%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT113^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/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 08:44:49 AM UTC 2026
% Result : Theorem 5.25s 2.15s
% Output : Refutation 5.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT113^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n014.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 06:07:42 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.48 ---- Original DTF file ---
% 0.30/0.48 thf(spec,logic,$$dhol).
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 % File : DAT113^1 : TPTP v9.2.1. Released v7.0.0.
% 0.30/0.48 % Domain : Data Structures
% 0.30/0.48 % Problem : Coinductive list 24
% 0.30/0.48 % Version : [Bla16] axioms : Especial.
% 0.30/0.48 % English :
% 0.30/0.48
% 0.30/0.48 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.30/0.48 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.30/0.48 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.30/0.48 % Source : [Bla16]
% 0.30/0.48 % Names : coinductive_list__24.p [Bla16]
% 0.30/0.48
% 0.30/0.48 % Status : Theorem
% 0.30/0.48 % Rating : 0.67 v8.1.0, 0.50 v7.5.0, 1.00 v7.2.0, 0.75 v7.1.0
% 0.30/0.48 % Syntax : Number of formulae : 289 ( 52 unt; 36 typ; 0 def)
% 0.30/0.48 % Number of atoms : 855 ( 133 equ; 0 cnn)
% 0.30/0.48 % Maximal formula atoms : 13 ( 3 avg)
% 0.30/0.48 % Number of connectives : 3387 ( 79 ~; 26 |; 59 &;2643 @)
% 0.30/0.48 % ( 0 <=>; 580 =>; 0 <=; 0 <~>)
% 0.30/0.48 % Maximal formula depth : 21 ( 10 avg)
% 0.30/0.48 % Number of types : 3 ( 2 usr)
% 0.30/0.48 % Number of type conns : 264 ( 264 >; 0 *; 0 +; 0 <<)
% 0.30/0.48 % Number of symbols : 35 ( 34 usr; 2 con; 0-4 aty)
% 0.30/0.48 % Number of variables : 1031 ( 46 ^; 909 !; 42 ?;1031 :)
% 0.30/0.48 % ( 34 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.48 % SPC : TH1_THM_EQU_NAR
% 0.30/0.48
% 0.30/0.48 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.48 % 2016-07-13 14:42:11.560
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 %----Could-be-implicit typings (4)
% 0.30/0.48 thf(ty_t_Set_Oset,type,
% 0.30/0.48 set: $tType > $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_t_Nat_Onat,type,
% 0.30/0.48 nat: $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_t_itself,type,
% 0.30/0.48 itself: $tType > $tType ).
% 0.30/0.48
% 0.30/0.48 thf(ty_tf_a,type,
% 0.30/0.48 a: $tType ).
% 0.30/0.48
% 0.30/0.48 %----Explicit typings (32)
% 0.30/0.48 thf(sy_cl_HOL_Otype,type,
% 0.30/0.48 type:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.48 ord:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.48 order:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/0.48 no_bot:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Ono__top,type,
% 0.30/0.48 no_top:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.48 linorder:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.48 preorder:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Owellorder,type,
% 0.30/0.48 wellorder:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Odense__order,type,
% 0.30/0.48 dense_order:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.30/0.48 linordered_idom:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Fields_Olinordered__field,type,
% 0.30/0.48 linordered_field:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.30/0.48 dense_linorder:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.30/0.48 comple187826305attice:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.30/0.48 condit1656338222tinuum:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.30/0.48 condit1037483654norder:
% 0.30/0.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Complete__Partial__Order_Ochain,type,
% 0.30/0.48 comple1193779247_chain:
% 0.30/0.48 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 0.30/0.48 hilbert_GreatestM:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 0.30/0.48 hilbert_LeastM:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oord__class_OLeast,type,
% 0.30/0.48 ord_Least:
% 0.30/0.48 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.48 ord_less:
% 0.30/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.48 ord_less_eq:
% 0.30/0.48 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.30/0.48 order_antimono:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 0.30/0.48 order_mono:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.30/0.48 order_strict_mono:
% 0.30/0.48 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Orderings_Oordering,type,
% 0.30/0.48 ordering:
% 0.30/0.48 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Pure_Otype,type,
% 0.30/0.48 type2:
% 0.30/0.48 !>[A: $tType] : ( itself @ A ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Set_OCollect,type,
% 0.30/0.48 collect:
% 0.30/0.48 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Topological__Spaces_Omonoseq,type,
% 0.30/0.48 topological_monoseq:
% 0.30/0.48 !>[A: $tType] : ( ( nat > A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member,type,
% 0.30/0.48 member:
% 0.30/0.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_P,type,
% 0.30/0.48 p: a > a > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_a,type,
% 0.30/0.48 a2: a ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_b,type,
% 0.30/0.48 b: a ).
% 0.30/0.48
% 0.30/0.48 %----Relevant facts (227)
% 0.30/0.48 thf(fact_0_local_Osym,axiom,
% 0.30/0.48 ( ( p @ b @ a2 )
% 0.30/0.48 => ( p @ a2 @ b ) ) ).
% 0.30/0.48
% 0.30/0.48 % local.sym
% 0.30/0.48 thf(fact_1_le,axiom,
% 0.30/0.48 ! [A2: a,B2: a] :
% 0.30/0.48 ( ( ord_less_eq @ a @ A2 @ B2 )
% 0.30/0.48 => ( p @ A2 @ B2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % le
% 0.30/0.48 thf(fact_2_order__refl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_refl
% 0.30/0.48 thf(fact_3_le__funD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_funD
% 0.30/0.48 thf(fact_4_le__funE,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_funE
% 0.30/0.48 thf(fact_5_le__funI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ! [F: A > B,G: A > B] :
% 0.30/0.48 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.30/0.48 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_funI
% 0.30/0.48 thf(fact_6_le__fun__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.48 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.48 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_fun_def
% 0.30/0.48 thf(fact_7_order__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y: B] :
% 0.30/0.48 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_subst1
% 0.30/0.48 thf(fact_8_order__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C2: $tType] :
% 0.30/0.48 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.48 => ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_subst2
% 0.30/0.48 thf(fact_9_ord__eq__le__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( A2
% 0.30/0.48 = ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y: B] :
% 0.30/0.48 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_le_subst
% 0.30/0.48 thf(fact_10_ord__le__eq__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ( F @ B2 )
% 0.30/0.48 = C )
% 0.30/0.48 => ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_le_eq_subst
% 0.30/0.48 thf(fact_11_eq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 0.30/0.48 = ( ^ [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % eq_iff
% 0.30/0.48 thf(fact_12_antisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym
% 0.30/0.48 thf(fact_13_linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linear
% 0.30/0.48 thf(fact_14_dual__order_Oantisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.antisym
% 0.30/0.48 thf(fact_15_dual__order_Otrans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A,C: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.trans
% 0.30/0.48 thf(fact_16_linorder__wlog,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.30/0.48 ( ! [A3: A,B3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.48 => ( P @ A3 @ B3 ) )
% 0.30/0.48 => ( ! [A3: A,B3: A] :
% 0.30/0.48 ( ( P @ B3 @ A3 )
% 0.30/0.48 => ( P @ A3 @ B3 ) )
% 0.30/0.48 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_wlog
% 0.30/0.48 thf(fact_17_dual__order_Orefl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.refl
% 0.30/0.48 thf(fact_18_order__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,Z2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_trans
% 0.30/0.48 thf(fact_19_order__class_Oorder_Oantisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_class.order.antisym
% 0.30/0.48 thf(fact_20_ord__le__eq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( B2 = C )
% 0.30/0.48 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_le_eq_trans
% 0.30/0.48 thf(fact_21_ord__eq__le__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( A2 = B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_le_trans
% 0.30/0.48 thf(fact_22_antisym__conv,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y4: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 = ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv
% 0.30/0.48 thf(fact_23_le__cases3,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,Z2: A] :
% 0.30/0.48 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 0.30/0.48 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.30/0.48 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_cases3
% 0.30/0.48 thf(fact_24_order_Otrans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.trans
% 0.30/0.48 thf(fact_25_le__cases,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_cases
% 0.30/0.48 thf(fact_26_eq__refl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( X = Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % eq_refl
% 0.30/0.48 thf(fact_27_lfp_Oleq__antisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfp.leq_antisym
% 0.30/0.48 thf(fact_28_gfp_Oleq__antisym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y4: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gfp.leq_antisym
% 0.30/0.48 thf(fact_29_lfp_Oleq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,Z2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfp.leq_trans
% 0.30/0.48 thf(fact_30_gfp_Oleq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y4: A,X: A,Z2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ Z2 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % gfp.leq_trans
% 0.30/0.48 thf(fact_31_lfp_Oleq__refl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfp.leq_refl
% 0.30/0.48 thf(fact_32_strict__mono__less__eq,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 0.30/0.48 = ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_mono_less_eq
% 0.30/0.48 thf(fact_33_antimono__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ( ( order_antimono @ A @ B )
% 0.30/0.48 = ( ^ [F2: A > B] :
% 0.30/0.48 ! [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antimono_def
% 0.30/0.48 thf(fact_34_antimonoI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B] :
% 0.30/0.48 ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) ) )
% 0.30/0.48 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antimonoI
% 0.30/0.48 thf(fact_35_antimonoE,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_antimono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antimonoE
% 0.30/0.48 thf(fact_36_antimonoD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_antimono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antimonoD
% 0.30/0.48 thf(fact_37_LeastI2__wellorder__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,Q: A > $o] :
% 0.30/0.48 ( ? [X1: A] : ( P @ X1 )
% 0.30/0.48 => ( ! [A3: A] :
% 0.30/0.48 ( ( P @ A3 )
% 0.30/0.48 => ( ! [B4: A] :
% 0.30/0.48 ( ( P @ B4 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 0.30/0.48 => ( Q @ A3 ) ) )
% 0.30/0.48 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastI2_wellorder_ex
% 0.30/0.48 thf(fact_38_LeastI2__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,Q: A > $o] :
% 0.30/0.48 ( ? [X1: A] : ( P @ X1 )
% 0.30/0.48 => ( ! [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 => ( Q @ X2 ) )
% 0.30/0.48 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastI2_ex
% 0.30/0.48 thf(fact_39_LeastI__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o] :
% 0.30/0.48 ( ? [X1: A] : ( P @ X1 )
% 0.30/0.48 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastI_ex
% 0.30/0.48 thf(fact_40_LeastI2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,A2: A,Q: A > $o] :
% 0.30/0.48 ( ( P @ A2 )
% 0.30/0.48 => ( ! [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 => ( Q @ X2 ) )
% 0.30/0.48 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastI2
% 0.30/0.48 thf(fact_41_strict__mono__eq,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ( F @ X )
% 0.30/0.48 = ( F @ Y4 ) )
% 0.30/0.48 = ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_mono_eq
% 0.30/0.48 thf(fact_42_LeastI2__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,X: A,Q: A > $o] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ! [Y: A] :
% 0.30/0.48 ( ( P @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y ) )
% 0.30/0.48 => ( ! [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 => ( ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 0.30/0.48 => ( Q @ X2 ) ) )
% 0.30/0.48 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastI2_order
% 0.30/0.48 thf(fact_43_Least__equality,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,X: A] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ! [Y: A] :
% 0.30/0.48 ( ( P @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y ) )
% 0.30/0.48 => ( ( ord_Least @ A @ P )
% 0.30/0.48 = X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Least_equality
% 0.30/0.48 thf(fact_44_LeastI2__wellorder,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,A2: A,Q: A > $o] :
% 0.30/0.48 ( ( P @ A2 )
% 0.30/0.48 => ( ! [A3: A] :
% 0.30/0.48 ( ( P @ A3 )
% 0.30/0.48 => ( ! [B4: A] :
% 0.30/0.48 ( ( P @ B4 )
% 0.30/0.48 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 0.30/0.48 => ( Q @ A3 ) ) )
% 0.30/0.48 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastI2_wellorder
% 0.30/0.48 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/0.48 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/0.48 = ( P @ A2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A: $tType,A4: set @ A] :
% 0.30/0.48 ( ( collect @ A
% 0.30/0.48 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.30/0.48 = A4 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_47_Collect__cong,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 = ( Q @ X2 ) )
% 0.30/0.48 => ( ( collect @ A @ P )
% 0.30/0.48 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_48_ext,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ( F @ X2 )
% 0.30/0.48 = ( G @ X2 ) )
% 0.30/0.48 => ( F = G ) ) ).
% 0.30/0.48
% 0.30/0.48 % ext
% 0.30/0.48 thf(fact_49_decseqD,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [F: nat > A,I: nat,J: nat] :
% 0.30/0.48 ( ( order_antimono @ nat @ A @ F )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % decseqD
% 0.30/0.48 thf(fact_50_decseq__def,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( order_antimono @ nat @ A )
% 0.30/0.48 = ( ^ [X4: nat > A] :
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % decseq_def
% 0.30/0.48 thf(fact_51_strict__mono__less,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 0.30/0.48 = ( ord_less @ A @ X @ Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_mono_less
% 0.30/0.48 thf(fact_52_strict__mono__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ( ( order_strict_mono @ A @ B )
% 0.30/0.48 = ( ^ [F2: A > B] :
% 0.30/0.48 ! [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.48 => ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_mono_def
% 0.30/0.48 thf(fact_53_strict__monoI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B] :
% 0.30/0.48 ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_monoI
% 0.30/0.48 thf(fact_54_strict__monoD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_monoD
% 0.30/0.48 thf(fact_55_strict__mono__mono,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B] :
% 0.30/0.48 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.48 => ( order_mono @ A @ B @ F ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_mono_mono
% 0.30/0.48 thf(fact_56_linorder__chain,axiom,
% 0.30/0.48 ! [B: $tType] :
% 0.30/0.48 ( ( linorder @ B @ ( type2 @ B ) )
% 0.30/0.48 => ! [Y6: set @ B] : ( comple1193779247_chain @ B @ ( ord_less_eq @ B ) @ Y6 ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_chain
% 0.30/0.48 thf(fact_57_LeastMI2,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ! [Y: A] :
% 0.30/0.48 ( ( P @ Y )
% 0.30/0.48 => ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y ) ) )
% 0.30/0.48 => ( ! [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 => ( ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
% 0.30/0.48 => ( Q @ X2 ) ) )
% 0.30/0.48 => ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastMI2
% 0.30/0.48 thf(fact_58_incseqD,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [F: nat > A,I: nat,J: nat] :
% 0.30/0.48 ( ( order_mono @ nat @ A @ F )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ I ) @ ( F @ J ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % incseqD
% 0.30/0.48 thf(fact_59_incseq__def,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( order_mono @ nat @ A )
% 0.30/0.48 = ( ^ [X4: nat > A] :
% 0.30/0.48 ! [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 => ( ord_less_eq @ A @ ( X4 @ M ) @ ( X4 @ N ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % incseq_def
% 0.30/0.48 thf(fact_60_ord__eq__less__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( A2
% 0.30/0.48 = ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y: B] :
% 0.30/0.48 ( ( ord_less @ B @ X2 @ Y )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_less_subst
% 0.30/0.48 thf(fact_61_ord__less__eq__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ( F @ B2 )
% 0.30/0.48 = C )
% 0.30/0.48 => ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_less_eq_subst
% 0.30/0.48 thf(fact_62_order__less__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y: B] :
% 0.30/0.48 ( ( ord_less @ B @ X2 @ Y )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_subst1
% 0.30/0.48 thf(fact_63_order__less__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C2: $tType] :
% 0.30/0.48 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.48 => ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_subst2
% 0.30/0.48 thf(fact_64_lt__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % lt_ex
% 0.30/0.48 thf(fact_65_gt__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( no_top @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% 0.30/0.48
% 0.30/0.48 % gt_ex
% 0.30/0.48 thf(fact_66_neqE,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( X != Y4 )
% 0.30/0.48 => ( ~ ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % neqE
% 0.30/0.48 thf(fact_67_neq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( X != Y4 )
% 0.30/0.48 = ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % neq_iff
% 0.30/0.48 thf(fact_68_order_Oasym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.asym
% 0.30/0.48 thf(fact_69_dense,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ? [Z3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Z3 )
% 0.30/0.48 & ( ord_less @ A @ Z3 @ Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense
% 0.30/0.48 thf(fact_70_less__imp__neq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( X != Y4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_neq
% 0.30/0.48 thf(fact_71_less__asym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_asym
% 0.30/0.48 thf(fact_72_less__asym_H,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_asym'
% 0.30/0.48 thf(fact_73_less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,Z2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less @ A @ Y4 @ Z2 )
% 0.30/0.48 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_trans
% 0.30/0.48 thf(fact_74_less__linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 | ( X = Y4 )
% 0.30/0.48 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_linear
% 0.30/0.48 thf(fact_75_less__irrefl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_irrefl
% 0.30/0.48 thf(fact_76_ord__eq__less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( A2 = B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_less_trans
% 0.30/0.48 thf(fact_77_ord__less__eq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( B2 = C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_less_eq_trans
% 0.30/0.48 thf(fact_78_dual__order_Oasym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.asym
% 0.30/0.48 thf(fact_79_less__imp__not__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( X != Y4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_not_eq
% 0.30/0.48 thf(fact_80_less__not__sym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_sym
% 0.30/0.48 thf(fact_81_less__induct,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,A2: A] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ! [Y5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Y5 @ X2 )
% 0.30/0.48 => ( P @ Y5 ) )
% 0.30/0.48 => ( P @ X2 ) )
% 0.30/0.48 => ( P @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_induct
% 0.30/0.48 thf(fact_82_antisym__conv3,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y4: A,X: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ Y4 @ X )
% 0.30/0.48 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.30/0.48 = ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv3
% 0.30/0.48 thf(fact_83_less__imp__not__eq2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( Y4 != X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_not_eq2
% 0.30/0.48 thf(fact_84_less__imp__triv,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,P: $o] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less @ A @ Y4 @ X )
% 0.30/0.48 => P ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_triv
% 0.30/0.48 thf(fact_85_linorder__cases,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ( X != Y4 )
% 0.30/0.48 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % linorder_cases
% 0.30/0.48 thf(fact_86_dual__order_Oirrefl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A] :
% 0.30/0.48 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.irrefl
% 0.30/0.48 thf(fact_87_order_Ostrict__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans
% 0.30/0.48 thf(fact_88_less__imp__not__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_not_less
% 0.30/0.48 thf(fact_89_dual__order_Ostrict__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ( ( ord_less @ A @ C @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans
% 0.30/0.48 thf(fact_90_not__less__iff__gr__or__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.30/0.48 = ( ( ord_less @ A @ Y4 @ X )
% 0.30/0.48 | ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less_iff_gr_or_eq
% 0.30/0.48 thf(fact_91_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( A2 != B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_implies_not_eq
% 0.30/0.48 thf(fact_92_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ( A2 != B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_implies_not_eq
% 0.30/0.48 thf(fact_93_mono__invE,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mono_invE
% 0.30/0.48 thf(fact_94_ex__has__greatest__nat,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ! [Y: A] :
% 0.30/0.48 ( ( P @ Y )
% 0.30/0.48 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 0.30/0.48 => ? [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 & ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_has_greatest_nat
% 0.30/0.48 thf(fact_95_ex__has__least__nat,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ? [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 & ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_has_least_nat
% 0.30/0.48 thf(fact_96_LeastM__nat__lemma,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
% 0.30/0.48 & ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastM_nat_lemma
% 0.30/0.48 thf(fact_97_LeastM__nat__le,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,X: A,M2: A > nat] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastM_nat_le
% 0.30/0.48 thf(fact_98_mono__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ( ( order_mono @ A @ B )
% 0.30/0.48 = ( ^ [F2: A > B] :
% 0.30/0.48 ! [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mono_def
% 0.30/0.48 thf(fact_99_monoI,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B] :
% 0.30/0.48 ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( order_mono @ A @ B @ F ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % monoI
% 0.30/0.48 thf(fact_100_monoE,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % monoE
% 0.30/0.48 thf(fact_101_monoD,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y4: A] :
% 0.30/0.48 ( ( order_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % monoD
% 0.30/0.48 thf(fact_102_order_Onot__eq__order__implies__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( A2 != B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.not_eq_order_implies_strict
% 0.30/0.48 thf(fact_103_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_implies_order
% 0.30/0.48 thf(fact_104_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [B5: A,A5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.30/0.48 & ( A5 != B5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_iff_order
% 0.30/0.48 thf(fact_105_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [B5: A,A5: A] :
% 0.30/0.48 ( ( ord_less @ A @ B5 @ A5 )
% 0.30/0.48 | ( A5 = B5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.order_iff_strict
% 0.30/0.48 thf(fact_106_order_Ostrict__implies__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_implies_order
% 0.30/0.48 thf(fact_107_dense__le__bounded,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,Z2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ! [W: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ W )
% 0.30/0.48 => ( ( ord_less @ A @ W @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_le_bounded
% 0.30/0.48 thf(fact_108_dense__ge__bounded,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Z2: A,X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z2 @ X )
% 0.30/0.48 => ( ! [W: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z2 @ W )
% 0.30/0.48 => ( ( ord_less @ A @ W @ X )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_ge_bounded
% 0.30/0.48 thf(fact_109_dual__order_Ostrict__trans2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans2
% 0.30/0.48 thf(fact_110_dual__order_Ostrict__trans1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A,C: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.48 => ( ( ord_less @ A @ C @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans1
% 0.30/0.48 thf(fact_111_order_Ostrict__iff__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [A5: A,B5: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.30/0.48 & ( A5 != B5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_iff_order
% 0.30/0.48 thf(fact_112_order_Oorder__iff__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [A5: A,B5: A] :
% 0.30/0.48 ( ( ord_less @ A @ A5 @ B5 )
% 0.30/0.48 | ( A5 = B5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.order_iff_strict
% 0.30/0.48 thf(fact_113_order_Ostrict__trans2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans2
% 0.30/0.48 thf(fact_114_order_Ostrict__trans1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans1
% 0.30/0.48 thf(fact_115_not__le__imp__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y4: A,X: A] :
% 0.30/0.48 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_le_imp_less
% 0.30/0.48 thf(fact_116_less__le__not__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_le_not_le
% 0.30/0.48 thf(fact_117_le__imp__less__or__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 | ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_imp_less_or_eq
% 0.30/0.48 thf(fact_118_le__less__linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less_linear
% 0.30/0.48 thf(fact_119_dense__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y4: A,Z2: A] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_le
% 0.30/0.48 thf(fact_120_dense__ge,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Z2: A,Y4: A] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z2 @ X2 )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_ge
% 0.30/0.48 thf(fact_121_less__le__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,Z2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 0.30/0.48 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_le_trans
% 0.30/0.48 thf(fact_122_le__less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A,Z2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less @ A @ Y4 @ Z2 )
% 0.30/0.48 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less_trans
% 0.30/0.48 thf(fact_123_antisym__conv2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.30/0.48 = ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv2
% 0.30/0.48 thf(fact_124_antisym__conv1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y4 )
% 0.30/0.48 = ( X = Y4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv1
% 0.30/0.48 thf(fact_125_less__imp__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_le
% 0.30/0.48 thf(fact_126_le__neq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( A2 != B2 )
% 0.30/0.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_neq_trans
% 0.30/0.48 thf(fact_127_not__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 0.30/0.48 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less
% 0.30/0.48 thf(fact_128_not__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 0.30/0.48 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_le
% 0.30/0.48 thf(fact_129_order__less__le__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C2: $tType] :
% 0.30/0.48 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.48 => ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le_subst2
% 0.30/0.48 thf(fact_130_order__less__le__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y: B] :
% 0.30/0.48 ( ( ord_less_eq @ B @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le_subst1
% 0.30/0.48 thf(fact_131_order__le__less__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C2: $tType] :
% 0.30/0.48 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.48 => ( ! [X2: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less_subst2
% 0.30/0.48 thf(fact_132_order__le__less__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y: B] :
% 0.30/0.48 ( ( ord_less @ B @ X2 @ Y )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less_subst1
% 0.30/0.48 thf(fact_133_less__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 & ( X3 != Y3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_le
% 0.30/0.48 thf(fact_134_le__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.48 | ( X3 = Y3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less
% 0.30/0.48 thf(fact_135_leI,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y4: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % leI
% 0.30/0.48 thf(fact_136_leD,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y4: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y4 @ X )
% 0.30/0.48 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % leD
% 0.30/0.49 thf(fact_137_less__fun__def,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.49 => ( ( ord_less @ ( A > B ) )
% 0.30/0.49 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.49 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.30/0.49 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_fun_def
% 0.30/0.49 thf(fact_138_minf_I8_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(8)
% 0.30/0.49 thf(fact_139_minf_I6_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(6)
% 0.30/0.49 thf(fact_140_pinf_I8_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(8)
% 0.30/0.49 thf(fact_141_pinf_I6_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(6)
% 0.30/0.49 thf(fact_142_complete__interval,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [A2: A,B2: A,P: A > $o] :
% 0.30/0.49 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.49 => ( ( P @ A2 )
% 0.30/0.49 => ( ~ ( P @ B2 )
% 0.30/0.49 => ? [C3: A] :
% 0.30/0.49 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.30/0.49 & ( ord_less_eq @ A @ C3 @ B2 )
% 0.30/0.49 & ! [X5: A] :
% 0.30/0.49 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 0.30/0.49 & ( ord_less @ A @ X5 @ C3 ) )
% 0.30/0.49 => ( P @ X5 ) )
% 0.30/0.49 & ! [D: A] :
% 0.30/0.49 ( ! [X2: A] :
% 0.30/0.49 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.30/0.49 & ( ord_less @ A @ X2 @ D ) )
% 0.30/0.49 => ( P @ X2 ) )
% 0.30/0.49 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % complete_interval
% 0.30/0.49 thf(fact_143_chain__equality,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A] :
% 0.30/0.49 ( ( comple1193779247_chain @ A
% 0.30/0.49 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 0.30/0.49 @ A4 )
% 0.30/0.49 = ( ! [X3: A] :
% 0.30/0.49 ( ( member @ A @ X3 @ A4 )
% 0.30/0.49 => ! [Y3: A] :
% 0.30/0.49 ( ( member @ A @ Y3 @ A4 )
% 0.30/0.49 => ( X3 = Y3 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % chain_equality
% 0.30/0.49 thf(fact_144_chain__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( comple1193779247_chain @ A )
% 0.30/0.49 = ( ^ [Ord: A > A > $o,S: set @ A] :
% 0.30/0.49 ! [X3: A] :
% 0.30/0.49 ( ( member @ A @ X3 @ S )
% 0.30/0.49 => ! [Y3: A] :
% 0.30/0.49 ( ( member @ A @ Y3 @ S )
% 0.30/0.49 => ( ( Ord @ X3 @ Y3 )
% 0.30/0.49 | ( Ord @ Y3 @ X3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % chain_def
% 0.30/0.49 thf(fact_145_LeastM__natI,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.30/0.49 ( ( P @ K )
% 0.30/0.49 => ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % LeastM_natI
% 0.30/0.49 thf(fact_146_minf_I11_J,axiom,
% 0.30/0.49 ! [C2: $tType,D2: $tType] :
% 0.30/0.49 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.30/0.49 => ! [F3: D2] :
% 0.30/0.49 ? [Z3: C2] :
% 0.30/0.49 ! [X5: C2] :
% 0.30/0.49 ( ( ord_less @ C2 @ X5 @ Z3 )
% 0.30/0.49 => ( F3 = F3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(11)
% 0.30/0.49 thf(fact_147_minf_I7_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(7)
% 0.30/0.49 thf(fact_148_minf_I5_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(5)
% 0.30/0.49 thf(fact_149_minf_I4_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ( X5 != T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(4)
% 0.30/0.49 thf(fact_150_minf_I3_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ( X5 != T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(3)
% 0.30/0.49 thf(fact_151_minf_I2_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.49 ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.49 => ( ( P @ X2 )
% 0.30/0.49 = ( P2 @ X2 ) ) )
% 0.30/0.49 => ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.49 => ( ( Q @ X2 )
% 0.30/0.49 = ( Q2 @ X2 ) ) )
% 0.30/0.49 => ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ( ( ( P @ X5 )
% 0.30/0.49 | ( Q @ X5 ) )
% 0.30/0.49 = ( ( P2 @ X5 )
% 0.30/0.49 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(2)
% 0.30/0.49 thf(fact_152_minf_I1_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.49 ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.49 => ( ( P @ X2 )
% 0.30/0.49 = ( P2 @ X2 ) ) )
% 0.30/0.49 => ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.49 => ( ( Q @ X2 )
% 0.30/0.49 = ( Q2 @ X2 ) ) )
% 0.30/0.49 => ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.49 => ( ( ( P @ X5 )
% 0.30/0.49 & ( Q @ X5 ) )
% 0.30/0.49 = ( ( P2 @ X5 )
% 0.30/0.49 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % minf(1)
% 0.30/0.49 thf(fact_153_pinf_I11_J,axiom,
% 0.30/0.49 ! [C2: $tType,D2: $tType] :
% 0.30/0.49 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.30/0.49 => ! [F3: D2] :
% 0.30/0.49 ? [Z3: C2] :
% 0.30/0.49 ! [X5: C2] :
% 0.30/0.49 ( ( ord_less @ C2 @ Z3 @ X5 )
% 0.30/0.49 => ( F3 = F3 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(11)
% 0.30/0.49 thf(fact_154_pinf_I7_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(7)
% 0.30/0.49 thf(fact_155_pinf_I5_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(5)
% 0.30/0.49 thf(fact_156_pinf_I4_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ( X5 != T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(4)
% 0.30/0.49 thf(fact_157_pinf_I3_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [T: A] :
% 0.30/0.49 ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ( X5 != T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(3)
% 0.30/0.49 thf(fact_158_pinf_I2_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.49 ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.49 => ( ( P @ X2 )
% 0.30/0.49 = ( P2 @ X2 ) ) )
% 0.30/0.49 => ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.49 => ( ( Q @ X2 )
% 0.30/0.49 = ( Q2 @ X2 ) ) )
% 0.30/0.49 => ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ( ( ( P @ X5 )
% 0.30/0.49 | ( Q @ X5 ) )
% 0.30/0.49 = ( ( P2 @ X5 )
% 0.30/0.49 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(2)
% 0.30/0.49 thf(fact_159_pinf_I1_J,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.49 ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.49 => ( ( P @ X2 )
% 0.30/0.49 = ( P2 @ X2 ) ) )
% 0.30/0.49 => ( ? [Z4: A] :
% 0.30/0.49 ! [X2: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.49 => ( ( Q @ X2 )
% 0.30/0.49 = ( Q2 @ X2 ) ) )
% 0.30/0.49 => ? [Z3: A] :
% 0.30/0.49 ! [X5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.49 => ( ( ( P @ X5 )
% 0.30/0.49 & ( Q @ X5 ) )
% 0.30/0.49 = ( ( P2 @ X5 )
% 0.30/0.49 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pinf(1)
% 0.30/0.49 thf(fact_160_ex__gt__or__lt,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [A2: A] :
% 0.30/0.49 ? [B3: A] :
% 0.30/0.49 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.49 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ex_gt_or_lt
% 0.30/0.49 thf(fact_161_chain__subset,axiom,
% 0.30/0.49 ! [A: $tType,Ord2: A > A > $o,A4: set @ A,B6: set @ A] :
% 0.30/0.49 ( ( comple1193779247_chain @ A @ Ord2 @ A4 )
% 0.30/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 0.30/0.49 => ( comple1193779247_chain @ A @ Ord2 @ B6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % chain_subset
% 0.30/0.49 thf(fact_162_chainD,axiom,
% 0.30/0.49 ! [A: $tType,Ord2: A > A > $o,S2: set @ A,X: A,Y4: A] :
% 0.30/0.49 ( ( comple1193779247_chain @ A @ Ord2 @ S2 )
% 0.30/0.49 => ( ( member @ A @ X @ S2 )
% 0.30/0.49 => ( ( member @ A @ Y4 @ S2 )
% 0.30/0.49 => ( ( Ord2 @ X @ Y4 )
% 0.30/0.49 | ( Ord2 @ Y4 @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % chainD
% 0.30/0.49 thf(fact_163_chainE,axiom,
% 0.30/0.49 ! [A: $tType,Ord2: A > A > $o,S2: set @ A,X: A,Y4: A] :
% 0.30/0.49 ( ( comple1193779247_chain @ A @ Ord2 @ S2 )
% 0.30/0.49 => ( ( member @ A @ X @ S2 )
% 0.30/0.49 => ( ( member @ A @ Y4 @ S2 )
% 0.30/0.49 => ( ~ ( Ord2 @ X @ Y4 )
% 0.30/0.49 => ( Ord2 @ Y4 @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % chainE
% 0.30/0.49 thf(fact_164_chainI,axiom,
% 0.30/0.49 ! [A: $tType,S2: set @ A,Ord2: A > A > $o] :
% 0.30/0.49 ( ! [X2: A,Y: A] :
% 0.30/0.49 ( ( member @ A @ X2 @ S2 )
% 0.30/0.49 => ( ( member @ A @ Y @ S2 )
% 0.30/0.49 => ( ( Ord2 @ X2 @ Y )
% 0.30/0.49 | ( Ord2 @ Y @ X2 ) ) ) )
% 0.30/0.49 => ( comple1193779247_chain @ A @ Ord2 @ S2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % chainI
% 0.30/0.49 thf(fact_165_nat__less__le,axiom,
% 0.30/0.49 ( ( ord_less @ nat )
% 0.30/0.49 = ( ^ [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 & ( M != N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_less_le
% 0.30/0.49 thf(fact_166_less__imp__le__nat,axiom,
% 0.30/0.49 ! [M2: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M2 @ N2 )
% 0.30/0.49 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_imp_le_nat
% 0.30/0.49 thf(fact_167_le__eq__less__or__eq,axiom,
% 0.30/0.49 ( ( ord_less_eq @ nat )
% 0.30/0.49 = ( ^ [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M @ N )
% 0.30/0.49 | ( M = N ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_eq_less_or_eq
% 0.30/0.49 thf(fact_168_nat__neq__iff,axiom,
% 0.30/0.49 ! [M2: nat,N2: nat] :
% 0.30/0.49 ( ( M2 != N2 )
% 0.30/0.49 = ( ( ord_less @ nat @ M2 @ N2 )
% 0.30/0.49 | ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_neq_iff
% 0.30/0.49 thf(fact_169_less__not__refl,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.30/0.49
% 0.30/0.49 % less_not_refl
% 0.30/0.49 thf(fact_170_less__not__refl2,axiom,
% 0.30/0.49 ! [N2: nat,M2: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ N2 @ M2 )
% 0.30/0.49 => ( M2 != N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_not_refl2
% 0.30/0.49 thf(fact_171_less__not__refl3,axiom,
% 0.30/0.49 ! [S3: nat,T: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ S3 @ T )
% 0.30/0.49 => ( S3 != T ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_not_refl3
% 0.30/0.49 thf(fact_172_measure__induct,axiom,
% 0.30/0.49 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.30/0.49 ( ! [X2: A] :
% 0.30/0.49 ( ! [Y5: A] :
% 0.30/0.49 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X2 ) )
% 0.30/0.49 => ( P @ Y5 ) )
% 0.30/0.49 => ( P @ X2 ) )
% 0.30/0.49 => ( P @ A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % measure_induct
% 0.30/0.49 thf(fact_173_less__irrefl__nat,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.30/0.49
% 0.30/0.49 % less_irrefl_nat
% 0.30/0.49 thf(fact_174_nat__less__induct,axiom,
% 0.30/0.49 ! [P: nat > $o,N2: nat] :
% 0.30/0.49 ( ! [N3: nat] :
% 0.30/0.49 ( ! [M3: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M3 @ N3 )
% 0.30/0.49 => ( P @ M3 ) )
% 0.30/0.49 => ( P @ N3 ) )
% 0.30/0.49 => ( P @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_less_induct
% 0.30/0.49 thf(fact_175_infinite__descent,axiom,
% 0.30/0.49 ! [P: nat > $o,N2: nat] :
% 0.30/0.49 ( ! [N3: nat] :
% 0.30/0.49 ( ~ ( P @ N3 )
% 0.30/0.49 => ? [M3: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ M3 @ N3 )
% 0.30/0.49 & ~ ( P @ M3 ) ) )
% 0.30/0.49 => ( P @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % infinite_descent
% 0.30/0.49 thf(fact_176_linorder__neqE__nat,axiom,
% 0.30/0.49 ! [X: nat,Y4: nat] :
% 0.30/0.49 ( ( X != Y4 )
% 0.30/0.49 => ( ~ ( ord_less @ nat @ X @ Y4 )
% 0.30/0.49 => ( ord_less @ nat @ Y4 @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % linorder_neqE_nat
% 0.30/0.49 thf(fact_177_measure__induct__rule,axiom,
% 0.30/0.49 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.30/0.49 ( ! [X2: A] :
% 0.30/0.49 ( ! [Y5: A] :
% 0.30/0.49 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X2 ) )
% 0.30/0.49 => ( P @ Y5 ) )
% 0.30/0.49 => ( P @ X2 ) )
% 0.30/0.49 => ( P @ A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % measure_induct_rule
% 0.30/0.49 thf(fact_178_infinite__descent__measure,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.30/0.49 ( ! [X2: A] :
% 0.30/0.49 ( ~ ( P @ X2 )
% 0.30/0.49 => ? [Y5: A] :
% 0.30/0.49 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X2 ) )
% 0.30/0.49 & ~ ( P @ Y5 ) ) )
% 0.30/0.49 => ( P @ X ) ) ).
% 0.30/0.49
% 0.30/0.49 % infinite_descent_measure
% 0.30/0.49 thf(fact_179_nat__le__linear,axiom,
% 0.30/0.49 ! [M2: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.30/0.49 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_le_linear
% 0.30/0.49 thf(fact_180_le__antisym,axiom,
% 0.30/0.49 ! [M2: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 0.30/0.49 => ( M2 = N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_antisym
% 0.30/0.49 thf(fact_181_eq__imp__le,axiom,
% 0.30/0.49 ! [M2: nat,N2: nat] :
% 0.30/0.49 ( ( M2 = N2 )
% 0.30/0.49 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % eq_imp_le
% 0.30/0.49 thf(fact_182_le__trans,axiom,
% 0.30/0.49 ! [I: nat,J: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ J @ K )
% 0.30/0.49 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_trans
% 0.30/0.49 thf(fact_183_le__refl,axiom,
% 0.30/0.49 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.30/0.49
% 0.30/0.49 % le_refl
% 0.30/0.49 thf(fact_184_less__mono__imp__le__mono,axiom,
% 0.30/0.49 ! [F: nat > nat,I: nat,J: nat] :
% 0.30/0.49 ( ! [I2: nat,J2: nat] :
% 0.30/0.49 ( ( ord_less @ nat @ I2 @ J2 )
% 0.30/0.49 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.30/0.49 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.49 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_mono_imp_le_mono
% 0.30/0.49 thf(fact_185_le__neq__implies__less,axiom,
% 0.30/0.49 ! [M2: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.30/0.49 => ( ( M2 != N2 )
% 0.30/0.49 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_neq_implies_less
% 0.30/0.49 thf(fact_186_less__or__eq__imp__le,axiom,
% 0.30/0.49 ! [M2: nat,N2: nat] :
% 0.30/0.49 ( ( ( ord_less @ nat @ M2 @ N2 )
% 0.30/0.49 | ( M2 = N2 ) )
% 0.30/0.49 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_or_eq_imp_le
% 0.30/0.49 thf(fact_187_psubsetI,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ( ( A4 != B6 )
% 0.30/0.49 => ( ord_less @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % psubsetI
% 0.30/0.49 thf(fact_188_GreatestM__nat__lemma,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 0.30/0.49 ( ( P @ K )
% 0.30/0.49 => ( ! [Y: A] :
% 0.30/0.49 ( ( P @ Y )
% 0.30/0.49 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 0.30/0.49 => ( ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) )
% 0.30/0.49 & ! [Y5: A] :
% 0.30/0.49 ( ( P @ Y5 )
% 0.30/0.49 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % GreatestM_nat_lemma
% 0.30/0.49 thf(fact_189_GreatestM__nat__le,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,X: A,M2: A > nat,B2: nat] :
% 0.30/0.49 ( ( P @ X )
% 0.30/0.49 => ( ! [Y: A] :
% 0.30/0.49 ( ( P @ Y )
% 0.30/0.49 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 0.30/0.49 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % GreatestM_nat_le
% 0.30/0.49 thf(fact_190_psubsetD,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A,B6: set @ A,C: A] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ( ( member @ A @ C @ A4 )
% 0.30/0.49 => ( member @ A @ C @ B6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % psubsetD
% 0.30/0.49 thf(fact_191_psubset__trans,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A,B6: set @ A,C4: set @ A] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 0.30/0.49 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % psubset_trans
% 0.30/0.49 thf(fact_192_GreatestM__natI,axiom,
% 0.30/0.49 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 0.30/0.49 ( ( P @ K )
% 0.30/0.49 => ( ! [Y: A] :
% 0.30/0.49 ( ( P @ Y )
% 0.30/0.49 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 0.30/0.49 => ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % GreatestM_natI
% 0.30/0.49 thf(fact_193_GreatestMI2,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.49 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 0.30/0.49 ( ( P @ X )
% 0.30/0.49 => ( ! [Y: A] :
% 0.30/0.49 ( ( P @ Y )
% 0.30/0.49 => ( ord_less_eq @ B @ ( M2 @ Y ) @ ( M2 @ X ) ) )
% 0.30/0.49 => ( ! [X2: A] :
% 0.30/0.49 ( ( P @ X2 )
% 0.30/0.49 => ( ! [Y5: A] :
% 0.30/0.49 ( ( P @ Y5 )
% 0.30/0.49 => ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
% 0.30/0.49 => ( Q @ X2 ) ) )
% 0.30/0.49 => ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % GreatestMI2
% 0.30/0.49 thf(fact_194_psubsetE,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % psubsetE
% 0.30/0.49 thf(fact_195_psubset__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) )
% 0.30/0.49 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.49 & ( A6 != B7 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % psubset_eq
% 0.30/0.49 thf(fact_196_psubset__imp__subset,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ( ord_less_eq @ ( set @ A ) @ A4 @ B6 ) ) ).
% 0.30/0.49
% 0.30/0.49 % psubset_imp_subset
% 0.30/0.49 thf(fact_197_psubset__subset__trans,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A,B6: set @ A,C4: set @ A] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.30/0.49 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % psubset_subset_trans
% 0.30/0.49 thf(fact_198_subset__not__subset__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) )
% 0.30/0.49 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.49 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_not_subset_eq
% 0.30/0.49 thf(fact_199_subset__psubset__trans,axiom,
% 0.30/0.49 ! [A: $tType,A4: set @ A,B6: set @ A,C4: set @ A] :
% 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.30/0.49 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 0.30/0.49 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_psubset_trans
% 0.30/0.49 thf(fact_200_subset__iff__psubset__eq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.49 = ( ^ [A6: set @ A,B7: set @ A] :
% 0.30/0.49 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 0.30/0.49 | ( A6 = B7 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_iff_psubset_eq
% 0.30/0.49 thf(fact_201_monoseq__iff,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.49 => ( ( topological_monoseq @ A )
% 0.30/0.49 = ( ^ [X4: nat > A] :
% 0.30/0.49 ( ( order_mono @ nat @ A @ X4 )
% 0.30/0.49 | ( order_antimono @ nat @ A @ X4 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % monoseq_iff
% 0.30/0.49 thf(fact_202_monoI1,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [X6: nat > A] :
% 0.30/0.49 ( ! [M4: nat,N3: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.30/0.49 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % monoI1
% 0.30/0.49 thf(fact_203_incseq__imp__monoseq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [X6: nat > A] :
% 0.30/0.49 ( ( order_mono @ nat @ A @ X6 )
% 0.30/0.49 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % incseq_imp_monoseq
% 0.30/0.49 thf(fact_204_decseq__imp__monoseq,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [X6: nat > A] :
% 0.30/0.49 ( ( order_antimono @ nat @ A @ X6 )
% 0.30/0.49 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % decseq_imp_monoseq
% 0.30/0.49 thf(fact_205_monoseq__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.49 => ( ( topological_monoseq @ A )
% 0.30/0.49 = ( ^ [X4: nat > A] :
% 0.30/0.49 ( ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq @ A @ ( X4 @ M ) @ ( X4 @ N ) ) )
% 0.30/0.49 | ! [M: nat,N: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.49 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % monoseq_def
% 0.30/0.49 thf(fact_206_monoI2,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [X6: nat > A] :
% 0.30/0.49 ( ! [M4: nat,N3: nat] :
% 0.30/0.49 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 0.30/0.49 => ( ord_less_eq @ A @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 0.30/0.49 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % monoI2
% 0.30/0.49 thf(fact_207_linorder__neqE__linordered__idom,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [X: A,Y4: A] :
% 0.30/0.49 ( ( X != Y4 )
% 0.30/0.49 => ( ~ ( ord_less @ A @ X @ Y4 )
% 0.30/0.49 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % linorder_neqE_linordered_idom
% 0.30/0.49 thf(fact_208_linordered__field__no__ub,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [X5: A] :
% 0.30/0.49 ? [X12: A] : ( ord_less @ A @ X5 @ X12 ) ) ).
% 0.30/0.49
% 0.30/0.49 % linordered_field_no_ub
% 0.30/0.49 thf(fact_209_linordered__field__no__lb,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linordered_field @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [X5: A] :
% 0.30/0.49 ? [Y: A] : ( ord_less @ A @ Y @ X5 ) ) ).
% 0.30/0.49
% 0.30/0.49 % linordered_field_no_lb
% 0.30/0.49 thf(fact_210_dependent__wellorder__choice,axiom,
% 0.30/0.49 ! [B: $tType,A: $tType] :
% 0.30/0.49 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ! [P: ( A > B ) > A > B > $o] :
% 0.30/0.49 ( ! [R: B,F4: A > B,G3: A > B,X2: A] :
% 0.30/0.49 ( ! [Y5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Y5 @ X2 )
% 0.30/0.49 => ( ( F4 @ Y5 )
% 0.30/0.49 = ( G3 @ Y5 ) ) )
% 0.30/0.49 => ( ( P @ F4 @ X2 @ R )
% 0.30/0.49 = ( P @ G3 @ X2 @ R ) ) )
% 0.30/0.49 => ( ! [X2: A,F4: A > B] :
% 0.30/0.49 ( ! [Y5: A] :
% 0.30/0.49 ( ( ord_less @ A @ Y5 @ X2 )
% 0.30/0.49 => ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
% 0.30/0.49 => ? [X1: B] : ( P @ F4 @ X2 @ X1 ) )
% 0.30/0.49 => ? [F4: A > B] :
% 0.30/0.49 ! [X5: A] : ( P @ F4 @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % dependent_wellorder_choice
% 0.30/0.49 thf(fact_211_order_Oordering__axioms,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.49 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % order.ordering_axioms
% 0.30/0.49 thf(fact_212_ordering__def,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( ordering @ A )
% 0.30/0.49 = ( ^ [Less_eq: A > A > $o,Less: A > A > $o] :
% 0.30/0.49 ( ! [A5: A,B5: A] :
% 0.30/0.49 ( ( Less @ A5 @ B5 )
% 0.30/0.49 = ( ( Less_eq @ A5 @ B5 )
% 0.30/0.49 & ( A5 != B5 ) ) )
% 0.30/0.49 & ! [A5: A] : ( Less_eq @ A5 @ A5 )
% 0.30/0.49 & ! [A5: A,B5: A] :
% 0.30/0.49 ( ( Less_eq @ A5 @ B5 )
% 0.30/0.49 => ( ( Less_eq @ B5 @ A5 )
% 0.30/0.49 => ( A5 = B5 ) ) )
% 0.30/0.49 & ! [A5: A,B5: A,C5: A] :
% 0.30/0.49 ( ( Less_eq @ A5 @ B5 )
% 0.30/0.49 => ( ( Less_eq @ B5 @ C5 )
% 0.30/0.49 => ( Less_eq @ A5 @ C5 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering_def
% 0.30/0.49 thf(fact_213_ordering_Oasym,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less2 @ A2 @ B2 )
% 0.30/0.49 => ~ ( Less2 @ B2 @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.asym
% 0.30/0.49 thf(fact_214_ordering_Orefl,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( Less_eq2 @ A2 @ A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.refl
% 0.30/0.49 thf(fact_215_ordering_Ointro,axiom,
% 0.30/0.49 ! [A: $tType,Less2: A > A > $o,Less_eq2: A > A > $o] :
% 0.30/0.49 ( ! [A3: A,B3: A] :
% 0.30/0.49 ( ( Less2 @ A3 @ B3 )
% 0.30/0.49 = ( ( Less_eq2 @ A3 @ B3 )
% 0.30/0.49 & ( A3 != B3 ) ) )
% 0.30/0.49 => ( ! [A3: A] : ( Less_eq2 @ A3 @ A3 )
% 0.30/0.49 => ( ! [A3: A,B3: A] :
% 0.30/0.49 ( ( Less_eq2 @ A3 @ B3 )
% 0.30/0.49 => ( ( Less_eq2 @ B3 @ A3 )
% 0.30/0.49 => ( A3 = B3 ) ) )
% 0.30/0.49 => ( ! [A3: A,B3: A,C3: A] :
% 0.30/0.49 ( ( Less_eq2 @ A3 @ B3 )
% 0.30/0.49 => ( ( Less_eq2 @ B3 @ C3 )
% 0.30/0.49 => ( Less_eq2 @ A3 @ C3 ) ) )
% 0.30/0.49 => ( ordering @ A @ Less_eq2 @ Less2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.intro
% 0.30/0.49 thf(fact_216_ordering_Otrans,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less_eq2 @ A2 @ B2 )
% 0.30/0.49 => ( ( Less_eq2 @ B2 @ C )
% 0.30/0.49 => ( Less_eq2 @ A2 @ C ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.trans
% 0.30/0.49 thf(fact_217_ordering_Oirrefl,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ~ ( Less2 @ A2 @ A2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.irrefl
% 0.30/0.49 thf(fact_218_ordering_Oantisym,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less_eq2 @ A2 @ B2 )
% 0.30/0.49 => ( ( Less_eq2 @ B2 @ A2 )
% 0.30/0.49 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.antisym
% 0.30/0.49 thf(fact_219_ordering_Ostrict__trans,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less2 @ A2 @ B2 )
% 0.30/0.49 => ( ( Less2 @ B2 @ C )
% 0.30/0.49 => ( Less2 @ A2 @ C ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.strict_trans
% 0.30/0.49 thf(fact_220_ordering_Ostrict__trans1,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less_eq2 @ A2 @ B2 )
% 0.30/0.49 => ( ( Less2 @ B2 @ C )
% 0.30/0.49 => ( Less2 @ A2 @ C ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.strict_trans1
% 0.30/0.49 thf(fact_221_ordering_Ostrict__trans2,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less2 @ A2 @ B2 )
% 0.30/0.49 => ( ( Less_eq2 @ B2 @ C )
% 0.30/0.49 => ( Less2 @ A2 @ C ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.strict_trans2
% 0.30/0.49 thf(fact_222_ordering_Oorder__iff__strict,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less_eq2 @ A2 @ B2 )
% 0.30/0.49 = ( ( Less2 @ A2 @ B2 )
% 0.30/0.49 | ( A2 = B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.order_iff_strict
% 0.30/0.49 thf(fact_223_ordering_Ostrict__iff__order,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less2 @ A2 @ B2 )
% 0.30/0.49 = ( ( Less_eq2 @ A2 @ B2 )
% 0.30/0.49 & ( A2 != B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.strict_iff_order
% 0.30/0.49 thf(fact_224_ordering_Ostrict__implies__order,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less2 @ A2 @ B2 )
% 0.30/0.49 => ( Less_eq2 @ A2 @ B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.strict_implies_order
% 0.30/0.49 thf(fact_225_ordering_Ostrict__implies__not__eq,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( Less2 @ A2 @ B2 )
% 0.30/0.49 => ( A2 != B2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.strict_implies_not_eq
% 0.30/0.49 thf(fact_226_ordering_Onot__eq__order__implies__strict,axiom,
% 0.30/0.49 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 0.30/0.49 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 0.30/0.49 => ( ( A2 != B2 )
% 0.30/0.49 => ( ( Less_eq2 @ A2 @ B2 )
% 0.30/0.49 => ( Less2 @ A2 @ B2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ordering.not_eq_order_implies_strict
% 0.30/0.49
% 0.30/0.49 %----Subclasses (4)
% 0.30/0.49 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ( type @ A @ ( type2 @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ( ord @ A @ ( type2 @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ( order @ A @ ( type2 @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 0.30/0.49 ! [A: $tType] :
% 0.30/0.49 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.49 => ( preorder @ A @ ( type2 @ A ) ) ) ).
% 0.30/0.49
% 0.30/0.49 %----Type constructors (20)
% 0.30/0.49 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.30/0.49 ! [A7: $tType,A8: $tType] :
% 0.30/0.49 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 0.30/0.49 => ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.49 ! [A7: $tType,A8: $tType] :
% 0.30/0.49 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.30/0.49 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.49 ! [A7: $tType,A8: $tType] :
% 0.30/0.49 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.30/0.49 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.49 ! [A7: $tType,A8: $tType] :
% 0.30/0.49 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.30/0.49 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.30/0.49 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.30/0.49 wellorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.30/0.49 preorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/0.49 linorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.30/0.49 no_top @ nat @ ( type2 @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.30/0.49 order @ nat @ ( type2 @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.30/0.49 ord @ nat @ ( type2 @ nat ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4,axiom,
% 0.30/0.49 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 0.30/0.49 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.30/0.49 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.30/0.49 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8,axiom,
% 0.30/0.49 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
% 0.30/0.49 preorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_HOL_Obool___Orderings_Olinorder_10,axiom,
% 0.30/0.49 linorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.30/0.49 order @ $o @ ( type2 @ $o ) ).
% 0.30/0.49
% 0.30/0.49 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 0.30/0.49 ord @ $o @ ( type2 @ $o ) ).
% 0.30/0.49
% 0.30/0.49 %----Free types (1)
% 0.30/0.49 thf(tfree_0,hypothesis,
% 0.30/0.49 linorder @ a @ ( type2 @ a ) ).
% 0.30/0.49
% 0.30/0.49 %----Conjectures (1)
% 0.30/0.49 thf(conj_0,conjecture,
% 0.30/0.49 p @ a2 @ b ).
% 0.30/0.49
% 0.30/0.49 %------------------------------------------------------------------------------
% 0.30/0.49 ------------------------
% 3.09/1.79 ---- Embedded in THF ---
% 3.09/1.79 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.09/1.79 %%% Generated on Tue Feb 24 06:07:43 EST 2026
% 3.09/1.79 %%% using '$$dhol' embedding, version 1.3.0.
% 3.09/1.79 %%% Logic specification used:
% 3.09/1.79 %%% thf(spec, logic, $$dhol).
% 3.09/1.79
% 3.09/1.79 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2DTF16938.p
% 3.09/1.79 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.09/1.79 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.09/1.79 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.09/1.79 thf(ty_tf_a, type, a: $tType).
% 3.09/1.79 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.09/1.79 thf(sy_c_Complete__Partial__Order_Ochain, type, comple1193779247_chain: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 3.09/1.79 thf(sy_c_Hilbert__Choice_OGreatestM, type, hilbert_GreatestM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 3.09/1.79 thf(sy_c_Hilbert__Choice_OLeastM, type, hilbert_LeastM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 3.09/1.79 thf(sy_c_Orderings_Oord__class_OLeast, type, ord_Least: (!> [A:$tType]: (((A > $o) > A)))).
% 3.09/1.79 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.09/1.79 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.09/1.79 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 3.09/1.79 thf(sy_c_Orderings_Oorder__class_Omono, type, order_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 3.09/1.79 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 3.09/1.79 thf(sy_c_Orderings_Oordering, type, ordering: (!> [A:$tType]: (((A > (A > $o)) > ((A > (A > $o)) > $o))))).
% 3.09/1.79 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 3.09/1.79 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.09/1.79 thf(sy_c_Topological__Spaces_Omonoseq, type, topological_monoseq: (!> [A:$tType]: (((nat > A) > $o)))).
% 3.09/1.79 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.09/1.79 thf(sy_v_P, type, p: (a > (a > $o))).
% 3.09/1.79 thf(sy_v_a, type, a2: a).
% 3.09/1.79 thf(sy_v_b, type, b: a).
% 3.09/1.79 thf(fact_0_local_Osym, axiom, (((p @ b) @ a2) => ((p @ a2) @ b))).
% 3.09/1.79 thf(fact_1_le, axiom, (! [A2:a,B2:a]: (((((ord_less_eq @ a) @ A2) @ B2) => ((p @ A2) @ B2))))).
% 3.09/1.79 thf(fact_2_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 3.09/1.79 thf(fact_3_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))))))))).
% 3.09/1.79 thf(fact_4_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))))))))).
% 3.09/1.79 thf(fact_5_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)))))))).
% 3.09/1.79 thf(fact_6_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))))))))))).
% 3.09/1.79 thf(fact_7_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))))))))))).
% 3.09/1.79 thf(fact_8_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)))))))))).
% 3.09/1.79 thf(fact_9_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))))))))))).
% 3.09/1.79 thf(fact_10_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)))))))))).
% 3.09/1.79 thf(fact_11_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y2:A,Z:A]: ((Y2 = Z))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 3.09/1.79 thf(fact_12_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X) => (X = Y4))))))))).
% 3.09/1.79 thf(fact_13_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_14_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 3.09/1.79 thf(fact_15_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 3.09/1.79 thf(fact_16_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((! [A3:A,B3:A]: ((((P @ B3) @ A3) => ((P @ A3) @ B3)))) => ((P @ A2) @ B2))))))))).
% 3.09/1.79 thf(fact_17_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 3.09/1.79 thf(fact_18_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 3.09/1.79 thf(fact_19_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 3.09/1.79 thf(fact_20_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 3.09/1.79 thf(fact_21_ord__eq__le__trans, axiom, (! [A:$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))))))))).
% 3.09/1.79 thf(fact_22_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 3.09/1.79 thf(fact_23_le__cases3, axiom, (! [A:$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))))))))))))))))).
% 3.09/1.79 thf(fact_24_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 3.09/1.79 thf(fact_25_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_26_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X = Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 3.09/1.79 thf(fact_27_lfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ X) => (X = Y4))))))))).
% 3.09/1.79 thf(fact_28_gfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => ((((ord_less_eq @ A) @ X) @ Y4) => (X = Y4))))))))).
% 3.09/1.79 thf(fact_29_lfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ 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))))))))).
% 3.09/1.79 thf(fact_30_gfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y4:A,X:A,Z2:A]: (((((ord_less_eq @ A) @ Y4) @ X) => ((((ord_less_eq @ A) @ Z2) @ Y4) => (((ord_less_eq @ A) @ Z2) @ X))))))))).
% 3.09/1.79 thf(fact_31_lfp_Oleq__refl, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 3.09/1.79 thf(fact_32_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))))))))).
% 3.09/1.79 thf(fact_33_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)))))))))))).
% 3.09/1.79 thf(fact_34_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)))))))).
% 3.09/1.79 thf(fact_35_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)))))))))).
% 3.09/1.79 thf(fact_36_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)))))))))).
% 3.09/1.79 thf(fact_37_LeastI2__wellorder__ex, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),Q:(A > $o)]: (((? [X1:A]: ((P @ X1))) => ((! [A3:A]: (((P @ A3) => ((! [B4:A]: (((P @ B4) => (((ord_less_eq @ A) @ A3) @ B4)))) => (Q @ A3))))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 3.09/1.79 thf(fact_38_LeastI2__ex, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),Q:(A > $o)]: (((? [X1:A]: ((P @ X1))) => ((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 3.09/1.79 thf(fact_39_LeastI__ex, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o)]: (((? [X1:A]: ((P @ X1))) => (P @ ((ord_Least @ A) @ P))))))))).
% 3.09/1.79 thf(fact_40_LeastI2, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 3.09/1.79 thf(fact_41_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X) = (F @ Y4)) = (X = Y4))))))))).
% 3.09/1.79 thf(fact_42_LeastI2__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [P:(A > $o),X:A,Q:(A > $o)]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ A) @ X) @ Y)))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ A) @ X2) @ Y5)))) => (Q @ X2))))) => (Q @ ((ord_Least @ A) @ P))))))))))).
% 3.09/1.79 thf(fact_43_Least__equality, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [P:(A > $o),X:A]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ A) @ X) @ Y)))) => (((ord_Least @ A) @ P) = X))))))))).
% 3.09/1.79 thf(fact_44_LeastI2__wellorder, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A,Q:(A > $o)]: (((P @ A2) => ((! [A3:A]: (((P @ A3) => ((! [B4:A]: (((P @ B4) => (((ord_less_eq @ A) @ A3) @ B4)))) => (Q @ A3))))) => (Q @ ((ord_Least @ A) @ P)))))))))).
% 3.09/1.79 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.09/1.79 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 3.09/1.79 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.09/1.79 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 3.09/1.79 thf(fact_49_decseqD, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),I:nat,J:nat]: (((((order_antimono @ nat) @ A) @ F) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ (F @ J)) @ (F @ I)))))))))).
% 3.09/1.79 thf(fact_50_decseq__def, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (((order_antimono @ nat) @ A) = (^ [X4:(nat > A)]: ((! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X4 @ N)) @ (X4 @ M)))))))))))).
% 3.09/1.79 thf(fact_51_strict__mono__less, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X)) @ (F @ Y4)) = (((ord_less @ A) @ X) @ Y4))))))))).
% 3.09/1.79 thf(fact_52_strict__mono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_strict_mono @ A) @ B) = (^ [F2:(A > B)]: ((! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less @ B) @ (F2 @ X3)) @ (F2 @ Y3)))))))))))).
% 3.09/1.79 thf(fact_53_strict__monoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y))))) => (((order_strict_mono @ A) @ B) @ F)))))))).
% 3.09/1.79 thf(fact_54_strict__monoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ A) @ X) @ Y4) => (((ord_less @ B) @ (F @ X)) @ (F @ Y4)))))))))).
% 3.09/1.79 thf(fact_55_strict__mono__mono, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((((order_strict_mono @ A) @ B) @ F) => (((order_mono @ A) @ B) @ F)))))))).
% 3.09/1.79 thf(fact_56_linorder__chain, axiom, (! [B:$tType]: ((((linorder @ B) @ (type2 @ B)) => (! [Y6:(set @ B)]: ((((comple1193779247_chain @ B) @ (ord_less_eq @ B)) @ Y6))))))).
% 3.09/1.79 thf(fact_57_LeastMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M2:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ B) @ (M2 @ X)) @ (M2 @ Y))))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (M2 @ X2)) @ (M2 @ Y5))))) => (Q @ X2))))) => (Q @ ((((hilbert_LeastM @ A) @ B) @ M2) @ P))))))))))).
% 3.09/1.79 thf(fact_58_incseqD, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),I:nat,J:nat]: (((((order_mono @ nat) @ A) @ F) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ (F @ I)) @ (F @ J)))))))))).
% 3.09/1.79 thf(fact_59_incseq__def, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (((order_mono @ nat) @ A) = (^ [X4:(nat > A)]: ((! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X4 @ M)) @ (X4 @ N)))))))))))).
% 3.09/1.79 thf(fact_60_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less @ B) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 3.09/1.79 thf(fact_61_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 3.09/1.79 thf(fact_62_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y:B]: (((((ord_less @ B) @ X2) @ Y) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 3.09/1.79 thf(fact_63_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 3.09/1.79 thf(fact_64_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y:A]: ((((ord_less @ A) @ Y) @ X))))))))).
% 3.09/1.79 thf(fact_65_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X12:A]: ((((ord_less @ A) @ X) @ X12))))))))).
% 3.09/1.79 thf(fact_66_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less @ A) @ Y4) @ X))))))))).
% 3.09/1.79 thf(fact_67_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) = ((((ord_less @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 3.09/1.79 thf(fact_68_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 3.09/1.79 thf(fact_69_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y4))))))))))).
% 3.09/1.79 thf(fact_70_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 3.09/1.79 thf(fact_71_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 3.09/1.79 thf(fact_72_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 3.09/1.79 thf(fact_73_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 3.09/1.79 thf(fact_74_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) | ((X = Y4) | (((ord_less @ A) @ Y4) @ X))))))))).
% 3.09/1.79 thf(fact_75_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 3.09/1.79 thf(fact_76_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.09/1.79 thf(fact_77_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.09/1.79 thf(fact_78_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 3.09/1.79 thf(fact_79_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (X != Y4)))))))).
% 3.09/1.79 thf(fact_80_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 3.09/1.79 thf(fact_81_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => (P @ Y5)))) => (P @ X2)))) => (P @ A2)))))))).
% 3.09/1.79 thf(fact_82_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((~ ((((ord_less @ A) @ Y4) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 3.09/1.79 thf(fact_83_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (Y4 != X)))))))).
% 3.09/1.79 thf(fact_84_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,P:$o]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less @ A) @ Y4) @ X) => P)))))))).
% 3.09/1.79 thf(fact_85_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((X != Y4) => (((ord_less @ A) @ Y4) @ X))))))))).
% 3.09/1.79 thf(fact_86_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 3.09/1.79 thf(fact_87_order_Ostrict__trans, axiom, (! [A:$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))))))))).
% 3.09/1.79 thf(fact_88_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (~ ((((ord_less @ A) @ Y4) @ X)))))))))).
% 3.09/1.79 thf(fact_89_dual__order_Ostrict__trans, axiom, (! [A:$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))))))))).
% 3.09/1.79 thf(fact_90_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = ((((ord_less @ A) @ Y4) @ X) | (X = Y4))))))))).
% 3.09/1.79 thf(fact_91_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 3.09/1.79 thf(fact_92_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 3.09/1.79 thf(fact_93_mono__invE, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X)) @ (F @ Y4)) => (((ord_less_eq @ A) @ X) @ Y4))))))))).
% 3.09/1.79 thf(fact_94_ex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat),B2:nat]: (((P @ K) => ((! [Y:A]: (((P @ Y) => (((ord_less @ nat) @ (M2 @ Y)) @ B2)))) => (? [X2:A]: (((P @ X2) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ Y5)) @ (M2 @ X2))))))))))))).
% 3.09/1.79 thf(fact_95_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => (? [X2:A]: (((P @ X2) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ X2)) @ (M2 @ Y5)))))))))))).
% 3.09/1.79 thf(fact_96_LeastM__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => ((P @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P)) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P))) @ (M2 @ Y5)))))))))).
% 3.09/1.79 thf(fact_97_LeastM__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M2:(A > nat)]: (((P @ X) => (((ord_less_eq @ nat) @ (M2 @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P))) @ (M2 @ X)))))).
% 3.09/1.79 thf(fact_98_mono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_mono @ A) @ B) = (^ [F2:(A > B)]: ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y3)))))))))))).
% 3.09/1.79 thf(fact_99_monoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y))))) => (((order_mono @ A) @ B) @ F)))))))).
% 3.09/1.79 thf(fact_100_monoE, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y4)))))))))).
% 3.09/1.79 thf(fact_101_monoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y4:A]: (((((order_mono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y4) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y4)))))))))).
% 3.09/1.79 thf(fact_102_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 3.09/1.79 thf(fact_103_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 3.09/1.79 thf(fact_104_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B5:A,A5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (A5 != B5))))))))).
% 3.09/1.79 thf(fact_105_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: (((((ord_less @ A) @ B5) @ A5) | (A5 = B5))))))))).
% 3.09/1.79 thf(fact_106_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 3.09/1.79 thf(fact_107_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y4) => (((ord_less_eq @ A) @ W) @ Z2))))) => (((ord_less_eq @ A) @ Y4) @ Z2))))))))).
% 3.09/1.79 thf(fact_108_dense__ge__bounded, axiom, (! [A:$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))))))))).
% 3.09/1.79 thf(fact_109_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 3.09/1.79 thf(fact_110_dual__order_Ostrict__trans1, axiom, (! [A:$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))))))))).
% 3.09/1.79 thf(fact_111_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (A5 != B5))))))))).
% 3.09/1.79 thf(fact_112_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: (((((ord_less @ A) @ A5) @ B5) | (A5 = B5))))))))).
% 3.09/1.79 thf(fact_113_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.09/1.79 thf(fact_114_order_Ostrict__trans1, axiom, (! [A:$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))))))))).
% 3.09/1.79 thf(fact_115_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y4) @ X))) => (((ord_less @ A) @ X) @ Y4)))))))).
% 3.09/1.79 thf(fact_116_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X3))))))))))).
% 3.09/1.79 thf(fact_117_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((((ord_less @ A) @ X) @ Y4) | (X = Y4))))))))).
% 3.09/1.79 thf(fact_118_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) | (((ord_less @ A) @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_119_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y4:A,Z2:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less_eq @ A) @ X2) @ Z2)))) => (((ord_less_eq @ A) @ Y4) @ Z2)))))))).
% 3.09/1.79 thf(fact_120_dense__ge, axiom, (! [A:$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)))))))).
% 3.09/1.79 thf(fact_121_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A,Z2:A]: (((((ord_less @ A) @ X) @ Y4) => ((((ord_less_eq @ A) @ Y4) @ Z2) => (((ord_less @ A) @ X) @ Z2))))))))).
% 3.09/1.79 thf(fact_122_le__less__trans, axiom, (! [A:$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))))))))).
% 3.09/1.79 thf(fact_123_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less_eq @ A) @ X) @ Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) = (X = Y4))))))))).
% 3.09/1.79 thf(fact_124_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => ((((ord_less_eq @ A) @ X) @ Y4) = (X = Y4))))))))).
% 3.09/1.79 thf(fact_125_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((((ord_less @ A) @ X) @ Y4) => (((ord_less_eq @ A) @ X) @ Y4)))))))).
% 3.09/1.79 thf(fact_126_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 3.09/1.79 thf(fact_127_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) = (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_128_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y4))) = (((ord_less @ A) @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_129_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)))))))))).
% 3.09/1.79 thf(fact_130_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))))))))))).
% 3.09/1.79 thf(fact_131_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)))))))))).
% 3.09/1.79 thf(fact_132_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))))))))))).
% 3.09/1.79 thf(fact_133_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (X3 != Y3))))))))).
% 3.09/1.79 thf(fact_134_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) | (X3 = Y3))))))))).
% 3.09/1.79 thf(fact_135_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less_eq @ A) @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_136_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y4:A,X:A]: (((((ord_less_eq @ A) @ Y4) @ X) => (~ ((((ord_less @ A) @ X) @ Y4)))))))))).
% 3.09/1.79 thf(fact_137_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))))))))))).
% 3.09/1.79 thf(fact_138_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X5)))))))))))))).
% 3.09/1.79 thf(fact_139_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less_eq @ A) @ X5) @ T)))))))))))).
% 3.09/1.79 thf(fact_140_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less_eq @ A) @ T) @ X5)))))))))))).
% 3.09/1.79 thf(fact_141_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T)))))))))))))).
% 3.09/1.79 thf(fact_142_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less @ A) @ X5) @ C3)) => (P @ X5)))) & (! [D:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ D)) => (P @ X2)))) => (((ord_less_eq @ A) @ D) @ C3)))))))))))))))))).
% 3.09/1.79 thf(fact_143_chain__equality, axiom, (! [A:$tType,A4:(set @ A)]: (((((comple1193779247_chain @ A) @ (^ [Y2:A,Z:A]: ((Y2 = Z)))) @ A4) = (! [X3:A]: (((((member @ A) @ X3) @ A4) => (! [Y3:A]: (((((member @ A) @ Y3) @ A4) => (X3 = Y3))))))))))).
% 3.09/1.79 thf(fact_144_chain__def, axiom, (! [A:$tType]: (((comple1193779247_chain @ A) = (^ [Ord:(A > (A > $o)),S:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ S) => (! [Y3:A]: (((((member @ A) @ Y3) @ S) => (((Ord @ X3) @ Y3) | ((Ord @ Y3) @ X3)))))))))))))).
% 3.09/1.79 thf(fact_145_LeastM__natI, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => (P @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P)))))).
% 3.09/1.79 thf(fact_146_minf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ X5) @ Z3) => (F3 = F3)))))))))))).
% 3.09/1.79 thf(fact_147_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less @ A) @ T) @ X5)))))))))))))).
% 3.09/1.79 thf(fact_148_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less @ A) @ X5) @ T)))))))))))).
% 3.09/1.79 thf(fact_149_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 3.09/1.79 thf(fact_150_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 3.09/1.79 thf(fact_151_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) | (Q @ X5)) = ((P2 @ X5) | (Q2 @ X5)))))))))))))))).
% 3.09/1.79 thf(fact_152_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) & (Q @ X5)) = ((P2 @ X5) & (Q2 @ X5)))))))))))))))).
% 3.09/1.79 thf(fact_153_pinf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ Z3) @ X5) => (F3 = F3)))))))))))).
% 3.09/1.79 thf(fact_154_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less @ A) @ T) @ X5)))))))))))).
% 3.09/1.79 thf(fact_155_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less @ A) @ X5) @ T)))))))))))))).
% 3.09/1.79 thf(fact_156_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 3.09/1.79 thf(fact_157_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 3.09/1.79 thf(fact_158_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) | (Q @ X5)) = ((P2 @ X5) | (Q2 @ X5)))))))))))))))).
% 3.09/1.79 thf(fact_159_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) & (Q @ X5)) = ((P2 @ X5) & (Q2 @ X5)))))))))))))))).
% 3.09/1.79 thf(fact_160_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B3:A]: (((((ord_less @ A) @ A2) @ B3) | (((ord_less @ A) @ B3) @ A2)))))))))).
% 3.09/1.79 thf(fact_161_chain__subset, axiom, (! [A:$tType,Ord2:(A > (A > $o)),A4:(set @ A),B6:(set @ A)]: (((((comple1193779247_chain @ A) @ Ord2) @ A4) => ((((ord_less_eq @ (set @ A)) @ B6) @ A4) => (((comple1193779247_chain @ A) @ Ord2) @ B6)))))).
% 3.09/1.79 thf(fact_162_chainD, axiom, (! [A:$tType,Ord2:(A > (A > $o)),S2:(set @ A),X:A,Y4:A]: (((((comple1193779247_chain @ A) @ Ord2) @ S2) => ((((member @ A) @ X) @ S2) => ((((member @ A) @ Y4) @ S2) => (((Ord2 @ X) @ Y4) | ((Ord2 @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_163_chainE, axiom, (! [A:$tType,Ord2:(A > (A > $o)),S2:(set @ A),X:A,Y4:A]: (((((comple1193779247_chain @ A) @ Ord2) @ S2) => ((((member @ A) @ X) @ S2) => ((((member @ A) @ Y4) @ S2) => ((~ (((Ord2 @ X) @ Y4))) => ((Ord2 @ Y4) @ X)))))))).
% 3.09/1.79 thf(fact_164_chainI, axiom, (! [A:$tType,S2:(set @ A),Ord2:(A > (A > $o))]: (((! [X2:A,Y:A]: (((((member @ A) @ X2) @ S2) => ((((member @ A) @ Y) @ S2) => (((Ord2 @ X2) @ Y) | ((Ord2 @ Y) @ X2)))))) => (((comple1193779247_chain @ A) @ Ord2) @ S2))))).
% 3.09/1.79 thf(fact_165_nat__less__le, axiom, ((ord_less @ nat) = (^ [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) & (M != N)))))).
% 3.09/1.79 thf(fact_166_less__imp__le__nat, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 3.09/1.79 thf(fact_167_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 3.09/1.79 thf(fact_168_nat__neq__iff, axiom, (! [M2:nat,N2:nat]: (((M2 != N2) = ((((ord_less @ nat) @ M2) @ N2) | (((ord_less @ nat) @ N2) @ M2)))))).
% 3.09/1.79 thf(fact_169_less__not__refl, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 3.09/1.79 thf(fact_170_less__not__refl2, axiom, (! [N2:nat,M2:nat]: (((((ord_less @ nat) @ N2) @ M2) => (M2 != N2))))).
% 3.09/1.79 thf(fact_171_less__not__refl3, axiom, (! [S3:nat,T:nat]: (((((ord_less @ nat) @ S3) @ T) => (S3 != T))))).
% 3.09/1.79 thf(fact_172_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y5:A]: (((((ord_less @ nat) @ (F @ Y5)) @ (F @ X2)) => (P @ Y5)))) => (P @ X2)))) => (P @ A2))))).
% 3.09/1.79 thf(fact_173_less__irrefl__nat, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 3.09/1.79 thf(fact_174_nat__less__induct, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P @ M3)))) => (P @ N3)))) => (P @ N2))))).
% 3.09/1.79 thf(fact_175_infinite__descent, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P @ M3))))))))) => (P @ N2))))).
% 3.09/1.79 thf(fact_176_linorder__neqE__nat, axiom, (! [X:nat,Y4:nat]: (((X != Y4) => ((~ ((((ord_less @ nat) @ X) @ Y4))) => (((ord_less @ nat) @ Y4) @ X)))))).
% 3.09/1.79 thf(fact_177_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y5:A]: (((((ord_less @ nat) @ (F @ Y5)) @ (F @ X2)) => (P @ Y5)))) => (P @ X2)))) => (P @ A2))))).
% 3.09/1.79 thf(fact_178_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X2:A]: (((~ ((P @ X2))) => (? [Y5:A]: (((((ord_less @ nat) @ (V @ Y5)) @ (V @ X2)) & (~ ((P @ Y5))))))))) => (P @ X))))).
% 3.09/1.79 thf(fact_179_nat__le__linear, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) | (((ord_less_eq @ nat) @ N2) @ M2))))).
% 3.09/1.79 thf(fact_180_le__antisym, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M2) => (M2 = N2)))))).
% 3.09/1.79 thf(fact_181_eq__imp__le, axiom, (! [M2:nat,N2:nat]: (((M2 = N2) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 3.09/1.79 thf(fact_182_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 3.09/1.79 thf(fact_183_le__refl, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))).
% 3.09/1.79 thf(fact_184_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 3.09/1.79 thf(fact_185_le__neq__implies__less, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((M2 != N2) => (((ord_less @ nat) @ M2) @ N2)))))).
% 3.09/1.79 thf(fact_186_less__or__eq__imp__le, axiom, (! [M2:nat,N2:nat]: ((((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 3.09/1.79 thf(fact_187_psubsetI, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B6) => ((A4 != B6) => (((ord_less @ (set @ A)) @ A4) @ B6)))))).
% 3.09/1.79 thf(fact_188_GreatestM__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat),B2:nat]: (((P @ K) => ((! [Y:A]: (((P @ Y) => (((ord_less @ nat) @ (M2 @ Y)) @ B2)))) => ((P @ ((((hilbert_GreatestM @ A) @ nat) @ M2) @ P)) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ Y5)) @ (M2 @ ((((hilbert_GreatestM @ A) @ nat) @ M2) @ P)))))))))))).
% 3.09/1.79 thf(fact_189_GreatestM__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M2:(A > nat),B2:nat]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less @ nat) @ (M2 @ Y)) @ B2)))) => (((ord_less_eq @ nat) @ (M2 @ X)) @ (M2 @ ((((hilbert_GreatestM @ A) @ nat) @ M2) @ P)))))))).
% 3.09/1.79 thf(fact_190_psubsetD, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A),C:A]: (((((ord_less @ (set @ A)) @ A4) @ B6) => ((((member @ A) @ C) @ A4) => (((member @ A) @ C) @ B6)))))).
% 3.09/1.79 thf(fact_191_psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A4) @ C4)))))).
% 3.09/1.79 thf(fact_192_GreatestM__natI, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat),B2:nat]: (((P @ K) => ((! [Y:A]: (((P @ Y) => (((ord_less @ nat) @ (M2 @ Y)) @ B2)))) => (P @ ((((hilbert_GreatestM @ A) @ nat) @ M2) @ P))))))).
% 3.09/1.79 thf(fact_193_GreatestMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M2:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y:A]: (((P @ Y) => (((ord_less_eq @ B) @ (M2 @ Y)) @ (M2 @ X))))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (M2 @ Y5)) @ (M2 @ X2))))) => (Q @ X2))))) => (Q @ ((((hilbert_GreatestM @ A) @ B) @ M2) @ P))))))))))).
% 3.09/1.79 thf(fact_194_psubsetE, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B6) => (~ (((((ord_less_eq @ (set @ A)) @ A4) @ B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A4)))))))).
% 3.09/1.79 thf(fact_195_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (A6 != B7)))))))).
% 3.09/1.79 thf(fact_196_psubset__imp__subset, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B6) => (((ord_less_eq @ (set @ A)) @ A4) @ B6))))).
% 3.09/1.79 thf(fact_197_psubset__subset__trans, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A4) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A4) @ C4)))))).
% 3.09/1.79 thf(fact_198_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B7) & (~ ((((ord_less_eq @ (set @ A)) @ B7) @ A6)))))))))).
% 3.09/1.79 thf(fact_199_subset__psubset__trans, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B6) => ((((ord_less @ (set @ A)) @ B6) @ C4) => (((ord_less @ (set @ A)) @ A4) @ C4)))))).
% 3.09/1.79 thf(fact_200_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A6) @ B7) | (A6 = B7)))))))).
% 3.09/1.79 thf(fact_201_monoseq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((topological_monoseq @ A) = (^ [X4:(nat > A)]: (((((order_mono @ nat) @ A) @ X4) | (((order_antimono @ nat) @ A) @ X4))))))))).
% 3.09/1.79 thf(fact_202_monoI1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X6:(nat > A)]: (((! [M4:nat,N3:nat]: (((((ord_less_eq @ nat) @ M4) @ N3) => (((ord_less_eq @ A) @ (X6 @ M4)) @ (X6 @ N3))))) => ((topological_monoseq @ A) @ X6)))))))).
% 3.09/1.79 thf(fact_203_incseq__imp__monoseq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X6:(nat > A)]: (((((order_mono @ nat) @ A) @ X6) => ((topological_monoseq @ A) @ X6)))))))).
% 3.09/1.79 thf(fact_204_decseq__imp__monoseq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X6:(nat > A)]: (((((order_antimono @ nat) @ A) @ X6) => ((topological_monoseq @ A) @ X6)))))))).
% 3.09/1.79 thf(fact_205_monoseq__def, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((topological_monoseq @ A) = (^ [X4:(nat > A)]: (((! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X4 @ M)) @ (X4 @ N))))) | (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X4 @ N)) @ (X4 @ M))))))))))))).
% 3.09/1.79 thf(fact_206_monoI2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X6:(nat > A)]: (((! [M4:nat,N3:nat]: (((((ord_less_eq @ nat) @ M4) @ N3) => (((ord_less_eq @ A) @ (X6 @ N3)) @ (X6 @ M4))))) => ((topological_monoseq @ A) @ X6)))))))).
% 3.09/1.79 thf(fact_207_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y4:A]: (((X != Y4) => ((~ ((((ord_less @ A) @ X) @ Y4))) => (((ord_less @ A) @ Y4) @ X))))))))).
% 3.09/1.79 thf(fact_208_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X5:A]: ((? [X12:A]: ((((ord_less @ A) @ X5) @ X12))))))))).
% 3.09/1.79 thf(fact_209_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type2 @ A)) => (! [X5:A]: ((? [Y:A]: ((((ord_less @ A) @ Y) @ X5))))))))).
% 3.09/1.79 thf(fact_210_dependent__wellorder__choice, axiom, (! [B:$tType,A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:((A > B) > (A > (B > $o)))]: (((! [R:B,F4:(A > B),G3:(A > B),X2:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => ((F4 @ Y5) = (G3 @ Y5))))) => ((((P @ F4) @ X2) @ R) = (((P @ G3) @ X2) @ R))))) => ((! [X2:A,F4:(A > B)]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => (((P @ F4) @ Y5) @ (F4 @ Y5))))) => (? [X1:B]: ((((P @ F4) @ X2) @ X1)))))) => (? [F4:(A > B)]: ((! [X5:A]: ((((P @ F4) @ X5) @ (F4 @ X5)))))))))))))).
% 3.09/1.79 thf(fact_211_order_Oordering__axioms, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (((ordering @ A) @ (ord_less_eq @ A)) @ (ord_less @ A)))))).
% 3.09/1.79 thf(fact_212_ordering__def, axiom, (! [A:$tType]: (((ordering @ A) = (^ [Less_eq:(A > (A > $o)),Less:(A > (A > $o))]: (((! [A5:A,B5:A]: ((((Less @ A5) @ B5) = (((Less_eq @ A5) @ B5) & (A5 != B5))))) & ((! [A5:A]: (((Less_eq @ A5) @ A5))) & ((! [A5:A,B5:A]: ((((Less_eq @ A5) @ B5) => (((Less_eq @ B5) @ A5) => (A5 = B5))))) & (! [A5:A,B5:A,C5:A]: ((((Less_eq @ A5) @ B5) => (((Less_eq @ B5) @ C5) => ((Less_eq @ A5) @ C5)))))))))))))).
% 3.09/1.79 thf(fact_213_ordering_Oasym, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less2 @ A2) @ B2) => (~ (((Less2 @ B2) @ A2)))))))).
% 3.09/1.79 thf(fact_214_ordering_Orefl, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => ((Less_eq2 @ A2) @ A2))))).
% 3.09/1.79 thf(fact_215_ordering_Ointro, axiom, (! [A:$tType,Less2:(A > (A > $o)),Less_eq2:(A > (A > $o))]: (((! [A3:A,B3:A]: ((((Less2 @ A3) @ B3) = (((Less_eq2 @ A3) @ B3) & (A3 != B3))))) => ((! [A3:A]: (((Less_eq2 @ A3) @ A3))) => ((! [A3:A,B3:A]: ((((Less_eq2 @ A3) @ B3) => (((Less_eq2 @ B3) @ A3) => (A3 = B3))))) => ((! [A3:A,B3:A,C3:A]: ((((Less_eq2 @ A3) @ B3) => (((Less_eq2 @ B3) @ C3) => ((Less_eq2 @ A3) @ C3))))) => (((ordering @ A) @ Less_eq2) @ Less2)))))))).
% 3.09/1.79 thf(fact_216_ordering_Otrans, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less_eq2 @ A2) @ B2) => (((Less_eq2 @ B2) @ C) => ((Less_eq2 @ A2) @ C))))))).
% 3.09/1.79 thf(fact_217_ordering_Oirrefl, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (~ (((Less2 @ A2) @ A2))))))).
% 3.09/1.79 thf(fact_218_ordering_Oantisym, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less_eq2 @ A2) @ B2) => (((Less_eq2 @ B2) @ A2) => (A2 = B2))))))).
% 3.09/1.79 thf(fact_219_ordering_Ostrict__trans, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less2 @ A2) @ B2) => (((Less2 @ B2) @ C) => ((Less2 @ A2) @ C))))))).
% 3.09/1.79 thf(fact_220_ordering_Ostrict__trans1, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less_eq2 @ A2) @ B2) => (((Less2 @ B2) @ C) => ((Less2 @ A2) @ C))))))).
% 3.09/1.79 thf(fact_221_ordering_Ostrict__trans2, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A,C:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less2 @ A2) @ B2) => (((Less_eq2 @ B2) @ C) => ((Less2 @ A2) @ C))))))).
% 3.09/1.79 thf(fact_222_ordering_Oorder__iff__strict, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less_eq2 @ A2) @ B2) = (((Less2 @ A2) @ B2) | (A2 = B2))))))).
% 3.09/1.79 thf(fact_223_ordering_Ostrict__iff__order, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less2 @ A2) @ B2) = (((Less_eq2 @ A2) @ B2) & (A2 != B2))))))).
% 3.09/1.79 thf(fact_224_ordering_Ostrict__implies__order, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less2 @ A2) @ B2) => ((Less_eq2 @ A2) @ B2)))))).
% 3.09/1.79 thf(fact_225_ordering_Ostrict__implies__not__eq, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => (((Less2 @ A2) @ B2) => (A2 != B2)))))).
% 3.09/1.79 thf(fact_226_ordering_Onot__eq__order__implies__strict, axiom, (! [A:$tType,Less_eq2:(A > (A > $o)),Less2:(A > (A > $o)),A2:A,B2:A]: (((((ordering @ A) @ Less_eq2) @ Less2) => ((A2 != B2) => (((Less_eq2 @ A2) @ B2) => ((Less2 @ A2) @ B2))))))).
% 3.09/1.79 thf(subcl_Orderings_Olinorder___HOL_Otype, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((type @ A) @ (type2 @ A)))))).
% 3.09/1.79 thf(subcl_Orderings_Olinorder___Orderings_Oord, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((ord @ A) @ (type2 @ A)))))).
% 3.09/1.79 thf(subcl_Orderings_Olinorder___Orderings_Oorder, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((order @ A) @ (type2 @ A)))))).
% 3.09/1.79 thf(subcl_Orderings_Olinorder___Orderings_Opreorder, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => ((preorder @ A) @ (type2 @ A)))))).
% 3.09/1.79 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A7:$tType,A8:$tType]: ((((comple187826305attice @ A8) @ (type2 @ A8)) => ((comple187826305attice @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.09/1.79 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.09/1.79 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.09/1.79 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 3.09/1.79 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 3.09/1.79 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 3.09/1.79 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 3.09/1.79 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 3.09/1.79 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 3.09/1.79 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 3.09/1.79 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 3.09/1.79 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4, axiom, (! [A7:$tType]: (((comple187826305attice @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.09/1.79 thf(tcon_Set_Oset___Orderings_Opreorder_5, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.09/1.79 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.09/1.79 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 3.09/1.79 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8, axiom, ((comple187826305attice @ $o) @ (type2 @ $o))).
% 3.09/1.79 thf(tcon_HOL_Obool___Orderings_Opreorder_9, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 3.09/1.79 thf(tcon_HOL_Obool___Orderings_Olinorder_10, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 3.09/1.79 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, ((order @ $o) @ (type2 @ $o))).
% 3.09/1.79 thf(tcon_HOL_Obool___Orderings_Oord_12, axiom, ((ord @ $o) @ (type2 @ $o))).
% 3.09/1.79 thf(tfree_0, hypothesis, ((linorder @ a) @ (type2 @ a))).
% 3.09/1.79 thf(conj_0, conjecture, ((p @ a2) @ b)).
% 3.09/1.79 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2DTF16938.p
% 3.09/1.79 ------------------------
% 3.38/1.83 ---- Cleaned THF ---
% 3.38/1.83 thf(ty_t_Set_Oset,type,
% 3.38/1.83 set: $tType > $tType ).
% 3.38/1.83
% 3.38/1.83 thf(ty_t_Nat_Onat,type,
% 3.38/1.83 nat: $tType ).
% 3.38/1.83
% 3.38/1.83 thf(ty_t_itself,type,
% 3.38/1.83 itself: $tType > $tType ).
% 3.38/1.83
% 3.38/1.83 thf(ty_tf_a,type,
% 3.38/1.83 a: $tType ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_HOL_Otype,type,
% 3.38/1.83 type:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Oord,type,
% 3.38/1.83 ord:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Oorder,type,
% 3.38/1.83 order:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Ono__bot,type,
% 3.38/1.83 no_bot:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Ono__top,type,
% 3.38/1.83 no_top:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Olinorder,type,
% 3.38/1.83 linorder:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Opreorder,type,
% 3.38/1.83 preorder:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Owellorder,type,
% 3.38/1.83 wellorder:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Odense__order,type,
% 3.38/1.83 dense_order:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Rings_Olinordered__idom,type,
% 3.38/1.83 linordered_idom:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Fields_Olinordered__field,type,
% 3.38/1.83 linordered_field:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Orderings_Odense__linorder,type,
% 3.38/1.83 dense_linorder:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 3.38/1.83 comple187826305attice:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 3.38/1.83 condit1656338222tinuum:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 3.38/1.83 condit1037483654norder:
% 3.38/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Complete__Partial__Order_Ochain,type,
% 3.38/1.83 comple1193779247_chain:
% 3.38/1.83 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 3.38/1.83 hilbert_GreatestM:
% 3.38/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 3.38/1.83 hilbert_LeastM:
% 3.38/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Orderings_Oord__class_OLeast,type,
% 3.38/1.83 ord_Least:
% 3.38/1.83 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.38/1.83 ord_less:
% 3.38/1.83 !>[A: $tType] : ( A > A > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.38/1.83 ord_less_eq:
% 3.38/1.83 !>[A: $tType] : ( A > A > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 3.38/1.83 order_antimono:
% 3.38/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 3.38/1.83 order_mono:
% 3.38/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 3.38/1.83 order_strict_mono:
% 3.38/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Orderings_Oordering,type,
% 3.38/1.83 ordering:
% 3.38/1.83 !>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Pure_Otype,type,
% 3.38/1.83 type2:
% 3.38/1.83 !>[A: $tType] : ( itself @ A ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Set_OCollect,type,
% 3.38/1.83 collect:
% 3.38/1.83 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_Topological__Spaces_Omonoseq,type,
% 3.38/1.83 topological_monoseq:
% 3.38/1.83 !>[A: $tType] : ( ( nat > A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_c_member,type,
% 3.38/1.83 member:
% 3.38/1.83 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(sy_v_P,type,
% 3.38/1.83 p: a > a > $o ).
% 3.38/1.83
% 3.38/1.83 thf(sy_v_a,type,
% 3.38/1.83 a2: a ).
% 3.38/1.83
% 3.38/1.83 thf(sy_v_b,type,
% 3.38/1.83 b: a ).
% 3.38/1.83
% 3.38/1.83 thf(fact_0_local_Osym,axiom,
% 3.38/1.83 ( ( p @ b @ a2 )
% 3.38/1.83 => ( p @ a2 @ b ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_1_le,axiom,
% 3.38/1.83 ! [A2: a,B2: a] :
% 3.38/1.83 ( ( ord_less_eq @ a @ A2 @ B2 )
% 3.38/1.83 => ( p @ A2 @ B2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_2_order__refl,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_3_le__funD,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 => ! [F: A > B,G: A > B,X: A] :
% 3.38/1.83 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_4_le__funE,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 => ! [F: A > B,G: A > B,X: A] :
% 3.38/1.83 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_5_le__funI,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 => ! [F: A > B,G: A > B] :
% 3.38/1.83 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 3.38/1.83 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_6_le__fun__def,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 => ( ( ord_less_eq @ ( A > B ) )
% 3.38/1.83 = ( ^ [F2: A > B,G2: A > B] :
% 3.38/1.83 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_7_order__subst1,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 3.38/1.83 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.38/1.83 => ( ! [X2: B,Y: B] :
% 3.38/1.83 ( ( ord_less_eq @ B @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_8_order__subst2,axiom,
% 3.38/1.83 ! [A: $tType,C2: $tType] :
% 3.38/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 3.38/1.83 => ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_9_ord__eq__le__subst,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.38/1.83 ( ( A2
% 3.38/1.83 = ( F @ B2 ) )
% 3.38/1.83 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.38/1.83 => ( ! [X2: B,Y: B] :
% 3.38/1.83 ( ( ord_less_eq @ B @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_10_ord__le__eq__subst,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,B2: A,F: A > B,C: B] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ( F @ B2 )
% 3.38/1.83 = C )
% 3.38/1.83 => ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_11_eq__iff,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ^ [Y2: A,Z: A] : ( Y2 = Z ) )
% 3.38/1.83 = ( ^ [X3: A,Y3: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.38/1.83 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_12_antisym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_13_linear,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 | ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_14_dual__order_Oantisym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( A2 = B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_15_dual__order_Otrans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A,C: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.38/1.83 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_16_linorder__wlog,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ! [A3: A,B3: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.38/1.83 => ( P @ A3 @ B3 ) )
% 3.38/1.83 => ( ! [A3: A,B3: A] :
% 3.38/1.83 ( ( P @ B3 @ A3 )
% 3.38/1.83 => ( P @ A3 @ B3 ) )
% 3.38/1.83 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_17_dual__order_Orefl,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_18_order__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,Z2: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 3.38/1.83 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_19_order__class_Oorder_Oantisym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.38/1.83 => ( A2 = B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_20_ord__le__eq__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( B2 = C )
% 3.38/1.83 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_21_ord__eq__le__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( A2 = B2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.38/1.83 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_22_antisym__conv,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Y4: A,X: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 = ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_23_le__cases3,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,Z2: A] :
% 3.38/1.83 ( ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ Y4 @ Z2 ) )
% 3.38/1.83 => ( ( ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 3.38/1.83 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ Z2 @ Y4 ) )
% 3.38/1.83 => ( ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ Y4 @ X ) )
% 3.38/1.83 => ( ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 3.38/1.83 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_24_order_Otrans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.38/1.83 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_25_le__cases,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ~ ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_26_eq__refl,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( X = Y4 )
% 3.38/1.83 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_27_lfp_Oleq__antisym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_28_gfp_Oleq__antisym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Y4: A,X: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_29_lfp_Oleq__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,Z2: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 3.38/1.83 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_30_gfp_Oleq__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Y4: A,X: A,Z2: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ( ( ord_less_eq @ A @ Z2 @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ A @ Z2 @ X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_31_lfp_Oleq__refl,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_32_strict__mono__less__eq,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 3.38/1.83 = ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_33_antimono__def,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ( ( order_antimono @ A @ B )
% 3.38/1.83 = ( ^ [F2: A > B] :
% 3.38/1.83 ! [X3: A,Y3: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_34_antimonoI,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B] :
% 3.38/1.83 ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) ) )
% 3.38/1.83 => ( order_antimono @ A @ B @ F ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_35_antimonoE,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_antimono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_36_antimonoD,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_antimono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_37_LeastI2__wellorder__ex,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,Q: A > $o] :
% 3.38/1.83 ( ? [X1: A] : ( P @ X1 )
% 3.38/1.83 => ( ! [A3: A] :
% 3.38/1.83 ( ( P @ A3 )
% 3.38/1.83 => ( ! [B4: A] :
% 3.38/1.83 ( ( P @ B4 )
% 3.38/1.83 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 3.38/1.83 => ( Q @ A3 ) ) )
% 3.38/1.83 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_38_LeastI2__ex,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,Q: A > $o] :
% 3.38/1.83 ( ? [X1: A] : ( P @ X1 )
% 3.38/1.83 => ( ! [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 => ( Q @ X2 ) )
% 3.38/1.83 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_39_LeastI__ex,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o] :
% 3.38/1.83 ( ? [X1: A] : ( P @ X1 )
% 3.38/1.83 => ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_40_LeastI2,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,A2: A,Q: A > $o] :
% 3.38/1.83 ( ( P @ A2 )
% 3.38/1.83 => ( ! [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 => ( Q @ X2 ) )
% 3.38/1.83 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_41_strict__mono__eq,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.38/1.83 => ( ( ( F @ X )
% 3.38/1.83 = ( F @ Y4 ) )
% 3.38/1.83 = ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_42_LeastI2__order,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,X: A,Q: A > $o] :
% 3.38/1.83 ( ( P @ X )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less_eq @ A @ X @ Y ) )
% 3.38/1.83 => ( ! [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 => ( ! [Y5: A] :
% 3.38/1.83 ( ( P @ Y5 )
% 3.38/1.83 => ( ord_less_eq @ A @ X2 @ Y5 ) )
% 3.38/1.83 => ( Q @ X2 ) ) )
% 3.38/1.83 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_43_Least__equality,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,X: A] :
% 3.38/1.83 ( ( P @ X )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less_eq @ A @ X @ Y ) )
% 3.38/1.83 => ( ( ord_Least @ A @ P )
% 3.38/1.83 = X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_44_LeastI2__wellorder,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,A2: A,Q: A > $o] :
% 3.38/1.83 ( ( P @ A2 )
% 3.38/1.83 => ( ! [A3: A] :
% 3.38/1.83 ( ( P @ A3 )
% 3.38/1.83 => ( ! [B4: A] :
% 3.38/1.83 ( ( P @ B4 )
% 3.38/1.83 => ( ord_less_eq @ A @ A3 @ B4 ) )
% 3.38/1.83 => ( Q @ A3 ) ) )
% 3.38/1.83 => ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_45_mem__Collect__eq,axiom,
% 3.38/1.83 ! [A: $tType,A2: A,P: A > $o] :
% 3.38/1.83 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.38/1.83 = ( P @ A2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_46_Collect__mem__eq,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A] :
% 3.38/1.83 ( ( collect @ A
% 3.38/1.83 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 3.38/1.83 = A4 ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_47_Collect__cong,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 = ( Q @ X2 ) )
% 3.38/1.83 => ( ( collect @ A @ P )
% 3.38/1.83 = ( collect @ A @ Q ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_48_ext,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ( F @ X2 )
% 3.38/1.83 = ( G @ X2 ) )
% 3.38/1.83 => ( F = G ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_49_decseqD,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [F: nat > A,I: nat,J: nat] :
% 3.38/1.83 ( ( order_antimono @ nat @ A @ F )
% 3.38/1.83 => ( ( ord_less_eq @ nat @ I @ J )
% 3.38/1.83 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_50_decseq__def,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( order_antimono @ nat @ A )
% 3.38/1.83 = ( ^ [X4: nat > A] :
% 3.38/1.83 ! [M: nat,N: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M @ N )
% 3.38/1.83 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_51_strict__mono__less,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 3.38/1.83 = ( ord_less @ A @ X @ Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_52_strict__mono__def,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ( ( order_strict_mono @ A @ B )
% 3.38/1.83 = ( ^ [F2: A > B] :
% 3.38/1.83 ! [X3: A,Y3: A] :
% 3.38/1.83 ( ( ord_less @ A @ X3 @ Y3 )
% 3.38/1.83 => ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_53_strict__monoI,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B] :
% 3.38/1.83 ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_54_strict__monoD,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_55_strict__mono__mono,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B] :
% 3.38/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.38/1.83 => ( order_mono @ A @ B @ F ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_56_linorder__chain,axiom,
% 3.38/1.83 ! [B: $tType] :
% 3.38/1.83 ( ( linorder @ B @ ( type2 @ B ) )
% 3.38/1.83 => ! [Y6: set @ B] : ( comple1193779247_chain @ B @ ( ord_less_eq @ B ) @ Y6 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_57_LeastMI2,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 3.38/1.83 ( ( P @ X )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y ) ) )
% 3.38/1.83 => ( ! [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 => ( ! [Y5: A] :
% 3.38/1.83 ( ( P @ Y5 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
% 3.38/1.83 => ( Q @ X2 ) ) )
% 3.38/1.83 => ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_58_incseqD,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [F: nat > A,I: nat,J: nat] :
% 3.38/1.83 ( ( order_mono @ nat @ A @ F )
% 3.38/1.83 => ( ( ord_less_eq @ nat @ I @ J )
% 3.38/1.83 => ( ord_less_eq @ A @ ( F @ I ) @ ( F @ J ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_59_incseq__def,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( order_mono @ nat @ A )
% 3.38/1.83 = ( ^ [X4: nat > A] :
% 3.38/1.83 ! [M: nat,N: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M @ N )
% 3.38/1.83 => ( ord_less_eq @ A @ ( X4 @ M ) @ ( X4 @ N ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_60_ord__eq__less__subst,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.38/1.83 ( ( A2
% 3.38/1.83 = ( F @ B2 ) )
% 3.38/1.83 => ( ( ord_less @ B @ B2 @ C )
% 3.38/1.83 => ( ! [X2: B,Y: B] :
% 3.38/1.83 ( ( ord_less @ B @ X2 @ Y )
% 3.38/1.83 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_61_ord__less__eq__subst,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,B2: A,F: A > B,C: B] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ( F @ B2 )
% 3.38/1.83 = C )
% 3.38/1.83 => ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_62_order__less__subst1,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 3.38/1.83 => ( ( ord_less @ B @ B2 @ C )
% 3.38/1.83 => ( ! [X2: B,Y: B] :
% 3.38/1.83 ( ( ord_less @ B @ X2 @ Y )
% 3.38/1.83 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_63_order__less__subst2,axiom,
% 3.38/1.83 ! [A: $tType,C2: $tType] :
% 3.38/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 3.38/1.83 => ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_64_lt__ex,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( no_bot @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A] :
% 3.38/1.83 ? [Y: A] : ( ord_less @ A @ Y @ X ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_65_gt__ex,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( no_top @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A] :
% 3.38/1.83 ? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_66_neqE,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( X != Y4 )
% 3.38/1.83 => ( ~ ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_67_neq__iff,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( X != Y4 )
% 3.38/1.83 = ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 | ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_68_order_Oasym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_69_dense,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( dense_order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ? [Z3: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Z3 )
% 3.38/1.83 & ( ord_less @ A @ Z3 @ Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_70_less__imp__neq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( X != Y4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_71_less__asym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_72_less__asym_H,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_73_less__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,Z2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less @ A @ Y4 @ Z2 )
% 3.38/1.83 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_74_less__linear,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 | ( X = Y4 )
% 3.38/1.83 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_75_less__irrefl,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A] :
% 3.38/1.83 ~ ( ord_less @ A @ X @ X ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_76_ord__eq__less__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( A2 = B2 )
% 3.38/1.83 => ( ( ord_less @ A @ B2 @ C )
% 3.38/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_77_ord__less__eq__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( B2 = C )
% 3.38/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_78_dual__order_Oasym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A] :
% 3.38/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.38/1.83 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_79_less__imp__not__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( X != Y4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_80_less__not__sym,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_81_less__induct,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,A2: A] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ! [Y5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Y5 @ X2 )
% 3.38/1.83 => ( P @ Y5 ) )
% 3.38/1.83 => ( P @ X2 ) )
% 3.38/1.83 => ( P @ A2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_82_antisym__conv3,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Y4: A,X: A] :
% 3.38/1.83 ( ~ ( ord_less @ A @ Y4 @ X )
% 3.38/1.83 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 3.38/1.83 = ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_83_less__imp__not__eq2,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( Y4 != X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_84_less__imp__triv,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,P: $o] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less @ A @ Y4 @ X )
% 3.38/1.83 => P ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_85_linorder__cases,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ~ ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ( X != Y4 )
% 3.38/1.83 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_86_dual__order_Oirrefl,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A] :
% 3.38/1.83 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_87_order_Ostrict__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less @ A @ B2 @ C )
% 3.38/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_88_less__imp__not__less,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ~ ( ord_less @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_89_dual__order_Ostrict__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A,C: A] :
% 3.38/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.38/1.83 => ( ( ord_less @ A @ C @ B2 )
% 3.38/1.83 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_90_not__less__iff__gr__or__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 3.38/1.83 = ( ( ord_less @ A @ Y4 @ X )
% 3.38/1.83 | ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_91_order_Ostrict__implies__not__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( A2 != B2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_92_dual__order_Ostrict__implies__not__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A] :
% 3.38/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.38/1.83 => ( A2 != B2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_93_mono__invE,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_mono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) )
% 3.38/1.83 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_94_ex__has__greatest__nat,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 3.38/1.83 ( ( P @ K )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 3.38/1.83 => ? [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 & ! [Y5: A] :
% 3.38/1.83 ( ( P @ Y5 )
% 3.38/1.83 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_95_ex__has__least__nat,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 3.38/1.83 ( ( P @ K )
% 3.38/1.83 => ? [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 & ! [Y5: A] :
% 3.38/1.83 ( ( P @ Y5 )
% 3.38/1.83 => ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_96_LeastM__nat__lemma,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 3.38/1.83 ( ( P @ K )
% 3.38/1.83 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
% 3.38/1.83 & ! [Y5: A] :
% 3.38/1.83 ( ( P @ Y5 )
% 3.38/1.83 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_97_LeastM__nat__le,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,X: A,M2: A > nat] :
% 3.38/1.83 ( ( P @ X )
% 3.38/1.83 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_98_mono__def,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ( ( order_mono @ A @ B )
% 3.38/1.83 = ( ^ [F2: A > B] :
% 3.38/1.83 ! [X3: A,Y3: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_99_monoI,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B] :
% 3.38/1.83 ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( order_mono @ A @ B @ F ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_100_monoE,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_mono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_101_monoD,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.38/1.83 => ! [F: A > B,X: A,Y4: A] :
% 3.38/1.83 ( ( order_mono @ A @ B @ F )
% 3.38/1.83 => ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_102_order_Onot__eq__order__implies__strict,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A] :
% 3.38/1.83 ( ( A2 != B2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_103_dual__order_Ostrict__implies__order,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A] :
% 3.38/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.38/1.83 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_104_dual__order_Ostrict__iff__order,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ord_less @ A )
% 3.38/1.83 = ( ^ [B5: A,A5: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ B5 @ A5 )
% 3.38/1.83 & ( A5 != B5 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_105_dual__order_Oorder__iff__strict,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ord_less_eq @ A )
% 3.38/1.83 = ( ^ [B5: A,A5: A] :
% 3.38/1.83 ( ( ord_less @ A @ B5 @ A5 )
% 3.38/1.83 | ( A5 = B5 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_106_order_Ostrict__implies__order,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_107_dense__le__bounded,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,Z2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ! [W: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ W )
% 3.38/1.83 => ( ( ord_less @ A @ W @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ A @ W @ Z2 ) ) )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_108_dense__ge__bounded,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Z2: A,X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z2 @ X )
% 3.38/1.83 => ( ! [W: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z2 @ W )
% 3.38/1.83 => ( ( ord_less @ A @ W @ X )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ W ) ) )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_109_dual__order_Ostrict__trans2,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A,C: A] :
% 3.38/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.38/1.83 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_110_dual__order_Ostrict__trans1,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [B2: A,A2: A,C: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.38/1.83 => ( ( ord_less @ A @ C @ B2 )
% 3.38/1.83 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_111_order_Ostrict__iff__order,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ord_less @ A )
% 3.38/1.83 = ( ^ [A5: A,B5: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A5 @ B5 )
% 3.38/1.83 & ( A5 != B5 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_112_order_Oorder__iff__strict,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ord_less_eq @ A )
% 3.38/1.83 = ( ^ [A5: A,B5: A] :
% 3.38/1.83 ( ( ord_less @ A @ A5 @ B5 )
% 3.38/1.83 | ( A5 = B5 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_113_order_Ostrict__trans2,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.38/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_114_order_Ostrict__trans1,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less @ A @ B2 @ C )
% 3.38/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_115_not__le__imp__less,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Y4: A,X: A] :
% 3.38/1.83 ( ~ ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ( ord_less @ A @ X @ Y4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_116_less__le__not__le,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ord_less @ A )
% 3.38/1.83 = ( ^ [X3: A,Y3: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.38/1.83 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_117_le__imp__less__or__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 | ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_118_le__less__linear,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 | ( ord_less @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_119_dense__le,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Y4: A,Z2: A] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ A @ X2 @ Z2 ) )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_120_dense__ge,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Z2: A,Y4: A] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z2 @ X2 )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ Z2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_121_less__le__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,Z2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ Y4 @ Z2 )
% 3.38/1.83 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_122_le__less__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A,Z2: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less @ A @ Y4 @ Z2 )
% 3.38/1.83 => ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_123_antisym__conv2,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 3.38/1.83 = ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_124_antisym__conv1,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ~ ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ( ord_less_eq @ A @ X @ Y4 )
% 3.38/1.83 = ( X = Y4 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_125_less__imp__le,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_126_le__neq__trans,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( A2 != B2 )
% 3.38/1.83 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_127_not__less,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ~ ( ord_less @ A @ X @ Y4 ) )
% 3.38/1.83 = ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_128_not__le,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( ~ ( ord_less_eq @ A @ X @ Y4 ) )
% 3.38/1.83 = ( ord_less @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_129_order__less__le__subst2,axiom,
% 3.38/1.83 ! [A: $tType,C2: $tType] :
% 3.38/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 3.38/1.83 => ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_130_order__less__le__subst1,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 3.38/1.83 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.38/1.83 => ( ! [X2: B,Y: B] :
% 3.38/1.83 ( ( ord_less_eq @ B @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_131_order__le__less__subst2,axiom,
% 3.38/1.83 ! [A: $tType,C2: $tType] :
% 3.38/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 3.38/1.83 => ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.38/1.83 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_132_order__le__less__subst1,axiom,
% 3.38/1.83 ! [A: $tType,B: $tType] :
% 3.38/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.38/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.38/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 3.38/1.83 => ( ( ord_less @ B @ B2 @ C )
% 3.38/1.83 => ( ! [X2: B,Y: B] :
% 3.38/1.83 ( ( ord_less @ B @ X2 @ Y )
% 3.38/1.83 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
% 3.38/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_133_less__le,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ord_less @ A )
% 3.38/1.83 = ( ^ [X3: A,Y3: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.38/1.83 & ( X3 != Y3 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_134_le__less,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( ord_less_eq @ A )
% 3.38/1.83 = ( ^ [X3: A,Y3: A] :
% 3.38/1.83 ( ( ord_less @ A @ X3 @ Y3 )
% 3.38/1.83 | ( X3 = Y3 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_135_leI,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ~ ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_136_leD,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [Y4: A,X: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ Y4 @ X )
% 3.38/1.83 => ~ ( ord_less @ A @ X @ Y4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_137_less__fun__def,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 => ( ( ord_less @ ( A > B ) )
% 3.38/1.83 = ( ^ [F2: A > B,G2: A > B] :
% 3.38/1.83 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 3.38/1.83 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_138_minf_I8_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_139_minf_I6_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_140_pinf_I8_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_141_pinf_I6_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_142_complete__interval,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A,B2: A,P: A > $o] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.38/1.83 => ( ( P @ A2 )
% 3.38/1.83 => ( ~ ( P @ B2 )
% 3.38/1.83 => ? [C3: A] :
% 3.38/1.83 ( ( ord_less_eq @ A @ A2 @ C3 )
% 3.38/1.83 & ( ord_less_eq @ A @ C3 @ B2 )
% 3.38/1.83 & ! [X5: A] :
% 3.38/1.83 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 3.38/1.83 & ( ord_less @ A @ X5 @ C3 ) )
% 3.38/1.83 => ( P @ X5 ) )
% 3.38/1.83 & ! [D: A] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 3.38/1.83 & ( ord_less @ A @ X2 @ D ) )
% 3.38/1.83 => ( P @ X2 ) )
% 3.38/1.83 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_143_chain__equality,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A] :
% 3.38/1.83 ( ( comple1193779247_chain @ A
% 3.38/1.83 @ ^ [Y2: A,Z: A] : ( Y2 = Z )
% 3.38/1.83 @ A4 )
% 3.38/1.83 = ( ! [X3: A] :
% 3.38/1.83 ( ( member @ A @ X3 @ A4 )
% 3.38/1.83 => ! [Y3: A] :
% 3.38/1.83 ( ( member @ A @ Y3 @ A4 )
% 3.38/1.83 => ( X3 = Y3 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_144_chain__def,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( comple1193779247_chain @ A )
% 3.38/1.83 = ( ^ [Ord: A > A > $o,S: set @ A] :
% 3.38/1.83 ! [X3: A] :
% 3.38/1.83 ( ( member @ A @ X3 @ S )
% 3.38/1.83 => ! [Y3: A] :
% 3.38/1.83 ( ( member @ A @ Y3 @ S )
% 3.38/1.83 => ( ( Ord @ X3 @ Y3 )
% 3.38/1.83 | ( Ord @ Y3 @ X3 ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_145_LeastM__natI,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 3.38/1.83 ( ( P @ K )
% 3.38/1.83 => ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_146_minf_I11_J,axiom,
% 3.38/1.83 ! [C2: $tType,D2: $tType] :
% 3.38/1.83 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 3.38/1.83 => ! [F3: D2] :
% 3.38/1.83 ? [Z3: C2] :
% 3.38/1.83 ! [X5: C2] :
% 3.38/1.83 ( ( ord_less @ C2 @ X5 @ Z3 )
% 3.38/1.83 => ( F3 = F3 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_147_minf_I7_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_148_minf_I5_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_149_minf_I4_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ( X5 != T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_150_minf_I3_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ( X5 != T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_151_minf_I2_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.38/1.83 ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.38/1.83 => ( ( P @ X2 )
% 3.38/1.83 = ( P2 @ X2 ) ) )
% 3.38/1.83 => ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.38/1.83 => ( ( Q @ X2 )
% 3.38/1.83 = ( Q2 @ X2 ) ) )
% 3.38/1.83 => ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ( ( ( P @ X5 )
% 3.38/1.83 | ( Q @ X5 ) )
% 3.38/1.83 = ( ( P2 @ X5 )
% 3.38/1.83 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_152_minf_I1_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.38/1.83 ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.38/1.83 => ( ( P @ X2 )
% 3.38/1.83 = ( P2 @ X2 ) ) )
% 3.38/1.83 => ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.38/1.83 => ( ( Q @ X2 )
% 3.38/1.83 = ( Q2 @ X2 ) ) )
% 3.38/1.83 => ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.38/1.83 => ( ( ( P @ X5 )
% 3.38/1.83 & ( Q @ X5 ) )
% 3.38/1.83 = ( ( P2 @ X5 )
% 3.38/1.83 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_153_pinf_I11_J,axiom,
% 3.38/1.83 ! [C2: $tType,D2: $tType] :
% 3.38/1.83 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 3.38/1.83 => ! [F3: D2] :
% 3.38/1.83 ? [Z3: C2] :
% 3.38/1.83 ! [X5: C2] :
% 3.38/1.83 ( ( ord_less @ C2 @ Z3 @ X5 )
% 3.38/1.83 => ( F3 = F3 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_154_pinf_I7_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_155_pinf_I5_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_156_pinf_I4_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ( X5 != T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_157_pinf_I3_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [T: A] :
% 3.38/1.83 ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ( X5 != T ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_158_pinf_I2_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.38/1.83 ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.38/1.83 => ( ( P @ X2 )
% 3.38/1.83 = ( P2 @ X2 ) ) )
% 3.38/1.83 => ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.38/1.83 => ( ( Q @ X2 )
% 3.38/1.83 = ( Q2 @ X2 ) ) )
% 3.38/1.83 => ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ( ( ( P @ X5 )
% 3.38/1.83 | ( Q @ X5 ) )
% 3.38/1.83 = ( ( P2 @ X5 )
% 3.38/1.83 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_159_pinf_I1_J,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.38/1.83 ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.38/1.83 => ( ( P @ X2 )
% 3.38/1.83 = ( P2 @ X2 ) ) )
% 3.38/1.83 => ( ? [Z4: A] :
% 3.38/1.83 ! [X2: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.38/1.83 => ( ( Q @ X2 )
% 3.38/1.83 = ( Q2 @ X2 ) ) )
% 3.38/1.83 => ? [Z3: A] :
% 3.38/1.83 ! [X5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.38/1.83 => ( ( ( P @ X5 )
% 3.38/1.83 & ( Q @ X5 ) )
% 3.38/1.83 = ( ( P2 @ X5 )
% 3.38/1.83 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_160_ex__gt__or__lt,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [A2: A] :
% 3.38/1.83 ? [B3: A] :
% 3.38/1.83 ( ( ord_less @ A @ A2 @ B3 )
% 3.38/1.83 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_161_chain__subset,axiom,
% 3.38/1.83 ! [A: $tType,Ord2: A > A > $o,A4: set @ A,B6: set @ A] :
% 3.38/1.83 ( ( comple1193779247_chain @ A @ Ord2 @ A4 )
% 3.38/1.83 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 3.38/1.83 => ( comple1193779247_chain @ A @ Ord2 @ B6 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_162_chainD,axiom,
% 3.38/1.83 ! [A: $tType,Ord2: A > A > $o,S2: set @ A,X: A,Y4: A] :
% 3.38/1.83 ( ( comple1193779247_chain @ A @ Ord2 @ S2 )
% 3.38/1.83 => ( ( member @ A @ X @ S2 )
% 3.38/1.83 => ( ( member @ A @ Y4 @ S2 )
% 3.38/1.83 => ( ( Ord2 @ X @ Y4 )
% 3.38/1.83 | ( Ord2 @ Y4 @ X ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_163_chainE,axiom,
% 3.38/1.83 ! [A: $tType,Ord2: A > A > $o,S2: set @ A,X: A,Y4: A] :
% 3.38/1.83 ( ( comple1193779247_chain @ A @ Ord2 @ S2 )
% 3.38/1.83 => ( ( member @ A @ X @ S2 )
% 3.38/1.83 => ( ( member @ A @ Y4 @ S2 )
% 3.38/1.83 => ( ~ ( Ord2 @ X @ Y4 )
% 3.38/1.83 => ( Ord2 @ Y4 @ X ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_164_chainI,axiom,
% 3.38/1.83 ! [A: $tType,S2: set @ A,Ord2: A > A > $o] :
% 3.38/1.83 ( ! [X2: A,Y: A] :
% 3.38/1.83 ( ( member @ A @ X2 @ S2 )
% 3.38/1.83 => ( ( member @ A @ Y @ S2 )
% 3.38/1.83 => ( ( Ord2 @ X2 @ Y )
% 3.38/1.83 | ( Ord2 @ Y @ X2 ) ) ) )
% 3.38/1.83 => ( comple1193779247_chain @ A @ Ord2 @ S2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_165_nat__less__le,axiom,
% 3.38/1.83 ( ( ord_less @ nat )
% 3.38/1.83 = ( ^ [M: nat,N: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M @ N )
% 3.38/1.83 & ( M != N ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_166_less__imp__le__nat,axiom,
% 3.38/1.83 ! [M2: nat,N2: nat] :
% 3.38/1.83 ( ( ord_less @ nat @ M2 @ N2 )
% 3.38/1.83 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_167_le__eq__less__or__eq,axiom,
% 3.38/1.83 ( ( ord_less_eq @ nat )
% 3.38/1.83 = ( ^ [M: nat,N: nat] :
% 3.38/1.83 ( ( ord_less @ nat @ M @ N )
% 3.38/1.83 | ( M = N ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_168_nat__neq__iff,axiom,
% 3.38/1.83 ! [M2: nat,N2: nat] :
% 3.38/1.83 ( ( M2 != N2 )
% 3.38/1.83 = ( ( ord_less @ nat @ M2 @ N2 )
% 3.38/1.83 | ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_169_less__not__refl,axiom,
% 3.38/1.83 ! [N2: nat] :
% 3.38/1.83 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_170_less__not__refl2,axiom,
% 3.38/1.83 ! [N2: nat,M2: nat] :
% 3.38/1.83 ( ( ord_less @ nat @ N2 @ M2 )
% 3.38/1.83 => ( M2 != N2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_171_less__not__refl3,axiom,
% 3.38/1.83 ! [S3: nat,T: nat] :
% 3.38/1.83 ( ( ord_less @ nat @ S3 @ T )
% 3.38/1.83 => ( S3 != T ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_172_measure__induct,axiom,
% 3.38/1.83 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ! [Y5: A] :
% 3.38/1.83 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X2 ) )
% 3.38/1.83 => ( P @ Y5 ) )
% 3.38/1.83 => ( P @ X2 ) )
% 3.38/1.83 => ( P @ A2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_173_less__irrefl__nat,axiom,
% 3.38/1.83 ! [N2: nat] :
% 3.38/1.83 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_174_nat__less__induct,axiom,
% 3.38/1.83 ! [P: nat > $o,N2: nat] :
% 3.38/1.83 ( ! [N3: nat] :
% 3.38/1.83 ( ! [M3: nat] :
% 3.38/1.83 ( ( ord_less @ nat @ M3 @ N3 )
% 3.38/1.83 => ( P @ M3 ) )
% 3.38/1.83 => ( P @ N3 ) )
% 3.38/1.83 => ( P @ N2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_175_infinite__descent,axiom,
% 3.38/1.83 ! [P: nat > $o,N2: nat] :
% 3.38/1.83 ( ! [N3: nat] :
% 3.38/1.83 ( ~ ( P @ N3 )
% 3.38/1.83 => ? [M3: nat] :
% 3.38/1.83 ( ( ord_less @ nat @ M3 @ N3 )
% 3.38/1.83 & ~ ( P @ M3 ) ) )
% 3.38/1.83 => ( P @ N2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_176_linorder__neqE__nat,axiom,
% 3.38/1.83 ! [X: nat,Y4: nat] :
% 3.38/1.83 ( ( X != Y4 )
% 3.38/1.83 => ( ~ ( ord_less @ nat @ X @ Y4 )
% 3.38/1.83 => ( ord_less @ nat @ Y4 @ X ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_177_measure__induct__rule,axiom,
% 3.38/1.83 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ! [Y5: A] :
% 3.38/1.83 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X2 ) )
% 3.38/1.83 => ( P @ Y5 ) )
% 3.38/1.83 => ( P @ X2 ) )
% 3.38/1.83 => ( P @ A2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_178_infinite__descent__measure,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 3.38/1.83 ( ! [X2: A] :
% 3.38/1.83 ( ~ ( P @ X2 )
% 3.38/1.83 => ? [Y5: A] :
% 3.38/1.83 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X2 ) )
% 3.38/1.83 & ~ ( P @ Y5 ) ) )
% 3.38/1.83 => ( P @ X ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_179_nat__le__linear,axiom,
% 3.38/1.83 ! [M2: nat,N2: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 3.38/1.83 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_180_le__antisym,axiom,
% 3.38/1.83 ! [M2: nat,N2: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 3.38/1.83 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 3.38/1.83 => ( M2 = N2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_181_eq__imp__le,axiom,
% 3.38/1.83 ! [M2: nat,N2: nat] :
% 3.38/1.83 ( ( M2 = N2 )
% 3.38/1.83 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_182_le__trans,axiom,
% 3.38/1.83 ! [I: nat,J: nat,K: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ I @ J )
% 3.38/1.83 => ( ( ord_less_eq @ nat @ J @ K )
% 3.38/1.83 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_183_le__refl,axiom,
% 3.38/1.83 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_184_less__mono__imp__le__mono,axiom,
% 3.38/1.83 ! [F: nat > nat,I: nat,J: nat] :
% 3.38/1.83 ( ! [I2: nat,J2: nat] :
% 3.38/1.83 ( ( ord_less @ nat @ I2 @ J2 )
% 3.38/1.83 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 3.38/1.83 => ( ( ord_less_eq @ nat @ I @ J )
% 3.38/1.83 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_185_le__neq__implies__less,axiom,
% 3.38/1.83 ! [M2: nat,N2: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 3.38/1.83 => ( ( M2 != N2 )
% 3.38/1.83 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_186_less__or__eq__imp__le,axiom,
% 3.38/1.83 ! [M2: nat,N2: nat] :
% 3.38/1.83 ( ( ( ord_less @ nat @ M2 @ N2 )
% 3.38/1.83 | ( M2 = N2 ) )
% 3.38/1.83 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_187_psubsetI,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 3.38/1.83 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ( ( A4 != B6 )
% 3.38/1.83 => ( ord_less @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_188_GreatestM__nat__lemma,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 3.38/1.83 ( ( P @ K )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 3.38/1.83 => ( ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) )
% 3.38/1.83 & ! [Y5: A] :
% 3.38/1.83 ( ( P @ Y5 )
% 3.38/1.83 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_189_GreatestM__nat__le,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,X: A,M2: A > nat,B2: nat] :
% 3.38/1.83 ( ( P @ X )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 3.38/1.83 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_190_psubsetD,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A,B6: set @ A,C: A] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ( ( member @ A @ C @ A4 )
% 3.38/1.83 => ( member @ A @ C @ B6 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_191_psubset__trans,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A,B6: set @ A,C4: set @ A] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 3.38/1.83 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_192_GreatestM__natI,axiom,
% 3.38/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 3.38/1.83 ( ( P @ K )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less @ nat @ ( M2 @ Y ) @ B2 ) )
% 3.38/1.83 => ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_193_GreatestMI2,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.38/1.83 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 3.38/1.83 ( ( P @ X )
% 3.38/1.83 => ( ! [Y: A] :
% 3.38/1.83 ( ( P @ Y )
% 3.38/1.83 => ( ord_less_eq @ B @ ( M2 @ Y ) @ ( M2 @ X ) ) )
% 3.38/1.83 => ( ! [X2: A] :
% 3.38/1.83 ( ( P @ X2 )
% 3.38/1.83 => ( ! [Y5: A] :
% 3.38/1.83 ( ( P @ Y5 )
% 3.38/1.83 => ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
% 3.38/1.83 => ( Q @ X2 ) ) )
% 3.38/1.83 => ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_194_psubsetE,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_195_psubset__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) )
% 3.38/1.83 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.38/1.83 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 3.38/1.83 & ( A6 != B7 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_196_psubset__imp__subset,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ( ord_less_eq @ ( set @ A ) @ A4 @ B6 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_197_psubset__subset__trans,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A,B6: set @ A,C4: set @ A] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 3.38/1.83 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_198_subset__not__subset__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) )
% 3.38/1.83 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.38/1.83 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
% 3.38/1.83 & ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_199_subset__psubset__trans,axiom,
% 3.38/1.83 ! [A: $tType,A4: set @ A,B6: set @ A,C4: set @ A] :
% 3.38/1.83 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 3.38/1.83 => ( ( ord_less @ ( set @ A ) @ B6 @ C4 )
% 3.38/1.83 => ( ord_less @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_200_subset__iff__psubset__eq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ord_less_eq @ ( set @ A ) )
% 3.38/1.83 = ( ^ [A6: set @ A,B7: set @ A] :
% 3.38/1.83 ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
% 3.38/1.83 | ( A6 = B7 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_201_monoseq__iff,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( topological_monoseq @ A )
% 3.38/1.83 = ( ^ [X4: nat > A] :
% 3.38/1.83 ( ( order_mono @ nat @ A @ X4 )
% 3.38/1.83 | ( order_antimono @ nat @ A @ X4 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_202_monoI1,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X6: nat > A] :
% 3.38/1.83 ( ! [M4: nat,N3: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 3.38/1.83 => ( ord_less_eq @ A @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 3.38/1.83 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_203_incseq__imp__monoseq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X6: nat > A] :
% 3.38/1.83 ( ( order_mono @ nat @ A @ X6 )
% 3.38/1.83 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_204_decseq__imp__monoseq,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X6: nat > A] :
% 3.38/1.83 ( ( order_antimono @ nat @ A @ X6 )
% 3.38/1.83 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_205_monoseq__def,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ( topological_monoseq @ A )
% 3.38/1.83 = ( ^ [X4: nat > A] :
% 3.38/1.83 ( ! [M: nat,N: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M @ N )
% 3.38/1.83 => ( ord_less_eq @ A @ ( X4 @ M ) @ ( X4 @ N ) ) )
% 3.38/1.83 | ! [M: nat,N: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M @ N )
% 3.38/1.83 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M ) ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_206_monoI2,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X6: nat > A] :
% 3.38/1.83 ( ! [M4: nat,N3: nat] :
% 3.38/1.83 ( ( ord_less_eq @ nat @ M4 @ N3 )
% 3.38/1.83 => ( ord_less_eq @ A @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 3.38/1.83 => ( topological_monoseq @ A @ X6 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_207_linorder__neqE__linordered__idom,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X: A,Y4: A] :
% 3.38/1.83 ( ( X != Y4 )
% 3.38/1.83 => ( ~ ( ord_less @ A @ X @ Y4 )
% 3.38/1.83 => ( ord_less @ A @ Y4 @ X ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_208_linordered__field__no__ub,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linordered_field @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X5: A] :
% 3.38/1.83 ? [X12: A] : ( ord_less @ A @ X5 @ X12 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_209_linordered__field__no__lb,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linordered_field @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [X5: A] :
% 3.38/1.83 ? [Y: A] : ( ord_less @ A @ Y @ X5 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_210_dependent__wellorder__choice,axiom,
% 3.38/1.83 ! [B: $tType,A: $tType] :
% 3.38/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ! [P: ( A > B ) > A > B > $o] :
% 3.38/1.83 ( ! [R: B,F4: A > B,G3: A > B,X2: A] :
% 3.38/1.83 ( ! [Y5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Y5 @ X2 )
% 3.38/1.83 => ( ( F4 @ Y5 )
% 3.38/1.83 = ( G3 @ Y5 ) ) )
% 3.38/1.83 => ( ( P @ F4 @ X2 @ R )
% 3.38/1.83 = ( P @ G3 @ X2 @ R ) ) )
% 3.38/1.83 => ( ! [X2: A,F4: A > B] :
% 3.38/1.83 ( ! [Y5: A] :
% 3.38/1.83 ( ( ord_less @ A @ Y5 @ X2 )
% 3.38/1.83 => ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
% 3.38/1.83 => ? [X1: B] : ( P @ F4 @ X2 @ X1 ) )
% 3.38/1.83 => ? [F4: A > B] :
% 3.38/1.83 ! [X5: A] : ( P @ F4 @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_211_order_Oordering__axioms,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_212_ordering__def,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( ordering @ A )
% 3.38/1.83 = ( ^ [Less_eq: A > A > $o,Less: A > A > $o] :
% 3.38/1.83 ( ! [A5: A,B5: A] :
% 3.38/1.83 ( ( Less @ A5 @ B5 )
% 3.38/1.83 = ( ( Less_eq @ A5 @ B5 )
% 3.38/1.83 & ( A5 != B5 ) ) )
% 3.38/1.83 & ! [A5: A] : ( Less_eq @ A5 @ A5 )
% 3.38/1.83 & ! [A5: A,B5: A] :
% 3.38/1.83 ( ( Less_eq @ A5 @ B5 )
% 3.38/1.83 => ( ( Less_eq @ B5 @ A5 )
% 3.38/1.83 => ( A5 = B5 ) ) )
% 3.38/1.83 & ! [A5: A,B5: A,C5: A] :
% 3.38/1.83 ( ( Less_eq @ A5 @ B5 )
% 3.38/1.83 => ( ( Less_eq @ B5 @ C5 )
% 3.38/1.83 => ( Less_eq @ A5 @ C5 ) ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_213_ordering_Oasym,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less2 @ A2 @ B2 )
% 3.38/1.83 => ~ ( Less2 @ B2 @ A2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_214_ordering_Orefl,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( Less_eq2 @ A2 @ A2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_215_ordering_Ointro,axiom,
% 3.38/1.83 ! [A: $tType,Less2: A > A > $o,Less_eq2: A > A > $o] :
% 3.38/1.83 ( ! [A3: A,B3: A] :
% 3.38/1.83 ( ( Less2 @ A3 @ B3 )
% 3.38/1.83 = ( ( Less_eq2 @ A3 @ B3 )
% 3.38/1.83 & ( A3 != B3 ) ) )
% 3.38/1.83 => ( ! [A3: A] : ( Less_eq2 @ A3 @ A3 )
% 3.38/1.83 => ( ! [A3: A,B3: A] :
% 3.38/1.83 ( ( Less_eq2 @ A3 @ B3 )
% 3.38/1.83 => ( ( Less_eq2 @ B3 @ A3 )
% 3.38/1.83 => ( A3 = B3 ) ) )
% 3.38/1.83 => ( ! [A3: A,B3: A,C3: A] :
% 3.38/1.83 ( ( Less_eq2 @ A3 @ B3 )
% 3.38/1.83 => ( ( Less_eq2 @ B3 @ C3 )
% 3.38/1.83 => ( Less_eq2 @ A3 @ C3 ) ) )
% 3.38/1.83 => ( ordering @ A @ Less_eq2 @ Less2 ) ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_216_ordering_Otrans,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less_eq2 @ A2 @ B2 )
% 3.38/1.83 => ( ( Less_eq2 @ B2 @ C )
% 3.38/1.83 => ( Less_eq2 @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_217_ordering_Oirrefl,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ~ ( Less2 @ A2 @ A2 ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_218_ordering_Oantisym,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less_eq2 @ A2 @ B2 )
% 3.38/1.83 => ( ( Less_eq2 @ B2 @ A2 )
% 3.38/1.83 => ( A2 = B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_219_ordering_Ostrict__trans,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less2 @ A2 @ B2 )
% 3.38/1.83 => ( ( Less2 @ B2 @ C )
% 3.38/1.83 => ( Less2 @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_220_ordering_Ostrict__trans1,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less_eq2 @ A2 @ B2 )
% 3.38/1.83 => ( ( Less2 @ B2 @ C )
% 3.38/1.83 => ( Less2 @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_221_ordering_Ostrict__trans2,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A,C: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less2 @ A2 @ B2 )
% 3.38/1.83 => ( ( Less_eq2 @ B2 @ C )
% 3.38/1.83 => ( Less2 @ A2 @ C ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_222_ordering_Oorder__iff__strict,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less_eq2 @ A2 @ B2 )
% 3.38/1.83 = ( ( Less2 @ A2 @ B2 )
% 3.38/1.83 | ( A2 = B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_223_ordering_Ostrict__iff__order,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less2 @ A2 @ B2 )
% 3.38/1.83 = ( ( Less_eq2 @ A2 @ B2 )
% 3.38/1.83 & ( A2 != B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_224_ordering_Ostrict__implies__order,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less2 @ A2 @ B2 )
% 3.38/1.83 => ( Less_eq2 @ A2 @ B2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_225_ordering_Ostrict__implies__not__eq,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( Less2 @ A2 @ B2 )
% 3.38/1.83 => ( A2 != B2 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(fact_226_ordering_Onot__eq__order__implies__strict,axiom,
% 3.38/1.83 ! [A: $tType,Less_eq2: A > A > $o,Less2: A > A > $o,A2: A,B2: A] :
% 3.38/1.83 ( ( ordering @ A @ Less_eq2 @ Less2 )
% 3.38/1.83 => ( ( A2 != B2 )
% 3.38/1.83 => ( ( Less_eq2 @ A2 @ B2 )
% 3.38/1.83 => ( Less2 @ A2 @ B2 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( type @ A @ ( type2 @ A ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( ord @ A @ ( type2 @ A ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( order @ A @ ( type2 @ A ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
% 3.38/1.83 ! [A: $tType] :
% 3.38/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.38/1.83 => ( preorder @ A @ ( type2 @ A ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 3.38/1.83 ! [A7: $tType,A8: $tType] :
% 3.38/1.83 ( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
% 3.38/1.83 => ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.38/1.83 ! [A7: $tType,A8: $tType] :
% 3.38/1.83 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 3.38/1.83 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.38/1.83 ! [A7: $tType,A8: $tType] :
% 3.38/1.83 ( ( order @ A8 @ ( type2 @ A8 ) )
% 3.38/1.83 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_fun___Orderings_Oord,axiom,
% 3.38/1.83 ! [A7: $tType,A8: $tType] :
% 3.38/1.83 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 3.38/1.83 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 3.38/1.83 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.38/1.83 wellorder @ nat @ ( type2 @ nat ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 3.38/1.83 preorder @ nat @ ( type2 @ nat ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.38/1.83 linorder @ nat @ ( type2 @ nat ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 3.38/1.83 no_top @ nat @ ( type2 @ nat ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 3.38/1.83 order @ nat @ ( type2 @ nat ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 3.38/1.83 ord @ nat @ ( type2 @ nat ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4,axiom,
% 3.38/1.83 ! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
% 3.38/1.83 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 3.38/1.83 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 3.38/1.83 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_8,axiom,
% 3.38/1.83 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
% 3.38/1.83 preorder @ $o @ ( type2 @ $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_HOL_Obool___Orderings_Olinorder_10,axiom,
% 3.38/1.83 linorder @ $o @ ( type2 @ $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 3.38/1.83 order @ $o @ ( type2 @ $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 3.38/1.83 ord @ $o @ ( type2 @ $o ) ).
% 3.38/1.83
% 3.38/1.83 thf(tfree_0,hypothesis,
% 3.38/1.83 linorder @ a @ ( type2 @ a ) ).
% 3.38/1.83
% 3.38/1.83 thf(conj_0,conjecture,
% 3.38/1.83 p @ a2 @ b ).
% 3.38/1.83 ------------------------
% 3.38/1.86 % (17076)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_16938 for (2999ds/27Mi)
% 3.38/1.86 % (17079)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_16938 for (2999ds/275Mi)
% 3.38/1.86 % (17080)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/18Mi)
% 3.38/1.86 % (17078)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_16938 for (2999ds/2Mi)
% 3.38/1.86 % (17077)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/2Mi)
% 3.38/1.86 % (17074)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/183Mi)
% 3.38/1.86 % (17075)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_16938 for (2999ds/4Mi)
% 3.38/1.87 % (17077)Instruction limit reached!
% 3.38/1.87 % (17077)------------------------------
% 3.38/1.87 % (17077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.87 % (17077)Termination reason: Unknown
% 3.38/1.87 % (17077)Termination phase: shuffling
% 3.38/1.87
% 3.38/1.87 % (17077)Memory used [KB]: 1279
% 3.38/1.87 % (17077)Time elapsed: 0.003 s
% 3.38/1.87 % (17077)Instructions burned: 2 (million)
% 3.38/1.87 % (17077)------------------------------
% 3.38/1.87 % (17077)------------------------------
% 3.38/1.87 % (17078)Instruction limit reached!
% 3.38/1.87 % (17078)------------------------------
% 3.38/1.87 % (17078)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.87 % (17078)Termination reason: Unknown
% 3.38/1.87 % (17078)Termination phase: shuffling
% 3.38/1.87
% 3.38/1.87 % (17078)Memory used [KB]: 1279
% 3.38/1.87 % (17078)Time elapsed: 0.003 s
% 3.38/1.87 % (17078)Instructions burned: 2 (million)
% 3.38/1.87 % (17078)------------------------------
% 3.38/1.87 % (17078)------------------------------
% 3.38/1.87 % (17075)Instruction limit reached!
% 3.38/1.87 % (17075)------------------------------
% 3.38/1.87 % (17075)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.87 % (17075)Termination reason: Unknown
% 3.38/1.87 % (17075)Termination phase: shuffling
% 3.38/1.87
% 3.38/1.87 % (17075)Memory used [KB]: 1407
% 3.38/1.87 % (17075)Time elapsed: 0.004 s
% 3.38/1.87 % (17075)Instructions burned: 4 (million)
% 3.38/1.87 % (17075)------------------------------
% 3.38/1.87 % (17075)------------------------------
% 3.38/1.87 % (17080)Instruction limit reached!
% 3.38/1.87 % (17080)------------------------------
% 3.38/1.87 % (17080)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.87 % (17080)Termination reason: Unknown
% 3.38/1.87 % (17080)Termination phase: shuffling
% 3.38/1.87
% 3.38/1.87 % (17080)Memory used [KB]: 1791
% 3.38/1.87 % (17080)Time elapsed: 0.011 s
% 3.38/1.87 % (17080)Instructions burned: 19 (million)
% 3.38/1.87 % (17080)------------------------------
% 3.38/1.87 % (17080)------------------------------
% 3.38/1.88 % (17076)Instruction limit reached!
% 3.38/1.88 % (17076)------------------------------
% 3.38/1.88 % (17076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.88 % (17076)Termination reason: Unknown
% 3.38/1.88 % (17076)Termination phase: Property scanning
% 3.38/1.88
% 3.38/1.88 % (17076)Memory used [KB]: 1791
% 3.38/1.88 % (17076)Time elapsed: 0.016 s
% 3.38/1.88 % (17076)Instructions burned: 28 (million)
% 3.38/1.88 % (17076)------------------------------
% 3.38/1.88 % (17076)------------------------------
% 3.38/1.88 % (17082)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_16938 for (2999ds/37Mi)
% 3.38/1.88 % (17081)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/3Mi)
% 3.38/1.88 % (17083)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_16938 for (2999ds/15Mi)
% 3.38/1.88 % (17081)Instruction limit reached!
% 3.38/1.88 % (17081)------------------------------
% 3.38/1.88 % (17081)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.88 % (17081)Termination reason: Unknown
% 3.38/1.88 % (17081)Termination phase: shuffling
% 3.38/1.88
% 3.38/1.88 % (17081)Memory used [KB]: 1279
% 3.38/1.88 % (17081)Time elapsed: 0.003 s
% 3.38/1.88 % (17081)Instructions burned: 3 (million)
% 3.38/1.88 % (17081)------------------------------
% 3.38/1.88 % (17081)------------------------------
% 3.38/1.89 % (17083)Instruction limit reached!
% 3.38/1.89 % (17083)------------------------------
% 3.38/1.89 % (17083)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.89 % (17083)Termination reason: Unknown
% 3.38/1.89 % (17083)Termination phase: shuffling
% 3.38/1.89
% 3.38/1.89 % (17083)Memory used [KB]: 1663
% 3.38/1.89 % (17083)Time elapsed: 0.009 s
% 3.38/1.89 % (17083)Instructions burned: 15 (million)
% 3.38/1.89 % (17083)------------------------------
% 3.38/1.89 % (17083)------------------------------
% 3.38/1.89 % (17084)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_16938 for (2999ds/3Mi)
% 3.38/1.89 % (17084)Instruction limit reached!
% 3.38/1.89 % (17084)------------------------------
% 3.38/1.89 % (17084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.89 % (17084)Termination reason: Unknown
% 3.38/1.89 % (17084)Termination phase: shuffling
% 3.38/1.89
% 3.38/1.89 % (17084)Memory used [KB]: 1279
% 3.38/1.89 % (17084)Time elapsed: 0.003 s
% 3.38/1.89 % (17084)Instructions burned: 3 (million)
% 3.38/1.89 % (17084)------------------------------
% 3.38/1.89 % (17084)------------------------------
% 3.38/1.89 % (17085)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_16938 for (2999ds/1041Mi)
% 3.41/1.90 % (17082)Instruction limit reached!
% 3.41/1.90 % (17082)------------------------------
% 3.41/1.90 % (17082)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.90 % (17086)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/7Mi)
% 3.41/1.90 % (17082)Termination reason: Unknown
% 3.41/1.90 % (17082)Termination phase: Property scanning
% 3.41/1.90
% 3.41/1.90 % (17082)Memory used [KB]: 2046
% 3.41/1.90 % (17082)Time elapsed: 0.019 s
% 3.41/1.90 % (17082)Instructions burned: 37 (million)
% 3.41/1.90 % (17082)------------------------------
% 3.41/1.90 % (17082)------------------------------
% 3.41/1.90 % (17079)Refutation not found, incomplete strategy
% 3.41/1.90 % (17079)------------------------------
% 3.41/1.90 % (17079)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.90 % (17079)Termination reason: Refutation not found, incomplete strategy
% 3.41/1.90
% 3.41/1.90
% 3.41/1.90 % (17079)Memory used [KB]: 6780
% 3.41/1.90 % (17079)Time elapsed: 0.039 s
% 3.41/1.90 % (17079)Instructions burned: 74 (million)
% 3.41/1.90 % (17079)------------------------------
% 3.41/1.90 % (17079)------------------------------
% 3.41/1.90 % (17086)Instruction limit reached!
% 3.41/1.90 % (17086)------------------------------
% 3.41/1.90 % (17086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.90 % (17086)Termination reason: Unknown
% 3.41/1.90 % (17086)Termination phase: shuffling
% 3.41/1.90
% 3.41/1.90 % (17086)Memory used [KB]: 1407
% 3.41/1.90 % (17086)Time elapsed: 0.005 s
% 3.41/1.90 % (17086)Instructions burned: 7 (million)
% 3.41/1.90 % (17086)------------------------------
% 3.41/1.90 % (17086)------------------------------
% 3.41/1.91 % (17087)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/16Mi)
% 3.41/1.91 % (17088)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/3Mi)
% 3.41/1.91 % (17088)Instruction limit reached!
% 3.41/1.91 % (17088)------------------------------
% 3.41/1.91 % (17088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.91 % (17088)Termination reason: Unknown
% 3.41/1.91 % (17088)Termination phase: shuffling
% 3.41/1.91
% 3.41/1.91 % (17088)Memory used [KB]: 1279
% 3.41/1.91 % (17088)Time elapsed: 0.003 s
% 3.41/1.91 % (17088)Instructions burned: 3 (million)
% 3.41/1.91 % (17088)------------------------------
% 3.41/1.91 % (17088)------------------------------
% 3.41/1.91 % (17087)Instruction limit reached!
% 3.41/1.91 % (17087)------------------------------
% 3.41/1.91 % (17087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.91 % (17087)Termination reason: Unknown
% 3.41/1.91 % (17087)Termination phase: shuffling
% 3.41/1.91
% 3.41/1.91 % (17087)Memory used [KB]: 1663
% 3.41/1.91 % (17087)Time elapsed: 0.010 s
% 3.41/1.91 % (17087)Instructions burned: 17 (million)
% 3.41/1.91 % (17087)------------------------------
% 3.41/1.91 % (17087)------------------------------
% 3.41/1.92 % (17089)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/3Mi)
% 3.41/1.92 % (17089)Instruction limit reached!
% 3.41/1.92 % (17089)------------------------------
% 3.41/1.92 % (17089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.92 % (17089)Termination reason: Unknown
% 3.41/1.92 % (17089)Termination phase: shuffling
% 3.41/1.92
% 3.41/1.92 % (17089)Memory used [KB]: 1279
% 3.41/1.92 % (17089)Time elapsed: 0.003 s
% 3.41/1.92 % (17089)Instructions burned: 3 (million)
% 3.41/1.92 % (17089)------------------------------
% 3.41/1.92 % (17089)------------------------------
% 3.41/1.92 % (17090)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/7Mi)
% 3.41/1.92 % (17091)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/3Mi)
% 3.41/1.92 % (17091)Instruction limit reached!
% 3.41/1.92 % (17091)------------------------------
% 3.41/1.92 % (17091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.92 % (17091)Termination reason: Unknown
% 3.41/1.92 % (17091)Termination phase: shuffling
% 3.41/1.92
% 3.41/1.92 % (17091)Memory used [KB]: 1407
% 3.41/1.92 % (17091)Time elapsed: 0.003 s
% 3.41/1.92 % (17091)Instructions burned: 3 (million)
% 3.41/1.92 % (17091)------------------------------
% 3.41/1.92 % (17091)------------------------------
% 3.41/1.92 % (17090)Instruction limit reached!
% 3.41/1.92 % (17090)------------------------------
% 3.41/1.92 % (17090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.92 % (17090)Termination reason: Unknown
% 3.41/1.92 % (17090)Termination phase: shuffling
% 3.41/1.92
% 3.41/1.92 % (17090)Memory used [KB]: 1407
% 3.41/1.92 % (17090)Time elapsed: 0.005 s
% 3.41/1.92 % (17090)Instructions burned: 7 (million)
% 3.41/1.92 % (17090)------------------------------
% 3.41/1.92 % (17090)------------------------------
% 3.41/1.93 % (17092)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/4Mi)
% 3.41/1.93 % (17092)Instruction limit reached!
% 3.41/1.93 % (17092)------------------------------
% 3.41/1.93 % (17092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.93 % (17092)Termination reason: Unknown
% 3.41/1.93 % (17092)Termination phase: shuffling
% 3.41/1.93
% 3.41/1.93 % (17092)Memory used [KB]: 1407
% 3.41/1.93 % (17092)Time elapsed: 0.004 s
% 3.41/1.93 % (17092)Instructions burned: 5 (million)
% 3.41/1.93 % (17092)------------------------------
% 3.41/1.93 % (17092)------------------------------
% 3.41/1.93 % (17093)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/18Mi)
% 3.41/1.93 % (17094)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/710Mi)
% 3.41/1.94 % (17095)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/6Mi)
% 3.41/1.94 % (17096)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/902Mi)
% 3.41/1.94 % (17095)Instruction limit reached!
% 3.41/1.94 % (17095)------------------------------
% 3.41/1.94 % (17095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.94 % (17095)Termination reason: Unknown
% 3.41/1.94 % (17095)Termination phase: shuffling
% 3.41/1.94
% 3.41/1.94 % (17095)Memory used [KB]: 1407
% 3.41/1.94 % (17095)Time elapsed: 0.005 s
% 3.41/1.94 % (17095)Instructions burned: 7 (million)
% 3.41/1.94 % (17095)------------------------------
% 3.41/1.94 % (17095)------------------------------
% 3.41/1.94 % (17093)Instruction limit reached!
% 3.41/1.94 % (17093)------------------------------
% 3.41/1.94 % (17093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.94 % (17093)Termination reason: Unknown
% 3.41/1.94 % (17093)Termination phase: shuffling
% 3.41/1.94
% 3.41/1.94 % (17093)Memory used [KB]: 1663
% 3.41/1.94 % (17093)Time elapsed: 0.011 s
% 3.41/1.94 % (17093)Instructions burned: 18 (million)
% 3.41/1.94 % (17093)------------------------------
% 3.41/1.94 % (17093)------------------------------
% 3.41/1.94 % (17097)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/21Mi)
% 3.41/1.95 % (17074)Instruction limit reached!
% 3.41/1.95 % (17074)------------------------------
% 3.41/1.95 % (17074)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.95 % (17074)Termination reason: Unknown
% 3.41/1.95 % (17074)Termination phase: Saturation
% 3.41/1.95
% 3.41/1.95 % (17074)Memory used [KB]: 7291
% 3.41/1.95 % (17074)Time elapsed: 0.088 s
% 3.41/1.95 % (17074)Instructions burned: 184 (million)
% 3.41/1.95 % (17074)------------------------------
% 3.41/1.95 % (17074)------------------------------
% 3.41/1.95 % (17097)Instruction limit reached!
% 3.41/1.95 % (17097)------------------------------
% 3.41/1.95 % (17097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.95 % (17097)Termination reason: Unknown
% 3.41/1.95 % (17097)Termination phase: Property scanning
% 3.41/1.95
% 3.41/1.95 % (17097)Memory used [KB]: 1791
% 3.41/1.95 % (17097)Time elapsed: 0.012 s
% 3.41/1.95 % (17097)Instructions burned: 21 (million)
% 3.41/1.95 % (17097)------------------------------
% 3.41/1.95 % (17097)------------------------------
% 3.41/1.96 % (17098)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/5Mi)
% 3.41/1.96 % (17099)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_16938 for (2999ds/6Mi)
% 3.41/1.96 % (17098)Instruction limit reached!
% 3.41/1.96 % (17098)------------------------------
% 3.41/1.96 % (17098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.96 % (17098)Termination reason: Unknown
% 3.41/1.96 % (17098)Termination phase: shuffling
% 3.41/1.96
% 3.41/1.96 % (17098)Memory used [KB]: 1407
% 3.41/1.96 % (17098)Time elapsed: 0.004 s
% 3.41/1.96 % (17098)Instructions burned: 5 (million)
% 3.41/1.96 % (17098)------------------------------
% 3.41/1.96 % (17098)------------------------------
% 3.41/1.96 % (17099)Instruction limit reached!
% 3.41/1.96 % (17099)------------------------------
% 3.41/1.96 % (17099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.96 % (17099)Termination reason: Unknown
% 3.41/1.96 % (17099)Termination phase: shuffling
% 3.41/1.96
% 3.41/1.96 % (17099)Memory used [KB]: 1407
% 3.41/1.96 % (17099)Time elapsed: 0.005 s
% 3.41/1.96 % (17099)Instructions burned: 7 (million)
% 3.41/1.96 % (17099)------------------------------
% 3.41/1.96 % (17099)------------------------------
% 3.41/1.97 % (17100)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_16938 for (2998ds/377Mi)
% 3.41/1.97 % (17101)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_16938 for (2998ds/779Mi)
% 3.41/1.97 % (17102)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_16938 for (2998ds/19Mi)
% 3.41/1.98 % (17103)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_16938 for (2998ds/879Mi)
% 3.41/1.98 % (17102)Instruction limit reached!
% 3.41/1.98 % (17102)------------------------------
% 3.41/1.98 % (17102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/1.98 % (17102)Termination reason: Unknown
% 3.41/1.98 % (17102)Termination phase: shuffling
% 3.41/1.98
% 3.41/1.98 % (17102)Memory used [KB]: 1663
% 3.41/1.98 % (17102)Time elapsed: 0.011 s
% 3.41/1.98 % (17102)Instructions burned: 20 (million)
% 3.41/1.98 % (17102)------------------------------
% 3.41/1.98 % (17102)------------------------------
% 3.41/2.00 % (17104)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_16938 for (2998ds/17Mi)
% 3.41/2.01 % (17104)Instruction limit reached!
% 3.41/2.01 % (17104)------------------------------
% 3.41/2.01 % (17104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.41/2.01 % (17104)Termination reason: Unknown
% 3.41/2.01 % (17104)Termination phase: shuffling
% 3.41/2.01
% 3.41/2.01 % (17104)Memory used [KB]: 1663
% 3.41/2.01 % (17104)Time elapsed: 0.011 s
% 3.41/2.01 % (17104)Instructions burned: 18 (million)
% 3.41/2.01 % (17104)------------------------------
% 3.41/2.01 % (17104)------------------------------
% 4.58/2.03 % (17105)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_16938 for (2998ds/3Mi)
% 4.58/2.03 % (17105)Instruction limit reached!
% 4.58/2.03 % (17105)------------------------------
% 4.58/2.03 % (17105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.58/2.03 % (17105)Termination reason: Unknown
% 4.58/2.03 % (17105)Termination phase: shuffling
% 4.58/2.03
% 4.58/2.03 % (17105)Memory used [KB]: 1279
% 4.58/2.03 % (17105)Time elapsed: 0.004 s
% 4.58/2.03 % (17105)Instructions burned: 3 (million)
% 4.58/2.03 % (17105)------------------------------
% 4.58/2.03 % (17105)------------------------------
% 4.58/2.04 % (17106)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_16938 for (2998ds/30Mi)
% 4.58/2.06 % (17106)Instruction limit reached!
% 4.58/2.06 % (17106)------------------------------
% 4.58/2.06 % (17106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.58/2.06 % (17106)Termination reason: Unknown
% 4.58/2.06 % (17106)Termination phase: Preprocessing 1
% 4.58/2.06
% 4.58/2.06 % (17106)Memory used [KB]: 1791
% 4.58/2.06 % (17106)Time elapsed: 0.017 s
% 4.58/2.06 % (17106)Instructions burned: 31 (million)
% 4.58/2.06 % (17106)------------------------------
% 4.58/2.06 % (17106)------------------------------
% 4.58/2.08 % (17107)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_16938 for (2997ds/127Mi)
% 5.25/2.14 % (17107)Instruction limit reached!
% 5.25/2.14 % (17107)------------------------------
% 5.25/2.14 % (17107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.25/2.14 % (17107)Termination reason: Unknown
% 5.25/2.14 % (17107)Termination phase: Saturation
% 5.25/2.14
% 5.25/2.14 % (17107)Memory used [KB]: 7036
% 5.25/2.14 % (17107)Time elapsed: 0.063 s
% 5.25/2.14 % (17107)Instructions burned: 128 (million)
% 5.25/2.14 % (17107)------------------------------
% 5.25/2.14 % (17107)------------------------------
% 5.25/2.14 % (17094)First to succeed.
% 5.25/2.14 % (17100)Instruction limit reached!
% 5.25/2.14 % (17100)------------------------------
% 5.25/2.14 % (17100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.25/2.14 % (17100)Termination reason: Unknown
% 5.25/2.14 % (17100)Termination phase: Saturation
% 5.25/2.14
% 5.25/2.14 % (17100)Memory used [KB]: 8315
% 5.25/2.14 % (17100)Time elapsed: 0.178 s
% 5.25/2.14 % (17100)Instructions burned: 377 (million)
% 5.25/2.14 % (17100)------------------------------
% 5.25/2.14 % (17100)------------------------------
% 5.25/2.15 % (17094)Refutation found. Thanks to Tanya!
% 5.25/2.15 % SZS status Theorem for DTF2THF_16938
% 5.25/2.15 % SZS output start Proof for DTF2THF_16938
% 5.25/2.15 thf(type_def_5, type, itself: ($tType) > $tType).
% 5.25/2.15 thf(type_def_7, type, set: ($tType) > $tType).
% 5.25/2.15 thf(type_def_8, type, nat: $tType).
% 5.25/2.15 thf(type_def_9, type, a: $tType).
% 5.25/2.15 thf(func_def_0, type, set: ($tType) > $tType).
% 5.25/2.15 thf(func_def_1, type, nat: $tType).
% 5.25/2.15 thf(func_def_2, type, itself: ($tType) > $tType).
% 5.25/2.15 thf(func_def_3, type, a: $tType).
% 5.25/2.15 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_5, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_6, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_7, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_8, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_9, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_10, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_11, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_12, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_13, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_14, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_15, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_16, type, comple187826305attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_17, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_18, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 5.25/2.15 thf(func_def_19, type, comple1193779247_chain: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 5.25/2.15 thf(func_def_20, type, hilbert_GreatestM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_21, type, hilbert_LeastM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_22, type, ord_Least: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_23, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.25/2.15 thf(func_def_24, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 5.25/2.15 thf(func_def_25, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 5.25/2.15 thf(func_def_26, type, order_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 5.25/2.15 thf(func_def_27, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 5.25/2.15 thf(func_def_28, type, ordering: !>[X0: $tType]:((X0 > X0 > $o) > (X0 > X0 > $o) > $o)).
% 5.25/2.15 thf(func_def_29, type, type2: !>[X0: $tType]:(itself @ X0)).
% 5.25/2.15 thf(func_def_30, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 5.25/2.15 thf(func_def_31, type, topological_monoseq: !>[X0: $tType]:((nat > X0) > $o)).
% 5.25/2.15 thf(func_def_32, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 5.25/2.15 thf(func_def_33, type, p: a > a > $o).
% 5.25/2.15 thf(func_def_34, type, a2: a).
% 5.25/2.15 thf(func_def_35, type, b: a).
% 5.25/2.15 thf(func_def_51, type, sP0: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 5.25/2.15 thf(func_def_52, type, sP1: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 5.25/2.15 thf(func_def_53, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.25/2.15 thf(func_def_54, type, sP3: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.25/2.15 thf(func_def_55, type, sP4: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 5.25/2.15 thf(func_def_56, type, sP5: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 5.25/2.15 thf(func_def_57, type, sP6: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > $o)).
% 5.25/2.15 thf(func_def_58, type, sK7: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_59, type, sK8: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_60, type, sK9: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_61, type, sK10: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_62, type, sK11: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_63, type, sK12: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_64, type, sK13: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_65, type, sK14: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_66, type, sK15: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_67, type, sK16: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_68, type, sK17: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_69, type, sK18: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_70, type, sK19: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_71, type, sK20: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 5.25/2.15 thf(func_def_72, type, sK21: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_73, type, sK22: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_74, type, sK23: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_75, type, sK24: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_76, type, sK25: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_77, type, sK26: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_78, type, sK27: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_79, type, sK28: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_80, type, sK29: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_81, type, sK30: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_82, type, sK31: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_83, type, sK32: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_84, type, sK33: !>[X0: $tType]:(X0 > (X0 > $o) > (X0 > $o) > X0)).
% 5.25/2.15 thf(func_def_85, type, sK34: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_86, type, sK35: !>[X0: $tType]:(X0 > X0)).
% 5.25/2.15 thf(func_def_88, type, ph37: !>[X0: $tType]:(X0)).
% 5.25/2.15 thf(f1605,plain,(
% 5.25/2.15 $false),
% 5.25/2.15 inference(subsumption_resolution,[status(thm)],[f1602,f1062])).
% 5.25/2.15 thf(f1062,plain,(
% 5.25/2.15 ((p @ b @ a2) != $true)),
% 5.25/2.15 inference(subsumption_resolution,[status(thm)],[f949,f975])).
% 5.25/2.15 thf(f975,plain,(
% 5.25/2.15 ((p @ a2 @ b) != $true)),
% 5.25/2.15 inference(cnf_transformation,[status(thm)],[f756])).
% 5.25/2.15 thf(f756,plain,(
% 5.25/2.15 ((p @ a2 @ b) != $true)),
% 5.25/2.15 inference(flattening,[status(thm)],[f741])).
% 5.25/2.15 thf(f741,plain,(
% 5.25/2.15 ~((p @ a2 @ b) = $true)),
% 5.25/2.15 inference(fool_elimination,[status(thm)],[f740])).
% 5.25/2.15 thf(f740,plain,(
% 5.25/2.15 ~(p @ a2 @ b)),
% 5.25/2.15 inference(rectify,[status(thm)],[f254])).
% 5.25/2.15 thf(f254,negated_conjecture,(
% 5.25/2.15 ~(p @ a2 @ b)),
% 5.25/2.15 inference(negated_conjecture,[status(cth)],[f253])).
% 5.25/2.15 thf(f253,conjecture,(
% 5.25/2.15 (p @ a2 @ b)),
% 5.25/2.15 file('/export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2THF_16938.p',conj_0)).
% 5.25/2.15 thf(f949,plain,(
% 5.25/2.15 ((p @ b @ a2) != $true) | ((p @ a2 @ b) = $true)),
% 5.25/2.15 inference(cnf_transformation,[status(thm)],[f788])).
% 5.25/2.15 thf(f788,plain,(
% 5.25/2.15 ((p @ b @ a2) != $true) | ((p @ a2 @ b) = $true)),
% 5.25/2.15 inference(ennf_transformation,[status(thm)],[f519])).
% 5.25/2.15 thf(f519,plain,(
% 5.25/2.15 ((p @ b @ a2) = $true) => ((p @ a2 @ b) = $true)),
% 5.25/2.15 inference(fool_elimination,[status(thm)],[f518])).
% 5.25/2.15 thf(f518,plain,(
% 5.25/2.15 (p @ b @ a2) => (p @ a2 @ b)),
% 5.25/2.15 inference(rectify,[status(thm)],[f1])).
% 5.25/2.15 thf(f1,axiom,(
% 5.25/2.15 (p @ b @ a2) => (p @ a2 @ b)),
% 5.25/2.15 file('/export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2THF_16938.p',fact_0_local_Osym)).
% 5.25/2.15 thf(f1602,plain,(
% 5.25/2.15 ((p @ b @ a2) = $true)),
% 5.25/2.15 inference(trivial_inequality_removal,[status(thm)],[f1576])).
% 5.25/2.15 thf(f1576,plain,(
% 5.25/2.15 ((p @ b @ a2) = $true) | ($true != $true)),
% 5.25/2.15 inference(superposition,[status(thm)],[f975,f1565])).
% 5.25/2.15 thf(f1565,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((p @ X1 @ X0) = $true) | ((p @ X0 @ X1) = $true)) )),
% 5.25/2.15 inference(trivial_inequality_removal,[status(thm)],[f1557])).
% 5.25/2.15 thf(f1557,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((p @ X1 @ X0) = $true) | ($true != $true) | ((p @ X0 @ X1) = $true)) )),
% 5.25/2.15 inference(superposition,[status(thm)],[f1198,f1084])).
% 5.25/2.15 thf(f1084,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((ord_less @ a @ X1 @ X0) = $true) | ((p @ X0 @ X1) = $true)) )),
% 5.25/2.15 inference(trivial_inequality_removal,[status(thm)],[f1081])).
% 5.25/2.15 thf(f1081,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((ord_less @ a @ X1 @ X0) = $true) | ((p @ X0 @ X1) = $true) | ($true != $true)) )),
% 5.25/2.15 inference(superposition,[status(thm)],[f911,f1074])).
% 5.25/2.15 thf(f1074,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((ord_less @ a @ X0 @ X1) = $true) | ((ord_less_eq @ a @ X1 @ X0) = $true)) )),
% 5.25/2.15 inference(trivial_inequality_removal,[status(thm)],[f1069])).
% 5.25/2.15 thf(f1069,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((ord_less @ a @ X0 @ X1) = $true) | ((ord_less_eq @ a @ X1 @ X0) = $true) | ($true != $true)) )),
% 5.25/2.15 inference(superposition,[status(thm)],[f913,f935])).
% 5.25/2.15 thf(f935,plain,(
% 5.25/2.15 ((linorder @ a @ type2 @ a) = $true)),
% 5.25/2.15 inference(cnf_transformation,[status(thm)],[f485])).
% 5.25/2.15 thf(f485,plain,(
% 5.25/2.15 ((linorder @ a @ type2 @ a) = $true)),
% 5.25/2.15 inference(fool_elimination,[status(thm)],[f484])).
% 5.25/2.15 thf(f484,plain,(
% 5.25/2.15 (linorder @ a @ type2 @ a)),
% 5.25/2.15 inference(rectify,[status(thm)],[f252])).
% 5.25/2.15 thf(f252,axiom,(
% 5.25/2.15 (linorder @ a @ type2 @ a)),
% 5.25/2.15 file('/export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2THF_16938.p',tfree_0)).
% 5.25/2.15 thf(f913,plain,(
% 5.25/2.15 ( ! [X0 : $tType,X2 : X0,X1 : X0] : (((linorder @ X0 @ type2 @ X0) != $true) | ((ord_less @ X0 @ X2 @ X1) = $true) | ((ord_less_eq @ X0 @ X1 @ X2) = $true)) )),
% 5.25/2.15 inference(cnf_transformation,[status(thm)],[f833])).
% 5.25/2.15 thf(f833,plain,(
% 5.25/2.15 ! [X0 : $tType] : (! [X1 : X0,X2 : X0] : ((((ord_less @ X0 @ X2 @ X1) = $true) | ((ord_less_eq @ X0 @ X1 @ X2) = $true)) & (((ord_less_eq @ X0 @ X1 @ X2) != $true) | ((ord_less @ X0 @ X2 @ X1) != $true))) | ((linorder @ X0 @ type2 @ X0) != $true))),
% 5.25/2.15 inference(nnf_transformation,[status(thm)],[f789])).
% 5.25/2.15 thf(f789,plain,(
% 5.25/2.15 ! [X0 : $tType] : (! [X1 : X0,X2 : X0] : (((ord_less @ X0 @ X2 @ X1) = $true) <=> ((ord_less_eq @ X0 @ X1 @ X2) != $true)) | ((linorder @ X0 @ type2 @ X0) != $true))),
% 5.25/2.15 inference(ennf_transformation,[status(thm)],[f772])).
% 5.25/2.15 thf(f772,plain,(
% 5.25/2.15 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (((ord_less @ X0 @ X2 @ X1) = $true) <=> ((ord_less_eq @ X0 @ X1 @ X2) != $true)))),
% 5.25/2.15 inference(flattening,[status(thm)],[f267])).
% 5.25/2.15 thf(f267,plain,(
% 5.25/2.15 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (~((ord_less_eq @ X0 @ X1 @ X2) = $true) <=> ((ord_less @ X0 @ X2 @ X1) = $true)))),
% 5.25/2.15 inference(fool_elimination,[status(thm)],[f266])).
% 5.25/2.15 thf(f266,plain,(
% 5.25/2.15 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X1 : X0,X2 : X0] : (~(ord_less_eq @ X0 @ X1 @ X2) = (ord_less @ X0 @ X2 @ X1)))),
% 5.25/2.15 inference(rectify,[status(thm)],[f129])).
% 5.25/2.15 thf(f129,axiom,(
% 5.25/2.15 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X4 : X0,X17 : X0] : (~(ord_less_eq @ X0 @ X4 @ X17) = (ord_less @ X0 @ X17 @ X4)))),
% 5.25/2.15 file('/export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2THF_16938.p',fact_128_not__le)).
% 5.25/2.15 thf(f911,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((ord_less_eq @ a @ X0 @ X1) != $true) | ((p @ X0 @ X1) = $true)) )),
% 5.25/2.15 inference(cnf_transformation,[status(thm)],[f802])).
% 5.25/2.15 thf(f802,plain,(
% 5.25/2.15 ! [X0 : a,X1 : a] : (((ord_less_eq @ a @ X0 @ X1) != $true) | ((p @ X0 @ X1) = $true))),
% 5.25/2.15 inference(ennf_transformation,[status(thm)],[f327])).
% 5.25/2.15 thf(f327,plain,(
% 5.25/2.15 ! [X0 : a,X1 : a] : (((ord_less_eq @ a @ X0 @ X1) = $true) => ((p @ X0 @ X1) = $true))),
% 5.25/2.15 inference(fool_elimination,[status(thm)],[f326])).
% 5.25/2.15 thf(f326,plain,(
% 5.25/2.15 ! [X0 : a,X1 : a] : ((ord_less_eq @ a @ X0 @ X1) => (p @ X0 @ X1))),
% 5.25/2.15 inference(rectify,[status(thm)],[f2])).
% 5.25/2.15 thf(f2,axiom,(
% 5.25/2.15 ! [X2 : a,X3 : a] : ((ord_less_eq @ a @ X2 @ X3) => (p @ X2 @ X3))),
% 5.25/2.15 file('/export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2THF_16938.p',fact_1_le)).
% 5.25/2.15 thf(f1198,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((ord_less @ a @ X1 @ X0) != $true) | ((p @ X1 @ X0) = $true)) )),
% 5.25/2.15 inference(subsumption_resolution,[status(thm)],[f1195,f935])).
% 5.25/2.15 thf(f1195,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((ord_less @ a @ X1 @ X0) != $true) | ((linorder @ a @ type2 @ a) != $true) | ((p @ X1 @ X0) = $true)) )),
% 5.25/2.15 inference(trivial_inequality_removal,[status(thm)],[f1182])).
% 5.25/2.15 thf(f1182,plain,(
% 5.25/2.15 ( ! [X0 : a,X1 : a] : (((linorder @ a @ type2 @ a) != $true) | ((ord_less @ a @ X1 @ X0) != $true) | ((p @ X1 @ X0) = $true) | ($true != $true)) )),
% 5.25/2.15 inference(superposition,[status(thm)],[f958,f1084])).
% 5.25/2.15 thf(f958,plain,(
% 5.25/2.15 ( ! [X0 : $tType,X2 : X0,X1 : X0] : (((ord_less @ X0 @ X2 @ X1) != $true) | ((linorder @ X0 @ type2 @ X0) != $true) | ((ord_less @ X0 @ X1 @ X2) != $true)) )),
% 5.25/2.15 inference(cnf_transformation,[status(thm)],[f872])).
% 5.25/2.15 thf(f872,plain,(
% 5.25/2.15 ! [X0 : $tType] : (! [X1 : X0,X2 : X0] : ((((ord_less @ X0 @ X1 @ X2) != $true) | ((X1 != X2) & ((ord_less @ X0 @ X2 @ X1) != $true))) & ((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true) | ((ord_less @ X0 @ X1 @ X2) = $true))) | ((linorder @ X0 @ type2 @ X0) != $true))),
% 5.25/2.15 inference(flattening,[status(thm)],[f871])).
% 5.25/2.15 thf(f871,plain,(
% 5.25/2.15 ! [X0 : $tType] : (! [X1 : X0,X2 : X0] : ((((ord_less @ X0 @ X1 @ X2) != $true) | ((X1 != X2) & ((ord_less @ X0 @ X2 @ X1) != $true))) & (((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true)) | ((ord_less @ X0 @ X1 @ X2) = $true))) | ((linorder @ X0 @ type2 @ X0) != $true))),
% 5.25/2.15 inference(nnf_transformation,[status(thm)],[f792])).
% 5.25/2.15 thf(f792,plain,(
% 5.25/2.15 ! [X0 : $tType] : (! [X1 : X0,X2 : X0] : (((ord_less @ X0 @ X1 @ X2) != $true) <=> ((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true))) | ((linorder @ X0 @ type2 @ X0) != $true))),
% 5.25/2.15 inference(ennf_transformation,[status(thm)],[f758])).
% 5.25/2.15 thf(f758,plain,(
% 5.25/2.15 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (((ord_less @ X0 @ X1 @ X2) != $true) <=> ((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true))))),
% 5.25/2.15 inference(flattening,[status(thm)],[f679])).
% 5.25/2.15 thf(f679,plain,(
% 5.25/2.15 ! [X0 : $tType] : (((linorder @ X0 @ type2 @ X0) = $true) => ! [X1 : X0,X2 : X0] : (~((ord_less @ X0 @ X1 @ X2) = $true) <=> ((X1 = X2) | ((ord_less @ X0 @ X2 @ X1) = $true))))),
% 5.25/2.15 inference(fool_elimination,[status(thm)],[f678])).
% 5.25/2.15 thf(f678,plain,(
% 5.25/2.15 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X1 : X0,X2 : X0] : (~(ord_less @ X0 @ X1 @ X2) = (ord_less @ X0 @ X2 @ X1) | (X1 = X2)))),
% 5.25/2.15 inference(rectify,[status(thm)],[f91])).
% 5.25/2.15 thf(f91,axiom,(
% 5.25/2.15 ! [X0 : $tType] : ((linorder @ X0 @ type2 @ X0) => ! [X4 : X0,X17 : X0] : (~(ord_less @ X0 @ X4 @ X17) = (ord_less @ X0 @ X17 @ X4) | (X4 = X17)))),
% 5.25/2.15 file('/export/starexec/sandbox/tmp/tmp.LBjh2JQvVe/DTF2THF_16938.p',fact_90_not__less__iff__gr__or__eq)).
% 5.25/2.15 % SZS output end Proof for DTF2THF_16938
% 5.25/2.15 % (17094)------------------------------
% 5.25/2.15 % (17094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.25/2.15 % (17094)Termination reason: Refutation
% 5.25/2.15
% 5.25/2.15 % (17094)Memory used [KB]: 7675
% 5.25/2.15 % (17094)Time elapsed: 0.212 s
% 5.25/2.15 % (17094)Instructions burned: 431 (million)
% 5.25/2.15 % (17094)------------------------------
% 5.25/2.15 % (17094)------------------------------
% 5.25/2.15 % (17073)Success in time 0.296 s
%------------------------------------------------------------------------------