%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT140^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:44:54 AM UTC 2026
% Result : Theorem 3.26s 1.89s
% Output : Refutation 3.26s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT140^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n014.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 06:08:57 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.47 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : DAT140^1 : TPTP v9.2.1. Released v7.0.0.
% 0.30/0.47 % Domain : Data Structures
% 0.30/0.47 % Problem : Coinductive list prefix 36
% 0.30/0.47 % Version : [Bla16] axioms : Especial.
% 0.30/0.47 % English :
% 0.30/0.47
% 0.30/0.47 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.30/0.47 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.30/0.47 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.30/0.47 % Source : [Bla16]
% 0.30/0.47 % Names : coinductive_list_prefix__36.p [Bla16]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : 0.33 v9.1.0, 0.00 v7.5.0, 0.33 v7.3.0, 0.00 v7.1.0
% 0.30/0.47 % Syntax : Number of formulae : 309 ( 76 unt; 35 typ; 0 def)
% 0.30/0.47 % Number of atoms : 929 ( 201 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 13 ( 3 avg)
% 0.30/0.47 % Number of connectives : 3616 ( 98 ~; 36 |; 69 &;2907 @)
% 0.30/0.47 % ( 0 <=>; 506 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 21 ( 9 avg)
% 0.30/0.47 % Number of types : 3 ( 2 usr)
% 0.30/0.47 % Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 34 ( 33 usr; 0 con; 1-4 aty)
% 0.30/0.47 % Number of variables : 1045 ( 40 ^; 910 !; 60 ?;1045 :)
% 0.30/0.47 % ( 35 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.47 % SPC : TH1_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.47 % 2016-07-13 14:40:41.558
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 %----Could-be-implicit typings (4)
% 0.30/0.47 thf(ty_t_Coinductive__List_Ollist,type,
% 0.30/0.47 coinductive_llist: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_Nat_Onat,type,
% 0.30/0.47 nat: $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_t_itself,type,
% 0.30/0.47 itself: $tType > $tType ).
% 0.30/0.47
% 0.30/0.47 thf(ty_tf_a,type,
% 0.30/0.47 a: $tType ).
% 0.30/0.47
% 0.30/0.47 %----Explicit typings (31)
% 0.30/0.47 thf(sy_cl_HOL_Otype,type,
% 0.30/0.47 type:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.30/0.47 ord:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.30/0.47 order:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.30/0.47 no_bot:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.30/0.47 no_top:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.47 linorder:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.30/0.47 preorder:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.30/0.47 wellorder:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.30/0.47 dense_order:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.30/0.47 dense_linorder:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 0.30/0.47 comple187826305attice:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 0.30/0.47 condit1656338222tinuum:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 0.30/0.47 condit1037483654norder:
% 0.30/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 0.30/0.47 coindu328551480prefix:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Olappend,type,
% 0.30/0.47 coinductive_lappend:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.30/0.47 coinductive_lfinite:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Ollast,type,
% 0.30/0.47 coinductive_llast:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 0.30/0.47 coinductive_LCons:
% 0.30/0.47 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.30/0.47 coinductive_LNil:
% 0.30/0.47 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.30/0.47 coinductive_lnull:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Olmember,type,
% 0.30/0.47 coinductive_lmember:
% 0.30/0.47 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Olprefix,type,
% 0.30/0.47 coinductive_lprefix:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 0.30/0.47 coindu1478340336prefix:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 0.30/0.47 coindu63249387sorted:
% 0.30/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 0.30/0.47 hilbert_GreatestM:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 0.30/0.47 hilbert_LeastM:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.47 ord_less:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.47 ord_less_eq:
% 0.30/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 0.30/0.47 order_antimono:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.30/0.47 order_strict_mono:
% 0.30/0.47 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Pure_Otype,type,
% 0.30/0.47 type2:
% 0.30/0.47 !>[A: $tType] : ( itself @ A ) ).
% 0.30/0.47
% 0.30/0.47 %----Relevant facts (254)
% 0.30/0.47 thf(fact_0_less__eq__llist__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less_eq @ ( coinductive_llist @ A ) )
% 0.30/0.47 = ( coinductive_lprefix @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_eq_llist_def
% 0.30/0.47 thf(fact_1_lprefix__refl,axiom,
% 0.30/0.47 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 0.30/0.47
% 0.30/0.47 % lprefix_refl
% 0.30/0.47 thf(fact_2_llist_Oleq__refl,axiom,
% 0.30/0.47 ! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% 0.30/0.47
% 0.30/0.47 % llist.leq_refl
% 0.30/0.47 thf(fact_3_order__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_refl
% 0.30/0.47 thf(fact_4_lprefix__trans,axiom,
% 0.30/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.30/0.47 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.47 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.30/0.47 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lprefix_trans
% 0.30/0.47 thf(fact_5_llist_Oleq__trans,axiom,
% 0.30/0.47 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,Z: coinductive_llist @ A] :
% 0.30/0.47 ( ( coinductive_lprefix @ A @ X @ Y )
% 0.30/0.47 => ( ( coinductive_lprefix @ A @ Y @ Z )
% 0.30/0.47 => ( coinductive_lprefix @ A @ X @ Z ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % llist.leq_trans
% 0.30/0.47 thf(fact_6_lprefix__antisym,axiom,
% 0.30/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.47 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.47 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 0.30/0.47 => ( Xs = Ys ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lprefix_antisym
% 0.30/0.47 thf(fact_7_llist_Oleq__antisym,axiom,
% 0.30/0.47 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
% 0.30/0.47 ( ( coinductive_lprefix @ A @ X @ Y )
% 0.30/0.47 => ( ( coinductive_lprefix @ A @ Y @ X )
% 0.30/0.47 => ( X = Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % llist.leq_antisym
% 0.30/0.47 thf(fact_8_lprefix__down__linear,axiom,
% 0.30/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.47 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 0.30/0.47 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.30/0.47 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.47 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lprefix_down_linear
% 0.30/0.47 thf(fact_9_le__funD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funD
% 0.30/0.47 thf(fact_10_le__funE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funE
% 0.30/0.47 thf(fact_11_le__funI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 => ! [F: A > B,G: A > B] :
% 0.30/0.47 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.30/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_funI
% 0.30/0.47 thf(fact_12_le__fun__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.30/0.47 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.47 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_fun_def
% 0.30/0.47 thf(fact_13_dual__order_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B2: A,A2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.antisym
% 0.30/0.47 thf(fact_14_dual__order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B2: A,A2: A,C: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.30/0.47 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.trans
% 0.30/0.47 thf(fact_15_linorder__wlog,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.30/0.47 ( ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( P @ A3 @ B3 ) )
% 0.30/0.47 => ( ! [A3: A,B3: A] :
% 0.30/0.47 ( ( P @ B3 @ A3 )
% 0.30/0.47 => ( P @ A3 @ B3 ) )
% 0.30/0.47 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_wlog
% 0.30/0.47 thf(fact_16_dual__order_Orefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.refl
% 0.30/0.47 thf(fact_17_order__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_trans
% 0.30/0.47 thf(fact_18_order__class_Oorder_Oantisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_class.order.antisym
% 0.30/0.47 thf(fact_19_ord__le__eq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A,C: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( B2 = C )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_trans
% 0.30/0.47 thf(fact_20_ord__eq__le__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A,C: A] :
% 0.30/0.47 ( ( A2 = B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_trans
% 0.30/0.47 thf(fact_21_antisym__conv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Y: A,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym_conv
% 0.30/0.47 thf(fact_22_le__cases3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.30/0.47 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.30/0.47 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.30/0.47 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases3
% 0.30/0.47 thf(fact_23_order_Otrans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A,C: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.trans
% 0.30/0.47 thf(fact_24_le__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % le_cases
% 0.30/0.47 thf(fact_25_wlog__linorder__le,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.30/0.47 ( ! [A3: A,B3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.30/0.47 => ( P @ A3 @ B3 ) )
% 0.30/0.47 => ( ( ( P @ B2 @ A2 )
% 0.30/0.47 => ( P @ A2 @ B2 ) )
% 0.30/0.47 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % wlog_linorder_le
% 0.30/0.47 thf(fact_26_eq__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( X = Y )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_refl
% 0.30/0.47 thf(fact_27_linear,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linear
% 0.30/0.47 thf(fact_28_antisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antisym
% 0.30/0.47 thf(fact_29_eq__iff,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.30/0.47 = ( ^ [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % eq_iff
% 0.30/0.47 thf(fact_30_ord__le__eq__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ( F @ B2 )
% 0.30/0.47 = C )
% 0.30/0.47 => ( ! [X2: A,Y4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_le_eq_subst
% 0.30/0.47 thf(fact_31_ord__eq__le__subst,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.47 ( ( A2
% 0.30/0.47 = ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.47 => ( ! [X2: B,Y4: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ord_eq_le_subst
% 0.30/0.47 thf(fact_32_order__subst2,axiom,
% 0.30/0.47 ! [A: $tType,C2: $tType] :
% 0.30/0.47 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.47 => ( ! [X2: A,Y4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst2
% 0.30/0.47 thf(fact_33_order__subst1,axiom,
% 0.30/0.47 ! [A: $tType,B: $tType] :
% 0.30/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.47 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.47 => ( ! [X2: B,Y4: B] :
% 0.30/0.47 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order_subst1
% 0.30/0.47 thf(fact_34_lfp_Oleq__antisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lfp.leq_antisym
% 0.30/0.47 thf(fact_35_gfp_Oleq__antisym,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Y: A,X: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % gfp.leq_antisym
% 0.30/0.47 thf(fact_36_lfp_Oleq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,Z: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.47 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lfp.leq_trans
% 0.30/0.47 thf(fact_37_gfp_Oleq__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [Y: A,X: A,Z: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.47 => ( ( ord_less_eq @ A @ Z @ Y )
% 0.30/0.47 => ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % gfp.leq_trans
% 0.30/0.47 thf(fact_38_lfp_Oleq__refl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.30/0.47
% 0.30/0.47 % lfp.leq_refl
% 0.30/0.47 thf(fact_39_Coinductive__List_Ofinite__lprefix__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( coindu328551480prefix @ A )
% 0.30/0.47 = ( coinductive_lprefix @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % Coinductive_List.finite_lprefix_def
% 0.30/0.47 thf(fact_40_strict__mono__less__eq,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.30/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.47 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 0.30/0.47 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % strict_mono_less_eq
% 0.30/0.47 thf(fact_41_antimono__def,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ( ( order_antimono @ A @ B )
% 0.30/0.47 = ( ^ [F2: A > B] :
% 0.30/0.47 ! [X3: A,Y3: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antimono_def
% 0.30/0.47 thf(fact_42_antimonoI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ! [F: A > B] :
% 0.30/0.47 ( ! [X2: A,Y4: A] :
% 0.30/0.47 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 0.30/0.47 => ( order_antimono @ A @ B @ F ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antimonoI
% 0.30/0.47 thf(fact_43_antimonoE,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.30/0.47 ( ( order_antimono @ A @ B @ F )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antimonoE
% 0.30/0.47 thf(fact_44_antimonoD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.30/0.47 ( ( order_antimono @ A @ B @ F )
% 0.30/0.47 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.47 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % antimonoD
% 0.30/0.47 thf(fact_45_ext,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.47 ( ! [X2: A] :
% 0.30/0.47 ( ( F @ X2 )
% 0.30/0.47 = ( G @ X2 ) )
% 0.30/0.47 => ( F = G ) ) ).
% 0.30/0.47
% 0.30/0.47 % ext
% 0.30/0.47 thf(fact_46_strict__mono__eq,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.30/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.47 => ( ( ( F @ X )
% 0.30/0.47 = ( F @ Y ) )
% 0.30/0.47 = ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % strict_mono_eq
% 0.30/0.47 thf(fact_47_decseqD,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [F: nat > A,I: nat,J: nat] :
% 0.30/0.47 ( ( order_antimono @ nat @ A @ F )
% 0.30/0.47 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.47 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % decseqD
% 0.30/0.47 thf(fact_48_decseq__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ( ( order_antimono @ nat @ A )
% 0.30/0.47 = ( ^ [X4: nat > A] :
% 0.30/0.47 ! [M: nat,N: nat] :
% 0.30/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.47 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % decseq_def
% 0.30/0.47 thf(fact_49_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( coinductive_lprefix @ A )
% 0.30/0.47 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 0.30/0.47 ( ( ( coinductive_lfinite @ A @ Xs2 )
% 0.30/0.47 => ( coindu328551480prefix @ A @ Xs2 @ Ys2 ) )
% 0.30/0.47 & ( ~ ( coinductive_lfinite @ A @ Xs2 )
% 0.30/0.47 => ( Xs2 = Ys2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Coinductive_List.lprefix_nitpick_simps
% 0.30/0.47 thf(fact_50_lstrict__prefix__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( coindu1478340336prefix @ A )
% 0.30/0.47 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 0.30/0.47 ( ( coinductive_lprefix @ A @ Xs2 @ Ys2 )
% 0.30/0.47 & ( Xs2 != Ys2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % lstrict_prefix_def
% 0.30/0.47 thf(fact_51_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 0.30/0.47 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 0.30/0.47
% 0.30/0.47 % Coinductive_List.finite_lprefix_nitpick_simps(2)
% 0.30/0.47 thf(fact_52_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.30/0.47 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.30/0.47 = ( Xs
% 0.30/0.47 = ( coinductive_LNil @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Coinductive_List.finite_lprefix_nitpick_simps(1)
% 0.30/0.47 thf(fact_53_strict__monoD,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ! [F: A > B,X: A,Y: A] :
% 0.30/0.47 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.47 => ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % strict_monoD
% 0.30/0.47 thf(fact_54_strict__monoI,axiom,
% 0.30/0.47 ! [B: $tType,A: $tType] :
% 0.30/0.47 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.47 => ! [F: A > B] :
% 0.30/0.47 ( ! [X2: A,Y4: A] :
% 0.30/0.47 ( ( ord_less @ A @ X2 @ Y4 )
% 0.30/0.47 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.47 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % strict_monoI
% 0.30/0.47 thf(fact_55_lprefix__code_I1_J,axiom,
% 0.30/0.47 ! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% 0.30/0.47
% 0.30/0.47 % lprefix_code(1)
% 0.30/0.47 thf(fact_56_lfinite__code_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % lfinite_code(1)
% 0.30/0.47 thf(fact_57_lstrict__prefix__code_I1_J,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % lstrict_prefix_code(1)
% 0.30/0.47 thf(fact_58_less__llist__def,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( ord_less @ ( coinductive_llist @ A ) )
% 0.30/0.47 = ( coindu1478340336prefix @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_llist_def
% 0.30/0.47 thf(fact_59_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B2: A,A2: A] :
% 0.30/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.47 => ( A2 != B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_implies_not_eq
% 0.30/0.47 thf(fact_60_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.47 => ( A2 != B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_implies_not_eq
% 0.30/0.47 thf(fact_61_not__less__iff__gr__or__eq,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.47 = ( ( ord_less @ A @ Y @ X )
% 0.30/0.47 | ( X = Y ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % not_less_iff_gr_or_eq
% 0.30/0.47 thf(fact_62_dual__order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [B2: A,A2: A,C: A] :
% 0.30/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.47 => ( ( ord_less @ A @ C @ B2 )
% 0.30/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.strict_trans
% 0.30/0.47 thf(fact_63_less__imp__not__less,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_less
% 0.30/0.47 thf(fact_64_order_Ostrict__trans,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A,B2: A,C: A] :
% 0.30/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.47 => ( ( ord_less @ A @ B2 @ C )
% 0.30/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % order.strict_trans
% 0.30/0.47 thf(fact_65_lstrict__prefix__lfinite1,axiom,
% 0.30/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.47 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 0.30/0.47 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.30/0.47
% 0.30/0.47 % lstrict_prefix_lfinite1
% 0.30/0.47 thf(fact_66_dual__order_Oirrefl,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [A2: A] :
% 0.30/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % dual_order.irrefl
% 0.30/0.47 thf(fact_67_linorder__cases,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( X != Y )
% 0.30/0.47 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % linorder_cases
% 0.30/0.47 thf(fact_68_less__imp__triv,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A,P: $o] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( ( ord_less @ A @ Y @ X )
% 0.30/0.47 => P ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_triv
% 0.30/0.47 thf(fact_69_less__imp__not__eq2,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.47 => ! [X: A,Y: A] :
% 0.30/0.47 ( ( ord_less @ A @ X @ Y )
% 0.30/0.47 => ( Y != X ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_imp_not_eq2
% 0.30/0.47 thf(fact_70_antisym__conv3,axiom,
% 0.30/0.47 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y: A,X: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ Y @ X )
% 0.30/0.48 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.48 = ( X = Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv3
% 0.30/0.48 thf(fact_71_less__induct,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,A2: A] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ! [Y5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Y5 @ X2 )
% 0.30/0.48 => ( P @ Y5 ) )
% 0.30/0.48 => ( P @ X2 ) )
% 0.30/0.48 => ( P @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_induct
% 0.30/0.48 thf(fact_72_less__not__sym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_not_sym
% 0.30/0.48 thf(fact_73_less__imp__not__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( X != Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_not_eq
% 0.30/0.48 thf(fact_74_dual__order_Oasym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.asym
% 0.30/0.48 thf(fact_75_ord__less__eq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( B2 = C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_less_eq_trans
% 0.30/0.48 thf(fact_76_ord__eq__less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( A2 = B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_less_trans
% 0.30/0.48 thf(fact_77_lfinite__LNil,axiom,
% 0.30/0.48 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite_LNil
% 0.30/0.48 thf(fact_78_less__irrefl,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_irrefl
% 0.30/0.48 thf(fact_79_less__linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 | ( X = Y )
% 0.30/0.48 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_linear
% 0.30/0.48 thf(fact_80_less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A,Z: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( ( ord_less @ A @ Y @ Z )
% 0.30/0.48 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_trans
% 0.30/0.48 thf(fact_81_less__asym_H,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_asym'
% 0.30/0.48 thf(fact_82_less__asym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_asym
% 0.30/0.48 thf(fact_83_less__imp__neq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( X != Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_neq
% 0.30/0.48 thf(fact_84_llist__less__induct,axiom,
% 0.30/0.48 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ! [Xs3: coinductive_llist @ A] :
% 0.30/0.48 ( ! [Ys3: coinductive_llist @ A] :
% 0.30/0.48 ( ( coindu1478340336prefix @ A @ Ys3 @ Xs3 )
% 0.30/0.48 => ( P @ Ys3 ) )
% 0.30/0.48 => ( P @ Xs3 ) )
% 0.30/0.48 => ( P @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist_less_induct
% 0.30/0.48 thf(fact_85_dense,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ? [Z3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Z3 )
% 0.30/0.48 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense
% 0.30/0.48 thf(fact_86_order_Oasym,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.asym
% 0.30/0.48 thf(fact_87_neq__iff,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( X != Y )
% 0.30/0.48 = ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % neq_iff
% 0.30/0.48 thf(fact_88_neqE,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( X != Y )
% 0.30/0.48 => ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % neqE
% 0.30/0.48 thf(fact_89_gt__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( no_top @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.30/0.48
% 0.30/0.48 % gt_ex
% 0.30/0.48 thf(fact_90_lt__ex,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] :
% 0.30/0.48 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % lt_ex
% 0.30/0.48 thf(fact_91_order__less__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C2: $tType] :
% 0.30/0.48 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.48 => ( ! [X2: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_subst2
% 0.30/0.48 thf(fact_92_order__less__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y4: B] :
% 0.30/0.48 ( ( ord_less @ B @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_subst1
% 0.30/0.48 thf(fact_93_ord__less__eq__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ( F @ B2 )
% 0.30/0.48 = C )
% 0.30/0.48 => ( ! [X2: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_less_eq_subst
% 0.30/0.48 thf(fact_94_ord__eq__less__subst,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( A2
% 0.30/0.48 = ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y4: B] :
% 0.30/0.48 ( ( ord_less @ B @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ord_eq_less_subst
% 0.30/0.48 thf(fact_95_not__lfinite__lprefix__conv__eq,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 = ( Xs = Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_lfinite_lprefix_conv_eq
% 0.30/0.48 thf(fact_96_lprefix__lfiniteD,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 => ( ( coinductive_lfinite @ A @ Ys )
% 0.30/0.48 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_lfiniteD
% 0.30/0.48 thf(fact_97_order_Onot__eq__order__implies__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( A2 != B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.not_eq_order_implies_strict
% 0.30/0.48 thf(fact_98_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_implies_order
% 0.30/0.48 thf(fact_99_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [B4: A,A4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.30/0.48 & ( A4 != B4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_iff_order
% 0.30/0.48 thf(fact_100_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [B4: A,A4: A] :
% 0.30/0.48 ( ( ord_less @ A @ B4 @ A4 )
% 0.30/0.48 | ( A4 = B4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.order_iff_strict
% 0.30/0.48 thf(fact_101_order_Ostrict__implies__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_implies_order
% 0.30/0.48 thf(fact_102_dense__le__bounded,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A,Z: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( ! [W: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ W )
% 0.30/0.48 => ( ( ord_less @ A @ W @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ W @ Z ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_le_bounded
% 0.30/0.48 thf(fact_103_dense__ge__bounded,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Z: A,X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z @ X )
% 0.30/0.48 => ( ! [W: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z @ W )
% 0.30/0.48 => ( ( ord_less @ A @ W @ X )
% 0.30/0.48 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_ge_bounded
% 0.30/0.48 thf(fact_104_dual__order_Ostrict__trans2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans2
% 0.30/0.48 thf(fact_105_dual__order_Ostrict__trans1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [B2: A,A2: A,C: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.48 => ( ( ord_less @ A @ C @ B2 )
% 0.30/0.48 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dual_order.strict_trans1
% 0.30/0.48 thf(fact_106_order_Ostrict__iff__order,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [A4: A,B4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.30/0.48 & ( A4 != B4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_iff_order
% 0.30/0.48 thf(fact_107_order_Oorder__iff__strict,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [A4: A,B4: A] :
% 0.30/0.48 ( ( ord_less @ A @ A4 @ B4 )
% 0.30/0.48 | ( A4 = B4 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.order_iff_strict
% 0.30/0.48 thf(fact_108_order_Ostrict__trans2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans2
% 0.30/0.48 thf(fact_109_order_Ostrict__trans1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,C: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less @ A @ B2 @ C )
% 0.30/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order.strict_trans1
% 0.30/0.48 thf(fact_110_not__le__imp__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y: A,X: A] :
% 0.30/0.48 ( ~ ( ord_less_eq @ A @ Y @ X )
% 0.30/0.48 => ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_le_imp_less
% 0.30/0.48 thf(fact_111_less__le__not__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_le_not_le
% 0.30/0.48 thf(fact_112_le__imp__less__or__eq,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.48 => ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 | ( X = Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_imp_less_or_eq
% 0.30/0.48 thf(fact_113_le__less__linear,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.48 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less_linear
% 0.30/0.48 thf(fact_114_dense__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y: A,Z: A] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ X2 @ Z ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_le
% 0.30/0.48 thf(fact_115_dense__ge,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Z: A,Y: A] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z @ X2 )
% 0.30/0.48 => ( ord_less_eq @ A @ Y @ X2 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % dense_ge
% 0.30/0.48 thf(fact_116_less__le__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A,Z: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.30/0.48 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_le_trans
% 0.30/0.48 thf(fact_117_le__less__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A,Z: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.48 => ( ( ord_less @ A @ Y @ Z )
% 0.30/0.48 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less_trans
% 0.30/0.48 thf(fact_118_antisym__conv2,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.48 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.48 = ( X = Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv2
% 0.30/0.48 thf(fact_119_antisym__conv1,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.48 = ( X = Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % antisym_conv1
% 0.30/0.48 thf(fact_120_less__imp__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_le
% 0.30/0.48 thf(fact_121_le__neq__trans,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( A2 != B2 )
% 0.30/0.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_neq_trans
% 0.30/0.48 thf(fact_122_not__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.30/0.48 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_less
% 0.30/0.48 thf(fact_123_not__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 0.30/0.48 = ( ord_less @ A @ Y @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_le
% 0.30/0.48 thf(fact_124_order__less__le__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C2: $tType] :
% 0.30/0.48 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.48 => ( ! [X2: A,Y4: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le_subst2
% 0.30/0.48 thf(fact_125_order__less__le__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y4: B] :
% 0.30/0.48 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_less_le_subst1
% 0.30/0.48 thf(fact_126_order__le__less__subst2,axiom,
% 0.30/0.48 ! [A: $tType,C2: $tType] :
% 0.30/0.48 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 0.30/0.48 => ( ! [X2: A,Y4: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less_subst2
% 0.30/0.48 thf(fact_127_order__le__less__subst1,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType] :
% 0.30/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.30/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.30/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.30/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.30/0.48 => ( ! [X2: B,Y4: B] :
% 0.30/0.48 ( ( ord_less @ B @ X2 @ Y4 )
% 0.30/0.48 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.30/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % order_le_less_subst1
% 0.30/0.48 thf(fact_128_less__le,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less @ A )
% 0.30/0.48 = ( ^ [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 & ( X3 != Y3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_le
% 0.30/0.48 thf(fact_129_le__less,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( ord_less_eq @ A )
% 0.30/0.48 = ( ^ [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.48 | ( X3 = Y3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_less
% 0.30/0.48 thf(fact_130_leI,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A] :
% 0.30/0.48 ( ~ ( ord_less @ A @ X @ Y )
% 0.30/0.48 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % leI
% 0.30/0.48 thf(fact_131_leD,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Y: A,X: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ Y @ X )
% 0.30/0.48 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % leD
% 0.30/0.48 thf(fact_132_LNil__lprefix,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 0.30/0.48
% 0.30/0.48 % LNil_lprefix
% 0.30/0.48 thf(fact_133_less__fun__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ( ( ord_less @ ( A > B ) )
% 0.30/0.48 = ( ^ [F2: A > B,G2: A > B] :
% 0.30/0.48 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.30/0.48 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_fun_def
% 0.30/0.48 thf(fact_134_strict__mono__less,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ! [F: A > B,X: A,Y: A] :
% 0.30/0.48 ( ( order_strict_mono @ A @ B @ F )
% 0.30/0.48 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
% 0.30/0.48 = ( ord_less @ A @ X @ Y ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_mono_less
% 0.30/0.48 thf(fact_135_strict__mono__def,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ( order @ A @ ( type2 @ A ) )
% 0.30/0.48 & ( order @ B @ ( type2 @ B ) ) )
% 0.30/0.48 => ( ( order_strict_mono @ A @ B )
% 0.30/0.48 = ( ^ [F2: A > B] :
% 0.30/0.48 ! [X3: A,Y3: A] :
% 0.30/0.48 ( ( ord_less @ A @ X3 @ Y3 )
% 0.30/0.48 => ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % strict_mono_def
% 0.30/0.48 thf(fact_136_minf_I8_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(8)
% 0.30/0.48 thf(fact_137_minf_I6_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(6)
% 0.30/0.48 thf(fact_138_pinf_I8_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(8)
% 0.30/0.48 thf(fact_139_pinf_I6_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(6)
% 0.30/0.48 thf(fact_140_complete__interval,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A,B2: A,P: A > $o] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.48 => ( ( P @ A2 )
% 0.30/0.48 => ( ~ ( P @ B2 )
% 0.30/0.48 => ? [C3: A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ A2 @ C3 )
% 0.30/0.48 & ( ord_less_eq @ A @ C3 @ B2 )
% 0.30/0.48 & ! [X5: A] :
% 0.30/0.48 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 0.30/0.48 & ( ord_less @ A @ X5 @ C3 ) )
% 0.30/0.48 => ( P @ X5 ) )
% 0.30/0.48 & ! [D: A] :
% 0.30/0.48 ( ! [X2: A] :
% 0.30/0.48 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.30/0.48 & ( ord_less @ A @ X2 @ D ) )
% 0.30/0.48 => ( P @ X2 ) )
% 0.30/0.48 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % complete_interval
% 0.30/0.48 thf(fact_141_lmember__code_I1_J,axiom,
% 0.30/0.48 ! [A: $tType,X: A] :
% 0.30/0.48 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % lmember_code(1)
% 0.30/0.48 thf(fact_142_llimit__induct,axiom,
% 0.30/0.48 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( P @ ( coinductive_LNil @ A ) )
% 0.30/0.48 => ( ! [X2: A,Xs3: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ Xs3 )
% 0.30/0.48 => ( ( P @ Xs3 )
% 0.30/0.48 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
% 0.30/0.48 => ( ( ! [Ys3: coinductive_llist @ A] :
% 0.30/0.48 ( ( coindu1478340336prefix @ A @ Ys3 @ Xs )
% 0.30/0.48 => ( P @ Ys3 ) )
% 0.30/0.48 => ( P @ Xs ) )
% 0.30/0.48 => ( P @ Xs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % llimit_induct
% 0.30/0.48 thf(fact_143_llist_Oinject,axiom,
% 0.30/0.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 0.30/0.48 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 0.30/0.48 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 0.30/0.48 = ( ( X21 = Y21 )
% 0.30/0.48 & ( X22 = Y22 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.inject
% 0.30/0.48 thf(fact_144_LCons__lprefix__LCons,axiom,
% 0.30/0.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.30/0.48 = ( ( X = Y )
% 0.30/0.48 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LCons_lprefix_LCons
% 0.30/0.48 thf(fact_145_lfinite__code_I2_J,axiom,
% 0.30/0.48 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.30/0.48 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 0.30/0.48 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite_code(2)
% 0.30/0.48 thf(fact_146_lfinite__LCons,axiom,
% 0.30/0.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.30/0.48 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite_LCons
% 0.30/0.48 thf(fact_147_lstrict__prefix__code_I4_J,axiom,
% 0.30/0.48 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 0.30/0.48 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 0.30/0.48 = ( ( X = Y )
% 0.30/0.48 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lstrict_prefix_code(4)
% 0.30/0.48 thf(fact_148_lstrict__prefix__code_I3_J,axiom,
% 0.30/0.48 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.30/0.48 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % lstrict_prefix_code(3)
% 0.30/0.48 thf(fact_149_lstrict__prefix__code_I2_J,axiom,
% 0.30/0.48 ! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% 0.30/0.48
% 0.30/0.48 % lstrict_prefix_code(2)
% 0.30/0.48 thf(fact_150_lmember__code_I2_J,axiom,
% 0.30/0.48 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.30/0.48 = ( ( X = Y )
% 0.30/0.48 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lmember_code(2)
% 0.30/0.48 thf(fact_151_Le__LCons,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Le_LCons
% 0.30/0.48 thf(fact_152_LCons__lprefix__conv,axiom,
% 0.30/0.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
% 0.30/0.48 = ( ? [Ys4: coinductive_llist @ A] :
% 0.30/0.48 ( ( Ys
% 0.30/0.48 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 0.30/0.48 & ( coinductive_lprefix @ A @ Xs @ Ys4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LCons_lprefix_conv
% 0.30/0.48 thf(fact_153_llist_Odistinct_I1_J,axiom,
% 0.30/0.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_LNil @ A )
% 0.30/0.48 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.distinct(1)
% 0.30/0.48 thf(fact_154_llist_Oexhaust,axiom,
% 0.30/0.48 ! [A: $tType,Y: coinductive_llist @ A] :
% 0.30/0.48 ( ( Y
% 0.30/0.48 != ( coinductive_LNil @ A ) )
% 0.30/0.48 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 0.30/0.48 ( Y
% 0.30/0.48 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.exhaust
% 0.30/0.48 thf(fact_155_neq__LNil__conv,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( Xs
% 0.30/0.48 != ( coinductive_LNil @ A ) )
% 0.30/0.48 = ( ? [X3: A,Xs4: coinductive_llist @ A] :
% 0.30/0.48 ( Xs
% 0.30/0.48 = ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % neq_LNil_conv
% 0.30/0.48 thf(fact_156_lfinite__LConsI,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite_LConsI
% 0.30/0.48 thf(fact_157_lprefix__code_I2_J,axiom,
% 0.30/0.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.30/0.48 ~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_code(2)
% 0.30/0.48 thf(fact_158_lprefix_Ocases,axiom,
% 0.30/0.48 ! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ A1 @ A22 )
% 0.30/0.48 => ( ( ( A1
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 => ! [Xs3: coinductive_llist @ A] : ( A22 != Xs3 ) )
% 0.30/0.48 => ~ ! [Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A,X2: A] :
% 0.30/0.48 ( ( A1
% 0.30/0.48 = ( coinductive_LCons @ A @ X2 @ Xs3 ) )
% 0.30/0.48 => ( ( A22
% 0.30/0.48 = ( coinductive_LCons @ A @ X2 @ Ys5 ) )
% 0.30/0.48 => ~ ( coinductive_lprefix @ A @ Xs3 @ Ys5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix.cases
% 0.30/0.48 thf(fact_159_lprefix_Osimps,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( coinductive_lprefix @ A )
% 0.30/0.48 = ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 0.30/0.48 ( ? [Xs2: coinductive_llist @ A] :
% 0.30/0.48 ( ( A12
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 & ( A23 = Xs2 ) )
% 0.30/0.48 | ? [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A,X3: A] :
% 0.30/0.48 ( ( A12
% 0.30/0.48 = ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 0.30/0.48 & ( A23
% 0.30/0.48 = ( coinductive_LCons @ A @ X3 @ Ys2 ) )
% 0.30/0.48 & ( coinductive_lprefix @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix.simps
% 0.30/0.48 thf(fact_160_lprefix_Ocoinduct,axiom,
% 0.30/0.48 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 0.30/0.48 ( ( X6 @ X @ Xa )
% 0.30/0.48 => ( ! [X2: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
% 0.30/0.48 ( ( X6 @ X2 @ Xa2 )
% 0.30/0.48 => ( ? [Xs5: coinductive_llist @ A] :
% 0.30/0.48 ( ( X2
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 & ( Xa2 = Xs5 ) )
% 0.30/0.48 | ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,Xb: A] :
% 0.30/0.48 ( ( X2
% 0.30/0.48 = ( coinductive_LCons @ A @ Xb @ Xs5 ) )
% 0.30/0.48 & ( Xa2
% 0.30/0.48 = ( coinductive_LCons @ A @ Xb @ Ys3 ) )
% 0.30/0.48 & ( ( X6 @ Xs5 @ Ys3 )
% 0.30/0.48 | ( coinductive_lprefix @ A @ Xs5 @ Ys3 ) ) ) ) )
% 0.30/0.48 => ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix.coinduct
% 0.30/0.48 thf(fact_161_lprefix__LCons__conv,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.30/0.48 = ( ( Xs
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 | ? [Xs4: coinductive_llist @ A] :
% 0.30/0.48 ( ( Xs
% 0.30/0.48 = ( coinductive_LCons @ A @ Y @ Xs4 ) )
% 0.30/0.48 & ( coinductive_lprefix @ A @ Xs4 @ Ys ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_LCons_conv
% 0.30/0.48 thf(fact_162_lfinite_Ocases,axiom,
% 0.30/0.48 ! [A: $tType,A2: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ A2 )
% 0.30/0.48 => ( ( A2
% 0.30/0.48 != ( coinductive_LNil @ A ) )
% 0.30/0.48 => ~ ! [Xs3: coinductive_llist @ A] :
% 0.30/0.48 ( ? [X2: A] :
% 0.30/0.48 ( A2
% 0.30/0.48 = ( coinductive_LCons @ A @ X2 @ Xs3 ) )
% 0.30/0.48 => ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite.cases
% 0.30/0.48 thf(fact_163_lfinite_Osimps,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( coinductive_lfinite @ A )
% 0.30/0.48 = ( ^ [A4: coinductive_llist @ A] :
% 0.30/0.48 ( ( A4
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 | ? [Xs2: coinductive_llist @ A,X3: A] :
% 0.30/0.48 ( ( A4
% 0.30/0.48 = ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 0.30/0.48 & ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite.simps
% 0.30/0.48 thf(fact_164_lfinite_Oinducts,axiom,
% 0.30/0.48 ! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ X )
% 0.30/0.48 => ( ( P @ ( coinductive_LNil @ A ) )
% 0.30/0.48 => ( ! [Xs3: coinductive_llist @ A,X2: A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ Xs3 )
% 0.30/0.48 => ( ( P @ Xs3 )
% 0.30/0.48 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
% 0.30/0.48 => ( P @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite.inducts
% 0.30/0.48 thf(fact_165_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.30/0.48 = ( ( Xs
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 | ? [Xs4: coinductive_llist @ A] :
% 0.30/0.48 ( ( Xs
% 0.30/0.48 = ( coinductive_LCons @ A @ Y @ Xs4 ) )
% 0.30/0.48 & ( coindu328551480prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Coinductive_List.finite_lprefix_nitpick_simps(3)
% 0.30/0.48 thf(fact_166_ex__gt__or__lt,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: A] :
% 0.30/0.48 ? [B3: A] :
% 0.30/0.48 ( ( ord_less @ A @ A2 @ B3 )
% 0.30/0.48 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_gt_or_lt
% 0.30/0.48 thf(fact_167_pinf_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.48 => ( ( P @ X2 )
% 0.30/0.48 = ( P2 @ X2 ) ) )
% 0.30/0.48 => ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.48 => ( ( Q @ X2 )
% 0.30/0.48 = ( Q2 @ X2 ) ) )
% 0.30/0.48 => ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ( ( ( P @ X5 )
% 0.30/0.48 & ( Q @ X5 ) )
% 0.30/0.48 = ( ( P2 @ X5 )
% 0.30/0.48 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(1)
% 0.30/0.48 thf(fact_168_pinf_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.48 => ( ( P @ X2 )
% 0.30/0.48 = ( P2 @ X2 ) ) )
% 0.30/0.48 => ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z4 @ X2 )
% 0.30/0.48 => ( ( Q @ X2 )
% 0.30/0.48 = ( Q2 @ X2 ) ) )
% 0.30/0.48 => ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ( ( ( P @ X5 )
% 0.30/0.48 | ( Q @ X5 ) )
% 0.30/0.48 = ( ( P2 @ X5 )
% 0.30/0.48 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(2)
% 0.30/0.48 thf(fact_169_pinf_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ( X5 != T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(3)
% 0.30/0.48 thf(fact_170_pinf_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ( X5 != T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(4)
% 0.30/0.48 thf(fact_171_pinf_I5_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(5)
% 0.30/0.48 thf(fact_172_pinf_I7_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ Z3 @ X5 )
% 0.30/0.48 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(7)
% 0.30/0.48 thf(fact_173_pinf_I11_J,axiom,
% 0.30/0.48 ! [C2: $tType,D2: $tType] :
% 0.30/0.48 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 => ! [F3: D2] :
% 0.30/0.48 ? [Z3: C2] :
% 0.30/0.48 ! [X5: C2] :
% 0.30/0.48 ( ( ord_less @ C2 @ Z3 @ X5 )
% 0.30/0.48 => ( F3 = F3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % pinf(11)
% 0.30/0.48 thf(fact_174_minf_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.48 => ( ( P @ X2 )
% 0.30/0.48 = ( P2 @ X2 ) ) )
% 0.30/0.48 => ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.48 => ( ( Q @ X2 )
% 0.30/0.48 = ( Q2 @ X2 ) ) )
% 0.30/0.48 => ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ( ( ( P @ X5 )
% 0.30/0.48 & ( Q @ X5 ) )
% 0.30/0.48 = ( ( P2 @ X5 )
% 0.30/0.48 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(1)
% 0.30/0.48 thf(fact_175_minf_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 0.30/0.48 ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.48 => ( ( P @ X2 )
% 0.30/0.48 = ( P2 @ X2 ) ) )
% 0.30/0.48 => ( ? [Z4: A] :
% 0.30/0.48 ! [X2: A] :
% 0.30/0.48 ( ( ord_less @ A @ X2 @ Z4 )
% 0.30/0.48 => ( ( Q @ X2 )
% 0.30/0.48 = ( Q2 @ X2 ) ) )
% 0.30/0.48 => ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ( ( ( P @ X5 )
% 0.30/0.48 | ( Q @ X5 ) )
% 0.30/0.48 = ( ( P2 @ X5 )
% 0.30/0.48 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(2)
% 0.30/0.48 thf(fact_176_minf_I3_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ( X5 != T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(3)
% 0.30/0.48 thf(fact_177_minf_I4_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ( X5 != T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(4)
% 0.30/0.48 thf(fact_178_minf_I5_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(5)
% 0.30/0.48 thf(fact_179_minf_I7_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [T: A] :
% 0.30/0.48 ? [Z3: A] :
% 0.30/0.48 ! [X5: A] :
% 0.30/0.48 ( ( ord_less @ A @ X5 @ Z3 )
% 0.30/0.48 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(7)
% 0.30/0.48 thf(fact_180_minf_I11_J,axiom,
% 0.30/0.48 ! [C2: $tType,D2: $tType] :
% 0.30/0.48 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 0.30/0.48 => ! [F3: D2] :
% 0.30/0.48 ? [Z3: C2] :
% 0.30/0.48 ! [X5: C2] :
% 0.30/0.48 ( ( ord_less @ C2 @ X5 @ Z3 )
% 0.30/0.48 => ( F3 = F3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % minf(11)
% 0.30/0.48 thf(fact_181_llast__singleton,axiom,
% 0.30/0.48 ! [A: $tType,X: A] :
% 0.30/0.48 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 0.30/0.48 = X ) ).
% 0.30/0.48
% 0.30/0.48 % llast_singleton
% 0.30/0.48 thf(fact_182_lsorted_Ocases,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [A2: coinductive_llist @ A] :
% 0.30/0.48 ( ( coindu63249387sorted @ A @ A2 )
% 0.30/0.48 => ( ( A2
% 0.30/0.48 != ( coinductive_LNil @ A ) )
% 0.30/0.48 => ( ! [X2: A] :
% 0.30/0.48 ( A2
% 0.30/0.48 != ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
% 0.30/0.48 => ~ ! [X2: A,Y4: A,Xs3: coinductive_llist @ A] :
% 0.30/0.48 ( ( A2
% 0.30/0.48 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) )
% 0.30/0.48 => ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.30/0.48 => ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lsorted.cases
% 0.30/0.48 thf(fact_183_lsorted_Osimps,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( ( coindu63249387sorted @ A )
% 0.30/0.48 = ( ^ [A4: coinductive_llist @ A] :
% 0.30/0.48 ( ( A4
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 | ? [X3: A] :
% 0.30/0.48 ( A4
% 0.30/0.48 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 0.30/0.48 | ? [X3: A,Y3: A,Xs2: coinductive_llist @ A] :
% 0.30/0.48 ( ( A4
% 0.30/0.48 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) )
% 0.30/0.48 & ( ord_less_eq @ A @ X3 @ Y3 )
% 0.30/0.48 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lsorted.simps
% 0.30/0.48 thf(fact_184_lsorted__code_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lsorted_code(1)
% 0.30/0.48 thf(fact_185_llast__LCons2,axiom,
% 0.30/0.48 ! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 0.30/0.48 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % llast_LCons2
% 0.30/0.48 thf(fact_186_lsorted__LCons__LCons,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 0.30/0.48 = ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.48 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lsorted_LCons_LCons
% 0.30/0.48 thf(fact_187_lsorted__code_I2_J,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lsorted_code(2)
% 0.30/0.48 thf(fact_188_lsorted__lprefixD,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 => ( ( coindu63249387sorted @ A @ Ys )
% 0.30/0.48 => ( coindu63249387sorted @ A @ Xs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lsorted_lprefixD
% 0.30/0.48 thf(fact_189_LNil,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LNil
% 0.30/0.48 thf(fact_190_LCons__LCons,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.30/0.48 => ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 0.30/0.48 => ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LCons_LCons
% 0.30/0.48 thf(fact_191_Singleton,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Singleton
% 0.30/0.48 thf(fact_192_lsorted_Ocoinduct,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.30/0.48 => ! [X6: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
% 0.30/0.48 ( ( X6 @ X )
% 0.30/0.48 => ( ! [X2: coinductive_llist @ A] :
% 0.30/0.48 ( ( X6 @ X2 )
% 0.30/0.48 => ( ( X2
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 | ? [Xa3: A] :
% 0.30/0.48 ( X2
% 0.30/0.48 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
% 0.30/0.48 | ? [Xa3: A,Y5: A,Xs5: coinductive_llist @ A] :
% 0.30/0.48 ( ( X2
% 0.30/0.48 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) )
% 0.30/0.48 & ( ord_less_eq @ A @ Xa3 @ Y5 )
% 0.30/0.48 & ( ( X6 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) )
% 0.30/0.48 | ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) ) ) ) )
% 0.30/0.48 => ( coindu63249387sorted @ A @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lsorted.coinduct
% 0.30/0.48 thf(fact_193_llast__lappend__LCons,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 0.30/0.48 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % llast_lappend_LCons
% 0.30/0.48 thf(fact_194_ex__has__greatest__nat,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ! [Y4: A] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
% 0.30/0.48 => ? [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 & ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_has_greatest_nat
% 0.30/0.48 thf(fact_195_ex__has__least__nat,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ? [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 & ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % ex_has_least_nat
% 0.30/0.48 thf(fact_196_lappend__code_I2_J,axiom,
% 0.30/0.48 ! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
% 0.30/0.48 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_code(2)
% 0.30/0.48 thf(fact_197_lappend__LNil2,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.30/0.48 = Xs ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_LNil2
% 0.30/0.48 thf(fact_198_lappend__code_I1_J,axiom,
% 0.30/0.48 ! [A: $tType,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 0.30/0.48 = Ys ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_code(1)
% 0.30/0.48 thf(fact_199_lfinite__lappend,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.30/0.48 = ( ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite_lappend
% 0.30/0.48 thf(fact_200_lprefix__lappend__same,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 0.30/0.48 = ( ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 => ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_lappend_same
% 0.30/0.48 thf(fact_201_lappend__snocL1__conv__LCons2,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
% 0.30/0.48 = ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_snocL1_conv_LCons2
% 0.30/0.48 thf(fact_202_lappend__assoc,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.30/0.48 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_assoc
% 0.30/0.48 thf(fact_203_lappend__inf,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.30/0.48 = Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_inf
% 0.30/0.48 thf(fact_204_lappend__eq__LNil__iff,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 = ( ( Xs
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 & ( Ys
% 0.30/0.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_eq_LNil_iff
% 0.30/0.48 thf(fact_205_LNil__eq__lappend__iff,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( ( coinductive_LNil @ A )
% 0.30/0.48 = ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.30/0.48 = ( ( Xs
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 & ( Ys
% 0.30/0.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LNil_eq_lappend_iff
% 0.30/0.48 thf(fact_206_lappend__LNil__LNil,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
% 0.30/0.48 = ( coinductive_LNil @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_LNil_LNil
% 0.30/0.48 thf(fact_207_lprefix__lappend__sameI,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 => ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_lappend_sameI
% 0.30/0.48 thf(fact_208_lprefix__conv__lappend,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( coinductive_lprefix @ A )
% 0.30/0.48 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 0.30/0.48 ? [Zs2: coinductive_llist @ A] :
% 0.30/0.48 ( Ys2
% 0.30/0.48 = ( coinductive_lappend @ A @ Xs2 @ Zs2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_conv_lappend
% 0.30/0.48 thf(fact_209_lprefix__lappendD,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
% 0.30/0.48 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_lappendD
% 0.30/0.48 thf(fact_210_lappend__lprefixE,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.30/0.48 => ~ ! [Zs3: coinductive_llist @ A] :
% 0.30/0.48 ( Zs
% 0.30/0.48 != ( coinductive_lappend @ A @ Xs @ Zs3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_lprefixE
% 0.30/0.48 thf(fact_211_lprefix__lappend,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_lappend
% 0.30/0.48 thf(fact_212_lfinite__rev__induct,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 => ( ( P @ ( coinductive_LNil @ A ) )
% 0.30/0.48 => ( ! [X2: A,Xs3: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lfinite @ A @ Xs3 )
% 0.30/0.48 => ( ( P @ Xs3 )
% 0.30/0.48 => ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
% 0.30/0.48 => ( P @ Xs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lfinite_rev_induct
% 0.30/0.48 thf(fact_213_nat__less__le,axiom,
% 0.30/0.48 ( ( ord_less @ nat )
% 0.30/0.48 = ( ^ [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M @ N )
% 0.30/0.48 & ( M != N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_less_le
% 0.30/0.48 thf(fact_214_less__imp__le__nat,axiom,
% 0.30/0.48 ! [M2: nat,N2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M2 @ N2 )
% 0.30/0.48 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_imp_le_nat
% 0.30/0.48 thf(fact_215_nat__le__linear,axiom,
% 0.30/0.48 ! [M2: nat,N2: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.30/0.48 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % nat_le_linear
% 0.30/0.48 thf(fact_216_le__antisym,axiom,
% 0.30/0.48 ! [M2: nat,N2: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 0.30/0.48 => ( M2 = N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_antisym
% 0.30/0.48 thf(fact_217_eq__imp__le,axiom,
% 0.30/0.48 ! [M2: nat,N2: nat] :
% 0.30/0.48 ( ( M2 = N2 )
% 0.30/0.48 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % eq_imp_le
% 0.30/0.48 thf(fact_218_le__trans,axiom,
% 0.30/0.48 ! [I: nat,J: nat,K: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ J @ K )
% 0.30/0.48 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_trans
% 0.30/0.48 thf(fact_219_le__refl,axiom,
% 0.30/0.48 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.30/0.48
% 0.30/0.48 % le_refl
% 0.30/0.48 thf(fact_220_less__mono__imp__le__mono,axiom,
% 0.30/0.48 ! [F: nat > nat,I: nat,J: nat] :
% 0.30/0.48 ( ! [I2: nat,J2: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ I2 @ J2 )
% 0.30/0.48 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.30/0.48 => ( ( ord_less_eq @ nat @ I @ J )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_mono_imp_le_mono
% 0.30/0.48 thf(fact_221_le__neq__implies__less,axiom,
% 0.30/0.48 ! [M2: nat,N2: nat] :
% 0.30/0.48 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.30/0.48 => ( ( M2 != N2 )
% 0.30/0.48 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_neq_implies_less
% 0.30/0.48 thf(fact_222_less__or__eq__imp__le,axiom,
% 0.30/0.48 ! [M2: nat,N2: nat] :
% 0.30/0.48 ( ( ( ord_less @ nat @ M2 @ N2 )
% 0.30/0.48 | ( M2 = N2 ) )
% 0.30/0.48 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_or_eq_imp_le
% 0.30/0.48 thf(fact_223_le__eq__less__or__eq,axiom,
% 0.30/0.48 ( ( ord_less_eq @ nat )
% 0.30/0.48 = ( ^ [M: nat,N: nat] :
% 0.30/0.48 ( ( ord_less @ nat @ M @ N )
% 0.30/0.48 | ( M = N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_eq_less_or_eq
% 0.30/0.48 thf(fact_224_GreatestM__nat__le,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,X: A,M2: A > nat,B2: nat] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ! [Y4: A] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % GreatestM_nat_le
% 0.30/0.48 thf(fact_225_GreatestM__nat__lemma,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ! [Y4: A] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
% 0.30/0.48 => ( ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) )
% 0.30/0.48 & ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % GreatestM_nat_lemma
% 0.30/0.48 thf(fact_226_GreatestMI2,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ! [Y4: A] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( M2 @ Y4 ) @ ( M2 @ X ) ) )
% 0.30/0.48 => ( ! [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 => ( ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
% 0.30/0.48 => ( Q @ X2 ) ) )
% 0.30/0.48 => ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % GreatestMI2
% 0.30/0.48 thf(fact_227_LeastMI2,axiom,
% 0.30/0.48 ! [B: $tType,A: $tType] :
% 0.30/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.30/0.48 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ! [Y4: A] :
% 0.30/0.48 ( ( P @ Y4 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y4 ) ) )
% 0.30/0.48 => ( ! [X2: A] :
% 0.30/0.48 ( ( P @ X2 )
% 0.30/0.48 => ( ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
% 0.30/0.48 => ( Q @ X2 ) ) )
% 0.30/0.48 => ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastMI2
% 0.30/0.48 thf(fact_228_lstrict__prefix__lappend__conv,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.30/0.48 = ( ( coinductive_lfinite @ A @ Xs )
% 0.30/0.48 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lstrict_prefix_lappend_conv
% 0.30/0.48 thf(fact_229_lappend_Odisc__iff_I2_J,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.30/0.48 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend.disc_iff(2)
% 0.30/0.48 thf(fact_230_lnull__lappend,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.30/0.48 = ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lnull_lappend
% 0.30/0.48 thf(fact_231_lprefix__LNil,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.30/0.48 = ( coinductive_lnull @ A @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_LNil
% 0.30/0.48 thf(fact_232_lappend_Octr_I1_J,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_lnull @ A @ Ys )
% 0.30/0.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.30/0.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend.ctr(1)
% 0.30/0.48 thf(fact_233_lappend_Odisc_I2_J,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 | ~ ( coinductive_lnull @ A @ Ys ) )
% 0.30/0.48 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend.disc(2)
% 0.30/0.48 thf(fact_234_lappend_Odisc_I1_J,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_lnull @ A @ Ys )
% 0.30/0.48 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend.disc(1)
% 0.30/0.48 thf(fact_235_lappend__lnull1,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.30/0.48 = Ys ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_lnull1
% 0.30/0.48 thf(fact_236_lappend__lnull2,axiom,
% 0.30/0.48 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Ys )
% 0.30/0.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.30/0.48 = Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend_lnull2
% 0.30/0.48 thf(fact_237_LeastM__nat__lemma,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 0.30/0.48 ( ( P @ K )
% 0.30/0.48 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
% 0.30/0.48 & ! [Y5: A] :
% 0.30/0.48 ( ( P @ Y5 )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastM_nat_lemma
% 0.30/0.48 thf(fact_238_LeastM__nat__le,axiom,
% 0.30/0.48 ! [A: $tType,P: A > $o,X: A,M2: A > nat] :
% 0.30/0.48 ( ( P @ X )
% 0.30/0.48 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % LeastM_nat_le
% 0.30/0.48 thf(fact_239_llast__LCons,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.30/0.48 ( ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.30/0.48 = X ) )
% 0.30/0.48 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.30/0.48 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % llast_LCons
% 0.30/0.48 thf(fact_240_llist_Odisc_I2_J,axiom,
% 0.30/0.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.30/0.48 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.disc(2)
% 0.30/0.48 thf(fact_241_llist_OdiscI_I2_J,axiom,
% 0.30/0.48 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 0.30/0.48 ( ( Llist
% 0.30/0.48 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.30/0.48 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.discI(2)
% 0.30/0.48 thf(fact_242_not__lnull__conv,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 0.30/0.48 = ( ? [X3: A,Xs4: coinductive_llist @ A] :
% 0.30/0.48 ( Xs
% 0.30/0.48 = ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_lnull_conv
% 0.30/0.48 thf(fact_243_lnull__lprefix,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 0.30/0.48
% 0.30/0.48 % lnull_lprefix
% 0.30/0.48 thf(fact_244_lprefix__lnull,axiom,
% 0.30/0.48 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Ys )
% 0.30/0.48 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_lnull
% 0.30/0.48 thf(fact_245_lprefix__lnullD,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 => ( ( coinductive_lnull @ A @ Ys )
% 0.30/0.48 => ( coinductive_lnull @ A @ Xs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_lnullD
% 0.30/0.48 thf(fact_246_lprefix__not__lnullD,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.30/0.48 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lprefix_not_lnullD
% 0.30/0.48 thf(fact_247_llist_Odisc_I1_J,axiom,
% 0.30/0.48 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.disc(1)
% 0.30/0.48 thf(fact_248_llist_OdiscI_I1_J,axiom,
% 0.30/0.48 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.30/0.48 ( ( Llist
% 0.30/0.48 = ( coinductive_LNil @ A ) )
% 0.30/0.48 => ( coinductive_lnull @ A @ Llist ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.discI(1)
% 0.30/0.48 thf(fact_249_llist_Ocollapse_I1_J,axiom,
% 0.30/0.48 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Llist )
% 0.30/0.48 => ( Llist
% 0.30/0.48 = ( coinductive_LNil @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % llist.collapse(1)
% 0.30/0.48 thf(fact_250_lnull__def,axiom,
% 0.30/0.48 ! [A: $tType] :
% 0.30/0.48 ( ( coinductive_lnull @ A )
% 0.30/0.48 = ( ^ [Llist2: coinductive_llist @ A] :
% 0.30/0.48 ( Llist2
% 0.30/0.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lnull_def
% 0.30/0.48 thf(fact_251_lnull__imp__lfinite,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.30/0.48 ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % lnull_imp_lfinite
% 0.30/0.48 thf(fact_252_lappend_Oexhaust,axiom,
% 0.30/0.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.30/0.48 ( ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.30/0.48 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lappend.exhaust
% 0.30/0.48 thf(fact_253_lzip_Oexhaust,axiom,
% 0.30/0.48 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.30/0.48 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 | ( coinductive_lnull @ B @ Ys ) )
% 0.30/0.48 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.30/0.48 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lzip.exhaust
% 0.30/0.48
% 0.30/0.48 %----Type constructors (19)
% 0.30/0.48 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.30/0.48 ! [A5: $tType,A6: $tType] :
% 0.30/0.48 ( ( comple187826305attice @ A6 @ ( type2 @ A6 ) )
% 0.30/0.48 => ( comple187826305attice @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.30/0.48 ! [A5: $tType,A6: $tType] :
% 0.30/0.48 ( ( preorder @ A6 @ ( type2 @ A6 ) )
% 0.30/0.48 => ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.30/0.48 ! [A5: $tType,A6: $tType] :
% 0.30/0.48 ( ( order @ A6 @ ( type2 @ A6 ) )
% 0.30/0.48 => ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.30/0.48 ! [A5: $tType,A6: $tType] :
% 0.30/0.48 ( ( ord @ A6 @ ( type2 @ A6 ) )
% 0.30/0.48 => ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 0.30/0.48 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.30/0.48 wellorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.30/0.48 preorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.30/0.48 linorder @ nat @ ( type2 @ nat ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.30/0.48 no_top @ nat @ ( type2 @ nat ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.30/0.48 order @ nat @ ( type2 @ nat ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.30/0.48 ord @ nat @ ( type2 @ nat ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_4,axiom,
% 0.30/0.48 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 0.30/0.48 preorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 0.30/0.48 linorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 0.30/0.48 order @ $o @ ( type2 @ $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 0.30/0.48 ord @ $o @ ( type2 @ $o ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_9,axiom,
% 0.30/0.48 ! [A5: $tType] : ( preorder @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_10,axiom,
% 0.30/0.48 ! [A5: $tType] : ( order @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(tcon_Coinductive__List_Ollist___Orderings_Oord_11,axiom,
% 0.30/0.48 ! [A5: $tType] : ( ord @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 %----Conjectures (1)
% 0.30/0.48 thf(conj_0,conjecture,
% 0.30/0.48 ( ( ord_less_eq @ ( coinductive_llist @ a ) )
% 0.30/0.48 = ( coinductive_lprefix @ a ) ) ).
% 0.30/0.48
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 ------------------------
% 2.97/1.78 ---- Embedded in THF ---
% 2.97/1.78 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.97/1.78 %%% Generated on Tue Feb 24 06:08:58 EST 2026
% 2.97/1.78 %%% using '$$dhol' embedding, version 1.3.0.
% 2.97/1.78 %%% Logic specification used:
% 2.97/1.78 %%% thf(spec, logic, $$dhol).
% 2.97/1.78
% 2.97/1.78 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.CZmJ4dWwbg/DTF2DTF20653.p
% 2.97/1.78 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.97/1.78 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.97/1.78 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.97/1.78 thf(ty_tf_a, type, a: $tType).
% 2.97/1.78 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple187826305attice: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum, type, condit1656338222tinuum: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder, type, condit1037483654norder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Ofinite__lprefix, type, coindu328551480prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Olappend, type, coinductive_lappend: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Ollast, type, coinductive_llast: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Ollist_OLCons, type, coinductive_LCons: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Ollist_Olnull, type, coinductive_lnull: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Olprefix, type, coinductive_lprefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Olstrict__prefix, type, coindu1478340336prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.97/1.78 thf(sy_c_Coinductive__List_Oord__class_Olsorted, type, coindu63249387sorted: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.97/1.78 thf(sy_c_Hilbert__Choice_OGreatestM, type, hilbert_GreatestM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.97/1.78 thf(sy_c_Hilbert__Choice_OLeastM, type, hilbert_LeastM: (!> [A:$tType,B:$tType]: (((A > B) > ((A > $o) > A))))).
% 2.97/1.78 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.97/1.78 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.97/1.78 thf(sy_c_Orderings_Oorder__class_Oantimono, type, order_antimono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.97/1.78 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 2.97/1.78 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.97/1.78 thf(fact_0_less__eq__llist__def, axiom, (! [A:$tType]: (((ord_less_eq @ (coinductive_llist @ A)) = (coinductive_lprefix @ A))))).
% 2.97/1.78 thf(fact_1_lprefix__refl, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ Xs) @ Xs)))).
% 2.97/1.78 thf(fact_2_llist_Oleq__refl, axiom, (! [A:$tType,X:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ X) @ X)))).
% 2.97/1.78 thf(fact_3_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.97/1.78 thf(fact_4_lprefix__trans, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lprefix @ A) @ Ys) @ Zs) => (((coinductive_lprefix @ A) @ Xs) @ Zs)))))).
% 2.97/1.78 thf(fact_5_llist_Oleq__trans, axiom, (! [A:$tType,X:(coinductive_llist @ A),Y:(coinductive_llist @ A),Z:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X) @ Y) => ((((coinductive_lprefix @ A) @ Y) @ Z) => (((coinductive_lprefix @ A) @ X) @ Z)))))).
% 2.97/1.78 thf(fact_6_lprefix__antisym, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lprefix @ A) @ Ys) @ Xs) => (Xs = Ys)))))).
% 2.97/1.78 thf(fact_7_llist_Oleq__antisym, axiom, (! [A:$tType,X:(coinductive_llist @ A),Y:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X) @ Y) => ((((coinductive_lprefix @ A) @ Y) @ X) => (X = Y)))))).
% 2.97/1.78 thf(fact_8_lprefix__down__linear, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Zs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Zs) => ((((coinductive_lprefix @ A) @ Ys) @ Zs) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) | (((coinductive_lprefix @ A) @ Ys) @ Xs))))))).
% 2.97/1.78 thf(fact_9_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.97/1.78 thf(fact_10_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.97/1.78 thf(fact_11_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.97/1.78 thf(fact_12_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G2 @ X3))))))))))).
% 2.97/1.78 thf(fact_13_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.97/1.78 thf(fact_14_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.97/1.78 thf(fact_15_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((! [A3:A,B3:A]: ((((P @ B3) @ A3) => ((P @ A3) @ B3)))) => ((P @ A2) @ B2))))))))).
% 2.97/1.78 thf(fact_16_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.97/1.78 thf(fact_17_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.97/1.78 thf(fact_18_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.97/1.78 thf(fact_19_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_20_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_21_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.97/1.78 thf(fact_22_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z)))) => (((((ord_less_eq @ A) @ X) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X)))) => (~ (((((ord_less_eq @ A) @ Z) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 2.97/1.78 thf(fact_23_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_24_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.78 thf(fact_25_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A2:A]: (((! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((P @ A3) @ B3)))) => ((((P @ B2) @ A2) => ((P @ A2) @ B2)) => ((P @ A2) @ B2))))))))).
% 2.97/1.78 thf(fact_26_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.97/1.78 thf(fact_27_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.78 thf(fact_28_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.97/1.78 thf(fact_29_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X3))))))))).
% 2.97/1.78 thf(fact_30_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ B) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.97/1.78 thf(fact_31_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.97/1.78 thf(fact_32_order__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ C2) @ (F @ A2)) @ C)))))))))).
% 2.97/1.78 thf(fact_33_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.97/1.78 thf(fact_34_lfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.97/1.78 thf(fact_35_gfp_Oleq__antisym, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) => (X = Y))))))))).
% 2.97/1.78 thf(fact_36_lfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.97/1.78 thf(fact_37_gfp_Oleq__trans, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [Y:A,X:A,Z:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ Z) @ Y) => (((ord_less_eq @ A) @ Z) @ X))))))))).
% 2.97/1.78 thf(fact_38_lfp_Oleq__refl, axiom, (! [A:$tType]: ((((comple187826305attice @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.97/1.78 thf(fact_39_Coinductive__List_Ofinite__lprefix__def, axiom, (! [A:$tType]: (((coindu328551480prefix @ A) = (coinductive_lprefix @ A))))).
% 2.97/1.78 thf(fact_40_strict__mono__less__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)) = (((ord_less_eq @ A) @ X) @ Y))))))))).
% 2.97/1.78 thf(fact_41_antimono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_antimono @ A) @ B) = (^ [F2:(A > B)]: ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F2 @ Y3)) @ (F2 @ X3)))))))))))).
% 2.97/1.78 thf(fact_42_antimonoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ B) @ (F @ Y4)) @ (F @ X2))))) => (((order_antimono @ A) @ B) @ F)))))))).
% 2.97/1.78 thf(fact_43_antimonoE, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 2.97/1.78 thf(fact_44_antimonoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_antimono @ A) @ B) @ F) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F @ Y)) @ (F @ X)))))))))).
% 2.97/1.78 thf(fact_45_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.97/1.78 thf(fact_46_strict__mono__eq, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => (((F @ X) = (F @ Y)) = (X = Y))))))))).
% 2.97/1.78 thf(fact_47_decseqD, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [F:(nat > A),I:nat,J:nat]: (((((order_antimono @ nat) @ A) @ F) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ A) @ (F @ J)) @ (F @ I)))))))))).
% 2.97/1.78 thf(fact_48_decseq__def, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (((order_antimono @ nat) @ A) = (^ [X4:(nat > A)]: ((! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => (((ord_less_eq @ A) @ (X4 @ N)) @ (X4 @ M)))))))))))).
% 2.97/1.78 thf(fact_49_Coinductive__List_Olprefix__nitpick__simps, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [Xs2:(coinductive_llist @ A),Ys2:(coinductive_llist @ A)]: (((((coinductive_lfinite @ A) @ Xs2) => (((coindu328551480prefix @ A) @ Xs2) @ Ys2)) & ((~ (((coinductive_lfinite @ A) @ Xs2))) => (Xs2 = Ys2))))))))).
% 2.97/1.78 thf(fact_50_lstrict__prefix__def, axiom, (! [A:$tType]: (((coindu1478340336prefix @ A) = (^ [Xs2:(coinductive_llist @ A),Ys2:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs2) @ Ys2) & (Xs2 != Ys2)))))))).
% 2.97/1.78 thf(fact_51_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu328551480prefix @ A) @ (coinductive_LNil @ A)) @ Xs)))).
% 2.97/1.78 thf(fact_52_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (coinductive_LNil @ A)) = (Xs = (coinductive_LNil @ A)))))).
% 2.97/1.78 thf(fact_53_strict__monoD, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ A) @ X) @ Y) => (((ord_less @ B) @ (F @ X)) @ (F @ Y)))))))))).
% 2.97/1.78 thf(fact_54_strict__monoI, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B)]: (((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y4))))) => (((order_strict_mono @ A) @ B) @ F)))))))).
% 2.97/1.78 thf(fact_55_lprefix__code_I1_J, axiom, (! [A:$tType,Ys:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (coinductive_LNil @ A)) @ Ys)))).
% 2.97/1.78 thf(fact_56_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.97/1.78 thf(fact_57_lstrict__prefix__code_I1_J, axiom, (! [A:$tType]: ((~ ((((coindu1478340336prefix @ A) @ (coinductive_LNil @ A)) @ (coinductive_LNil @ A))))))).
% 2.97/1.78 thf(fact_58_less__llist__def, axiom, (! [A:$tType]: (((ord_less @ (coinductive_llist @ A)) = (coindu1478340336prefix @ A))))).
% 2.97/1.78 thf(fact_59_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.97/1.78 thf(fact_60_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.97/1.78 thf(fact_61_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 2.97/1.78 thf(fact_62_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.97/1.78 thf(fact_63_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.97/1.78 thf(fact_64_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_65_lstrict__prefix__lfinite1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Xs) @ Ys) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.97/1.78 thf(fact_66_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.97/1.78 thf(fact_67_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.78 thf(fact_68_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 2.97/1.78 thf(fact_69_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 2.97/1.78 thf(fact_70_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.97/1.78 thf(fact_71_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X2:A]: (((! [Y5:A]: (((((ord_less @ A) @ Y5) @ X2) => (P @ Y5)))) => (P @ X2)))) => (P @ A2)))))))).
% 2.97/1.78 thf(fact_72_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.97/1.78 thf(fact_73_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.97/1.78 thf(fact_74_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 2.97/1.78 thf(fact_75_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_76_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_77_lfinite__LNil, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.97/1.78 thf(fact_78_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 2.97/1.78 thf(fact_79_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.78 thf(fact_80_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.97/1.78 thf(fact_81_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.97/1.78 thf(fact_82_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 2.97/1.78 thf(fact_83_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 2.97/1.78 thf(fact_84_llist__less__induct, axiom, (! [A:$tType,P:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((! [Xs3:(coinductive_llist @ A)]: (((! [Ys3:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys3) @ Xs3) => (P @ Ys3)))) => (P @ Xs3)))) => (P @ Xs))))).
% 2.97/1.78 thf(fact_85_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z3:A]: (((((ord_less @ A) @ X) @ Z3) & (((ord_less @ A) @ Z3) @ Y))))))))))).
% 2.97/1.78 thf(fact_86_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 2.97/1.78 thf(fact_87_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.78 thf(fact_88_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 2.97/1.78 thf(fact_89_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 2.97/1.78 thf(fact_90_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y4:A]: ((((ord_less @ A) @ Y4) @ X))))))))).
% 2.97/1.78 thf(fact_91_order__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.97/1.78 thf(fact_92_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y4:B]: (((((ord_less @ B) @ X2) @ Y4) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.97/1.78 thf(fact_93_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ B) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 2.97/1.78 thf(fact_94_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y4:B]: (((((ord_less @ B) @ X2) @ Y4) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.97/1.78 thf(fact_95_not__lfinite__lprefix__conv__eq, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) = (Xs = Ys)))))).
% 2.97/1.78 thf(fact_96_lprefix__lfiniteD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lfinite @ A) @ Ys) => ((coinductive_lfinite @ A) @ Xs)))))).
% 2.97/1.78 thf(fact_97_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.97/1.78 thf(fact_98_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 2.97/1.78 thf(fact_99_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B4:A,A4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (A4 != B4))))))))).
% 2.97/1.78 thf(fact_100_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((ord_less @ A) @ B4) @ A4) | (A4 = B4))))))))).
% 2.97/1.78 thf(fact_101_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.97/1.78 thf(fact_102_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.97/1.78 thf(fact_103_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,X:A,Y:A]: (((((ord_less @ A) @ Z) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 2.97/1.78 thf(fact_104_dual__order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.97/1.78 thf(fact_105_dual__order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.97/1.78 thf(fact_106_order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (A4 != B4))))))))).
% 2.97/1.78 thf(fact_107_order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((ord_less @ A) @ A4) @ B4) | (A4 = B4))))))))).
% 2.97/1.78 thf(fact_108_order_Ostrict__trans2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_109_order_Ostrict__trans1, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.97/1.78 thf(fact_110_not__le__imp__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less_eq @ A) @ Y) @ X))) => (((ord_less @ A) @ X) @ Y)))))))).
% 2.97/1.78 thf(fact_111_less__le__not__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (~ ((((ord_less_eq @ A) @ Y3) @ X3))))))))))).
% 2.97/1.78 thf(fact_112_le__imp__less__or__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ X) @ Y) | (X = Y))))))))).
% 2.97/1.78 thf(fact_113_le__less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X)))))))).
% 2.97/1.78 thf(fact_114_dense__le, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Y:A,Z:A]: (((! [X2:A]: (((((ord_less @ A) @ X2) @ Y) => (((ord_less_eq @ A) @ X2) @ Z)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.97/1.78 thf(fact_115_dense__ge, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z:A,Y:A]: (((! [X2:A]: (((((ord_less @ A) @ Z) @ X2) => (((ord_less_eq @ A) @ Y) @ X2)))) => (((ord_less_eq @ A) @ Y) @ Z)))))))).
% 2.97/1.78 thf(fact_116_less__le__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.97/1.78 thf(fact_117_le__less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z) => (((ord_less @ A) @ X) @ Z))))))))).
% 2.97/1.78 thf(fact_118_antisym__conv2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 2.97/1.78 thf(fact_119_antisym__conv1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.97/1.78 thf(fact_120_less__imp__le, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.97/1.78 thf(fact_121_le__neq__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((A2 != B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 2.97/1.78 thf(fact_122_not__less, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.78 thf(fact_123_not__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) = (((ord_less @ A) @ Y) @ X)))))))).
% 2.97/1.78 thf(fact_124_order__less__le__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => (((ord_less @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.97/1.78 thf(fact_125_order__less__le__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X2:B,Y4:B]: (((((ord_less_eq @ B) @ X2) @ Y4) => (((ord_less_eq @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.97/1.78 thf(fact_126_order__le__less__subst2, axiom, (! [A:$tType,C2:$tType]: (((((order @ C2) @ (type2 @ C2)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ C2) @ (F @ B2)) @ C) => ((! [X2:A,Y4:A]: (((((ord_less_eq @ A) @ X2) @ Y4) => (((ord_less_eq @ C2) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ C2) @ (F @ A2)) @ C)))))))))).
% 2.97/1.78 thf(fact_127_order__le__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X2:B,Y4:B]: (((((ord_less @ B) @ X2) @ Y4) => (((ord_less @ A) @ (F @ X2)) @ (F @ Y4))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 2.97/1.78 thf(fact_128_less__le, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) & (X3 != Y3))))))))).
% 2.97/1.78 thf(fact_129_le__less, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) | (X3 = Y3))))))))).
% 2.97/1.78 thf(fact_130_leI, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.97/1.78 thf(fact_131_leD, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less @ A) @ X) @ Y)))))))))).
% 2.97/1.78 thf(fact_132_LNil__lprefix, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (coinductive_LNil @ A)) @ Xs)))).
% 2.97/1.78 thf(fact_133_less__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) & (~ ((((ord_less_eq @ (A > B)) @ G2) @ F2))))))))))).
% 2.97/1.78 thf(fact_134_strict__mono__less, axiom, (! [B:$tType,A:$tType]: (((((linorder @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (! [F:(A > B),X:A,Y:A]: (((((order_strict_mono @ A) @ B) @ F) => ((((ord_less @ B) @ (F @ X)) @ (F @ Y)) = (((ord_less @ A) @ X) @ Y))))))))).
% 2.97/1.78 thf(fact_135_strict__mono__def, axiom, (! [B:$tType,A:$tType]: (((((order @ A) @ (type2 @ A)) & ((order @ B) @ (type2 @ B))) => (((order_strict_mono @ A) @ B) = (^ [F2:(A > B)]: ((! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less @ B) @ (F2 @ X3)) @ (F2 @ Y3)))))))))))).
% 2.97/1.78 thf(fact_136_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less_eq @ A) @ T) @ X5)))))))))))))).
% 2.97/1.78 thf(fact_137_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less_eq @ A) @ X5) @ T)))))))))))).
% 2.97/1.78 thf(fact_138_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less_eq @ A) @ T) @ X5)))))))))))).
% 2.97/1.78 thf(fact_139_pinf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less_eq @ A) @ X5) @ T)))))))))))))).
% 2.97/1.78 thf(fact_140_complete__interval, axiom, (! [A:$tType]: ((((condit1037483654norder @ A) @ (type2 @ A)) => (! [A2:A,B2:A,P:(A > $o)]: (((((ord_less @ A) @ A2) @ B2) => ((P @ A2) => ((~ ((P @ B2))) => (? [C3:A]: (((((ord_less_eq @ A) @ A2) @ C3) & ((((ord_less_eq @ A) @ C3) @ B2) & ((! [X5:A]: ((((((ord_less_eq @ A) @ A2) @ X5) & (((ord_less @ A) @ X5) @ C3)) => (P @ X5)))) & (! [D:A]: (((! [X2:A]: ((((((ord_less_eq @ A) @ A2) @ X2) & (((ord_less @ A) @ X2) @ D)) => (P @ X2)))) => (((ord_less_eq @ A) @ D) @ C3)))))))))))))))))).
% 2.97/1.78 thf(fact_141_lmember__code_I1_J, axiom, (! [A:$tType,X:A]: ((~ ((((coinductive_lmember @ A) @ X) @ (coinductive_LNil @ A))))))).
% 2.97/1.78 thf(fact_142_llimit__induct, axiom, (! [A:$tType,P:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((P @ (coinductive_LNil @ A)) => ((! [X2:A,Xs3:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs3) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X2) @ Xs3)))))) => (((! [Ys3:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys3) @ Xs) => (P @ Ys3)))) => (P @ Xs)) => (P @ Xs))))))).
% 2.97/1.78 thf(fact_143_llist_Oinject, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A),Y21:A,Y22:(coinductive_llist @ A)]: ((((((coinductive_LCons @ A) @ X21) @ X22) = (((coinductive_LCons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.97/1.78 thf(fact_144_LCons__lprefix__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((X = Y) & (((coinductive_lprefix @ A) @ Xs) @ Ys)))))).
% 2.97/1.78 thf(fact_145_lfinite__code_I2_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 2.97/1.78 thf(fact_146_lfinite__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.97/1.78 thf(fact_147_lstrict__prefix__code_I4_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B),Y:B,Ys:(coinductive_llist @ B)]: (((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) @ (((coinductive_LCons @ B) @ Y) @ Ys)) = ((X = Y) & (((coindu1478340336prefix @ B) @ Xs) @ Ys)))))).
% 2.97/1.78 thf(fact_148_lstrict__prefix__code_I3_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((~ ((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) @ (coinductive_LNil @ B))))))).
% 2.97/1.78 thf(fact_149_lstrict__prefix__code_I2_J, axiom, (! [B:$tType,Y:B,Ys:(coinductive_llist @ B)]: ((((coindu1478340336prefix @ B) @ (coinductive_LNil @ B)) @ (((coinductive_LCons @ B) @ Y) @ Ys))))).
% 2.97/1.78 thf(fact_150_lmember__code_I2_J, axiom, (! [A:$tType,X:A,Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lmember @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((X = Y) | (((coinductive_lmember @ A) @ X) @ Ys)))))).
% 2.97/1.78 thf(fact_151_Le__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),X:A]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (((coinductive_LCons @ A) @ X) @ Ys)))))).
% 2.97/1.78 thf(fact_152_LCons__lprefix__conv, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ Ys) = (? [Ys4:(coinductive_llist @ A)]: (((Ys = (((coinductive_LCons @ A) @ X) @ Ys4)) & (((coinductive_lprefix @ A) @ Xs) @ Ys4)))))))).
% 2.97/1.78 thf(fact_153_llist_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: (((coinductive_LNil @ A) != (((coinductive_LCons @ A) @ X21) @ X22))))).
% 2.97/1.78 thf(fact_154_llist_Oexhaust, axiom, (! [A:$tType,Y:(coinductive_llist @ A)]: (((Y != (coinductive_LNil @ A)) => (~ ((! [X212:A,X222:(coinductive_llist @ A)]: ((Y != (((coinductive_LCons @ A) @ X212) @ X222)))))))))).
% 2.97/1.78 thf(fact_155_neq__LNil__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((Xs != (coinductive_LNil @ A)) = (? [X3:A,Xs4:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X3) @ Xs4)))))))).
% 2.97/1.78 thf(fact_156_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.97/1.78 thf(fact_157_lprefix__code_I2_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((~ ((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (coinductive_LNil @ A))))))).
% 2.97/1.78 thf(fact_158_lprefix_Ocases, axiom, (! [A:$tType,A1:(coinductive_llist @ A),A22:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ A1) @ A22) => (((A1 = (coinductive_LNil @ A)) => (! [Xs3:(coinductive_llist @ A)]: ((A22 != Xs3)))) => (~ ((! [Xs3:(coinductive_llist @ A),Ys5:(coinductive_llist @ A),X2:A]: (((A1 = (((coinductive_LCons @ A) @ X2) @ Xs3)) => ((A22 = (((coinductive_LCons @ A) @ X2) @ Ys5)) => (~ ((((coinductive_lprefix @ A) @ Xs3) @ Ys5)))))))))))))).
% 2.97/1.78 thf(fact_159_lprefix_Osimps, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [A12:(coinductive_llist @ A),A23:(coinductive_llist @ A)]: (((? [Xs2:(coinductive_llist @ A)]: (((A12 = (coinductive_LNil @ A)) & (A23 = Xs2)))) | (? [Xs2:(coinductive_llist @ A),Ys2:(coinductive_llist @ A),X3:A]: (((A12 = (((coinductive_LCons @ A) @ X3) @ Xs2)) & ((A23 = (((coinductive_LCons @ A) @ X3) @ Ys2)) & (((coinductive_lprefix @ A) @ Xs2) @ Ys2)))))))))))).
% 2.97/1.78 thf(fact_160_lprefix_Ocoinduct, axiom, (! [A:$tType,X6:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),X:(coinductive_llist @ A),Xa:(coinductive_llist @ A)]: ((((X6 @ X) @ Xa) => ((! [X2:(coinductive_llist @ A),Xa2:(coinductive_llist @ A)]: ((((X6 @ X2) @ Xa2) => ((? [Xs5:(coinductive_llist @ A)]: (((X2 = (coinductive_LNil @ A)) & (Xa2 = Xs5)))) | (? [Xs5:(coinductive_llist @ A),Ys3:(coinductive_llist @ A),Xb:A]: (((X2 = (((coinductive_LCons @ A) @ Xb) @ Xs5)) & ((Xa2 = (((coinductive_LCons @ A) @ Xb) @ Ys3)) & (((X6 @ Xs5) @ Ys3) | (((coinductive_lprefix @ A) @ Xs5) @ Ys3)))))))))) => (((coinductive_lprefix @ A) @ X) @ Xa)))))).
% 2.97/1.78 thf(fact_161_lprefix__LCons__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((Xs = (coinductive_LNil @ A)) | (? [Xs4:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y) @ Xs4)) & (((coinductive_lprefix @ A) @ Xs4) @ Ys))))))))).
% 2.97/1.78 thf(fact_162_lfinite_Ocases, axiom, (! [A:$tType,A2:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ A2) => ((A2 != (coinductive_LNil @ A)) => (~ ((! [Xs3:(coinductive_llist @ A)]: (((? [X2:A]: ((A2 = (((coinductive_LCons @ A) @ X2) @ Xs3)))) => (~ (((coinductive_lfinite @ A) @ Xs3))))))))))))).
% 2.97/1.78 thf(fact_163_lfinite_Osimps, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) = (^ [A4:(coinductive_llist @ A)]: (((A4 = (coinductive_LNil @ A)) | (? [Xs2:(coinductive_llist @ A),X3:A]: (((A4 = (((coinductive_LCons @ A) @ X3) @ Xs2)) & ((coinductive_lfinite @ A) @ Xs2))))))))))).
% 2.97/1.78 thf(fact_164_lfinite_Oinducts, axiom, (! [A:$tType,X:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ X) => ((P @ (coinductive_LNil @ A)) => ((! [Xs3:(coinductive_llist @ A),X2:A]: ((((coinductive_lfinite @ A) @ Xs3) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X2) @ Xs3)))))) => (P @ X))))))).
% 2.97/1.78 thf(fact_165_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((Xs = (coinductive_LNil @ A)) | (? [Xs4:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y) @ Xs4)) & (((coindu328551480prefix @ A) @ Xs4) @ Ys))))))))).
% 2.97/1.78 thf(fact_166_ex__gt__or__lt, axiom, (! [A:$tType]: ((((condit1656338222tinuum @ A) @ (type2 @ A)) => (! [A2:A]: ((? [B3:A]: (((((ord_less @ A) @ A2) @ B3) | (((ord_less @ A) @ B3) @ A2)))))))))).
% 2.97/1.78 thf(fact_167_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) & (Q @ X5)) = ((P2 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.97/1.78 thf(fact_168_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ Z4) @ X2) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((P @ X5) | (Q @ X5)) = ((P2 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.97/1.78 thf(fact_169_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.97/1.78 thf(fact_170_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (X5 != T)))))))))))).
% 2.97/1.78 thf(fact_171_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (~ ((((ord_less @ A) @ X5) @ T)))))))))))))).
% 2.97/1.78 thf(fact_172_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ Z3) @ X5) => (((ord_less @ A) @ T) @ X5)))))))))))).
% 2.97/1.78 thf(fact_173_pinf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ Z3) @ X5) => (F3 = F3)))))))))))).
% 2.97/1.78 thf(fact_174_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) & (Q @ X5)) = ((P2 @ X5) & (Q2 @ X5)))))))))))))))).
% 2.97/1.78 thf(fact_175_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P2:(A > $o),Q:(A > $o),Q2:(A > $o)]: (((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((P @ X2) = (P2 @ X2))))))) => ((? [Z4:A]: ((! [X2:A]: (((((ord_less @ A) @ X2) @ Z4) => ((Q @ X2) = (Q2 @ X2))))))) => (? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((P @ X5) | (Q @ X5)) = ((P2 @ X5) | (Q2 @ X5)))))))))))))))).
% 2.97/1.78 thf(fact_176_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.97/1.78 thf(fact_177_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (X5 != T)))))))))))).
% 2.97/1.78 thf(fact_178_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (((ord_less @ A) @ X5) @ T)))))))))))).
% 2.97/1.78 thf(fact_179_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z3:A]: ((! [X5:A]: (((((ord_less @ A) @ X5) @ Z3) => (~ ((((ord_less @ A) @ T) @ X5)))))))))))))).
% 2.97/1.78 thf(fact_180_minf_I11_J, axiom, (! [C2:$tType,D2:$tType]: ((((ord @ C2) @ (type2 @ C2)) => (! [F3:D2]: ((? [Z3:C2]: ((! [X5:C2]: (((((ord_less @ C2) @ X5) @ Z3) => (F3 = F3)))))))))))).
% 2.97/1.78 thf(fact_181_llast__singleton, axiom, (! [A:$tType,X:A]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))) = X)))).
% 2.97/1.78 thf(fact_182_lsorted_Ocases, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ A2) => ((A2 != (coinductive_LNil @ A)) => ((! [X2:A]: ((A2 != (((coinductive_LCons @ A) @ X2) @ (coinductive_LNil @ A))))) => (~ ((! [X2:A,Y4:A,Xs3:(coinductive_llist @ A)]: (((A2 = (((coinductive_LCons @ A) @ X2) @ (((coinductive_LCons @ A) @ Y4) @ Xs3))) => ((((ord_less_eq @ A) @ X2) @ Y4) => (~ (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y4) @ Xs3))))))))))))))))))).
% 2.97/1.78 thf(fact_183_lsorted_Osimps, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) = (^ [A4:(coinductive_llist @ A)]: (((A4 = (coinductive_LNil @ A)) | ((? [X3:A]: ((A4 = (((coinductive_LCons @ A) @ X3) @ (coinductive_LNil @ A))))) | (? [X3:A,Y3:A,Xs2:(coinductive_llist @ A)]: (((A4 = (((coinductive_LCons @ A) @ X3) @ (((coinductive_LCons @ A) @ Y3) @ Xs2))) & ((((ord_less_eq @ A) @ X3) @ Y3) & ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y3) @ Xs2))))))))))))))).
% 2.97/1.78 thf(fact_184_lsorted__code_I1_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 2.97/1.78 thf(fact_185_llast__LCons2, axiom, (! [A:$tType,X:A,Y:A,Xs:(coinductive_llist @ A)]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))) = ((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)))))).
% 2.97/1.78 thf(fact_186_lsorted__LCons__LCons, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Y:A,Xs:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))) = ((((ord_less_eq @ A) @ X) @ Y) & ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)))))))))).
% 2.97/1.78 thf(fact_187_lsorted__code_I2_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A]: (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))))))))).
% 2.97/1.78 thf(fact_188_lsorted__lprefixD, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coindu63249387sorted @ A) @ Ys) => ((coindu63249387sorted @ A) @ Xs))))))))).
% 2.97/1.78 thf(fact_189_LNil, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 2.97/1.78 thf(fact_190_LCons__LCons, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Y:A,Xs:(coinductive_llist @ A)]: (((((ord_less_eq @ A) @ X) @ Y) => (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)) => ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))))))))))).
% 2.97/1.78 thf(fact_191_Singleton, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A]: (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))))))))).
% 2.97/1.78 thf(fact_192_lsorted_Ocoinduct, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X6:((coinductive_llist @ A) > $o),X:(coinductive_llist @ A)]: (((X6 @ X) => ((! [X2:(coinductive_llist @ A)]: (((X6 @ X2) => ((X2 = (coinductive_LNil @ A)) | ((? [Xa3:A]: ((X2 = (((coinductive_LCons @ A) @ Xa3) @ (coinductive_LNil @ A))))) | (? [Xa3:A,Y5:A,Xs5:(coinductive_llist @ A)]: (((X2 = (((coinductive_LCons @ A) @ Xa3) @ (((coinductive_LCons @ A) @ Y5) @ Xs5))) & ((((ord_less_eq @ A) @ Xa3) @ Y5) & ((X6 @ (((coinductive_LCons @ A) @ Y5) @ Xs5)) | ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y5) @ Xs5)))))))))))) => ((coindu63249387sorted @ A) @ X))))))))).
% 2.97/1.78 thf(fact_193_llast__lappend__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys))) = ((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ Y) @ Ys))))))).
% 2.97/1.78 thf(fact_194_ex__has__greatest__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat),B2:nat]: (((P @ K) => ((! [Y4:A]: (((P @ Y4) => (((ord_less @ nat) @ (M2 @ Y4)) @ B2)))) => (? [X2:A]: (((P @ X2) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ Y5)) @ (M2 @ X2))))))))))))).
% 2.97/1.78 thf(fact_195_ex__has__least__nat, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => (? [X2:A]: (((P @ X2) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ X2)) @ (M2 @ Y5)))))))))))).
% 2.97/1.78 thf(fact_196_lappend__code_I2_J, axiom, (! [A:$tType,Xa:A,X:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_LCons @ A) @ Xa) @ X)) @ Ys) = (((coinductive_LCons @ A) @ Xa) @ (((coinductive_lappend @ A) @ X) @ Ys)))))).
% 2.97/1.78 thf(fact_197_lappend__LNil2, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ Xs) @ (coinductive_LNil @ A)) = Xs)))).
% 2.97/1.78 thf(fact_198_lappend__code_I1_J, axiom, (! [A:$tType,Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (coinductive_LNil @ A)) @ Ys) = Ys)))).
% 2.97/1.78 thf(fact_199_lfinite__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) & ((coinductive_lfinite @ A) @ Ys)))))).
% 2.97/1.78 thf(fact_200_lprefix__lappend__same, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ (((coinductive_lappend @ A) @ Xs) @ Zs)) = (((coinductive_lfinite @ A) @ Xs) => (((coinductive_lprefix @ A) @ Ys) @ Zs)))))).
% 2.97/1.78 thf(fact_201_lappend__snocL1__conv__LCons2, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_lappend @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ (coinductive_LNil @ A)))) @ Ys) = (((coinductive_lappend @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys)))))).
% 2.97/1.78 thf(fact_202_lappend__assoc, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ Zs) = (((coinductive_lappend @ A) @ Xs) @ (((coinductive_lappend @ A) @ Ys) @ Zs)))))).
% 2.97/1.78 thf(fact_203_lappend__inf, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.97/1.78 thf(fact_204_lappend__eq__LNil__iff, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((((coinductive_lappend @ A) @ Xs) @ Ys) = (coinductive_LNil @ A)) = ((Xs = (coinductive_LNil @ A)) & (Ys = (coinductive_LNil @ A))))))).
% 2.97/1.78 thf(fact_205_LNil__eq__lappend__iff, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_LNil @ A) = (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((Xs = (coinductive_LNil @ A)) & (Ys = (coinductive_LNil @ A))))))).
% 2.97/1.78 thf(fact_206_lappend__LNil__LNil, axiom, (! [A:$tType]: (((((coinductive_lappend @ A) @ (coinductive_LNil @ A)) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.97/1.78 thf(fact_207_lprefix__lappend__sameI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Zs) @ Xs)) @ (((coinductive_lappend @ A) @ Zs) @ Ys)))))).
% 2.97/1.78 thf(fact_208_lprefix__conv__lappend, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [Xs2:(coinductive_llist @ A),Ys2:(coinductive_llist @ A)]: ((? [Zs2:(coinductive_llist @ A)]: ((Ys2 = (((coinductive_lappend @ A) @ Xs2) @ Zs2)))))))))).
% 2.97/1.78 thf(fact_209_lprefix__lappendD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_lappend @ A) @ Ys) @ Zs)) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) | (((coinductive_lprefix @ A) @ Ys) @ Xs)))))).
% 2.97/1.78 thf(fact_210_lappend__lprefixE, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ Zs) => (~ ((! [Zs3:(coinductive_llist @ A)]: ((Zs != (((coinductive_lappend @ A) @ Xs) @ Zs3)))))))))).
% 2.97/1.78 thf(fact_211_lprefix__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_lappend @ A) @ Xs) @ Ys))))).
% 2.97/1.78 thf(fact_212_lfinite__rev__induct, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((P @ (coinductive_LNil @ A)) => ((! [X2:A,Xs3:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs3) => ((P @ Xs3) => (P @ (((coinductive_lappend @ A) @ Xs3) @ (((coinductive_LCons @ A) @ X2) @ (coinductive_LNil @ A)))))))) => (P @ Xs))))))).
% 2.97/1.78 thf(fact_213_nat__less__le, axiom, ((ord_less @ nat) = (^ [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) & (M != N)))))).
% 2.97/1.78 thf(fact_214_less__imp__le__nat, axiom, (! [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.97/1.78 thf(fact_215_nat__le__linear, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) | (((ord_less_eq @ nat) @ N2) @ M2))))).
% 2.97/1.78 thf(fact_216_le__antisym, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M2) => (M2 = N2)))))).
% 2.97/1.78 thf(fact_217_eq__imp__le, axiom, (! [M2:nat,N2:nat]: (((M2 = N2) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.97/1.78 thf(fact_218_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.97/1.78 thf(fact_219_le__refl, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))).
% 2.97/1.78 thf(fact_220_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 2.97/1.78 thf(fact_221_le__neq__implies__less, axiom, (! [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) => ((M2 != N2) => (((ord_less @ nat) @ M2) @ N2)))))).
% 2.97/1.78 thf(fact_222_less__or__eq__imp__le, axiom, (! [M2:nat,N2:nat]: ((((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)) => (((ord_less_eq @ nat) @ M2) @ N2))))).
% 2.97/1.78 thf(fact_223_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) | (M = N)))))).
% 2.97/1.78 thf(fact_224_GreatestM__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M2:(A > nat),B2:nat]: (((P @ X) => ((! [Y4:A]: (((P @ Y4) => (((ord_less @ nat) @ (M2 @ Y4)) @ B2)))) => (((ord_less_eq @ nat) @ (M2 @ X)) @ (M2 @ ((((hilbert_GreatestM @ A) @ nat) @ M2) @ P)))))))).
% 2.97/1.78 thf(fact_225_GreatestM__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat),B2:nat]: (((P @ K) => ((! [Y4:A]: (((P @ Y4) => (((ord_less @ nat) @ (M2 @ Y4)) @ B2)))) => ((P @ ((((hilbert_GreatestM @ A) @ nat) @ M2) @ P)) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ Y5)) @ (M2 @ ((((hilbert_GreatestM @ A) @ nat) @ M2) @ P)))))))))))).
% 2.97/1.78 thf(fact_226_GreatestMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M2:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ B) @ (M2 @ Y4)) @ (M2 @ X))))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (M2 @ Y5)) @ (M2 @ X2))))) => (Q @ X2))))) => (Q @ ((((hilbert_GreatestM @ A) @ B) @ M2) @ P))))))))))).
% 2.97/1.78 thf(fact_227_LeastMI2, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [P:(A > $o),X:A,M2:(A > B),Q:(A > $o)]: (((P @ X) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ B) @ (M2 @ X)) @ (M2 @ Y4))))) => ((! [X2:A]: (((P @ X2) => ((! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ B) @ (M2 @ X2)) @ (M2 @ Y5))))) => (Q @ X2))))) => (Q @ ((((hilbert_LeastM @ A) @ B) @ M2) @ P))))))))))).
% 2.97/1.78 thf(fact_228_lstrict__prefix__lappend__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Xs) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) & (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.97/1.78 thf(fact_229_lappend_Odisc__iff_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))) = ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.97/1.78 thf(fact_230_lnull__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lnull @ A) @ Xs) & ((coinductive_lnull @ A) @ Ys)))))).
% 2.97/1.78 thf(fact_231_lprefix__LNil, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (coinductive_LNil @ A)) = ((coinductive_lnull @ A) @ Xs))))).
% 2.97/1.78 thf(fact_232_lappend_Octr_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lnull @ A) @ Ys) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = (coinductive_LNil @ A))))))).
% 2.97/1.78 thf(fact_233_lappend_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))) => (~ (((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))))))).
% 2.97/1.78 thf(fact_234_lappend_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lnull @ A) @ Ys) => ((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys))))))).
% 2.97/1.78 thf(fact_235_lappend__lnull1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Ys))))).
% 2.97/1.78 thf(fact_236_lappend__lnull2, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.97/1.78 thf(fact_237_LeastM__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,M2:(A > nat)]: (((P @ K) => ((P @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P)) & (! [Y5:A]: (((P @ Y5) => (((ord_less_eq @ nat) @ (M2 @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P))) @ (M2 @ Y5)))))))))).
% 2.97/1.78 thf(fact_238_LeastM__nat__le, axiom, (! [A:$tType,P:(A > $o),X:A,M2:(A > nat)]: (((P @ X) => (((ord_less_eq @ nat) @ (M2 @ ((((hilbert_LeastM @ A) @ nat) @ M2) @ P))) @ (M2 @ X)))))).
% 2.97/1.78 thf(fact_239_llast__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: (((((coinductive_lnull @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = X)) & ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((coinductive_llast @ A) @ Xs))))))).
% 2.97/1.78 thf(fact_240_llist_Odisc_I2_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((~ (((coinductive_lnull @ A) @ (((coinductive_LCons @ A) @ X21) @ X22))))))).
% 2.97/1.78 thf(fact_241_llist_OdiscI_I2_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A),X21:A,X22:(coinductive_llist @ A)]: (((Llist = (((coinductive_LCons @ A) @ X21) @ X22)) => (~ (((coinductive_lnull @ A) @ Llist))))))).
% 2.97/1.78 thf(fact_242_not__lnull__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs))) = (? [X3:A,Xs4:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X3) @ Xs4)))))))).
% 2.97/1.78 thf(fact_243_lnull__lprefix, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lprefix @ A) @ Xs) @ Ys))))).
% 2.97/1.78 thf(fact_244_lprefix__lnull, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) = ((coinductive_lnull @ A) @ Xs)))))).
% 2.97/1.78 thf(fact_245_lprefix__lnullD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lnull @ A) @ Ys) => ((coinductive_lnull @ A) @ Xs)))))).
% 2.97/1.78 thf(fact_246_lprefix__not__lnullD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((~ (((coinductive_lnull @ A) @ Xs))) => (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.97/1.78 thf(fact_247_llist_Odisc_I1_J, axiom, (! [A:$tType]: (((coinductive_lnull @ A) @ (coinductive_LNil @ A))))).
% 2.97/1.78 thf(fact_248_llist_OdiscI_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((Llist = (coinductive_LNil @ A)) => ((coinductive_lnull @ A) @ Llist))))).
% 2.97/1.78 thf(fact_249_llist_Ocollapse_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Llist) => (Llist = (coinductive_LNil @ A)))))).
% 2.97/1.78 thf(fact_250_lnull__def, axiom, (! [A:$tType]: (((coinductive_lnull @ A) = (^ [Llist2:(coinductive_llist @ A)]: ((Llist2 = (coinductive_LNil @ A)))))))).
% 2.97/1.78 thf(fact_251_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.97/1.78 thf(fact_252_lappend_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (~ (((coinductive_lnull @ A) @ Ys)))) => ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.97/1.78 thf(fact_253_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ B) @ Ys)))))))).
% 2.97/1.78 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A5:$tType,A6:$tType]: ((((comple187826305attice @ A6) @ (type2 @ A6)) => ((comple187826305attice @ (A5 > A6)) @ (type2 @ (A5 > A6))))))).
% 2.97/1.78 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A5:$tType,A6:$tType]: ((((preorder @ A6) @ (type2 @ A6)) => ((preorder @ (A5 > A6)) @ (type2 @ (A5 > A6))))))).
% 2.97/1.78 thf(tcon_fun___Orderings_Oorder, axiom, (! [A5:$tType,A6:$tType]: ((((order @ A6) @ (type2 @ A6)) => ((order @ (A5 > A6)) @ (type2 @ (A5 > A6))))))).
% 2.97/1.78 thf(tcon_fun___Orderings_Oord, axiom, (! [A5:$tType,A6:$tType]: ((((ord @ A6) @ (type2 @ A6)) => ((ord @ (A5 > A6)) @ (type2 @ (A5 > A6))))))).
% 2.97/1.78 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, axiom, ((condit1037483654norder @ nat) @ (type2 @ nat))).
% 2.97/1.78 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 2.97/1.78 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 2.97/1.78 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 2.97/1.78 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 2.97/1.78 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 2.97/1.78 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 2.97/1.78 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_4, axiom, ((comple187826305attice @ $o) @ (type2 @ $o))).
% 2.97/1.78 thf(tcon_HOL_Obool___Orderings_Opreorder_5, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.97/1.78 thf(tcon_HOL_Obool___Orderings_Olinorder_6, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.97/1.78 thf(tcon_HOL_Obool___Orderings_Oorder_7, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.97/1.78 thf(tcon_HOL_Obool___Orderings_Oord_8, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.97/1.78 thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_9, axiom, (! [A5:$tType]: (((preorder @ (coinductive_llist @ A5)) @ (type2 @ (coinductive_llist @ A5)))))).
% 2.97/1.78 thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_10, axiom, (! [A5:$tType]: (((order @ (coinductive_llist @ A5)) @ (type2 @ (coinductive_llist @ A5)))))).
% 2.97/1.78 thf(tcon_Coinductive__List_Ollist___Orderings_Oord_11, axiom, (! [A5:$tType]: (((ord @ (coinductive_llist @ A5)) @ (type2 @ (coinductive_llist @ A5)))))).
% 2.97/1.78 thf(conj_0, conjecture, ((ord_less_eq @ (coinductive_llist @ a)) = (coinductive_lprefix @ a))).
% 2.97/1.78 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.CZmJ4dWwbg/DTF2DTF20653.p
% 2.97/1.78 ------------------------
% 3.26/1.83 ---- Cleaned THF ---
% 3.26/1.83 thf(ty_t_Coinductive__List_Ollist,type,
% 3.26/1.83 coinductive_llist: $tType > $tType ).
% 3.26/1.83
% 3.26/1.83 thf(ty_t_Nat_Onat,type,
% 3.26/1.83 nat: $tType ).
% 3.26/1.83
% 3.26/1.83 thf(ty_t_itself,type,
% 3.26/1.83 itself: $tType > $tType ).
% 3.26/1.83
% 3.26/1.83 thf(ty_tf_a,type,
% 3.26/1.83 a: $tType ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_HOL_Otype,type,
% 3.26/1.83 type:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Oord,type,
% 3.26/1.83 ord:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Oorder,type,
% 3.26/1.83 order:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Ono__bot,type,
% 3.26/1.83 no_bot:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Ono__top,type,
% 3.26/1.83 no_top:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Olinorder,type,
% 3.26/1.83 linorder:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Opreorder,type,
% 3.26/1.83 preorder:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Owellorder,type,
% 3.26/1.83 wellorder:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Odense__order,type,
% 3.26/1.83 dense_order:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Orderings_Odense__linorder,type,
% 3.26/1.83 dense_linorder:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 3.26/1.83 comple187826305attice:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
% 3.26/1.83 condit1656338222tinuum:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
% 3.26/1.83 condit1037483654norder:
% 3.26/1.83 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 3.26/1.83 coindu328551480prefix:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Olappend,type,
% 3.26/1.83 coinductive_lappend:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Olfinite,type,
% 3.26/1.83 coinductive_lfinite:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Ollast,type,
% 3.26/1.83 coinductive_llast:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 3.26/1.83 coinductive_LCons:
% 3.26/1.83 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 3.26/1.83 coinductive_LNil:
% 3.26/1.83 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 3.26/1.83 coinductive_lnull:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Olmember,type,
% 3.26/1.83 coinductive_lmember:
% 3.26/1.83 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Olprefix,type,
% 3.26/1.83 coinductive_lprefix:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 3.26/1.83 coindu1478340336prefix:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 3.26/1.83 coindu63249387sorted:
% 3.26/1.83 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Hilbert__Choice_OGreatestM,type,
% 3.26/1.83 hilbert_GreatestM:
% 3.26/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Hilbert__Choice_OLeastM,type,
% 3.26/1.83 hilbert_LeastM:
% 3.26/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Orderings_Oord__class_Oless,type,
% 3.26/1.83 ord_less:
% 3.26/1.83 !>[A: $tType] : ( A > A > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.26/1.83 ord_less_eq:
% 3.26/1.83 !>[A: $tType] : ( A > A > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Orderings_Oorder__class_Oantimono,type,
% 3.26/1.83 order_antimono:
% 3.26/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 3.26/1.83 order_strict_mono:
% 3.26/1.83 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(sy_c_Pure_Otype,type,
% 3.26/1.83 type2:
% 3.26/1.83 !>[A: $tType] : ( itself @ A ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_0_less__eq__llist__def,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord_less_eq @ ( coinductive_llist @ A ) )
% 3.26/1.83 = ( coinductive_lprefix @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_1_lprefix__refl,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_2_llist_Oleq__refl,axiom,
% 3.26/1.83 ! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_3_order__refl,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_4_lprefix__trans,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 3.26/1.83 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_5_llist_Oleq__trans,axiom,
% 3.26/1.83 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,Z: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ X @ Y )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Y @ Z )
% 3.26/1.83 => ( coinductive_lprefix @ A @ X @ Z ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_6_lprefix__antisym,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 3.26/1.83 => ( Xs = Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_7_llist_Oleq__antisym,axiom,
% 3.26/1.83 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ X @ Y )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Y @ X )
% 3.26/1.83 => ( X = Y ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_8_lprefix__down__linear,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_9_le__funD,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 => ! [F: A > B,G: A > B,X: A] :
% 3.26/1.83 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_10_le__funE,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 => ! [F: A > B,G: A > B,X: A] :
% 3.26/1.83 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_11_le__funI,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 => ! [F: A > B,G: A > B] :
% 3.26/1.83 ( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
% 3.26/1.83 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_12_le__fun__def,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 => ( ( ord_less_eq @ ( A > B ) )
% 3.26/1.83 = ( ^ [F2: A > B,G2: A > B] :
% 3.26/1.83 ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_13_dual__order_Oantisym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( A2 = B2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_14_dual__order_Otrans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A,C: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.26/1.83 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_15_linorder__wlog,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [P: A > A > $o,A2: A,B2: A] :
% 3.26/1.83 ( ! [A3: A,B3: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.26/1.83 => ( P @ A3 @ B3 ) )
% 3.26/1.83 => ( ! [A3: A,B3: A] :
% 3.26/1.83 ( ( P @ B3 @ A3 )
% 3.26/1.83 => ( P @ A3 @ B3 ) )
% 3.26/1.83 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_16_dual__order_Orefl,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_17_order__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Z: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.26/1.83 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_18_order__class_Oorder_Oantisym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.26/1.83 => ( A2 = B2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_19_ord__le__eq__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( B2 = C )
% 3.26/1.83 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_20_ord__eq__le__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( A2 = B2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.26/1.83 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_21_antisym__conv,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Y: A,X: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 = ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_22_le__cases3,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Z: A] :
% 3.26/1.83 ( ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 3.26/1.83 => ( ( ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 3.26/1.83 => ( ( ( ord_less_eq @ A @ X @ Z )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 3.26/1.83 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 3.26/1.83 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 3.26/1.83 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_23_order_Otrans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.26/1.83 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_24_le__cases,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ~ ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_25_wlog__linorder__le,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [P: A > A > $o,B2: A,A2: A] :
% 3.26/1.83 ( ! [A3: A,B3: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A3 @ B3 )
% 3.26/1.83 => ( P @ A3 @ B3 ) )
% 3.26/1.83 => ( ( ( P @ B2 @ A2 )
% 3.26/1.83 => ( P @ A2 @ B2 ) )
% 3.26/1.83 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_26_eq__refl,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( X = Y )
% 3.26/1.83 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_27_linear,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_28_antisym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_29_eq__iff,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 3.26/1.83 = ( ^ [X3: A,Y3: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.26/1.83 & ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_30_ord__le__eq__subst,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,B2: A,F: A > B,C: B] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ( F @ B2 )
% 3.26/1.83 = C )
% 3.26/1.83 => ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_31_ord__eq__le__subst,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.26/1.83 ( ( A2
% 3.26/1.83 = ( F @ B2 ) )
% 3.26/1.83 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.26/1.83 => ( ! [X2: B,Y4: B] :
% 3.26/1.83 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_32_order__subst2,axiom,
% 3.26/1.83 ! [A: $tType,C2: $tType] :
% 3.26/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 3.26/1.83 => ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_33_order__subst1,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 3.26/1.83 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.26/1.83 => ( ! [X2: B,Y4: B] :
% 3.26/1.83 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_34_lfp_Oleq__antisym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_35_gfp_Oleq__antisym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Y: A,X: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_36_lfp_Oleq__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Z: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.26/1.83 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_37_gfp_Oleq__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Y: A,X: A,Z: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ( ( ord_less_eq @ A @ Z @ Y )
% 3.26/1.83 => ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_38_lfp_Oleq__refl,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( comple187826305attice @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_39_Coinductive__List_Ofinite__lprefix__def,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coindu328551480prefix @ A )
% 3.26/1.83 = ( coinductive_lprefix @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_40_strict__mono__less__eq,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B,X: A,Y: A] :
% 3.26/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.26/1.83 => ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
% 3.26/1.83 = ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_41_antimono__def,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ( ( order_antimono @ A @ B )
% 3.26/1.83 = ( ^ [F2: A > B] :
% 3.26/1.83 ! [X3: A,Y3: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_42_antimonoI,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B] :
% 3.26/1.83 ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 3.26/1.83 => ( order_antimono @ A @ B @ F ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_43_antimonoE,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B,X: A,Y: A] :
% 3.26/1.83 ( ( order_antimono @ A @ B @ F )
% 3.26/1.83 => ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_44_antimonoD,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B,X: A,Y: A] :
% 3.26/1.83 ( ( order_antimono @ A @ B @ F )
% 3.26/1.83 => ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_45_ext,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 3.26/1.83 ( ! [X2: A] :
% 3.26/1.83 ( ( F @ X2 )
% 3.26/1.83 = ( G @ X2 ) )
% 3.26/1.83 => ( F = G ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_46_strict__mono__eq,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B,X: A,Y: A] :
% 3.26/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.26/1.83 => ( ( ( F @ X )
% 3.26/1.83 = ( F @ Y ) )
% 3.26/1.83 = ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_47_decseqD,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [F: nat > A,I: nat,J: nat] :
% 3.26/1.83 ( ( order_antimono @ nat @ A @ F )
% 3.26/1.83 => ( ( ord_less_eq @ nat @ I @ J )
% 3.26/1.83 => ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_48_decseq__def,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( order_antimono @ nat @ A )
% 3.26/1.83 = ( ^ [X4: nat > A] :
% 3.26/1.83 ! [M: nat,N: nat] :
% 3.26/1.83 ( ( ord_less_eq @ nat @ M @ N )
% 3.26/1.83 => ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_49_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coinductive_lprefix @ A )
% 3.26/1.83 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 3.26/1.83 ( ( ( coinductive_lfinite @ A @ Xs2 )
% 3.26/1.83 => ( coindu328551480prefix @ A @ Xs2 @ Ys2 ) )
% 3.26/1.83 & ( ~ ( coinductive_lfinite @ A @ Xs2 )
% 3.26/1.83 => ( Xs2 = Ys2 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_50_lstrict__prefix__def,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coindu1478340336prefix @ A )
% 3.26/1.83 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs2 @ Ys2 )
% 3.26/1.83 & ( Xs2 != Ys2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_51_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_52_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 3.26/1.83 = ( Xs
% 3.26/1.83 = ( coinductive_LNil @ A ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_53_strict__monoD,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B,X: A,Y: A] :
% 3.26/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.26/1.83 => ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_54_strict__monoI,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B] :
% 3.26/1.83 ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( order_strict_mono @ A @ B @ F ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_55_lprefix__code_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_56_lfinite__code_I1_J,axiom,
% 3.26/1.83 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_57_lstrict__prefix__code_I1_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_58_less__llist__def,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord_less @ ( coinductive_llist @ A ) )
% 3.26/1.83 = ( coindu1478340336prefix @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_59_dual__order_Ostrict__implies__not__eq,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A] :
% 3.26/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.26/1.83 => ( A2 != B2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_60_order_Ostrict__implies__not__eq,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( A2 != B2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_61_not__less__iff__gr__or__eq,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.26/1.83 = ( ( ord_less @ A @ Y @ X )
% 3.26/1.83 | ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_62_dual__order_Ostrict__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A,C: A] :
% 3.26/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.26/1.83 => ( ( ord_less @ A @ C @ B2 )
% 3.26/1.83 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_63_less__imp__not__less,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_64_order_Ostrict__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less @ A @ B2 @ C )
% 3.26/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_65_lstrict__prefix__lfinite1,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_66_dual__order_Oirrefl,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A] :
% 3.26/1.83 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_67_linorder__cases,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ~ ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ( X != Y )
% 3.26/1.83 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_68_less__imp__triv,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,P: $o] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less @ A @ Y @ X )
% 3.26/1.83 => P ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_69_less__imp__not__eq2,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( Y != X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_70_antisym__conv3,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Y: A,X: A] :
% 3.26/1.83 ( ~ ( ord_less @ A @ Y @ X )
% 3.26/1.83 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.26/1.83 = ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_71_less__induct,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( wellorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [P: A > $o,A2: A] :
% 3.26/1.83 ( ! [X2: A] :
% 3.26/1.83 ( ! [Y5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Y5 @ X2 )
% 3.26/1.83 => ( P @ Y5 ) )
% 3.26/1.83 => ( P @ X2 ) )
% 3.26/1.83 => ( P @ A2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_72_less__not__sym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_73_less__imp__not__eq,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( X != Y ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_74_dual__order_Oasym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A] :
% 3.26/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.26/1.83 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_75_ord__less__eq__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( B2 = C )
% 3.26/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_76_ord__eq__less__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( A2 = B2 )
% 3.26/1.83 => ( ( ord_less @ A @ B2 @ C )
% 3.26/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_77_lfinite__LNil,axiom,
% 3.26/1.83 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_78_less__irrefl,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A] :
% 3.26/1.83 ~ ( ord_less @ A @ X @ X ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_79_less__linear,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 | ( X = Y )
% 3.26/1.83 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_80_less__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Z: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less @ A @ Y @ Z )
% 3.26/1.83 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_81_less__asym_H,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_82_less__asym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_83_less__imp__neq,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( X != Y ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_84_llist__less__induct,axiom,
% 3.26/1.83 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ! [Xs3: coinductive_llist @ A] :
% 3.26/1.83 ( ! [Ys3: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu1478340336prefix @ A @ Ys3 @ Xs3 )
% 3.26/1.83 => ( P @ Ys3 ) )
% 3.26/1.83 => ( P @ Xs3 ) )
% 3.26/1.83 => ( P @ Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_85_dense,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( dense_order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ? [Z3: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Z3 )
% 3.26/1.83 & ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_86_order_Oasym,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_87_neq__iff,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( X != Y )
% 3.26/1.83 = ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_88_neqE,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( X != Y )
% 3.26/1.83 => ( ~ ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_89_gt__ex,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( no_top @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A] :
% 3.26/1.83 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_90_lt__ex,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( no_bot @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A] :
% 3.26/1.83 ? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_91_order__less__subst2,axiom,
% 3.26/1.83 ! [A: $tType,C2: $tType] :
% 3.26/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 3.26/1.83 => ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_92_order__less__subst1,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 3.26/1.83 => ( ( ord_less @ B @ B2 @ C )
% 3.26/1.83 => ( ! [X2: B,Y4: B] :
% 3.26/1.83 ( ( ord_less @ B @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_93_ord__less__eq__subst,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,B2: A,F: A > B,C: B] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ( F @ B2 )
% 3.26/1.83 = C )
% 3.26/1.83 => ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_94_ord__eq__less__subst,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( ord @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.26/1.83 ( ( A2
% 3.26/1.83 = ( F @ B2 ) )
% 3.26/1.83 => ( ( ord_less @ B @ B2 @ C )
% 3.26/1.83 => ( ! [X2: B,Y4: B] :
% 3.26/1.83 ( ( ord_less @ B @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_95_not__lfinite__lprefix__conv__eq,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 = ( Xs = Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_96_lprefix__lfiniteD,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( ( coinductive_lfinite @ A @ Ys )
% 3.26/1.83 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_97_order_Onot__eq__order__implies__strict,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A] :
% 3.26/1.83 ( ( A2 != B2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_98_dual__order_Ostrict__implies__order,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A] :
% 3.26/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.26/1.83 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_99_dual__order_Ostrict__iff__order,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ord_less @ A )
% 3.26/1.83 = ( ^ [B4: A,A4: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ B4 @ A4 )
% 3.26/1.83 & ( A4 != B4 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_100_dual__order_Oorder__iff__strict,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ord_less_eq @ A )
% 3.26/1.83 = ( ^ [B4: A,A4: A] :
% 3.26/1.83 ( ( ord_less @ A @ B4 @ A4 )
% 3.26/1.83 | ( A4 = B4 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_101_order_Ostrict__implies__order,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_102_dense__le__bounded,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Z: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ! [W: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ W )
% 3.26/1.83 => ( ( ord_less @ A @ W @ Y )
% 3.26/1.83 => ( ord_less_eq @ A @ W @ Z ) ) )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_103_dense__ge__bounded,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Z: A,X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z @ X )
% 3.26/1.83 => ( ! [W: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z @ W )
% 3.26/1.83 => ( ( ord_less @ A @ W @ X )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ W ) ) )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_104_dual__order_Ostrict__trans2,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A,C: A] :
% 3.26/1.83 ( ( ord_less @ A @ B2 @ A2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ C @ B2 )
% 3.26/1.83 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_105_dual__order_Ostrict__trans1,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [B2: A,A2: A,C: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.26/1.83 => ( ( ord_less @ A @ C @ B2 )
% 3.26/1.83 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_106_order_Ostrict__iff__order,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ord_less @ A )
% 3.26/1.83 = ( ^ [A4: A,B4: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A4 @ B4 )
% 3.26/1.83 & ( A4 != B4 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_107_order_Oorder__iff__strict,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ord_less_eq @ A )
% 3.26/1.83 = ( ^ [A4: A,B4: A] :
% 3.26/1.83 ( ( ord_less @ A @ A4 @ B4 )
% 3.26/1.83 | ( A4 = B4 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_108_order_Ostrict__trans2,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less_eq @ A @ B2 @ C )
% 3.26/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_109_order_Ostrict__trans1,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,C: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less @ A @ B2 @ C )
% 3.26/1.83 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_110_not__le__imp__less,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Y: A,X: A] :
% 3.26/1.83 ( ~ ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ( ord_less @ A @ X @ Y ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_111_less__le__not__le,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ord_less @ A )
% 3.26/1.83 = ( ^ [X3: A,Y3: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.26/1.83 & ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_112_le__imp__less__or__eq,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 | ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_113_le__less__linear,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 | ( ord_less @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_114_dense__le,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Y: A,Z: A] :
% 3.26/1.83 ( ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Y )
% 3.26/1.83 => ( ord_less_eq @ A @ X2 @ Z ) )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_115_dense__ge,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Z: A,Y: A] :
% 3.26/1.83 ( ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z @ X2 )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ X2 ) )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_116_less__le__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Z: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.26/1.83 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_117_le__less__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Z: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less @ A @ Y @ Z )
% 3.26/1.83 => ( ord_less @ A @ X @ Z ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_118_antisym__conv2,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.26/1.83 = ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_119_antisym__conv1,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ~ ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 = ( X = Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_120_less__imp__le,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( preorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_121_le__neq__trans,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( A2 != B2 )
% 3.26/1.83 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_122_not__less,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 3.26/1.83 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_123_not__le,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ( ~ ( ord_less_eq @ A @ X @ Y ) )
% 3.26/1.83 = ( ord_less @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_124_order__less__le__subst2,axiom,
% 3.26/1.83 ! [A: $tType,C2: $tType] :
% 3.26/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
% 3.26/1.83 => ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_125_order__less__le__subst1,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 3.26/1.83 => ( ( ord_less_eq @ B @ B2 @ C )
% 3.26/1.83 => ( ! [X2: B,Y4: B] :
% 3.26/1.83 ( ( ord_less_eq @ B @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_126_order__le__less__subst2,axiom,
% 3.26/1.83 ! [A: $tType,C2: $tType] :
% 3.26/1.83 ( ( ( order @ C2 @ ( type2 @ C2 ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,B2: A,F: A > C2,C: C2] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
% 3.26/1.83 => ( ! [X2: A,Y4: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_127_order__le__less__subst1,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType] :
% 3.26/1.83 ( ( ( order @ B @ ( type2 @ B ) )
% 3.26/1.83 & ( order @ A @ ( type2 @ A ) ) )
% 3.26/1.83 => ! [A2: A,F: B > A,B2: B,C: B] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 3.26/1.83 => ( ( ord_less @ B @ B2 @ C )
% 3.26/1.83 => ( ! [X2: B,Y4: B] :
% 3.26/1.83 ( ( ord_less @ B @ X2 @ Y4 )
% 3.26/1.83 => ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 3.26/1.83 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_128_less__le,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ord_less @ A )
% 3.26/1.83 = ( ^ [X3: A,Y3: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 3.26/1.83 & ( X3 != Y3 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_129_le__less,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( ord_less_eq @ A )
% 3.26/1.83 = ( ^ [X3: A,Y3: A] :
% 3.26/1.83 ( ( ord_less @ A @ X3 @ Y3 )
% 3.26/1.83 | ( X3 = Y3 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_130_leI,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A] :
% 3.26/1.83 ( ~ ( ord_less @ A @ X @ Y )
% 3.26/1.83 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_131_leD,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Y: A,X: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ Y @ X )
% 3.26/1.83 => ~ ( ord_less @ A @ X @ Y ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_132_LNil__lprefix,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_133_less__fun__def,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 => ( ( ord_less @ ( A > B ) )
% 3.26/1.83 = ( ^ [F2: A > B,G2: A > B] :
% 3.26/1.83 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 3.26/1.83 & ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_134_strict__mono__less,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ! [F: A > B,X: A,Y: A] :
% 3.26/1.83 ( ( order_strict_mono @ A @ B @ F )
% 3.26/1.83 => ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
% 3.26/1.83 = ( ord_less @ A @ X @ Y ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_135_strict__mono__def,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ( order @ A @ ( type2 @ A ) )
% 3.26/1.83 & ( order @ B @ ( type2 @ B ) ) )
% 3.26/1.83 => ( ( order_strict_mono @ A @ B )
% 3.26/1.83 = ( ^ [F2: A > B] :
% 3.26/1.83 ! [X3: A,Y3: A] :
% 3.26/1.83 ( ( ord_less @ A @ X3 @ Y3 )
% 3.26/1.83 => ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_136_minf_I8_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_137_minf_I6_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_138_pinf_I8_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_139_pinf_I6_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_140_complete__interval,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( condit1037483654norder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A,B2: A,P: A > $o] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B2 )
% 3.26/1.83 => ( ( P @ A2 )
% 3.26/1.83 => ( ~ ( P @ B2 )
% 3.26/1.83 => ? [C3: A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ A2 @ C3 )
% 3.26/1.83 & ( ord_less_eq @ A @ C3 @ B2 )
% 3.26/1.83 & ! [X5: A] :
% 3.26/1.83 ( ( ( ord_less_eq @ A @ A2 @ X5 )
% 3.26/1.83 & ( ord_less @ A @ X5 @ C3 ) )
% 3.26/1.83 => ( P @ X5 ) )
% 3.26/1.83 & ! [D: A] :
% 3.26/1.83 ( ! [X2: A] :
% 3.26/1.83 ( ( ( ord_less_eq @ A @ A2 @ X2 )
% 3.26/1.83 & ( ord_less @ A @ X2 @ D ) )
% 3.26/1.83 => ( P @ X2 ) )
% 3.26/1.83 => ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_141_lmember__code_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,X: A] :
% 3.26/1.83 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_142_llimit__induct,axiom,
% 3.26/1.83 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( P @ ( coinductive_LNil @ A ) )
% 3.26/1.83 => ( ! [X2: A,Xs3: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ Xs3 )
% 3.26/1.83 => ( ( P @ Xs3 )
% 3.26/1.83 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
% 3.26/1.83 => ( ( ! [Ys3: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu1478340336prefix @ A @ Ys3 @ Xs )
% 3.26/1.83 => ( P @ Ys3 ) )
% 3.26/1.83 => ( P @ Xs ) )
% 3.26/1.83 => ( P @ Xs ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_143_llist_Oinject,axiom,
% 3.26/1.83 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 3.26/1.83 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 3.26/1.83 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 3.26/1.83 = ( ( X21 = Y21 )
% 3.26/1.83 & ( X22 = Y22 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_144_LCons__lprefix__LCons,axiom,
% 3.26/1.83 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 3.26/1.83 = ( ( X = Y )
% 3.26/1.83 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_145_lfinite__code_I2_J,axiom,
% 3.26/1.83 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 3.26/1.83 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 3.26/1.83 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_146_lfinite__LCons,axiom,
% 3.26/1.83 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 3.26/1.83 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_147_lstrict__prefix__code_I4_J,axiom,
% 3.26/1.83 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 3.26/1.83 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 3.26/1.83 = ( ( X = Y )
% 3.26/1.83 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_148_lstrict__prefix__code_I3_J,axiom,
% 3.26/1.83 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 3.26/1.83 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_149_lstrict__prefix__code_I2_J,axiom,
% 3.26/1.83 ! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_150_lmember__code_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 3.26/1.83 = ( ( X = Y )
% 3.26/1.83 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_151_Le__LCons,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_152_LCons__lprefix__conv,axiom,
% 3.26/1.83 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
% 3.26/1.83 = ( ? [Ys4: coinductive_llist @ A] :
% 3.26/1.83 ( ( Ys
% 3.26/1.83 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 3.26/1.83 & ( coinductive_lprefix @ A @ Xs @ Ys4 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_153_llist_Odistinct_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_LNil @ A )
% 3.26/1.83 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_154_llist_Oexhaust,axiom,
% 3.26/1.83 ! [A: $tType,Y: coinductive_llist @ A] :
% 3.26/1.83 ( ( Y
% 3.26/1.83 != ( coinductive_LNil @ A ) )
% 3.26/1.83 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 3.26/1.83 ( Y
% 3.26/1.83 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_155_neq__LNil__conv,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( Xs
% 3.26/1.83 != ( coinductive_LNil @ A ) )
% 3.26/1.83 = ( ? [X3: A,Xs4: coinductive_llist @ A] :
% 3.26/1.83 ( Xs
% 3.26/1.83 = ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_156_lfinite__LConsI,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_157_lprefix__code_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 3.26/1.83 ~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_158_lprefix_Ocases,axiom,
% 3.26/1.83 ! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ A1 @ A22 )
% 3.26/1.83 => ( ( ( A1
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 => ! [Xs3: coinductive_llist @ A] : ( A22 != Xs3 ) )
% 3.26/1.83 => ~ ! [Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A,X2: A] :
% 3.26/1.83 ( ( A1
% 3.26/1.83 = ( coinductive_LCons @ A @ X2 @ Xs3 ) )
% 3.26/1.83 => ( ( A22
% 3.26/1.83 = ( coinductive_LCons @ A @ X2 @ Ys5 ) )
% 3.26/1.83 => ~ ( coinductive_lprefix @ A @ Xs3 @ Ys5 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_159_lprefix_Osimps,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coinductive_lprefix @ A )
% 3.26/1.83 = ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 3.26/1.83 ( ? [Xs2: coinductive_llist @ A] :
% 3.26/1.83 ( ( A12
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 & ( A23 = Xs2 ) )
% 3.26/1.83 | ? [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A,X3: A] :
% 3.26/1.83 ( ( A12
% 3.26/1.83 = ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 3.26/1.83 & ( A23
% 3.26/1.83 = ( coinductive_LCons @ A @ X3 @ Ys2 ) )
% 3.26/1.83 & ( coinductive_lprefix @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_160_lprefix_Ocoinduct,axiom,
% 3.26/1.83 ! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 3.26/1.83 ( ( X6 @ X @ Xa )
% 3.26/1.83 => ( ! [X2: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
% 3.26/1.83 ( ( X6 @ X2 @ Xa2 )
% 3.26/1.83 => ( ? [Xs5: coinductive_llist @ A] :
% 3.26/1.83 ( ( X2
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 & ( Xa2 = Xs5 ) )
% 3.26/1.83 | ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,Xb: A] :
% 3.26/1.83 ( ( X2
% 3.26/1.83 = ( coinductive_LCons @ A @ Xb @ Xs5 ) )
% 3.26/1.83 & ( Xa2
% 3.26/1.83 = ( coinductive_LCons @ A @ Xb @ Ys3 ) )
% 3.26/1.83 & ( ( X6 @ Xs5 @ Ys3 )
% 3.26/1.83 | ( coinductive_lprefix @ A @ Xs5 @ Ys3 ) ) ) ) )
% 3.26/1.83 => ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_161_lprefix__LCons__conv,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 3.26/1.83 = ( ( Xs
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 | ? [Xs4: coinductive_llist @ A] :
% 3.26/1.83 ( ( Xs
% 3.26/1.83 = ( coinductive_LCons @ A @ Y @ Xs4 ) )
% 3.26/1.83 & ( coinductive_lprefix @ A @ Xs4 @ Ys ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_162_lfinite_Ocases,axiom,
% 3.26/1.83 ! [A: $tType,A2: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ A2 )
% 3.26/1.83 => ( ( A2
% 3.26/1.83 != ( coinductive_LNil @ A ) )
% 3.26/1.83 => ~ ! [Xs3: coinductive_llist @ A] :
% 3.26/1.83 ( ? [X2: A] :
% 3.26/1.83 ( A2
% 3.26/1.83 = ( coinductive_LCons @ A @ X2 @ Xs3 ) )
% 3.26/1.83 => ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_163_lfinite_Osimps,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coinductive_lfinite @ A )
% 3.26/1.83 = ( ^ [A4: coinductive_llist @ A] :
% 3.26/1.83 ( ( A4
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 | ? [Xs2: coinductive_llist @ A,X3: A] :
% 3.26/1.83 ( ( A4
% 3.26/1.83 = ( coinductive_LCons @ A @ X3 @ Xs2 ) )
% 3.26/1.83 & ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_164_lfinite_Oinducts,axiom,
% 3.26/1.83 ! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ X )
% 3.26/1.83 => ( ( P @ ( coinductive_LNil @ A ) )
% 3.26/1.83 => ( ! [Xs3: coinductive_llist @ A,X2: A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ Xs3 )
% 3.26/1.83 => ( ( P @ Xs3 )
% 3.26/1.83 => ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
% 3.26/1.83 => ( P @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_165_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 3.26/1.83 = ( ( Xs
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 | ? [Xs4: coinductive_llist @ A] :
% 3.26/1.83 ( ( Xs
% 3.26/1.83 = ( coinductive_LCons @ A @ Y @ Xs4 ) )
% 3.26/1.83 & ( coindu328551480prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_166_ex__gt__or__lt,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: A] :
% 3.26/1.83 ? [B3: A] :
% 3.26/1.83 ( ( ord_less @ A @ A2 @ B3 )
% 3.26/1.83 | ( ord_less @ A @ B3 @ A2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_167_pinf_I1_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.26/1.83 ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.26/1.83 => ( ( P @ X2 )
% 3.26/1.83 = ( P2 @ X2 ) ) )
% 3.26/1.83 => ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.26/1.83 => ( ( Q @ X2 )
% 3.26/1.83 = ( Q2 @ X2 ) ) )
% 3.26/1.83 => ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ( ( ( P @ X5 )
% 3.26/1.83 & ( Q @ X5 ) )
% 3.26/1.83 = ( ( P2 @ X5 )
% 3.26/1.83 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_168_pinf_I2_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.26/1.83 ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.26/1.83 => ( ( P @ X2 )
% 3.26/1.83 = ( P2 @ X2 ) ) )
% 3.26/1.83 => ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z4 @ X2 )
% 3.26/1.83 => ( ( Q @ X2 )
% 3.26/1.83 = ( Q2 @ X2 ) ) )
% 3.26/1.83 => ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ( ( ( P @ X5 )
% 3.26/1.83 | ( Q @ X5 ) )
% 3.26/1.83 = ( ( P2 @ X5 )
% 3.26/1.83 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_169_pinf_I3_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ( X5 != T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_170_pinf_I4_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ( X5 != T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_171_pinf_I5_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_172_pinf_I7_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ Z3 @ X5 )
% 3.26/1.83 => ( ord_less @ A @ T @ X5 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_173_pinf_I11_J,axiom,
% 3.26/1.83 ! [C2: $tType,D2: $tType] :
% 3.26/1.83 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 3.26/1.83 => ! [F3: D2] :
% 3.26/1.83 ? [Z3: C2] :
% 3.26/1.83 ! [X5: C2] :
% 3.26/1.83 ( ( ord_less @ C2 @ Z3 @ X5 )
% 3.26/1.83 => ( F3 = F3 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_174_minf_I1_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.26/1.83 ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.26/1.83 => ( ( P @ X2 )
% 3.26/1.83 = ( P2 @ X2 ) ) )
% 3.26/1.83 => ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.26/1.83 => ( ( Q @ X2 )
% 3.26/1.83 = ( Q2 @ X2 ) ) )
% 3.26/1.83 => ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ( ( ( P @ X5 )
% 3.26/1.83 & ( Q @ X5 ) )
% 3.26/1.83 = ( ( P2 @ X5 )
% 3.26/1.83 & ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_175_minf_I2_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
% 3.26/1.83 ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.26/1.83 => ( ( P @ X2 )
% 3.26/1.83 = ( P2 @ X2 ) ) )
% 3.26/1.83 => ( ? [Z4: A] :
% 3.26/1.83 ! [X2: A] :
% 3.26/1.83 ( ( ord_less @ A @ X2 @ Z4 )
% 3.26/1.83 => ( ( Q @ X2 )
% 3.26/1.83 = ( Q2 @ X2 ) ) )
% 3.26/1.83 => ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ( ( ( P @ X5 )
% 3.26/1.83 | ( Q @ X5 ) )
% 3.26/1.83 = ( ( P2 @ X5 )
% 3.26/1.83 | ( Q2 @ X5 ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_176_minf_I3_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ( X5 != T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_177_minf_I4_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ( X5 != T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_178_minf_I5_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ( ord_less @ A @ X5 @ T ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_179_minf_I7_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( linorder @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [T: A] :
% 3.26/1.83 ? [Z3: A] :
% 3.26/1.83 ! [X5: A] :
% 3.26/1.83 ( ( ord_less @ A @ X5 @ Z3 )
% 3.26/1.83 => ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_180_minf_I11_J,axiom,
% 3.26/1.83 ! [C2: $tType,D2: $tType] :
% 3.26/1.83 ( ( ord @ C2 @ ( type2 @ C2 ) )
% 3.26/1.83 => ! [F3: D2] :
% 3.26/1.83 ? [Z3: C2] :
% 3.26/1.83 ! [X5: C2] :
% 3.26/1.83 ( ( ord_less @ C2 @ X5 @ Z3 )
% 3.26/1.83 => ( F3 = F3 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_181_llast__singleton,axiom,
% 3.26/1.83 ! [A: $tType,X: A] :
% 3.26/1.83 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 3.26/1.83 = X ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_182_lsorted_Ocases,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [A2: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu63249387sorted @ A @ A2 )
% 3.26/1.83 => ( ( A2
% 3.26/1.83 != ( coinductive_LNil @ A ) )
% 3.26/1.83 => ( ! [X2: A] :
% 3.26/1.83 ( A2
% 3.26/1.83 != ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
% 3.26/1.83 => ~ ! [X2: A,Y4: A,Xs3: coinductive_llist @ A] :
% 3.26/1.83 ( ( A2
% 3.26/1.83 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) )
% 3.26/1.83 => ( ( ord_less_eq @ A @ X2 @ Y4 )
% 3.26/1.83 => ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_183_lsorted_Osimps,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( ( coindu63249387sorted @ A )
% 3.26/1.83 = ( ^ [A4: coinductive_llist @ A] :
% 3.26/1.83 ( ( A4
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 | ? [X3: A] :
% 3.26/1.83 ( A4
% 3.26/1.83 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 3.26/1.83 | ? [X3: A,Y3: A,Xs2: coinductive_llist @ A] :
% 3.26/1.83 ( ( A4
% 3.26/1.83 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) )
% 3.26/1.83 & ( ord_less_eq @ A @ X3 @ Y3 )
% 3.26/1.83 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_184_lsorted__code_I1_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_185_llast__LCons2,axiom,
% 3.26/1.83 ! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 3.26/1.83 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_186_lsorted__LCons__LCons,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 3.26/1.83 = ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_187_lsorted__code_I2_J,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_188_lsorted__lprefixD,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( ( coindu63249387sorted @ A @ Ys )
% 3.26/1.83 => ( coindu63249387sorted @ A @ Xs ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_189_LNil,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_190_LCons__LCons,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( ord_less_eq @ A @ X @ Y )
% 3.26/1.83 => ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 3.26/1.83 => ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_191_Singleton,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_192_lsorted_Ocoinduct,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( ord @ A @ ( type2 @ A ) )
% 3.26/1.83 => ! [X6: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
% 3.26/1.83 ( ( X6 @ X )
% 3.26/1.83 => ( ! [X2: coinductive_llist @ A] :
% 3.26/1.83 ( ( X6 @ X2 )
% 3.26/1.83 => ( ( X2
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 | ? [Xa3: A] :
% 3.26/1.83 ( X2
% 3.26/1.83 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
% 3.26/1.83 | ? [Xa3: A,Y5: A,Xs5: coinductive_llist @ A] :
% 3.26/1.83 ( ( X2
% 3.26/1.83 = ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) )
% 3.26/1.83 & ( ord_less_eq @ A @ Xa3 @ Y5 )
% 3.26/1.83 & ( ( X6 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) )
% 3.26/1.83 | ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) ) ) ) )
% 3.26/1.83 => ( coindu63249387sorted @ A @ X ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_193_llast__lappend__LCons,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 3.26/1.83 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_194_ex__has__greatest__nat,axiom,
% 3.26/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 3.26/1.83 ( ( P @ K )
% 3.26/1.83 => ( ! [Y4: A] :
% 3.26/1.83 ( ( P @ Y4 )
% 3.26/1.83 => ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
% 3.26/1.83 => ? [X2: A] :
% 3.26/1.83 ( ( P @ X2 )
% 3.26/1.83 & ! [Y5: A] :
% 3.26/1.83 ( ( P @ Y5 )
% 3.26/1.83 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_195_ex__has__least__nat,axiom,
% 3.26/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 3.26/1.83 ( ( P @ K )
% 3.26/1.83 => ? [X2: A] :
% 3.26/1.83 ( ( P @ X2 )
% 3.26/1.83 & ! [Y5: A] :
% 3.26/1.83 ( ( P @ Y5 )
% 3.26/1.83 => ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_196_lappend__code_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
% 3.26/1.83 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_197_lappend__LNil2,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 3.26/1.83 = Xs ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_198_lappend__code_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 3.26/1.83 = Ys ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_199_lfinite__lappend,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.26/1.83 = ( ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_200_lprefix__lappend__same,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 3.26/1.83 = ( ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 => ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_201_lappend__snocL1__conv__LCons2,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
% 3.26/1.83 = ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_202_lappend__assoc,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 3.26/1.83 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_203_lappend__inf,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ~ ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 3.26/1.83 = Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_204_lappend__eq__LNil__iff,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 = ( ( Xs
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 & ( Ys
% 3.26/1.83 = ( coinductive_LNil @ A ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_205_LNil__eq__lappend__iff,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( ( coinductive_LNil @ A )
% 3.26/1.83 = ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.26/1.83 = ( ( Xs
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 & ( Ys
% 3.26/1.83 = ( coinductive_LNil @ A ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_206_lappend__LNil__LNil,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
% 3.26/1.83 = ( coinductive_LNil @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_207_lprefix__lappend__sameI,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_208_lprefix__conv__lappend,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coinductive_lprefix @ A )
% 3.26/1.83 = ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
% 3.26/1.83 ? [Zs2: coinductive_llist @ A] :
% 3.26/1.83 ( Ys2
% 3.26/1.83 = ( coinductive_lappend @ A @ Xs2 @ Zs2 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_209_lprefix__lappendD,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_210_lappend__lprefixE,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 3.26/1.83 => ~ ! [Zs3: coinductive_llist @ A] :
% 3.26/1.83 ( Zs
% 3.26/1.83 != ( coinductive_lappend @ A @ Xs @ Zs3 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_211_lprefix__lappend,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_212_lfinite__rev__induct,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 => ( ( P @ ( coinductive_LNil @ A ) )
% 3.26/1.83 => ( ! [X2: A,Xs3: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lfinite @ A @ Xs3 )
% 3.26/1.83 => ( ( P @ Xs3 )
% 3.26/1.83 => ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
% 3.26/1.83 => ( P @ Xs ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_213_nat__less__le,axiom,
% 3.26/1.83 ( ( ord_less @ nat )
% 3.26/1.83 = ( ^ [M: nat,N: nat] :
% 3.26/1.83 ( ( ord_less_eq @ nat @ M @ N )
% 3.26/1.83 & ( M != N ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_214_less__imp__le__nat,axiom,
% 3.26/1.83 ! [M2: nat,N2: nat] :
% 3.26/1.83 ( ( ord_less @ nat @ M2 @ N2 )
% 3.26/1.83 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_215_nat__le__linear,axiom,
% 3.26/1.83 ! [M2: nat,N2: nat] :
% 3.26/1.83 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 3.26/1.83 | ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_216_le__antisym,axiom,
% 3.26/1.83 ! [M2: nat,N2: nat] :
% 3.26/1.83 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 3.26/1.83 => ( ( ord_less_eq @ nat @ N2 @ M2 )
% 3.26/1.83 => ( M2 = N2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_217_eq__imp__le,axiom,
% 3.26/1.83 ! [M2: nat,N2: nat] :
% 3.26/1.83 ( ( M2 = N2 )
% 3.26/1.83 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_218_le__trans,axiom,
% 3.26/1.83 ! [I: nat,J: nat,K: nat] :
% 3.26/1.83 ( ( ord_less_eq @ nat @ I @ J )
% 3.26/1.83 => ( ( ord_less_eq @ nat @ J @ K )
% 3.26/1.83 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_219_le__refl,axiom,
% 3.26/1.83 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_220_less__mono__imp__le__mono,axiom,
% 3.26/1.83 ! [F: nat > nat,I: nat,J: nat] :
% 3.26/1.83 ( ! [I2: nat,J2: nat] :
% 3.26/1.83 ( ( ord_less @ nat @ I2 @ J2 )
% 3.26/1.83 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 3.26/1.83 => ( ( ord_less_eq @ nat @ I @ J )
% 3.26/1.83 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_221_le__neq__implies__less,axiom,
% 3.26/1.83 ! [M2: nat,N2: nat] :
% 3.26/1.83 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 3.26/1.83 => ( ( M2 != N2 )
% 3.26/1.83 => ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_222_less__or__eq__imp__le,axiom,
% 3.26/1.83 ! [M2: nat,N2: nat] :
% 3.26/1.83 ( ( ( ord_less @ nat @ M2 @ N2 )
% 3.26/1.83 | ( M2 = N2 ) )
% 3.26/1.83 => ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_223_le__eq__less__or__eq,axiom,
% 3.26/1.83 ( ( ord_less_eq @ nat )
% 3.26/1.83 = ( ^ [M: nat,N: nat] :
% 3.26/1.83 ( ( ord_less @ nat @ M @ N )
% 3.26/1.83 | ( M = N ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_224_GreatestM__nat__le,axiom,
% 3.26/1.83 ! [A: $tType,P: A > $o,X: A,M2: A > nat,B2: nat] :
% 3.26/1.83 ( ( P @ X )
% 3.26/1.83 => ( ! [Y4: A] :
% 3.26/1.83 ( ( P @ Y4 )
% 3.26/1.83 => ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
% 3.26/1.83 => ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_225_GreatestM__nat__lemma,axiom,
% 3.26/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
% 3.26/1.83 ( ( P @ K )
% 3.26/1.83 => ( ! [Y4: A] :
% 3.26/1.83 ( ( P @ Y4 )
% 3.26/1.83 => ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
% 3.26/1.83 => ( ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) )
% 3.26/1.83 & ! [Y5: A] :
% 3.26/1.83 ( ( P @ Y5 )
% 3.26/1.83 => ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_226_GreatestMI2,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 3.26/1.83 ( ( P @ X )
% 3.26/1.83 => ( ! [Y4: A] :
% 3.26/1.83 ( ( P @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ B @ ( M2 @ Y4 ) @ ( M2 @ X ) ) )
% 3.26/1.83 => ( ! [X2: A] :
% 3.26/1.83 ( ( P @ X2 )
% 3.26/1.83 => ( ! [Y5: A] :
% 3.26/1.83 ( ( P @ Y5 )
% 3.26/1.83 => ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
% 3.26/1.83 => ( Q @ X2 ) ) )
% 3.26/1.83 => ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_227_LeastMI2,axiom,
% 3.26/1.83 ! [B: $tType,A: $tType] :
% 3.26/1.83 ( ( ord @ B @ ( type2 @ B ) )
% 3.26/1.83 => ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
% 3.26/1.83 ( ( P @ X )
% 3.26/1.83 => ( ! [Y4: A] :
% 3.26/1.83 ( ( P @ Y4 )
% 3.26/1.83 => ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y4 ) ) )
% 3.26/1.83 => ( ! [X2: A] :
% 3.26/1.83 ( ( P @ X2 )
% 3.26/1.83 => ( ! [Y5: A] :
% 3.26/1.83 ( ( P @ Y5 )
% 3.26/1.83 => ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
% 3.26/1.83 => ( Q @ X2 ) ) )
% 3.26/1.83 => ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_228_lstrict__prefix__lappend__conv,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.26/1.83 = ( ( coinductive_lfinite @ A @ Xs )
% 3.26/1.83 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_229_lappend_Odisc__iff_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 3.26/1.83 = ( ~ ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_230_lnull__lappend,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 3.26/1.83 = ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_231_lprefix__LNil,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 3.26/1.83 = ( coinductive_lnull @ A @ Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_232_lappend_Octr_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_lnull @ A @ Ys )
% 3.26/1.83 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 3.26/1.83 = ( coinductive_LNil @ A ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_233_lappend_Odisc_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 | ~ ( coinductive_lnull @ A @ Ys ) )
% 3.26/1.83 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_234_lappend_Odisc_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_lnull @ A @ Ys )
% 3.26/1.83 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_235_lappend__lnull1,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 3.26/1.83 = Ys ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_236_lappend__lnull2,axiom,
% 3.26/1.83 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Ys )
% 3.26/1.83 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 3.26/1.83 = Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_237_LeastM__nat__lemma,axiom,
% 3.26/1.83 ! [A: $tType,P: A > $o,K: A,M2: A > nat] :
% 3.26/1.83 ( ( P @ K )
% 3.26/1.83 => ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
% 3.26/1.83 & ! [Y5: A] :
% 3.26/1.83 ( ( P @ Y5 )
% 3.26/1.83 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_238_LeastM__nat__le,axiom,
% 3.26/1.83 ! [A: $tType,P: A > $o,X: A,M2: A > nat] :
% 3.26/1.83 ( ( P @ X )
% 3.26/1.83 => ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_239_llast__LCons,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 3.26/1.83 ( ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 3.26/1.83 = X ) )
% 3.26/1.83 & ( ~ ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 3.26/1.83 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_240_llist_Odisc_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 3.26/1.83 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_241_llist_OdiscI_I2_J,axiom,
% 3.26/1.83 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 3.26/1.83 ( ( Llist
% 3.26/1.83 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 3.26/1.83 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_242_not__lnull__conv,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 3.26/1.83 = ( ? [X3: A,Xs4: coinductive_llist @ A] :
% 3.26/1.83 ( Xs
% 3.26/1.83 = ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_243_lnull__lprefix,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_244_lprefix__lnull,axiom,
% 3.26/1.83 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Ys )
% 3.26/1.83 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_245_lprefix__lnullD,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( ( coinductive_lnull @ A @ Ys )
% 3.26/1.83 => ( coinductive_lnull @ A @ Xs ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_246_lprefix__not__lnullD,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 3.26/1.83 => ( ~ ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_247_llist_Odisc_I1_J,axiom,
% 3.26/1.83 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_248_llist_OdiscI_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,Llist: coinductive_llist @ A] :
% 3.26/1.83 ( ( Llist
% 3.26/1.83 = ( coinductive_LNil @ A ) )
% 3.26/1.83 => ( coinductive_lnull @ A @ Llist ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_249_llist_Ocollapse_I1_J,axiom,
% 3.26/1.83 ! [A: $tType,Llist: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Llist )
% 3.26/1.83 => ( Llist
% 3.26/1.83 = ( coinductive_LNil @ A ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_250_lnull__def,axiom,
% 3.26/1.83 ! [A: $tType] :
% 3.26/1.83 ( ( coinductive_lnull @ A )
% 3.26/1.83 = ( ^ [Llist2: coinductive_llist @ A] :
% 3.26/1.83 ( Llist2
% 3.26/1.83 = ( coinductive_LNil @ A ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_251_lnull__imp__lfinite,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A] :
% 3.26/1.83 ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_252_lappend_Oexhaust,axiom,
% 3.26/1.83 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 3.26/1.83 ( ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ~ ( coinductive_lnull @ A @ Ys ) )
% 3.26/1.83 => ( ~ ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(fact_253_lzip_Oexhaust,axiom,
% 3.26/1.83 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 3.26/1.83 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 | ( coinductive_lnull @ B @ Ys ) )
% 3.26/1.83 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 3.26/1.83 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 3.26/1.83 ! [A5: $tType,A6: $tType] :
% 3.26/1.83 ( ( comple187826305attice @ A6 @ ( type2 @ A6 ) )
% 3.26/1.83 => ( comple187826305attice @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.26/1.83 ! [A5: $tType,A6: $tType] :
% 3.26/1.83 ( ( preorder @ A6 @ ( type2 @ A6 ) )
% 3.26/1.83 => ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.26/1.83 ! [A5: $tType,A6: $tType] :
% 3.26/1.83 ( ( order @ A6 @ ( type2 @ A6 ) )
% 3.26/1.83 => ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_fun___Orderings_Oord,axiom,
% 3.26/1.83 ! [A5: $tType,A6: $tType] :
% 3.26/1.83 ( ( ord @ A6 @ ( type2 @ A6 ) )
% 3.26/1.83 => ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
% 3.26/1.83 condit1037483654norder @ nat @ ( type2 @ nat ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 3.26/1.83 wellorder @ nat @ ( type2 @ nat ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 3.26/1.83 preorder @ nat @ ( type2 @ nat ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 3.26/1.83 linorder @ nat @ ( type2 @ nat ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 3.26/1.83 no_top @ nat @ ( type2 @ nat ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 3.26/1.83 order @ nat @ ( type2 @ nat ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 3.26/1.83 ord @ nat @ ( type2 @ nat ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_4,axiom,
% 3.26/1.83 comple187826305attice @ $o @ ( type2 @ $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
% 3.26/1.83 preorder @ $o @ ( type2 @ $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
% 3.26/1.83 linorder @ $o @ ( type2 @ $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
% 3.26/1.83 order @ $o @ ( type2 @ $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 3.26/1.83 ord @ $o @ ( type2 @ $o ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_9,axiom,
% 3.26/1.83 ! [A5: $tType] : ( preorder @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_10,axiom,
% 3.26/1.83 ! [A5: $tType] : ( order @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(tcon_Coinductive__List_Ollist___Orderings_Oord_11,axiom,
% 3.26/1.83 ! [A5: $tType] : ( ord @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
% 3.26/1.83
% 3.26/1.83 thf(conj_0,conjecture,
% 3.26/1.83 ( ( ord_less_eq @ ( coinductive_llist @ a ) )
% 3.26/1.83 = ( coinductive_lprefix @ a ) ) ).
% 3.26/1.83 ------------------------
% 3.26/1.86 % (20812)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_20653 for (2999ds/27Mi)
% 3.26/1.86 % (20813)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20653 for (2999ds/2Mi)
% 3.26/1.86 % (20814)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_20653 for (2999ds/2Mi)
% 3.26/1.86 % (20810)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20653 for (2999ds/183Mi)
% 3.26/1.86 % (20811)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_20653 for (2999ds/4Mi)
% 3.26/1.86 % (20815)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_20653 for (2999ds/275Mi)
% 3.26/1.86 % (20816)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20653 for (2999ds/18Mi)
% 3.26/1.86 % (20813)Instruction limit reached!
% 3.26/1.86 % (20813)------------------------------
% 3.26/1.86 % (20813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.86 % (20813)Termination reason: Unknown
% 3.26/1.86 % (20813)Termination phase: shuffling
% 3.26/1.86 % (20814)Instruction limit reached!
% 3.26/1.86 % (20814)------------------------------
% 3.26/1.86 % (20814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.86 % (20814)Termination reason: Unknown
% 3.26/1.86 % (20814)Termination phase: shuffling
% 3.26/1.86
% 3.26/1.86 % (20814)Memory used [KB]: 1279
% 3.26/1.86 % (20814)Time elapsed: 0.003 s
% 3.26/1.86 % (20814)Instructions burned: 3 (million)
% 3.26/1.87 % (20814)------------------------------
% 3.26/1.87 % (20814)------------------------------
% 3.26/1.87
% 3.26/1.87 % (20813)Memory used [KB]: 1279
% 3.26/1.87 % (20813)Time elapsed: 0.003 s
% 3.26/1.87 % (20813)Instructions burned: 2 (million)
% 3.26/1.87 % (20813)------------------------------
% 3.26/1.87 % (20813)------------------------------
% 3.26/1.87 % (20811)Instruction limit reached!
% 3.26/1.87 % (20811)------------------------------
% 3.26/1.87 % (20811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.87 % (20811)Termination reason: Unknown
% 3.26/1.87 % (20811)Termination phase: shuffling
% 3.26/1.87
% 3.26/1.87 % (20811)Memory used [KB]: 1407
% 3.26/1.87 % (20811)Time elapsed: 0.004 s
% 3.26/1.87 % (20811)Instructions burned: 4 (million)
% 3.26/1.87 % (20811)------------------------------
% 3.26/1.87 % (20811)------------------------------
% 3.26/1.87 % (20816)Instruction limit reached!
% 3.26/1.87 % (20816)------------------------------
% 3.26/1.87 % (20816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.87 % (20816)Termination reason: Unknown
% 3.26/1.87 % (20816)Termination phase: shuffling
% 3.26/1.87
% 3.26/1.87 % (20816)Memory used [KB]: 1791
% 3.26/1.87 % (20816)Time elapsed: 0.011 s
% 3.26/1.87 % (20816)Instructions burned: 18 (million)
% 3.26/1.87 % (20816)------------------------------
% 3.26/1.87 % (20816)------------------------------
% 3.26/1.88 % (20812)Instruction limit reached!
% 3.26/1.88 % (20812)------------------------------
% 3.26/1.88 % (20812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.88 % (20812)Termination reason: Unknown
% 3.26/1.88 % (20812)Termination phase: Property scanning
% 3.26/1.88
% 3.26/1.88 % (20812)Memory used [KB]: 1791
% 3.26/1.88 % (20812)Time elapsed: 0.015 s
% 3.26/1.88 % (20812)Instructions burned: 27 (million)
% 3.26/1.88 % (20812)------------------------------
% 3.26/1.88 % (20812)------------------------------
% 3.26/1.88 % (20817)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20653 for (2999ds/3Mi)
% 3.26/1.88 % (20818)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_20653 for (2999ds/37Mi)
% 3.26/1.88 % (20819)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_20653 for (2999ds/15Mi)
% 3.26/1.88 % (20817)Instruction limit reached!
% 3.26/1.88 % (20817)------------------------------
% 3.26/1.88 % (20817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.88 % (20817)Termination reason: Unknown
% 3.26/1.88 % (20817)Termination phase: shuffling
% 3.26/1.88
% 3.26/1.88 % (20817)Memory used [KB]: 1279
% 3.26/1.88 % (20817)Time elapsed: 0.003 s
% 3.26/1.88 % (20817)Instructions burned: 3 (million)
% 3.26/1.88 % (20817)------------------------------
% 3.26/1.88 % (20817)------------------------------
% 3.26/1.89 % (20815)First to succeed.
% 3.26/1.89 % (20815)Refutation found. Thanks to Tanya!
% 3.26/1.89 % SZS status Theorem for DTF2THF_20653
% 3.26/1.89 % SZS output start Proof for DTF2THF_20653
% 3.26/1.89 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.26/1.89 thf(type_def_7, type, coinductive_llist: ($tType) > $tType).
% 3.26/1.89 thf(type_def_8, type, nat: $tType).
% 3.26/1.89 thf(type_def_9, type, a: $tType).
% 3.26/1.89 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 3.26/1.89 thf(func_def_1, type, nat: $tType).
% 3.26/1.89 thf(func_def_2, type, itself: ($tType) > $tType).
% 3.26/1.89 thf(func_def_3, type, a: $tType).
% 3.26/1.89 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_5, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_6, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_7, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_8, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_9, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_10, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_11, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_12, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_13, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_14, type, comple187826305attice: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_15, type, condit1656338222tinuum: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_16, type, condit1037483654norder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.26/1.89 thf(func_def_17, type, coindu328551480prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 3.26/1.89 thf(func_def_18, type, coinductive_lappend: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_19, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.26/1.89 thf(func_def_20, type, coinductive_llast: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.26/1.89 thf(func_def_21, type, coinductive_LCons: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_22, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_23, type, coinductive_lnull: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.26/1.89 thf(func_def_24, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 3.26/1.89 thf(func_def_25, type, coinductive_lprefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 3.26/1.89 thf(func_def_26, type, coindu1478340336prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 3.26/1.89 thf(func_def_27, type, coindu63249387sorted: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 3.26/1.89 thf(func_def_28, type, hilbert_GreatestM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 3.26/1.89 thf(func_def_29, type, hilbert_LeastM: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > X0)).
% 3.26/1.89 thf(func_def_30, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.26/1.89 thf(func_def_31, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.26/1.89 thf(func_def_32, type, order_antimono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 3.26/1.89 thf(func_def_33, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 3.26/1.89 thf(func_def_34, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.26/1.89 thf(func_def_51, type, sK0: !>[X0: $tType]:((coinductive_llist @ X0 > coinductive_llist @ X0 > $o) > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_52, type, sK1: !>[X0: $tType]:((coinductive_llist @ X0 > coinductive_llist @ X0 > $o) > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_53, type, sK2: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.26/1.89 thf(func_def_54, type, sK3: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_55, type, sK4: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_56, type, sK5: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 3.26/1.89 thf(func_def_57, type, sK6: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_58, type, sK7: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_59, type, sK8: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_60, type, sK9: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_61, type, sK10: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > X0)).
% 3.26/1.89 thf(func_def_62, type, sK11: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 3.26/1.89 thf(func_def_63, type, sK12: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.26/1.89 thf(func_def_65, type, ph14: !>[X0: $tType]:(X0)).
% 3.26/1.89 thf(f920,plain,(
% 3.26/1.89 $false),
% 3.26/1.89 inference(trivial_inequality_removal,[status(thm)],[f892])).
% 3.26/1.89 thf(f892,plain,(
% 3.26/1.89 (ord_less_eq @ coinductive_llist @ a != ord_less_eq @ coinductive_llist @ a)),
% 3.26/1.89 inference(definition_unfolding,[status(thm)],[f856,f847])).
% 3.26/1.89 thf(f847,plain,(
% 3.26/1.89 ( ! [X0 : $tType] : ((ord_less_eq @ coinductive_llist @ X0 = coinductive_lprefix @ X0)) )),
% 3.26/1.89 inference(cnf_transformation,[status(thm)],[f1])).
% 3.26/1.89 thf(f1,axiom,(
% 3.26/1.89 ! [X0 : $tType] : (ord_less_eq @ coinductive_llist @ X0 = coinductive_lprefix @ X0)),
% 3.26/1.89 file('/export/starexec/sandbox/tmp/tmp.CZmJ4dWwbg/DTF2THF_20653.p',fact_0_less__eq__llist__def)).
% 3.26/1.89 thf(f856,plain,(
% 3.26/1.89 (ord_less_eq @ coinductive_llist @ a != coinductive_lprefix @ a)),
% 3.26/1.89 inference(cnf_transformation,[status(thm)],[f789])).
% 3.26/1.89 thf(f789,plain,(
% 3.26/1.89 (ord_less_eq @ coinductive_llist @ a != coinductive_lprefix @ a)),
% 3.26/1.89 inference(flattening,[status(thm)],[f275])).
% 3.26/1.89 thf(f275,negated_conjecture,(
% 3.26/1.89 ~(ord_less_eq @ coinductive_llist @ a = coinductive_lprefix @ a)),
% 3.26/1.89 inference(negated_conjecture,[status(cth)],[f274])).
% 3.26/1.89 thf(f274,conjecture,(
% 3.26/1.89 (ord_less_eq @ coinductive_llist @ a = coinductive_lprefix @ a)),
% 3.26/1.89 file('/export/starexec/sandbox/tmp/tmp.CZmJ4dWwbg/DTF2THF_20653.p',conj_0)).
% 3.26/1.89 % SZS output end Proof for DTF2THF_20653
% 3.26/1.89 % (20815)------------------------------
% 3.26/1.89 % (20815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.26/1.89 % (20815)Termination reason: Refutation
% 3.26/1.89
% 3.26/1.89 % (20815)Memory used [KB]: 6396
% 3.26/1.89 % (20815)Time elapsed: 0.023 s
% 3.26/1.89 % (20815)Instructions burned: 41 (million)
% 3.26/1.89 % (20815)------------------------------
% 3.26/1.89 % (20815)------------------------------
% 3.26/1.89 % (20809)Success in time 0.049 s
%------------------------------------------------------------------------------