.95/1.61 one_one:
% 2.95/1.61 !>[A: $tType] : A ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 2.95/1.61 sgn_sgn:
% 2.95/1.61 !>[A: $tType] : ( A > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.95/1.61 uminus_uminus:
% 2.95/1.61 !>[A: $tType] : ( A > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.95/1.61 zero_zero:
% 2.95/1.61 !>[A: $tType] : A ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_If,type,
% 2.95/1.61 if:
% 2.95/1.61 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.95/1.61 size_size:
% 2.95/1.61 !>[A: $tType] : ( A > nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Oord__class_OLeast,type,
% 2.95/1.61 ord_Least:
% 2.95/1.61 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.95/1.61 ord_less:
% 2.95/1.61 !>[A: $tType] : ( A > A > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.95/1.61 ord_less_eq:
% 2.95/1.61 !>[A: $tType] : ( A > A > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 2.95/1.61 order_antimono:
% 2.95/1.61 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 2.95/1.61 order_strict_mono:
% 2.95/1.61 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Orderings_Oordering,type,
% 2.95/1.61 ordering:
% 2.95/1.61 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Pure_Otype,type,
% 2.95/1.61 type2:
% 2.95/1.61 !>[A: $tType] : ( itself @ A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 2.95/1.61 divide_divide:
% 2.95/1.61 !>[A: $tType] : ( A > A > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 2.95/1.61 zero_neq_one_of_bool:
% 2.95/1.61 !>[A: $tType] : ( $o > A ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_Set_OCollect,type,
% 2.95/1.61 collect:
% 2.95/1.61 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_c_member,type,
% 2.95/1.61 member:
% 2.95/1.61 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_x,type,
% 2.95/1.61 x: binDag_Mirabelle_dag ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_y,type,
% 2.95/1.61 y: binDag_Mirabelle_dag ).
% 2.95/1.61
% 2.95/1.61 thf(sy_v_z,type,
% 2.95/1.61 z: binDag_Mirabelle_dag ).
% 2.95/1.61
% 2.95/1.61 thf(fact_0_True,axiom,
% 2.95/1.61 x = y ).
% 2.95/1.61
% 2.95/1.61 thf(fact_1_y__z,axiom,
% 2.95/1.61 ord_less_eq @ binDag_Mirabelle_dag @ y @ z ).
% 2.95/1.61
% 2.95/1.61 thf(fact_2_x__y,axiom,
% 2.95/1.61 ord_less_eq @ binDag_Mirabelle_dag @ x @ y ).
% 2.95/1.61
% 2.95/1.61 thf(fact_3_le__dag__refl,axiom,
% 2.95/1.61 ! [X: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X @ X ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_4_order__refl,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_5_le__funD,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 => ! [F: A > B,G: A > B,X: A] :
% 2.95/1.61 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_6_le__funE,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 => ! [F: A > B,G: A > B,X: A] :
% 2.95/1.61 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_7_le__funI,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 => ! [F: A > B,G: A > B] :
% 2.95/1.61 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 2.95/1.61 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_8_le__fun__def,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 => ( ( ord_less_eq @ ( A > B ) )
% 2.95/1.61 = ( ^ [F2: A > B,G2: A > B] :
% 2.95/1.61 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_9_order__subst1,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( order @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.95/1.61 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.95/1.61 => ( ! [X2: B,Y: B] :
% 2.95/1.61 ( ( ord_less_eq @ B @ X2 @ Y )
% 2.95/1.61 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_10_order__subst2,axiom,
% 2.95/1.61 ! [A: $tType,C2: $tType] :
% 2.95/1.61 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.95/1.61 => ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_11_ord__eq__le__subst,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( ord @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.95/1.61 ( ( A2
% 2.95/1.61 = ( F @ B2 ) )
% 2.95/1.61 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.95/1.61 => ( ! [X2: B,Y: B] :
% 2.95/1.61 ( ( ord_less_eq @ B @ X2 @ Y )
% 2.95/1.61 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_12_ord__le__eq__subst,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( ord @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ( F @ B2 )
% 2.95/1.61 = C )
% 2.95/1.61 => ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_13_eq__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 2.95/1.61 = ( ^ [X3: A,Y3: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.95/1.61 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_14_antisym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ Y4 @ X )
% 2.95/1.61 => ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_15_linear,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_16_dual__order_Oantisym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_17_dual__order_Otrans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.95/1.61 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_18_linorder__wlog,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ! [A3: A,B3: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.95/1.61 => ( P @ A3 @ B3 ) )
% 2.95/1.61 => ( ! [A3: A,B3: A] :
% 2.95/1.61 ( ( P @ B3 @ A3 )
% 2.95/1.61 => ( P @ A3 @ B3 ) )
% 2.95/1.61 => ( P @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_19_dual__order_Orefl,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_20_order__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A,Z2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.95/1.61 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_21_order__class_Oorder_Oantisym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.95/1.61 => ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_22_ord__le__eq__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ord @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( B2 = C )
% 2.95/1.61 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_23_ord__eq__le__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ord @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( A2 = B2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.95/1.61 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_24_antisym__conv,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [Y4: A,X: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ Y4 @ X )
% 2.95/1.61 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 = ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_25_le__cases3,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A,Z2: A] :
% 2.95/1.61 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 2.95/1.61 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.95/1.61 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 2.95/1.61 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 2.95/1.61 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.95/1.61 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_26_order_Otrans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.95/1.61 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_27_le__cases,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_28_eq__refl,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( X = Y4 )
% 2.95/1.61 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_29_antimono__def,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ( ( order_antimono @ A @ B )
% 2.95/1.61 = ( ^ [F2: A > B] :
% 2.95/1.61 ! [X3: A,Y3: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_30_antimonoI,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B] :
% 2.95/1.61 ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) ) )
% 2.95/1.61 => ( order_antimono @ A @ B @ F ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_31_antimonoE,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B,X: A,Y4: A] :
% 2.95/1.61 ( ( order_antimono @ A @ B @ F )
% 2.95/1.61 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_32_antimonoD,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B,X: A,Y4: A] :
% 2.95/1.61 ( ( order_antimono @ A @ B @ F )
% 2.95/1.61 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_33_strict__mono__less__eq,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B,X: A,Y4: A] :
% 2.95/1.61 ( ( order_strict_mono @ A @ B @ F )
% 2.95/1.61 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_34_le__dag__def,axiom,
% 2.95/1.61 ( ( ord_less_eq @ binDag_Mirabelle_dag )
% 2.95/1.61 = ( ^ [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( S = T )
% 2.95/1.61 | ( ord_less @ binDag_Mirabelle_dag @ S @ T ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_35_LeastI2__wellorder__ex,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,Q: A > $o] :
% 2.95/1.61 ( ? [X1: A] : ( P @ X1 )
% 2.95/1.61 => ( ! [A3: A] :
% 2.95/1.61 ( ( P @ A3 )
% 2.95/1.61 => ( ! [B4: A] :
% 2.95/1.61 ( ( P @ B4 )
% 2.95/1.61 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 2.95/1.61 => ( Q @ A3 ) ) )
% 2.95/1.61 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_36_LeastI2__wellorder,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,A2: A,Q: A > $o] :
% 2.95/1.61 ( ( P @ A2 )
% 2.95/1.61 => ( ! [A3: A] :
% 2.95/1.61 ( ( P @ A3 )
% 2.95/1.61 => ( ! [B4: A] :
% 2.95/1.61 ( ( P @ B4 )
% 2.95/1.61 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 2.95/1.61 => ( Q @ A3 ) ) )
% 2.95/1.61 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_37_Least__equality,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,X: A] :
% 2.95/1.61 ( ( P @ X )
% 2.95/1.61 => ( ! [Y: A] :
% 2.95/1.61 ( ( P @ Y )
% 2.95/1.61 => ( ord_less_eq @ A @ X @ Y ) )
% 2.95/1.61 => ( ( ord_Least @ A @ P )
% 2.95/1.61 = X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_38_LeastI2__order,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,X: A,Q: A > $o] :
% 2.95/1.61 ( ( P @ X )
% 2.95/1.61 => ( ! [Y: A] :
% 2.95/1.61 ( ( P @ Y )
% 2.95/1.61 => ( ord_less_eq @ A @ X @ Y ) )
% 2.95/1.61 => ( ! [X2: A] :
% 2.95/1.61 ( ( P @ X2 )
% 2.95/1.61 => ( ! [Y5: A] :
% 2.95/1.61 ( ( P @ Y5 )
% 2.95/1.61 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 2.95/1.61 => ( Q @ X2 ) ) )
% 2.95/1.61 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_39_ord__eq__less__subst,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( ord @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.95/1.61 ( ( A2
% 2.95/1.61 = ( F @ B2 ) )
% 2.95/1.61 => ( ( ord_less @ B @ B2 @ C )
% 2.95/1.61 => ( ! [X2: B,Y: B] :
% 2.95/1.61 ( ( ord_less @ B @ X2 @ Y )
% 2.95/1.61 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_40_ord__less__eq__subst,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( ord @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ( F @ B2 )
% 2.95/1.61 = C )
% 2.95/1.61 => ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_41_order__less__subst1,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( order @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.95/1.61 => ( ( ord_less @ B @ B2 @ C )
% 2.95/1.61 => ( ! [X2: B,Y: B] :
% 2.95/1.61 ( ( ord_less @ B @ X2 @ Y )
% 2.95/1.61 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_42_order__less__subst2,axiom,
% 2.95/1.61 ! [A: $tType,C2: $tType] :
% 2.95/1.61 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 2.95/1.61 => ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_43_lt__ex,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( no_bot @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A] :
% 2.95/1.61 ? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_44_gt__ex,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( no_top @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A] :
% 2.95/1.61 ? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_45_mem__Collect__eq,axiom,
% 2.95/1.61 ! [A: $tType,A2: A,P: A > $o] :
% 2.95/1.61 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.95/1.61 = ( P @ A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_46_Collect__mem__eq,axiom,
% 2.95/1.61 ! [A: $tType,A4: set @ A] :
% 2.95/1.61 ( ( collect @ A
% 2.95/1.61 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 2.95/1.61 = A4 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_47_Collect__cong,axiom,
% 2.95/1.61 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( P @ X2 )
% 2.95/1.61 = ( Q @ X2 ) )
% 2.95/1.61 => ( ( collect @ A @ P )
% 2.95/1.61 = ( collect @ A @ Q ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_48_ext,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( F @ X2 )
% 2.95/1.61 = ( G @ X2 ) )
% 2.95/1.61 => ( F = G ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_49_neqE,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( X != Y4 )
% 2.95/1.61 => ( ~ ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_50_neq__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( X != Y4 )
% 2.95/1.61 = ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_51_order_Oasym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_52_dense,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( dense_order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ? [Z3: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Z3 )
% 2.95/1.61 & ( ord_less @ A @ Z3 @ Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_53_LeastI2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,A2: A,Q: A > $o] :
% 2.95/1.61 ( ( P @ A2 )
% 2.95/1.61 => ( ! [X2: A] :
% 2.95/1.61 ( ( P @ X2 )
% 2.95/1.61 => ( Q @ X2 ) )
% 2.95/1.61 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_54_less__imp__neq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( X != Y4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_55_strict__monoD,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B,X: A,Y4: A] :
% 2.95/1.61 ( ( order_strict_mono @ A @ B @ F )
% 2.95/1.61 => ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_56_strict__monoI,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B] :
% 2.95/1.61 ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_57_less__asym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_58_less__asym_H,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_59_less__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A,Z2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less @ A @ Y4 @ Z2 )
% 2.95/1.61 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_60_LeastI__ex,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o] :
% 2.95/1.61 ( ? [X1: A] : ( P @ X1 )
% 2.95/1.61 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_61_less__linear,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 | ( X = Y4 )
% 2.95/1.61 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_62_less__irrefl,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A] :
% 2.95/1.61 ~ ( ord_less @ A @ X @ X ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_63_LeastI2__ex,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,Q: A > $o] :
% 2.95/1.61 ( ? [X1: A] : ( P @ X1 )
% 2.95/1.61 => ( ! [X2: A] :
% 2.95/1.61 ( ( P @ X2 )
% 2.95/1.61 => ( Q @ X2 ) )
% 2.95/1.61 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_64_ord__eq__less__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ord @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( A2 = B2 )
% 2.95/1.61 => ( ( ord_less @ A @ B2 @ C )
% 2.95/1.61 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_65_ord__less__eq__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ord @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( B2 = C )
% 2.95/1.61 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_66_dual__order_Oasym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ B2 @ A2 )
% 2.95/1.61 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_67_less__imp__not__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( X != Y4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_68_strict__mono__def,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ( ( order_strict_mono @ A @ B )
% 2.95/1.61 = ( ^ [F2: A > B] :
% 2.95/1.61 ! [X3: A,Y3: A] :
% 2.95/1.61 ( ( ord_less @ A @ X3 @ Y3 )
% 2.95/1.61 => ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_69_less__not__sym,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_70_less__induct,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,A2: A] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ! [Y5: A] :
% 2.95/1.61 ( ( ord_less @ A @ Y5 @ X2 )
% 2.95/1.61 => ( P @ Y5 ) )
% 2.95/1.61 => ( P @ X2 ) )
% 2.95/1.61 => ( P @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_71_antisym__conv3,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [Y4: A,X: A] :
% 2.95/1.61 ( ~ ( ord_less @ A @ Y4 @ X )
% 2.95/1.61 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.95/1.61 = ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_72_less__imp__not__eq2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( Y4 != X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_73_less__imp__triv,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A,P: $o] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less @ A @ Y4 @ X )
% 2.95/1.61 => P ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_74_linorder__cases,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ~ ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ( X != Y4 )
% 2.95/1.61 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_75_dual__order_Oirrefl,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_76_order_Ostrict__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less @ A @ B2 @ C )
% 2.95/1.61 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_77_strict__mono__less,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B,X: A,Y4: A] :
% 2.95/1.61 ( ( order_strict_mono @ A @ B @ F )
% 2.95/1.61 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 2.95/1.61 = ( ord_less @ A @ X @ Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_78_less__imp__not__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_79_dual__order_Ostrict__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ B2 @ A2 )
% 2.95/1.61 => ( ( ord_less @ A @ C @ B2 )
% 2.95/1.61 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_80_not__less__iff__gr__or__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.95/1.61 = ( ( ord_less @ A @ Y4 @ X )
% 2.95/1.61 | ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_81_order_Ostrict__implies__not__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( A2 != B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_82_dual__order_Ostrict__implies__not__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ B2 @ A2 )
% 2.95/1.61 => ( A2 != B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_83_strict__mono__eq,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 & ( order @ B @ ( type2 @ B ) ) )
% 2.95/1.61 => ! [F: A > B,X: A,Y4: A] :
% 2.95/1.61 ( ( order_strict_mono @ A @ B @ F )
% 2.95/1.61 => ( ( ( F @ X )
% 2.95/1.61 = ( F @ Y4 ) )
% 2.95/1.61 = ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_84_leD,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [Y4: A,X: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ Y4 @ X )
% 2.95/1.61 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_85_leI,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ~ ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_86_le__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ord_less_eq @ A )
% 2.95/1.61 = ( ^ [X3: A,Y3: A] :
% 2.95/1.61 ( ( ord_less @ A @ X3 @ Y3 )
% 2.95/1.61 | ( X3 = Y3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_87_less__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ord_less @ A )
% 2.95/1.61 = ( ^ [X3: A,Y3: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.95/1.61 & ( X3 != Y3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_88_order__le__less__subst1,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( order @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.95/1.61 => ( ( ord_less @ B @ B2 @ C )
% 2.95/1.61 => ( ! [X2: B,Y: B] :
% 2.95/1.61 ( ( ord_less @ B @ X2 @ Y )
% 2.95/1.61 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_89_order__le__less__subst2,axiom,
% 2.95/1.61 ! [A: $tType,C2: $tType] :
% 2.95/1.61 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 2.95/1.61 => ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_90_order__less__le__subst1,axiom,
% 2.95/1.61 ! [A: $tType,B: $tType] :
% 2.95/1.61 ( ( ( order @ B @ ( type2 @ B ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 2.95/1.61 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.95/1.61 => ( ! [X2: B,Y: B] :
% 2.95/1.61 ( ( ord_less_eq @ B @ X2 @ Y )
% 2.95/1.61 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_91_order__less__le__subst2,axiom,
% 2.95/1.61 ! [A: $tType,C2: $tType] :
% 2.95/1.61 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 2.95/1.61 & ( order @ A @ ( type2 @ A ) ) )
% 2.95/1.61 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 2.95/1.61 => ( ! [X2: A,Y: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Y )
% 2.95/1.61 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 2.95/1.61 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_92_not__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 2.95/1.61 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_93_not__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_94_le__neq__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( A2 != B2 )
% 2.95/1.61 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_95_less__imp__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_96_antisym__conv1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ~ ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 = ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_97_antisym__conv2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 2.95/1.61 = ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_98_le__less__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A,Z2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less @ A @ Y4 @ Z2 )
% 2.95/1.61 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_99_less__le__trans,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A,Z2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 2.95/1.61 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_100_dense__ge,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [Z2: A,Y4: A] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z2 @ X2 )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_101_dense__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [Y4: A,Z2: A] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Y4 )
% 2.95/1.61 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_102_le__less__linear,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_103_le__imp__less__or__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X @ Y4 )
% 2.95/1.61 => ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 | ( X = Y4 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_104_less__le__not__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( preorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ord_less @ A )
% 2.95/1.61 = ( ^ [X3: A,Y3: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.95/1.61 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_105_not__le__imp__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [Y4: A,X: A] :
% 2.95/1.61 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 2.95/1.61 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_106_order_Ostrict__trans1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less @ A @ B2 @ C )
% 2.95/1.61 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_107_order_Ostrict__trans2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.95/1.61 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_108_order_Oorder__iff__strict,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ord_less_eq @ A )
% 2.95/1.61 = ( ^ [A5: A,B5: A] :
% 2.95/1.61 ( ( ord_less @ A @ A5 @ B5 )
% 2.95/1.61 | ( A5 = B5 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_109_order_Ostrict__iff__order,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ord_less @ A )
% 2.95/1.61 = ( ^ [A5: A,B5: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.95/1.61 & ( A5 != B5 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_110_dual__order_Ostrict__trans1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.95/1.61 => ( ( ord_less @ A @ C @ B2 )
% 2.95/1.61 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_111_dual__order_Ostrict__trans2,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A,C: A] :
% 2.95/1.61 ( ( ord_less @ A @ B2 @ A2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.95/1.61 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_112_dense__ge__bounded,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [Z2: A,X: A,Y4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z2 @ X )
% 2.95/1.61 => ( ! [W: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z2 @ W )
% 2.95/1.61 => ( ( ord_less @ A @ W @ X )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_113_dense__le__bounded,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A,Z2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ! [W: A] :
% 2.95/1.61 ( ( ord_less @ A @ X @ W )
% 2.95/1.61 => ( ( ord_less @ A @ W @ Y4 )
% 2.95/1.61 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 2.95/1.61 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_114_order_Ostrict__implies__order,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_115_dual__order_Oorder__iff__strict,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ord_less_eq @ A )
% 2.95/1.61 = ( ^ [B5: A,A5: A] :
% 2.95/1.61 ( ( ord_less @ A @ B5 @ A5 )
% 2.95/1.61 | ( A5 = B5 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_116_dual__order_Ostrict__iff__order,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( ord_less @ A )
% 2.95/1.61 = ( ^ [B5: A,A5: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ B5 @ A5 )
% 2.95/1.61 & ( A5 != B5 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_117_dual__order_Ostrict__implies__order,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ B2 @ A2 )
% 2.95/1.61 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_118_order_Onot__eq__order__implies__strict,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( A2 != B2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.95/1.61 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_119_minf_I8_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_120_minf_I6_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_121_pinf_I8_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_122_pinf_I6_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_123_complete__interval,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A,P: A > $o] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B2 )
% 2.95/1.61 => ( ( P @ A2 )
% 2.95/1.61 => ( ~ ( P @ B2 )
% 2.95/1.61 => ? [C3: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ C3 )
% 2.95/1.61 & ( ord_less_eq @ A @ C3 @ B2 )
% 2.95/1.61 & ! [X4: A] :
% 2.95/1.61 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 2.95/1.61 & ( ord_less @ A @ X4 @ C3 ) )
% 2.95/1.61 => ( P @ X4 ) )
% 2.95/1.61 & ! [D: A] :
% 2.95/1.61 ( ! [X2: A] :
% 2.95/1.61 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 2.95/1.61 & ( ord_less @ A @ X2 @ D ) )
% 2.95/1.61 => ( P @ X2 ) )
% 2.95/1.61 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_124_less__dag__def,axiom,
% 2.95/1.61 ( ( ord_less @ binDag_Mirabelle_dag )
% 2.95/1.61 = ( ^ [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T @ S ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_125_ex__gt__or__lt,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ? [B3: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ B3 )
% 2.95/1.61 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_126_linorder__neqE__linordered__idom,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A,Y4: A] :
% 2.95/1.61 ( ( X != Y4 )
% 2.95/1.61 => ( ~ ( ord_less @ A @ X @ Y4 )
% 2.95/1.61 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_127_subdag__neq,axiom,
% 2.95/1.61 ! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag786255756subdag @ T2 @ S2 )
% 2.95/1.61 => ( T2 != S2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_128_subdag__trans,axiom,
% 2.95/1.61 ! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag786255756subdag @ T2 @ S2 )
% 2.95/1.61 => ( ( binDag786255756subdag @ S2 @ R )
% 2.95/1.61 => ( binDag786255756subdag @ T2 @ R ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_129_subdag__not__sym,axiom,
% 2.95/1.61 ! [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag786255756subdag @ S2 @ T2 )
% 2.95/1.61 => ~ ( binDag786255756subdag @ T2 @ S2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_130_less__fun__def,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( ord @ B @ ( type2 @ B ) )
% 2.95/1.61 => ( ( ord_less @ ( A > B ) )
% 2.95/1.61 = ( ^ [F2: A > B,G2: A > B] :
% 2.95/1.61 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 2.95/1.61 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_131_minf_I11_J,axiom,
% 2.95/1.61 ! [C2: $tType,D2: $tType] :
% 2.95/1.61 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.95/1.61 => ! [F3: D2] :
% 2.95/1.61 ? [Z3: C2] :
% 2.95/1.61 ! [X4: C2] :
% 2.95/1.61 ( ( ord_less @ C2 @ X4 @ Z3 )
% 2.95/1.61 => ( F3 = F3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_132_minf_I7_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_133_minf_I5_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_134_minf_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ( X4 != T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_135_minf_I3_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ( X4 != T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_136_minf_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.95/1.61 ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Z4 )
% 2.95/1.61 => ( ( P @ X2 )
% 2.95/1.61 = ( P2 @ X2 ) ) )
% 2.95/1.61 => ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Z4 )
% 2.95/1.61 => ( ( Q @ X2 )
% 2.95/1.61 = ( Q2 @ X2 ) ) )
% 2.95/1.61 => ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ( ( ( P @ X4 )
% 2.95/1.61 | ( Q @ X4 ) )
% 2.95/1.61 = ( ( P2 @ X4 )
% 2.95/1.61 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_137_minf_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.95/1.61 ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Z4 )
% 2.95/1.61 => ( ( P @ X2 )
% 2.95/1.61 = ( P2 @ X2 ) ) )
% 2.95/1.61 => ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ X2 @ Z4 )
% 2.95/1.61 => ( ( Q @ X2 )
% 2.95/1.61 = ( Q2 @ X2 ) ) )
% 2.95/1.61 => ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ X4 @ Z3 )
% 2.95/1.61 => ( ( ( P @ X4 )
% 2.95/1.61 & ( Q @ X4 ) )
% 2.95/1.61 = ( ( P2 @ X4 )
% 2.95/1.61 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_138_pinf_I11_J,axiom,
% 2.95/1.61 ! [C2: $tType,D2: $tType] :
% 2.95/1.61 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 2.95/1.61 => ! [F3: D2] :
% 2.95/1.61 ? [Z3: C2] :
% 2.95/1.61 ! [X4: C2] :
% 2.95/1.61 ( ( ord_less @ C2 @ Z3 @ X4 )
% 2.95/1.61 => ( F3 = F3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_139_pinf_I7_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_140_pinf_I5_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_141_pinf_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ( X4 != T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_142_pinf_I3_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [T2: A] :
% 2.95/1.61 ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ( X4 != T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_143_pinf_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.95/1.61 ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z4 @ X2 )
% 2.95/1.61 => ( ( P @ X2 )
% 2.95/1.61 = ( P2 @ X2 ) ) )
% 2.95/1.61 => ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z4 @ X2 )
% 2.95/1.61 => ( ( Q @ X2 )
% 2.95/1.61 = ( Q2 @ X2 ) ) )
% 2.95/1.61 => ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ( ( ( P @ X4 )
% 2.95/1.61 | ( Q @ X4 ) )
% 2.95/1.61 = ( ( P2 @ X4 )
% 2.95/1.61 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_144_pinf_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 2.95/1.61 ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z4 @ X2 )
% 2.95/1.61 => ( ( P @ X2 )
% 2.95/1.61 = ( P2 @ X2 ) ) )
% 2.95/1.61 => ( ? [Z4: A] :
% 2.95/1.61 ! [X2: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z4 @ X2 )
% 2.95/1.61 => ( ( Q @ X2 )
% 2.95/1.61 = ( Q2 @ X2 ) ) )
% 2.95/1.61 => ? [Z3: A] :
% 2.95/1.61 ! [X4: A] :
% 2.95/1.61 ( ( ord_less @ A @ Z3 @ X4 )
% 2.95/1.61 => ( ( ( P @ X4 )
% 2.95/1.61 & ( Q @ X4 ) )
% 2.95/1.61 = ( ( P2 @ X4 )
% 2.95/1.61 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_145_dependent__wellorder__choice,axiom,
% 2.95/1.61 ! [B: $tType,A: $tType] :
% 2.95/1.61 ( ( wellorder @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: ( A > B ) > A > B > $o] :
% 2.95/1.61 ( ! [R2: B,F4: A > B,G3: A > B,X2: A] :
% 2.95/1.61 ( ! [Y5: A] :
% 2.95/1.61 ( ( ord_less @ A @ Y5 @ X2 )
% 2.95/1.61 => ( ( F4 @ Y5 )
% 2.95/1.61 = ( G3 @ Y5 ) ) )
% 2.95/1.61 => ( ( P @ F4 @ X2 @ R2 )
% 2.95/1.61 = ( P @ G3 @ X2 @ R2 ) ) )
% 2.95/1.61 => ( ! [X2: A,F4: A > B] :
% 2.95/1.61 ( ! [Y5: A] :
% 2.95/1.61 ( ( ord_less @ A @ Y5 @ X2 )
% 2.95/1.61 => ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
% 2.95/1.61 => ? [X1: B] : ( P @ F4 @ X2 @ X1 ) )
% 2.95/1.61 => ? [F4: A > B] :
% 2.95/1.61 ! [X4: A] : ( P @ F4 @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_146_linordered__field__no__lb,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X4: A] :
% 2.95/1.61 ? [Y: A] : ( ord_less @ A @ Y @ X4 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_147_linordered__field__no__ub,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X4: A] :
% 2.95/1.61 ? [X12: A] : ( ord_less @ A @ X4 @ X12 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_148_subdag_Osimps_I1_J,axiom,
% 2.95/1.61 ! [T2: binDag_Mirabelle_dag] :
% 2.95/1.61 ~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T2 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_149_DAG_Osimps_I1_J,axiom,
% 2.95/1.61 binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% 2.95/1.61
% 2.95/1.61 thf(fact_150_dag_Osimps_I4_J,axiom,
% 2.95/1.61 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
% 2.95/1.61 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 2.95/1.61 = F1 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_151_dag_Osimps_I6_J,axiom,
% 2.95/1.61 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
% 2.95/1.61 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
% 2.95/1.61 = F1 ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_152_subdag__size,axiom,
% 2.95/1.61 ! [T2: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag786255756subdag @ T2 @ S2 )
% 2.95/1.61 => ( ord_less @ nat @ ( size_size @ binDag_Mirabelle_dag @ S2 ) @ ( size_size @ binDag_Mirabelle_dag @ T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_153_dag_Osimps_I7_J,axiom,
% 2.95/1.61 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.95/1.61 = ( F22 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X23 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_154_dag_Osimps_I5_J,axiom,
% 2.95/1.61 ! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag1297733282se_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
% 2.95/1.61 = ( F22 @ X21 @ X22 @ X23 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_155_dag_Oinject,axiom,
% 2.95/1.61 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
% 2.95/1.61 = ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
% 2.95/1.61 = ( ( X21 = Y21 )
% 2.95/1.61 & ( X22 = Y22 )
% 2.95/1.61 & ( X23 = Y23 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_156_subdag__NodeD,axiom,
% 2.95/1.61 ! [T2: binDag_Mirabelle_dag,Lt: binDag_Mirabelle_dag,A2: simpl_ref,Rt: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag786255756subdag @ T2 @ ( binDag476092410e_Node @ Lt @ A2 @ Rt ) )
% 2.95/1.61 => ( ( binDag786255756subdag @ T2 @ Lt )
% 2.95/1.61 & ( binDag786255756subdag @ T2 @ Rt ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_157_subdag_Osimps_I2_J,axiom,
% 2.95/1.61 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T2 )
% 2.95/1.61 = ( ( T2 = L )
% 2.95/1.61 | ( T2 = R )
% 2.95/1.61 | ( binDag786255756subdag @ L @ T2 )
% 2.95/1.61 | ( binDag786255756subdag @ R @ T2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_158_dag_Oexhaust,axiom,
% 2.95/1.61 ! [Y4: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( Y4 != binDag_Mirabelle_Tip )
% 2.95/1.61 => ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
% 2.95/1.61 ( Y4
% 2.95/1.61 != ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_159_dag_Oinduct,axiom,
% 2.95/1.61 ! [P: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( P @ binDag_Mirabelle_Tip )
% 2.95/1.61 => ( ! [X12: binDag_Mirabelle_dag,X24: simpl_ref,X32: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( P @ X12 )
% 2.95/1.61 => ( ( P @ X32 )
% 2.95/1.61 => ( P @ ( binDag476092410e_Node @ X12 @ X24 @ X32 ) ) ) )
% 2.95/1.61 => ( P @ Dag ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_160_dag_Odistinct_I1_J,axiom,
% 2.95/1.61 ! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
% 2.95/1.61 ( binDag_Mirabelle_Tip
% 2.95/1.61 != ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_161_DAG_Osimps_I2_J,axiom,
% 2.95/1.61 ! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
% 2.95/1.61 ( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
% 2.95/1.61 = ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
% 2.95/1.61 & ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
% 2.95/1.61 & ( binDag_Mirabelle_DAG @ L )
% 2.95/1.61 & ( binDag_Mirabelle_DAG @ R ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_162_dag_Osize_I3_J,axiom,
% 2.95/1.61 ( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
% 2.95/1.61 = ( zero_zero @ nat ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_163_order_Oordering__axioms,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( order @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_164_ordering_Onot__eq__order__implies__strict,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( A2 != B2 )
% 2.95/1.61 => ( ( Less_eq @ A2 @ B2 )
% 2.95/1.61 => ( Less @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_165_ordering_Ostrict__implies__not__eq,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less @ A2 @ B2 )
% 2.95/1.61 => ( A2 != B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_166_ordering_Ostrict__implies__order,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less @ A2 @ B2 )
% 2.95/1.61 => ( Less_eq @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_167_ordering_Ostrict__iff__order,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less @ A2 @ B2 )
% 2.95/1.61 = ( ( Less_eq @ A2 @ B2 )
% 2.95/1.61 & ( A2 != B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_168_ordering_Oorder__iff__strict,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less_eq @ A2 @ B2 )
% 2.95/1.61 = ( ( Less @ A2 @ B2 )
% 2.95/1.61 | ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_169_ordering_Ostrict__trans2,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less @ A2 @ B2 )
% 2.95/1.61 => ( ( Less_eq @ B2 @ C )
% 2.95/1.61 => ( Less @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_170_ordering_Ostrict__trans1,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less_eq @ A2 @ B2 )
% 2.95/1.61 => ( ( Less @ B2 @ C )
% 2.95/1.61 => ( Less @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_171_ordering_Ostrict__trans,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less @ A2 @ B2 )
% 2.95/1.61 => ( ( Less @ B2 @ C )
% 2.95/1.61 => ( Less @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_172_ordering_Oantisym,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less_eq @ A2 @ B2 )
% 2.95/1.61 => ( ( Less_eq @ B2 @ A2 )
% 2.95/1.61 => ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_173_ordering_Oirrefl,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ~ ( Less @ A2 @ A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_174_ordering_Otrans,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less_eq @ A2 @ B2 )
% 2.95/1.61 => ( ( Less_eq @ B2 @ C )
% 2.95/1.61 => ( Less_eq @ A2 @ C ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_175_ordering_Ointro,axiom,
% 2.95/1.61 ! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
% 2.95/1.61 ( ! [A3: A,B3: A] :
% 2.95/1.61 ( ( Less @ A3 @ B3 )
% 2.95/1.61 = ( ( Less_eq @ A3 @ B3 )
% 2.95/1.61 & ( A3 != B3 ) ) )
% 2.95/1.61 => ( ! [A3: A] : ( Less_eq @ A3 @ A3 )
% 2.95/1.61 => ( ! [A3: A,B3: A] :
% 2.95/1.61 ( ( Less_eq @ A3 @ B3 )
% 2.95/1.61 => ( ( Less_eq @ B3 @ A3 )
% 2.95/1.61 => ( A3 = B3 ) ) )
% 2.95/1.61 => ( ! [A3: A,B3: A,C3: A] :
% 2.95/1.61 ( ( Less_eq @ A3 @ B3 )
% 2.95/1.61 => ( ( Less_eq @ B3 @ C3 )
% 2.95/1.61 => ( Less_eq @ A3 @ C3 ) ) )
% 2.95/1.61 => ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_176_ordering_Orefl,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( Less_eq @ A2 @ A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_177_ordering_Oasym,axiom,
% 2.95/1.61 ! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
% 2.95/1.61 ( ( ordering @ A @ Less_eq @ Less )
% 2.95/1.61 => ( ( Less @ A2 @ B2 )
% 2.95/1.61 => ~ ( Less @ B2 @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_178_ordering__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordering @ A )
% 2.95/1.61 = ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
% 2.95/1.61 ( ! [A5: A,B5: A] :
% 2.95/1.61 ( ( Less2 @ A5 @ B5 )
% 2.95/1.61 = ( ( Less_eq2 @ A5 @ B5 )
% 2.95/1.61 & ( A5 != B5 ) ) )
% 2.95/1.61 & ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
% 2.95/1.61 & ! [A5: A,B5: A] :
% 2.95/1.61 ( ( Less_eq2 @ A5 @ B5 )
% 2.95/1.61 => ( ( Less_eq2 @ B5 @ A5 )
% 2.95/1.61 => ( A5 = B5 ) ) )
% 2.95/1.61 & ! [A5: A,B5: A,C4: A] :
% 2.95/1.61 ( ( Less_eq2 @ A5 @ B5 )
% 2.95/1.61 => ( ( Less_eq2 @ B5 @ C4 )
% 2.95/1.61 => ( Less_eq2 @ A5 @ C4 ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_179_not__gr__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [N: A] :
% 2.95/1.61 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.95/1.61 = ( N
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_180_le__zero__eq,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [N: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.95/1.61 = ( N
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_181_dag_Osize__gen_I1_J,axiom,
% 2.95/1.61 ( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
% 2.95/1.61 = ( zero_zero @ nat ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_182_zero__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_183_zero__less__iff__neq__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [N: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.95/1.61 = ( N
% 2.95/1.61 != ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_184_gr__implies__not__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [M: A,N: A] :
% 2.95/1.61 ( ( ord_less @ A @ M @ N )
% 2.95/1.61 => ( N
% 2.95/1.61 != ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_185_not__less__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [N: A] :
% 2.95/1.61 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_186_gr__zeroI,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( canoni770627133id_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [N: A] :
% 2.95/1.61 ( ( N
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_187_less__numeral__extra_I3_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_188_le__numeral__extra_I3_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_189_sgn__greater,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_190_sgn__sgn,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
% 2.95/1.61 = ( sgn_sgn @ A @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_191_sgn__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_192_sgn__0__0,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ( sgn_sgn @ A @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 = ( A2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_193_sgn__pos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 => ( ( sgn_sgn @ A @ A2 )
% 2.95/1.61 = ( one_one @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_194_of__bool__eq_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( zero_neq_one_of_bool @ A @ $false )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_195_of__bool__eq_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( zero_neq_one_of_bool @ A @ $true )
% 2.95/1.61 = ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_196_less__numeral__extra_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_197_less__numeral__extra_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_198_le__numeral__extra_I1_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_199_le__numeral__extra_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_200_less__numeral__extra_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_201_le__numeral__extra_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_202_of__bool__eq__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P3: $o,Q3: $o] :
% 2.95/1.61 ( ( ( zero_neq_one_of_bool @ A @ P3 )
% 2.95/1.61 = ( zero_neq_one_of_bool @ A @ Q3 ) )
% 2.95/1.61 = ( P3 = Q3 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_203_zero__less__one,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_less_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_204_not__one__less__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_205_zero__le__one,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_206_not__one__le__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord1659791738miring @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_207_of__bool__def,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( zero_neq_one_of_bool @ A )
% 2.95/1.61 = ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_208_zero__neq__one,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ( zero_zero @ A )
% 2.95/1.61 != ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_209_split__of__bool,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,P3: $o] :
% 2.95/1.61 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 2.95/1.61 = ( ( P3
% 2.95/1.61 => ( P @ ( one_one @ A ) ) )
% 2.95/1.61 & ( ~ P3
% 2.95/1.61 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_210_split__of__bool__asm,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( zero_neq_one @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [P: A > $o,P3: $o] :
% 2.95/1.61 ( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
% 2.95/1.61 = ( ~ ( ( P3
% 2.95/1.61 & ~ ( P @ ( one_one @ A ) ) )
% 2.95/1.61 | ( ~ P3
% 2.95/1.61 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_211_sgn__1__pos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ( sgn_sgn @ A @ A2 )
% 2.95/1.61 = ( one_one @ A ) )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_212_sgn__neg,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 => ( ( sgn_sgn @ A @ A2 )
% 2.95/1.61 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_213_divide__le__eq__1__neg,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 2.95/1.61 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_214_neg__le__iff__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_215_neg__less__iff__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_216_division__ring__divide__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_217_divide__cancel__right,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,C: A,B2: A] :
% 2.95/1.61 ( ( ( divide_divide @ A @ A2 @ C )
% 2.95/1.61 = ( divide_divide @ A @ B2 @ C ) )
% 2.95/1.61 = ( ( C
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 | ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_218_divide__cancel__left,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [C: A,A2: A,B2: A] :
% 2.95/1.61 ( ( ( divide_divide @ A @ C @ A2 )
% 2.95/1.61 = ( divide_divide @ A @ C @ B2 ) )
% 2.95/1.61 = ( ( C
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 | ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_219_divide__eq__0__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ( divide_divide @ A @ A2 @ B2 )
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 = ( ( A2
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 | ( B2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_220_divide__zero__left,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_221_divide__zero,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 = ( zero_zero @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_222_divide__1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
% 2.95/1.61 = A2 ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_223_neg__0__le__iff__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_224_neg__le__0__iff__le,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.95/1.61 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_225_less__eq__neg__nonpos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_226_neg__less__eq__nonneg,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 2.95/1.61 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_227_neg__less__0__iff__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_228_neg__0__less__iff__less,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_229_neg__less__pos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_230_less__neg__neg,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linord219039673up_add @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_231_zero__eq__1__divide__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ( zero_zero @ A )
% 2.95/1.61 = ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 2.95/1.61 = ( A2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_232_one__divide__eq__0__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 = ( A2
% 2.95/1.61 = ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_233_eq__divide__eq__1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ( one_one @ A )
% 2.95/1.61 = ( divide_divide @ A @ B2 @ A2 ) )
% 2.95/1.61 = ( ( A2
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 & ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_234_divide__eq__eq__1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [B2: A,A2: A] :
% 2.95/1.61 ( ( ( divide_divide @ A @ B2 @ A2 )
% 2.95/1.61 = ( one_one @ A ) )
% 2.95/1.61 = ( ( A2
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 & ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_235_divide__self__if,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ( A2
% 2.95/1.61 = ( zero_zero @ A ) )
% 2.95/1.61 => ( ( divide_divide @ A @ A2 @ A2 )
% 2.95/1.61 = ( zero_zero @ A ) ) )
% 2.95/1.61 & ( ( A2
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 => ( ( divide_divide @ A @ A2 @ A2 )
% 2.95/1.61 = ( one_one @ A ) ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_236_divide__self,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( division_ring @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( A2
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 => ( ( divide_divide @ A @ A2 @ A2 )
% 2.95/1.61 = ( one_one @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_237_one__eq__divide__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ( one_one @ A )
% 2.95/1.61 = ( divide_divide @ A @ A2 @ B2 ) )
% 2.95/1.61 = ( ( B2
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 & ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_238_divide__eq__1__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ( divide_divide @ A @ A2 @ B2 )
% 2.95/1.61 = ( one_one @ A ) )
% 2.95/1.61 = ( ( B2
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 & ( A2 = B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_239_div__self,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( semidom_divide @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( A2
% 2.95/1.61 != ( zero_zero @ A ) )
% 2.95/1.61 => ( ( divide_divide @ A @ A2 @ A2 )
% 2.95/1.61 = ( one_one @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_240_divide__minus1,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [X: A] :
% 2.95/1.61 ( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 2.95/1.61 = ( uminus_uminus @ A @ X ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_241_divide__le__0__1__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 2.95/1.61 = ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_242_zero__le__divide__1__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_243_divide__less__0__1__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_244_divide__less__eq__1__neg,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_245_divide__less__eq__1__pos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 2.95/1.61 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_246_less__divide__eq__1__neg,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_247_less__divide__eq__1__pos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_248_zero__less__divide__1__iff,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
% 2.95/1.61 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_249_le__divide__eq__1__pos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_250_le__divide__eq__1__neg,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.95/1.61 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
% 2.95/1.61 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_251_divide__le__eq__1__pos,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_field @ A @ ( type2 @ A ) )
% 2.95/1.61 => ! [A2: A,B2: A] :
% 2.95/1.61 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.95/1.61 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
% 2.95/1.61 = ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_252_less__minus__one__simps_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_253_less__minus__one__simps_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_254_le__minus__one__simps_I2_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(fact_255_le__minus__one__simps_I4_J,axiom,
% 2.95/1.61 ! [A: $tType] :
% 2.95/1.61 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 2.95/1.61 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.95/1.61 ! [A6: $tType,A7: $tType] :
% 2.95/1.61 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 2.95/1.61 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.95/1.61 ! [A6: $tType,A7: $tType] :
% 2.95/1.61 ( ( order @ A7 @ ( type2 @ A7 ) )
% 2.95/1.61 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_fun___Orderings_Oord,axiom,
% 2.95/1.61 ! [A6: $tType,A7: $tType] :
% 2.95/1.61 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 2.95/1.61 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 2.95/1.61 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.95/1.61 canoni770627133id_add @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.95/1.61 linord1659791738miring @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 2.95/1.61 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Rings_Osemidom__divide,axiom,
% 2.95/1.61 semidom_divide @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
% 2.95/1.61 zero_less_one @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.95/1.61 wellorder @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
% 2.95/1.61 zero_neq_one @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.95/1.61 preorder @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.95/1.61 linorder @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 2.95/1.61 no_top @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.95/1.61 order @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.95/1.61 ord @ nat @ ( type2 @ nat ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.95/1.61 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.95/1.61 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 2.95/1.61 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.95/1.61 preorder @ $o @ ( type2 @ $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 2.95/1.61 linorder @ $o @ ( type2 @ $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 2.95/1.61 order @ $o @ ( type2 @ $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.95/1.61 ord @ $o @ ( type2 @ $o ) ).
% 2.95/1.61
% 2.95/1.61 thf(help_If_3_1_T,axiom,
% 2.95/1.61 ! [P: $o] :
% 2.95/1.61 ( ( P = $true )
% 2.95/1.61 | ( P = $false ) ) ).
% 2.95/1.61
% 2.95/1.61 thf(help_If_2_1_T,axiom,
% 2.95/1.61 ! [A: $tType,X: A,Y4: A] :
% 2.95/1.61 ( ( if @ A @ $false @ X @ Y4 )
% 2.95/1.61 = Y4 ) ).
% 2.95/1.61
% 2.95/1.61 thf(help_If_1_1_T,axiom,
% 2.95/1.61 ! [A: $tType,X: A,Y4: A] :
% 2.95/1.61 ( ( if @ A @ $true @ X @ Y4 )
% 2.95/1.61 = X ) ).
% 2.95/1.61
% 2.95/1.61 thf(conj_0,conjecture,
% 2.95/1.61 ord_less_eq @ binDag_Mirabelle_dag @ x @ z ).
% 2.95/1.61 ------------------------
% 2.95/1.63 % (19744)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_19573 for (2999ds/2Mi)
% 2.95/1.63 % (19743)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_19573 for (2999ds/2Mi)
% 2.95/1.63 % (19740)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_19573 for (2999ds/183Mi)
% 2.95/1.63 % (19746)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_19573 for (2999ds/18Mi)
% 2.95/1.63 % (19743)Instruction limit reached!
% 2.95/1.63 % (19743)------------------------------
% 2.95/1.63 % (19743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (19743)Termination reason: Unknown
% 2.95/1.63 % (19743)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (19745)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_19573 for (2999ds/275Mi)
% 2.95/1.63 % (19743)Memory used [KB]: 1279
% 2.95/1.63 % (19743)Time elapsed: 0.002 s
% 2.95/1.63 % (19743)Instructions burned: 2 (million)
% 2.95/1.63 % (19743)------------------------------
% 2.95/1.63 % (19743)------------------------------
% 2.95/1.63 % (19744)Instruction limit reached!
% 2.95/1.63 % (19744)------------------------------
% 2.95/1.63 % (19744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (19744)Termination reason: Unknown
% 2.95/1.63 % (19744)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (19744)Memory used [KB]: 1279
% 2.95/1.63 % (19744)Time elapsed: 0.002 s
% 2.95/1.63 % (19744)Instructions burned: 3 (million)
% 2.95/1.63 % (19744)------------------------------
% 2.95/1.63 % (19744)------------------------------
% 2.95/1.64 % (19746)Instruction limit reached!
% 2.95/1.64 % (19746)------------------------------
% 2.95/1.64 % (19746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.64 % (19746)Termination reason: Unknown
% 2.95/1.64 % (19746)Termination phase: shuffling
% 2.95/1.64
% 2.95/1.64 % (19746)Memory used [KB]: 1791
% 2.95/1.64 % (19746)Time elapsed: 0.007 s
% 2.95/1.64 % (19746)Instructions burned: 18 (million)
% 2.95/1.64 % (19746)------------------------------
% 2.95/1.64 % (19746)------------------------------
% 2.95/1.64 % (19741)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_19573 for (2999ds/4Mi)
% 2.95/1.64 % (19742)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_19573 for (2999ds/27Mi)
% 2.95/1.64 % (19747)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_19573 for (2999ds/3Mi)
% 2.95/1.64 % (19748)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_19573 for (2999ds/37Mi)
% 2.95/1.64 % (19741)Instruction limit reached!
% 2.95/1.64 % (19741)------------------------------
% 2.95/1.64 % (19741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.64 % (19741)Termination reason: Unknown
% 2.95/1.64 % (19741)Termination phase: shuffling
% 2.95/1.64
% 2.95/1.64 % (19741)Memory used [KB]: 1407
% 2.95/1.64 % (19741)Time elapsed: 0.005 s
% 2.95/1.64 % (19741)Instructions burned: 5 (million)
% 2.95/1.64 % (19741)------------------------------
% 2.95/1.64 % (19741)------------------------------
% 2.95/1.64 % (19747)Instruction limit reached!
% 2.95/1.64 % (19747)------------------------------
% 2.95/1.64 % (19747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.64 % (19747)Termination reason: Unknown
% 2.95/1.64 % (19747)Termination phase: shuffling
% 2.95/1.64
% 2.95/1.64 % (19747)Memory used [KB]: 1279
% 2.95/1.64 % (19747)Time elapsed: 0.002 s
% 2.95/1.64 % (19747)Instructions burned: 3 (million)
% 2.95/1.64 % (19747)------------------------------
% 2.95/1.64 % (19747)------------------------------
% 2.95/1.64 % (19745)First to succeed.
% 2.95/1.65 % (19748)Instruction limit reached!
% 2.95/1.65 % (19748)------------------------------
% 2.95/1.65 % (19748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.65 % (19748)Termination reason: Unknown
% 2.95/1.65 % (19748)Termination phase: Property scanning
% 2.95/1.65
% 2.95/1.65 % (19748)Memory used [KB]: 2046
% 2.95/1.65 % (19748)Time elapsed: 0.011 s
% 2.95/1.65 % (19748)Instructions burned: 37 (million)
% 2.95/1.65 % (19748)------------------------------
% 2.95/1.65 % (19748)------------------------------
% 2.95/1.65 % (19751)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_19573 for (2999ds/1041Mi)
% 2.95/1.66 % (19742)Instruction limit reached!
% 2.95/1.66 % (19742)------------------------------
% 2.95/1.66 % (19742)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (19742)Termination reason: Unknown
% 2.95/1.66 % (19742)Termination phase: Property scanning
% 2.95/1.66
% 2.95/1.66 % (19742)Memory used [KB]: 1791
% 2.95/1.66 % (19742)Time elapsed: 0.019 s
% 2.95/1.66 % (19742)Instructions burned: 28 (million)
% 2.95/1.66 % (19742)------------------------------
% 2.95/1.66 % (19742)------------------------------
% 2.95/1.66 % (19750)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_19573 for (2999ds/3Mi)
% 2.95/1.66 % (19749)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_19573 for (2999ds/15Mi)
% 2.95/1.66 % (19752)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_19573 for (2999ds/7Mi)
% 2.95/1.66 % (19750)Instruction limit reached!
% 2.95/1.66 % (19750)------------------------------
% 2.95/1.66 % (19750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (19750)Termination reason: Unknown
% 2.95/1.66 % (19750)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (19750)Memory used [KB]: 1279
% 2.95/1.66 % (19750)Time elapsed: 0.004 s
% 2.95/1.66 % (19750)Instructions burned: 3 (million)
% 2.95/1.66 % (19750)------------------------------
% 2.95/1.66 % (19750)------------------------------
% 2.95/1.66 % (19752)Instruction limit reached!
% 2.95/1.66 % (19752)------------------------------
% 2.95/1.66 % (19752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.66 % (19752)Termination reason: Unknown
% 2.95/1.66 % (19752)Termination phase: shuffling
% 2.95/1.66
% 2.95/1.66 % (19752)Memory used [KB]: 1407
% 2.95/1.66 % (19752)Time elapsed: 0.004 s
% 2.95/1.66 % (19752)Instructions burned: 7 (million)
% 2.95/1.66 % (19752)------------------------------
% 2.95/1.66 % (19752)------------------------------
% 2.95/1.67 % (19745)Refutation found. Thanks to Tanya!
% 2.95/1.67 % SZS status Theorem for DTF2THF_19573
% 2.95/1.67 % SZS output start Proof for DTF2THF_19573
% 2.95/1.67 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.95/1.67 thf(type_def_7, type, binDag_Mirabelle_dag: $tType).
% 2.95/1.67 thf(type_def_8, type, simpl_ref: $tType).
% 2.95/1.67 thf(type_def_9, type, nat: $tType).
% 2.95/1.67 thf(type_def_10, type, set: ($tType) > $tType).
% 2.95/1.67 thf(func_def_0, type, binDag_Mirabelle_dag: $tType).
% 2.95/1.67 thf(func_def_1, type, simpl_ref: $tType).
% 2.95/1.67 thf(func_def_2, type, set: ($tType) > $tType).
% 2.95/1.67 thf(func_def_3, type, nat: $tType).
% 2.95/1.67 thf(func_def_4, type, itself: ($tType) > $tType).
% 2.95/1.67 thf(func_def_5, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_6, type, field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_9, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_10, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_13, type, zero_neq_one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_14, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_15, type, zero_less_one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_16, type, division_ring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_17, type, semidom_divide: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_18, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_19, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_20, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_21, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_22, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_23, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_24, type, linord219039673up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_25, type, linord1659791738miring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_26, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_27, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_28, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.95/1.67 thf(func_def_29, type, binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o).
% 2.95/1.67 thf(func_def_30, type, binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag).
% 2.95/1.67 thf(func_def_31, type, binDag_Mirabelle_Tip: binDag_Mirabelle_dag).
% 2.95/1.67 thf(func_def_32, type, binDag1297733282se_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0) > binDag_Mirabelle_dag > X0)).
% 2.95/1.67 thf(func_def_33, type, binDag1442713106ec_dag: !>[X0: $tType]:(X0 > (binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > X0 > X0 > X0) > binDag_Mirabelle_dag > X0)).
% 2.95/1.67 thf(func_def_34, type, binDag1924123185ze_dag: binDag_Mirabelle_dag > nat).
% 2.95/1.67 thf(func_def_35, type, binDag1380252983set_of: binDag_Mirabelle_dag > set @ simpl_ref).
% 2.95/1.67 thf(func_def_36, type, binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o).
% 2.95/1.67 thf(func_def_37, type, one_one: !>[X0: $tType]:(X0)).
% 2.95/1.67 thf(func_def_38, type, sgn_sgn: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.67 thf(func_def_39, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.95/1.67 thf(func_def_40, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.95/1.67 thf(func_def_41, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.95/1.67 thf(func_def_42, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.95/1.67 thf(func_def_43, type, ord_Least: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.95/1.67 thf(func_def_44, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.95/1.67 thf(func_def_45, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.95/1.67 thf(func_def_46, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.95/1.67 thf(func_def_47, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 2.95/1.67 thf(func_def_48, type, ordering: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 2.95/1.67 thf(func_def_49, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.95/1.67 thf(func_def_50, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.95/1.67 thf(func_def_51, type, zero_neq_one_of_bool: !>[X0: $tType]:($o > X0)).
% 2.95/1.67 thf(func_def_52, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.95/1.67 thf(func_def_53, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.95/1.67 thf(func_def_54, type, x: binDag_Mirabelle_dag).
% 2.95/1.67 thf(func_def_55, type, y: binDag_Mirabelle_dag).
% 2.95/1.67 thf(func_def_56, type, z: binDag_Mirabelle_dag).
% 2.95/1.67 thf(func_def_72, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.95/1.67 thf(f845,plain,(
% 2.95/1.67 $false),
% 2.95/1.67 inference(subsumption_resolution,[status(thm)],[f842,f836])).
% 2.95/1.67 thf(f836,plain,(
% 2.95/1.67 ((ord_less_eq @ binDag_Mirabelle_dag @ y @ z) = $true)),
% 2.95/1.67 inference(cnf_transformation,[status(thm)],[f423])).
% 2.95/1.67 thf(f423,plain,(
% 2.95/1.67 ((ord_less_eq @ binDag_Mirabelle_dag @ y @ z) = $true)),
% 2.95/1.67 inference(fool_elimination,[status(thm)],[f422])).
% 2.95/1.67 thf(f422,plain,(
% 2.95/1.67 (ord_less_eq @ binDag_Mirabelle_dag @ y @ z)),
% 2.95/1.67 inference(rectify,[status(thm)],[f2])).
% 2.95/1.67 thf(f2,axiom,(
% 2.95/1.67 (ord_less_eq @ binDag_Mirabelle_dag @ y @ z)),
% 2.95/1.67 file('/export/starexec/sandbox2/tmp/tmp.EGMR65SasR/DTF2THF_19573.p',fact_1_y__z)).
% 2.95/1.67 thf(f842,plain,(
% 2.95/1.67 ((ord_less_eq @ binDag_Mirabelle_dag @ y @ z) != $true)),
% 2.95/1.67 inference(definition_unfolding,[status(thm)],[f837,f839])).
% 2.95/1.67 thf(f839,plain,(
% 2.95/1.67 (x = y)),
% 2.95/1.67 inference(cnf_transformation,[status(thm)],[f1])).
% 2.95/1.67 thf(f1,axiom,(
% 2.95/1.67 (x = y)),
% 2.95/1.67 file('/export/starexec/sandbox2/tmp/tmp.EGMR65SasR/DTF2THF_19573.p',fact_0_True)).
% 2.95/1.67 thf(f837,plain,(
% 2.95/1.67 ((ord_less_eq @ binDag_Mirabelle_dag @ x @ z) != $true)),
% 2.95/1.67 inference(cnf_transformation,[status(thm)],[f828])).
% 2.95/1.67 thf(f828,plain,(
% 2.95/1.67 ((ord_less_eq @ binDag_Mirabelle_dag @ x @ z) != $true)),
% 2.95/1.67 inference(flattening,[status(thm)],[f383])).
% 2.95/1.67 thf(f383,plain,(
% 2.95/1.67 ~((ord_less_eq @ binDag_Mirabelle_dag @ x @ z) = $true)),
% 2.95/1.67 inference(fool_elimination,[status(thm)],[f382])).
% 2.95/1.67 thf(f382,plain,(
% 2.95/1.67 ~(ord_less_eq @ binDag_Mirabelle_dag @ x @ z)),
% 2.95/1.67 inference(rectify,[status(thm)],[f284])).
% 2.95/1.67 thf(f284,negated_conjecture,(
% 2.95/1.67 ~(ord_less_eq @ binDag_Mirabelle_dag @ x @ z)),
% 2.95/1.67 inference(negated_conjecture,[status(cth)],[f283])).
% 2.95/1.67 thf(f283,conjecture,(
% 2.95/1.67 (ord_less_eq @ binDag_Mirabelle_dag @ x @ z)),
% 2.95/1.67 file('/export/starexec/sandbox2/tmp/tmp.EGMR65SasR/DTF2THF_19573.p',conj_0)).
% 2.95/1.67 % SZS output end Proof for DTF2THF_19573
% 2.95/1.67 % (19745)------------------------------
% 2.95/1.67 % (19745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.67 % (19745)Termination reason: Refutation
% 2.95/1.67
% 2.95/1.67 % (19745)Memory used [KB]: 6268
% 2.95/1.67 % (19745)Time elapsed: 0.015 s
% 2.95/1.67 % (19745)Instructions burned: 33 (million)
% 2.95/1.67 % (19745)------------------------------
% 2.95/1.67 % (19745)------------------------------
% 2.95/1.67 % (19739)Success in time 0.047 s
%------------------------------------------------------------------------------