%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT123^1 : 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 : n009.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 08:44:51 AM UTC 2026
% Result : Theorem 4.62s 2.07s
% Output : Refutation 4.62s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT123^1 : 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 : n009.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 06:08:25 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
% 0.34/0.50 ---- Original DTF file ---
% 0.34/0.50 thf(spec,logic,$$dhol).
% 0.34/0.50 %------------------------------------------------------------------------------
% 0.34/0.50 % File : DAT123^1 : TPTP v9.2.1. Released v7.0.0.
% 0.34/0.50 % Domain : Data Structures
% 0.34/0.50 % Problem : Coinductive list 2041
% 0.34/0.50 % Version : [Bla16] axioms : Especial.
% 0.34/0.50 % English :
% 0.34/0.50
% 0.34/0.50 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.34/0.50 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.34/0.50 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.34/0.50 % Source : [Bla16]
% 0.34/0.50 % Names : coinductive_list__2041.p [Bla16]
% 0.34/0.50
% 0.34/0.50 % Status : Theorem
% 0.34/0.50 % Rating : 0.33 v8.1.0, 0.25 v7.5.0, 1.00 v7.2.0, 0.75 v7.1.0
% 0.34/0.50 % Syntax : Number of formulae : 323 ( 91 unt; 41 typ; 0 def)
% 0.34/0.50 % Number of atoms : 857 ( 196 equ; 0 cnn)
% 0.34/0.50 % Maximal formula atoms : 13 ( 3 avg)
% 0.34/0.50 % Number of connectives : 3148 ( 86 ~; 21 |; 37 &;2541 @)
% 0.34/0.50 % ( 0 <=>; 463 =>; 0 <=; 0 <~>)
% 0.34/0.50 % Maximal formula depth : 21 ( 8 avg)
% 0.34/0.50 % Number of types : 4 ( 3 usr)
% 0.34/0.50 % Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% 0.34/0.50 % Number of symbols : 39 ( 38 usr; 4 con; 0-4 aty)
% 0.34/0.50 % Number of variables : 932 ( 31 ^; 828 !; 46 ?; 932 :)
% 0.34/0.50 % ( 27 !>; 0 ?*; 0 @-; 0 @+)
% 0.34/0.50 % SPC : TH1_THM_EQU_NAR
% 0.34/0.50
% 0.34/0.50 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.34/0.50 % 2016-07-13 14:54:22.534
% 0.34/0.50 %------------------------------------------------------------------------------
% 0.34/0.50 %----Could-be-implicit typings (6)
% 0.34/0.50 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 0.34/0.50 coindu1593790203_llist: $tType > $tType ).
% 0.34/0.50
% 0.34/0.50 thf(ty_t_Extended__Nat_Oenat,type,
% 0.34/0.50 extended_enat: $tType ).
% 0.34/0.50
% 0.34/0.50 thf(ty_t_Set_Oset,type,
% 0.34/0.50 set: $tType > $tType ).
% 0.34/0.50
% 0.34/0.50 thf(ty_t_Nat_Onat,type,
% 0.34/0.50 nat: $tType ).
% 0.34/0.50
% 0.34/0.50 thf(ty_t_itself,type,
% 0.34/0.50 itself: $tType > $tType ).
% 0.34/0.50
% 0.34/0.50 thf(ty_tf_a,type,
% 0.34/0.50 a: $tType ).
% 0.34/0.50
% 0.34/0.50 %----Explicit typings (35)
% 0.34/0.50 thf(sy_cl_HOL_Otype,type,
% 0.34/0.50 type:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Oord,type,
% 0.34/0.50 ord:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Oorder,type,
% 0.34/0.50 order:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Ono__bot,type,
% 0.34/0.50 no_bot:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Ono__top,type,
% 0.34/0.50 no_top:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Olinorder,type,
% 0.34/0.50 linorder:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Opreorder,type,
% 0.34/0.50 preorder:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Owellorder,type,
% 0.34/0.50 wellorder:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Odense__order,type,
% 0.34/0.50 dense_order:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.34/0.50 linordered_idom:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.34/0.50 dense_linorder:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.34/0.50 cancel_semigroup_add:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.34/0.50 ordere236663937imp_le:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.34/0.50 condit1656338222tinuum:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.34/0.50 condit1037483654norder:
% 0.34/0.50 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
% 0.34/0.50 coindu749330388e_lnth:
% 0.34/0.50 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
% 0.34/0.50 coindu1802687541_ltake:
% 0.34/0.50 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Coinductive__Nat_Oco_Oenat_Ocase__enat,type,
% 0.34/0.50 coindu440805660e_enat:
% 0.34/0.50 !>[A: $tType] : ( A > ( extended_enat > A ) > extended_enat > A ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Coinductive__Nat_Oenat__set,type,
% 0.34/0.50 coinductive_enat_set: set @ extended_enat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Coinductive__Nat_Oenat__setp,type,
% 0.34/0.50 coindu530039314t_setp: extended_enat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
% 0.34/0.50 coindu1491768222unfold:
% 0.34/0.50 !>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Extended__Nat_OeSuc,type,
% 0.34/0.50 extended_eSuc: extended_enat > extended_enat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Extended__Nat_Oenat,type,
% 0.34/0.50 extended_enat2: nat > extended_enat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 0.34/0.50 extended_case_enat:
% 0.34/0.50 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 0.34/0.50 extended_rec_enat:
% 0.34/0.50 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.34/0.50 plus_plus:
% 0.34/0.50 !>[A: $tType] : ( A > A > A ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_OSuc,type,
% 0.34/0.50 suc: nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.34/0.50 ord_less:
% 0.34/0.50 !>[A: $tType] : ( A > A > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.34/0.50 ord_less_eq:
% 0.34/0.50 !>[A: $tType] : ( A > A > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Pure_Otype,type,
% 0.34/0.50 type2:
% 0.34/0.50 !>[A: $tType] : ( itself @ A ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Set_OCollect,type,
% 0.34/0.50 collect:
% 0.34/0.50 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member,type,
% 0.34/0.50 member:
% 0.34/0.50 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_ma____,type,
% 0.34/0.50 ma: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_na____,type,
% 0.34/0.50 na: extended_enat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_thesis____,type,
% 0.34/0.50 thesis: $o ).
% 0.34/0.50
% 0.34/0.50 %----Relevant facts (255)
% 0.34/0.50 thf(fact_0_Suc_Oprems,axiom,
% 0.34/0.50 ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ ma ) ) @ na ).
% 0.34/0.50
% 0.34/0.50 % Suc.prems
% 0.34/0.50 thf(fact_1_eSuc__inject,axiom,
% 0.34/0.50 ! [M: extended_enat,N: extended_enat] :
% 0.34/0.50 ( ( ( extended_eSuc @ M )
% 0.34/0.50 = ( extended_eSuc @ N ) )
% 0.34/0.50 = ( M = N ) ) ).
% 0.34/0.50
% 0.34/0.50 % eSuc_inject
% 0.34/0.50 thf(fact_2_co_Oenat_Oinject,axiom,
% 0.34/0.50 ! [X2: extended_enat,Y2: extended_enat] :
% 0.34/0.50 ( ( ( extended_eSuc @ X2 )
% 0.34/0.50 = ( extended_eSuc @ Y2 ) )
% 0.34/0.50 = ( X2 = Y2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % co.enat.inject
% 0.34/0.50 thf(fact_3_Extended__Nat_OeSuc__mono,axiom,
% 0.34/0.50 ! [N: extended_enat,M: extended_enat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.34/0.50 = ( ord_less @ extended_enat @ N @ M ) ) ).
% 0.34/0.50
% 0.34/0.50 % Extended_Nat.eSuc_mono
% 0.34/0.50 thf(fact_4_eSuc__enat,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 0.34/0.50 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % eSuc_enat
% 0.34/0.50 thf(fact_5_eSuc__enat__iff,axiom,
% 0.34/0.50 ! [X: extended_enat,Y: nat] :
% 0.34/0.50 ( ( ( extended_eSuc @ X )
% 0.34/0.50 = ( extended_enat2 @ Y ) )
% 0.34/0.50 = ( ? [N2: nat] :
% 0.34/0.50 ( ( Y
% 0.34/0.50 = ( suc @ N2 ) )
% 0.34/0.50 & ( X
% 0.34/0.50 = ( extended_enat2 @ N2 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % eSuc_enat_iff
% 0.34/0.50 thf(fact_6_enat__eSuc__iff,axiom,
% 0.34/0.50 ! [Y: nat,X: extended_enat] :
% 0.34/0.50 ( ( ( extended_enat2 @ Y )
% 0.34/0.50 = ( extended_eSuc @ X ) )
% 0.34/0.50 = ( ? [N2: nat] :
% 0.34/0.50 ( ( Y
% 0.34/0.50 = ( suc @ N2 ) )
% 0.34/0.50 & ( ( extended_enat2 @ N2 )
% 0.34/0.50 = X ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_eSuc_iff
% 0.34/0.50 thf(fact_7_enat__set_Ointros_I2_J,axiom,
% 0.34/0.50 ! [N: extended_enat] :
% 0.34/0.50 ( ( member @ extended_enat @ N @ coinductive_enat_set )
% 0.34/0.50 => ( member @ extended_enat @ ( extended_eSuc @ N ) @ coinductive_enat_set ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_set.intros(2)
% 0.34/0.50 thf(fact_8_enat__setp_Ointros_I2_J,axiom,
% 0.34/0.50 ! [N: extended_enat] :
% 0.34/0.50 ( ( coindu530039314t_setp @ N )
% 0.34/0.50 => ( coindu530039314t_setp @ ( extended_eSuc @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_setp.intros(2)
% 0.34/0.50 thf(fact_9_enat__unfold__next,axiom,
% 0.34/0.50 ! [A: $tType,Stop: A > $o,A2: A,Next: A > A] :
% 0.34/0.50 ( ~ ( Stop @ A2 )
% 0.34/0.50 => ( ( coindu1491768222unfold @ A @ Stop @ Next @ A2 )
% 0.34/0.50 = ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A2 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_unfold_next
% 0.34/0.50 thf(fact_10_co_Oenat_Ocase_I2_J,axiom,
% 0.34/0.50 ! [A: $tType,F1: A,F2: extended_enat > A,X2: extended_enat] :
% 0.34/0.50 ( ( coindu440805660e_enat @ A @ F1 @ F2 @ ( extended_eSuc @ X2 ) )
% 0.34/0.50 = ( F2 @ X2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % co.enat.case(2)
% 0.34/0.50 thf(fact_11_enat__cocase__eSuc,axiom,
% 0.34/0.50 ! [A: $tType,Z: A,S: extended_enat > A,N: extended_enat] :
% 0.34/0.50 ( ( coindu440805660e_enat @ A @ Z @ S @ ( extended_eSuc @ N ) )
% 0.34/0.50 = ( S @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_cocase_eSuc
% 0.34/0.50 thf(fact_12_eSuc__ile__mono,axiom,
% 0.34/0.50 ! [N: extended_enat,M: extended_enat] :
% 0.34/0.50 ( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.34/0.50 = ( ord_less_eq @ extended_enat @ N @ M ) ) ).
% 0.34/0.50
% 0.34/0.50 % eSuc_ile_mono
% 0.34/0.50 thf(fact_13_enat_Oinject,axiom,
% 0.34/0.50 ! [Nat: nat,Nat2: nat] :
% 0.34/0.50 ( ( ( extended_enat2 @ Nat )
% 0.34/0.50 = ( extended_enat2 @ Nat2 ) )
% 0.34/0.50 = ( Nat = Nat2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat.inject
% 0.34/0.50 thf(fact_14_iless__Suc__eq,axiom,
% 0.34/0.50 ! [M: nat,N: extended_enat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
% 0.34/0.50 = ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % iless_Suc_eq
% 0.34/0.50 thf(fact_15_enat__ile,axiom,
% 0.34/0.50 ! [N: extended_enat,M: nat] :
% 0.34/0.50 ( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.34/0.50 => ? [K: nat] :
% 0.34/0.50 ( N
% 0.34/0.50 = ( extended_enat2 @ K ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_ile
% 0.34/0.50 thf(fact_16_Suc__ile__eq,axiom,
% 0.34/0.50 ! [M: nat,N: extended_enat] :
% 0.34/0.50 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 0.34/0.50 = ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_ile_eq
% 0.34/0.50 thf(fact_17_chain__incr,axiom,
% 0.34/0.50 ! [A: $tType,Y3: A > extended_enat,K2: nat] :
% 0.34/0.50 ( ! [I: A] :
% 0.34/0.50 ? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I ) @ ( Y3 @ J ) )
% 0.34/0.50 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y3 @ J2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % chain_incr
% 0.34/0.50 thf(fact_18_enat__iless,axiom,
% 0.34/0.50 ! [N: extended_enat,M: nat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.34/0.50 => ? [K: nat] :
% 0.34/0.50 ( N
% 0.34/0.50 = ( extended_enat2 @ K ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_iless
% 0.34/0.50 thf(fact_19_enat__less__induct,axiom,
% 0.34/0.50 ! [P: extended_enat > $o,N: extended_enat] :
% 0.34/0.50 ( ! [N3: extended_enat] :
% 0.34/0.50 ( ! [M2: extended_enat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ M2 @ N3 )
% 0.34/0.50 => ( P @ M2 ) )
% 0.34/0.50 => ( P @ N3 ) )
% 0.34/0.50 => ( P @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_less_induct
% 0.34/0.50 thf(fact_20_enat__less__imp__le,axiom,
% 0.34/0.50 ! [N: extended_enat,M: extended_enat] :
% 0.34/0.50 ( ! [K: nat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ K ) )
% 0.34/0.50 => ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K ) ) )
% 0.34/0.50 => ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_less_imp_le
% 0.34/0.50 thf(fact_21_enat__setp__enat__set__eq,axiom,
% 0.34/0.50 ( coindu530039314t_setp
% 0.34/0.50 = ( ^ [X3: extended_enat] : ( member @ extended_enat @ X3 @ coinductive_enat_set ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_setp_enat_set_eq
% 0.34/0.50 thf(fact_22_wlog__linorder__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [P: A > A > $o,B: A,A2: A] :
% 0.34/0.50 ( ! [A3: A,B2: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.50 => ( P @ A3 @ B2 ) )
% 0.34/0.50 => ( ( ( P @ B @ A2 )
% 0.34/0.50 => ( P @ A2 @ B ) )
% 0.34/0.50 => ( P @ A2 @ B ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % wlog_linorder_le
% 0.34/0.50 thf(fact_23_ileI1,axiom,
% 0.34/0.50 ! [M: extended_enat,N: extended_enat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ M @ N )
% 0.34/0.50 => ( ord_less_eq @ extended_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % ileI1
% 0.34/0.50 thf(fact_24_eSuc__le__iff,axiom,
% 0.34/0.50 ! [X: extended_enat,Y: extended_enat] :
% 0.34/0.50 ( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ X ) @ Y )
% 0.34/0.50 = ( ? [Y4: extended_enat] :
% 0.34/0.50 ( ( Y
% 0.34/0.50 = ( extended_eSuc @ Y4 ) )
% 0.34/0.50 & ( ord_less_eq @ extended_enat @ X @ Y4 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % eSuc_le_iff
% 0.34/0.50 thf(fact_25_ile__eSuc,axiom,
% 0.34/0.50 ! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % ile_eSuc
% 0.34/0.50 thf(fact_26_nat_Oinject,axiom,
% 0.34/0.50 ! [X2: nat,Y2: nat] :
% 0.34/0.50 ( ( ( suc @ X2 )
% 0.34/0.50 = ( suc @ Y2 ) )
% 0.34/0.50 = ( X2 = Y2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % nat.inject
% 0.34/0.50 thf(fact_27_old_Onat_Oinject,axiom,
% 0.34/0.50 ! [Nat: nat,Nat2: nat] :
% 0.34/0.50 ( ( ( suc @ Nat )
% 0.34/0.50 = ( suc @ Nat2 ) )
% 0.34/0.50 = ( Nat = Nat2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % old.nat.inject
% 0.34/0.50 thf(fact_28_order__refl,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_refl
% 0.34/0.50 thf(fact_29_lift__Suc__mono__less,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [F: nat > A,N: nat,N4: nat] :
% 0.34/0.50 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.50 => ( ( ord_less @ nat @ N @ N4 )
% 0.34/0.50 => ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % lift_Suc_mono_less
% 0.34/0.50 thf(fact_30_lift__Suc__mono__less__iff,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [F: nat > A,N: nat,M: nat] :
% 0.34/0.50 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.50 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 0.34/0.50 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % lift_Suc_mono_less_iff
% 0.34/0.50 thf(fact_31_lift__Suc__mono__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [F: nat > A,N: nat,N4: nat] :
% 0.34/0.50 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.50 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.34/0.50 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % lift_Suc_mono_le
% 0.34/0.50 thf(fact_32_lift__Suc__antimono__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [F: nat > A,N: nat,N4: nat] :
% 0.34/0.50 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.34/0.50 => ( ( ord_less_eq @ nat @ N @ N4 )
% 0.34/0.50 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % lift_Suc_antimono_le
% 0.34/0.50 thf(fact_33_minf_I8_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(8)
% 0.34/0.50 thf(fact_34_minf_I6_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(6)
% 0.34/0.50 thf(fact_35_pinf_I8_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(8)
% 0.34/0.50 thf(fact_36_Suc__less__eq,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.34/0.50 = ( ord_less @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_less_eq
% 0.34/0.50 thf(fact_37_Suc__mono,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_mono
% 0.34/0.50 thf(fact_38_lessI,axiom,
% 0.34/0.50 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % lessI
% 0.34/0.50 thf(fact_39_Suc__le__mono,axiom,
% 0.34/0.50 ! [N: nat,M: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.34/0.50 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_le_mono
% 0.34/0.50 thf(fact_40_enat__ord__simps_I1_J,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.34/0.50 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_ord_simps(1)
% 0.34/0.50 thf(fact_41_enat__ord__simps_I2_J,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.34/0.50 = ( ord_less @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % enat_ord_simps(2)
% 0.34/0.50 thf(fact_42_Suc__leI,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 => ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_leI
% 0.34/0.50 thf(fact_43_le__refl,axiom,
% 0.34/0.50 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.34/0.50
% 0.34/0.50 % le_refl
% 0.34/0.50 thf(fact_44_mem__Collect__eq,axiom,
% 0.34/0.50 ! [A: $tType,A2: A,P: A > $o] :
% 0.34/0.50 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.34/0.50 = ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % mem_Collect_eq
% 0.34/0.50 thf(fact_45_Collect__mem__eq,axiom,
% 0.34/0.50 ! [A: $tType,A4: set @ A] :
% 0.34/0.50 ( ( collect @ A
% 0.34/0.50 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.34/0.50 = A4 ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_mem_eq
% 0.34/0.50 thf(fact_46_Collect__cong,axiom,
% 0.34/0.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ( P @ X5 )
% 0.34/0.50 = ( Q @ X5 ) )
% 0.34/0.50 => ( ( collect @ A @ P )
% 0.34/0.50 = ( collect @ A @ Q ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_cong
% 0.34/0.50 thf(fact_47_ext,axiom,
% 0.34/0.50 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ( F @ X5 )
% 0.34/0.50 = ( G @ X5 ) )
% 0.34/0.50 => ( F = G ) ) ).
% 0.34/0.50
% 0.34/0.50 % ext
% 0.34/0.50 thf(fact_48_le__trans,axiom,
% 0.34/0.50 ! [I2: nat,J3: nat,K2: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.50 => ( ( ord_less_eq @ nat @ J3 @ K2 )
% 0.34/0.50 => ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_trans
% 0.34/0.50 thf(fact_49_Suc__le__eq,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.34/0.50 = ( ord_less @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_le_eq
% 0.34/0.50 thf(fact_50_eq__imp__le,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( M = N )
% 0.34/0.50 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % eq_imp_le
% 0.34/0.50 thf(fact_51_dec__induct,axiom,
% 0.34/0.50 ! [I2: nat,J3: nat,P: nat > $o] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.50 => ( ( P @ I2 )
% 0.34/0.50 => ( ! [N3: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ I2 @ N3 )
% 0.34/0.50 => ( ( ord_less @ nat @ N3 @ J3 )
% 0.34/0.50 => ( ( P @ N3 )
% 0.34/0.50 => ( P @ ( suc @ N3 ) ) ) ) )
% 0.34/0.50 => ( P @ J3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dec_induct
% 0.34/0.50 thf(fact_52_inc__induct,axiom,
% 0.34/0.50 ! [I2: nat,J3: nat,P: nat > $o] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.50 => ( ( P @ J3 )
% 0.34/0.50 => ( ! [N3: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ I2 @ N3 )
% 0.34/0.50 => ( ( ord_less @ nat @ N3 @ J3 )
% 0.34/0.50 => ( ( P @ ( suc @ N3 ) )
% 0.34/0.50 => ( P @ N3 ) ) ) )
% 0.34/0.50 => ( P @ I2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % inc_induct
% 0.34/0.50 thf(fact_53_le__antisym,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 => ( ( ord_less_eq @ nat @ N @ M )
% 0.34/0.50 => ( M = N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_antisym
% 0.34/0.50 thf(fact_54_nat__less__le,axiom,
% 0.34/0.50 ( ( ord_less @ nat )
% 0.34/0.50 = ( ^ [M3: nat,N2: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M3 @ N2 )
% 0.34/0.50 & ( M3 != N2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % nat_less_le
% 0.34/0.50 thf(fact_55_nat__neq__iff,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( M != N )
% 0.34/0.50 = ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % nat_neq_iff
% 0.34/0.50 thf(fact_56_Suc__le__lessD,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.34/0.50 => ( ord_less @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_le_lessD
% 0.34/0.50 thf(fact_57_less__not__refl,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ~ ( ord_less @ nat @ N @ N ) ).
% 0.34/0.50
% 0.34/0.50 % less_not_refl
% 0.34/0.50 thf(fact_58_nat__le__linear,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.34/0.50
% 0.34/0.50 % nat_le_linear
% 0.34/0.50 thf(fact_59_le__less__Suc__eq,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.34/0.50 = ( N = M ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_less_Suc_eq
% 0.34/0.50 thf(fact_60_less__Suc__eq__le,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.34/0.50 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_Suc_eq_le
% 0.34/0.50 thf(fact_61_less__eq__Suc__le,axiom,
% 0.34/0.50 ( ( ord_less @ nat )
% 0.34/0.50 = ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_eq_Suc_le
% 0.34/0.50 thf(fact_62_less__not__refl2,axiom,
% 0.34/0.50 ! [N: nat,M: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ N @ M )
% 0.34/0.50 => ( M != N ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_not_refl2
% 0.34/0.50 thf(fact_63_less__not__refl3,axiom,
% 0.34/0.50 ! [S: nat,T2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ S @ T2 )
% 0.34/0.50 => ( S != T2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_not_refl3
% 0.34/0.50 thf(fact_64_measure__induct,axiom,
% 0.34/0.50 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ! [Y5: A] :
% 0.34/0.50 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X5 ) )
% 0.34/0.50 => ( P @ Y5 ) )
% 0.34/0.50 => ( P @ X5 ) )
% 0.34/0.50 => ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % measure_induct
% 0.34/0.50 thf(fact_65_le__imp__less__Suc,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_imp_less_Suc
% 0.34/0.50 thf(fact_66_less__imp__le__nat,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_imp_le_nat
% 0.34/0.50 thf(fact_67_less__irrefl__nat,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ~ ( ord_less @ nat @ N @ N ) ).
% 0.34/0.50
% 0.34/0.50 % less_irrefl_nat
% 0.34/0.50 thf(fact_68_nat__less__induct,axiom,
% 0.34/0.50 ! [P: nat > $o,N: nat] :
% 0.34/0.50 ( ! [N3: nat] :
% 0.34/0.50 ( ! [M2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M2 @ N3 )
% 0.34/0.50 => ( P @ M2 ) )
% 0.34/0.50 => ( P @ N3 ) )
% 0.34/0.50 => ( P @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % nat_less_induct
% 0.34/0.50 thf(fact_69_infinite__descent,axiom,
% 0.34/0.50 ! [P: nat > $o,N: nat] :
% 0.34/0.50 ( ! [N3: nat] :
% 0.34/0.50 ( ~ ( P @ N3 )
% 0.34/0.50 => ? [M2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M2 @ N3 )
% 0.34/0.50 & ~ ( P @ M2 ) ) )
% 0.34/0.50 => ( P @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % infinite_descent
% 0.34/0.50 thf(fact_70_le__eq__less__or__eq,axiom,
% 0.34/0.50 ( ( ord_less_eq @ nat )
% 0.34/0.50 = ( ^ [M3: nat,N2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M3 @ N2 )
% 0.34/0.50 | ( M3 = N2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_eq_less_or_eq
% 0.34/0.50 thf(fact_71_less__or__eq__imp__le,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 | ( M = N ) )
% 0.34/0.50 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_or_eq_imp_le
% 0.34/0.50 thf(fact_72_linorder__neqE__nat,axiom,
% 0.34/0.50 ! [X: nat,Y: nat] :
% 0.34/0.50 ( ( X != Y )
% 0.34/0.50 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.34/0.50 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % linorder_neqE_nat
% 0.34/0.50 thf(fact_73_le__neq__implies__less,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 => ( ( M != N )
% 0.34/0.50 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_neq_implies_less
% 0.34/0.50 thf(fact_74_measure__induct__rule,axiom,
% 0.34/0.50 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ! [Y5: A] :
% 0.34/0.50 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X5 ) )
% 0.34/0.50 => ( P @ Y5 ) )
% 0.34/0.50 => ( P @ X5 ) )
% 0.34/0.50 => ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % measure_induct_rule
% 0.34/0.50 thf(fact_75_less__mono__imp__le__mono,axiom,
% 0.34/0.50 ! [F: nat > nat,I2: nat,J3: nat] :
% 0.34/0.50 ( ! [I: nat,J2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ I @ J2 )
% 0.34/0.50 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J2 ) ) )
% 0.34/0.50 => ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.50 => ( ord_less_eq @ nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_mono_imp_le_mono
% 0.34/0.50 thf(fact_76_infinite__descent__measure,axiom,
% 0.34/0.50 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ~ ( P @ X5 )
% 0.34/0.50 => ? [Y5: A] :
% 0.34/0.50 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) )
% 0.34/0.50 & ~ ( P @ Y5 ) ) )
% 0.34/0.50 => ( P @ X ) ) ).
% 0.34/0.50
% 0.34/0.50 % infinite_descent_measure
% 0.34/0.50 thf(fact_77_not__less__less__Suc__eq,axiom,
% 0.34/0.50 ! [N: nat,M: nat] :
% 0.34/0.50 ( ~ ( ord_less @ nat @ N @ M )
% 0.34/0.50 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.34/0.50 = ( N = M ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_less_less_Suc_eq
% 0.34/0.50 thf(fact_78_strict__inc__induct,axiom,
% 0.34/0.50 ! [I2: nat,J3: nat,P: nat > $o] :
% 0.34/0.50 ( ( ord_less @ nat @ I2 @ J3 )
% 0.34/0.50 => ( ! [I: nat] :
% 0.34/0.50 ( ( J3
% 0.34/0.50 = ( suc @ I ) )
% 0.34/0.50 => ( P @ I ) )
% 0.34/0.50 => ( ! [I: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ I @ J3 )
% 0.34/0.50 => ( ( P @ ( suc @ I ) )
% 0.34/0.50 => ( P @ I ) ) )
% 0.34/0.50 => ( P @ I2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % strict_inc_induct
% 0.34/0.50 thf(fact_79_less__Suc__induct,axiom,
% 0.34/0.50 ! [I2: nat,J3: nat,P: nat > nat > $o] :
% 0.34/0.50 ( ( ord_less @ nat @ I2 @ J3 )
% 0.34/0.50 => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
% 0.34/0.50 => ( ! [I: nat,J2: nat,K: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ I @ J2 )
% 0.34/0.50 => ( ( ord_less @ nat @ J2 @ K )
% 0.34/0.50 => ( ( P @ I @ J2 )
% 0.34/0.50 => ( ( P @ J2 @ K )
% 0.34/0.50 => ( P @ I @ K ) ) ) ) )
% 0.34/0.50 => ( P @ I2 @ J3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_Suc_induct
% 0.34/0.50 thf(fact_80_less__trans__Suc,axiom,
% 0.34/0.50 ! [I2: nat,J3: nat,K2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ I2 @ J3 )
% 0.34/0.50 => ( ( ord_less @ nat @ J3 @ K2 )
% 0.34/0.50 => ( ord_less @ nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_trans_Suc
% 0.34/0.50 thf(fact_81_Suc__less__SucD,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.34/0.50 => ( ord_less @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_less_SucD
% 0.34/0.50 thf(fact_82_less__antisym,axiom,
% 0.34/0.50 ! [N: nat,M: nat] :
% 0.34/0.50 ( ~ ( ord_less @ nat @ N @ M )
% 0.34/0.50 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 0.34/0.50 => ( M = N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_antisym
% 0.34/0.50 thf(fact_83_Suc__less__eq2,axiom,
% 0.34/0.50 ! [N: nat,M: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 0.34/0.50 = ( ? [M4: nat] :
% 0.34/0.50 ( ( M
% 0.34/0.50 = ( suc @ M4 ) )
% 0.34/0.50 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_less_eq2
% 0.34/0.50 thf(fact_84_not__less__eq,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 0.34/0.50 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_less_eq
% 0.34/0.50 thf(fact_85_less__Suc__eq,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.34/0.50 = ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 | ( M = N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_Suc_eq
% 0.34/0.50 thf(fact_86_less__SucI,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_SucI
% 0.34/0.50 thf(fact_87_less__SucE,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 0.34/0.50 => ( ~ ( ord_less @ nat @ M @ N )
% 0.34/0.50 => ( M = N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_SucE
% 0.34/0.50 thf(fact_88_Suc__lessI,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ M @ N )
% 0.34/0.50 => ( ( ( suc @ M )
% 0.34/0.50 != N )
% 0.34/0.50 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_lessI
% 0.34/0.50 thf(fact_89_Suc__lessE,axiom,
% 0.34/0.50 ! [I2: nat,K2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ ( suc @ I2 ) @ K2 )
% 0.34/0.50 => ~ ! [J2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ I2 @ J2 )
% 0.34/0.50 => ( K2
% 0.34/0.50 != ( suc @ J2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_lessE
% 0.34/0.50 thf(fact_90_Suc__lessD,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 0.34/0.50 => ( ord_less @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_lessD
% 0.34/0.50 thf(fact_91_lessE,axiom,
% 0.34/0.50 ! [I2: nat,K2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ I2 @ K2 )
% 0.34/0.50 => ( ( K2
% 0.34/0.50 != ( suc @ I2 ) )
% 0.34/0.50 => ~ ! [J2: nat] :
% 0.34/0.50 ( ( ord_less @ nat @ I2 @ J2 )
% 0.34/0.50 => ( K2
% 0.34/0.50 != ( suc @ J2 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % lessE
% 0.34/0.50 thf(fact_92_full__nat__induct,axiom,
% 0.34/0.50 ! [P: nat > $o,N: nat] :
% 0.34/0.50 ( ! [N3: nat] :
% 0.34/0.50 ( ! [M2: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N3 )
% 0.34/0.50 => ( P @ M2 ) )
% 0.34/0.50 => ( P @ N3 ) )
% 0.34/0.50 => ( P @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % full_nat_induct
% 0.34/0.50 thf(fact_93_not__less__eq__eq,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 0.34/0.50 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_less_eq_eq
% 0.34/0.50 thf(fact_94_Suc__n__not__le__n,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_n_not_le_n
% 0.34/0.50 thf(fact_95_le__Suc__eq,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.34/0.50 = ( ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 | ( M
% 0.34/0.50 = ( suc @ N ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_Suc_eq
% 0.34/0.50 thf(fact_96_Suc__le__D,axiom,
% 0.34/0.50 ! [N: nat,M5: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
% 0.34/0.50 => ? [M6: nat] :
% 0.34/0.50 ( M5
% 0.34/0.50 = ( suc @ M6 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_le_D
% 0.34/0.50 thf(fact_97_le__SucI,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_SucI
% 0.34/0.50 thf(fact_98_le__SucE,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 0.34/0.50 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 0.34/0.50 => ( M
% 0.34/0.50 = ( suc @ N ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_SucE
% 0.34/0.50 thf(fact_99_Suc__leD,axiom,
% 0.34/0.50 ! [M: nat,N: nat] :
% 0.34/0.50 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 0.34/0.50 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_leD
% 0.34/0.50 thf(fact_100_less__enatE,axiom,
% 0.34/0.50 ! [N: extended_enat,M: nat] :
% 0.34/0.50 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.34/0.50 => ~ ! [K: nat] :
% 0.34/0.50 ( ( N
% 0.34/0.50 = ( extended_enat2 @ K ) )
% 0.34/0.50 => ~ ( ord_less @ nat @ K @ M ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_enatE
% 0.34/0.50 thf(fact_101_dual__order_Oantisym,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ B @ A2 )
% 0.34/0.50 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( A2 = B ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.antisym
% 0.34/0.50 thf(fact_102_dual__order_Otrans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A,C: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ B @ A2 )
% 0.34/0.50 => ( ( ord_less_eq @ A @ C @ B )
% 0.34/0.50 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.trans
% 0.34/0.50 thf(fact_103_linorder__wlog,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [P: A > A > $o,A2: A,B: A] :
% 0.34/0.50 ( ! [A3: A,B2: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.34/0.50 => ( P @ A3 @ B2 ) )
% 0.34/0.50 => ( ! [A3: A,B2: A] :
% 0.34/0.50 ( ( P @ B2 @ A3 )
% 0.34/0.50 => ( P @ A3 @ B2 ) )
% 0.34/0.50 => ( P @ A2 @ B ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % linorder_wlog
% 0.34/0.50 thf(fact_104_dual__order_Orefl,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.refl
% 0.34/0.50 thf(fact_105_order__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A,Z: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.34/0.50 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_trans
% 0.34/0.50 thf(fact_106_order__class_Oorder_Oantisym,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less_eq @ A @ B @ A2 )
% 0.34/0.50 => ( A2 = B ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_class.order.antisym
% 0.34/0.50 thf(fact_107_ord__le__eq__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( ord @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( B = C )
% 0.34/0.50 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_le_eq_trans
% 0.34/0.50 thf(fact_108_ord__eq__le__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( ord @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( A2 = B )
% 0.34/0.50 => ( ( ord_less_eq @ A @ B @ C )
% 0.34/0.50 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_eq_le_trans
% 0.34/0.50 thf(fact_109_antisym__conv,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [Y: A,X: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ Y @ X )
% 0.34/0.50 => ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 = ( X = Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % antisym_conv
% 0.34/0.50 thf(fact_110_le__cases3,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A,Z: A] :
% 0.34/0.50 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.34/0.50 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.34/0.50 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.34/0.50 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.34/0.50 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.34/0.50 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_cases3
% 0.34/0.50 thf(fact_111_order_Otrans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less_eq @ A @ B @ C )
% 0.34/0.50 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.trans
% 0.34/0.50 thf(fact_112_le__cases,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_cases
% 0.34/0.50 thf(fact_113_eq__refl,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( X = Y )
% 0.34/0.50 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % eq_refl
% 0.34/0.50 thf(fact_114_linear,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % linear
% 0.34/0.50 thf(fact_115_antisym,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less_eq @ A @ Y @ X )
% 0.34/0.50 => ( X = Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % antisym
% 0.34/0.50 thf(fact_116_eq__iff,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ^ [Y6: A,Z3: A] : ( Y6 = Z3 ) )
% 0.34/0.50 = ( ^ [X3: A,Y7: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 0.34/0.50 & ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % eq_iff
% 0.34/0.50 thf(fact_117_ord__le__eq__subst,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( ord @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( ( F @ B )
% 0.34/0.50 = C )
% 0.34/0.50 => ( ! [X5: A,Y8: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less_eq @ B3 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_le_eq_subst
% 0.34/0.50 thf(fact_118_ord__eq__le__subst,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( ord @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.34/0.50 ( ( A2
% 0.34/0.50 = ( F @ B ) )
% 0.34/0.50 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.34/0.50 => ( ! [X5: B3,Y8: B3] :
% 0.34/0.50 ( ( ord_less_eq @ B3 @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_eq_le_subst
% 0.34/0.50 thf(fact_119_order__subst2,axiom,
% 0.34/0.50 ! [A: $tType,C2: $tType] :
% 0.34/0.50 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.34/0.50 => ( ! [X5: A,Y8: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_subst2
% 0.34/0.50 thf(fact_120_order__subst1,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.34/0.50 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.34/0.50 => ( ! [X5: B3,Y8: B3] :
% 0.34/0.50 ( ( ord_less_eq @ B3 @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_subst1
% 0.34/0.50 thf(fact_121_le__fun__def,axiom,
% 0.34/0.50 ! [B3: $tType,A: $tType] :
% 0.34/0.50 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 => ( ( ord_less_eq @ ( A > B3 ) )
% 0.34/0.50 = ( ^ [F3: A > B3,G2: A > B3] :
% 0.34/0.50 ! [X3: A] : ( ord_less_eq @ B3 @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_fun_def
% 0.34/0.50 thf(fact_122_le__funI,axiom,
% 0.34/0.50 ! [B3: $tType,A: $tType] :
% 0.34/0.50 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 => ! [F: A > B3,G: A > B3] :
% 0.34/0.50 ( ! [X5: A] : ( ord_less_eq @ B3 @ ( F @ X5 ) @ ( G @ X5 ) )
% 0.34/0.50 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_funI
% 0.34/0.50 thf(fact_123_le__funE,axiom,
% 0.34/0.50 ! [B3: $tType,A: $tType] :
% 0.34/0.50 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 => ! [F: A > B3,G: A > B3,X: A] :
% 0.34/0.50 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.34/0.50 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_funE
% 0.34/0.50 thf(fact_124_le__funD,axiom,
% 0.34/0.50 ! [B3: $tType,A: $tType] :
% 0.34/0.50 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 => ! [F: A > B3,G: A > B3,X: A] :
% 0.34/0.50 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 0.34/0.50 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_funD
% 0.34/0.50 thf(fact_125_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A] :
% 0.34/0.50 ( ( ord_less @ A @ B @ A2 )
% 0.34/0.50 => ( A2 != B ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.strict_implies_not_eq
% 0.34/0.50 thf(fact_126_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( A2 != B ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.strict_implies_not_eq
% 0.34/0.50 thf(fact_127_not__less__iff__gr__or__eq,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.34/0.50 = ( ( ord_less @ A @ Y @ X )
% 0.34/0.50 | ( X = Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_less_iff_gr_or_eq
% 0.34/0.50 thf(fact_128_dual__order_Ostrict__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A,C: A] :
% 0.34/0.50 ( ( ord_less @ A @ B @ A2 )
% 0.34/0.50 => ( ( ord_less @ A @ C @ B )
% 0.34/0.50 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.strict_trans
% 0.34/0.50 thf(fact_129_less__imp__not__less,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_imp_not_less
% 0.34/0.50 thf(fact_130_order_Ostrict__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less @ A @ B @ C )
% 0.34/0.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.strict_trans
% 0.34/0.50 thf(fact_131_dual__order_Oirrefl,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A] :
% 0.34/0.50 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.irrefl
% 0.34/0.50 thf(fact_132_linorder__cases,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ~ ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ( X != Y )
% 0.34/0.50 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % linorder_cases
% 0.34/0.50 thf(fact_133_less__imp__triv,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A,P: $o] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less @ A @ Y @ X )
% 0.34/0.50 => P ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_imp_triv
% 0.34/0.50 thf(fact_134_less__imp__not__eq2,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( Y != X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_imp_not_eq2
% 0.34/0.50 thf(fact_135_antisym__conv3,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [Y: A,X: A] :
% 0.34/0.50 ( ~ ( ord_less @ A @ Y @ X )
% 0.34/0.50 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.34/0.50 = ( X = Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % antisym_conv3
% 0.34/0.50 thf(fact_136_less__induct,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [P: A > $o,A2: A] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ! [Y5: A] :
% 0.34/0.50 ( ( ord_less @ A @ Y5 @ X5 )
% 0.34/0.50 => ( P @ Y5 ) )
% 0.34/0.50 => ( P @ X5 ) )
% 0.34/0.50 => ( P @ A2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_induct
% 0.34/0.50 thf(fact_137_less__not__sym,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_not_sym
% 0.34/0.50 thf(fact_138_less__imp__not__eq,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( X != Y ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_imp_not_eq
% 0.34/0.50 thf(fact_139_dual__order_Oasym,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A] :
% 0.34/0.50 ( ( ord_less @ A @ B @ A2 )
% 0.34/0.50 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.asym
% 0.34/0.50 thf(fact_140_ord__less__eq__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( ord @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ( B = C )
% 0.34/0.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_less_eq_trans
% 0.34/0.50 thf(fact_141_ord__eq__less__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( ord @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( A2 = B )
% 0.34/0.50 => ( ( ord_less @ A @ B @ C )
% 0.34/0.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_eq_less_trans
% 0.34/0.50 thf(fact_142_less__irrefl,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A] :
% 0.34/0.50 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_irrefl
% 0.34/0.50 thf(fact_143_less__linear,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 | ( X = Y )
% 0.34/0.50 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_linear
% 0.34/0.50 thf(fact_144_less__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A,Z: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less @ A @ Y @ Z )
% 0.34/0.50 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_trans
% 0.34/0.50 thf(fact_145_less__asym_H,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_asym'
% 0.34/0.50 thf(fact_146_less__asym,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_asym
% 0.34/0.50 thf(fact_147_less__imp__neq,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( X != Y ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_imp_neq
% 0.34/0.50 thf(fact_148_dense,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ? [Z2: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Z2 )
% 0.34/0.50 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dense
% 0.34/0.50 thf(fact_149_order_Oasym,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.asym
% 0.34/0.50 thf(fact_150_neq__iff,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( X != Y )
% 0.34/0.50 = ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % neq_iff
% 0.34/0.50 thf(fact_151_neqE,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( X != Y )
% 0.34/0.50 => ( ~ ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % neqE
% 0.34/0.50 thf(fact_152_gt__ex,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( no_top @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A] :
% 0.34/0.50 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.34/0.50
% 0.34/0.50 % gt_ex
% 0.34/0.50 thf(fact_153_lt__ex,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A] :
% 0.34/0.50 ? [Y8: A] : ( ord_less @ A @ Y8 @ X ) ) ).
% 0.34/0.50
% 0.34/0.50 % lt_ex
% 0.34/0.50 thf(fact_154_order__less__subst2,axiom,
% 0.34/0.50 ! [A: $tType,C2: $tType] :
% 0.34/0.50 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.34/0.50 => ( ! [X5: A,Y8: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_less_subst2
% 0.34/0.50 thf(fact_155_order__less__subst1,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.34/0.50 => ( ( ord_less @ B3 @ B @ C )
% 0.34/0.50 => ( ! [X5: B3,Y8: B3] :
% 0.34/0.50 ( ( ord_less @ B3 @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_less_subst1
% 0.34/0.50 thf(fact_156_ord__less__eq__subst,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( ord @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ( ( F @ B )
% 0.34/0.50 = C )
% 0.34/0.50 => ( ! [X5: A,Y8: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less @ B3 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_less_eq_subst
% 0.34/0.50 thf(fact_157_ord__eq__less__subst,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( ord @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.34/0.50 ( ( A2
% 0.34/0.50 = ( F @ B ) )
% 0.34/0.50 => ( ( ord_less @ B3 @ B @ C )
% 0.34/0.50 => ( ! [X5: B3,Y8: B3] :
% 0.34/0.50 ( ( ord_less @ B3 @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ord_eq_less_subst
% 0.34/0.50 thf(fact_158_pinf_I1_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.50 ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z4 @ X5 )
% 0.34/0.50 => ( ( P @ X5 )
% 0.34/0.50 = ( P2 @ X5 ) ) )
% 0.34/0.50 => ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z4 @ X5 )
% 0.34/0.50 => ( ( Q @ X5 )
% 0.34/0.50 = ( Q2 @ X5 ) ) )
% 0.34/0.50 => ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ( ( ( P @ X4 )
% 0.34/0.50 & ( Q @ X4 ) )
% 0.34/0.50 = ( ( P2 @ X4 )
% 0.34/0.50 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(1)
% 0.34/0.50 thf(fact_159_pinf_I2_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.50 ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z4 @ X5 )
% 0.34/0.50 => ( ( P @ X5 )
% 0.34/0.50 = ( P2 @ X5 ) ) )
% 0.34/0.50 => ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z4 @ X5 )
% 0.34/0.50 => ( ( Q @ X5 )
% 0.34/0.50 = ( Q2 @ X5 ) ) )
% 0.34/0.50 => ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ( ( ( P @ X4 )
% 0.34/0.50 | ( Q @ X4 ) )
% 0.34/0.50 = ( ( P2 @ X4 )
% 0.34/0.50 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(2)
% 0.34/0.50 thf(fact_160_pinf_I3_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ( X4 != T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(3)
% 0.34/0.50 thf(fact_161_pinf_I4_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ( X4 != T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(4)
% 0.34/0.50 thf(fact_162_pinf_I5_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(5)
% 0.34/0.50 thf(fact_163_pinf_I7_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(7)
% 0.34/0.50 thf(fact_164_pinf_I11_J,axiom,
% 0.34/0.50 ! [C2: $tType,D: $tType] :
% 0.34/0.50 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.34/0.50 => ! [F4: D] :
% 0.34/0.50 ? [Z2: C2] :
% 0.34/0.50 ! [X4: C2] :
% 0.34/0.50 ( ( ord_less @ C2 @ Z2 @ X4 )
% 0.34/0.50 => ( F4 = F4 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(11)
% 0.34/0.50 thf(fact_165_minf_I1_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.50 ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Z4 )
% 0.34/0.50 => ( ( P @ X5 )
% 0.34/0.50 = ( P2 @ X5 ) ) )
% 0.34/0.50 => ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Z4 )
% 0.34/0.50 => ( ( Q @ X5 )
% 0.34/0.50 = ( Q2 @ X5 ) ) )
% 0.34/0.50 => ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ( ( ( P @ X4 )
% 0.34/0.50 & ( Q @ X4 ) )
% 0.34/0.50 = ( ( P2 @ X4 )
% 0.34/0.50 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(1)
% 0.34/0.50 thf(fact_166_minf_I2_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.34/0.50 ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Z4 )
% 0.34/0.50 => ( ( P @ X5 )
% 0.34/0.50 = ( P2 @ X5 ) ) )
% 0.34/0.50 => ( ? [Z4: A] :
% 0.34/0.50 ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Z4 )
% 0.34/0.50 => ( ( Q @ X5 )
% 0.34/0.50 = ( Q2 @ X5 ) ) )
% 0.34/0.50 => ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ( ( ( P @ X4 )
% 0.34/0.50 | ( Q @ X4 ) )
% 0.34/0.50 = ( ( P2 @ X4 )
% 0.34/0.50 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(2)
% 0.34/0.50 thf(fact_167_minf_I3_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ( X4 != T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(3)
% 0.34/0.50 thf(fact_168_minf_I4_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ( X4 != T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(4)
% 0.34/0.50 thf(fact_169_minf_I5_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(5)
% 0.34/0.50 thf(fact_170_minf_I7_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ X4 @ Z2 )
% 0.34/0.50 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(7)
% 0.34/0.50 thf(fact_171_minf_I11_J,axiom,
% 0.34/0.50 ! [C2: $tType,D: $tType] :
% 0.34/0.50 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.34/0.50 => ! [F4: D] :
% 0.34/0.50 ? [Z2: C2] :
% 0.34/0.50 ! [X4: C2] :
% 0.34/0.50 ( ( ord_less @ C2 @ X4 @ Z2 )
% 0.34/0.50 => ( F4 = F4 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % minf(11)
% 0.34/0.50 thf(fact_172_n__not__Suc__n,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ( N
% 0.34/0.50 != ( suc @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % n_not_Suc_n
% 0.34/0.50 thf(fact_173_Suc__inject,axiom,
% 0.34/0.50 ! [X: nat,Y: nat] :
% 0.34/0.50 ( ( ( suc @ X )
% 0.34/0.50 = ( suc @ Y ) )
% 0.34/0.50 => ( X = Y ) ) ).
% 0.34/0.50
% 0.34/0.50 % Suc_inject
% 0.34/0.50 thf(fact_174_order_Onot__eq__order__implies__strict,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A] :
% 0.34/0.50 ( ( A2 != B )
% 0.34/0.50 => ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.not_eq_order_implies_strict
% 0.34/0.50 thf(fact_175_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A] :
% 0.34/0.50 ( ( ord_less @ A @ B @ A2 )
% 0.34/0.50 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.strict_implies_order
% 0.34/0.50 thf(fact_176_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ord_less @ A )
% 0.34/0.50 = ( ^ [B4: A,A5: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ B4 @ A5 )
% 0.34/0.50 & ( A5 != B4 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.strict_iff_order
% 0.34/0.50 thf(fact_177_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ord_less_eq @ A )
% 0.34/0.50 = ( ^ [B4: A,A5: A] :
% 0.34/0.50 ( ( ord_less @ A @ B4 @ A5 )
% 0.34/0.50 | ( A5 = B4 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.order_iff_strict
% 0.34/0.50 thf(fact_178_order_Ostrict__implies__order,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.strict_implies_order
% 0.34/0.50 thf(fact_179_dense__le__bounded,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A,Z: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ! [W: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ W )
% 0.34/0.50 => ( ( ord_less @ A @ W @ Y )
% 0.34/0.50 => ( ord_less_eq @ A @ W @ Z ) ) )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dense_le_bounded
% 0.34/0.50 thf(fact_180_dense__ge__bounded,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [Z: A,X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z @ X )
% 0.34/0.50 => ( ! [W: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z @ W )
% 0.34/0.50 => ( ( ord_less @ A @ W @ X )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dense_ge_bounded
% 0.34/0.50 thf(fact_181_dual__order_Ostrict__trans2,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A,C: A] :
% 0.34/0.50 ( ( ord_less @ A @ B @ A2 )
% 0.34/0.50 => ( ( ord_less_eq @ A @ C @ B )
% 0.34/0.50 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.strict_trans2
% 0.34/0.50 thf(fact_182_dual__order_Ostrict__trans1,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [B: A,A2: A,C: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ B @ A2 )
% 0.34/0.50 => ( ( ord_less @ A @ C @ B )
% 0.34/0.50 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dual_order.strict_trans1
% 0.34/0.50 thf(fact_183_order_Ostrict__iff__order,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ord_less @ A )
% 0.34/0.50 = ( ^ [A5: A,B4: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A5 @ B4 )
% 0.34/0.50 & ( A5 != B4 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.strict_iff_order
% 0.34/0.50 thf(fact_184_order_Oorder__iff__strict,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ord_less_eq @ A )
% 0.34/0.50 = ( ^ [A5: A,B4: A] :
% 0.34/0.50 ( ( ord_less @ A @ A5 @ B4 )
% 0.34/0.50 | ( A5 = B4 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.order_iff_strict
% 0.34/0.50 thf(fact_185_order_Ostrict__trans2,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less_eq @ A @ B @ C )
% 0.34/0.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.strict_trans2
% 0.34/0.50 thf(fact_186_order_Ostrict__trans1,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,C: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less @ A @ B @ C )
% 0.34/0.50 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order.strict_trans1
% 0.34/0.50 thf(fact_187_not__le__imp__less,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [Y: A,X: A] :
% 0.34/0.50 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.34/0.50 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_le_imp_less
% 0.34/0.50 thf(fact_188_less__le__not__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ord_less @ A )
% 0.34/0.50 = ( ^ [X3: A,Y7: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 0.34/0.50 & ~ ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_le_not_le
% 0.34/0.50 thf(fact_189_le__imp__less__or__eq,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 | ( X = Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_imp_less_or_eq
% 0.34/0.50 thf(fact_190_le__less__linear,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_less_linear
% 0.34/0.50 thf(fact_191_dense__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [Y: A,Z: A] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Y )
% 0.34/0.50 => ( ord_less_eq @ A @ X5 @ Z ) )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dense_le
% 0.34/0.50 thf(fact_192_dense__ge,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [Z: A,Y: A] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z @ X5 )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ X5 ) )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % dense_ge
% 0.34/0.50 thf(fact_193_less__le__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A,Z: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.34/0.50 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_le_trans
% 0.34/0.50 thf(fact_194_le__less__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A,Z: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less @ A @ Y @ Z )
% 0.34/0.50 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_less_trans
% 0.34/0.50 thf(fact_195_antisym__conv2,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.34/0.50 = ( X = Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % antisym_conv2
% 0.34/0.50 thf(fact_196_antisym__conv1,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ~ ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ( ord_less_eq @ A @ X @ Y )
% 0.34/0.50 = ( X = Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % antisym_conv1
% 0.34/0.50 thf(fact_197_less__imp__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( preorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_imp_le
% 0.34/0.50 thf(fact_198_le__neq__trans,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( A2 != B )
% 0.34/0.50 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_neq_trans
% 0.34/0.50 thf(fact_199_not__less,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.34/0.50 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_less
% 0.34/0.50 thf(fact_200_not__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.34/0.50 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_le
% 0.34/0.50 thf(fact_201_order__less__le__subst2,axiom,
% 0.34/0.50 ! [A: $tType,C2: $tType] :
% 0.34/0.50 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 0.34/0.50 => ( ! [X5: A,Y8: A] :
% 0.34/0.50 ( ( ord_less @ A @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_less_le_subst2
% 0.34/0.50 thf(fact_202_order__less__le__subst1,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 0.34/0.50 => ( ( ord_less_eq @ B3 @ B @ C )
% 0.34/0.50 => ( ! [X5: B3,Y8: B3] :
% 0.34/0.50 ( ( ord_less_eq @ B3 @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_less_le_subst1
% 0.34/0.50 thf(fact_203_order__le__less__subst2,axiom,
% 0.34/0.50 ! [A: $tType,C2: $tType] :
% 0.34/0.50 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ B )
% 0.34/0.50 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 0.34/0.50 => ( ! [X5: A,Y8: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_le_less_subst2
% 0.34/0.50 thf(fact_204_order__le__less__subst1,axiom,
% 0.34/0.50 ! [A: $tType,B3: $tType] :
% 0.34/0.50 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 0.34/0.50 & ( order @ A @ ( type2 @ A ) ) )
% 0.34/0.50 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 0.34/0.50 => ( ( ord_less @ B3 @ B @ C )
% 0.34/0.50 => ( ! [X5: B3,Y8: B3] :
% 0.34/0.50 ( ( ord_less @ B3 @ X5 @ Y8 )
% 0.34/0.50 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 0.34/0.50 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % order_le_less_subst1
% 0.34/0.50 thf(fact_205_less__le,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ord_less @ A )
% 0.34/0.50 = ( ^ [X3: A,Y7: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 0.34/0.50 & ( X3 != Y7 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_le
% 0.34/0.50 thf(fact_206_le__less,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( order @ A @ ( type2 @ A ) )
% 0.34/0.50 => ( ( ord_less_eq @ A )
% 0.34/0.50 = ( ^ [X3: A,Y7: A] :
% 0.34/0.50 ( ( ord_less @ A @ X3 @ Y7 )
% 0.34/0.50 | ( X3 = Y7 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % le_less
% 0.34/0.50 thf(fact_207_leI,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [X: A,Y: A] :
% 0.34/0.50 ( ~ ( ord_less @ A @ X @ Y )
% 0.34/0.50 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % leI
% 0.34/0.50 thf(fact_208_leD,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [Y: A,X: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ Y @ X )
% 0.34/0.50 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % leD
% 0.34/0.50 thf(fact_209_pinf_I6_J,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( linorder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [T2: A] :
% 0.34/0.50 ? [Z2: A] :
% 0.34/0.50 ! [X4: A] :
% 0.34/0.50 ( ( ord_less @ A @ Z2 @ X4 )
% 0.34/0.50 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pinf(6)
% 0.34/0.50 thf(fact_210_ex__has__greatest__nat,axiom,
% 0.34/0.50 ! [A: $tType,P: A > $o,K2: A,M: A > nat,B: nat] :
% 0.34/0.50 ( ( P @ K2 )
% 0.34/0.50 => ( ! [Y8: A] :
% 0.34/0.50 ( ( P @ Y8 )
% 0.34/0.50 => ( ord_less @ nat @ ( M @ Y8 ) @ B ) )
% 0.34/0.50 => ? [X5: A] :
% 0.34/0.50 ( ( P @ X5 )
% 0.34/0.50 & ! [Y5: A] :
% 0.34/0.50 ( ( P @ Y5 )
% 0.34/0.50 => ( ord_less_eq @ nat @ ( M @ Y5 ) @ ( M @ X5 ) ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_has_greatest_nat
% 0.34/0.50 thf(fact_211_complete__interval,axiom,
% 0.34/0.50 ! [A: $tType] :
% 0.34/0.50 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.34/0.50 => ! [A2: A,B: A,P: A > $o] :
% 0.34/0.50 ( ( ord_less @ A @ A2 @ B )
% 0.34/0.50 => ( ( P @ A2 )
% 0.34/0.50 => ( ~ ( P @ B )
% 0.34/0.50 => ? [C3: A] :
% 0.34/0.50 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.34/0.50 & ( ord_less_eq @ A @ C3 @ B )
% 0.34/0.50 & ! [X4: A] :
% 0.34/0.50 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 0.34/0.50 & ( ord_less @ A @ X4 @ C3 ) )
% 0.34/0.50 => ( P @ X4 ) )
% 0.34/0.50 & ! [D2: A] :
% 0.34/0.50 ( ! [X5: A] :
% 0.34/0.50 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 0.34/0.50 & ( ord_less @ A @ X5 @ D2 ) )
% 0.34/0.50 => ( P @ X5 ) )
% 0.34/0.51 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % complete_interval
% 0.34/0.51 thf(fact_212_enat_Osimps_I6_J,axiom,
% 0.34/0.51 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.34/0.51 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.34/0.51 = ( F1 @ Nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat.simps(6)
% 0.34/0.51 thf(fact_213_enat_Osimps_I4_J,axiom,
% 0.34/0.51 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.34/0.51 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.34/0.51 = ( F1 @ Nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat.simps(4)
% 0.34/0.51 thf(fact_214_Suc_Ohyps,axiom,
% 0.34/0.51 ! [N: extended_enat,Xs: coindu1593790203_llist @ a] :
% 0.34/0.51 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ma ) @ N )
% 0.34/0.51 => ( ( coindu749330388e_lnth @ a @ ( coindu1802687541_ltake @ a @ N @ Xs ) @ ma )
% 0.34/0.51 = ( coindu749330388e_lnth @ a @ Xs @ ma ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Suc.hyps
% 0.34/0.51 thf(fact_215_ex__has__least__nat,axiom,
% 0.34/0.51 ! [A: $tType,P: A > $o,K2: A,M: A > nat] :
% 0.34/0.51 ( ( P @ K2 )
% 0.34/0.51 => ? [X5: A] :
% 0.34/0.51 ( ( P @ X5 )
% 0.34/0.51 & ! [Y5: A] :
% 0.34/0.51 ( ( P @ Y5 )
% 0.34/0.51 => ( ord_less_eq @ nat @ ( M @ X5 ) @ ( M @ Y5 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_has_least_nat
% 0.34/0.51 thf(fact_216_enat__less__enat__plusI,axiom,
% 0.34/0.51 ! [X: nat,Y: nat,Z: extended_enat] :
% 0.34/0.51 ( ( ord_less @ nat @ X @ Y )
% 0.34/0.51 => ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat_less_enat_plusI
% 0.34/0.51 thf(fact_217_ex__gt__or__lt,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [A2: A] :
% 0.34/0.51 ? [B2: A] :
% 0.34/0.51 ( ( ord_less @ A @ A2 @ B2 )
% 0.34/0.51 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_gt_or_lt
% 0.34/0.51 thf(fact_218_linorder__neqE__linordered__idom,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [X: A,Y: A] :
% 0.34/0.51 ( ( X != Y )
% 0.34/0.51 => ( ~ ( ord_less @ A @ X @ Y )
% 0.34/0.51 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neqE_linordered_idom
% 0.34/0.51 thf(fact_219_enat__add1__eq,axiom,
% 0.34/0.51 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 0.34/0.51 ( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
% 0.34/0.51 = ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 0.34/0.51 = ( Y = Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat_add1_eq
% 0.34/0.51 thf(fact_220_enat__add2__eq,axiom,
% 0.34/0.51 ! [Y: extended_enat,X: nat,Z: extended_enat] :
% 0.34/0.51 ( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
% 0.34/0.51 = ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X ) ) )
% 0.34/0.51 = ( Y = Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat_add2_eq
% 0.34/0.51 thf(fact_221_enat__add__mono,axiom,
% 0.34/0.51 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 0.34/0.51 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 0.34/0.51 = ( ord_less @ extended_enat @ Y @ Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat_add_mono
% 0.34/0.51 thf(fact_222_less__fun__def,axiom,
% 0.34/0.51 ! [B3: $tType,A: $tType] :
% 0.34/0.51 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 0.34/0.51 => ( ( ord_less @ ( A > B3 ) )
% 0.34/0.51 = ( ^ [F3: A > B3,G2: A > B3] :
% 0.34/0.51 ( ( ord_less_eq @ ( A > B3 ) @ F3 @ G2 )
% 0.34/0.51 & ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F3 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_fun_def
% 0.34/0.51 thf(fact_223_ltake__eq__ltake__antimono,axiom,
% 0.34/0.51 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
% 0.34/0.51 ( ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 0.34/0.51 = ( coindu1802687541_ltake @ A @ N @ Ys ) )
% 0.34/0.51 => ( ( ord_less_eq @ extended_enat @ M @ N )
% 0.34/0.51 => ( ( coindu1802687541_ltake @ A @ M @ Xs )
% 0.34/0.51 = ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ltake_eq_ltake_antimono
% 0.34/0.51 thf(fact_224_enat__le__plus__same_I2_J,axiom,
% 0.34/0.51 ! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ Y @ X ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat_le_plus_same(2)
% 0.34/0.51 thf(fact_225_enat__le__plus__same_I1_J,axiom,
% 0.34/0.51 ! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ X @ Y ) ) ).
% 0.34/0.51
% 0.34/0.51 % enat_le_plus_same(1)
% 0.34/0.51 thf(fact_226_iadd__Suc__right,axiom,
% 0.34/0.51 ! [M: extended_enat,N: extended_enat] :
% 0.34/0.51 ( ( plus_plus @ extended_enat @ M @ ( extended_eSuc @ N ) )
% 0.34/0.51 = ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % iadd_Suc_right
% 0.34/0.51 thf(fact_227_eSuc__plus,axiom,
% 0.34/0.51 ! [M: extended_enat,N: extended_enat] :
% 0.34/0.51 ( ( plus_plus @ extended_enat @ ( extended_eSuc @ M ) @ N )
% 0.34/0.51 = ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % eSuc_plus
% 0.34/0.51 thf(fact_228_add__less__cancel__left,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [C: A,A2: A,B: A] :
% 0.34/0.51 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.34/0.51 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_less_cancel_left
% 0.34/0.51 thf(fact_229_add__less__cancel__right,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [A2: A,C: A,B: A] :
% 0.34/0.51 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.34/0.51 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_less_cancel_right
% 0.34/0.51 thf(fact_230_add__right__cancel,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [B: A,A2: A,C: A] :
% 0.34/0.51 ( ( ( plus_plus @ A @ B @ A2 )
% 0.34/0.51 = ( plus_plus @ A @ C @ A2 ) )
% 0.34/0.51 = ( B = C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_right_cancel
% 0.34/0.51 thf(fact_231_add__left__cancel,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [A2: A,B: A,C: A] :
% 0.34/0.51 ( ( ( plus_plus @ A @ A2 @ B )
% 0.34/0.51 = ( plus_plus @ A @ A2 @ C ) )
% 0.34/0.51 = ( B = C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_left_cancel
% 0.34/0.51 thf(fact_232_add__Suc__right,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( plus_plus @ nat @ M @ ( suc @ N ) )
% 0.34/0.51 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_Suc_right
% 0.34/0.51 thf(fact_233_nat__add__left__cancel__less,axiom,
% 0.34/0.51 ! [K2: nat,M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
% 0.34/0.51 = ( ord_less @ nat @ M @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % nat_add_left_cancel_less
% 0.34/0.51 thf(fact_234_nat__add__left__cancel__le,axiom,
% 0.34/0.51 ! [K2: nat,M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
% 0.34/0.51 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % nat_add_left_cancel_le
% 0.34/0.51 thf(fact_235_add__le__cancel__right,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [A2: A,C: A,B: A] :
% 0.34/0.51 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 0.34/0.51 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_le_cancel_right
% 0.34/0.51 thf(fact_236_add__le__cancel__left,axiom,
% 0.34/0.51 ! [A: $tType] :
% 0.34/0.51 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 0.34/0.51 => ! [C: A,A2: A,B: A] :
% 0.34/0.51 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 0.34/0.51 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_le_cancel_left
% 0.34/0.51 thf(fact_237_plus__enat__simps_I1_J,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.34/0.51 = ( extended_enat2 @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % plus_enat_simps(1)
% 0.34/0.51 thf(fact_238_add__Suc__shift,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 0.34/0.51 = ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_Suc_shift
% 0.34/0.51 thf(fact_239_add__Suc,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 0.34/0.51 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_Suc
% 0.34/0.51 thf(fact_240_less__add__eq__less,axiom,
% 0.34/0.51 ! [K2: nat,L: nat,M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less @ nat @ K2 @ L )
% 0.34/0.51 => ( ( ( plus_plus @ nat @ M @ L )
% 0.34/0.51 = ( plus_plus @ nat @ K2 @ N ) )
% 0.34/0.51 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_add_eq_less
% 0.34/0.51 thf(fact_241_trans__less__add2,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,M: nat] :
% 0.34/0.51 ( ( ord_less @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % trans_less_add2
% 0.34/0.51 thf(fact_242_trans__less__add1,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,M: nat] :
% 0.34/0.51 ( ( ord_less @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % trans_less_add1
% 0.34/0.51 thf(fact_243_add__less__mono1,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,K2: nat] :
% 0.34/0.51 ( ( ord_less @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_less_mono1
% 0.34/0.51 thf(fact_244_not__add__less2,axiom,
% 0.34/0.51 ! [J3: nat,I2: nat] :
% 0.34/0.51 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J3 @ I2 ) @ I2 ) ).
% 0.34/0.51
% 0.34/0.51 % not_add_less2
% 0.34/0.51 thf(fact_245_not__add__less1,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat] :
% 0.34/0.51 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ I2 ) ).
% 0.34/0.51
% 0.34/0.51 % not_add_less1
% 0.34/0.51 thf(fact_246_add__less__mono,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,K2: nat,L: nat] :
% 0.34/0.51 ( ( ord_less @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ( ord_less @ nat @ K2 @ L )
% 0.34/0.51 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_less_mono
% 0.34/0.51 thf(fact_247_add__lessD1,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,K2: nat] :
% 0.34/0.51 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ K2 )
% 0.34/0.51 => ( ord_less @ nat @ I2 @ K2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_lessD1
% 0.34/0.51 thf(fact_248_nat__le__iff__add,axiom,
% 0.34/0.51 ( ( ord_less_eq @ nat )
% 0.34/0.51 = ( ^ [M3: nat,N2: nat] :
% 0.34/0.51 ? [K3: nat] :
% 0.34/0.51 ( N2
% 0.34/0.51 = ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % nat_le_iff_add
% 0.34/0.51 thf(fact_249_trans__le__add2,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,M: nat] :
% 0.34/0.51 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % trans_le_add2
% 0.34/0.51 thf(fact_250_trans__le__add1,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,M: nat] :
% 0.34/0.51 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % trans_le_add1
% 0.34/0.51 thf(fact_251_add__le__mono1,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,K2: nat] :
% 0.34/0.51 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_le_mono1
% 0.34/0.51 thf(fact_252_add__le__mono,axiom,
% 0.34/0.51 ! [I2: nat,J3: nat,K2: nat,L: nat] :
% 0.34/0.51 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 0.34/0.51 => ( ( ord_less_eq @ nat @ K2 @ L )
% 0.34/0.51 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_le_mono
% 0.34/0.51 thf(fact_253_le__Suc__ex,axiom,
% 0.34/0.51 ! [K2: nat,L: nat] :
% 0.34/0.51 ( ( ord_less_eq @ nat @ K2 @ L )
% 0.34/0.51 => ? [N3: nat] :
% 0.34/0.51 ( L
% 0.34/0.51 = ( plus_plus @ nat @ K2 @ N3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_Suc_ex
% 0.34/0.51 thf(fact_254_add__leD2,axiom,
% 0.34/0.51 ! [M: nat,K2: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
% 0.34/0.51 => ( ord_less_eq @ nat @ K2 @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % add_leD2
% 0.34/0.51
% 0.34/0.51 %----Type constructors (25)
% 0.34/0.51 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.34/0.51 ! [A6: $tType,A7: $tType] :
% 0.34/0.51 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.34/0.51 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.34/0.51 ! [A6: $tType,A7: $tType] :
% 0.34/0.51 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.34/0.51 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_fun___Orderings_Oord,axiom,
% 0.34/0.51 ! [A6: $tType,A7: $tType] :
% 0.34/0.51 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.34/0.51 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.34/0.51 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.34/0.51 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.34/0.51 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.34/0.51 wellorder @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.34/0.51 preorder @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.34/0.51 linorder @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.34/0.51 no_top @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.34/0.51 order @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.34/0.51 ord @ nat @ ( type2 @ nat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.34/0.51 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.34/0.51 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.34/0.51 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.34/0.51 preorder @ $o @ ( type2 @ $o ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.34/0.51 linorder @ $o @ ( type2 @ $o ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.34/0.51 order @ $o @ ( type2 @ $o ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.34/0.51 ord @ $o @ ( type2 @ $o ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
% 0.34/0.51 condit1037483654norder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_12,axiom,
% 0.34/0.51 wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_13,axiom,
% 0.34/0.51 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_14,axiom,
% 0.34/0.51 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_15,axiom,
% 0.34/0.51 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.34/0.51
% 0.34/0.51 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_16,axiom,
% 0.34/0.51 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 0.34/0.51
% 0.34/0.51 %----Conjectures (2)
% 0.34/0.51 thf(conj_0,hypothesis,
% 0.34/0.51 ! [N5: extended_enat] :
% 0.34/0.51 ( ( na
% 0.34/0.51 = ( extended_eSuc @ N5 ) )
% 0.34/0.51 => thesis ) ).
% 0.34/0.51
% 0.34/0.51 thf(conj_1,conjecture,
% 0.34/0.51 thesis ).
% 0.34/0.51
% 0.34/0.51 %------------------------------------------------------------------------------
% 0.34/0.51 ------------------------
% 3.00/1.74 ---- Embedded in THF ---
% 3.00/1.75 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.00/1.75 %%% Generated on Tue Feb 24 06:08:26 EST 2026
% 3.00/1.75 %%% using '$$dhol' embedding, version 1.3.0.
% 3.00/1.75 %%% Logic specification used:
% 3.00/1.75 %%% thf(spec, logic, $$dhol).
% 3.00/1.75
% 3.00/1.75 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2DTF30566.p
% 3.00/1.75 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist, type, coindu1593790203_llist: ($tType > $tType)).
% 3.00/1.75 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 3.00/1.75 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.00/1.75 thf(ty_t_Nat_Onat, type, nat: $tType).
% 3.00/1.75 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.00/1.75 thf(ty_tf_a, type, a: $tType).
% 3.00/1.75 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.00/1.75 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth, type, coindu749330388e_lnth: (!> [A:$tType]: (((coindu1593790203_llist @ A) > (nat > A))))).
% 3.00/1.75 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake, type, coindu1802687541_ltake: (!> [A:$tType]: ((extended_enat > ((coindu1593790203_llist @ A) > (coindu1593790203_llist @ A)))))).
% 3.00/1.75 thf(sy_c_Coinductive__Nat_Oco_Oenat_Ocase__enat, type, coindu440805660e_enat: (!> [A:$tType]: ((A > ((extended_enat > A) > (extended_enat > A)))))).
% 3.00/1.75 thf(sy_c_Coinductive__Nat_Oenat__set, type, coinductive_enat_set: (set @ extended_enat)).
% 3.00/1.75 thf(sy_c_Coinductive__Nat_Oenat__setp, type, coindu530039314t_setp: (extended_enat > $o)).
% 3.00/1.75 thf(sy_c_Coinductive__Nat_Oenat__unfold, type, coindu1491768222unfold: (!> [A:$tType]: (((A > $o) > ((A > A) > (A > extended_enat)))))).
% 3.00/1.75 thf(sy_c_Extended__Nat_OeSuc, type, extended_eSuc: (extended_enat > extended_enat)).
% 3.00/1.75 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 3.00/1.75 thf(sy_c_Extended__Nat_Oenat_Ocase__enat, type, extended_case_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 3.00/1.75 thf(sy_c_Extended__Nat_Oenat_Orec__enat, type, extended_rec_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 3.00/1.75 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 3.00/1.75 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 3.00/1.75 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 3.00/1.75 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.00/1.75 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 3.00/1.75 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.00/1.75 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.00/1.75 thf(sy_v_ma____, type, ma: nat).
% 3.00/1.75 thf(sy_v_na____, type, na: extended_enat).
% 3.00/1.75 thf(sy_v_thesis____, type, thesis: $o).
% 3.00/1.75 thf(fact_0_Suc_Oprems, axiom, (((ord_less @ extended_enat) @ (extended_enat2 @ (suc @ ma))) @ na)).
% 3.00/1.75 thf(fact_1_eSuc__inject, axiom, (! [M:extended_enat,N:extended_enat]: ((((extended_eSuc @ M) = (extended_eSuc @ N)) = (M = N))))).
% 3.00/1.75 thf(fact_2_co_Oenat_Oinject, axiom, (! [X2:extended_enat,Y2:extended_enat]: ((((extended_eSuc @ X2) = (extended_eSuc @ Y2)) = (X2 = Y2))))).
% 3.00/1.75 thf(fact_3_Extended__Nat_OeSuc__mono, axiom, (! [N:extended_enat,M:extended_enat]: (((((ord_less @ extended_enat) @ (extended_eSuc @ N)) @ (extended_eSuc @ M)) = (((ord_less @ extended_enat) @ N) @ M))))).
% 3.00/1.75 thf(fact_4_eSuc__enat, axiom, (! [N:nat]: (((extended_eSuc @ (extended_enat2 @ N)) = (extended_enat2 @ (suc @ N)))))).
% 3.00/1.75 thf(fact_5_eSuc__enat__iff, axiom, (! [X:extended_enat,Y:nat]: ((((extended_eSuc @ X) = (extended_enat2 @ Y)) = (? [N2:nat]: (((Y = (suc @ N2)) & (X = (extended_enat2 @ N2))))))))).
% 3.00/1.75 thf(fact_6_enat__eSuc__iff, axiom, (! [Y:nat,X:extended_enat]: ((((extended_enat2 @ Y) = (extended_eSuc @ X)) = (? [N2:nat]: (((Y = (suc @ N2)) & ((extended_enat2 @ N2) = X)))))))).
% 3.00/1.75 thf(fact_7_enat__set_Ointros_I2_J, axiom, (! [N:extended_enat]: (((((member @ extended_enat) @ N) @ coinductive_enat_set) => (((member @ extended_enat) @ (extended_eSuc @ N)) @ coinductive_enat_set))))).
% 3.00/1.75 thf(fact_8_enat__setp_Ointros_I2_J, axiom, (! [N:extended_enat]: (((coindu530039314t_setp @ N) => (coindu530039314t_setp @ (extended_eSuc @ N)))))).
% 3.00/1.75 thf(fact_9_enat__unfold__next, axiom, (! [A:$tType,Stop:(A > $o),A2:A,Next:(A > A)]: (((~ ((Stop @ A2))) => (((((coindu1491768222unfold @ A) @ Stop) @ Next) @ A2) = (extended_eSuc @ ((((coindu1491768222unfold @ A) @ Stop) @ Next) @ (Next @ A2)))))))).
% 3.00/1.75 thf(fact_10_co_Oenat_Ocase_I2_J, axiom, (! [A:$tType,F1:A,F2:(extended_enat > A),X2:extended_enat]: ((((((coindu440805660e_enat @ A) @ F1) @ F2) @ (extended_eSuc @ X2)) = (F2 @ X2))))).
% 3.00/1.75 thf(fact_11_enat__cocase__eSuc, axiom, (! [A:$tType,Z:A,S:(extended_enat > A),N:extended_enat]: ((((((coindu440805660e_enat @ A) @ Z) @ S) @ (extended_eSuc @ N)) = (S @ N))))).
% 3.00/1.75 thf(fact_12_eSuc__ile__mono, axiom, (! [N:extended_enat,M:extended_enat]: (((((ord_less_eq @ extended_enat) @ (extended_eSuc @ N)) @ (extended_eSuc @ M)) = (((ord_less_eq @ extended_enat) @ N) @ M))))).
% 3.00/1.75 thf(fact_13_enat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((extended_enat2 @ Nat) = (extended_enat2 @ Nat2)) = (Nat = Nat2))))).
% 3.00/1.75 thf(fact_14_iless__Suc__eq, axiom, (! [M:nat,N:extended_enat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ (extended_eSuc @ N)) = (((ord_less_eq @ extended_enat) @ (extended_enat2 @ M)) @ N))))).
% 3.00/1.75 thf(fact_15_enat__ile, axiom, (! [N:extended_enat,M:nat]: (((((ord_less_eq @ extended_enat) @ N) @ (extended_enat2 @ M)) => (? [K:nat]: ((N = (extended_enat2 @ K)))))))).
% 3.00/1.75 thf(fact_16_Suc__ile__eq, axiom, (! [M:nat,N:extended_enat]: (((((ord_less_eq @ extended_enat) @ (extended_enat2 @ (suc @ M))) @ N) = (((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ N))))).
% 3.00/1.75 thf(fact_17_chain__incr, axiom, (! [A:$tType,Y3:(A > extended_enat),K2:nat]: (((! [I:A]: ((? [J:A]: ((((ord_less @ extended_enat) @ (Y3 @ I)) @ (Y3 @ J)))))) => (? [J2:A]: ((((ord_less @ extended_enat) @ (extended_enat2 @ K2)) @ (Y3 @ J2)))))))).
% 3.00/1.75 thf(fact_18_enat__iless, axiom, (! [N:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N) @ (extended_enat2 @ M)) => (? [K:nat]: ((N = (extended_enat2 @ K)))))))).
% 3.00/1.75 thf(fact_19_enat__less__induct, axiom, (! [P:(extended_enat > $o),N:extended_enat]: (((! [N3:extended_enat]: (((! [M2:extended_enat]: (((((ord_less @ extended_enat) @ M2) @ N3) => (P @ M2)))) => (P @ N3)))) => (P @ N))))).
% 3.00/1.75 thf(fact_20_enat__less__imp__le, axiom, (! [N:extended_enat,M:extended_enat]: (((! [K:nat]: (((((ord_less @ extended_enat) @ N) @ (extended_enat2 @ K)) => (((ord_less @ extended_enat) @ M) @ (extended_enat2 @ K))))) => (((ord_less_eq @ extended_enat) @ M) @ N))))).
% 3.00/1.75 thf(fact_21_enat__setp__enat__set__eq, axiom, (coindu530039314t_setp = (^ [X3:extended_enat]: ((((member @ extended_enat) @ X3) @ coinductive_enat_set))))).
% 3.00/1.75 thf(fact_22_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B:A,A2:A]: (((! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((P @ A3) @ B2)))) => ((((P @ B) @ A2) => ((P @ A2) @ B)) => ((P @ A2) @ B))))))))).
% 3.00/1.75 thf(fact_23_ileI1, axiom, (! [M:extended_enat,N:extended_enat]: (((((ord_less @ extended_enat) @ M) @ N) => (((ord_less_eq @ extended_enat) @ (extended_eSuc @ M)) @ N))))).
% 3.00/1.75 thf(fact_24_eSuc__le__iff, axiom, (! [X:extended_enat,Y:extended_enat]: (((((ord_less_eq @ extended_enat) @ (extended_eSuc @ X)) @ Y) = (? [Y4:extended_enat]: (((Y = (extended_eSuc @ Y4)) & (((ord_less_eq @ extended_enat) @ X) @ Y4)))))))).
% 3.00/1.75 thf(fact_25_ile__eSuc, axiom, (! [N:extended_enat]: ((((ord_less_eq @ extended_enat) @ N) @ (extended_eSuc @ N))))).
% 3.00/1.75 thf(fact_26_nat_Oinject, axiom, (! [X2:nat,Y2:nat]: ((((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))).
% 3.00/1.75 thf(fact_27_old_Onat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))).
% 3.00/1.75 thf(fact_28_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 3.00/1.75 thf(fact_29_lift__Suc__mono__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less @ nat) @ N) @ N4) => (((ord_less @ A) @ (F @ N)) @ (F @ N4)))))))))).
% 3.00/1.75 thf(fact_30_lift__Suc__mono__less__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N:nat,M:nat]: (((! [N3:nat]: ((((ord_less @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less @ A) @ (F @ N)) @ (F @ M)) = (((ord_less @ nat) @ N) @ M))))))))).
% 3.00/1.75 thf(fact_31_lift__Suc__mono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F @ N3)) @ (F @ (suc @ N3))))) => ((((ord_less_eq @ nat) @ N) @ N4) => (((ord_less_eq @ A) @ (F @ N)) @ (F @ N4)))))))))).
% 3.00/1.75 thf(fact_32_lift__Suc__antimono__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),N:nat,N4:nat]: (((! [N3:nat]: ((((ord_less_eq @ A) @ (F @ (suc @ N3))) @ (F @ N3)))) => ((((ord_less_eq @ nat) @ N) @ N4) => (((ord_less_eq @ A) @ (F @ N4)) @ (F @ N)))))))))).
% 3.00/1.75 thf(fact_33_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (~ ((((ord_less_eq @ A) @ T2) @ X4)))))))))))))).
% 3.00/1.75 thf(fact_34_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (((ord_less_eq @ A) @ X4) @ T2)))))))))))).
% 3.00/1.75 thf(fact_35_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((ord_less_eq @ A) @ T2) @ X4)))))))))))).
% 3.00/1.75 thf(fact_36_Suc__less__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_37_Suc__mono, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ (suc @ M)) @ (suc @ N)))))).
% 3.00/1.75 thf(fact_38_lessI, axiom, (! [N:nat]: ((((ord_less @ nat) @ N) @ (suc @ N))))).
% 3.00/1.75 thf(fact_39_Suc__le__mono, axiom, (! [N:nat,M:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ (suc @ M)) = (((ord_less_eq @ nat) @ N) @ M))))).
% 3.00/1.75 thf(fact_40_enat__ord__simps_I1_J, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_41_enat__ord__simps_I2_J, axiom, (! [M:nat,N:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_42_Suc__leI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ (suc @ M)) @ N))))).
% 3.00/1.75 thf(fact_43_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 3.00/1.75 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.00/1.75 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 3.00/1.75 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X5:A]: (((P @ X5) = (Q @ X5)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.00/1.75 thf(fact_47_ext, axiom, (! [B3:$tType,A:$tType,F:(A > B3),G:(A > B3)]: (((! [X5:A]: (((F @ X5) = (G @ X5)))) => (F = G))))).
% 3.00/1.75 thf(fact_48_le__trans, axiom, (! [I2:nat,J3:nat,K2:nat]: (((((ord_less_eq @ nat) @ I2) @ J3) => ((((ord_less_eq @ nat) @ J3) @ K2) => (((ord_less_eq @ nat) @ I2) @ K2)))))).
% 3.00/1.75 thf(fact_49_Suc__le__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) = (((ord_less @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_50_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_51_dec__induct, axiom, (! [I2:nat,J3:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I2) @ J3) => ((P @ I2) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I2) @ N3) => ((((ord_less @ nat) @ N3) @ J3) => ((P @ N3) => (P @ (suc @ N3))))))) => (P @ J3))))))).
% 3.00/1.75 thf(fact_52_inc__induct, axiom, (! [I2:nat,J3:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ I2) @ J3) => ((P @ J3) => ((! [N3:nat]: (((((ord_less_eq @ nat) @ I2) @ N3) => ((((ord_less @ nat) @ N3) @ J3) => ((P @ (suc @ N3)) => (P @ N3)))))) => (P @ I2))))))).
% 3.00/1.75 thf(fact_53_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 3.00/1.75 thf(fact_54_nat__less__le, axiom, ((ord_less @ nat) = (^ [M3:nat,N2:nat]: (((((ord_less_eq @ nat) @ M3) @ N2) & (M3 != N2)))))).
% 3.00/1.75 thf(fact_55_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 3.00/1.75 thf(fact_56_Suc__le__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_57_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 3.00/1.75 thf(fact_58_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 3.00/1.75 thf(fact_59_le__less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 3.00/1.75 thf(fact_60_less__Suc__eq__le, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_61_less__eq__Suc__le, axiom, ((ord_less @ nat) = (^ [N2:nat]: (((ord_less_eq @ nat) @ (suc @ N2)))))).
% 3.00/1.75 thf(fact_62_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 3.00/1.75 thf(fact_63_less__not__refl3, axiom, (! [S:nat,T2:nat]: (((((ord_less @ nat) @ S) @ T2) => (S != T2))))).
% 3.00/1.75 thf(fact_64_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ nat) @ (F @ Y5)) @ (F @ X5)) => (P @ Y5)))) => (P @ X5)))) => (P @ A2))))).
% 3.00/1.75 thf(fact_65_le__imp__less__Suc, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 3.00/1.75 thf(fact_66_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_67_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 3.00/1.75 thf(fact_68_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N3) => (P @ M2)))) => (P @ N3)))) => (P @ N))))).
% 3.00/1.75 thf(fact_69_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N3) & (~ ((P @ M2))))))))) => (P @ N))))).
% 3.00/1.75 thf(fact_70_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M3:nat,N2:nat]: (((((ord_less @ nat) @ M3) @ N2) | (M3 = N2)))))).
% 3.00/1.75 thf(fact_71_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_72_linorder__neqE__nat, axiom, (! [X:nat,Y:nat]: (((X != Y) => ((~ ((((ord_less @ nat) @ X) @ Y))) => (((ord_less @ nat) @ Y) @ X)))))).
% 3.00/1.75 thf(fact_73_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 3.00/1.75 thf(fact_74_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ nat) @ (F @ Y5)) @ (F @ X5)) => (P @ Y5)))) => (P @ X5)))) => (P @ A2))))).
% 3.00/1.75 thf(fact_75_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I2:nat,J3:nat]: (((! [I:nat,J2:nat]: (((((ord_less @ nat) @ I) @ J2) => (((ord_less @ nat) @ (F @ I)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I2) @ J3) => (((ord_less_eq @ nat) @ (F @ I2)) @ (F @ J3))))))).
% 3.00/1.75 thf(fact_76_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X5:A]: (((~ ((P @ X5))) => (? [Y5:A]: (((((ord_less @ nat) @ (V @ Y5)) @ (V @ X5)) & (~ ((P @ Y5))))))))) => (P @ X))))).
% 3.00/1.75 thf(fact_77_not__less__less__Suc__eq, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) = (N = M)))))).
% 3.00/1.75 thf(fact_78_strict__inc__induct, axiom, (! [I2:nat,J3:nat,P:(nat > $o)]: (((((ord_less @ nat) @ I2) @ J3) => ((! [I:nat]: (((J3 = (suc @ I)) => (P @ I)))) => ((! [I:nat]: (((((ord_less @ nat) @ I) @ J3) => ((P @ (suc @ I)) => (P @ I))))) => (P @ I2))))))).
% 3.00/1.75 thf(fact_79_less__Suc__induct, axiom, (! [I2:nat,J3:nat,P:(nat > (nat > $o))]: (((((ord_less @ nat) @ I2) @ J3) => ((! [I:nat]: (((P @ I) @ (suc @ I)))) => ((! [I:nat,J2:nat,K:nat]: (((((ord_less @ nat) @ I) @ J2) => ((((ord_less @ nat) @ J2) @ K) => (((P @ I) @ J2) => (((P @ J2) @ K) => ((P @ I) @ K))))))) => ((P @ I2) @ J3))))))).
% 3.00/1.75 thf(fact_80_less__trans__Suc, axiom, (! [I2:nat,J3:nat,K2:nat]: (((((ord_less @ nat) @ I2) @ J3) => ((((ord_less @ nat) @ J3) @ K2) => (((ord_less @ nat) @ (suc @ I2)) @ K2)))))).
% 3.00/1.75 thf(fact_81_Suc__less__SucD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ (suc @ N)) => (((ord_less @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_82_less__antisym, axiom, (! [N:nat,M:nat]: (((~ ((((ord_less @ nat) @ N) @ M))) => ((((ord_less @ nat) @ N) @ (suc @ M)) => (M = N)))))).
% 3.00/1.75 thf(fact_83_Suc__less__eq2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ (suc @ N)) @ M) = (? [M4:nat]: (((M = (suc @ M4)) & (((ord_less @ nat) @ N) @ M4)))))))).
% 3.00/1.75 thf(fact_84_not__less__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less @ nat) @ M) @ N))) = (((ord_less @ nat) @ N) @ (suc @ M)))))).
% 3.00/1.75 thf(fact_85_less__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) = ((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 3.00/1.75 thf(fact_86_less__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ nat) @ M) @ (suc @ N)))))).
% 3.00/1.75 thf(fact_87_less__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less @ nat) @ M) @ N))) => (M = N)))))).
% 3.00/1.75 thf(fact_88_Suc__lessI, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((suc @ M) != N) => (((ord_less @ nat) @ (suc @ M)) @ N)))))).
% 3.00/1.75 thf(fact_89_Suc__lessE, axiom, (! [I2:nat,K2:nat]: (((((ord_less @ nat) @ (suc @ I2)) @ K2) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (K2 != (suc @ J2))))))))))).
% 3.00/1.75 thf(fact_90_Suc__lessD, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (suc @ M)) @ N) => (((ord_less @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_91_lessE, axiom, (! [I2:nat,K2:nat]: (((((ord_less @ nat) @ I2) @ K2) => ((K2 != (suc @ I2)) => (~ ((! [J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (K2 != (suc @ J2)))))))))))).
% 3.00/1.75 thf(fact_92_full__nat__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M2:nat]: (((((ord_less_eq @ nat) @ (suc @ M2)) @ N3) => (P @ M2)))) => (P @ N3)))) => (P @ N))))).
% 3.00/1.75 thf(fact_93_not__less__eq__eq, axiom, (! [M:nat,N:nat]: (((~ ((((ord_less_eq @ nat) @ M) @ N))) = (((ord_less_eq @ nat) @ (suc @ N)) @ M))))).
% 3.00/1.75 thf(fact_94_Suc__n__not__le__n, axiom, (! [N:nat]: ((~ ((((ord_less_eq @ nat) @ (suc @ N)) @ N)))))).
% 3.00/1.75 thf(fact_95_le__Suc__eq, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) = ((((ord_less_eq @ nat) @ M) @ N) | (M = (suc @ N))))))).
% 3.00/1.75 thf(fact_96_Suc__le__D, axiom, (! [N:nat,M5:nat]: (((((ord_less_eq @ nat) @ (suc @ N)) @ M5) => (? [M6:nat]: ((M5 = (suc @ M6)))))))).
% 3.00/1.75 thf(fact_97_le__SucI, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ (suc @ N)))))).
% 3.00/1.75 thf(fact_98_le__SucE, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ (suc @ N)) => ((~ ((((ord_less_eq @ nat) @ M) @ N))) => (M = (suc @ N))))))).
% 3.00/1.75 thf(fact_99_Suc__leD, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ (suc @ M)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_100_less__enatE, axiom, (! [N:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N) @ (extended_enat2 @ M)) => (~ ((! [K:nat]: (((N = (extended_enat2 @ K)) => (~ ((((ord_less @ nat) @ K) @ M)))))))))))).
% 3.00/1.75 thf(fact_101_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ A2) @ B) => (A2 = B))))))))).
% 3.00/1.75 thf(fact_102_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 3.00/1.75 thf(fact_103_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B:A]: (((! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((P @ A3) @ B2)))) => ((! [A3:A,B2:A]: ((((P @ B2) @ A3) => ((P @ A3) @ B2)))) => ((P @ A2) @ B))))))))).
% 3.00/1.75 thf(fact_104_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 3.00/1.75 thf(fact_105_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 3.00/1.75 thf(fact_106_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ A2) => (A2 = B))))))))).
% 3.00/1.75 thf(fact_107_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((B = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_108_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_109_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 3.00/1.75 thf(fact_110_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 3.00/1.75 thf(fact_111_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_112_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.00/1.75 thf(fact_113_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.00/1.75 thf(fact_114_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.00/1.75 thf(fact_115_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 3.00/1.75 thf(fact_116_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y6:A,Z3:A]: ((Y6 = Z3))) = (^ [X3:A,Y7:A]: (((((ord_less_eq @ A) @ X3) @ Y7) & (((ord_less_eq @ A) @ Y7) @ X3))))))))).
% 3.00/1.75 thf(fact_117_ord__le__eq__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > B3),C:B3]: (((((ord_less_eq @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X5:A,Y8:A]: (((((ord_less_eq @ A) @ X5) @ Y8) => (((ord_less_eq @ B3) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less_eq @ B3) @ (F @ A2)) @ C)))))))))).
% 3.00/1.75 thf(fact_118_ord__eq__le__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((A2 = (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X5:B3,Y8:B3]: (((((ord_less_eq @ B3) @ X5) @ Y8) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 3.00/1.75 thf(fact_119_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X5:A,Y8:A]: (((((ord_less_eq @ A) @ X5) @ Y8) => (((ord_less_eq @ C2) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 3.00/1.75 thf(fact_120_order__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X5:B3,Y8:B3]: (((((ord_less_eq @ B3) @ X5) @ Y8) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 3.00/1.75 thf(fact_121_le__fun__def, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => ((ord_less_eq @ (A > B3)) = (^ [F3:(A > B3),G2:(A > B3)]: ((! [X3:A]: ((((ord_less_eq @ B3) @ (F3 @ X3)) @ (G2 @ X3))))))))))).
% 3.00/1.75 thf(fact_122_le__funI, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G:(A > B3)]: (((! [X5:A]: ((((ord_less_eq @ B3) @ (F @ X5)) @ (G @ X5)))) => (((ord_less_eq @ (A > B3)) @ F) @ G)))))))).
% 3.00/1.75 thf(fact_123_le__funE, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G:(A > B3),X:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X)) @ (G @ X))))))))).
% 3.00/1.75 thf(fact_124_le__funD, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => (! [F:(A > B3),G:(A > B3),X:A]: (((((ord_less_eq @ (A > B3)) @ F) @ G) => (((ord_less_eq @ B3) @ (F @ X)) @ (G @ X))))))))).
% 3.00/1.75 thf(fact_125_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (A2 != B)))))))).
% 3.00/1.75 thf(fact_126_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (A2 != B)))))))).
% 3.00/1.75 thf(fact_127_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 3.00/1.75 thf(fact_128_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 3.00/1.75 thf(fact_129_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.00/1.75 thf(fact_130_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_131_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 3.00/1.75 thf(fact_132_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.00/1.75 thf(fact_133_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 3.00/1.75 thf(fact_134_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 3.00/1.75 thf(fact_135_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 3.00/1.75 thf(fact_136_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X5:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X5) => (P @ Y5)))) => (P @ X5)))) => (P @ A2)))))))).
% 3.00/1.75 thf(fact_137_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.00/1.75 thf(fact_138_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 3.00/1.75 thf(fact_139_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (~ ((((ord_less @ A) @ A2) @ B)))))))))).
% 3.00/1.75 thf(fact_140_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((B = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_141_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((A2 = B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_142_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 3.00/1.75 thf(fact_143_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 3.00/1.75 thf(fact_144_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 3.00/1.75 thf(fact_145_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 3.00/1.75 thf(fact_146_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 3.00/1.75 thf(fact_147_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 3.00/1.75 thf(fact_148_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z2:A]: (((((ord_less @ A) @ X) @ Z2) & (((ord_less @ A) @ Z2) @ Y))))))))))).
% 3.00/1.75 thf(fact_149_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (~ ((((ord_less @ A) @ B) @ A2)))))))))).
% 3.00/1.75 thf(fact_150_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 3.00/1.75 thf(fact_151_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.00/1.75 thf(fact_152_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 3.00/1.75 thf(fact_153_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y8:A]: ((((ord_less @ A) @ Y8) @ X))))))))).
% 3.00/1.75 thf(fact_154_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X5:A,Y8:A]: (((((ord_less @ A) @ X5) @ Y8) => (((ord_less @ C2) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 3.00/1.75 thf(fact_155_order__less__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less @ B3) @ B) @ C) => ((! [X5:B3,Y8:B3]: (((((ord_less @ B3) @ X5) @ Y8) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 3.00/1.75 thf(fact_156_ord__less__eq__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > B3),C:B3]: (((((ord_less @ A) @ A2) @ B) => (((F @ B) = C) => ((! [X5:A,Y8:A]: (((((ord_less @ A) @ X5) @ Y8) => (((ord_less @ B3) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ B3) @ (F @ A2)) @ C)))))))))).
% 3.00/1.75 thf(fact_157_ord__eq__less__subst, axiom, (! [A:$tType,B3:$tType]: (((((ord @ B3) @ (type2 @ B3)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((A2 = (F @ B)) => ((((ord_less @ B3) @ B) @ C) => ((! [X5:B3,Y8:B3]: (((((ord_less @ B3) @ X5) @ Y8) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 3.00/1.75 thf(fact_158_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 3.00/1.75 thf(fact_159_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ Z4) @ X5) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 3.00/1.75 thf(fact_160_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (X4 != T2)))))))))))).
% 3.00/1.75 thf(fact_161_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (X4 != T2)))))))))))).
% 3.00/1.75 thf(fact_162_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (~ ((((ord_less @ A) @ X4) @ T2)))))))))))))).
% 3.00/1.75 thf(fact_163_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (((ord_less @ A) @ T2) @ X4)))))))))))).
% 3.00/1.75 thf(fact_164_pinf_I11_J, axiom, (! [C2:$tType,D:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F4:D]: ((? [Z2:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ Z2) @ X4) => (F4 = F4)))))))))))).
% 3.00/1.75 thf(fact_165_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (((P @ X4) & (Q @ X4)) = ((P2 @ X4) & (Q2 @ X4)))))))))))))))).
% 3.00/1.75 thf(fact_166_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((P @ X5) = (P2 @ X5))))))) => ((? [Z4:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z4) => ((Q @ X5) = (Q2 @ X5))))))) => (? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (((P @ X4) | (Q @ X4)) = ((P2 @ X4) | (Q2 @ X4)))))))))))))))).
% 3.00/1.75 thf(fact_167_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (X4 != T2)))))))))))).
% 3.00/1.75 thf(fact_168_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (X4 != T2)))))))))))).
% 3.00/1.75 thf(fact_169_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (((ord_less @ A) @ X4) @ T2)))))))))))).
% 3.00/1.75 thf(fact_170_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z2) => (~ ((((ord_less @ A) @ T2) @ X4)))))))))))))).
% 3.00/1.75 thf(fact_171_minf_I11_J, axiom, (! [C2:$tType,D:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F4:D]: ((? [Z2:C2]: ((! [X4:C2]: (((((ord_less @ C2) @ X4) @ Z2) => (F4 = F4)))))))))))).
% 3.00/1.75 thf(fact_172_n__not__Suc__n, axiom, (! [N:nat]: ((N != (suc @ N))))).
% 3.00/1.75 thf(fact_173_Suc__inject, axiom, (! [X:nat,Y:nat]: ((((suc @ X) = (suc @ Y)) => (X = Y))))).
% 3.00/1.75 thf(fact_174_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((A2 != B) => ((((ord_less_eq @ A) @ A2) @ B) => (((ord_less @ A) @ A2) @ B))))))))).
% 3.00/1.75 thf(fact_175_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A]: (((((ord_less @ A) @ B) @ A2) => (((ord_less_eq @ A) @ B) @ A2)))))))).
% 3.00/1.75 thf(fact_176_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B4:A,A5:A]: (((((ord_less_eq @ A) @ B4) @ A5) & (A5 != B4))))))))).
% 3.00/1.75 thf(fact_177_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B4:A,A5:A]: (((((ord_less @ A) @ B4) @ A5) | (A5 = B4))))))))).
% 3.00/1.75 thf(fact_178_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less @ A) @ A2) @ B) => (((ord_less_eq @ A) @ A2) @ B)))))))).
% 3.00/1.75 thf(fact_179_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 3.00/1.75 thf(fact_180_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,X:A,Y:A]: (((((ord_less @ A) @ Z) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 3.00/1.75 thf(fact_181_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: (((((ord_less @ A) @ B) @ A2) => ((((ord_less_eq @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 3.00/1.75 thf(fact_182_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B) @ A2) => ((((ord_less @ A) @ C) @ B) => (((ord_less @ A) @ C) @ A2))))))))).
% 3.00/1.75 thf(fact_183_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A5:A,B4:A]: (((((ord_less_eq @ A) @ A5) @ B4) & (A5 != B4))))))))).
% 3.00/1.75 thf(fact_184_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A5:A,B4:A]: (((((ord_less @ A) @ A5) @ B4) | (A5 = B4))))))))).
% 3.00/1.75 thf(fact_185_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_186_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ A) @ B) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 3.00/1.75 thf(fact_187_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 3.00/1.75 thf(fact_188_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y7:A]: (((((ord_less_eq @ A) @ X3) @ Y7) & (~ ((((ord_less_eq @ A) @ Y7) @ X3))))))))))).
% 3.00/1.75 thf(fact_189_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 3.00/1.75 thf(fact_190_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 3.00/1.75 thf(fact_191_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y:A,Z:A]: (((! [X5:A]: (((((ord_less @ A) @ X5) @ Y) => (((ord_less_eq @ A) @ X5) @ Z)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 3.00/1.75 thf(fact_192_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,Y:A]: (((! [X5:A]: (((((ord_less @ A) @ Z) @ X5) => (((ord_less_eq @ A) @ Y) @ X5)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 3.00/1.75 thf(fact_193_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 3.00/1.75 thf(fact_194_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 3.00/1.75 thf(fact_195_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 3.00/1.75 thf(fact_196_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 3.00/1.75 thf(fact_197_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 3.00/1.75 thf(fact_198_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B:A]: (((((ord_less_eq @ A) @ A2) @ B) => ((A2 != B) => (((ord_less @ A) @ A2) @ B))))))))).
% 3.00/1.75 thf(fact_199_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.00/1.75 thf(fact_200_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 3.00/1.75 thf(fact_201_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B) => ((((ord_less_eq @ C2) @ (F @ B)) @ C) => ((! [X5:A,Y8:A]: (((((ord_less @ A) @ X5) @ Y8) => (((ord_less @ C2) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 3.00/1.75 thf(fact_202_order__less__le__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less @ A) @ A2) @ (F @ B)) => ((((ord_less_eq @ B3) @ B) @ C) => ((! [X5:B3,Y8:B3]: (((((ord_less_eq @ B3) @ X5) @ Y8) => (((ord_less_eq @ A) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 3.00/1.75 thf(fact_203_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B) => ((((ord_less @ C2) @ (F @ B)) @ C) => ((! [X5:A,Y8:A]: (((((ord_less_eq @ A) @ X5) @ Y8) => (((ord_less_eq @ C2) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 3.00/1.75 thf(fact_204_order__le__less__subst1, axiom, (! [A:$tType,B3:$tType]: (((((order @ B3) @ (type2 @ B3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B3 > A),B:B3,C:B3]: (((((ord_less_eq @ A) @ A2) @ (F @ B)) => ((((ord_less @ B3) @ B) @ C) => ((! [X5:B3,Y8:B3]: (((((ord_less @ B3) @ X5) @ Y8) => (((ord_less @ A) @ (F @ X5)) @ (F @ Y8))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 3.00/1.75 thf(fact_205_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y7:A]: (((((ord_less_eq @ A) @ X3) @ Y7) & (X3 != Y7))))))))).
% 3.00/1.75 thf(fact_206_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y7:A]: (((((ord_less @ A) @ X3) @ Y7) | (X3 = Y7))))))))).
% 3.00/1.75 thf(fact_207_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 3.00/1.75 thf(fact_208_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 3.00/1.75 thf(fact_209_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T2:A]: ((? [Z2:A]: ((! [X4:A]: (((((ord_less @ A) @ Z2) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ T2)))))))))))))).
% 3.00/1.75 thf(fact_210_ex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K2:A,M:(A > nat),B:nat]: (((P @ K2) => ((! [Y8:A]: (((P @ Y8) => (((ord_less @ nat) @ (M @ Y8)) @ B)))) => (? [X5:A]: (((P @ X5) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M @ Y5)) @ (M @ X5))))))))))))).
% 3.00/1.75 thf(fact_211_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B) => ((P @ A2) => ((~ ((P @ B))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B) & ((! [X4:A]: ((((((ord_less_eq @ A) @ A2) @ X4) & (((ord_less @ A) @ X4) @ C3)) => (P @ X4)))) & (! [D2:A]: (((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less @ A) @ X5) @ D2)) => (P @ X5)))) => (((ord_less_eq @ A) @ D2) @ C3)))))))))))))))))).
% 3.00/1.75 thf(fact_212_enat_Osimps_I6_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_rec_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 3.00/1.75 thf(fact_213_enat_Osimps_I4_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_case_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 3.00/1.75 thf(fact_214_Suc_Ohyps, axiom, (! [N:extended_enat,Xs:(coindu1593790203_llist @ a)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ ma)) @ N) => ((((coindu749330388e_lnth @ a) @ (((coindu1802687541_ltake @ a) @ N) @ Xs)) @ ma) = (((coindu749330388e_lnth @ a) @ Xs) @ ma)))))).
% 3.00/1.75 thf(fact_215_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K2:A,M:(A > nat)]: (((P @ K2) => (? [X5:A]: (((P @ X5) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M @ X5)) @ (M @ Y5)))))))))))).
% 3.00/1.75 thf(fact_216_enat__less__enat__plusI, axiom, (! [X:nat,Y:nat,Z:extended_enat]: (((((ord_less @ nat) @ X) @ Y) => (((ord_less @ extended_enat) @ (extended_enat2 @ X)) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ Y)) @ Z)))))).
% 3.00/1.75 thf(fact_217_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B2:A]: (((((ord_less @ A) @ A2) @ B2) | (((ord_less @ A) @ B2) @ A2)))))))))).
% 3.00/1.75 thf(fact_218_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 3.00/1.75 thf(fact_219_enat__add1__eq, axiom, (! [X:nat,Y:extended_enat,Z:extended_enat]: ((((((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Y) = (((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Z)) = (Y = Z))))).
% 3.00/1.75 thf(fact_220_enat__add2__eq, axiom, (! [Y:extended_enat,X:nat,Z:extended_enat]: ((((((plus_plus @ extended_enat) @ Y) @ (extended_enat2 @ X)) = (((plus_plus @ extended_enat) @ Z) @ (extended_enat2 @ X))) = (Y = Z))))).
% 3.00/1.75 thf(fact_221_enat__add__mono, axiom, (! [X:nat,Y:extended_enat,Z:extended_enat]: (((((ord_less @ extended_enat) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Y)) @ (((plus_plus @ extended_enat) @ (extended_enat2 @ X)) @ Z)) = (((ord_less @ extended_enat) @ Y) @ Z))))).
% 3.00/1.75 thf(fact_222_less__fun__def, axiom, (! [B3:$tType,A:$tType]: ((((ord @ B3) @ (type2 @ B3)) => ((ord_less @ (A > B3)) = (^ [F3:(A > B3),G2:(A > B3)]: (((((ord_less_eq @ (A > B3)) @ F3) @ G2) & (~ ((((ord_less_eq @ (A > B3)) @ G2) @ F3))))))))))).
% 3.00/1.75 thf(fact_223_ltake__eq__ltake__antimono, axiom, (! [A:$tType,N:extended_enat,Xs:(coindu1593790203_llist @ A),Ys:(coindu1593790203_llist @ A),M:extended_enat]: ((((((coindu1802687541_ltake @ A) @ N) @ Xs) = (((coindu1802687541_ltake @ A) @ N) @ Ys)) => ((((ord_less_eq @ extended_enat) @ M) @ N) => ((((coindu1802687541_ltake @ A) @ M) @ Xs) = (((coindu1802687541_ltake @ A) @ M) @ Ys))))))).
% 3.00/1.75 thf(fact_224_enat__le__plus__same_I2_J, axiom, (! [X:extended_enat,Y:extended_enat]: ((((ord_less_eq @ extended_enat) @ X) @ (((plus_plus @ extended_enat) @ Y) @ X))))).
% 3.00/1.75 thf(fact_225_enat__le__plus__same_I1_J, axiom, (! [X:extended_enat,Y:extended_enat]: ((((ord_less_eq @ extended_enat) @ X) @ (((plus_plus @ extended_enat) @ X) @ Y))))).
% 3.00/1.75 thf(fact_226_iadd__Suc__right, axiom, (! [M:extended_enat,N:extended_enat]: (((((plus_plus @ extended_enat) @ M) @ (extended_eSuc @ N)) = (extended_eSuc @ (((plus_plus @ extended_enat) @ M) @ N)))))).
% 3.00/1.75 thf(fact_227_eSuc__plus, axiom, (! [M:extended_enat,N:extended_enat]: (((((plus_plus @ extended_enat) @ (extended_eSuc @ M)) @ N) = (extended_eSuc @ (((plus_plus @ extended_enat) @ M) @ N)))))).
% 3.00/1.75 thf(fact_228_add__less__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less @ A) @ A2) @ B)))))))).
% 3.00/1.75 thf(fact_229_add__less__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less @ A) @ A2) @ B)))))))).
% 3.00/1.75 thf(fact_230_add__right__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [B:A,A2:A,C:A]: ((((((plus_plus @ A) @ B) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B = C)))))))).
% 3.00/1.75 thf(fact_231_add__left__cancel, axiom, (! [A:$tType]: ((((cancel_semigroup_add @ A) @ (type2 @ A)) => (! [A2:A,B:A,C:A]: ((((((plus_plus @ A) @ A2) @ B) = (((plus_plus @ A) @ A2) @ C)) = (B = C)))))))).
% 3.00/1.75 thf(fact_232_add__Suc__right, axiom, (! [M:nat,N:nat]: (((((plus_plus @ nat) @ M) @ (suc @ N)) = (suc @ (((plus_plus @ nat) @ M) @ N)))))).
% 3.00/1.75 thf(fact_233_nat__add__left__cancel__less, axiom, (! [K2:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K2) @ M)) @ (((plus_plus @ nat) @ K2) @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_234_nat__add__left__cancel__le, axiom, (! [K2:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K2) @ M)) @ (((plus_plus @ nat) @ K2) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 3.00/1.75 thf(fact_235_add__le__cancel__right, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [A2:A,C:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B) @ C)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 3.00/1.75 thf(fact_236_add__le__cancel__left, axiom, (! [A:$tType]: ((((ordere236663937imp_le @ A) @ (type2 @ A)) => (! [C:A,A2:A,B:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B)) = (((ord_less_eq @ A) @ A2) @ B)))))))).
% 3.00/1.75 thf(fact_237_plus__enat__simps_I1_J, axiom, (! [M:nat,N:nat]: (((((plus_plus @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (extended_enat2 @ (((plus_plus @ nat) @ M) @ N)))))).
% 3.00/1.75 thf(fact_238_add__Suc__shift, axiom, (! [M:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M)) @ N) = (((plus_plus @ nat) @ M) @ (suc @ N)))))).
% 3.00/1.75 thf(fact_239_add__Suc, axiom, (! [M:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M)) @ N) = (suc @ (((plus_plus @ nat) @ M) @ N)))))).
% 3.00/1.75 thf(fact_240_less__add__eq__less, axiom, (! [K2:nat,L:nat,M:nat,N:nat]: (((((ord_less @ nat) @ K2) @ L) => (((((plus_plus @ nat) @ M) @ L) = (((plus_plus @ nat) @ K2) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 3.00/1.75 thf(fact_241_trans__less__add2, axiom, (! [I2:nat,J3:nat,M:nat]: (((((ord_less @ nat) @ I2) @ J3) => (((ord_less @ nat) @ I2) @ (((plus_plus @ nat) @ M) @ J3)))))).
% 3.00/1.75 thf(fact_242_trans__less__add1, axiom, (! [I2:nat,J3:nat,M:nat]: (((((ord_less @ nat) @ I2) @ J3) => (((ord_less @ nat) @ I2) @ (((plus_plus @ nat) @ J3) @ M)))))).
% 3.00/1.75 thf(fact_243_add__less__mono1, axiom, (! [I2:nat,J3:nat,K2:nat]: (((((ord_less @ nat) @ I2) @ J3) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ K2)) @ (((plus_plus @ nat) @ J3) @ K2)))))).
% 3.00/1.75 thf(fact_244_not__add__less2, axiom, (! [J3:nat,I2:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J3) @ I2)) @ I2)))))).
% 3.00/1.75 thf(fact_245_not__add__less1, axiom, (! [I2:nat,J3:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ J3)) @ I2)))))).
% 3.00/1.75 thf(fact_246_add__less__mono, axiom, (! [I2:nat,J3:nat,K2:nat,L:nat]: (((((ord_less @ nat) @ I2) @ J3) => ((((ord_less @ nat) @ K2) @ L) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ K2)) @ (((plus_plus @ nat) @ J3) @ L))))))).
% 3.00/1.75 thf(fact_247_add__lessD1, axiom, (! [I2:nat,J3:nat,K2:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I2) @ J3)) @ K2) => (((ord_less @ nat) @ I2) @ K2))))).
% 3.00/1.75 thf(fact_248_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M3:nat,N2:nat]: ((? [K3:nat]: ((N2 = (((plus_plus @ nat) @ M3) @ K3)))))))).
% 3.00/1.75 thf(fact_249_trans__le__add2, axiom, (! [I2:nat,J3:nat,M:nat]: (((((ord_less_eq @ nat) @ I2) @ J3) => (((ord_less_eq @ nat) @ I2) @ (((plus_plus @ nat) @ M) @ J3)))))).
% 3.00/1.75 thf(fact_250_trans__le__add1, axiom, (! [I2:nat,J3:nat,M:nat]: (((((ord_less_eq @ nat) @ I2) @ J3) => (((ord_less_eq @ nat) @ I2) @ (((plus_plus @ nat) @ J3) @ M)))))).
% 3.00/1.75 thf(fact_251_add__le__mono1, axiom, (! [I2:nat,J3:nat,K2:nat]: (((((ord_less_eq @ nat) @ I2) @ J3) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I2) @ K2)) @ (((plus_plus @ nat) @ J3) @ K2)))))).
% 3.00/1.75 thf(fact_252_add__le__mono, axiom, (! [I2:nat,J3:nat,K2:nat,L:nat]: (((((ord_less_eq @ nat) @ I2) @ J3) => ((((ord_less_eq @ nat) @ K2) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I2) @ K2)) @ (((plus_plus @ nat) @ J3) @ L))))))).
% 3.00/1.75 thf(fact_253_le__Suc__ex, axiom, (! [K2:nat,L:nat]: (((((ord_less_eq @ nat) @ K2) @ L) => (? [N3:nat]: ((L = (((plus_plus @ nat) @ K2) @ N3)))))))).
% 3.00/1.75 thf(fact_254_add__leD2, axiom, (! [M:nat,K2:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K2)) @ N) => (((ord_less_eq @ nat) @ K2) @ N))))).
% 3.00/1.75 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.00/1.75 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.00/1.75 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 3.00/1.75 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, ((ordere236663937imp_le @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, ((cancel_semigroup_add @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 3.00/1.75 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A6:$tType]: (((preorder @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.00/1.75 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.00/1.75 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 3.00/1.75 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 3.00/1.75 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 3.00/1.75 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, ((order @ $o) @ (type2 @ $o))).
% 3.00/1.75 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, ((ord @ $o) @ (type2 @ $o))).
% 3.00/1.75 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11, axiom, ((condit1037483654norder @ extended_enat) @ (type2 @ extended_enat))).
% 3.00/1.75 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_12, axiom, ((wellorder @ extended_enat) @ (type2 @ extended_enat))).
% 3.00/1.75 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_13, axiom, ((preorder @ extended_enat) @ (type2 @ extended_enat))).
% 3.00/1.75 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_14, axiom, ((linorder @ extended_enat) @ (type2 @ extended_enat))).
% 3.00/1.75 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_15, axiom, ((order @ extended_enat) @ (type2 @ extended_enat))).
% 3.00/1.75 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_16, axiom, ((ord @ extended_enat) @ (type2 @ extended_enat))).
% 3.00/1.75 thf(conj_0, hypothesis, (! [N5:extended_enat]: (((na = (extended_eSuc @ N5)) => thesis)))).
% 3.00/1.75 thf(conj_1, conjecture, thesis).
% 3.00/1.75 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2DTF30566.p
% 3.00/1.75 ------------------------
% 3.00/1.79 ---- Cleaned THF ---
% 3.00/1.79 thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
% 3.00/1.79 coindu1593790203_llist: $tType > $tType ).
% 3.00/1.79
% 3.00/1.79 thf(ty_t_Extended__Nat_Oenat,type,
% 3.00/1.79 extended_enat: $tType ).
% 3.00/1.79
% 3.00/1.79 thf(ty_t_Set_Oset,type,
% 3.00/1.79 set: $tType > $tType ).
% 3.00/1.79
% 3.00/1.79 thf(ty_t_Nat_Onat,type,
% 3.00/1.79 nat: $tType ).
% 3.00/1.79
% 3.00/1.79 thf(ty_t_itself,type,
% 3.00/1.79 itself: $tType > $tType ).
% 3.00/1.79
% 3.00/1.79 thf(ty_tf_a,type,
% 3.00/1.79 a: $tType ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_HOL_Otype,type,
% 3.00/1.79 type:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Oord,type,
% 3.00/1.79 ord:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Oorder,type,
% 3.00/1.79 order:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Ono__bot,type,
% 3.00/1.79 no_bot:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Ono__top,type,
% 3.00/1.79 no_top:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Olinorder,type,
% 3.00/1.79 linorder:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Opreorder,type,
% 3.00/1.79 preorder:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Owellorder,type,
% 3.00/1.79 wellorder:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Odense__order,type,
% 3.00/1.79 dense_order:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Rings_Olinordered__idom,type,
% 3.00/1.79 linordered_idom:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Orderings_Odense__linorder,type,
% 3.00/1.79 dense_linorder:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 3.00/1.79 cancel_semigroup_add:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 3.00/1.79 ordere236663937imp_le:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 3.00/1.79 condit1656338222tinuum:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 3.00/1.79 condit1037483654norder:
% 3.00/1.79 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
% 3.00/1.79 coindu749330388e_lnth:
% 3.00/1.79 !>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
% 3.00/1.79 coindu1802687541_ltake:
% 3.00/1.79 !>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Coinductive__Nat_Oco_Oenat_Ocase__enat,type,
% 3.00/1.79 coindu440805660e_enat:
% 3.00/1.79 !>[A: $tType] : ( A > ( extended_enat > A ) > extended_enat > A ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Coinductive__Nat_Oenat__set,type,
% 3.00/1.79 coinductive_enat_set: set @ extended_enat ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Coinductive__Nat_Oenat__setp,type,
% 3.00/1.79 coindu530039314t_setp: extended_enat > $o ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
% 3.00/1.79 coindu1491768222unfold:
% 3.00/1.79 !>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Extended__Nat_OeSuc,type,
% 3.00/1.79 extended_eSuc: extended_enat > extended_enat ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Extended__Nat_Oenat,type,
% 3.00/1.79 extended_enat2: nat > extended_enat ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 3.00/1.79 extended_case_enat:
% 3.00/1.79 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 3.00/1.79 extended_rec_enat:
% 3.00/1.79 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 3.00/1.79 plus_plus:
% 3.00/1.79 !>[A: $tType] : ( A > A > A ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Nat_OSuc,type,
% 3.00/1.79 suc: nat > nat ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.00/1.79 ord_less:
% 3.00/1.79 !>[A: $tType] : ( A > A > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.00/1.79 ord_less_eq:
% 3.00/1.79 !>[A: $tType] : ( A > A > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Pure_Otype,type,
% 3.00/1.79 type2:
% 3.00/1.79 !>[A: $tType] : ( itself @ A ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_Set_OCollect,type,
% 3.00/1.79 collect:
% 3.00/1.79 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_c_member,type,
% 3.00/1.79 member:
% 3.00/1.79 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(sy_v_ma____,type,
% 3.00/1.79 ma: nat ).
% 3.00/1.79
% 3.00/1.79 thf(sy_v_na____,type,
% 3.00/1.79 na: extended_enat ).
% 3.00/1.79
% 3.00/1.79 thf(sy_v_thesis____,type,
% 3.00/1.79 thesis: $o ).
% 3.00/1.79
% 3.00/1.79 thf(fact_0_Suc_Oprems,axiom,
% 3.00/1.79 ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ ma ) ) @ na ).
% 3.00/1.79
% 3.00/1.79 thf(fact_1_eSuc__inject,axiom,
% 3.00/1.79 ! [M: extended_enat,N: extended_enat] :
% 3.00/1.79 ( ( ( extended_eSuc @ M )
% 3.00/1.79 = ( extended_eSuc @ N ) )
% 3.00/1.79 = ( M = N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_2_co_Oenat_Oinject,axiom,
% 3.00/1.79 ! [X2: extended_enat,Y2: extended_enat] :
% 3.00/1.79 ( ( ( extended_eSuc @ X2 )
% 3.00/1.79 = ( extended_eSuc @ Y2 ) )
% 3.00/1.79 = ( X2 = Y2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_3_Extended__Nat_OeSuc__mono,axiom,
% 3.00/1.79 ! [N: extended_enat,M: extended_enat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 3.00/1.79 = ( ord_less @ extended_enat @ N @ M ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_4_eSuc__enat,axiom,
% 3.00/1.79 ! [N: nat] :
% 3.00/1.79 ( ( extended_eSuc @ ( extended_enat2 @ N ) )
% 3.00/1.79 = ( extended_enat2 @ ( suc @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_5_eSuc__enat__iff,axiom,
% 3.00/1.79 ! [X: extended_enat,Y: nat] :
% 3.00/1.79 ( ( ( extended_eSuc @ X )
% 3.00/1.79 = ( extended_enat2 @ Y ) )
% 3.00/1.79 = ( ? [N2: nat] :
% 3.00/1.79 ( ( Y
% 3.00/1.79 = ( suc @ N2 ) )
% 3.00/1.79 & ( X
% 3.00/1.79 = ( extended_enat2 @ N2 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_6_enat__eSuc__iff,axiom,
% 3.00/1.79 ! [Y: nat,X: extended_enat] :
% 3.00/1.79 ( ( ( extended_enat2 @ Y )
% 3.00/1.79 = ( extended_eSuc @ X ) )
% 3.00/1.79 = ( ? [N2: nat] :
% 3.00/1.79 ( ( Y
% 3.00/1.79 = ( suc @ N2 ) )
% 3.00/1.79 & ( ( extended_enat2 @ N2 )
% 3.00/1.79 = X ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_7_enat__set_Ointros_I2_J,axiom,
% 3.00/1.79 ! [N: extended_enat] :
% 3.00/1.79 ( ( member @ extended_enat @ N @ coinductive_enat_set )
% 3.00/1.79 => ( member @ extended_enat @ ( extended_eSuc @ N ) @ coinductive_enat_set ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_8_enat__setp_Ointros_I2_J,axiom,
% 3.00/1.79 ! [N: extended_enat] :
% 3.00/1.79 ( ( coindu530039314t_setp @ N )
% 3.00/1.79 => ( coindu530039314t_setp @ ( extended_eSuc @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_9_enat__unfold__next,axiom,
% 3.00/1.79 ! [A: $tType,Stop: A > $o,A2: A,Next: A > A] :
% 3.00/1.79 ( ~ ( Stop @ A2 )
% 3.00/1.79 => ( ( coindu1491768222unfold @ A @ Stop @ Next @ A2 )
% 3.00/1.79 = ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A2 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_10_co_Oenat_Ocase_I2_J,axiom,
% 3.00/1.79 ! [A: $tType,F1: A,F2: extended_enat > A,X2: extended_enat] :
% 3.00/1.79 ( ( coindu440805660e_enat @ A @ F1 @ F2 @ ( extended_eSuc @ X2 ) )
% 3.00/1.79 = ( F2 @ X2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_11_enat__cocase__eSuc,axiom,
% 3.00/1.79 ! [A: $tType,Z: A,S: extended_enat > A,N: extended_enat] :
% 3.00/1.79 ( ( coindu440805660e_enat @ A @ Z @ S @ ( extended_eSuc @ N ) )
% 3.00/1.79 = ( S @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_12_eSuc__ile__mono,axiom,
% 3.00/1.79 ! [N: extended_enat,M: extended_enat] :
% 3.00/1.79 ( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 3.00/1.79 = ( ord_less_eq @ extended_enat @ N @ M ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_13_enat_Oinject,axiom,
% 3.00/1.79 ! [Nat: nat,Nat2: nat] :
% 3.00/1.79 ( ( ( extended_enat2 @ Nat )
% 3.00/1.79 = ( extended_enat2 @ Nat2 ) )
% 3.00/1.79 = ( Nat = Nat2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_14_iless__Suc__eq,axiom,
% 3.00/1.79 ! [M: nat,N: extended_enat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
% 3.00/1.79 = ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_15_enat__ile,axiom,
% 3.00/1.79 ! [N: extended_enat,M: nat] :
% 3.00/1.79 ( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 3.00/1.79 => ? [K: nat] :
% 3.00/1.79 ( N
% 3.00/1.79 = ( extended_enat2 @ K ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_16_Suc__ile__eq,axiom,
% 3.00/1.79 ! [M: nat,N: extended_enat] :
% 3.00/1.79 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 3.00/1.79 = ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_17_chain__incr,axiom,
% 3.00/1.79 ! [A: $tType,Y3: A > extended_enat,K2: nat] :
% 3.00/1.79 ( ! [I: A] :
% 3.00/1.79 ? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I ) @ ( Y3 @ J ) )
% 3.00/1.79 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y3 @ J2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_18_enat__iless,axiom,
% 3.00/1.79 ! [N: extended_enat,M: nat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 3.00/1.79 => ? [K: nat] :
% 3.00/1.79 ( N
% 3.00/1.79 = ( extended_enat2 @ K ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_19_enat__less__induct,axiom,
% 3.00/1.79 ! [P: extended_enat > $o,N: extended_enat] :
% 3.00/1.79 ( ! [N3: extended_enat] :
% 3.00/1.79 ( ! [M2: extended_enat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ M2 @ N3 )
% 3.00/1.79 => ( P @ M2 ) )
% 3.00/1.79 => ( P @ N3 ) )
% 3.00/1.79 => ( P @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_20_enat__less__imp__le,axiom,
% 3.00/1.79 ! [N: extended_enat,M: extended_enat] :
% 3.00/1.79 ( ! [K: nat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ K ) )
% 3.00/1.79 => ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K ) ) )
% 3.00/1.79 => ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_21_enat__setp__enat__set__eq,axiom,
% 3.00/1.79 ( coindu530039314t_setp
% 3.00/1.79 = ( ^ [X3: extended_enat] : ( member @ extended_enat @ X3 @ coinductive_enat_set ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_22_wlog__linorder__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [P: A > A > $o,B: A,A2: A] :
% 3.00/1.79 ( ! [A3: A,B2: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.00/1.79 => ( P @ A3 @ B2 ) )
% 3.00/1.79 => ( ( ( P @ B @ A2 )
% 3.00/1.79 => ( P @ A2 @ B ) )
% 3.00/1.79 => ( P @ A2 @ B ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_23_ileI1,axiom,
% 3.00/1.79 ! [M: extended_enat,N: extended_enat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ M @ N )
% 3.00/1.79 => ( ord_less_eq @ extended_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_24_eSuc__le__iff,axiom,
% 3.00/1.79 ! [X: extended_enat,Y: extended_enat] :
% 3.00/1.79 ( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ X ) @ Y )
% 3.00/1.79 = ( ? [Y4: extended_enat] :
% 3.00/1.79 ( ( Y
% 3.00/1.79 = ( extended_eSuc @ Y4 ) )
% 3.00/1.79 & ( ord_less_eq @ extended_enat @ X @ Y4 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_25_ile__eSuc,axiom,
% 3.00/1.79 ! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_26_nat_Oinject,axiom,
% 3.00/1.79 ! [X2: nat,Y2: nat] :
% 3.00/1.79 ( ( ( suc @ X2 )
% 3.00/1.79 = ( suc @ Y2 ) )
% 3.00/1.79 = ( X2 = Y2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_27_old_Onat_Oinject,axiom,
% 3.00/1.79 ! [Nat: nat,Nat2: nat] :
% 3.00/1.79 ( ( ( suc @ Nat )
% 3.00/1.79 = ( suc @ Nat2 ) )
% 3.00/1.79 = ( Nat = Nat2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_28_order__refl,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_29_lift__Suc__mono__less,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [F: nat > A,N: nat,N4: nat] :
% 3.00/1.79 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 3.00/1.79 => ( ( ord_less @ nat @ N @ N4 )
% 3.00/1.79 => ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_30_lift__Suc__mono__less__iff,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [F: nat > A,N: nat,M: nat] :
% 3.00/1.79 ( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 3.00/1.79 => ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
% 3.00/1.79 = ( ord_less @ nat @ N @ M ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_31_lift__Suc__mono__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [F: nat > A,N: nat,N4: nat] :
% 3.00/1.79 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 3.00/1.79 => ( ( ord_less_eq @ nat @ N @ N4 )
% 3.00/1.79 => ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_32_lift__Suc__antimono__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [F: nat > A,N: nat,N4: nat] :
% 3.00/1.79 ( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 3.00/1.79 => ( ( ord_less_eq @ nat @ N @ N4 )
% 3.00/1.79 => ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_33_minf_I8_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_34_minf_I6_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_35_pinf_I8_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_36_Suc__less__eq,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.00/1.79 = ( ord_less @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_37_Suc__mono,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 => ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_38_lessI,axiom,
% 3.00/1.79 ! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_39_Suc__le__mono,axiom,
% 3.00/1.79 ! [N: nat,M: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
% 3.00/1.79 = ( ord_less_eq @ nat @ N @ M ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_40_enat__ord__simps_I1_J,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 3.00/1.79 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_41_enat__ord__simps_I2_J,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 3.00/1.79 = ( ord_less @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_42_Suc__leI,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 => ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_43_le__refl,axiom,
% 3.00/1.79 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_44_mem__Collect__eq,axiom,
% 3.00/1.79 ! [A: $tType,A2: A,P: A > $o] :
% 3.00/1.79 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.00/1.79 = ( P @ A2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_45_Collect__mem__eq,axiom,
% 3.00/1.79 ! [A: $tType,A4: set @ A] :
% 3.00/1.79 ( ( collect @ A
% 3.00/1.79 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 3.00/1.79 = A4 ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_46_Collect__cong,axiom,
% 3.00/1.79 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ( P @ X5 )
% 3.00/1.79 = ( Q @ X5 ) )
% 3.00/1.79 => ( ( collect @ A @ P )
% 3.00/1.79 = ( collect @ A @ Q ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_47_ext,axiom,
% 3.00/1.79 ! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ( F @ X5 )
% 3.00/1.79 = ( G @ X5 ) )
% 3.00/1.79 => ( F = G ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_48_le__trans,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,K2: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ( ord_less_eq @ nat @ J3 @ K2 )
% 3.00/1.79 => ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_49_Suc__le__eq,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 3.00/1.79 = ( ord_less @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_50_eq__imp__le,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( M = N )
% 3.00/1.79 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_51_dec__induct,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,P: nat > $o] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ( P @ I2 )
% 3.00/1.79 => ( ! [N3: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ N3 )
% 3.00/1.79 => ( ( ord_less @ nat @ N3 @ J3 )
% 3.00/1.79 => ( ( P @ N3 )
% 3.00/1.79 => ( P @ ( suc @ N3 ) ) ) ) )
% 3.00/1.79 => ( P @ J3 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_52_inc__induct,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,P: nat > $o] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ( P @ J3 )
% 3.00/1.79 => ( ! [N3: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ N3 )
% 3.00/1.79 => ( ( ord_less @ nat @ N3 @ J3 )
% 3.00/1.79 => ( ( P @ ( suc @ N3 ) )
% 3.00/1.79 => ( P @ N3 ) ) ) )
% 3.00/1.79 => ( P @ I2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_53_le__antisym,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 => ( ( ord_less_eq @ nat @ N @ M )
% 3.00/1.79 => ( M = N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_54_nat__less__le,axiom,
% 3.00/1.79 ( ( ord_less @ nat )
% 3.00/1.79 = ( ^ [M3: nat,N2: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M3 @ N2 )
% 3.00/1.79 & ( M3 != N2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_55_nat__neq__iff,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( M != N )
% 3.00/1.79 = ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 | ( ord_less @ nat @ N @ M ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_56_Suc__le__lessD,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 3.00/1.79 => ( ord_less @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_57_less__not__refl,axiom,
% 3.00/1.79 ! [N: nat] :
% 3.00/1.79 ~ ( ord_less @ nat @ N @ N ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_58_nat__le__linear,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_59_le__less__Suc__eq,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 3.00/1.79 = ( N = M ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_60_less__Suc__eq__le,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 3.00/1.79 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_61_less__eq__Suc__le,axiom,
% 3.00/1.79 ( ( ord_less @ nat )
% 3.00/1.79 = ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_62_less__not__refl2,axiom,
% 3.00/1.79 ! [N: nat,M: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ N @ M )
% 3.00/1.79 => ( M != N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_63_less__not__refl3,axiom,
% 3.00/1.79 ! [S: nat,T2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ S @ T2 )
% 3.00/1.79 => ( S != T2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_64_measure__induct,axiom,
% 3.00/1.79 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ! [Y5: A] :
% 3.00/1.79 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X5 ) )
% 3.00/1.79 => ( P @ Y5 ) )
% 3.00/1.79 => ( P @ X5 ) )
% 3.00/1.79 => ( P @ A2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_65_le__imp__less__Suc,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_66_less__imp__le__nat,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_67_less__irrefl__nat,axiom,
% 3.00/1.79 ! [N: nat] :
% 3.00/1.79 ~ ( ord_less @ nat @ N @ N ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_68_nat__less__induct,axiom,
% 3.00/1.79 ! [P: nat > $o,N: nat] :
% 3.00/1.79 ( ! [N3: nat] :
% 3.00/1.79 ( ! [M2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M2 @ N3 )
% 3.00/1.79 => ( P @ M2 ) )
% 3.00/1.79 => ( P @ N3 ) )
% 3.00/1.79 => ( P @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_69_infinite__descent,axiom,
% 3.00/1.79 ! [P: nat > $o,N: nat] :
% 3.00/1.79 ( ! [N3: nat] :
% 3.00/1.79 ( ~ ( P @ N3 )
% 3.00/1.79 => ? [M2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M2 @ N3 )
% 3.00/1.79 & ~ ( P @ M2 ) ) )
% 3.00/1.79 => ( P @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_70_le__eq__less__or__eq,axiom,
% 3.00/1.79 ( ( ord_less_eq @ nat )
% 3.00/1.79 = ( ^ [M3: nat,N2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M3 @ N2 )
% 3.00/1.79 | ( M3 = N2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_71_less__or__eq__imp__le,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 | ( M = N ) )
% 3.00/1.79 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_72_linorder__neqE__nat,axiom,
% 3.00/1.79 ! [X: nat,Y: nat] :
% 3.00/1.79 ( ( X != Y )
% 3.00/1.79 => ( ~ ( ord_less @ nat @ X @ Y )
% 3.00/1.79 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_73_le__neq__implies__less,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 => ( ( M != N )
% 3.00/1.79 => ( ord_less @ nat @ M @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_74_measure__induct__rule,axiom,
% 3.00/1.79 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ! [Y5: A] :
% 3.00/1.79 ( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X5 ) )
% 3.00/1.79 => ( P @ Y5 ) )
% 3.00/1.79 => ( P @ X5 ) )
% 3.00/1.79 => ( P @ A2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_75_less__mono__imp__le__mono,axiom,
% 3.00/1.79 ! [F: nat > nat,I2: nat,J3: nat] :
% 3.00/1.79 ( ! [I: nat,J2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I @ J2 )
% 3.00/1.79 => ( ord_less @ nat @ ( F @ I ) @ ( F @ J2 ) ) )
% 3.00/1.79 => ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ord_less_eq @ nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_76_infinite__descent__measure,axiom,
% 3.00/1.79 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ~ ( P @ X5 )
% 3.00/1.79 => ? [Y5: A] :
% 3.00/1.79 ( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) )
% 3.00/1.79 & ~ ( P @ Y5 ) ) )
% 3.00/1.79 => ( P @ X ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_77_not__less__less__Suc__eq,axiom,
% 3.00/1.79 ! [N: nat,M: nat] :
% 3.00/1.79 ( ~ ( ord_less @ nat @ N @ M )
% 3.00/1.79 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 3.00/1.79 = ( N = M ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_78_strict__inc__induct,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,P: nat > $o] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ! [I: nat] :
% 3.00/1.79 ( ( J3
% 3.00/1.79 = ( suc @ I ) )
% 3.00/1.79 => ( P @ I ) )
% 3.00/1.79 => ( ! [I: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I @ J3 )
% 3.00/1.79 => ( ( P @ ( suc @ I ) )
% 3.00/1.79 => ( P @ I ) ) )
% 3.00/1.79 => ( P @ I2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_79_less__Suc__induct,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,P: nat > nat > $o] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
% 3.00/1.79 => ( ! [I: nat,J2: nat,K: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I @ J2 )
% 3.00/1.79 => ( ( ord_less @ nat @ J2 @ K )
% 3.00/1.79 => ( ( P @ I @ J2 )
% 3.00/1.79 => ( ( P @ J2 @ K )
% 3.00/1.79 => ( P @ I @ K ) ) ) ) )
% 3.00/1.79 => ( P @ I2 @ J3 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_80_less__trans__Suc,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,K2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ( ord_less @ nat @ J3 @ K2 )
% 3.00/1.79 => ( ord_less @ nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_81_Suc__less__SucD,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.00/1.79 => ( ord_less @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_82_less__antisym,axiom,
% 3.00/1.79 ! [N: nat,M: nat] :
% 3.00/1.79 ( ~ ( ord_less @ nat @ N @ M )
% 3.00/1.79 => ( ( ord_less @ nat @ N @ ( suc @ M ) )
% 3.00/1.79 => ( M = N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_83_Suc__less__eq2,axiom,
% 3.00/1.79 ! [N: nat,M: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ ( suc @ N ) @ M )
% 3.00/1.79 = ( ? [M4: nat] :
% 3.00/1.79 ( ( M
% 3.00/1.79 = ( suc @ M4 ) )
% 3.00/1.79 & ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_84_not__less__eq,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ~ ( ord_less @ nat @ M @ N ) )
% 3.00/1.79 = ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_85_less__Suc__eq,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 3.00/1.79 = ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 | ( M = N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_86_less__SucI,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 => ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_87_less__SucE,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ ( suc @ N ) )
% 3.00/1.79 => ( ~ ( ord_less @ nat @ M @ N )
% 3.00/1.79 => ( M = N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_88_Suc__lessI,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ M @ N )
% 3.00/1.79 => ( ( ( suc @ M )
% 3.00/1.79 != N )
% 3.00/1.79 => ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_89_Suc__lessE,axiom,
% 3.00/1.79 ! [I2: nat,K2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ ( suc @ I2 ) @ K2 )
% 3.00/1.79 => ~ ! [J2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J2 )
% 3.00/1.79 => ( K2
% 3.00/1.79 != ( suc @ J2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_90_Suc__lessD,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ ( suc @ M ) @ N )
% 3.00/1.79 => ( ord_less @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_91_lessE,axiom,
% 3.00/1.79 ! [I2: nat,K2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ K2 )
% 3.00/1.79 => ( ( K2
% 3.00/1.79 != ( suc @ I2 ) )
% 3.00/1.79 => ~ ! [J2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J2 )
% 3.00/1.79 => ( K2
% 3.00/1.79 != ( suc @ J2 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_92_full__nat__induct,axiom,
% 3.00/1.79 ! [P: nat > $o,N: nat] :
% 3.00/1.79 ( ! [N3: nat] :
% 3.00/1.79 ( ! [M2: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N3 )
% 3.00/1.79 => ( P @ M2 ) )
% 3.00/1.79 => ( P @ N3 ) )
% 3.00/1.79 => ( P @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_93_not__less__eq__eq,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ~ ( ord_less_eq @ nat @ M @ N ) )
% 3.00/1.79 = ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_94_Suc__n__not__le__n,axiom,
% 3.00/1.79 ! [N: nat] :
% 3.00/1.79 ~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_95_le__Suc__eq,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 3.00/1.79 = ( ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 | ( M
% 3.00/1.79 = ( suc @ N ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_96_Suc__le__D,axiom,
% 3.00/1.79 ! [N: nat,M5: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
% 3.00/1.79 => ? [M6: nat] :
% 3.00/1.79 ( M5
% 3.00/1.79 = ( suc @ M6 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_97_le__SucI,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 => ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_98_le__SucE,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
% 3.00/1.79 => ( ~ ( ord_less_eq @ nat @ M @ N )
% 3.00/1.79 => ( M
% 3.00/1.79 = ( suc @ N ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_99_Suc__leD,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
% 3.00/1.79 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_100_less__enatE,axiom,
% 3.00/1.79 ! [N: extended_enat,M: nat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 3.00/1.79 => ~ ! [K: nat] :
% 3.00/1.79 ( ( N
% 3.00/1.79 = ( extended_enat2 @ K ) )
% 3.00/1.79 => ~ ( ord_less @ nat @ K @ M ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_101_dual__order_Oantisym,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ B @ A2 )
% 3.00/1.79 => ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( A2 = B ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_102_dual__order_Otrans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A,C: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ B @ A2 )
% 3.00/1.79 => ( ( ord_less_eq @ A @ C @ B )
% 3.00/1.79 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_103_linorder__wlog,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [P: A > A > $o,A2: A,B: A] :
% 3.00/1.79 ( ! [A3: A,B2: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A3 @ B2 )
% 3.00/1.79 => ( P @ A3 @ B2 ) )
% 3.00/1.79 => ( ! [A3: A,B2: A] :
% 3.00/1.79 ( ( P @ B2 @ A3 )
% 3.00/1.79 => ( P @ A3 @ B2 ) )
% 3.00/1.79 => ( P @ A2 @ B ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_104_dual__order_Orefl,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_105_order__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A,Z: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.00/1.79 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_106_order__class_Oorder_Oantisym,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less_eq @ A @ B @ A2 )
% 3.00/1.79 => ( A2 = B ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_107_ord__le__eq__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ord @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( B = C )
% 3.00/1.79 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_108_ord__eq__le__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ord @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( A2 = B )
% 3.00/1.79 => ( ( ord_less_eq @ A @ B @ C )
% 3.00/1.79 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_109_antisym__conv,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [Y: A,X: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ Y @ X )
% 3.00/1.79 => ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 = ( X = Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_110_le__cases3,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A,Z: A] :
% 3.00/1.79 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 3.00/1.79 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 3.00/1.79 => ( ( ( ord_less_eq @ A @ X @ Z )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 3.00/1.79 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.00/1.79 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 3.00/1.79 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_111_order_Otrans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less_eq @ A @ B @ C )
% 3.00/1.79 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_112_le__cases,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_113_eq__refl,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( X = Y )
% 3.00/1.79 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_114_linear,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_115_antisym,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less_eq @ A @ Y @ X )
% 3.00/1.79 => ( X = Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_116_eq__iff,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ^ [Y6: A,Z3: A] : ( Y6 = Z3 ) )
% 3.00/1.79 = ( ^ [X3: A,Y7: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 3.00/1.79 & ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_117_ord__le__eq__subst,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( ord @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( ( F @ B )
% 3.00/1.79 = C )
% 3.00/1.79 => ( ! [X5: A,Y8: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less_eq @ B3 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_118_ord__eq__le__subst,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( ord @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 3.00/1.79 ( ( A2
% 3.00/1.79 = ( F @ B ) )
% 3.00/1.79 => ( ( ord_less_eq @ B3 @ B @ C )
% 3.00/1.79 => ( ! [X5: B3,Y8: B3] :
% 3.00/1.79 ( ( ord_less_eq @ B3 @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_119_order__subst2,axiom,
% 3.00/1.79 ! [A: $tType,C2: $tType] :
% 3.00/1.79 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 3.00/1.79 => ( ! [X5: A,Y8: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_120_order__subst1,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 3.00/1.79 => ( ( ord_less_eq @ B3 @ B @ C )
% 3.00/1.79 => ( ! [X5: B3,Y8: B3] :
% 3.00/1.79 ( ( ord_less_eq @ B3 @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_121_le__fun__def,axiom,
% 3.00/1.79 ! [B3: $tType,A: $tType] :
% 3.00/1.79 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 => ( ( ord_less_eq @ ( A > B3 ) )
% 3.00/1.79 = ( ^ [F3: A > B3,G2: A > B3] :
% 3.00/1.79 ! [X3: A] : ( ord_less_eq @ B3 @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_122_le__funI,axiom,
% 3.00/1.79 ! [B3: $tType,A: $tType] :
% 3.00/1.79 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 => ! [F: A > B3,G: A > B3] :
% 3.00/1.79 ( ! [X5: A] : ( ord_less_eq @ B3 @ ( F @ X5 ) @ ( G @ X5 ) )
% 3.00/1.79 => ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_123_le__funE,axiom,
% 3.00/1.79 ! [B3: $tType,A: $tType] :
% 3.00/1.79 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 => ! [F: A > B3,G: A > B3,X: A] :
% 3.00/1.79 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 3.00/1.79 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_124_le__funD,axiom,
% 3.00/1.79 ! [B3: $tType,A: $tType] :
% 3.00/1.79 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 => ! [F: A > B3,G: A > B3,X: A] :
% 3.00/1.79 ( ( ord_less_eq @ ( A > B3 ) @ F @ G )
% 3.00/1.79 => ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_125_dual__order_Ostrict__implies__not__eq,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A] :
% 3.00/1.79 ( ( ord_less @ A @ B @ A2 )
% 3.00/1.79 => ( A2 != B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_126_order_Ostrict__implies__not__eq,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( A2 != B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_127_not__less__iff__gr__or__eq,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.00/1.79 = ( ( ord_less @ A @ Y @ X )
% 3.00/1.79 | ( X = Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_128_dual__order_Ostrict__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A,C: A] :
% 3.00/1.79 ( ( ord_less @ A @ B @ A2 )
% 3.00/1.79 => ( ( ord_less @ A @ C @ B )
% 3.00/1.79 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_129_less__imp__not__less,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_130_order_Ostrict__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less @ A @ B @ C )
% 3.00/1.79 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_131_dual__order_Oirrefl,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A] :
% 3.00/1.79 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_132_linorder__cases,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ~ ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ( X != Y )
% 3.00/1.79 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_133_less__imp__triv,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A,P: $o] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less @ A @ Y @ X )
% 3.00/1.79 => P ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_134_less__imp__not__eq2,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( Y != X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_135_antisym__conv3,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [Y: A,X: A] :
% 3.00/1.79 ( ~ ( ord_less @ A @ Y @ X )
% 3.00/1.79 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.00/1.79 = ( X = Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_136_less__induct,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [P: A > $o,A2: A] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ! [Y5: A] :
% 3.00/1.79 ( ( ord_less @ A @ Y5 @ X5 )
% 3.00/1.79 => ( P @ Y5 ) )
% 3.00/1.79 => ( P @ X5 ) )
% 3.00/1.79 => ( P @ A2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_137_less__not__sym,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_138_less__imp__not__eq,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( X != Y ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_139_dual__order_Oasym,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A] :
% 3.00/1.79 ( ( ord_less @ A @ B @ A2 )
% 3.00/1.79 => ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_140_ord__less__eq__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ord @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ( B = C )
% 3.00/1.79 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_141_ord__eq__less__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ord @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( A2 = B )
% 3.00/1.79 => ( ( ord_less @ A @ B @ C )
% 3.00/1.79 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_142_less__irrefl,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A] :
% 3.00/1.79 ~ ( ord_less @ A @ X @ X ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_143_less__linear,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 | ( X = Y )
% 3.00/1.79 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_144_less__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A,Z: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less @ A @ Y @ Z )
% 3.00/1.79 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_145_less__asym_H,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_146_less__asym,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_147_less__imp__neq,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( X != Y ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_148_dense,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( dense_order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ? [Z2: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Z2 )
% 3.00/1.79 & ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_149_order_Oasym,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_150_neq__iff,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( X != Y )
% 3.00/1.79 = ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_151_neqE,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( X != Y )
% 3.00/1.79 => ( ~ ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_152_gt__ex,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( no_top @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A] :
% 3.00/1.79 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_153_lt__ex,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( no_bot @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A] :
% 3.00/1.79 ? [Y8: A] : ( ord_less @ A @ Y8 @ X ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_154_order__less__subst2,axiom,
% 3.00/1.79 ! [A: $tType,C2: $tType] :
% 3.00/1.79 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 3.00/1.79 => ( ! [X5: A,Y8: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_155_order__less__subst1,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 3.00/1.79 => ( ( ord_less @ B3 @ B @ C )
% 3.00/1.79 => ( ! [X5: B3,Y8: B3] :
% 3.00/1.79 ( ( ord_less @ B3 @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_156_ord__less__eq__subst,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( ord @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,B: A,F: A > B3,C: B3] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ( ( F @ B )
% 3.00/1.79 = C )
% 3.00/1.79 => ( ! [X5: A,Y8: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less @ B3 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_157_ord__eq__less__subst,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( ord @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 3.00/1.79 ( ( A2
% 3.00/1.79 = ( F @ B ) )
% 3.00/1.79 => ( ( ord_less @ B3 @ B @ C )
% 3.00/1.79 => ( ! [X5: B3,Y8: B3] :
% 3.00/1.79 ( ( ord_less @ B3 @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_158_pinf_I1_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.00/1.79 ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z4 @ X5 )
% 3.00/1.79 => ( ( P @ X5 )
% 3.00/1.79 = ( P2 @ X5 ) ) )
% 3.00/1.79 => ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z4 @ X5 )
% 3.00/1.79 => ( ( Q @ X5 )
% 3.00/1.79 = ( Q2 @ X5 ) ) )
% 3.00/1.79 => ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ( ( ( P @ X4 )
% 3.00/1.79 & ( Q @ X4 ) )
% 3.00/1.79 = ( ( P2 @ X4 )
% 3.00/1.79 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_159_pinf_I2_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.00/1.79 ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z4 @ X5 )
% 3.00/1.79 => ( ( P @ X5 )
% 3.00/1.79 = ( P2 @ X5 ) ) )
% 3.00/1.79 => ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z4 @ X5 )
% 3.00/1.79 => ( ( Q @ X5 )
% 3.00/1.79 = ( Q2 @ X5 ) ) )
% 3.00/1.79 => ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ( ( ( P @ X4 )
% 3.00/1.79 | ( Q @ X4 ) )
% 3.00/1.79 = ( ( P2 @ X4 )
% 3.00/1.79 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_160_pinf_I3_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ( X4 != T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_161_pinf_I4_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ( X4 != T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_162_pinf_I5_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_163_pinf_I7_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_164_pinf_I11_J,axiom,
% 3.00/1.79 ! [C2: $tType,D: $tType] :
% 3.00/1.79 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 3.00/1.79 => ! [F4: D] :
% 3.00/1.79 ? [Z2: C2] :
% 3.00/1.79 ! [X4: C2] :
% 3.00/1.79 ( ( ord_less @ C2 @ Z2 @ X4 )
% 3.00/1.79 => ( F4 = F4 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_165_minf_I1_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.00/1.79 ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Z4 )
% 3.00/1.79 => ( ( P @ X5 )
% 3.00/1.79 = ( P2 @ X5 ) ) )
% 3.00/1.79 => ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Z4 )
% 3.00/1.79 => ( ( Q @ X5 )
% 3.00/1.79 = ( Q2 @ X5 ) ) )
% 3.00/1.79 => ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ( ( ( P @ X4 )
% 3.00/1.79 & ( Q @ X4 ) )
% 3.00/1.79 = ( ( P2 @ X4 )
% 3.00/1.79 & ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_166_minf_I2_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.00/1.79 ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Z4 )
% 3.00/1.79 => ( ( P @ X5 )
% 3.00/1.79 = ( P2 @ X5 ) ) )
% 3.00/1.79 => ( ? [Z4: A] :
% 3.00/1.79 ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Z4 )
% 3.00/1.79 => ( ( Q @ X5 )
% 3.00/1.79 = ( Q2 @ X5 ) ) )
% 3.00/1.79 => ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ( ( ( P @ X4 )
% 3.00/1.79 | ( Q @ X4 ) )
% 3.00/1.79 = ( ( P2 @ X4 )
% 3.00/1.79 | ( Q2 @ X4 ) ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_167_minf_I3_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ( X4 != T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_168_minf_I4_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ( X4 != T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_169_minf_I5_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ( ord_less @ A @ X4 @ T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_170_minf_I7_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ X4 @ Z2 )
% 3.00/1.79 => ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_171_minf_I11_J,axiom,
% 3.00/1.79 ! [C2: $tType,D: $tType] :
% 3.00/1.79 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 3.00/1.79 => ! [F4: D] :
% 3.00/1.79 ? [Z2: C2] :
% 3.00/1.79 ! [X4: C2] :
% 3.00/1.79 ( ( ord_less @ C2 @ X4 @ Z2 )
% 3.00/1.79 => ( F4 = F4 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_172_n__not__Suc__n,axiom,
% 3.00/1.79 ! [N: nat] :
% 3.00/1.79 ( N
% 3.00/1.79 != ( suc @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_173_Suc__inject,axiom,
% 3.00/1.79 ! [X: nat,Y: nat] :
% 3.00/1.79 ( ( ( suc @ X )
% 3.00/1.79 = ( suc @ Y ) )
% 3.00/1.79 => ( X = Y ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_174_order_Onot__eq__order__implies__strict,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A] :
% 3.00/1.79 ( ( A2 != B )
% 3.00/1.79 => ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_175_dual__order_Ostrict__implies__order,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A] :
% 3.00/1.79 ( ( ord_less @ A @ B @ A2 )
% 3.00/1.79 => ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_176_dual__order_Ostrict__iff__order,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ord_less @ A )
% 3.00/1.79 = ( ^ [B4: A,A5: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ B4 @ A5 )
% 3.00/1.79 & ( A5 != B4 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_177_dual__order_Oorder__iff__strict,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ord_less_eq @ A )
% 3.00/1.79 = ( ^ [B4: A,A5: A] :
% 3.00/1.79 ( ( ord_less @ A @ B4 @ A5 )
% 3.00/1.79 | ( A5 = B4 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_178_order_Ostrict__implies__order,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_179_dense__le__bounded,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A,Z: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ! [W: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ W )
% 3.00/1.79 => ( ( ord_less @ A @ W @ Y )
% 3.00/1.79 => ( ord_less_eq @ A @ W @ Z ) ) )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_180_dense__ge__bounded,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [Z: A,X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z @ X )
% 3.00/1.79 => ( ! [W: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z @ W )
% 3.00/1.79 => ( ( ord_less @ A @ W @ X )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ W ) ) )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_181_dual__order_Ostrict__trans2,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A,C: A] :
% 3.00/1.79 ( ( ord_less @ A @ B @ A2 )
% 3.00/1.79 => ( ( ord_less_eq @ A @ C @ B )
% 3.00/1.79 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_182_dual__order_Ostrict__trans1,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A,C: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ B @ A2 )
% 3.00/1.79 => ( ( ord_less @ A @ C @ B )
% 3.00/1.79 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_183_order_Ostrict__iff__order,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ord_less @ A )
% 3.00/1.79 = ( ^ [A5: A,B4: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A5 @ B4 )
% 3.00/1.79 & ( A5 != B4 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_184_order_Oorder__iff__strict,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ord_less_eq @ A )
% 3.00/1.79 = ( ^ [A5: A,B4: A] :
% 3.00/1.79 ( ( ord_less @ A @ A5 @ B4 )
% 3.00/1.79 | ( A5 = B4 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_185_order_Ostrict__trans2,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less_eq @ A @ B @ C )
% 3.00/1.79 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_186_order_Ostrict__trans1,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less @ A @ B @ C )
% 3.00/1.79 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_187_not__le__imp__less,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [Y: A,X: A] :
% 3.00/1.79 ( ~ ( ord_less_eq @ A @ Y @ X )
% 3.00/1.79 => ( ord_less @ A @ X @ Y ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_188_less__le__not__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ord_less @ A )
% 3.00/1.79 = ( ^ [X3: A,Y7: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 3.00/1.79 & ~ ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_189_le__imp__less__or__eq,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 | ( X = Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_190_le__less__linear,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_191_dense__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [Y: A,Z: A] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Y )
% 3.00/1.79 => ( ord_less_eq @ A @ X5 @ Z ) )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_192_dense__ge,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [Z: A,Y: A] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z @ X5 )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ X5 ) )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_193_less__le__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A,Z: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.00/1.79 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_194_le__less__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A,Z: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less @ A @ Y @ Z )
% 3.00/1.79 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_195_antisym__conv2,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.00/1.79 = ( X = Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_196_antisym__conv1,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ~ ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ( ord_less_eq @ A @ X @ Y )
% 3.00/1.79 = ( X = Y ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_197_less__imp__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( preorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_198_le__neq__trans,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( A2 != B )
% 3.00/1.79 => ( ord_less @ A @ A2 @ B ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_199_not__less,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.00/1.79 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_200_not__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 3.00/1.79 = ( ord_less @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_201_order__less__le__subst2,axiom,
% 3.00/1.79 ! [A: $tType,C2: $tType] :
% 3.00/1.79 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
% 3.00/1.79 => ( ! [X5: A,Y8: A] :
% 3.00/1.79 ( ( ord_less @ A @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_202_order__less__le__subst1,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ ( F @ B ) )
% 3.00/1.79 => ( ( ord_less_eq @ B3 @ B @ C )
% 3.00/1.79 => ( ! [X5: B3,Y8: B3] :
% 3.00/1.79 ( ( ord_less_eq @ B3 @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_203_order__le__less__subst2,axiom,
% 3.00/1.79 ! [A: $tType,C2: $tType] :
% 3.00/1.79 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,B: A,F: A > C2,C: C2] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ B )
% 3.00/1.79 => ( ( ord_less @ C2 @ ( F @ B ) @ C )
% 3.00/1.79 => ( ! [X5: A,Y8: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_204_order__le__less__subst1,axiom,
% 3.00/1.79 ! [A: $tType,B3: $tType] :
% 3.00/1.79 ( ( ( order @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 & ( order @ A @ ( type2 @ A ) ) )
% 3.00/1.79 => ! [A2: A,F: B3 > A,B: B3,C: B3] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
% 3.00/1.79 => ( ( ord_less @ B3 @ B @ C )
% 3.00/1.79 => ( ! [X5: B3,Y8: B3] :
% 3.00/1.79 ( ( ord_less @ B3 @ X5 @ Y8 )
% 3.00/1.79 => ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
% 3.00/1.79 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_205_less__le,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ord_less @ A )
% 3.00/1.79 = ( ^ [X3: A,Y7: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ X3 @ Y7 )
% 3.00/1.79 & ( X3 != Y7 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_206_le__less,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( order @ A @ ( type2 @ A ) )
% 3.00/1.79 => ( ( ord_less_eq @ A )
% 3.00/1.79 = ( ^ [X3: A,Y7: A] :
% 3.00/1.79 ( ( ord_less @ A @ X3 @ Y7 )
% 3.00/1.79 | ( X3 = Y7 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_207_leI,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ~ ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_208_leD,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [Y: A,X: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ Y @ X )
% 3.00/1.79 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_209_pinf_I6_J,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linorder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [T2: A] :
% 3.00/1.79 ? [Z2: A] :
% 3.00/1.79 ! [X4: A] :
% 3.00/1.79 ( ( ord_less @ A @ Z2 @ X4 )
% 3.00/1.79 => ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_210_ex__has__greatest__nat,axiom,
% 3.00/1.79 ! [A: $tType,P: A > $o,K2: A,M: A > nat,B: nat] :
% 3.00/1.79 ( ( P @ K2 )
% 3.00/1.79 => ( ! [Y8: A] :
% 3.00/1.79 ( ( P @ Y8 )
% 3.00/1.79 => ( ord_less @ nat @ ( M @ Y8 ) @ B ) )
% 3.00/1.79 => ? [X5: A] :
% 3.00/1.79 ( ( P @ X5 )
% 3.00/1.79 & ! [Y5: A] :
% 3.00/1.79 ( ( P @ Y5 )
% 3.00/1.79 => ( ord_less_eq @ nat @ ( M @ Y5 ) @ ( M @ X5 ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_211_complete__interval,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,P: A > $o] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B )
% 3.00/1.79 => ( ( P @ A2 )
% 3.00/1.79 => ( ~ ( P @ B )
% 3.00/1.79 => ? [C3: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ A2 @ C3 )
% 3.00/1.79 & ( ord_less_eq @ A @ C3 @ B )
% 3.00/1.79 & ! [X4: A] :
% 3.00/1.79 ( ( ( ord_less_eq @ A @ A2 @ X4 )
% 3.00/1.79 & ( ord_less @ A @ X4 @ C3 ) )
% 3.00/1.79 => ( P @ X4 ) )
% 3.00/1.79 & ! [D2: A] :
% 3.00/1.79 ( ! [X5: A] :
% 3.00/1.79 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 3.00/1.79 & ( ord_less @ A @ X5 @ D2 ) )
% 3.00/1.79 => ( P @ X5 ) )
% 3.00/1.79 => ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_212_enat_Osimps_I6_J,axiom,
% 3.00/1.79 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 3.00/1.79 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 3.00/1.79 = ( F1 @ Nat ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_213_enat_Osimps_I4_J,axiom,
% 3.00/1.79 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 3.00/1.79 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 3.00/1.79 = ( F1 @ Nat ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_214_Suc_Ohyps,axiom,
% 3.00/1.79 ! [N: extended_enat,Xs: coindu1593790203_llist @ a] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ma ) @ N )
% 3.00/1.79 => ( ( coindu749330388e_lnth @ a @ ( coindu1802687541_ltake @ a @ N @ Xs ) @ ma )
% 3.00/1.79 = ( coindu749330388e_lnth @ a @ Xs @ ma ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_215_ex__has__least__nat,axiom,
% 3.00/1.79 ! [A: $tType,P: A > $o,K2: A,M: A > nat] :
% 3.00/1.79 ( ( P @ K2 )
% 3.00/1.79 => ? [X5: A] :
% 3.00/1.79 ( ( P @ X5 )
% 3.00/1.79 & ! [Y5: A] :
% 3.00/1.79 ( ( P @ Y5 )
% 3.00/1.79 => ( ord_less_eq @ nat @ ( M @ X5 ) @ ( M @ Y5 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_216_enat__less__enat__plusI,axiom,
% 3.00/1.79 ! [X: nat,Y: nat,Z: extended_enat] :
% 3.00/1.79 ( ( ord_less @ nat @ X @ Y )
% 3.00/1.79 => ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_217_ex__gt__or__lt,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A] :
% 3.00/1.79 ? [B2: A] :
% 3.00/1.79 ( ( ord_less @ A @ A2 @ B2 )
% 3.00/1.79 | ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_218_linorder__neqE__linordered__idom,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [X: A,Y: A] :
% 3.00/1.79 ( ( X != Y )
% 3.00/1.79 => ( ~ ( ord_less @ A @ X @ Y )
% 3.00/1.79 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_219_enat__add1__eq,axiom,
% 3.00/1.79 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 3.00/1.79 ( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
% 3.00/1.79 = ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 3.00/1.79 = ( Y = Z ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_220_enat__add2__eq,axiom,
% 3.00/1.79 ! [Y: extended_enat,X: nat,Z: extended_enat] :
% 3.00/1.79 ( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
% 3.00/1.79 = ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X ) ) )
% 3.00/1.79 = ( Y = Z ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_221_enat__add__mono,axiom,
% 3.00/1.79 ! [X: nat,Y: extended_enat,Z: extended_enat] :
% 3.00/1.79 ( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
% 3.00/1.79 = ( ord_less @ extended_enat @ Y @ Z ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_222_less__fun__def,axiom,
% 3.00/1.79 ! [B3: $tType,A: $tType] :
% 3.00/1.79 ( ( ord @ B3 @ ( type2 @ B3 ) )
% 3.00/1.79 => ( ( ord_less @ ( A > B3 ) )
% 3.00/1.79 = ( ^ [F3: A > B3,G2: A > B3] :
% 3.00/1.79 ( ( ord_less_eq @ ( A > B3 ) @ F3 @ G2 )
% 3.00/1.79 & ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F3 ) ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_223_ltake__eq__ltake__antimono,axiom,
% 3.00/1.79 ! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
% 3.00/1.79 ( ( ( coindu1802687541_ltake @ A @ N @ Xs )
% 3.00/1.79 = ( coindu1802687541_ltake @ A @ N @ Ys ) )
% 3.00/1.79 => ( ( ord_less_eq @ extended_enat @ M @ N )
% 3.00/1.79 => ( ( coindu1802687541_ltake @ A @ M @ Xs )
% 3.00/1.79 = ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_224_enat__le__plus__same_I2_J,axiom,
% 3.00/1.79 ! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ Y @ X ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_225_enat__le__plus__same_I1_J,axiom,
% 3.00/1.79 ! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ X @ Y ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_226_iadd__Suc__right,axiom,
% 3.00/1.79 ! [M: extended_enat,N: extended_enat] :
% 3.00/1.79 ( ( plus_plus @ extended_enat @ M @ ( extended_eSuc @ N ) )
% 3.00/1.79 = ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_227_eSuc__plus,axiom,
% 3.00/1.79 ! [M: extended_enat,N: extended_enat] :
% 3.00/1.79 ( ( plus_plus @ extended_enat @ ( extended_eSuc @ M ) @ N )
% 3.00/1.79 = ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_228_add__less__cancel__left,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [C: A,A2: A,B: A] :
% 3.00/1.79 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 3.00/1.79 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_229_add__less__cancel__right,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,C: A,B: A] :
% 3.00/1.79 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 3.00/1.79 = ( ord_less @ A @ A2 @ B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_230_add__right__cancel,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [B: A,A2: A,C: A] :
% 3.00/1.79 ( ( ( plus_plus @ A @ B @ A2 )
% 3.00/1.79 = ( plus_plus @ A @ C @ A2 ) )
% 3.00/1.79 = ( B = C ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_231_add__left__cancel,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,B: A,C: A] :
% 3.00/1.79 ( ( ( plus_plus @ A @ A2 @ B )
% 3.00/1.79 = ( plus_plus @ A @ A2 @ C ) )
% 3.00/1.79 = ( B = C ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_232_add__Suc__right,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( plus_plus @ nat @ M @ ( suc @ N ) )
% 3.00/1.79 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_233_nat__add__left__cancel__less,axiom,
% 3.00/1.79 ! [K2: nat,M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
% 3.00/1.79 = ( ord_less @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_234_nat__add__left__cancel__le,axiom,
% 3.00/1.79 ! [K2: nat,M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
% 3.00/1.79 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_235_add__le__cancel__right,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [A2: A,C: A,B: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
% 3.00/1.79 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_236_add__le__cancel__left,axiom,
% 3.00/1.79 ! [A: $tType] :
% 3.00/1.79 ( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
% 3.00/1.79 => ! [C: A,A2: A,B: A] :
% 3.00/1.79 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
% 3.00/1.79 = ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_237_plus__enat__simps_I1_J,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 3.00/1.79 = ( extended_enat2 @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_238_add__Suc__shift,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 3.00/1.79 = ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_239_add__Suc,axiom,
% 3.00/1.79 ! [M: nat,N: nat] :
% 3.00/1.79 ( ( plus_plus @ nat @ ( suc @ M ) @ N )
% 3.00/1.79 = ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_240_less__add__eq__less,axiom,
% 3.00/1.79 ! [K2: nat,L: nat,M: nat,N: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ K2 @ L )
% 3.00/1.79 => ( ( ( plus_plus @ nat @ M @ L )
% 3.00/1.79 = ( plus_plus @ nat @ K2 @ N ) )
% 3.00/1.79 => ( ord_less @ nat @ M @ N ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_241_trans__less__add2,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,M: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_242_trans__less__add1,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,M: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_243_add__less__mono1,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,K2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_244_not__add__less2,axiom,
% 3.00/1.79 ! [J3: nat,I2: nat] :
% 3.00/1.79 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J3 @ I2 ) @ I2 ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_245_not__add__less1,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat] :
% 3.00/1.79 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ I2 ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_246_add__less__mono,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,K2: nat,L: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ( ord_less @ nat @ K2 @ L )
% 3.00/1.79 => ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_247_add__lessD1,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,K2: nat] :
% 3.00/1.79 ( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ K2 )
% 3.00/1.79 => ( ord_less @ nat @ I2 @ K2 ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_248_nat__le__iff__add,axiom,
% 3.00/1.79 ( ( ord_less_eq @ nat )
% 3.00/1.79 = ( ^ [M3: nat,N2: nat] :
% 3.00/1.79 ? [K3: nat] :
% 3.00/1.79 ( N2
% 3.00/1.79 = ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_249_trans__le__add2,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,M: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_250_trans__le__add1,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,M: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_251_add__le__mono1,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,K2: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_252_add__le__mono,axiom,
% 3.00/1.79 ! [I2: nat,J3: nat,K2: nat,L: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ I2 @ J3 )
% 3.00/1.79 => ( ( ord_less_eq @ nat @ K2 @ L )
% 3.00/1.79 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_253_le__Suc__ex,axiom,
% 3.00/1.79 ! [K2: nat,L: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ K2 @ L )
% 3.00/1.79 => ? [N3: nat] :
% 3.00/1.79 ( L
% 3.00/1.79 = ( plus_plus @ nat @ K2 @ N3 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(fact_254_add__leD2,axiom,
% 3.00/1.79 ! [M: nat,K2: nat,N: nat] :
% 3.00/1.79 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
% 3.00/1.79 => ( ord_less_eq @ nat @ K2 @ N ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.00/1.79 ! [A6: $tType,A7: $tType] :
% 3.00/1.79 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 3.00/1.79 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.00/1.79 ! [A6: $tType,A7: $tType] :
% 3.00/1.79 ( ( order @ A7 @ ( type2 @ A7 ) )
% 3.00/1.79 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_fun___Orderings_Oord,axiom,
% 3.00/1.79 ! [A6: $tType,A7: $tType] :
% 3.00/1.79 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 3.00/1.79 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 3.00/1.79 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 3.00/1.79 ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 3.00/1.79 cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.00/1.79 wellorder @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 3.00/1.79 preorder @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.00/1.79 linorder @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 3.00/1.79 no_top @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 3.00/1.79 order @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 3.00/1.79 ord @ nat @ ( type2 @ nat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 3.00/1.79 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 3.00/1.79 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 3.00/1.79 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 3.00/1.79 preorder @ $o @ ( type2 @ $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 3.00/1.79 linorder @ $o @ ( type2 @ $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 3.00/1.79 order @ $o @ ( type2 @ $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 3.00/1.79 ord @ $o @ ( type2 @ $o ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
% 3.00/1.79 condit1037483654norder @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_12,axiom,
% 3.00/1.79 wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_13,axiom,
% 3.00/1.79 preorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_14,axiom,
% 3.00/1.79 linorder @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_15,axiom,
% 3.00/1.79 order @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.00/1.79
% 3.00/1.79 thf(tcon_Extended__Nat_Oenat___Orderings_Oord_16,axiom,
% 3.00/1.79 ord @ extended_enat @ ( type2 @ extended_enat ) ).
% 3.00/1.79
% 3.00/1.79 thf(conj_0,hypothesis,
% 3.00/1.79 ! [N5: extended_enat] :
% 3.00/1.79 ( ( na
% 3.00/1.79 = ( extended_eSuc @ N5 ) )
% 3.00/1.79 => thesis ) ).
% 3.00/1.79
% 3.00/1.79 thf(conj_1,conjecture,
% 3.00/1.79 thesis ).
% 3.00/1.79 ------------------------
% 3.18/1.82 % (30698)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/2Mi)
% 3.18/1.82 % (30697)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_30566 for (2999ds/27Mi)
% 3.18/1.82 % (30696)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_30566 for (2999ds/4Mi)
% 3.18/1.82 % (30695)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/183Mi)
% 3.18/1.82 % (30699)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_30566 for (2999ds/2Mi)
% 3.18/1.82 % (30700)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_30566 for (2999ds/275Mi)
% 3.18/1.82 % (30701)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/18Mi)
% 3.18/1.82 % (30698)Instruction limit reached!
% 3.18/1.82 % (30698)------------------------------
% 3.18/1.82 % (30698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.82 % (30698)Termination reason: Unknown
% 3.18/1.82 % (30698)Termination phase: shuffling
% 3.18/1.82
% 3.18/1.82 % (30699)Instruction limit reached!
% 3.18/1.82 % (30699)------------------------------
% 3.18/1.82 % (30699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.82 % (30698)Memory used [KB]: 1279
% 3.18/1.82 % (30698)Time elapsed: 0.003 s
% 3.18/1.82 % (30698)Instructions burned: 2 (million)
% 3.18/1.82 % (30698)------------------------------
% 3.18/1.82 % (30698)------------------------------
% 3.18/1.82 % (30699)Termination reason: Unknown
% 3.18/1.82 % (30699)Termination phase: shuffling
% 3.18/1.82
% 3.18/1.82 % (30699)Memory used [KB]: 1279
% 3.18/1.82 % (30699)Time elapsed: 0.003 s
% 3.18/1.82 % (30699)Instructions burned: 3 (million)
% 3.18/1.82 % (30699)------------------------------
% 3.18/1.82 % (30699)------------------------------
% 3.18/1.82 % (30696)Instruction limit reached!
% 3.18/1.82 % (30696)------------------------------
% 3.18/1.82 % (30696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.82 % (30696)Termination reason: Unknown
% 3.18/1.82 % (30696)Termination phase: shuffling
% 3.18/1.82
% 3.18/1.82 % (30696)Memory used [KB]: 1407
% 3.18/1.82 % (30696)Time elapsed: 0.004 s
% 3.18/1.82 % (30696)Instructions burned: 5 (million)
% 3.18/1.82 % (30696)------------------------------
% 3.18/1.82 % (30696)------------------------------
% 3.18/1.83 % (30701)Instruction limit reached!
% 3.18/1.83 % (30701)------------------------------
% 3.18/1.83 % (30701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.83 % (30701)Termination reason: Unknown
% 3.18/1.83 % (30701)Termination phase: Property scanning
% 3.18/1.83
% 3.18/1.83 % (30701)Memory used [KB]: 1663
% 3.18/1.83 % (30701)Time elapsed: 0.011 s
% 3.18/1.83 % (30701)Instructions burned: 18 (million)
% 3.18/1.83 % (30701)------------------------------
% 3.18/1.83 % (30701)------------------------------
% 3.18/1.84 % (30697)Instruction limit reached!
% 3.18/1.84 % (30697)------------------------------
% 3.18/1.84 % (30697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.84 % (30697)Termination reason: Unknown
% 3.18/1.84 % (30697)Termination phase: Property scanning
% 3.18/1.84
% 3.18/1.84 % (30697)Memory used [KB]: 1663
% 3.18/1.84 % (30697)Time elapsed: 0.015 s
% 3.18/1.84 % (30697)Instructions burned: 27 (million)
% 3.18/1.84 % (30697)------------------------------
% 3.18/1.84 % (30697)------------------------------
% 3.18/1.84 % (30702)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/3Mi)
% 3.18/1.84 % (30703)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_30566 for (2999ds/37Mi)
% 3.18/1.84 % (30704)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_30566 for (2999ds/15Mi)
% 3.18/1.84 % (30702)Instruction limit reached!
% 3.18/1.84 % (30702)------------------------------
% 3.18/1.84 % (30702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.84 % (30702)Termination reason: Unknown
% 3.18/1.84 % (30702)Termination phase: shuffling
% 3.18/1.84
% 3.18/1.84 % (30702)Memory used [KB]: 1279
% 3.18/1.84 % (30702)Time elapsed: 0.003 s
% 3.18/1.84 % (30702)Instructions burned: 3 (million)
% 3.18/1.84 % (30702)------------------------------
% 3.18/1.84 % (30702)------------------------------
% 3.18/1.85 % (30704)Instruction limit reached!
% 3.18/1.85 % (30704)------------------------------
% 3.18/1.85 % (30704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.85 % (30704)Termination reason: Unknown
% 3.18/1.85 % (30704)Termination phase: shuffling
% 3.18/1.85
% 3.18/1.85 % (30704)Memory used [KB]: 1663
% 3.18/1.85 % (30704)Time elapsed: 0.009 s
% 3.18/1.85 % (30704)Instructions burned: 15 (million)
% 3.18/1.85 % (30704)------------------------------
% 3.18/1.85 % (30704)------------------------------
% 3.18/1.85 % (30705)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_30566 for (2999ds/3Mi)
% 3.18/1.85 % (30705)Instruction limit reached!
% 3.18/1.85 % (30705)------------------------------
% 3.18/1.85 % (30705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.85 % (30705)Termination reason: Unknown
% 3.18/1.85 % (30705)Termination phase: shuffling
% 3.18/1.85
% 3.18/1.85 % (30705)Memory used [KB]: 1407
% 3.18/1.85 % (30705)Time elapsed: 0.003 s
% 3.18/1.85 % (30705)Instructions burned: 3 (million)
% 3.18/1.85 % (30705)------------------------------
% 3.18/1.85 % (30705)------------------------------
% 3.18/1.85 % (30706)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_30566 for (2999ds/1041Mi)
% 3.18/1.86 % (30707)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/7Mi)
% 3.18/1.86 % (30703)Instruction limit reached!
% 3.18/1.86 % (30703)------------------------------
% 3.18/1.86 % (30703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.86 % (30703)Termination reason: Unknown
% 3.18/1.86 % (30703)Termination phase: Property scanning
% 3.18/1.86
% 3.18/1.86 % (30703)Memory used [KB]: 2046
% 3.18/1.86 % (30703)Time elapsed: 0.019 s
% 3.18/1.86 % (30703)Instructions burned: 38 (million)
% 3.18/1.86 % (30703)------------------------------
% 3.18/1.86 % (30703)------------------------------
% 3.18/1.86 % (30707)Instruction limit reached!
% 3.18/1.86 % (30707)------------------------------
% 3.18/1.86 % (30707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.86 % (30707)Termination reason: Unknown
% 3.18/1.86 % (30707)Termination phase: shuffling
% 3.18/1.86
% 3.18/1.86 % (30707)Memory used [KB]: 1407
% 3.18/1.86 % (30707)Time elapsed: 0.005 s
% 3.18/1.86 % (30707)Instructions burned: 7 (million)
% 3.18/1.86 % (30707)------------------------------
% 3.18/1.86 % (30707)------------------------------
% 3.18/1.86 % (30708)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/16Mi)
% 3.18/1.87 % (30709)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_30566 for (2999ds/3Mi)
% 3.18/1.87 % (30709)Instruction limit reached!
% 3.18/1.87 % (30709)------------------------------
% 3.18/1.87 % (30709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.87 % (30709)Termination reason: Unknown
% 3.18/1.87 % (30709)Termination phase: shuffling
% 3.18/1.87
% 3.18/1.87 % (30709)Memory used [KB]: 1279
% 3.18/1.87 % (30709)Time elapsed: 0.003 s
% 3.18/1.87 % (30709)Instructions burned: 3 (million)
% 3.18/1.87 % (30709)------------------------------
% 3.18/1.87 % (30709)------------------------------
% 3.18/1.87 % (30708)Instruction limit reached!
% 3.18/1.87 % (30708)------------------------------
% 3.18/1.87 % (30708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.87 % (30708)Termination reason: Unknown
% 3.18/1.87 % (30708)Termination phase: shuffling
% 3.18/1.87
% 3.18/1.87 % (30708)Memory used [KB]: 1663
% 3.18/1.87 % (30708)Time elapsed: 0.010 s
% 3.18/1.87 % (30708)Instructions burned: 17 (million)
% 3.18/1.87 % (30708)------------------------------
% 3.18/1.87 % (30708)------------------------------
% 3.18/1.87 % (30710)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_30566 for (2999ds/3Mi)
% 3.18/1.87 % (30710)Instruction limit reached!
% 3.18/1.87 % (30710)------------------------------
% 3.18/1.87 % (30710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.87 % (30710)Termination reason: Unknown
% 3.18/1.88 % (30710)Termination phase: shuffling
% 3.18/1.88
% 3.18/1.88 % (30710)Memory used [KB]: 1279
% 3.18/1.88 % (30710)Time elapsed: 0.003 s
% 3.18/1.88 % (30710)Instructions burned: 3 (million)
% 3.18/1.88 % (30710)------------------------------
% 3.18/1.88 % (30710)------------------------------
% 3.18/1.88 % (30711)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/7Mi)
% 3.18/1.88 % (30711)Instruction limit reached!
% 3.18/1.88 % (30711)------------------------------
% 3.18/1.88 % (30711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.88 % (30711)Termination reason: Unknown
% 3.18/1.88 % (30711)Termination phase: shuffling
% 3.18/1.88
% 3.18/1.88 % (30711)Memory used [KB]: 1407
% 3.18/1.88 % (30711)Time elapsed: 0.005 s
% 3.18/1.88 % (30711)Instructions burned: 7 (million)
% 3.18/1.88 % (30711)------------------------------
% 3.18/1.88 % (30711)------------------------------
% 3.18/1.88 % (30712)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/3Mi)
% 3.18/1.88 % (30712)Instruction limit reached!
% 3.18/1.88 % (30712)------------------------------
% 3.18/1.88 % (30712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.88 % (30712)Termination reason: Unknown
% 3.18/1.88 % (30712)Termination phase: shuffling
% 3.18/1.88
% 3.18/1.88 % (30712)Memory used [KB]: 1407
% 3.18/1.88 % (30712)Time elapsed: 0.003 s
% 3.18/1.88 % (30712)Instructions burned: 3 (million)
% 3.18/1.88 % (30712)------------------------------
% 3.18/1.88 % (30712)------------------------------
% 3.18/1.89 % (30713)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/4Mi)
% 3.18/1.89 % (30713)Instruction limit reached!
% 3.18/1.89 % (30713)------------------------------
% 3.18/1.89 % (30713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.89 % (30713)Termination reason: Unknown
% 3.18/1.89 % (30713)Termination phase: shuffling
% 3.18/1.89
% 3.18/1.89 % (30713)Memory used [KB]: 1407
% 3.18/1.89 % (30713)Time elapsed: 0.004 s
% 3.18/1.89 % (30714)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/18Mi)
% 3.18/1.89 % (30713)Instructions burned: 5 (million)
% 3.18/1.89 % (30713)------------------------------
% 3.18/1.89 % (30713)------------------------------
% 3.18/1.90 % (30715)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/710Mi)
% 3.18/1.90 % (30716)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_30566 for (2999ds/6Mi)
% 3.18/1.90 % (30714)Instruction limit reached!
% 3.18/1.90 % (30714)------------------------------
% 3.18/1.90 % (30714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.90 % (30714)Termination reason: Unknown
% 3.18/1.90 % (30714)Termination phase: Property scanning
% 3.18/1.90
% 3.18/1.90 % (30714)Memory used [KB]: 1663
% 3.18/1.90 % (30714)Time elapsed: 0.011 s
% 3.18/1.90 % (30714)Instructions burned: 19 (million)
% 3.18/1.90 % (30714)------------------------------
% 3.18/1.90 % (30714)------------------------------
% 3.18/1.90 % (30716)Instruction limit reached!
% 3.18/1.90 % (30716)------------------------------
% 3.18/1.90 % (30716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.90 % (30716)Termination reason: Unknown
% 3.18/1.90 % (30716)Termination phase: shuffling
% 3.18/1.90
% 3.18/1.90 % (30716)Memory used [KB]: 1407
% 3.18/1.90 % (30716)Time elapsed: 0.005 s
% 3.18/1.90 % (30716)Instructions burned: 7 (million)
% 3.18/1.90 % (30716)------------------------------
% 3.18/1.90 % (30716)------------------------------
% 3.18/1.91 % (30717)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_30566 for (2999ds/902Mi)
% 3.18/1.91 % (30695)Instruction limit reached!
% 3.18/1.91 % (30695)------------------------------
% 3.18/1.91 % (30695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.91 % (30695)Termination reason: Unknown
% 3.18/1.91 % (30695)Termination phase: Saturation
% 3.18/1.91
% 3.18/1.91 % (30695)Memory used [KB]: 6908
% 3.18/1.91 % (30695)Time elapsed: 0.089 s
% 3.18/1.91 % (30695)Instructions burned: 184 (million)
% 3.18/1.91 % (30695)------------------------------
% 3.18/1.91 % (30695)------------------------------
% 3.18/1.92 % (30718)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_30566 for (2998ds/21Mi)
% 3.18/1.92 % (30719)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_30566 for (2998ds/5Mi)
% 3.18/1.92 % (30719)Instruction limit reached!
% 3.18/1.92 % (30719)------------------------------
% 3.18/1.92 % (30719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.92 % (30719)Termination reason: Unknown
% 3.18/1.92 % (30719)Termination phase: shuffling
% 3.18/1.92
% 3.18/1.92 % (30719)Memory used [KB]: 1407
% 3.18/1.92 % (30719)Time elapsed: 0.004 s
% 3.18/1.92 % (30719)Instructions burned: 5 (million)
% 3.18/1.92 % (30719)------------------------------
% 3.18/1.92 % (30719)------------------------------
% 3.18/1.93 % (30720)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_30566 for (2998ds/6Mi)
% 3.18/1.93 % (30718)Instruction limit reached!
% 3.18/1.93 % (30718)------------------------------
% 3.18/1.93 % (30718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.93 % (30718)Termination reason: Unknown
% 3.18/1.93 % (30718)Termination phase: Property scanning
% 3.18/1.93
% 3.18/1.93 % (30718)Memory used [KB]: 1791
% 3.18/1.93 % (30718)Time elapsed: 0.013 s
% 3.18/1.93 % (30718)Instructions burned: 23 (million)
% 3.18/1.93 % (30718)------------------------------
% 3.18/1.93 % (30718)------------------------------
% 3.18/1.93 % (30720)Instruction limit reached!
% 3.18/1.93 % (30720)------------------------------
% 3.18/1.93 % (30720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.93 % (30720)Termination reason: Unknown
% 3.18/1.93 % (30720)Termination phase: shuffling
% 3.18/1.93
% 3.18/1.93 % (30720)Memory used [KB]: 1407
% 3.18/1.93 % (30720)Time elapsed: 0.005 s
% 3.18/1.93 % (30720)Instructions burned: 7 (million)
% 3.18/1.93 % (30720)------------------------------
% 3.18/1.93 % (30720)------------------------------
% 3.18/1.94 % (30721)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_30566 for (2998ds/377Mi)
% 3.18/1.94 % (30722)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_30566 for (2998ds/779Mi)
% 3.18/1.95 % (30723)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_30566 for (2998ds/19Mi)
% 3.18/1.95 % (30723)Instruction limit reached!
% 3.18/1.95 % (30723)------------------------------
% 3.18/1.95 % (30723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.95 % (30723)Termination reason: Unknown
% 3.18/1.95 % (30723)Termination phase: shuffling
% 3.18/1.95
% 3.18/1.95 % (30723)Memory used [KB]: 1663
% 3.18/1.95 % (30723)Time elapsed: 0.010 s
% 3.18/1.95 % (30723)Instructions burned: 19 (million)
% 3.18/1.95 % (30723)------------------------------
% 3.18/1.95 % (30723)------------------------------
% 3.18/1.96 % (30700)Instruction limit reached!
% 3.18/1.96 % (30700)------------------------------
% 3.18/1.96 % (30700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.96 % (30700)Termination reason: Unknown
% 3.18/1.96 % (30700)Termination phase: Saturation
% 3.18/1.96
% 3.18/1.96 % (30700)Memory used [KB]: 7164
% 3.18/1.96 % (30700)Time elapsed: 0.137 s
% 3.18/1.96 % (30700)Instructions burned: 276 (million)
% 3.18/1.96 % (30700)------------------------------
% 3.18/1.96 % (30700)------------------------------
% 3.18/1.97 % (30724)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_30566 for (2998ds/879Mi)
% 3.18/1.97 % (30725)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_30566 for (2998ds/17Mi)
% 3.18/1.98 % (30725)Instruction limit reached!
% 3.18/1.98 % (30725)------------------------------
% 3.18/1.98 % (30725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.98 % (30725)Termination reason: Unknown
% 3.18/1.98 % (30725)Termination phase: Property scanning
% 3.18/1.98
% 3.18/1.98 % (30725)Memory used [KB]: 1663
% 3.18/1.98 % (30725)Time elapsed: 0.011 s
% 3.18/1.98 % (30725)Instructions burned: 18 (million)
% 3.18/1.98 % (30725)------------------------------
% 3.18/1.98 % (30725)------------------------------
% 3.18/2.00 % (30726)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_30566 for (2998ds/3Mi)
% 3.18/2.00 % (30726)Instruction limit reached!
% 3.18/2.00 % (30726)------------------------------
% 3.18/2.00 % (30726)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/2.00 % (30726)Termination reason: Unknown
% 3.18/2.00 % (30726)Termination phase: shuffling
% 3.18/2.00
% 3.18/2.00 % (30726)Memory used [KB]: 1407
% 3.18/2.00 % (30726)Time elapsed: 0.004 s
% 3.18/2.00 % (30726)Instructions burned: 4 (million)
% 3.18/2.00 % (30726)------------------------------
% 3.18/2.00 % (30726)------------------------------
% 3.18/2.02 % (30727)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_30566 for (2997ds/30Mi)
% 4.62/2.03 % (30727)Instruction limit reached!
% 4.62/2.03 % (30727)------------------------------
% 4.62/2.03 % (30727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.62/2.03 % (30727)Termination reason: Unknown
% 4.62/2.03 % (30727)Termination phase: Preprocessing 1
% 4.62/2.03
% 4.62/2.03 % (30727)Memory used [KB]: 1791
% 4.62/2.03 % (30727)Time elapsed: 0.017 s
% 4.62/2.03 % (30727)Instructions burned: 30 (million)
% 4.62/2.03 % (30727)------------------------------
% 4.62/2.03 % (30727)------------------------------
% 4.62/2.05 % (30728)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_30566 for (2997ds/127Mi)
% 4.62/2.06 % (30715)First to succeed.
% 4.62/2.07 % (30715)Refutation found. Thanks to Tanya!
% 4.62/2.07 % SZS status Theorem for DTF2THF_30566
% 4.62/2.07 % SZS output start Proof for DTF2THF_30566
% 4.62/2.07 thf(type_def_5, type, itself: ($tType) > $tType).
% 4.62/2.07 thf(type_def_7, type, coindu1593790203_llist: ($tType) > $tType).
% 4.62/2.07 thf(type_def_8, type, nat: $tType).
% 4.62/2.07 thf(type_def_9, type, extended_enat: $tType).
% 4.62/2.07 thf(type_def_10, type, set: ($tType) > $tType).
% 4.62/2.07 thf(type_def_11, type, a: $tType).
% 4.62/2.07 thf(func_def_0, type, coindu1593790203_llist: ($tType) > $tType).
% 4.62/2.07 thf(func_def_1, type, extended_enat: $tType).
% 4.62/2.07 thf(func_def_2, type, set: ($tType) > $tType).
% 4.62/2.07 thf(func_def_3, type, nat: $tType).
% 4.62/2.07 thf(func_def_4, type, itself: ($tType) > $tType).
% 4.62/2.07 thf(func_def_5, type, a: $tType).
% 4.62/2.07 thf(func_def_6, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_9, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_10, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_12, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_13, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_14, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_15, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_16, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_17, type, cancel_semigroup_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_18, type, ordere236663937imp_le: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_19, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_20, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 4.62/2.07 thf(func_def_21, type, coindu749330388e_lnth: !>[X0: $tType]:(coindu1593790203_llist @ X0 > nat > X0)).
% 4.62/2.07 thf(func_def_22, type, coindu1802687541_ltake: !>[X0: $tType]:(extended_enat > coindu1593790203_llist @ X0 > coindu1593790203_llist @ X0)).
% 4.62/2.07 thf(func_def_23, type, coindu440805660e_enat: !>[X0: $tType]:(X0 > (extended_enat > X0) > extended_enat > X0)).
% 4.62/2.07 thf(func_def_24, type, coinductive_enat_set: set @ extended_enat).
% 4.62/2.07 thf(func_def_25, type, coindu530039314t_setp: extended_enat > $o).
% 4.62/2.07 thf(func_def_26, type, coindu1491768222unfold: !>[X0: $tType]:((X0 > $o) > (X0 > X0) > X0 > extended_enat)).
% 4.62/2.07 thf(func_def_27, type, extended_eSuc: extended_enat > extended_enat).
% 4.62/2.07 thf(func_def_28, type, extended_enat2: nat > extended_enat).
% 4.62/2.07 thf(func_def_29, type, extended_case_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 4.62/2.07 thf(func_def_30, type, extended_rec_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 4.62/2.07 thf(func_def_31, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 4.62/2.07 thf(func_def_32, type, suc: nat > nat).
% 4.62/2.07 thf(func_def_33, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 4.62/2.07 thf(func_def_34, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 4.62/2.07 thf(func_def_35, type, type2: !>[X0: $tType]:(itself @ X0)).
% 4.62/2.07 thf(func_def_36, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 4.62/2.07 thf(func_def_37, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 4.62/2.07 thf(func_def_38, type, ma: nat).
% 4.62/2.07 thf(func_def_39, type, na: extended_enat).
% 4.62/2.07 thf(func_def_54, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 4.62/2.07 thf(func_def_55, type, sK1: nat > extended_enat > nat).
% 4.62/2.07 thf(func_def_56, type, sK2: extended_enat > extended_enat > extended_enat).
% 4.62/2.07 thf(func_def_57, type, sK3: extended_enat > nat > nat).
% 4.62/2.07 thf(func_def_58, type, sK4: (extended_enat > $o) > extended_enat).
% 4.62/2.07 thf(func_def_60, type, ph6: !>[X0: $tType]:(X0)).
% 4.62/2.07 thf(f1790,plain,(
% 4.62/2.07 $false),
% 4.62/2.07 inference(subsumption_resolution,[status(thm)],[f1778,f862])).
% 4.62/2.07 thf(f862,plain,(
% 4.62/2.07 ( ! [X0 : extended_enat] : ((na != (extended_eSuc @ X0))) )),
% 4.62/2.07 inference(subsumption_resolution,[status(thm)],[f824,f841])).
% 4.62/2.07 thf(f841,plain,(
% 4.62/2.07 (thesis != $true)),
% 4.62/2.07 inference(cnf_transformation,[status(thm)],[f791])).
% 4.62/2.07 thf(f791,plain,(
% 4.62/2.07 (thesis != $true)),
% 4.62/2.07 inference(flattening,[status(thm)],[f606])).
% 4.62/2.07 thf(f606,plain,(
% 4.62/2.07 ~(thesis = $true)),
% 4.62/2.07 inference(fool_elimination,[status(thm)],[f605])).
% 4.62/2.07 thf(f605,plain,(
% 4.62/2.07 ~thesis),
% 4.62/2.07 inference(rectify,[status(thm)],[f283])).
% 4.62/2.07 thf(f283,negated_conjecture,(
% 4.62/2.07 ~thesis),
% 4.62/2.07 inference(negated_conjecture,[status(cth)],[f282])).
% 4.62/2.07 thf(f282,conjecture,(
% 4.62/2.07 thesis),
% 4.62/2.07 file('/export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2THF_30566.p',conj_1)).
% 4.62/2.07 thf(f824,plain,(
% 4.62/2.07 ( ! [X0 : extended_enat] : ((thesis = $true) | (na != (extended_eSuc @ X0))) )),
% 4.62/2.07 inference(cnf_transformation,[status(thm)],[f800])).
% 4.62/2.07 thf(f800,plain,(
% 4.62/2.07 ! [X0 : extended_enat] : ((na != (extended_eSuc @ X0)) | (thesis = $true))),
% 4.62/2.07 inference(ennf_transformation,[status(thm)],[f372])).
% 4.62/2.07 thf(f372,plain,(
% 4.62/2.07 ! [X0 : extended_enat] : ((na = (extended_eSuc @ X0)) => (thesis = $true))),
% 4.62/2.07 inference(fool_elimination,[status(thm)],[f371])).
% 4.62/2.07 thf(f371,plain,(
% 4.62/2.07 ! [X0 : extended_enat] : ((na = (extended_eSuc @ X0)) => thesis)),
% 4.62/2.07 inference(rectify,[status(thm)],[f281])).
% 4.62/2.07 thf(f281,axiom,(
% 4.62/2.07 ! [X75 : extended_enat] : ((na = (extended_eSuc @ X75)) => thesis)),
% 4.62/2.07 file('/export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2THF_30566.p',conj_0)).
% 4.62/2.07 thf(f1778,plain,(
% 4.62/2.07 (na = (extended_eSuc @ (sK2 @ na @ (extended_enat2 @ (suc @ ma)))))),
% 4.62/2.07 inference(trivial_inequality_removal,[status(thm)],[f1769])).
% 4.62/2.07 thf(f1769,plain,(
% 4.62/2.07 ($true != $true) | (na = (extended_eSuc @ (sK2 @ na @ (extended_enat2 @ (suc @ ma)))))),
% 4.62/2.07 inference(superposition,[status(thm)],[f931,f898])).
% 4.62/2.07 thf(f898,plain,(
% 4.62/2.07 ((ord_less_eq @ extended_enat @ (extended_enat2 @ (suc @ (suc @ ma))) @ na) = $true)),
% 4.62/2.07 inference(forward_demodulation,[status(thm)],[f897,f851])).
% 4.62/2.07 thf(f851,plain,(
% 4.62/2.07 ( ! [X2 : nat] : (((extended_enat2 @ (suc @ X2)) = (extended_eSuc @ (extended_enat2 @ X2)))) )),
% 4.62/2.07 inference(equality_resolution,[status(thm)],[f850])).
% 4.62/2.07 thf(f850,plain,(
% 4.62/2.07 ( ! [X2 : nat,X1 : nat] : (((extended_enat2 @ X1) = (extended_eSuc @ (extended_enat2 @ X2))) | ((suc @ X2) != X1)) )),
% 4.62/2.07 inference(equality_resolution,[status(thm)],[f830])).
% 4.62/2.07 thf(f830,plain,(
% 4.62/2.07 ( ! [X2 : nat,X0 : extended_enat,X1 : nat] : (((extended_enat2 @ X1) = (extended_eSuc @ X0)) | ((extended_enat2 @ X2) != X0) | ((suc @ X2) != X1)) )),
% 4.62/2.07 inference(cnf_transformation,[status(thm)],[f809])).
% 4.62/2.07 thf(f809,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : nat] : ((((extended_enat2 @ X1) = (extended_eSuc @ X0)) | ! [X2 : nat] : (((extended_enat2 @ X2) != X0) | ((suc @ X2) != X1))) & ((((extended_enat2 @ (sK1 @ X1 @ X0)) = X0) & ((suc @ (sK1 @ X1 @ X0)) = X1)) | ((extended_enat2 @ X1) != (extended_eSuc @ X0))))),
% 4.62/2.07 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1])],[f807,f808])).
% 4.62/2.07 thf(f808,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : nat] : (? [X3 : nat] : (((extended_enat2 @ X3) = X0) & ((suc @ X3) = X1)) => (((extended_enat2 @ (sK1 @ X1 @ X0)) = X0) & ((suc @ (sK1 @ X1 @ X0)) = X1)))),
% 4.62/2.07 introduced(definition,[],[choice_axiom])).
% 4.62/2.07 thf(f807,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : nat] : ((((extended_enat2 @ X1) = (extended_eSuc @ X0)) | ! [X2 : nat] : (((extended_enat2 @ X2) != X0) | ((suc @ X2) != X1))) & (? [X3 : nat] : (((extended_enat2 @ X3) = X0) & ((suc @ X3) = X1)) | ((extended_enat2 @ X1) != (extended_eSuc @ X0))))),
% 4.62/2.07 inference(rectify,[status(thm)],[f806])).
% 4.62/2.07 thf(f806,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : nat] : ((((extended_enat2 @ X1) = (extended_eSuc @ X0)) | ! [X2 : nat] : (((extended_enat2 @ X2) != X0) | ((suc @ X2) != X1))) & (? [X2 : nat] : (((extended_enat2 @ X2) = X0) & ((suc @ X2) = X1)) | ((extended_enat2 @ X1) != (extended_eSuc @ X0))))),
% 4.62/2.07 inference(nnf_transformation,[status(thm)],[f560])).
% 4.62/2.07 thf(f560,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : nat] : (((extended_enat2 @ X1) = (extended_eSuc @ X0)) <=> ? [X2 : nat] : (((extended_enat2 @ X2) = X0) & ((suc @ X2) = X1)))),
% 4.62/2.07 inference(fool_elimination,[status(thm)],[f559])).
% 4.62/2.07 thf(f559,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : nat] : (((extended_enat2 @ X1) = (extended_eSuc @ X0)) = ? [X2 : nat] : (((extended_enat2 @ X2) = X0) & ((suc @ X2) = X1)))),
% 4.62/2.07 inference(rectify,[status(thm)],[f6])).
% 4.62/2.07 thf(f6,axiom,(
% 4.62/2.07 ! [X6 : extended_enat,X7 : nat] : (((extended_eSuc @ X6) = (extended_enat2 @ X7)) = ? [X8 : nat] : (((extended_enat2 @ X8) = X6) & ((suc @ X8) = X7)))),
% 4.62/2.07 file('/export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2THF_30566.p',fact_5_eSuc__enat__iff)).
% 4.62/2.07 thf(f897,plain,(
% 4.62/2.07 ((ord_less_eq @ extended_enat @ (extended_eSuc @ (extended_enat2 @ (suc @ ma))) @ na) = $true)),
% 4.62/2.07 inference(trivial_inequality_removal,[status(thm)],[f896])).
% 4.62/2.07 thf(f896,plain,(
% 4.62/2.07 ($true != $true) | ((ord_less_eq @ extended_enat @ (extended_eSuc @ (extended_enat2 @ (suc @ ma))) @ na) = $true)),
% 4.62/2.07 inference(superposition,[status(thm)],[f822,f834])).
% 4.62/2.07 thf(f834,plain,(
% 4.62/2.07 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ ma)) @ na) = $true)),
% 4.62/2.07 inference(cnf_transformation,[status(thm)],[f568])).
% 4.62/2.07 thf(f568,plain,(
% 4.62/2.07 ((ord_less @ extended_enat @ (extended_enat2 @ (suc @ ma)) @ na) = $true)),
% 4.62/2.07 inference(fool_elimination,[status(thm)],[f567])).
% 4.62/2.07 thf(f567,plain,(
% 4.62/2.07 (ord_less @ extended_enat @ (extended_enat2 @ (suc @ ma)) @ na)),
% 4.62/2.07 inference(rectify,[status(thm)],[f1])).
% 4.62/2.07 thf(f1,axiom,(
% 4.62/2.07 (ord_less @ extended_enat @ (extended_enat2 @ (suc @ ma)) @ na)),
% 4.62/2.07 file('/export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2THF_30566.p',fact_0_Suc_Oprems)).
% 4.62/2.07 thf(f822,plain,(
% 4.62/2.07 ( ! [X0 : extended_enat,X1 : extended_enat] : (($true != (ord_less @ extended_enat @ X0 @ X1)) | ((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = $true)) )),
% 4.62/2.07 inference(cnf_transformation,[status(thm)],[f799])).
% 4.62/2.07 thf(f799,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : (((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = $true) | ($true != (ord_less @ extended_enat @ X0 @ X1)))),
% 4.62/2.07 inference(ennf_transformation,[status(thm)],[f780])).
% 4.62/2.07 thf(f780,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : (($true = (ord_less @ extended_enat @ X0 @ X1)) => ((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = $true))),
% 4.62/2.07 inference(fool_elimination,[status(thm)],[f779])).
% 4.62/2.07 thf(f779,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : ((ord_less @ extended_enat @ X0 @ X1) => (ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1))),
% 4.62/2.07 inference(rectify,[status(thm)],[f24])).
% 4.62/2.07 thf(f24,axiom,(
% 4.62/2.07 ! [X2 : extended_enat,X3 : extended_enat] : ((ord_less @ extended_enat @ X2 @ X3) => (ord_less_eq @ extended_enat @ (extended_eSuc @ X2) @ X3))),
% 4.62/2.07 file('/export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2THF_30566.p',fact_23_ileI1)).
% 4.62/2.07 thf(f931,plain,(
% 4.62/2.07 ( ! [X0 : nat,X1 : extended_enat] : (((ord_less_eq @ extended_enat @ (extended_enat2 @ (suc @ X0)) @ X1) != $true) | ((extended_eSuc @ (sK2 @ X1 @ (extended_enat2 @ X0))) = X1)) )),
% 4.62/2.07 inference(superposition,[status(thm)],[f832,f851])).
% 4.62/2.07 thf(f832,plain,(
% 4.62/2.07 ( ! [X0 : extended_enat,X1 : extended_enat] : (((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) != $true) | ((extended_eSuc @ (sK2 @ X1 @ X0)) = X1)) )),
% 4.62/2.07 inference(cnf_transformation,[status(thm)],[f813])).
% 4.62/2.07 thf(f813,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : ((((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = $true) | ! [X2 : extended_enat] : (((extended_eSuc @ X2) != X1) | ((ord_less_eq @ extended_enat @ X0 @ X2) != $true))) & ((((extended_eSuc @ (sK2 @ X1 @ X0)) = X1) & ((ord_less_eq @ extended_enat @ X0 @ (sK2 @ X1 @ X0)) = $true)) | ((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) != $true)))),
% 4.62/2.07 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2])],[f811,f812])).
% 4.62/2.07 thf(f812,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : (? [X3 : extended_enat] : (((extended_eSuc @ X3) = X1) & ((ord_less_eq @ extended_enat @ X0 @ X3) = $true)) => (((extended_eSuc @ (sK2 @ X1 @ X0)) = X1) & ((ord_less_eq @ extended_enat @ X0 @ (sK2 @ X1 @ X0)) = $true)))),
% 4.62/2.07 introduced(definition,[],[choice_axiom])).
% 4.62/2.07 thf(f811,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : ((((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = $true) | ! [X2 : extended_enat] : (((extended_eSuc @ X2) != X1) | ((ord_less_eq @ extended_enat @ X0 @ X2) != $true))) & (? [X3 : extended_enat] : (((extended_eSuc @ X3) = X1) & ((ord_less_eq @ extended_enat @ X0 @ X3) = $true)) | ((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) != $true)))),
% 4.62/2.07 inference(rectify,[status(thm)],[f810])).
% 4.62/2.07 thf(f810,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : ((((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = $true) | ! [X2 : extended_enat] : (((extended_eSuc @ X2) != X1) | ((ord_less_eq @ extended_enat @ X0 @ X2) != $true))) & (? [X2 : extended_enat] : (((extended_eSuc @ X2) = X1) & ((ord_less_eq @ extended_enat @ X0 @ X2) = $true)) | ((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) != $true)))),
% 4.62/2.07 inference(nnf_transformation,[status(thm)],[f740])).
% 4.62/2.07 thf(f740,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : (((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = $true) <=> ? [X2 : extended_enat] : (((extended_eSuc @ X2) = X1) & ((ord_less_eq @ extended_enat @ X0 @ X2) = $true)))),
% 4.62/2.07 inference(fool_elimination,[status(thm)],[f739])).
% 4.62/2.07 thf(f739,plain,(
% 4.62/2.07 ! [X0 : extended_enat,X1 : extended_enat] : ((ord_less_eq @ extended_enat @ (extended_eSuc @ X0) @ X1) = ? [X2 : extended_enat] : (((extended_eSuc @ X2) = X1) & (ord_less_eq @ extended_enat @ X0 @ X2)))),
% 4.62/2.07 inference(rectify,[status(thm)],[f25])).
% 4.62/2.07 thf(f25,axiom,(
% 4.62/2.07 ! [X6 : extended_enat,X7 : extended_enat] : ((ord_less_eq @ extended_enat @ (extended_eSuc @ X6) @ X7) = ? [X31 : extended_enat] : (((extended_eSuc @ X31) = X7) & (ord_less_eq @ extended_enat @ X6 @ X31)))),
% 4.62/2.07 file('/export/starexec/sandbox/tmp/tmp.Oklx7XWLnY/DTF2THF_30566.p',fact_24_eSuc__le__iff)).
% 4.62/2.07 % SZS output end Proof for DTF2THF_30566
% 4.62/2.07 % (30715)------------------------------
% 4.62/2.07 % (30715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.62/2.07 % (30715)Termination reason: Refutation
% 4.62/2.07
% 4.62/2.07 % (30715)Memory used [KB]: 7291
% 4.62/2.07 % (30715)Time elapsed: 0.169 s
% 4.62/2.07 % (30715)Instructions burned: 330 (million)
% 4.62/2.07 % (30715)------------------------------
% 4.62/2.07 % (30715)------------------------------
% 4.62/2.07 % (30694)Success in time 0.281 s
%------------------------------------------------------------------------------