%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT194^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n016.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 08:45:06 AM UTC 2026
% Result : Theorem 2.84s 1.54s
% Output : Refutation 2.84s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT194^1 : TPTP v9.2.1. Released v7.0.0.
% 0.13/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n016.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 06:11:06 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : DAT194^1 : TPTP v9.2.1. Released v7.0.0.
% 0.29/0.46 % Domain : Data Structures
% 0.29/0.46 % Problem : Lazy list mirror 89
% 0.29/0.46 % Version : [Bla16] axioms : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.29/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.29/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Bla16]
% 0.29/0.46 % Names : lmirror__89.p [Bla16]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.1.0
% 0.29/0.46 % Syntax : Number of formulae : 306 ( 128 unt; 38 typ; 0 def)
% 0.29/0.46 % Number of atoms : 736 ( 243 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.29/0.46 % Number of connectives : 3667 ( 109 ~; 26 |; 60 &;3150 @)
% 0.29/0.46 % ( 0 <=>; 322 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.29/0.46 % Number of types : 2 ( 1 usr)
% 0.29/0.46 % Number of type conns : 187 ( 187 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 40 ( 37 usr; 4 con; 0-8 aty)
% 0.29/0.46 % Number of variables : 1038 ( 40 ^; 927 !; 37 ?;1038 :)
% 0.29/0.46 % ( 34 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.46 % SPC : TH1_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.29/0.46 % 2016-07-13 14:41:25.281
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 %----Could-be-implicit typings (4)
% 0.29/0.46 thf(ty_t_Coinductive__List_Ollist,type,
% 0.29/0.46 coinductive_llist: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Set_Oset,type,
% 0.29/0.46 set: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_itself,type,
% 0.29/0.46 itself: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_tf_a,type,
% 0.29/0.46 a: $tType ).
% 0.29/0.46
% 0.29/0.46 %----Explicit typings (34)
% 0.29/0.46 thf(sy_cl_HOL_Otype,type,
% 0.29/0.46 type:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.46 ord:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.46 order:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.46 linorder:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.46 preorder:
% 0.29/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 0.29/0.46 coindu328551480prefix:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 0.29/0.46 coinductive_gen_lset:
% 0.29/0.46 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Olappend,type,
% 0.29/0.46 coinductive_lappend:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Olfilter,type,
% 0.29/0.46 coinductive_lfilter:
% 0.29/0.46 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.29/0.46 coinductive_lfinite:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ollast,type,
% 0.29/0.46 coinductive_llast:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 0.29/0.46 coinductive_LCons:
% 0.29/0.46 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.29/0.46 coinductive_LNil:
% 0.29/0.46 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 0.29/0.46 coindu1259883913_llist:
% 0.29/0.46 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 0.29/0.46 coinductive_lhd:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.29/0.46 coinductive_lnull:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.29/0.46 coinductive_lset:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Olmember,type,
% 0.29/0.46 coinductive_lmember:
% 0.29/0.46 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Olprefix,type,
% 0.29/0.46 coinductive_lprefix:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 0.29/0.46 coindu1478340336prefix:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 0.29/0.46 coindu63249387sorted:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 0.29/0.46 coindu1441602521_llist:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_HOL_Oundefined,type,
% 0.29/0.46 undefined:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_If,type,
% 0.29/0.46 if:
% 0.29/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror,type,
% 0.29/0.46 lMirro427583474mirror:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
% 0.29/0.46 lMirro999291890or_aux:
% 0.29/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.29/0.46 sup_sup:
% 0.29/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.46 ord_less_eq:
% 0.29/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Pure_Otype,type,
% 0.29/0.46 type2:
% 0.29/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect,type,
% 0.29/0.46 collect:
% 0.29/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_Oinsert,type,
% 0.29/0.46 insert:
% 0.29/0.46 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member,type,
% 0.29/0.46 member:
% 0.29/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_x,type,
% 0.29/0.46 x: a ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_xs,type,
% 0.29/0.46 xs: coinductive_llist @ a ).
% 0.29/0.46
% 0.29/0.46 %----Relevant facts (254)
% 0.29/0.46 thf(fact_0_lmirror__LNil,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lMirro427583474mirror @ A @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_LNil
% 0.29/0.46 thf(fact_1_lappend__code_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 0.29/0.46 = Ys ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_code(1)
% 0.29/0.46 thf(fact_2_lappend__LNil2,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = Xs ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_LNil2
% 0.29/0.46 thf(fact_3_lappend__code_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
% 0.29/0.46 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_code(2)
% 0.29/0.46 thf(fact_4_lappend__snocL1__conv__LCons2,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
% 0.29/0.46 = ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_snocL1_conv_LCons2
% 0.29/0.46 thf(fact_5_llist_Oinject,axiom,
% 0.29/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 0.29/0.46 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 0.29/0.46 = ( ( X21 = Y21 )
% 0.29/0.46 & ( X22 = Y22 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.inject
% 0.29/0.46 thf(fact_6_lappend__LNil__LNil,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_LNil_LNil
% 0.29/0.46 thf(fact_7_LNil__eq__lappend__iff,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_LNil @ A )
% 0.29/0.46 = ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( ( Xs
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 & ( Ys
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LNil_eq_lappend_iff
% 0.29/0.46 thf(fact_8_lappend__eq__LNil__iff,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( ( Xs
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 & ( Ys
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_eq_LNil_iff
% 0.29/0.46 thf(fact_9_llist_Odistinct_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_LNil @ A )
% 0.29/0.46 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.distinct(1)
% 0.29/0.46 thf(fact_10_llist_Oexhaust,axiom,
% 0.29/0.46 ! [A: $tType,Y: coinductive_llist @ A] :
% 0.29/0.46 ( ( Y
% 0.29/0.46 != ( coinductive_LNil @ A ) )
% 0.29/0.46 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 0.29/0.46 ( Y
% 0.29/0.46 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.exhaust
% 0.29/0.46 thf(fact_11_neq__LNil__conv,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( Xs
% 0.29/0.46 != ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( ? [X2: A,Xs2: coinductive_llist @ A] :
% 0.29/0.46 ( Xs
% 0.29/0.46 = ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neq_LNil_conv
% 0.29/0.46 thf(fact_12_lmirror__aux__LCons,axiom,
% 0.29/0.46 ! [A: $tType,Acc: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_LCons @ A @ X @ ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) @ Xs ) @ ( coinductive_LCons @ A @ X @ Acc ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux_LCons
% 0.29/0.46 thf(fact_13_lmirror__aux__simps_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X: coinductive_llist @ A] :
% 0.29/0.46 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X ) )
% 0.29/0.46 = ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux_simps(2)
% 0.29/0.46 thf(fact_14_lmirror__aux__simps_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Acc: coinductive_llist @ A] :
% 0.29/0.46 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = Acc ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux_simps(1)
% 0.29/0.46 thf(fact_15_lmirror__aux__acc,axiom,
% 0.29/0.46 ! [A: $tType,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( lMirro999291890or_aux @ A @ ( coinductive_lappend @ A @ Ys @ Zs ) @ Xs )
% 0.29/0.46 = ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ Ys @ Xs ) @ Zs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux_acc
% 0.29/0.46 thf(fact_16_lappend__assoc,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.29/0.46 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_assoc
% 0.29/0.46 thf(fact_17_lmirror__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( lMirro427583474mirror @ A )
% 0.29/0.46 = ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_def
% 0.29/0.46 thf(fact_18_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.29/0.46 = ( ( Xs
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 | ? [Xs2: coinductive_llist @ A] :
% 0.29/0.46 ( ( Xs
% 0.29/0.46 = ( coinductive_LCons @ A @ Y @ Xs2 ) )
% 0.29/0.46 & ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Coinductive_List.finite_lprefix_nitpick_simps(3)
% 0.29/0.46 thf(fact_19_lstrict__prefix__code_I2_J,axiom,
% 0.29/0.46 ! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% 0.29/0.46
% 0.29/0.46 % lstrict_prefix_code(2)
% 0.29/0.46 thf(fact_20_lstrict__prefix__code_I3_J,axiom,
% 0.29/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.29/0.46 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % lstrict_prefix_code(3)
% 0.29/0.46 thf(fact_21_llast__singleton,axiom,
% 0.29/0.46 ! [A: $tType,X: A] :
% 0.29/0.46 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 0.29/0.46 = X ) ).
% 0.29/0.46
% 0.29/0.46 % llast_singleton
% 0.29/0.46 thf(fact_22_lmember__code_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,X: A] :
% 0.29/0.46 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmember_code(1)
% 0.29/0.46 thf(fact_23_lmember__code_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.29/0.46 = ( ( X = Y )
% 0.29/0.46 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmember_code(2)
% 0.29/0.46 thf(fact_24_gen__lset__code_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( coinductive_gen_lset @ A @ A2 @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % gen_lset_code(1)
% 0.29/0.46 thf(fact_25_lfinite__rev__induct,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( P @ ( coinductive_LNil @ A ) )
% 0.29/0.46 => ( ! [X3: A,Xs3: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs3 )
% 0.29/0.46 => ( ( P @ Xs3 )
% 0.29/0.46 => ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) )
% 0.29/0.46 => ( P @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_rev_induct
% 0.29/0.46 thf(fact_26_lsorted__code_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted_code(2)
% 0.29/0.46 thf(fact_27_llist_Ocorec__code,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType] :
% 0.29/0.46 ( ( coindu1259883913_llist @ C @ A )
% 0.29/0.46 = ( ^ [P2: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C,A3: C] : ( if @ ( coinductive_llist @ A ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P2 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.corec_code
% 0.29/0.46 thf(fact_28_unfold__llist_Ocode,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( coindu1441602521_llist @ A @ B )
% 0.29/0.46 = ( ^ [P2: A > $o,G21: A > B,G22: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist.code
% 0.29/0.46 thf(fact_29_lfinite__code_I2_J,axiom,
% 0.29/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 0.29/0.46 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_code(2)
% 0.29/0.46 thf(fact_30_lfinite__LCons,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_LCons
% 0.29/0.46 thf(fact_31_lfinite__code_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_code(1)
% 0.29/0.46 thf(fact_32_lfinite__lappend,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_lappend
% 0.29/0.46 thf(fact_33_lsorted__code_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted_code(1)
% 0.29/0.46 thf(fact_34_unfold__llist__eq__LCons,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 0.29/0.46 = ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( ~ ( IS_LNIL @ B2 )
% 0.29/0.46 & ( X
% 0.29/0.46 = ( LHD @ B2 ) )
% 0.29/0.46 & ( Xs
% 0.29/0.46 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist_eq_LCons
% 0.29/0.46 thf(fact_35_llast__LCons2,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 0.29/0.46 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llast_LCons2
% 0.29/0.46 thf(fact_36_lfinite__lmirror__aux,axiom,
% 0.29/0.46 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 0.29/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 & ( coinductive_lfinite @ A @ Acc ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_lmirror_aux
% 0.29/0.46 thf(fact_37_lstrict__prefix__code_I4_J,axiom,
% 0.29/0.46 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 0.29/0.46 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 0.29/0.46 = ( ( X = Y )
% 0.29/0.46 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lstrict_prefix_code(4)
% 0.29/0.46 thf(fact_38_lstrict__prefix__code_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lstrict_prefix_code(1)
% 0.29/0.46 thf(fact_39_llast__lappend__LCons,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 0.29/0.46 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llast_lappend_LCons
% 0.29/0.46 thf(fact_40_llist__less__induct,axiom,
% 0.29/0.46 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ! [Xs3: coinductive_llist @ A] :
% 0.29/0.46 ( ! [Ys2: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs3 )
% 0.29/0.46 => ( P @ Ys2 ) )
% 0.29/0.46 => ( P @ Xs3 ) )
% 0.29/0.46 => ( P @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist_less_induct
% 0.29/0.46 thf(fact_41_lstrict__prefix__lfinite1,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lstrict_prefix_lfinite1
% 0.29/0.46 thf(fact_42_lfinite__LConsI,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_LConsI
% 0.29/0.46 thf(fact_43_lfinite__LNil,axiom,
% 0.29/0.46 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_LNil
% 0.29/0.46 thf(fact_44_lappend__inf,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.29/0.46 = Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_inf
% 0.29/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A: $tType,A4: A,P: A > $o] :
% 0.29/0.46 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 0.29/0.46 = ( P @ A4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] :
% 0.29/0.46 ( ( collect @ A
% 0.29/0.46 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.29/0.46 = A2 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_47_Collect__cong,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 = ( Q @ X3 ) )
% 0.29/0.46 => ( ( collect @ A @ P )
% 0.29/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_48_ext,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( F @ X3 )
% 0.29/0.46 = ( G @ X3 ) )
% 0.29/0.46 => ( F = G ) ) ).
% 0.29/0.46
% 0.29/0.46 % ext
% 0.29/0.46 thf(fact_49_llimit__induct,axiom,
% 0.29/0.46 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( P @ ( coinductive_LNil @ A ) )
% 0.29/0.46 => ( ! [X3: A,Xs3: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs3 )
% 0.29/0.46 => ( ( P @ Xs3 )
% 0.29/0.46 => ( P @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
% 0.29/0.46 => ( ( ! [Ys2: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
% 0.29/0.46 => ( P @ Ys2 ) )
% 0.29/0.46 => ( P @ Xs ) )
% 0.29/0.46 => ( P @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llimit_induct
% 0.29/0.46 thf(fact_50_lmirror__aux__inf,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 0.29/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
% 0.29/0.46 = Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux_inf
% 0.29/0.46 thf(fact_51_LNil,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LNil
% 0.29/0.46 thf(fact_52_unfold__llist_Octr_I2_J,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 0.29/0.46 ( ~ ( P3 @ A4 )
% 0.29/0.46 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 )
% 0.29/0.46 = ( coinductive_LCons @ B @ ( G212 @ A4 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ ( G223 @ A4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist.ctr(2)
% 0.29/0.46 thf(fact_53_unfold__llist_Octr_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 0.29/0.46 ( ( P3 @ A4 )
% 0.29/0.46 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 )
% 0.29/0.46 = ( coinductive_LNil @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist.ctr(1)
% 0.29/0.46 thf(fact_54_llist_Ocorec_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.29/0.46 ( ~ ( P3 @ A4 )
% 0.29/0.46 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 )
% 0.29/0.46 = ( coinductive_LCons @ A @ ( G212 @ A4 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q222 @ A4 ) @ ( G2212 @ A4 ) @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A4 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.corec(2)
% 0.29/0.46 thf(fact_55_llist_Ocorec_I1_J,axiom,
% 0.29/0.46 ! [C: $tType,A: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.29/0.46 ( ( P3 @ A4 )
% 0.29/0.46 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 )
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.corec(1)
% 0.29/0.46 thf(fact_56_lfinite_Ocases,axiom,
% 0.29/0.46 ! [A: $tType,A4: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ A4 )
% 0.29/0.46 => ( ( A4
% 0.29/0.46 != ( coinductive_LNil @ A ) )
% 0.29/0.46 => ~ ! [Xs3: coinductive_llist @ A] :
% 0.29/0.46 ( ? [X3: A] :
% 0.29/0.46 ( A4
% 0.29/0.46 = ( coinductive_LCons @ A @ X3 @ Xs3 ) )
% 0.29/0.46 => ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite.cases
% 0.29/0.46 thf(fact_57_lfinite_Osimps,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coinductive_lfinite @ A )
% 0.29/0.46 = ( ^ [A3: coinductive_llist @ A] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 | ? [Xs4: coinductive_llist @ A,X2: A] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( coinductive_LCons @ A @ X2 @ Xs4 ) )
% 0.29/0.46 & ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite.simps
% 0.29/0.46 thf(fact_58_lfinite_Oinducts,axiom,
% 0.29/0.46 ! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ X )
% 0.29/0.46 => ( ( P @ ( coinductive_LNil @ A ) )
% 0.29/0.46 => ( ! [Xs3: coinductive_llist @ A,X3: A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs3 )
% 0.29/0.46 => ( ( P @ Xs3 )
% 0.29/0.46 => ( P @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
% 0.29/0.46 => ( P @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite.inducts
% 0.29/0.46 thf(fact_59_Singleton,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Singleton
% 0.29/0.46 thf(fact_60_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 0.29/0.46
% 0.29/0.46 % Coinductive_List.finite_lprefix_nitpick_simps(2)
% 0.29/0.46 thf(fact_61_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( Xs
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Coinductive_List.finite_lprefix_nitpick_simps(1)
% 0.29/0.46 thf(fact_62_lsorted_Ocases,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A4: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu63249387sorted @ A @ A4 )
% 0.29/0.46 => ( ( A4
% 0.29/0.46 != ( coinductive_LNil @ A ) )
% 0.29/0.46 => ( ! [X3: A] :
% 0.29/0.46 ( A4
% 0.29/0.46 != ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 0.29/0.46 => ~ ! [X3: A,Y2: A,Xs3: coinductive_llist @ A] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.46 => ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted.cases
% 0.29/0.46 thf(fact_63_lsorted_Osimps,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( coindu63249387sorted @ A )
% 0.29/0.46 = ( ^ [A3: coinductive_llist @ A] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 | ? [X2: A] :
% 0.29/0.46 ( A3
% 0.29/0.46 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
% 0.29/0.46 | ? [X2: A,Y3: A,Xs4: coinductive_llist @ A] :
% 0.29/0.46 ( ( A3
% 0.29/0.46 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) )
% 0.29/0.46 & ( ord_less_eq @ A @ X2 @ Y3 )
% 0.29/0.46 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted.simps
% 0.29/0.46 thf(fact_64_lsorted_Ocoinduct,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
% 0.29/0.46 ( ( X4 @ X )
% 0.29/0.46 => ( ! [X3: coinductive_llist @ A] :
% 0.29/0.46 ( ( X4 @ X3 )
% 0.29/0.46 => ( ( X3
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 | ? [Xa2: A] :
% 0.29/0.46 ( X3
% 0.29/0.46 = ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LNil @ A ) ) )
% 0.29/0.46 | ? [Xa2: A,Y4: A,Xs5: coinductive_llist @ A] :
% 0.29/0.46 ( ( X3
% 0.29/0.46 = ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
% 0.29/0.46 & ( ord_less_eq @ A @ Xa2 @ Y4 )
% 0.29/0.46 & ( ( X4 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) )
% 0.29/0.46 | ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) ) ) ) )
% 0.29/0.46 => ( coindu63249387sorted @ A @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted.coinduct
% 0.29/0.46 thf(fact_65_lstrict__prefix__lappend__conv,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lstrict_prefix_lappend_conv
% 0.29/0.46 thf(fact_66_split__llist,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 0.29/0.46 ( ( Xs
% 0.29/0.46 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 0.29/0.46 & ( coinductive_lfinite @ A @ Ys3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % split_llist
% 0.29/0.46 thf(fact_67_split__llist__first,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 0.29/0.46 ( ( Xs
% 0.29/0.46 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 0.29/0.46 & ( coinductive_lfinite @ A @ Ys3 )
% 0.29/0.46 & ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % split_llist_first
% 0.29/0.46 thf(fact_68_lsorted__LCons__LCons,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 0.29/0.46 = ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted_LCons_LCons
% 0.29/0.46 thf(fact_69_gen__lset__code_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_gen_lset @ A @ A2 @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_gen_lset @ A @ ( insert @ A @ X @ A2 ) @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % gen_lset_code(2)
% 0.29/0.46 thf(fact_70_lprefix__lappend__same,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 0.29/0.46 = ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lappend_same
% 0.29/0.46 thf(fact_71_llist_Oleq__refl,axiom,
% 0.29/0.46 ! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% 0.29/0.46
% 0.29/0.46 % llist.leq_refl
% 0.29/0.46 thf(fact_72_lprefix__refl,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_refl
% 0.29/0.46 thf(fact_73_LCons__lprefix__LCons,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.29/0.46 = ( ( X = Y )
% 0.29/0.46 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LCons_lprefix_LCons
% 0.29/0.46 thf(fact_74_lprefix__code_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_code(1)
% 0.29/0.46 thf(fact_75_lnull__lappend,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lnull_lappend
% 0.29/0.46 thf(fact_76_lappend_Odisc__iff_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.29/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend.disc_iff(2)
% 0.29/0.46 thf(fact_77_lmirror__aux_Odisc__iff_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) )
% 0.29/0.46 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 | ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux.disc_iff(2)
% 0.29/0.46 thf(fact_78_lnull__lmirror__aux,axiom,
% 0.29/0.46 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 0.29/0.46 = ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 & ( coinductive_lnull @ A @ Acc ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lnull_lmirror_aux
% 0.29/0.46 thf(fact_79_unfold__llist_Odisc__iff_I1_J,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A4: A] :
% 0.29/0.46 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) )
% 0.29/0.46 = ( P3 @ A4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist.disc_iff(1)
% 0.29/0.46 thf(fact_80_unfold__llist_Odisc__iff_I2_J,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A4: A] :
% 0.29/0.46 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) ) )
% 0.29/0.46 = ( ~ ( P3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist.disc_iff(2)
% 0.29/0.46 thf(fact_81_llist_Ocorec__disc__iff_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A4: C] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) )
% 0.29/0.46 = ( P3 @ A4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.corec_disc_iff(1)
% 0.29/0.46 thf(fact_82_llist_Ocorec__disc__iff_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A4: C] :
% 0.29/0.46 ( ( ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) ) )
% 0.29/0.46 = ( ~ ( P3 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.corec_disc_iff(2)
% 0.29/0.46 thf(fact_83_lnull__lmirror,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ ( lMirro427583474mirror @ A @ Xs ) )
% 0.29/0.46 = ( coinductive_lnull @ A @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lnull_lmirror
% 0.29/0.46 thf(fact_84_lprefix__LNil,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( coinductive_lnull @ A @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_LNil
% 0.29/0.46 thf(fact_85_lset__LCons,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( insert @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_LCons
% 0.29/0.46 thf(fact_86_llist_Osimps_I19_J,axiom,
% 0.29/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.29/0.46 = ( insert @ A @ X21 @ ( coinductive_lset @ A @ X22 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.simps(19)
% 0.29/0.46 thf(fact_87_lset__lappend1,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_lappend1
% 0.29/0.46 thf(fact_88_lmirror__aux_Oexhaust,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ~ ( coinductive_lnull @ A @ Acc ) )
% 0.29/0.46 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 | ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux.exhaust
% 0.29/0.46 thf(fact_89_lprefix__down__linear,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_down_linear
% 0.29/0.46 thf(fact_90_lprefix__not__lnullD,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_not_lnullD
% 0.29/0.46 thf(fact_91_llist_Oleq__antisym,axiom,
% 0.29/0.46 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ X @ Y )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Y @ X )
% 0.29/0.46 => ( X = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.leq_antisym
% 0.29/0.46 thf(fact_92_wlog__linorder__le,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > A > $o,B2: A,A4: A] :
% 0.29/0.46 ( ! [A5: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.29/0.46 => ( P @ A5 @ B3 ) )
% 0.29/0.46 => ( ( ( P @ B2 @ A4 )
% 0.29/0.46 => ( P @ A4 @ B2 ) )
% 0.29/0.46 => ( P @ A4 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % wlog_linorder_le
% 0.29/0.46 thf(fact_93_lprefix__antisym,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 0.29/0.46 => ( Xs = Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_antisym
% 0.29/0.46 thf(fact_94_llist_Oleq__trans,axiom,
% 0.29/0.46 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,Z: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ X @ Y )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Y @ Z )
% 0.29/0.46 => ( coinductive_lprefix @ A @ X @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.leq_trans
% 0.29/0.46 thf(fact_95_lappend_Oexhaust,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.29/0.46 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend.exhaust
% 0.29/0.46 thf(fact_96_lprefix__lnullD,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( ( coinductive_lnull @ A @ Ys )
% 0.29/0.46 => ( coinductive_lnull @ A @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lnullD
% 0.29/0.46 thf(fact_97_lprefix__trans,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 0.29/0.46 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_trans
% 0.29/0.46 thf(fact_98_lprefix__lsetD,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lsetD
% 0.29/0.46 thf(fact_99_lprefix__lnull,axiom,
% 0.29/0.46 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Ys )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lnull
% 0.29/0.46 thf(fact_100_lnull__lprefix,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 0.29/0.46
% 0.29/0.46 % lnull_lprefix
% 0.29/0.46 thf(fact_101_lzip_Oexhaust,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.29/0.46 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 | ( coinductive_lnull @ B @ Ys ) )
% 0.29/0.46 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lzip.exhaust
% 0.29/0.46 thf(fact_102_lsorted__LCons,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( ( coindu63249387sorted @ A @ Xs )
% 0.29/0.46 & ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted_LCons
% 0.29/0.46 thf(fact_103_llist_Oset__induct,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A4: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ A4 ) )
% 0.29/0.46 => ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 0.29/0.46 => ( ! [Z1: A,Z2: coinductive_llist @ A,Xa3: A] :
% 0.29/0.46 ( ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Z2 ) )
% 0.29/0.46 => ( ( P @ Xa3 @ Z2 )
% 0.29/0.46 => ( P @ Xa3 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
% 0.29/0.46 => ( P @ X @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.set_induct
% 0.29/0.46 thf(fact_104_llist_Oset__cases,axiom,
% 0.29/0.46 ! [A: $tType,E: A,A4: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ E @ ( coinductive_lset @ A @ A4 ) )
% 0.29/0.46 => ( ! [Z2: coinductive_llist @ A] :
% 0.29/0.46 ( A4
% 0.29/0.46 != ( coinductive_LCons @ A @ E @ Z2 ) )
% 0.29/0.46 => ~ ! [Z1: A,Z2: coinductive_llist @ A] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 0.29/0.46 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.set_cases
% 0.29/0.46 thf(fact_105_lset__induct_H,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 0.29/0.46 => ( ! [X5: A,Xs3: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 0.29/0.46 => ( ( P @ Xs3 )
% 0.29/0.46 => ( P @ ( coinductive_LCons @ A @ X5 @ Xs3 ) ) ) )
% 0.29/0.46 => ( P @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_induct'
% 0.29/0.46 thf(fact_106_lset__induct,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 0.29/0.46 => ( ! [X5: A,Xs3: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 0.29/0.46 => ( ( X != X5 )
% 0.29/0.46 => ( ( P @ Xs3 )
% 0.29/0.46 => ( P @ ( coinductive_LCons @ A @ X5 @ Xs3 ) ) ) ) )
% 0.29/0.46 => ( P @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_induct
% 0.29/0.46 thf(fact_107_lset__cases,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ( ! [Xs6: coinductive_llist @ A] :
% 0.29/0.46 ( Xs
% 0.29/0.46 != ( coinductive_LCons @ A @ X @ Xs6 ) )
% 0.29/0.46 => ~ ! [X5: A,Xs6: coinductive_llist @ A] :
% 0.29/0.46 ( ( Xs
% 0.29/0.46 = ( coinductive_LCons @ A @ X5 @ Xs6 ) )
% 0.29/0.46 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs6 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_cases
% 0.29/0.46 thf(fact_108_llist_Oset__intros_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.set_intros(1)
% 0.29/0.46 thf(fact_109_llist_Oset__intros_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 0.29/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.set_intros(2)
% 0.29/0.46 thf(fact_110_lset__intros_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_intros(1)
% 0.29/0.46 thf(fact_111_lset__intros_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X6: A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X6 @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_intros(2)
% 0.29/0.46 thf(fact_112_not__lnull__conv,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 0.29/0.46 = ( ? [X2: A,Xs2: coinductive_llist @ A] :
% 0.29/0.46 ( Xs
% 0.29/0.46 = ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_lnull_conv
% 0.29/0.46 thf(fact_113_llist_OdiscI_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 0.29/0.46 ( ( Llist
% 0.29/0.46 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 0.29/0.46 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.discI(2)
% 0.29/0.46 thf(fact_114_llist_Odisc_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 0.29/0.46 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.disc(2)
% 0.29/0.46 thf(fact_115_lnull__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coinductive_lnull @ A )
% 0.29/0.46 = ( ^ [Llist2: coinductive_llist @ A] :
% 0.29/0.46 ( Llist2
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lnull_def
% 0.29/0.46 thf(fact_116_llist_Ocollapse_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Llist )
% 0.29/0.46 => ( Llist
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.collapse(1)
% 0.29/0.46 thf(fact_117_llist_OdiscI_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.29/0.46 ( ( Llist
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 => ( coinductive_lnull @ A @ Llist ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.discI(1)
% 0.29/0.46 thf(fact_118_llist_Odisc_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.disc(1)
% 0.29/0.46 thf(fact_119_LCons__lprefix__conv,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
% 0.29/0.46 = ( ? [Ys4: coinductive_llist @ A] :
% 0.29/0.46 ( ( Ys
% 0.29/0.46 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 0.29/0.46 & ( coinductive_lprefix @ A @ Xs @ Ys4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LCons_lprefix_conv
% 0.29/0.46 thf(fact_120_Le__LCons,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Le_LCons
% 0.29/0.46 thf(fact_121_lappend__lnull2,axiom,
% 0.29/0.46 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Ys )
% 0.29/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.29/0.46 = Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_lnull2
% 0.29/0.46 thf(fact_122_lappend__lnull1,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.29/0.46 = Ys ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_lnull1
% 0.29/0.46 thf(fact_123_lappend_Odisc_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lnull @ A @ Ys )
% 0.29/0.46 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend.disc(1)
% 0.29/0.46 thf(fact_124_lappend_Odisc_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 | ~ ( coinductive_lnull @ A @ Ys ) )
% 0.29/0.46 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend.disc(2)
% 0.29/0.46 thf(fact_125_lnull__imp__lfinite,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lnull_imp_lfinite
% 0.29/0.46 thf(fact_126_LNil__lprefix,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 0.29/0.46
% 0.29/0.46 % LNil_lprefix
% 0.29/0.46 thf(fact_127_lprefix__lappend__sameI,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lappend_sameI
% 0.29/0.46 thf(fact_128_lprefix__conv__lappend,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coinductive_lprefix @ A )
% 0.29/0.46 = ( ^ [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 0.29/0.46 ? [Zs3: coinductive_llist @ A] :
% 0.29/0.46 ( Ys5
% 0.29/0.46 = ( coinductive_lappend @ A @ Xs4 @ Zs3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_conv_lappend
% 0.29/0.46 thf(fact_129_lprefix__lappendD,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lappendD
% 0.29/0.46 thf(fact_130_lappend__lprefixE,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 0.29/0.46 => ~ ! [Zs4: coinductive_llist @ A] :
% 0.29/0.46 ( Zs
% 0.29/0.46 != ( coinductive_lappend @ A @ Xs @ Zs4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend_lprefixE
% 0.29/0.46 thf(fact_131_lprefix__lappend,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lappend
% 0.29/0.46 thf(fact_132_not__lfinite__lprefix__conv__eq,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 = ( Xs = Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_lfinite_lprefix_conv_eq
% 0.29/0.46 thf(fact_133_lprefix__lfiniteD,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( ( coinductive_lfinite @ A @ Ys )
% 0.29/0.46 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lfiniteD
% 0.29/0.46 thf(fact_134_lsorted__lprefixD,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( ( coindu63249387sorted @ A @ Ys )
% 0.29/0.46 => ( coindu63249387sorted @ A @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted_lprefixD
% 0.29/0.46 thf(fact_135_lmirror__aux_Odisc_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 0.29/0.46 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 | ~ ( coinductive_lnull @ A @ Acc ) )
% 0.29/0.46 => ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux.disc(2)
% 0.29/0.46 thf(fact_136_lmirror__aux_Odisc_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lnull @ A @ Acc )
% 0.29/0.46 => ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux.disc(1)
% 0.29/0.46 thf(fact_137_unfold__llist_Odisc_I1_J,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 0.29/0.46 ( ( P3 @ A4 )
% 0.29/0.46 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist.disc(1)
% 0.29/0.46 thf(fact_138_unfold__llist_Odisc_I2_J,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 0.29/0.46 ( ~ ( P3 @ A4 )
% 0.29/0.46 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % unfold_llist.disc(2)
% 0.29/0.46 thf(fact_139_llist_Ocorec__disc_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.29/0.46 ( ( P3 @ A4 )
% 0.29/0.46 => ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.corec_disc(1)
% 0.29/0.46 thf(fact_140_llist_Ocorec__disc_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 0.29/0.46 ( ~ ( P3 @ A4 )
% 0.29/0.46 => ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llist.corec_disc(2)
% 0.29/0.46 thf(fact_141_lstrict__prefix__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coindu1478340336prefix @ A )
% 0.29/0.46 = ( ^ [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs4 @ Ys5 )
% 0.29/0.46 & ( Xs4 != Ys5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lstrict_prefix_def
% 0.29/0.46 thf(fact_142_Coinductive__List_Ofinite__lprefix__def,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coindu328551480prefix @ A )
% 0.29/0.46 = ( coinductive_lprefix @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % Coinductive_List.finite_lprefix_def
% 0.29/0.46 thf(fact_143_lset__lmember,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lset_lmember
% 0.29/0.46 thf(fact_144_in__lset__lappend__iff,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 0.29/0.46 = ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 | ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 & ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % in_lset_lappend_iff
% 0.29/0.46 thf(fact_145_lappend_Octr_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lnull @ A @ Ys )
% 0.29/0.46 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lappend.ctr(1)
% 0.29/0.46 thf(fact_146_lprefix__LCons__conv,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 0.29/0.46 = ( ( Xs
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 | ? [Xs2: coinductive_llist @ A] :
% 0.29/0.46 ( ( Xs
% 0.29/0.46 = ( coinductive_LCons @ A @ Y @ Xs2 ) )
% 0.29/0.46 & ( coinductive_lprefix @ A @ Xs2 @ Ys ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_LCons_conv
% 0.29/0.46 thf(fact_147_lprefix_Ocoinduct,axiom,
% 0.29/0.46 ! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 0.29/0.46 ( ( X4 @ X @ Xa )
% 0.29/0.46 => ( ! [X3: coinductive_llist @ A,Xa3: coinductive_llist @ A] :
% 0.29/0.46 ( ( X4 @ X3 @ Xa3 )
% 0.29/0.46 => ( ? [Xs5: coinductive_llist @ A] :
% 0.29/0.46 ( ( X3
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 & ( Xa3 = Xs5 ) )
% 0.29/0.46 | ? [Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
% 0.29/0.46 ( ( X3
% 0.29/0.46 = ( coinductive_LCons @ A @ Xb @ Xs5 ) )
% 0.29/0.46 & ( Xa3
% 0.29/0.46 = ( coinductive_LCons @ A @ Xb @ Ys2 ) )
% 0.29/0.46 & ( ( X4 @ Xs5 @ Ys2 )
% 0.29/0.46 | ( coinductive_lprefix @ A @ Xs5 @ Ys2 ) ) ) ) )
% 0.29/0.46 => ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix.coinduct
% 0.29/0.46 thf(fact_148_lprefix_Osimps,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coinductive_lprefix @ A )
% 0.29/0.46 = ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 0.29/0.46 ( ? [Xs4: coinductive_llist @ A] :
% 0.29/0.46 ( ( A12
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 & ( A23 = Xs4 ) )
% 0.29/0.46 | ? [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A,X2: A] :
% 0.29/0.46 ( ( A12
% 0.29/0.46 = ( coinductive_LCons @ A @ X2 @ Xs4 ) )
% 0.29/0.46 & ( A23
% 0.29/0.46 = ( coinductive_LCons @ A @ X2 @ Ys5 ) )
% 0.29/0.46 & ( coinductive_lprefix @ A @ Xs4 @ Ys5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix.simps
% 0.29/0.46 thf(fact_149_lprefix_Ocases,axiom,
% 0.29/0.46 ! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ A1 @ A22 )
% 0.29/0.46 => ( ( ( A1
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 => ! [Xs3: coinductive_llist @ A] : ( A22 != Xs3 ) )
% 0.29/0.46 => ~ ! [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A,X3: A] :
% 0.29/0.46 ( ( A1
% 0.29/0.46 = ( coinductive_LCons @ A @ X3 @ Xs3 ) )
% 0.29/0.46 => ( ( A22
% 0.29/0.46 = ( coinductive_LCons @ A @ X3 @ Ys3 ) )
% 0.29/0.46 => ~ ( coinductive_lprefix @ A @ Xs3 @ Ys3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix.cases
% 0.29/0.46 thf(fact_150_lprefix__code_I2_J,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_code(2)
% 0.29/0.46 thf(fact_151_lmirror__aux_Octr_I1_J,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lnull @ A @ Acc )
% 0.29/0.46 => ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lmirror_aux.ctr(1)
% 0.29/0.46 thf(fact_152_llast__LCons,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.29/0.46 ( ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = X ) )
% 0.29/0.46 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llast_LCons
% 0.29/0.46 thf(fact_153_LCons__LCons,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 0.29/0.46 => ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % LCons_LCons
% 0.29/0.46 thf(fact_154_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coinductive_lprefix @ A )
% 0.29/0.46 = ( ^ [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lfinite @ A @ Xs4 )
% 0.29/0.46 => ( coindu328551480prefix @ A @ Xs4 @ Ys5 ) )
% 0.29/0.46 & ( ~ ( coinductive_lfinite @ A @ Xs4 )
% 0.29/0.46 => ( Xs4 = Ys5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Coinductive_List.lprefix_nitpick_simps
% 0.29/0.46 thf(fact_155_insertCI,axiom,
% 0.29/0.46 ! [A: $tType,A4: A,B4: set @ A,B2: A] :
% 0.29/0.46 ( ( ~ ( member @ A @ A4 @ B4 )
% 0.29/0.46 => ( A4 = B2 ) )
% 0.29/0.46 => ( member @ A @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insertCI
% 0.29/0.46 thf(fact_156_insert__iff,axiom,
% 0.29/0.46 ! [A: $tType,A4: A,B2: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A4 @ ( insert @ A @ B2 @ A2 ) )
% 0.29/0.46 = ( ( A4 = B2 )
% 0.29/0.46 | ( member @ A @ A4 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_iff
% 0.29/0.46 thf(fact_157_insert__subset,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B4 )
% 0.29/0.46 = ( ( member @ A @ X @ B4 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_subset
% 0.29/0.46 thf(fact_158_insert__absorb2,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A] :
% 0.29/0.46 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 0.29/0.46 = ( insert @ A @ X @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb2
% 0.29/0.46 thf(fact_159_order__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_160_llast__lappend,axiom,
% 0.29/0.46 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lnull @ A @ Ys )
% 0.29/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( coinductive_llast @ A @ Xs ) ) )
% 0.29/0.46 & ( ~ ( coinductive_lnull @ A @ Ys )
% 0.29/0.46 => ( ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( coinductive_llast @ A @ Ys ) ) )
% 0.29/0.46 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( undefined @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llast_lappend
% 0.29/0.46 thf(fact_161_lfilter__eq__LConsD,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 0.29/0.46 = ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 0.29/0.46 ( ( Ys
% 0.29/0.46 = ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
% 0.29/0.46 & ( coinductive_lfinite @ A @ Us )
% 0.29/0.46 & ! [X7: A] :
% 0.29/0.46 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Us ) )
% 0.29/0.46 => ~ ( P @ X7 ) )
% 0.29/0.46 & ( P @ X )
% 0.29/0.46 & ( Xs
% 0.29/0.46 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_eq_LConsD
% 0.29/0.46 thf(fact_162_subsetI,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ A2 )
% 0.29/0.46 => ( member @ A @ X3 @ B4 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetI
% 0.29/0.46 thf(fact_163_subset__antisym,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
% 0.29/0.46 => ( A2 = B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_antisym
% 0.29/0.46 thf(fact_164_lfilter__idem,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.29/0.46 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_idem
% 0.29/0.46 thf(fact_165_lfilter__LCons,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( P @ X )
% 0.29/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 0.29/0.46 & ( ~ ( P @ X )
% 0.29/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_LCons
% 0.29/0.46 thf(fact_166_lfilter__LNil,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o] :
% 0.29/0.46 ( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_LNil
% 0.29/0.46 thf(fact_167_lnull__lfilter,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ~ ( P @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lnull_lfilter
% 0.29/0.46 thf(fact_168_diverge__lfilter__LNil,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ~ ( P @ X3 ) )
% 0.29/0.46 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.29/0.46 = ( coinductive_LNil @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % diverge_lfilter_LNil
% 0.29/0.46 thf(fact_169_lfilter__lappend__lfinite,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.46 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_lappend_lfinite
% 0.29/0.46 thf(fact_170_set__mp,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( member @ A @ X @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_mp
% 0.29/0.46 thf(fact_171_in__mono,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( member @ A @ X @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % in_mono
% 0.29/0.46 thf(fact_172_subsetD,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ( member @ A @ C2 @ A2 )
% 0.29/0.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetD
% 0.29/0.46 thf(fact_173_subsetCE,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ( member @ A @ C2 @ A2 )
% 0.29/0.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subsetCE
% 0.29/0.46 thf(fact_174_equalityE,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( A2 = B4 )
% 0.29/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityE
% 0.29/0.46 thf(fact_175_subset__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.29/0.46 ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ A6 )
% 0.29/0.46 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_eq
% 0.29/0.46 thf(fact_176_equalityD1,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( A2 = B4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD1
% 0.29/0.46 thf(fact_177_equalityD2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( A2 = B4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % equalityD2
% 0.29/0.46 thf(fact_178_set__rev__mp,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( member @ A @ X @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_rev_mp
% 0.29/0.46 thf(fact_179_subset__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.29/0.46 ! [T: A] :
% 0.29/0.46 ( ( member @ A @ T @ A6 )
% 0.29/0.46 => ( member @ A @ T @ B5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_iff
% 0.29/0.46 thf(fact_180_rev__subsetD,axiom,
% 0.29/0.46 ! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ( member @ A @ C2 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( member @ A @ C2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % rev_subsetD
% 0.29/0.46 thf(fact_181_subset__refl,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.29/0.46
% 0.29/0.46 % subset_refl
% 0.29/0.46 thf(fact_182_Collect__mono,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( P @ X3 )
% 0.29/0.46 => ( Q @ X3 ) )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono
% 0.29/0.46 thf(fact_183_subset__trans,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_trans
% 0.29/0.46 thf(fact_184_set__eq__subset,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ^ [Y5: set @ A,Z3: set @ A] : ( Y5 = Z3 ) )
% 0.29/0.46 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 0.29/0.46 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % set_eq_subset
% 0.29/0.46 thf(fact_185_contra__subsetD,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ~ ( member @ A @ C2 @ B4 )
% 0.29/0.46 => ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % contra_subsetD
% 0.29/0.46 thf(fact_186_Collect__mono__iff,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ( ( P @ X2 )
% 0.29/0.46 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mono_iff
% 0.29/0.46 thf(fact_187_lprefix__lfilterI,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o] :
% 0.29/0.46 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 0.29/0.46 => ( coinductive_lprefix @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lprefix_lfilterI
% 0.29/0.46 thf(fact_188_lfilter__LCons__seek,axiom,
% 0.29/0.46 ! [A: $tType,P3: A > $o,X: A,L: coinductive_llist @ A] :
% 0.29/0.46 ( ~ ( P3 @ X )
% 0.29/0.46 => ( ( coinductive_lfilter @ A @ P3 @ ( coinductive_LCons @ A @ X @ L ) )
% 0.29/0.46 = ( coinductive_lfilter @ A @ P3 @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_LCons_seek
% 0.29/0.46 thf(fact_189_lfilter__LCons__found,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.46 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_LCons_found
% 0.29/0.46 thf(fact_190_lfilter__cong,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 0.29/0.46 ( ( Xs = Ys )
% 0.29/0.46 => ( ! [X3: A] :
% 0.29/0.46 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 0.29/0.46 => ( ( P @ X3 )
% 0.29/0.46 = ( Q @ X3 ) ) )
% 0.29/0.46 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.29/0.46 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_cong
% 0.29/0.46 thf(fact_191_lfilter__id__conv,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.29/0.46 = Xs )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ( P @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_id_conv
% 0.29/0.46 thf(fact_192_lfinite__lfilterI,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 0.29/0.46 ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfinite_lfilterI
% 0.29/0.46 thf(fact_193_lsorted__lfilterI,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Xs: coinductive_llist @ A,P: A > $o] :
% 0.29/0.46 ( ( coindu63249387sorted @ A @ Xs )
% 0.29/0.46 => ( coindu63249387sorted @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lsorted_lfilterI
% 0.29/0.46 thf(fact_194_lfilter__empty__conv,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.29/0.46 = ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( ! [X2: A] :
% 0.29/0.46 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 0.29/0.46 => ~ ( P @ X2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_empty_conv
% 0.29/0.46 thf(fact_195_lfilter__eq__lappend__lfiniteD,axiom,
% 0.29/0.46 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 0.29/0.46 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 0.29/0.46 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 0.29/0.46 => ( ( coinductive_lfinite @ A @ Ys )
% 0.29/0.46 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 0.29/0.46 ( ( Xs
% 0.29/0.46 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 0.29/0.46 & ( coinductive_lfinite @ A @ Us )
% 0.29/0.46 & ( Ys
% 0.29/0.46 = ( coinductive_lfilter @ A @ P @ Us ) )
% 0.29/0.46 & ( Zs
% 0.29/0.46 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % lfilter_eq_lappend_lfiniteD
% 0.29/0.46 thf(fact_196_llast__LNil,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
% 0.29/0.46 = ( undefined @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % llast_LNil
% 0.29/0.46 thf(fact_197_llast__linfinite,axiom,
% 0.29/0.46 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.29/0.46 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.29/0.46 => ( ( coinductive_llast @ A @ Xs )
% 0.29/0.46 = ( undefined @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % llast_linfinite
% 0.29/0.46 thf(fact_198_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,A4: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A4 @ B2 )
% 0.29/0.46 => ( A4 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_199_dual__order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [B2: A,A4: A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A4 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_200_linorder__wlog,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [P: A > A > $o,A4: A,B2: A] :
% 0.29/0.46 ( ! [A5: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A5 @ B3 )
% 0.29/0.46 => ( P @ A5 @ B3 ) )
% 0.29/0.46 => ( ! [A5: A,B3: A] :
% 0.29/0.46 ( ( P @ B3 @ A5 )
% 0.29/0.46 => ( P @ A5 @ B3 ) )
% 0.29/0.46 => ( P @ A4 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_wlog
% 0.29/0.46 thf(fact_201_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_202_order__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_203_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A4: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ A4 )
% 0.29/0.46 => ( A4 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_204_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A4: A,B2: A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B2 )
% 0.29/0.46 => ( ( B2 = C2 )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_205_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A4: A,B2: A,C2: A] :
% 0.29/0.46 ( ( A4 = B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_206_antisym__conv,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [Y: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 = ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_207_le__cases3,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A,Z: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ X @ Z )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_208_order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [A4: A,B2: A,C2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_209_le__cases,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_210_eq__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( X = Y )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_refl
% 0.29/0.46 thf(fact_211_linear,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linear
% 0.29/0.46 thf(fact_212_antisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ! [X: A,Y: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y @ X )
% 0.29/0.46 => ( X = Y ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym
% 0.29/0.46 thf(fact_213_eq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A @ ( type2 @ A ) )
% 0.29/0.46 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 0.29/0.46 = ( ^ [X2: A,Y3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 0.29/0.46 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_iff
% 0.29/0.46 thf(fact_214_ord__le__eq__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A4: A,B2: A,F: A > B,C2: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B2 )
% 0.29/0.46 => ( ( ( F @ B2 )
% 0.29/0.46 = C2 )
% 0.29/0.46 => ( ! [X3: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_subst
% 0.29/0.46 thf(fact_215_ord__eq__le__subst,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( ord @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A4: A,F: B > A,B2: B,C2: B] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( F @ B2 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.29/0.46 => ( ! [X3: B,Y2: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_subst
% 0.29/0.46 thf(fact_216_order__subst2,axiom,
% 0.29/0.46 ! [A: $tType,C: $tType] :
% 0.29/0.46 ( ( ( order @ C @ ( type2 @ C ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A4: A,B2: A,F: A > C,C2: C] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 0.29/0.46 => ( ! [X3: A,Y2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst2
% 0.29/0.46 thf(fact_217_order__subst1,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType] :
% 0.29/0.46 ( ( ( order @ B @ ( type2 @ B ) )
% 0.29/0.46 & ( order @ A @ ( type2 @ A ) ) )
% 0.29/0.46 => ! [A4: A,F: B > A,B2: B,C2: B] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A4 @ ( F @ B2 ) )
% 0.29/0.46 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.29/0.46 => ( ! [X3: B,Y2: B] :
% 0.29/0.46 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_subst1
% 0.29/0.46 thf(fact_218_le__fun__def,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.46 = ( ^ [F2: A > B,G2: A > B] :
% 0.29/0.46 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_fun_def
% 0.29/0.46 thf(fact_219_le__funI,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [F: A > B,G: A > B] :
% 0.29/0.46 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.29/0.46 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funI
% 0.29/0.46 thf(fact_220_le__funE,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funE
% 0.29/0.46 thf(fact_221_le__funD,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( ord @ B @ ( type2 @ B ) )
% 0.29/0.46 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.46 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_funD
% 0.29/0.46 thf(fact_222_mk__disjoint__insert,axiom,
% 0.29/0.46 ! [A: $tType,A4: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A4 @ A2 )
% 0.29/0.46 => ? [B6: set @ A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( insert @ A @ A4 @ B6 ) )
% 0.29/0.46 & ~ ( member @ A @ A4 @ B6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mk_disjoint_insert
% 0.29/0.46 thf(fact_223_subset__insertI2,axiom,
% 0.29/0.46 ! [A: $tType,A2: set @ A,B4: set @ A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_insertI2
% 0.29/0.46 thf(fact_224_subset__insertI,axiom,
% 0.29/0.46 ! [A: $tType,B4: set @ A,A4: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_insertI
% 0.29/0.46 thf(fact_225_insert__commute,axiom,
% 0.29/0.46 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 0.29/0.46 ( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
% 0.29/0.46 = ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_commute
% 0.29/0.46 thf(fact_226_subset__insert,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B4 ) )
% 0.29/0.46 = ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % subset_insert
% 0.29/0.46 thf(fact_227_insert__eq__iff,axiom,
% 0.29/0.46 ! [A: $tType,A4: A,A2: set @ A,B2: A,B4: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ A4 @ A2 )
% 0.29/0.46 => ( ~ ( member @ A @ B2 @ B4 )
% 0.29/0.46 => ( ( ( insert @ A @ A4 @ A2 )
% 0.29/0.46 = ( insert @ A @ B2 @ B4 ) )
% 0.29/0.46 = ( ( ( A4 = B2 )
% 0.29/0.46 => ( A2 = B4 ) )
% 0.29/0.46 & ( ( A4 != B2 )
% 0.29/0.46 => ? [C4: set @ A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( insert @ A @ B2 @ C4 ) )
% 0.29/0.46 & ~ ( member @ A @ B2 @ C4 )
% 0.29/0.46 & ( B4
% 0.29/0.46 = ( insert @ A @ A4 @ C4 ) )
% 0.29/0.46 & ~ ( member @ A @ A4 @ C4 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_eq_iff
% 0.29/0.46 thf(fact_228_insert__absorb,axiom,
% 0.29/0.46 ! [A: $tType,A4: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ A4 @ A2 )
% 0.29/0.46 => ( ( insert @ A @ A4 @ A2 )
% 0.29/0.46 = A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_absorb
% 0.29/0.46 thf(fact_229_insert__ident,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 0.29/0.46 ( ~ ( member @ A @ X @ A2 )
% 0.29/0.46 => ( ~ ( member @ A @ X @ B4 )
% 0.29/0.46 => ( ( ( insert @ A @ X @ A2 )
% 0.29/0.46 = ( insert @ A @ X @ B4 ) )
% 0.29/0.46 = ( A2 = B4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % insert_ident
% 0.29/0.46 thf(fact_230_Set_Oinsert__mono,axiom,
% 0.29/0.46 ! [A: $tType,C3: set @ A,D: set @ A,A4: A] :
% 0.29/0.46 ( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
% 0.29/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A4 @ C3 ) @ ( insert @ A @ A4 @ D ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Set.insert_mono
% 0.29/0.46 thf(fact_231_Set_Oset__insert,axiom,
% 0.29/0.46 ! [A: $tType,X: A,A2: set @ A] :
% 0.29/0.46 ( ( member @ A @ X @ A2 )
% 0.29/0.46 => ~ ! [B6: set @ A] :
% 0.29/0.46 ( ( A2
% 0.29/0.46 = ( insert @ A @ X @ B6 ) )
% 0.29/0.47 => ( member @ A @ X @ B6 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Set.set_insert
% 0.29/0.47 thf(fact_232_insertI2,axiom,
% 0.29/0.47 ! [A: $tType,A4: A,B4: set @ A,B2: A] :
% 0.29/0.47 ( ( member @ A @ A4 @ B4 )
% 0.29/0.47 => ( member @ A @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % insertI2
% 0.29/0.47 thf(fact_233_insertI1,axiom,
% 0.29/0.47 ! [A: $tType,A4: A,B4: set @ A] : ( member @ A @ A4 @ ( insert @ A @ A4 @ B4 ) ) ).
% 0.29/0.47
% 0.29/0.47 % insertI1
% 0.29/0.47 thf(fact_234_insertE,axiom,
% 0.29/0.47 ! [A: $tType,A4: A,B2: A,A2: set @ A] :
% 0.29/0.47 ( ( member @ A @ A4 @ ( insert @ A @ B2 @ A2 ) )
% 0.29/0.47 => ( ( A4 != B2 )
% 0.29/0.47 => ( member @ A @ A4 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % insertE
% 0.29/0.47 thf(fact_235_insert__subsetI,axiom,
% 0.29/0.47 ! [A: $tType,X: A,A2: set @ A,X4: set @ A] :
% 0.29/0.47 ( ( member @ A @ X @ A2 )
% 0.29/0.47 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ A2 )
% 0.29/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X4 ) @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % insert_subsetI
% 0.29/0.47 thf(fact_236_lset__lappend__lfinite,axiom,
% 0.29/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.47 ( ( coinductive_lfinite @ A @ Xs )
% 0.29/0.47 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % lset_lappend_lfinite
% 0.29/0.47 thf(fact_237_lsorted__LCons_H,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.29/0.47 => ! [X: A,Xs: coinductive_llist @ A] :
% 0.29/0.47 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 0.29/0.47 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.47 => ( ( ord_less_eq @ A @ X @ ( coinductive_lhd @ A @ Xs ) )
% 0.29/0.47 & ( coindu63249387sorted @ A @ Xs ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % lsorted_LCons'
% 0.29/0.47 thf(fact_238_Un__iff,axiom,
% 0.29/0.47 ! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
% 0.29/0.47 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) )
% 0.29/0.47 = ( ( member @ A @ C2 @ A2 )
% 0.29/0.47 | ( member @ A @ C2 @ B4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_iff
% 0.29/0.47 thf(fact_239_UnCI,axiom,
% 0.29/0.47 ! [A: $tType,C2: A,B4: set @ A,A2: set @ A] :
% 0.29/0.47 ( ( ~ ( member @ A @ C2 @ B4 )
% 0.29/0.47 => ( member @ A @ C2 @ A2 ) )
% 0.29/0.47 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % UnCI
% 0.29/0.47 thf(fact_240_Un__subset__iff,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
% 0.29/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) @ C3 )
% 0.29/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.29/0.47 & ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_subset_iff
% 0.29/0.47 thf(fact_241_Un__insert__left,axiom,
% 0.29/0.47 ! [A: $tType,A4: A,B4: set @ A,C3: set @ A] :
% 0.29/0.47 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A4 @ B4 ) @ C3 )
% 0.29/0.47 = ( insert @ A @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_insert_left
% 0.29/0.47 thf(fact_242_Un__insert__right,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,A4: A,B4: set @ A] :
% 0.29/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A4 @ B4 ) )
% 0.29/0.47 = ( insert @ A @ A4 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_insert_right
% 0.29/0.47 thf(fact_243_lhd__lappend,axiom,
% 0.29/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.29/0.47 ( ( ( coinductive_lnull @ A @ Xs )
% 0.29/0.47 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( coinductive_lhd @ A @ Ys ) ) )
% 0.29/0.47 & ( ~ ( coinductive_lnull @ A @ Xs )
% 0.29/0.47 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 0.29/0.47 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % lhd_lappend
% 0.29/0.47 thf(fact_244_Un__mono,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,C3: set @ A,B4: set @ A,D: set @ A] :
% 0.29/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.29/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ D )
% 0.29/0.47 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_mono
% 0.29/0.47 thf(fact_245_Un__least,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,C3: set @ A,B4: set @ A] :
% 0.29/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.29/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 0.29/0.47 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) @ C3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_least
% 0.29/0.47 thf(fact_246_Un__upper1,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_upper1
% 0.29/0.47 thf(fact_247_Un__upper2,axiom,
% 0.29/0.47 ! [A: $tType,B4: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_upper2
% 0.29/0.47 thf(fact_248_Un__absorb1,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 0.29/0.47 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.47 => ( ( sup_sup @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.47 = B4 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_absorb1
% 0.29/0.47 thf(fact_249_Un__absorb2,axiom,
% 0.29/0.47 ! [A: $tType,B4: set @ A,A2: set @ A] :
% 0.29/0.47 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
% 0.29/0.47 => ( ( sup_sup @ ( set @ A ) @ A2 @ B4 )
% 0.29/0.47 = A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_absorb2
% 0.29/0.47 thf(fact_250_subset__Un__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.29/0.47 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.29/0.47 ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
% 0.29/0.47 = B5 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_Un_eq
% 0.29/0.47 thf(fact_251_Un__left__commute,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
% 0.29/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
% 0.29/0.47 = ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_left_commute
% 0.29/0.47 thf(fact_252_Un__left__absorb,axiom,
% 0.29/0.47 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 0.29/0.47 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) )
% 0.29/0.47 = ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_left_absorb
% 0.29/0.47 thf(fact_253_Un__commute,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( sup_sup @ ( set @ A ) )
% 0.29/0.47 = ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Un_commute
% 0.29/0.47
% 0.29/0.47 %----Type constructors (10)
% 0.29/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.47 ! [A7: $tType,A8: $tType] :
% 0.29/0.47 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 0.29/0.47 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.47 ! [A7: $tType,A8: $tType] :
% 0.29/0.47 ( ( order @ A8 @ ( type2 @ A8 ) )
% 0.29/0.47 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.47 ! [A7: $tType,A8: $tType] :
% 0.29/0.47 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 0.29/0.47 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 0.29/0.47 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 0.29/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 0.29/0.47 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 0.29/0.47 preorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.29/0.47 linorder @ $o @ ( type2 @ $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 0.29/0.47 order @ $o @ ( type2 @ $o ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 0.29/0.47 ord @ $o @ ( type2 @ $o ) ).
% 0.29/0.47
% 0.29/0.47 %----Helper facts (3)
% 0.29/0.47 thf(help_If_3_1_T,axiom,
% 0.29/0.47 ! [P: $o] :
% 0.29/0.47 ( ( P = $true )
% 0.29/0.47 | ( P = $false ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_2_1_T,axiom,
% 0.29/0.47 ! [A: $tType,X: A,Y: A] :
% 0.29/0.47 ( ( if @ A @ $false @ X @ Y )
% 0.29/0.47 = Y ) ).
% 0.29/0.47
% 0.29/0.47 thf(help_If_1_1_T,axiom,
% 0.29/0.47 ! [A: $tType,X: A,Y: A] :
% 0.29/0.47 ( ( if @ A @ $true @ X @ Y )
% 0.29/0.47 = X ) ).
% 0.29/0.47
% 0.29/0.47 %----Conjectures (1)
% 0.29/0.47 thf(conj_0,conjecture,
% 0.29/0.47 ( ( lMirro427583474mirror @ a @ ( coinductive_LCons @ a @ x @ xs ) )
% 0.29/0.47 = ( coinductive_LCons @ a @ x @ ( coinductive_lappend @ a @ ( lMirro427583474mirror @ a @ xs ) @ ( coinductive_LCons @ a @ x @ ( coinductive_LNil @ a ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.70/1.45 ---- Embedded in THF ---
% 2.70/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.70/1.45 %%% Generated on Tue Feb 24 06:11:07 EST 2026
% 2.70/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.70/1.45 %%% Logic specification used:
% 2.70/1.45 %%% thf(spec, logic, $$dhol).
% 2.70/1.45
% 2.70/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Mre2OmccfH/DTF2DTF7148.p
% 2.70/1.45 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.70/1.45 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.70/1.45 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.70/1.45 thf(ty_tf_a, type, a: $tType).
% 2.70/1.45 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.70/1.45 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.70/1.45 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.70/1.45 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.70/1.45 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ofinite__lprefix, type, coindu328551480prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ogen__lset, type, coinductive_gen_lset: (!> [A:$tType]: (((set @ A) > ((coinductive_llist @ A) > (set @ A)))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Olappend, type, coinductive_lappend: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Olfilter, type, coinductive_lfilter: (!> [A:$tType]: (((A > $o) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ollast, type, coinductive_llast: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ollist_OLCons, type, coinductive_LCons: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist, type, coindu1259883913_llist: (!> [C:$tType,A:$tType]: (((C > $o) > ((C > A) > ((C > $o) > ((C > (coinductive_llist @ A)) > ((C > C) > (C > (coinductive_llist @ A)))))))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ollist_Olhd, type, coinductive_lhd: (!> [A:$tType]: (((coinductive_llist @ A) > A)))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ollist_Olnull, type, coinductive_lnull: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Olprefix, type, coinductive_lprefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Olstrict__prefix, type, coindu1478340336prefix: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > $o))))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Oord__class_Olsorted, type, coindu63249387sorted: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.70/1.45 thf(sy_c_Coinductive__List_Ounfold__llist, type, coindu1441602521_llist: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > B) > ((A > A) > (A > (coinductive_llist @ B)))))))).
% 2.70/1.45 thf(sy_c_HOL_Oundefined, type, undefined: (!> [A:$tType]: (A))).
% 2.70/1.45 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.70/1.45 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror, type, lMirro427583474mirror: (!> [A:$tType]: (((coinductive_llist @ A) > (coinductive_llist @ A))))).
% 2.70/1.45 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux, type, lMirro999291890or_aux: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 2.70/1.45 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.70/1.45 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.70/1.45 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.70/1.45 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.70/1.45 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.70/1.45 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.70/1.45 thf(sy_v_x, type, x: a).
% 2.70/1.45 thf(sy_v_xs, type, xs: (coinductive_llist @ a)).
% 2.70/1.45 thf(fact_0_lmirror__LNil, axiom, (! [A:$tType]: ((((lMirro427583474mirror @ A) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.70/1.45 thf(fact_1_lappend__code_I1_J, axiom, (! [A:$tType,Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (coinductive_LNil @ A)) @ Ys) = Ys)))).
% 2.70/1.45 thf(fact_2_lappend__LNil2, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ Xs) @ (coinductive_LNil @ A)) = Xs)))).
% 2.70/1.45 thf(fact_3_lappend__code_I2_J, axiom, (! [A:$tType,Xa:A,X:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lappend @ A) @ (((coinductive_LCons @ A) @ Xa) @ X)) @ Ys) = (((coinductive_LCons @ A) @ Xa) @ (((coinductive_lappend @ A) @ X) @ Ys)))))).
% 2.70/1.45 thf(fact_4_lappend__snocL1__conv__LCons2, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_5_llist_Oinject, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A),Y21:A,Y22:(coinductive_llist @ A)]: ((((((coinductive_LCons @ A) @ X21) @ X22) = (((coinductive_LCons @ A) @ Y21) @ Y22)) = ((X21 = Y21) & (X22 = Y22)))))).
% 2.70/1.45 thf(fact_6_lappend__LNil__LNil, axiom, (! [A:$tType]: (((((coinductive_lappend @ A) @ (coinductive_LNil @ A)) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.70/1.45 thf(fact_7_LNil__eq__lappend__iff, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_LNil @ A) = (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((Xs = (coinductive_LNil @ A)) & (Ys = (coinductive_LNil @ A))))))).
% 2.70/1.45 thf(fact_8_lappend__eq__LNil__iff, axiom, (! [A:$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))))))).
% 2.70/1.45 thf(fact_9_llist_Odistinct_I1_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: (((coinductive_LNil @ A) != (((coinductive_LCons @ A) @ X21) @ X22))))).
% 2.70/1.45 thf(fact_10_llist_Oexhaust, axiom, (! [A:$tType,Y:(coinductive_llist @ A)]: (((Y != (coinductive_LNil @ A)) => (~ ((! [X212:A,X222:(coinductive_llist @ A)]: ((Y != (((coinductive_LCons @ A) @ X212) @ X222)))))))))).
% 2.70/1.45 thf(fact_11_neq__LNil__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((Xs != (coinductive_LNil @ A)) = (? [X2:A,Xs2:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X2) @ Xs2)))))))).
% 2.70/1.45 thf(fact_12_lmirror__aux__LCons, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),X:A,Xs:(coinductive_llist @ A)]: (((((lMirro999291890or_aux @ A) @ Acc) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coinductive_lappend @ A) @ (((lMirro999291890or_aux @ A) @ (coinductive_LNil @ A)) @ Xs)) @ (((coinductive_LCons @ A) @ X) @ Acc))))))).
% 2.70/1.45 thf(fact_13_lmirror__aux__simps_I2_J, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),Xa:A,X:(coinductive_llist @ A)]: (((((lMirro999291890or_aux @ A) @ Acc) @ (((coinductive_LCons @ A) @ Xa) @ X)) = (((coinductive_LCons @ A) @ Xa) @ (((lMirro999291890or_aux @ A) @ (((coinductive_LCons @ A) @ Xa) @ Acc)) @ X)))))).
% 2.70/1.45 thf(fact_14_lmirror__aux__simps_I1_J, axiom, (! [A:$tType,Acc:(coinductive_llist @ A)]: (((((lMirro999291890or_aux @ A) @ Acc) @ (coinductive_LNil @ A)) = Acc)))).
% 2.70/1.45 thf(fact_15_lmirror__aux__acc, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: (((((lMirro999291890or_aux @ A) @ (((coinductive_lappend @ A) @ Ys) @ Zs)) @ Xs) = (((coinductive_lappend @ A) @ (((lMirro999291890or_aux @ A) @ Ys) @ Xs)) @ Zs))))).
% 2.70/1.45 thf(fact_16_lappend__assoc, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_17_lmirror__def, axiom, (! [A:$tType]: (((lMirro427583474mirror @ A) = ((lMirro999291890or_aux @ A) @ (coinductive_LNil @ A)))))).
% 2.70/1.45 thf(fact_18_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((Xs = (coinductive_LNil @ A)) | (? [Xs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y) @ Xs2)) & (((coindu328551480prefix @ A) @ Xs2) @ Ys))))))))).
% 2.70/1.45 thf(fact_19_lstrict__prefix__code_I2_J, axiom, (! [B:$tType,Y:B,Ys:(coinductive_llist @ B)]: ((((coindu1478340336prefix @ B) @ (coinductive_LNil @ B)) @ (((coinductive_LCons @ B) @ Y) @ Ys))))).
% 2.70/1.45 thf(fact_20_lstrict__prefix__code_I3_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((~ ((((coindu1478340336prefix @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) @ (coinductive_LNil @ B))))))).
% 2.70/1.45 thf(fact_21_llast__singleton, axiom, (! [A:$tType,X:A]: ((((coinductive_llast @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))) = X)))).
% 2.70/1.45 thf(fact_22_lmember__code_I1_J, axiom, (! [A:$tType,X:A]: ((~ ((((coinductive_lmember @ A) @ X) @ (coinductive_LNil @ A))))))).
% 2.70/1.45 thf(fact_23_lmember__code_I2_J, axiom, (! [A:$tType,X:A,Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lmember @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((X = Y) | (((coinductive_lmember @ A) @ X) @ Ys)))))).
% 2.70/1.45 thf(fact_24_gen__lset__code_I1_J, axiom, (! [A:$tType,A2:(set @ A)]: (((((coinductive_gen_lset @ A) @ A2) @ (coinductive_LNil @ A)) = A2)))).
% 2.70/1.45 thf(fact_25_lfinite__rev__induct, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((P @ (coinductive_LNil @ A)) => ((! [X3:A,Xs3:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs3) => ((P @ Xs3) => (P @ (((coinductive_lappend @ A) @ Xs3) @ (((coinductive_LCons @ A) @ X3) @ (coinductive_LNil @ A)))))))) => (P @ Xs))))))).
% 2.70/1.45 thf(fact_26_lsorted__code_I2_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A]: (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))))))))).
% 2.70/1.45 thf(fact_27_llist_Ocorec__code, axiom, (! [A:$tType,C:$tType]: ((((coindu1259883913_llist @ C) @ A) = (^ [P2:(C > $o),G21:(C > A),Q22:(C > $o),G221:(C > (coinductive_llist @ A)),G222:(C > C),A3:C]: (((((if @ (coinductive_llist @ A)) @ (P2 @ A3)) @ (coinductive_LNil @ A)) @ (((coinductive_LCons @ A) @ (G21 @ A3)) @ ((((if @ (coinductive_llist @ A)) @ (Q22 @ A3)) @ (G221 @ A3)) @ ((((((((coindu1259883913_llist @ C) @ A) @ P2) @ G21) @ Q22) @ G221) @ G222) @ (G222 @ A3))))))))))).
% 2.70/1.45 thf(fact_28_unfold__llist_Ocode, axiom, (! [B:$tType,A:$tType]: ((((coindu1441602521_llist @ A) @ B) = (^ [P2:(A > $o),G21:(A > B),G22:(A > A),A3:A]: (((((if @ (coinductive_llist @ B)) @ (P2 @ A3)) @ (coinductive_LNil @ B)) @ (((coinductive_LCons @ B) @ (G21 @ A3)) @ ((((((coindu1441602521_llist @ A) @ B) @ P2) @ G21) @ G22) @ (G22 @ A3)))))))))).
% 2.70/1.45 thf(fact_29_lfinite__code_I2_J, axiom, (! [B:$tType,X:B,Xs:(coinductive_llist @ B)]: ((((coinductive_lfinite @ B) @ (((coinductive_LCons @ B) @ X) @ Xs)) = ((coinductive_lfinite @ B) @ Xs))))).
% 2.70/1.45 thf(fact_30_lfinite__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((coinductive_lfinite @ A) @ Xs))))).
% 2.70/1.45 thf(fact_31_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.70/1.45 thf(fact_32_lfinite__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lfinite @ A) @ Xs) & ((coinductive_lfinite @ A) @ Ys)))))).
% 2.70/1.45 thf(fact_33_lsorted__code_I1_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 2.70/1.45 thf(fact_34_unfold__llist__eq__LCons, axiom, (! [A:$tType,B:$tType,IS_LNIL:(B > $o),LHD:(B > A),LTL:(B > B),B2:B,X:A,Xs:(coinductive_llist @ A)]: (((((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ B2) = (((coinductive_LCons @ A) @ X) @ Xs)) = ((~ ((IS_LNIL @ B2))) & ((X = (LHD @ B2)) & (Xs = ((((((coindu1441602521_llist @ B) @ A) @ IS_LNIL) @ LHD) @ LTL) @ (LTL @ B2))))))))).
% 2.70/1.45 thf(fact_35_llast__LCons2, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_36_lfinite__lmirror__aux, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs)) = (((coinductive_lfinite @ A) @ Xs) & ((coinductive_lfinite @ A) @ Acc)))))).
% 2.70/1.45 thf(fact_37_lstrict__prefix__code_I4_J, axiom, (! [B:$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)))))).
% 2.70/1.45 thf(fact_38_lstrict__prefix__code_I1_J, axiom, (! [A:$tType]: ((~ ((((coindu1478340336prefix @ A) @ (coinductive_LNil @ A)) @ (coinductive_LNil @ A))))))).
% 2.70/1.45 thf(fact_39_llast__lappend__LCons, axiom, (! [A:$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))))))).
% 2.70/1.45 thf(fact_40_llist__less__induct, axiom, (! [A:$tType,P:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((! [Xs3:(coinductive_llist @ A)]: (((! [Ys2:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys2) @ Xs3) => (P @ Ys2)))) => (P @ Xs3)))) => (P @ Xs))))).
% 2.70/1.45 thf(fact_41_lstrict__prefix__lfinite1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Xs) @ Ys) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.70/1.45 thf(fact_42_lfinite__LConsI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.70/1.45 thf(fact_43_lfinite__LNil, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 2.70/1.45 thf(fact_44_lappend__inf, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.70/1.45 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A4:A,P:(A > $o)]: (((((member @ A) @ A4) @ ((collect @ A) @ P)) = (P @ A4))))).
% 2.70/1.45 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 2.70/1.45 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.70/1.45 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.70/1.45 thf(fact_49_llimit__induct, axiom, (! [A:$tType,P:((coinductive_llist @ A) > $o),Xs:(coinductive_llist @ A)]: (((P @ (coinductive_LNil @ A)) => ((! [X3:A,Xs3:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs3) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X3) @ Xs3)))))) => (((! [Ys2:(coinductive_llist @ A)]: (((((coindu1478340336prefix @ A) @ Ys2) @ Xs) => (P @ Ys2)))) => (P @ Xs)) => (P @ Xs))))))).
% 2.70/1.45 thf(fact_50_lmirror__aux__inf, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Acc:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((lMirro999291890or_aux @ A) @ Acc) @ Xs) = Xs))))).
% 2.70/1.45 thf(fact_51_LNil, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 2.70/1.45 thf(fact_52_unfold__llist_Octr_I2_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),A4:A,G212:(A > B),G223:(A > A)]: (((~ ((P3 @ A4))) => (((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A4) = (((coinductive_LCons @ B) @ (G212 @ A4)) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ (G223 @ A4)))))))).
% 2.70/1.45 thf(fact_53_unfold__llist_Octr_I1_J, axiom, (! [A:$tType,B:$tType,P3:(A > $o),A4:A,G212:(A > B),G223:(A > A)]: (((P3 @ A4) => (((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A4) = (coinductive_LNil @ B)))))).
% 2.70/1.45 thf(fact_54_llist_Ocorec_I2_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),A4:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((~ ((P3 @ A4))) => (((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A4) = (((coinductive_LCons @ A) @ (G212 @ A4)) @ ((((if @ (coinductive_llist @ A)) @ (Q222 @ A4)) @ (G2212 @ A4)) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ (G2222 @ A4))))))))).
% 2.70/1.45 thf(fact_55_llist_Ocorec_I1_J, axiom, (! [C:$tType,A:$tType,P3:(C > $o),A4:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((P3 @ A4) => (((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A4) = (coinductive_LNil @ A)))))).
% 2.70/1.45 thf(fact_56_lfinite_Ocases, axiom, (! [A:$tType,A4:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ A4) => ((A4 != (coinductive_LNil @ A)) => (~ ((! [Xs3:(coinductive_llist @ A)]: (((? [X3:A]: ((A4 = (((coinductive_LCons @ A) @ X3) @ Xs3)))) => (~ (((coinductive_lfinite @ A) @ Xs3))))))))))))).
% 2.70/1.45 thf(fact_57_lfinite_Osimps, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) = (^ [A3:(coinductive_llist @ A)]: (((A3 = (coinductive_LNil @ A)) | (? [Xs4:(coinductive_llist @ A),X2:A]: (((A3 = (((coinductive_LCons @ A) @ X2) @ Xs4)) & ((coinductive_lfinite @ A) @ Xs4))))))))))).
% 2.70/1.45 thf(fact_58_lfinite_Oinducts, axiom, (! [A:$tType,X:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: ((((coinductive_lfinite @ A) @ X) => ((P @ (coinductive_LNil @ A)) => ((! [Xs3:(coinductive_llist @ A),X3:A]: ((((coinductive_lfinite @ A) @ Xs3) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X3) @ Xs3)))))) => (P @ X))))))).
% 2.70/1.45 thf(fact_59_Singleton, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A]: (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (coinductive_LNil @ A))))))))).
% 2.70/1.45 thf(fact_60_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coindu328551480prefix @ A) @ (coinductive_LNil @ A)) @ Xs)))).
% 2.70/1.45 thf(fact_61_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coindu328551480prefix @ A) @ Xs) @ (coinductive_LNil @ A)) = (Xs = (coinductive_LNil @ A)))))).
% 2.70/1.45 thf(fact_62_lsorted_Ocases, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ A4) => ((A4 != (coinductive_LNil @ A)) => ((! [X3:A]: ((A4 != (((coinductive_LCons @ A) @ X3) @ (coinductive_LNil @ A))))) => (~ ((! [X3:A,Y2:A,Xs3:(coinductive_llist @ A)]: (((A4 = (((coinductive_LCons @ A) @ X3) @ (((coinductive_LCons @ A) @ Y2) @ Xs3))) => ((((ord_less_eq @ A) @ X3) @ Y2) => (~ (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y2) @ Xs3))))))))))))))))))).
% 2.70/1.45 thf(fact_63_lsorted_Osimps, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) = (^ [A3:(coinductive_llist @ A)]: (((A3 = (coinductive_LNil @ A)) | ((? [X2:A]: ((A3 = (((coinductive_LCons @ A) @ X2) @ (coinductive_LNil @ A))))) | (? [X2:A,Y3:A,Xs4:(coinductive_llist @ A)]: (((A3 = (((coinductive_LCons @ A) @ X2) @ (((coinductive_LCons @ A) @ Y3) @ Xs4))) & ((((ord_less_eq @ A) @ X2) @ Y3) & ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y3) @ Xs4))))))))))))))).
% 2.70/1.45 thf(fact_64_lsorted_Ocoinduct, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X4:((coinductive_llist @ A) > $o),X:(coinductive_llist @ A)]: (((X4 @ X) => ((! [X3:(coinductive_llist @ A)]: (((X4 @ X3) => ((X3 = (coinductive_LNil @ A)) | ((? [Xa2:A]: ((X3 = (((coinductive_LCons @ A) @ Xa2) @ (coinductive_LNil @ A))))) | (? [Xa2:A,Y4:A,Xs5:(coinductive_llist @ A)]: (((X3 = (((coinductive_LCons @ A) @ Xa2) @ (((coinductive_LCons @ A) @ Y4) @ Xs5))) & ((((ord_less_eq @ A) @ Xa2) @ Y4) & ((X4 @ (((coinductive_LCons @ A) @ Y4) @ Xs5)) | ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y4) @ Xs5)))))))))))) => ((coindu63249387sorted @ A) @ X))))))))).
% 2.70/1.45 thf(fact_65_lstrict__prefix__lappend__conv, axiom, (! [A:$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)))))))).
% 2.70/1.45 thf(fact_66_split__llist, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys3:(coinductive_llist @ A),Zs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys3) @ (((coinductive_LCons @ A) @ X) @ Zs2))) & ((coinductive_lfinite @ A) @ Ys3)))))))).
% 2.70/1.45 thf(fact_67_split__llist__first, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (? [Ys3:(coinductive_llist @ A),Zs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Ys3) @ (((coinductive_LCons @ A) @ X) @ Zs2))) & (((coinductive_lfinite @ A) @ Ys3) & (~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys3)))))))))))).
% 2.70/1.45 thf(fact_68_lsorted__LCons__LCons, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Y:A,Xs:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))) = ((((ord_less_eq @ A) @ X) @ Y) & ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)))))))))).
% 2.70/1.45 thf(fact_69_gen__lset__code_I2_J, axiom, (! [A:$tType,A2:(set @ A),X:A,Xs:(coinductive_llist @ A)]: (((((coinductive_gen_lset @ A) @ A2) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_gen_lset @ A) @ (((insert @ A) @ X) @ A2)) @ Xs))))).
% 2.70/1.45 thf(fact_70_lprefix__lappend__same, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ (((coinductive_lappend @ A) @ Xs) @ Zs)) = (((coinductive_lfinite @ A) @ Xs) => (((coinductive_lprefix @ A) @ Ys) @ Zs)))))).
% 2.70/1.45 thf(fact_71_llist_Oleq__refl, axiom, (! [A:$tType,X:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ X) @ X)))).
% 2.70/1.45 thf(fact_72_lprefix__refl, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ Xs) @ Xs)))).
% 2.70/1.45 thf(fact_73_LCons__lprefix__LCons, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_74_lprefix__code_I1_J, axiom, (! [A:$tType,Ys:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (coinductive_LNil @ A)) @ Ys)))).
% 2.70/1.45 thf(fact_75_lnull__lappend, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_76_lappend_Odisc__iff_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))) = ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.70/1.45 thf(fact_77_lmirror__aux_Odisc__iff_I2_J, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs)))) = ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Acc)))))))).
% 2.70/1.45 thf(fact_78_lnull__lmirror__aux, axiom, (! [A:$tType,Acc:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs)) = (((coinductive_lnull @ A) @ Xs) & ((coinductive_lnull @ A) @ Acc)))))).
% 2.70/1.45 thf(fact_79_unfold__llist_Odisc__iff_I1_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),G212:(A > B),G223:(A > A),A4:A]: ((((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A4)) = (P3 @ A4))))).
% 2.70/1.45 thf(fact_80_unfold__llist_Odisc__iff_I2_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),G212:(A > B),G223:(A > A),A4:A]: (((~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A4)))) = (~ ((P3 @ A4))))))).
% 2.70/1.45 thf(fact_81_llist_Ocorec__disc__iff_I1_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C),A4:C]: ((((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A4)) = (P3 @ A4))))).
% 2.70/1.45 thf(fact_82_llist_Ocorec__disc__iff_I2_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C),A4:C]: (((~ (((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A4)))) = (~ ((P3 @ A4))))))).
% 2.70/1.45 thf(fact_83_lnull__lmirror, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ ((lMirro427583474mirror @ A) @ Xs)) = ((coinductive_lnull @ A) @ Xs))))).
% 2.70/1.45 thf(fact_84_lprefix__LNil, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (coinductive_LNil @ A)) = ((coinductive_lnull @ A) @ Xs))))).
% 2.70/1.45 thf(fact_85_lset__LCons, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((insert @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.70/1.45 thf(fact_86_llist_Osimps_I19_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X21) @ X22)) = (((insert @ A) @ X21) @ ((coinductive_lset @ A) @ X22)))))).
% 2.70/1.45 thf(fact_87_lset__lappend1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((ord_less_eq @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))))).
% 2.70/1.45 thf(fact_88_lmirror__aux_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Acc:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (~ (((coinductive_lnull @ A) @ Acc)))) => ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Acc)))))))).
% 2.70/1.45 thf(fact_89_lprefix__down__linear, axiom, (! [A:$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))))))).
% 2.70/1.45 thf(fact_90_lprefix__not__lnullD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((~ (((coinductive_lnull @ A) @ Xs))) => (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.70/1.45 thf(fact_91_llist_Oleq__antisym, axiom, (! [A:$tType,X:(coinductive_llist @ A),Y:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X) @ Y) => ((((coinductive_lprefix @ A) @ Y) @ X) => (X = Y)))))).
% 2.70/1.45 thf(fact_92_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A4:A]: (((! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((P @ A5) @ B3)))) => ((((P @ B2) @ A4) => ((P @ A4) @ B2)) => ((P @ A4) @ B2))))))))).
% 2.70/1.45 thf(fact_93_lprefix__antisym, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => ((((coinductive_lprefix @ A) @ Ys) @ Xs) => (Xs = Ys)))))).
% 2.70/1.45 thf(fact_94_llist_Oleq__trans, axiom, (! [A:$tType,X:(coinductive_llist @ A),Y:(coinductive_llist @ A),Z:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ X) @ Y) => ((((coinductive_lprefix @ A) @ Y) @ Z) => (((coinductive_lprefix @ A) @ X) @ Z)))))).
% 2.70/1.45 thf(fact_95_lappend_Oexhaust, axiom, (! [A:$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)))))))).
% 2.70/1.45 thf(fact_96_lprefix__lnullD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lnull @ A) @ Ys) => ((coinductive_lnull @ A) @ Xs)))))).
% 2.70/1.45 thf(fact_97_lprefix__trans, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_98_lprefix__lsetD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((ord_less_eq @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ Ys)))))).
% 2.70/1.45 thf(fact_99_lprefix__lnull, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) = ((coinductive_lnull @ A) @ Xs)))))).
% 2.70/1.45 thf(fact_100_lnull__lprefix, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lprefix @ A) @ Xs) @ Ys))))).
% 2.70/1.45 thf(fact_101_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ B) @ Ys)))))))).
% 2.70/1.45 thf(fact_102_lsorted__LCons, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Xs:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coindu63249387sorted @ A) @ Xs) & (! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (((ord_less_eq @ A) @ X) @ X2)))))))))))).
% 2.70/1.45 thf(fact_103_llist_Oset__induct, axiom, (! [A:$tType,X:A,A4:(coinductive_llist @ A),P:(A > ((coinductive_llist @ A) > $o))]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ A4)) => ((! [Z1:A,Z2:(coinductive_llist @ A)]: (((P @ Z1) @ (((coinductive_LCons @ A) @ Z1) @ Z2)))) => ((! [Z1:A,Z2:(coinductive_llist @ A),Xa3:A]: (((((member @ A) @ Xa3) @ ((coinductive_lset @ A) @ Z2)) => (((P @ Xa3) @ Z2) => ((P @ Xa3) @ (((coinductive_LCons @ A) @ Z1) @ Z2)))))) => ((P @ X) @ A4))))))).
% 2.70/1.45 thf(fact_104_llist_Oset__cases, axiom, (! [A:$tType,E:A,A4:(coinductive_llist @ A)]: (((((member @ A) @ E) @ ((coinductive_lset @ A) @ A4)) => ((! [Z2:(coinductive_llist @ A)]: ((A4 != (((coinductive_LCons @ A) @ E) @ Z2)))) => (~ ((! [Z1:A,Z2:(coinductive_llist @ A)]: (((A4 = (((coinductive_LCons @ A) @ Z1) @ Z2)) => (~ ((((member @ A) @ E) @ ((coinductive_lset @ A) @ Z2)))))))))))))).
% 2.70/1.45 thf(fact_105_lset__induct_H, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs3:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs3)))) => ((! [X5:A,Xs3:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs3)) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X5) @ Xs3)))))) => (P @ Xs))))))).
% 2.70/1.45 thf(fact_106_lset__induct, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),P:((coinductive_llist @ A) > $o)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs3:(coinductive_llist @ A)]: ((P @ (((coinductive_LCons @ A) @ X) @ Xs3)))) => ((! [X5:A,Xs3:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs3)) => ((X != X5) => ((P @ Xs3) => (P @ (((coinductive_LCons @ A) @ X5) @ Xs3))))))) => (P @ Xs))))))).
% 2.70/1.45 thf(fact_107_lset__cases, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => ((! [Xs6:(coinductive_llist @ A)]: ((Xs != (((coinductive_LCons @ A) @ X) @ Xs6)))) => (~ ((! [X5:A,Xs6:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ X5) @ Xs6)) => (~ ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs6)))))))))))))).
% 2.70/1.45 thf(fact_108_llist_Oset__intros_I1_J, axiom, (! [A:$tType,A1:A,A22:(coinductive_llist @ A)]: ((((member @ A) @ A1) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22)))))).
% 2.70/1.45 thf(fact_109_llist_Oset__intros_I2_J, axiom, (! [A:$tType,X:A,A22:(coinductive_llist @ A),A1:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ A22)) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ A1) @ A22))))))).
% 2.70/1.45 thf(fact_110_lset__intros_I1_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)))))).
% 2.70/1.45 thf(fact_111_lset__intros_I2_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),X6:A]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_LCons @ A) @ X6) @ Xs))))))).
% 2.70/1.45 thf(fact_112_not__lnull__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs))) = (? [X2:A,Xs2:(coinductive_llist @ A)]: ((Xs = (((coinductive_LCons @ A) @ X2) @ Xs2)))))))).
% 2.70/1.45 thf(fact_113_llist_OdiscI_I2_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A),X21:A,X22:(coinductive_llist @ A)]: (((Llist = (((coinductive_LCons @ A) @ X21) @ X22)) => (~ (((coinductive_lnull @ A) @ Llist))))))).
% 2.70/1.45 thf(fact_114_llist_Odisc_I2_J, axiom, (! [A:$tType,X21:A,X22:(coinductive_llist @ A)]: ((~ (((coinductive_lnull @ A) @ (((coinductive_LCons @ A) @ X21) @ X22))))))).
% 2.70/1.45 thf(fact_115_lnull__def, axiom, (! [A:$tType]: (((coinductive_lnull @ A) = (^ [Llist2:(coinductive_llist @ A)]: ((Llist2 = (coinductive_LNil @ A)))))))).
% 2.70/1.45 thf(fact_116_llist_Ocollapse_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Llist) => (Llist = (coinductive_LNil @ A)))))).
% 2.70/1.45 thf(fact_117_llist_OdiscI_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((Llist = (coinductive_LNil @ A)) => ((coinductive_lnull @ A) @ Llist))))).
% 2.70/1.45 thf(fact_118_llist_Odisc_I1_J, axiom, (! [A:$tType]: (((coinductive_lnull @ A) @ (coinductive_LNil @ A))))).
% 2.70/1.45 thf(fact_119_LCons__lprefix__conv, axiom, (! [A:$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)))))))).
% 2.70/1.45 thf(fact_120_Le__LCons, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),X:A]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (((coinductive_LCons @ A) @ X) @ Ys)))))).
% 2.70/1.45 thf(fact_121_lappend__lnull2, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Ys) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Xs))))).
% 2.70/1.45 thf(fact_122_lappend__lnull1, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((((coinductive_lappend @ A) @ Xs) @ Ys) = Ys))))).
% 2.70/1.45 thf(fact_123_lappend_Odisc_I1_J, axiom, (! [A:$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))))))).
% 2.70/1.45 thf(fact_124_lappend_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))) => (~ (((coinductive_lnull @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)))))))).
% 2.70/1.45 thf(fact_125_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.70/1.45 thf(fact_126_LNil__lprefix, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ (coinductive_LNil @ A)) @ Xs)))).
% 2.70/1.45 thf(fact_127_lprefix__lappend__sameI, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_128_lprefix__conv__lappend, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [Xs4:(coinductive_llist @ A),Ys5:(coinductive_llist @ A)]: ((? [Zs3:(coinductive_llist @ A)]: ((Ys5 = (((coinductive_lappend @ A) @ Xs4) @ Zs3)))))))))).
% 2.70/1.45 thf(fact_129_lprefix__lappendD, axiom, (! [A:$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)))))).
% 2.70/1.45 thf(fact_130_lappend__lprefixE, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) @ Zs) => (~ ((! [Zs4:(coinductive_llist @ A)]: ((Zs != (((coinductive_lappend @ A) @ Xs) @ Zs4)))))))))).
% 2.70/1.45 thf(fact_131_lprefix__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_lappend @ A) @ Xs) @ Ys))))).
% 2.70/1.45 thf(fact_132_not__lfinite__lprefix__conv__eq, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((((coinductive_lprefix @ A) @ Xs) @ Ys) = (Xs = Ys)))))).
% 2.70/1.45 thf(fact_133_lprefix__lfiniteD, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lfinite @ A) @ Ys) => ((coinductive_lfinite @ A) @ Xs)))))).
% 2.70/1.45 thf(fact_134_lsorted__lprefixD, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coindu63249387sorted @ A) @ Ys) => ((coindu63249387sorted @ A) @ Xs))))))))).
% 2.70/1.45 thf(fact_135_lmirror__aux_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Acc:(coinductive_llist @ A)]: ((((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Acc)))) => (~ (((coinductive_lnull @ A) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs)))))))).
% 2.70/1.45 thf(fact_136_lmirror__aux_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Acc:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lnull @ A) @ Acc) => ((coinductive_lnull @ A) @ (((lMirro999291890or_aux @ A) @ Acc) @ Xs))))))).
% 2.70/1.45 thf(fact_137_unfold__llist_Odisc_I1_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),A4:A,G212:(A > B),G223:(A > A)]: (((P3 @ A4) => ((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A4)))))).
% 2.70/1.45 thf(fact_138_unfold__llist_Odisc_I2_J, axiom, (! [B:$tType,A:$tType,P3:(A > $o),A4:A,G212:(A > B),G223:(A > A)]: (((~ ((P3 @ A4))) => (~ (((coinductive_lnull @ B) @ ((((((coindu1441602521_llist @ A) @ B) @ P3) @ G212) @ G223) @ A4)))))))).
% 2.70/1.45 thf(fact_139_llist_Ocorec__disc_I1_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),A4:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((P3 @ A4) => ((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A4)))))).
% 2.70/1.45 thf(fact_140_llist_Ocorec__disc_I2_J, axiom, (! [A:$tType,C:$tType,P3:(C > $o),A4:C,G212:(C > A),Q222:(C > $o),G2212:(C > (coinductive_llist @ A)),G2222:(C > C)]: (((~ ((P3 @ A4))) => (~ (((coinductive_lnull @ A) @ ((((((((coindu1259883913_llist @ C) @ A) @ P3) @ G212) @ Q222) @ G2212) @ G2222) @ A4)))))))).
% 2.70/1.45 thf(fact_141_lstrict__prefix__def, axiom, (! [A:$tType]: (((coindu1478340336prefix @ A) = (^ [Xs4:(coinductive_llist @ A),Ys5:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs4) @ Ys5) & (Xs4 != Ys5)))))))).
% 2.70/1.45 thf(fact_142_Coinductive__List_Ofinite__lprefix__def, axiom, (! [A:$tType]: (((coindu328551480prefix @ A) = (coinductive_lprefix @ A))))).
% 2.70/1.45 thf(fact_143_lset__lmember, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) = (((coinductive_lmember @ A) @ X) @ Xs))))).
% 2.70/1.45 thf(fact_144_in__lset__lappend__iff, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys))) = ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) | (((coinductive_lfinite @ A) @ Xs) & (((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)))))))).
% 2.70/1.45 thf(fact_145_lappend_Octr_I1_J, axiom, (! [A:$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))))))).
% 2.70/1.45 thf(fact_146_lprefix__LCons__conv, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Y:A,Ys:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ Xs) @ (((coinductive_LCons @ A) @ Y) @ Ys)) = ((Xs = (coinductive_LNil @ A)) | (? [Xs2:(coinductive_llist @ A)]: (((Xs = (((coinductive_LCons @ A) @ Y) @ Xs2)) & (((coinductive_lprefix @ A) @ Xs2) @ Ys))))))))).
% 2.70/1.45 thf(fact_147_lprefix_Ocoinduct, axiom, (! [A:$tType,X4:((coinductive_llist @ A) > ((coinductive_llist @ A) > $o)),X:(coinductive_llist @ A),Xa:(coinductive_llist @ A)]: ((((X4 @ X) @ Xa) => ((! [X3:(coinductive_llist @ A),Xa3:(coinductive_llist @ A)]: ((((X4 @ X3) @ Xa3) => ((? [Xs5:(coinductive_llist @ A)]: (((X3 = (coinductive_LNil @ A)) & (Xa3 = Xs5)))) | (? [Xs5:(coinductive_llist @ A),Ys2:(coinductive_llist @ A),Xb:A]: (((X3 = (((coinductive_LCons @ A) @ Xb) @ Xs5)) & ((Xa3 = (((coinductive_LCons @ A) @ Xb) @ Ys2)) & (((X4 @ Xs5) @ Ys2) | (((coinductive_lprefix @ A) @ Xs5) @ Ys2)))))))))) => (((coinductive_lprefix @ A) @ X) @ Xa)))))).
% 2.70/1.45 thf(fact_148_lprefix_Osimps, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [A12:(coinductive_llist @ A),A23:(coinductive_llist @ A)]: (((? [Xs4:(coinductive_llist @ A)]: (((A12 = (coinductive_LNil @ A)) & (A23 = Xs4)))) | (? [Xs4:(coinductive_llist @ A),Ys5:(coinductive_llist @ A),X2:A]: (((A12 = (((coinductive_LCons @ A) @ X2) @ Xs4)) & ((A23 = (((coinductive_LCons @ A) @ X2) @ Ys5)) & (((coinductive_lprefix @ A) @ Xs4) @ Ys5)))))))))))).
% 2.70/1.45 thf(fact_149_lprefix_Ocases, axiom, (! [A:$tType,A1:(coinductive_llist @ A),A22:(coinductive_llist @ A)]: (((((coinductive_lprefix @ A) @ A1) @ A22) => (((A1 = (coinductive_LNil @ A)) => (! [Xs3:(coinductive_llist @ A)]: ((A22 != Xs3)))) => (~ ((! [Xs3:(coinductive_llist @ A),Ys3:(coinductive_llist @ A),X3:A]: (((A1 = (((coinductive_LCons @ A) @ X3) @ Xs3)) => ((A22 = (((coinductive_LCons @ A) @ X3) @ Ys3)) => (~ ((((coinductive_lprefix @ A) @ Xs3) @ Ys3)))))))))))))).
% 2.70/1.45 thf(fact_150_lprefix__code_I2_J, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: ((~ ((((coinductive_lprefix @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) @ (coinductive_LNil @ A))))))).
% 2.70/1.45 thf(fact_151_lmirror__aux_Octr_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Acc:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lnull @ A) @ Acc) => ((((lMirro999291890or_aux @ A) @ Acc) @ Xs) = (coinductive_LNil @ A))))))).
% 2.70/1.45 thf(fact_152_llast__LCons, axiom, (! [A:$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))))))).
% 2.70/1.45 thf(fact_153_LCons__LCons, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Y:A,Xs:(coinductive_llist @ A)]: (((((ord_less_eq @ A) @ X) @ Y) => (((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ Y) @ Xs)) => ((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ (((coinductive_LCons @ A) @ Y) @ Xs))))))))))).
% 2.70/1.45 thf(fact_154_Coinductive__List_Olprefix__nitpick__simps, axiom, (! [A:$tType]: (((coinductive_lprefix @ A) = (^ [Xs4:(coinductive_llist @ A),Ys5:(coinductive_llist @ A)]: (((((coinductive_lfinite @ A) @ Xs4) => (((coindu328551480prefix @ A) @ Xs4) @ Ys5)) & ((~ (((coinductive_lfinite @ A) @ Xs4))) => (Xs4 = Ys5))))))))).
% 2.70/1.45 thf(fact_155_insertCI, axiom, (! [A:$tType,A4:A,B4:(set @ A),B2:A]: ((((~ ((((member @ A) @ A4) @ B4))) => (A4 = B2)) => (((member @ A) @ A4) @ (((insert @ A) @ B2) @ B4)))))).
% 2.70/1.45 thf(fact_156_insert__iff, axiom, (! [A:$tType,A4:A,B2:A,A2:(set @ A)]: (((((member @ A) @ A4) @ (((insert @ A) @ B2) @ A2)) = ((A4 = B2) | (((member @ A) @ A4) @ A2)))))).
% 2.70/1.45 thf(fact_157_insert__subset, axiom, (! [A:$tType,X:A,A2:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A2)) @ B4) = ((((member @ A) @ X) @ B4) & (((ord_less_eq @ (set @ A)) @ A2) @ B4)))))).
% 2.70/1.45 thf(fact_158_insert__absorb2, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A2)) = (((insert @ A) @ X) @ A2))))).
% 2.70/1.45 thf(fact_159_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.70/1.45 thf(fact_160_llast__lappend, axiom, (! [A:$tType,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Ys) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_llast @ A) @ Xs))) & ((~ (((coinductive_lnull @ A) @ Ys))) => ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_llast @ A) @ Ys))) & ((~ (((coinductive_lfinite @ A) @ Xs))) => (((coinductive_llast @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (undefined @ A))))))))).
% 2.70/1.45 thf(fact_161_lfilter__eq__LConsD, axiom, (! [A:$tType,P:(A > $o),Ys:(coinductive_llist @ A),X:A,Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Ys) = (((coinductive_LCons @ A) @ X) @ Xs)) => (? [Us:(coinductive_llist @ A),Vs:(coinductive_llist @ A)]: (((Ys = (((coinductive_lappend @ A) @ Us) @ (((coinductive_LCons @ A) @ X) @ Vs))) & (((coinductive_lfinite @ A) @ Us) & ((! [X7:A]: (((((member @ A) @ X7) @ ((coinductive_lset @ A) @ Us)) => (~ ((P @ X7)))))) & ((P @ X) & (Xs = (((coinductive_lfilter @ A) @ P) @ Vs)))))))))))).
% 2.70/1.45 thf(fact_162_subsetI, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B4)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B4))))).
% 2.70/1.45 thf(fact_163_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ A2) => (A2 = B4)))))).
% 2.70/1.45 thf(fact_164_lfilter__idem, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: (((((coinductive_lfilter @ A) @ P) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))).
% 2.70/1.45 thf(fact_165_lfilter__LCons, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: ((((P @ X) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coinductive_lfilter @ A) @ P) @ Xs)))) & ((~ ((P @ X))) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 2.70/1.45 thf(fact_166_lfilter__LNil, axiom, (! [A:$tType,P:(A > $o)]: (((((coinductive_lfilter @ A) @ P) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 2.70/1.45 thf(fact_167_lnull__lfilter, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) = (! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (~ ((P @ X2)))))))))).
% 2.70/1.45 thf(fact_168_diverge__lfilter__LNil, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Xs)) => (~ ((P @ X3)))))) => ((((coinductive_lfilter @ A) @ P) @ Xs) = (coinductive_LNil @ A)))))).
% 2.70/1.45 thf(fact_169_lfilter__lappend__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((coinductive_lappend @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) @ (((coinductive_lfilter @ A) @ P) @ Ys))))))).
% 2.70/1.45 thf(fact_170_set__mp, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B4)))))).
% 2.70/1.45 thf(fact_171_in__mono, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B4)))))).
% 2.70/1.45 thf(fact_172_subsetD, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B4)))))).
% 2.70/1.45 thf(fact_173_subsetCE, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B4)))))).
% 2.70/1.45 thf(fact_174_equalityE, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: (((A2 = B4) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => (~ ((((ord_less_eq @ (set @ A)) @ B4) @ A2)))))))))).
% 2.70/1.45 thf(fact_175_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A6) => (((member @ A) @ X2) @ B5)))))))))).
% 2.70/1.45 thf(fact_176_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: (((A2 = B4) => (((ord_less_eq @ (set @ A)) @ A2) @ B4))))).
% 2.70/1.45 thf(fact_177_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: (((A2 = B4) => (((ord_less_eq @ (set @ A)) @ B4) @ A2))))).
% 2.70/1.45 thf(fact_178_set__rev__mp, axiom, (! [A:$tType,X:A,A2:(set @ A),B4:(set @ A)]: (((((member @ A) @ X) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B4) => (((member @ A) @ X) @ B4)))))).
% 2.70/1.45 thf(fact_179_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A6) => (((member @ A) @ T) @ B5)))))))))).
% 2.70/1.45 thf(fact_180_rev__subsetD, axiom, (! [A:$tType,C2:A,A2:(set @ A),B4:(set @ A)]: (((((member @ A) @ C2) @ A2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B4) => (((member @ A) @ C2) @ B4)))))).
% 2.70/1.45 thf(fact_181_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.70/1.45 thf(fact_182_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.70/1.45 thf(fact_183_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less_eq @ (set @ A)) @ A2) @ C3)))))).
% 2.70/1.45 thf(fact_184_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y5:(set @ A),Z3:(set @ A)]: ((Y5 = Z3))) = (^ [A6:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A6)))))))).
% 2.70/1.45 thf(fact_185_contra__subsetD, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((~ ((((member @ A) @ C2) @ B4))) => (~ ((((member @ A) @ C2) @ A2)))))))).
% 2.70/1.45 thf(fact_186_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.70/1.45 thf(fact_187_lprefix__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o)]: (((((coinductive_lprefix @ A) @ Xs) @ Ys) => (((coinductive_lprefix @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)) @ (((coinductive_lfilter @ A) @ P) @ Ys)))))).
% 2.70/1.45 thf(fact_188_lfilter__LCons__seek, axiom, (! [A:$tType,P3:(A > $o),X:A,L:(coinductive_llist @ A)]: (((~ ((P3 @ X))) => ((((coinductive_lfilter @ A) @ P3) @ (((coinductive_LCons @ A) @ X) @ L)) = (((coinductive_lfilter @ A) @ P3) @ L)))))).
% 2.70/1.45 thf(fact_189_lfilter__LCons__found, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(coinductive_llist @ A)]: (((P @ X) => ((((coinductive_lfilter @ A) @ P) @ (((coinductive_LCons @ A) @ X) @ Xs)) = (((coinductive_LCons @ A) @ X) @ (((coinductive_lfilter @ A) @ P) @ Xs))))))).
% 2.70/1.45 thf(fact_190_lfilter__cong, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),P:(A > $o),Q:(A > $o)]: (((Xs = Ys) => ((! [X3:A]: (((((member @ A) @ X3) @ ((coinductive_lset @ A) @ Ys)) => ((P @ X3) = (Q @ X3))))) => ((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lfilter @ A) @ Q) @ Ys))))))).
% 2.70/1.45 thf(fact_191_lfilter__id__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = Xs) = (! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (P @ X2)))))))).
% 2.70/1.45 thf(fact_192_lfinite__lfilterI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),P:(A > $o)]: ((((coinductive_lfinite @ A) @ Xs) => ((coinductive_lfinite @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs)))))).
% 2.70/1.45 thf(fact_193_lsorted__lfilterI, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),P:(A > $o)]: ((((coindu63249387sorted @ A) @ Xs) => ((coindu63249387sorted @ A) @ (((coinductive_lfilter @ A) @ P) @ Xs))))))))).
% 2.70/1.45 thf(fact_194_lfilter__empty__conv, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = (coinductive_LNil @ A)) = (! [X2:A]: (((((member @ A) @ X2) @ ((coinductive_lset @ A) @ Xs)) => (~ ((P @ X2)))))))))).
% 2.70/1.45 thf(fact_195_lfilter__eq__lappend__lfiniteD, axiom, (! [A:$tType,P:(A > $o),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Zs:(coinductive_llist @ A)]: ((((((coinductive_lfilter @ A) @ P) @ Xs) = (((coinductive_lappend @ A) @ Ys) @ Zs)) => (((coinductive_lfinite @ A) @ Ys) => (? [Us:(coinductive_llist @ A),Vs:(coinductive_llist @ A)]: (((Xs = (((coinductive_lappend @ A) @ Us) @ Vs)) & (((coinductive_lfinite @ A) @ Us) & ((Ys = (((coinductive_lfilter @ A) @ P) @ Us)) & (Zs = (((coinductive_lfilter @ A) @ P) @ Vs)))))))))))).
% 2.70/1.45 thf(fact_196_llast__LNil, axiom, (! [A:$tType]: ((((coinductive_llast @ A) @ (coinductive_LNil @ A)) = (undefined @ A))))).
% 2.70/1.45 thf(fact_197_llast__linfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => (((coinductive_llast @ A) @ Xs) = (undefined @ A)))))).
% 2.70/1.45 thf(fact_198_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A4:A]: (((((ord_less_eq @ A) @ B2) @ A4) => ((((ord_less_eq @ A) @ A4) @ B2) => (A4 = B2))))))))).
% 2.70/1.45 thf(fact_199_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A4:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A4) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A4))))))))).
% 2.70/1.45 thf(fact_200_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A4:A,B2:A]: (((! [A5:A,B3:A]: (((((ord_less_eq @ A) @ A5) @ B3) => ((P @ A5) @ B3)))) => ((! [A5:A,B3:A]: ((((P @ B3) @ A5) => ((P @ A5) @ B3)))) => ((P @ A4) @ B2))))))))).
% 2.70/1.45 thf(fact_201_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A]: ((((ord_less_eq @ A) @ A4) @ A4))))))).
% 2.70/1.45 thf(fact_202_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X) @ Z))))))))).
% 2.70/1.45 thf(fact_203_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B2:A]: (((((ord_less_eq @ A) @ A4) @ B2) => ((((ord_less_eq @ A) @ B2) @ A4) => (A4 = B2))))))))).
% 2.70/1.45 thf(fact_204_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.70/1.45 thf(fact_205_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A4:A,B2:A,C2:A]: (((A4 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.70/1.45 thf(fact_206_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 2.70/1.45 thf(fact_207_le__cases3, axiom, (! [A:$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))))))))))))))))).
% 2.70/1.45 thf(fact_208_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A4:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A4) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A4) @ C2))))))))).
% 2.70/1.45 thf(fact_209_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.70/1.45 thf(fact_210_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 2.70/1.45 thf(fact_211_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 2.70/1.45 thf(fact_212_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 2.70/1.45 thf(fact_213_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y5:A,Z3:A]: ((Y5 = Z3))) = (^ [X2:A,Y3:A]: (((((ord_less_eq @ A) @ X2) @ Y3) & (((ord_less_eq @ A) @ Y3) @ X2))))))))).
% 2.70/1.45 thf(fact_214_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,B2:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A4) @ B2) => (((F @ B2) = C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ B) @ (F @ A4)) @ C2)))))))))).
% 2.70/1.45 thf(fact_215_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B2:B,C2:B]: (((A4 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.70/1.45 thf(fact_216_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,B2:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A4) @ B2) => ((((ord_less_eq @ C) @ (F @ B2)) @ C2) => ((! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ C) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ C) @ (F @ A4)) @ C2)))))))))).
% 2.70/1.45 thf(fact_217_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A4:A,F:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A4) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y2:B]: (((((ord_less_eq @ B) @ X3) @ Y2) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y2))))) => (((ord_less_eq @ A) @ A4) @ (F @ C2))))))))))).
% 2.70/1.45 thf(fact_218_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 2.70/1.45 thf(fact_219_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.70/1.45 thf(fact_220_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))))))))).
% 2.70/1.45 thf(fact_221_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))))))))).
% 2.70/1.45 thf(fact_222_mk__disjoint__insert, axiom, (! [A:$tType,A4:A,A2:(set @ A)]: (((((member @ A) @ A4) @ A2) => (? [B6:(set @ A)]: (((A2 = (((insert @ A) @ A4) @ B6)) & (~ ((((member @ A) @ A4) @ B6)))))))))).
% 2.70/1.45 thf(fact_223_subset__insertI2, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => (((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ B2) @ B4)))))).
% 2.70/1.45 thf(fact_224_subset__insertI, axiom, (! [A:$tType,B4:(set @ A),A4:A]: ((((ord_less_eq @ (set @ A)) @ B4) @ (((insert @ A) @ A4) @ B4))))).
% 2.70/1.45 thf(fact_225_insert__commute, axiom, (! [A:$tType,X:A,Y:A,A2:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y) @ A2)) = (((insert @ A) @ Y) @ (((insert @ A) @ X) @ A2)))))).
% 2.70/1.45 thf(fact_226_subset__insert, axiom, (! [A:$tType,X:A,A2:(set @ A),B4:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((((ord_less_eq @ (set @ A)) @ A2) @ (((insert @ A) @ X) @ B4)) = (((ord_less_eq @ (set @ A)) @ A2) @ B4)))))).
% 2.70/1.45 thf(fact_227_insert__eq__iff, axiom, (! [A:$tType,A4:A,A2:(set @ A),B2:A,B4:(set @ A)]: (((~ ((((member @ A) @ A4) @ A2))) => ((~ ((((member @ A) @ B2) @ B4))) => (((((insert @ A) @ A4) @ A2) = (((insert @ A) @ B2) @ B4)) = (((A4 = B2) => (A2 = B4)) & ((A4 != B2) => (? [C4:(set @ A)]: (((A2 = (((insert @ A) @ B2) @ C4)) & ((~ ((((member @ A) @ B2) @ C4))) & ((B4 = (((insert @ A) @ A4) @ C4)) & (~ ((((member @ A) @ A4) @ C4)))))))))))))))).
% 2.70/1.45 thf(fact_228_insert__absorb, axiom, (! [A:$tType,A4:A,A2:(set @ A)]: (((((member @ A) @ A4) @ A2) => ((((insert @ A) @ A4) @ A2) = A2))))).
% 2.70/1.45 thf(fact_229_insert__ident, axiom, (! [A:$tType,X:A,A2:(set @ A),B4:(set @ A)]: (((~ ((((member @ A) @ X) @ A2))) => ((~ ((((member @ A) @ X) @ B4))) => (((((insert @ A) @ X) @ A2) = (((insert @ A) @ X) @ B4)) = (A2 = B4))))))).
% 2.70/1.45 thf(fact_230_Set_Oinsert__mono, axiom, (! [A:$tType,C3:(set @ A),D:(set @ A),A4:A]: (((((ord_less_eq @ (set @ A)) @ C3) @ D) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A4) @ C3)) @ (((insert @ A) @ A4) @ D)))))).
% 2.70/1.45 thf(fact_231_Set_Oset__insert, axiom, (! [A:$tType,X:A,A2:(set @ A)]: (((((member @ A) @ X) @ A2) => (~ ((! [B6:(set @ A)]: (((A2 = (((insert @ A) @ X) @ B6)) => (((member @ A) @ X) @ B6)))))))))).
% 2.70/1.45 thf(fact_232_insertI2, axiom, (! [A:$tType,A4:A,B4:(set @ A),B2:A]: (((((member @ A) @ A4) @ B4) => (((member @ A) @ A4) @ (((insert @ A) @ B2) @ B4)))))).
% 2.70/1.45 thf(fact_233_insertI1, axiom, (! [A:$tType,A4:A,B4:(set @ A)]: ((((member @ A) @ A4) @ (((insert @ A) @ A4) @ B4))))).
% 2.70/1.45 thf(fact_234_insertE, axiom, (! [A:$tType,A4:A,B2:A,A2:(set @ A)]: (((((member @ A) @ A4) @ (((insert @ A) @ B2) @ A2)) => ((A4 != B2) => (((member @ A) @ A4) @ A2)))))).
% 2.70/1.45 thf(fact_235_insert__subsetI, axiom, (! [A:$tType,X:A,A2:(set @ A),X4:(set @ A)]: (((((member @ A) @ X) @ A2) => ((((ord_less_eq @ (set @ A)) @ X4) @ A2) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ X4)) @ A2)))))).
% 2.70/1.45 thf(fact_236_lset__lappend__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lset @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = (((sup_sup @ (set @ A)) @ ((coinductive_lset @ A) @ Xs)) @ ((coinductive_lset @ A) @ Ys))))))).
% 2.70/1.45 thf(fact_237_lsorted__LCons_H, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Xs:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ (((coinductive_LCons @ A) @ X) @ Xs)) = ((~ (((coinductive_lnull @ A) @ Xs))) => ((((ord_less_eq @ A) @ X) @ ((coinductive_lhd @ A) @ Xs)) & ((coindu63249387sorted @ A) @ Xs)))))))))).
% 2.70/1.45 thf(fact_238_Un__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A),B4:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B4)) = ((((member @ A) @ C2) @ A2) | (((member @ A) @ C2) @ B4)))))).
% 2.70/1.45 thf(fact_239_UnCI, axiom, (! [A:$tType,C2:A,B4:(set @ A),A2:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B4))) => (((member @ A) @ C2) @ A2)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B4)))))).
% 2.70/1.45 thf(fact_240_Un__subset__iff, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B4)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A2) @ C3) & (((ord_less_eq @ (set @ A)) @ B4) @ C3)))))).
% 2.70/1.45 thf(fact_241_Un__insert__left, axiom, (! [A:$tType,A4:A,B4:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((insert @ A) @ A4) @ B4)) @ C3) = (((insert @ A) @ A4) @ (((sup_sup @ (set @ A)) @ B4) @ C3)))))).
% 2.70/1.45 thf(fact_242_Un__insert__right, axiom, (! [A:$tType,A2:(set @ A),A4:A,B4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((insert @ A) @ A4) @ B4)) = (((insert @ A) @ A4) @ (((sup_sup @ (set @ A)) @ A2) @ B4)))))).
% 2.70/1.45 thf(fact_243_lhd__lappend, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (((coinductive_lhd @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_lhd @ A) @ Ys))) & ((~ (((coinductive_lnull @ A) @ Xs))) => (((coinductive_lhd @ A) @ (((coinductive_lappend @ A) @ Xs) @ Ys)) = ((coinductive_lhd @ A) @ Xs))))))).
% 2.70/1.45 thf(fact_244_Un__mono, axiom, (! [A:$tType,A2:(set @ A),C3:(set @ A),B4:(set @ A),D:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C3) => ((((ord_less_eq @ (set @ A)) @ B4) @ D) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B4)) @ (((sup_sup @ (set @ A)) @ C3) @ D))))))).
% 2.70/1.45 thf(fact_245_Un__least, axiom, (! [A:$tType,A2:(set @ A),C3:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C3) => ((((ord_less_eq @ (set @ A)) @ B4) @ C3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B4)) @ C3)))))).
% 2.70/1.45 thf(fact_246_Un__upper1, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B4))))).
% 2.70/1.45 thf(fact_247_Un__upper2, axiom, (! [A:$tType,B4:(set @ A),A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B4) @ (((sup_sup @ (set @ A)) @ A2) @ B4))))).
% 2.70/1.45 thf(fact_248_Un__absorb1, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B4) => ((((sup_sup @ (set @ A)) @ A2) @ B4) = B4))))).
% 2.70/1.45 thf(fact_249_Un__absorb2, axiom, (! [A:$tType,B4:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B4) @ A2) => ((((sup_sup @ (set @ A)) @ A2) @ B4) = A2))))).
% 2.70/1.45 thf(fact_250_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ B5) = B5))))))).
% 2.70/1.45 thf(fact_251_Un__left__commute, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B4) @ C3)) = (((sup_sup @ (set @ A)) @ B4) @ (((sup_sup @ (set @ A)) @ A2) @ C3)))))).
% 2.70/1.45 thf(fact_252_Un__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B4)) = (((sup_sup @ (set @ A)) @ A2) @ B4))))).
% 2.70/1.45 thf(fact_253_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((((sup_sup @ (set @ A)) @ B5) @ A6))))))).
% 2.70/1.45 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: ((((preorder @ A8) @ (type2 @ A8)) => ((preorder @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.70/1.45 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: ((((order @ A8) @ (type2 @ A8)) => ((order @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.70/1.45 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: ((((ord @ A8) @ (type2 @ A8)) => ((ord @ (A7 > A8)) @ (type2 @ (A7 > A8))))))).
% 2.70/1.45 thf(tcon_Set_Oset___Orderings_Opreorder_1, axiom, (! [A7:$tType]: (((preorder @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.70/1.45 thf(tcon_Set_Oset___Orderings_Oorder_2, axiom, (! [A7:$tType]: (((order @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.70/1.45 thf(tcon_Set_Oset___Orderings_Oord_3, axiom, (! [A7:$tType]: (((ord @ (set @ A7)) @ (type2 @ (set @ A7)))))).
% 2.70/1.45 thf(tcon_HOL_Obool___Orderings_Opreorder_4, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.70/1.45 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.70/1.45 thf(tcon_HOL_Obool___Orderings_Oorder_5, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.70/1.45 thf(tcon_HOL_Obool___Orderings_Oord_6, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.70/1.45 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.70/1.45 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.70/1.45 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.70/1.45 thf(conj_0, conjecture, (((lMirro427583474mirror @ a) @ (((coinductive_LCons @ a) @ x) @ xs)) = (((coinductive_LCons @ a) @ x) @ (((coinductive_lappend @ a) @ ((lMirro427583474mirror @ a) @ xs)) @ (((coinductive_LCons @ a) @ x) @ (coinductive_LNil @ a)))))).
% 2.70/1.45 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Mre2OmccfH/DTF2DTF7148.p
% 2.70/1.45 ------------------------
% 2.70/1.48 ---- Cleaned THF ---
% 2.70/1.48 thf(ty_t_Coinductive__List_Ollist,type,
% 2.70/1.48 coinductive_llist: $tType > $tType ).
% 2.70/1.48
% 2.70/1.48 thf(ty_t_Set_Oset,type,
% 2.70/1.48 set: $tType > $tType ).
% 2.70/1.48
% 2.70/1.48 thf(ty_t_itself,type,
% 2.70/1.48 itself: $tType > $tType ).
% 2.70/1.48
% 2.70/1.48 thf(ty_tf_a,type,
% 2.70/1.48 a: $tType ).
% 2.70/1.48
% 2.70/1.48 thf(sy_cl_HOL_Otype,type,
% 2.70/1.48 type:
% 2.70/1.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_cl_Orderings_Oord,type,
% 2.70/1.48 ord:
% 2.70/1.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_cl_Orderings_Oorder,type,
% 2.70/1.48 order:
% 2.70/1.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_cl_Orderings_Olinorder,type,
% 2.70/1.48 linorder:
% 2.70/1.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_cl_Orderings_Opreorder,type,
% 2.70/1.48 preorder:
% 2.70/1.48 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
% 2.70/1.48 coindu328551480prefix:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 2.70/1.48 coinductive_gen_lset:
% 2.70/1.48 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Olappend,type,
% 2.70/1.48 coinductive_lappend:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Olfilter,type,
% 2.70/1.48 coinductive_lfilter:
% 2.70/1.48 !>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Olfinite,type,
% 2.70/1.48 coinductive_lfinite:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ollast,type,
% 2.70/1.48 coinductive_llast:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ollist_OLCons,type,
% 2.70/1.48 coinductive_LCons:
% 2.70/1.48 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 2.70/1.48 coinductive_LNil:
% 2.70/1.48 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
% 2.70/1.48 coindu1259883913_llist:
% 2.70/1.48 !>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ollist_Olhd,type,
% 2.70/1.48 coinductive_lhd:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 2.70/1.48 coinductive_lnull:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 2.70/1.48 coinductive_lset:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Olmember,type,
% 2.70/1.48 coinductive_lmember:
% 2.70/1.48 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Olprefix,type,
% 2.70/1.48 coinductive_lprefix:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Olstrict__prefix,type,
% 2.70/1.48 coindu1478340336prefix:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 2.70/1.48 coindu63249387sorted:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Coinductive__List_Ounfold__llist,type,
% 2.70/1.48 coindu1441602521_llist:
% 2.70/1.48 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_HOL_Oundefined,type,
% 2.70/1.48 undefined:
% 2.70/1.48 !>[A: $tType] : A ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_If,type,
% 2.70/1.48 if:
% 2.70/1.48 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror,type,
% 2.70/1.48 lMirro427583474mirror:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
% 2.70/1.48 lMirro999291890or_aux:
% 2.70/1.48 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.70/1.48 sup_sup:
% 2.70/1.48 !>[A: $tType] : ( A > A > A ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.70/1.48 ord_less_eq:
% 2.70/1.48 !>[A: $tType] : ( A > A > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Pure_Otype,type,
% 2.70/1.48 type2:
% 2.70/1.48 !>[A: $tType] : ( itself @ A ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Set_OCollect,type,
% 2.70/1.48 collect:
% 2.70/1.48 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_Set_Oinsert,type,
% 2.70/1.48 insert:
% 2.70/1.48 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_c_member,type,
% 2.70/1.48 member:
% 2.70/1.48 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(sy_v_x,type,
% 2.70/1.48 x: a ).
% 2.70/1.48
% 2.70/1.48 thf(sy_v_xs,type,
% 2.70/1.48 xs: coinductive_llist @ a ).
% 2.70/1.48
% 2.70/1.48 thf(fact_0_lmirror__LNil,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( lMirro427583474mirror @ A @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_1_lappend__code_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
% 2.70/1.48 = Ys ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_2_lappend__LNil2,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = Xs ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_3_lappend__code_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
% 2.70/1.48 = ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_4_lappend__snocL1__conv__LCons2,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
% 2.70/1.48 = ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_5_llist_Oinject,axiom,
% 2.70/1.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_LCons @ A @ X21 @ X22 )
% 2.70/1.48 = ( coinductive_LCons @ A @ Y21 @ Y22 ) )
% 2.70/1.48 = ( ( X21 = Y21 )
% 2.70/1.48 & ( X22 = Y22 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_6_lappend__LNil__LNil,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_7_LNil__eq__lappend__iff,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_LNil @ A )
% 2.70/1.48 = ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( ( Xs
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 & ( Ys
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_8_lappend__eq__LNil__iff,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( ( Xs
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 & ( Ys
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_9_llist_Odistinct_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_LNil @ A )
% 2.70/1.48 != ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_10_llist_Oexhaust,axiom,
% 2.70/1.48 ! [A: $tType,Y: coinductive_llist @ A] :
% 2.70/1.48 ( ( Y
% 2.70/1.48 != ( coinductive_LNil @ A ) )
% 2.70/1.48 => ~ ! [X212: A,X222: coinductive_llist @ A] :
% 2.70/1.48 ( Y
% 2.70/1.48 != ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_11_neq__LNil__conv,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( Xs
% 2.70/1.48 != ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( ? [X2: A,Xs2: coinductive_llist @ A] :
% 2.70/1.48 ( Xs
% 2.70/1.48 = ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_12_lmirror__aux__LCons,axiom,
% 2.70/1.48 ! [A: $tType,Acc: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_LCons @ A @ X @ ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) @ Xs ) @ ( coinductive_LCons @ A @ X @ Acc ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_13_lmirror__aux__simps_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X: coinductive_llist @ A] :
% 2.70/1.48 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X ) )
% 2.70/1.48 = ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_14_lmirror__aux__simps_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Acc: coinductive_llist @ A] :
% 2.70/1.48 ( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = Acc ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_15_lmirror__aux__acc,axiom,
% 2.70/1.48 ! [A: $tType,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( lMirro999291890or_aux @ A @ ( coinductive_lappend @ A @ Ys @ Zs ) @ Xs )
% 2.70/1.48 = ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ Ys @ Xs ) @ Zs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_16_lappend__assoc,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 2.70/1.48 = ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_17_lmirror__def,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( lMirro427583474mirror @ A )
% 2.70/1.48 = ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_18_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.70/1.48 = ( ( Xs
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 | ? [Xs2: coinductive_llist @ A] :
% 2.70/1.48 ( ( Xs
% 2.70/1.48 = ( coinductive_LCons @ A @ Y @ Xs2 ) )
% 2.70/1.48 & ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_19_lstrict__prefix__code_I2_J,axiom,
% 2.70/1.48 ! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_20_lstrict__prefix__code_I3_J,axiom,
% 2.70/1.48 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 2.70/1.48 ~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_21_llast__singleton,axiom,
% 2.70/1.48 ! [A: $tType,X: A] :
% 2.70/1.48 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
% 2.70/1.48 = X ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_22_lmember__code_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,X: A] :
% 2.70/1.48 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_23_lmember__code_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.70/1.48 = ( ( X = Y )
% 2.70/1.48 | ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_24_gen__lset__code_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A] :
% 2.70/1.48 ( ( coinductive_gen_lset @ A @ A2 @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = A2 ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_25_lfinite__rev__induct,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( P @ ( coinductive_LNil @ A ) )
% 2.70/1.48 => ( ! [X3: A,Xs3: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs3 )
% 2.70/1.48 => ( ( P @ Xs3 )
% 2.70/1.48 => ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) )
% 2.70/1.48 => ( P @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_26_lsorted__code_I2_J,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_27_llist_Ocorec__code,axiom,
% 2.70/1.48 ! [A: $tType,C: $tType] :
% 2.70/1.48 ( ( coindu1259883913_llist @ C @ A )
% 2.70/1.48 = ( ^ [P2: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C,A3: C] : ( if @ ( coinductive_llist @ A ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P2 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_28_unfold__llist_Ocode,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType] :
% 2.70/1.48 ( ( coindu1441602521_llist @ A @ B )
% 2.70/1.48 = ( ^ [P2: A > $o,G21: A > B,G22: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_29_lfinite__code_I2_J,axiom,
% 2.70/1.48 ! [B: $tType,X: B,Xs: coinductive_llist @ B] :
% 2.70/1.48 ( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_lfinite @ B @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_30_lfinite__LCons,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_31_lfinite__code_I1_J,axiom,
% 2.70/1.48 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_32_lfinite__lappend,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 & ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_33_lsorted__code_I1_J,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_34_unfold__llist__eq__LCons,axiom,
% 2.70/1.48 ! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
% 2.70/1.48 = ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( ~ ( IS_LNIL @ B2 )
% 2.70/1.48 & ( X
% 2.70/1.48 = ( LHD @ B2 ) )
% 2.70/1.48 & ( Xs
% 2.70/1.48 = ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_35_llast__LCons2,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 2.70/1.48 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_36_lfinite__lmirror__aux,axiom,
% 2.70/1.48 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 2.70/1.48 = ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 & ( coinductive_lfinite @ A @ Acc ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_37_lstrict__prefix__code_I4_J,axiom,
% 2.70/1.48 ! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
% 2.70/1.48 ( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
% 2.70/1.48 = ( ( X = Y )
% 2.70/1.48 & ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_38_lstrict__prefix__code_I1_J,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_39_llast__lappend__LCons,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
% 2.70/1.48 = ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_40_llist__less__induct,axiom,
% 2.70/1.48 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ! [Xs3: coinductive_llist @ A] :
% 2.70/1.48 ( ! [Ys2: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs3 )
% 2.70/1.48 => ( P @ Ys2 ) )
% 2.70/1.48 => ( P @ Xs3 ) )
% 2.70/1.48 => ( P @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_41_lstrict__prefix__lfinite1,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu1478340336prefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_42_lfinite__LConsI,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_43_lfinite__LNil,axiom,
% 2.70/1.48 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_44_lappend__inf,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.70/1.48 = Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_45_mem__Collect__eq,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,P: A > $o] :
% 2.70/1.48 ( ( member @ A @ A4 @ ( collect @ A @ P ) )
% 2.70/1.48 = ( P @ A4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_46_Collect__mem__eq,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A] :
% 2.70/1.48 ( ( collect @ A
% 2.70/1.48 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.70/1.48 = A2 ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_47_Collect__cong,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.48 ( ! [X3: A] :
% 2.70/1.48 ( ( P @ X3 )
% 2.70/1.48 = ( Q @ X3 ) )
% 2.70/1.48 => ( ( collect @ A @ P )
% 2.70/1.48 = ( collect @ A @ Q ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_48_ext,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.70/1.48 ( ! [X3: A] :
% 2.70/1.48 ( ( F @ X3 )
% 2.70/1.48 = ( G @ X3 ) )
% 2.70/1.48 => ( F = G ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_49_llimit__induct,axiom,
% 2.70/1.48 ! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( P @ ( coinductive_LNil @ A ) )
% 2.70/1.48 => ( ! [X3: A,Xs3: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs3 )
% 2.70/1.48 => ( ( P @ Xs3 )
% 2.70/1.48 => ( P @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
% 2.70/1.48 => ( ( ! [Ys2: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
% 2.70/1.48 => ( P @ Ys2 ) )
% 2.70/1.48 => ( P @ Xs ) )
% 2.70/1.48 => ( P @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_50_lmirror__aux__inf,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 2.70/1.48 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
% 2.70/1.48 = Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_51_LNil,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_52_unfold__llist_Octr_I2_J,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 2.70/1.48 ( ~ ( P3 @ A4 )
% 2.70/1.48 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 )
% 2.70/1.48 = ( coinductive_LCons @ B @ ( G212 @ A4 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ ( G223 @ A4 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_53_unfold__llist_Octr_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,B: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 2.70/1.48 ( ( P3 @ A4 )
% 2.70/1.48 => ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 )
% 2.70/1.48 = ( coinductive_LNil @ B ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_54_llist_Ocorec_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,C: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.70/1.48 ( ~ ( P3 @ A4 )
% 2.70/1.48 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 )
% 2.70/1.48 = ( coinductive_LCons @ A @ ( G212 @ A4 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q222 @ A4 ) @ ( G2212 @ A4 ) @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A4 ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_55_llist_Ocorec_I1_J,axiom,
% 2.70/1.48 ! [C: $tType,A: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.70/1.48 ( ( P3 @ A4 )
% 2.70/1.48 => ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 )
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_56_lfinite_Ocases,axiom,
% 2.70/1.48 ! [A: $tType,A4: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ A4 )
% 2.70/1.48 => ( ( A4
% 2.70/1.48 != ( coinductive_LNil @ A ) )
% 2.70/1.48 => ~ ! [Xs3: coinductive_llist @ A] :
% 2.70/1.48 ( ? [X3: A] :
% 2.70/1.48 ( A4
% 2.70/1.48 = ( coinductive_LCons @ A @ X3 @ Xs3 ) )
% 2.70/1.48 => ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_57_lfinite_Osimps,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coinductive_lfinite @ A )
% 2.70/1.48 = ( ^ [A3: coinductive_llist @ A] :
% 2.70/1.48 ( ( A3
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 | ? [Xs4: coinductive_llist @ A,X2: A] :
% 2.70/1.48 ( ( A3
% 2.70/1.48 = ( coinductive_LCons @ A @ X2 @ Xs4 ) )
% 2.70/1.48 & ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_58_lfinite_Oinducts,axiom,
% 2.70/1.48 ! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ X )
% 2.70/1.48 => ( ( P @ ( coinductive_LNil @ A ) )
% 2.70/1.48 => ( ! [Xs3: coinductive_llist @ A,X3: A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs3 )
% 2.70/1.48 => ( ( P @ Xs3 )
% 2.70/1.48 => ( P @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
% 2.70/1.48 => ( P @ X ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_59_Singleton,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_60_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_61_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( Xs
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_62_lsorted_Ocases,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [A4: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu63249387sorted @ A @ A4 )
% 2.70/1.48 => ( ( A4
% 2.70/1.48 != ( coinductive_LNil @ A ) )
% 2.70/1.48 => ( ! [X3: A] :
% 2.70/1.48 ( A4
% 2.70/1.48 != ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
% 2.70/1.48 => ~ ! [X3: A,Y2: A,Xs3: coinductive_llist @ A] :
% 2.70/1.48 ( ( A4
% 2.70/1.48 = ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) )
% 2.70/1.48 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.70/1.48 => ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_63_lsorted_Osimps,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ( ( coindu63249387sorted @ A )
% 2.70/1.48 = ( ^ [A3: coinductive_llist @ A] :
% 2.70/1.48 ( ( A3
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 | ? [X2: A] :
% 2.70/1.48 ( A3
% 2.70/1.48 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
% 2.70/1.48 | ? [X2: A,Y3: A,Xs4: coinductive_llist @ A] :
% 2.70/1.48 ( ( A3
% 2.70/1.48 = ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) )
% 2.70/1.48 & ( ord_less_eq @ A @ X2 @ Y3 )
% 2.70/1.48 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_64_lsorted_Ocoinduct,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
% 2.70/1.48 ( ( X4 @ X )
% 2.70/1.48 => ( ! [X3: coinductive_llist @ A] :
% 2.70/1.48 ( ( X4 @ X3 )
% 2.70/1.48 => ( ( X3
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 | ? [Xa2: A] :
% 2.70/1.48 ( X3
% 2.70/1.48 = ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LNil @ A ) ) )
% 2.70/1.48 | ? [Xa2: A,Y4: A,Xs5: coinductive_llist @ A] :
% 2.70/1.48 ( ( X3
% 2.70/1.48 = ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
% 2.70/1.48 & ( ord_less_eq @ A @ Xa2 @ Y4 )
% 2.70/1.48 & ( ( X4 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) )
% 2.70/1.48 | ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) ) ) ) )
% 2.70/1.48 => ( coindu63249387sorted @ A @ X ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_65_lstrict__prefix__lappend__conv,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_66_split__llist,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 2.70/1.48 ( ( Xs
% 2.70/1.48 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 2.70/1.48 & ( coinductive_lfinite @ A @ Ys3 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_67_split__llist__first,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ? [Ys3: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
% 2.70/1.48 ( ( Xs
% 2.70/1.48 = ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
% 2.70/1.48 & ( coinductive_lfinite @ A @ Ys3 )
% 2.70/1.48 & ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys3 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_68_lsorted__LCons__LCons,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
% 2.70/1.48 = ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 & ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_69_gen__lset__code_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_gen_lset @ A @ A2 @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_gen_lset @ A @ ( insert @ A @ X @ A2 ) @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_70_lprefix__lappend__same,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
% 2.70/1.48 = ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_71_llist_Oleq__refl,axiom,
% 2.70/1.48 ! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_72_lprefix__refl,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_73_LCons__lprefix__LCons,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.70/1.48 = ( ( X = Y )
% 2.70/1.48 & ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_74_lprefix__code_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_75_lnull__lappend,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 & ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_76_lappend_Odisc__iff_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 2.70/1.48 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_77_lmirror__aux_Odisc__iff_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) )
% 2.70/1.48 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 | ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_78_lnull__lmirror__aux,axiom,
% 2.70/1.48 ! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
% 2.70/1.48 = ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 & ( coinductive_lnull @ A @ Acc ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_79_unfold__llist_Odisc__iff_I1_J,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A4: A] :
% 2.70/1.48 ( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) )
% 2.70/1.48 = ( P3 @ A4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_80_unfold__llist_Odisc__iff_I2_J,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A4: A] :
% 2.70/1.48 ( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) ) )
% 2.70/1.48 = ( ~ ( P3 @ A4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_81_llist_Ocorec__disc__iff_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A4: C] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) )
% 2.70/1.48 = ( P3 @ A4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_82_llist_Ocorec__disc__iff_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A4: C] :
% 2.70/1.48 ( ( ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) ) )
% 2.70/1.48 = ( ~ ( P3 @ A4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_83_lnull__lmirror,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ ( lMirro427583474mirror @ A @ Xs ) )
% 2.70/1.48 = ( coinductive_lnull @ A @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_84_lprefix__LNil,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( coinductive_lnull @ A @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_85_lset__LCons,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( insert @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_86_llist_Osimps_I19_J,axiom,
% 2.70/1.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.70/1.48 = ( insert @ A @ X21 @ ( coinductive_lset @ A @ X22 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_87_lset__lappend1,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_88_lmirror__aux_Oexhaust,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ~ ( coinductive_lnull @ A @ Acc ) )
% 2.70/1.48 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 | ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_89_lprefix__down__linear,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Zs )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_90_lprefix__not__lnullD,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_91_llist_Oleq__antisym,axiom,
% 2.70/1.48 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ X @ Y )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Y @ X )
% 2.70/1.48 => ( X = Y ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_92_wlog__linorder__le,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( linorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [P: A > A > $o,B2: A,A4: A] :
% 2.70/1.48 ( ! [A5: A,B3: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 2.70/1.48 => ( P @ A5 @ B3 ) )
% 2.70/1.48 => ( ( ( P @ B2 @ A4 )
% 2.70/1.48 => ( P @ A4 @ B2 ) )
% 2.70/1.48 => ( P @ A4 @ B2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_93_lprefix__antisym,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Ys @ Xs )
% 2.70/1.48 => ( Xs = Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_94_llist_Oleq__trans,axiom,
% 2.70/1.48 ! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,Z: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ X @ Y )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Y @ Z )
% 2.70/1.48 => ( coinductive_lprefix @ A @ X @ Z ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_95_lappend_Oexhaust,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ~ ( coinductive_lnull @ A @ Ys ) )
% 2.70/1.48 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_96_lprefix__lnullD,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( ( coinductive_lnull @ A @ Ys )
% 2.70/1.48 => ( coinductive_lnull @ A @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_97_lprefix__trans,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Ys @ Zs )
% 2.70/1.48 => ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_98_lprefix__lsetD,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_99_lprefix__lnull,axiom,
% 2.70/1.48 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Ys )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 = ( coinductive_lnull @ A @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_100_lnull__lprefix,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_101_lzip_Oexhaust,axiom,
% 2.70/1.48 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.70/1.48 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 | ( coinductive_lnull @ B @ Ys ) )
% 2.70/1.48 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_102_lsorted__LCons,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( preorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( ( coindu63249387sorted @ A @ Xs )
% 2.70/1.48 & ! [X2: A] :
% 2.70/1.48 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_103_llist_Oset__induct,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A4: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ A4 ) )
% 2.70/1.48 => ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 2.70/1.48 => ( ! [Z1: A,Z2: coinductive_llist @ A,Xa3: A] :
% 2.70/1.48 ( ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Z2 ) )
% 2.70/1.48 => ( ( P @ Xa3 @ Z2 )
% 2.70/1.48 => ( P @ Xa3 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
% 2.70/1.48 => ( P @ X @ A4 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_104_llist_Oset__cases,axiom,
% 2.70/1.48 ! [A: $tType,E: A,A4: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ E @ ( coinductive_lset @ A @ A4 ) )
% 2.70/1.48 => ( ! [Z2: coinductive_llist @ A] :
% 2.70/1.48 ( A4
% 2.70/1.48 != ( coinductive_LCons @ A @ E @ Z2 ) )
% 2.70/1.48 => ~ ! [Z1: A,Z2: coinductive_llist @ A] :
% 2.70/1.48 ( ( A4
% 2.70/1.48 = ( coinductive_LCons @ A @ Z1 @ Z2 ) )
% 2.70/1.48 => ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_105_lset__induct_H,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 2.70/1.48 => ( ! [X5: A,Xs3: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 2.70/1.48 => ( ( P @ Xs3 )
% 2.70/1.48 => ( P @ ( coinductive_LCons @ A @ X5 @ Xs3 ) ) ) )
% 2.70/1.48 => ( P @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_106_lset__induct,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
% 2.70/1.48 => ( ! [X5: A,Xs3: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
% 2.70/1.48 => ( ( X != X5 )
% 2.70/1.48 => ( ( P @ Xs3 )
% 2.70/1.48 => ( P @ ( coinductive_LCons @ A @ X5 @ Xs3 ) ) ) ) )
% 2.70/1.48 => ( P @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_107_lset__cases,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ( ! [Xs6: coinductive_llist @ A] :
% 2.70/1.48 ( Xs
% 2.70/1.48 != ( coinductive_LCons @ A @ X @ Xs6 ) )
% 2.70/1.48 => ~ ! [X5: A,Xs6: coinductive_llist @ A] :
% 2.70/1.48 ( ( Xs
% 2.70/1.48 = ( coinductive_LCons @ A @ X5 @ Xs6 ) )
% 2.70/1.48 => ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs6 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_108_llist_Oset__intros_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_109_llist_Oset__intros_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
% 2.70/1.48 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_110_lset__intros_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_111_lset__intros_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,X6: A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X6 @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_112_not__lnull__conv,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ~ ( coinductive_lnull @ A @ Xs ) )
% 2.70/1.48 = ( ? [X2: A,Xs2: coinductive_llist @ A] :
% 2.70/1.48 ( Xs
% 2.70/1.48 = ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_113_llist_OdiscI_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
% 2.70/1.48 ( ( Llist
% 2.70/1.48 = ( coinductive_LCons @ A @ X21 @ X22 ) )
% 2.70/1.48 => ~ ( coinductive_lnull @ A @ Llist ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_114_llist_Odisc_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,X21: A,X22: coinductive_llist @ A] :
% 2.70/1.48 ~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_115_lnull__def,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coinductive_lnull @ A )
% 2.70/1.48 = ( ^ [Llist2: coinductive_llist @ A] :
% 2.70/1.48 ( Llist2
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_116_llist_Ocollapse_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Llist )
% 2.70/1.48 => ( Llist
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_117_llist_OdiscI_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Llist: coinductive_llist @ A] :
% 2.70/1.48 ( ( Llist
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 => ( coinductive_lnull @ A @ Llist ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_118_llist_Odisc_I1_J,axiom,
% 2.70/1.48 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_119_LCons__lprefix__conv,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
% 2.70/1.48 = ( ? [Ys4: coinductive_llist @ A] :
% 2.70/1.48 ( ( Ys
% 2.70/1.48 = ( coinductive_LCons @ A @ X @ Ys4 ) )
% 2.70/1.48 & ( coinductive_lprefix @ A @ Xs @ Ys4 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_120_Le__LCons,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_121_lappend__lnull2,axiom,
% 2.70/1.48 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Ys )
% 2.70/1.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.70/1.48 = Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_122_lappend__lnull1,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.70/1.48 = Ys ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_123_lappend_Odisc_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lnull @ A @ Ys )
% 2.70/1.48 => ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_124_lappend_Odisc_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 | ~ ( coinductive_lnull @ A @ Ys ) )
% 2.70/1.48 => ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_125_lnull__imp__lfinite,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_126_LNil__lprefix,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_127_lprefix__lappend__sameI,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_128_lprefix__conv__lappend,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coinductive_lprefix @ A )
% 2.70/1.48 = ( ^ [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 2.70/1.48 ? [Zs3: coinductive_llist @ A] :
% 2.70/1.48 ( Ys5
% 2.70/1.48 = ( coinductive_lappend @ A @ Xs4 @ Zs3 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_129_lprefix__lappendD,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 | ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_130_lappend__lprefixE,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
% 2.70/1.48 => ~ ! [Zs4: coinductive_llist @ A] :
% 2.70/1.48 ( Zs
% 2.70/1.48 != ( coinductive_lappend @ A @ Xs @ Zs4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_131_lprefix__lappend,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_132_not__lfinite__lprefix__conv__eq,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 = ( Xs = Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_133_lprefix__lfiniteD,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( ( coinductive_lfinite @ A @ Ys )
% 2.70/1.48 => ( coinductive_lfinite @ A @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_134_lsorted__lprefixD,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( ( coindu63249387sorted @ A @ Ys )
% 2.70/1.48 => ( coindu63249387sorted @ A @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_135_lmirror__aux_Odisc_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 2.70/1.48 ( ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 | ~ ( coinductive_lnull @ A @ Acc ) )
% 2.70/1.48 => ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_136_lmirror__aux_Odisc_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lnull @ A @ Acc )
% 2.70/1.48 => ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_137_unfold__llist_Odisc_I1_J,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 2.70/1.48 ( ( P3 @ A4 )
% 2.70/1.48 => ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_138_unfold__llist_Odisc_I2_J,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType,P3: A > $o,A4: A,G212: A > B,G223: A > A] :
% 2.70/1.48 ( ~ ( P3 @ A4 )
% 2.70/1.48 => ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_139_llist_Ocorec__disc_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,C: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.70/1.48 ( ( P3 @ A4 )
% 2.70/1.48 => ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_140_llist_Ocorec__disc_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,C: $tType,P3: C > $o,A4: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
% 2.70/1.48 ( ~ ( P3 @ A4 )
% 2.70/1.48 => ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_141_lstrict__prefix__def,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coindu1478340336prefix @ A )
% 2.70/1.48 = ( ^ [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs4 @ Ys5 )
% 2.70/1.48 & ( Xs4 != Ys5 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_142_Coinductive__List_Ofinite__lprefix__def,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coindu328551480prefix @ A )
% 2.70/1.48 = ( coinductive_lprefix @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_143_lset__lmember,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_144_in__lset__lappend__iff,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
% 2.70/1.48 = ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 | ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 & ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_145_lappend_Octr_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lnull @ A @ Ys )
% 2.70/1.48 => ( ( coinductive_lappend @ A @ Xs @ Ys )
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_146_lprefix__LCons__conv,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
% 2.70/1.48 = ( ( Xs
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 | ? [Xs2: coinductive_llist @ A] :
% 2.70/1.48 ( ( Xs
% 2.70/1.48 = ( coinductive_LCons @ A @ Y @ Xs2 ) )
% 2.70/1.48 & ( coinductive_lprefix @ A @ Xs2 @ Ys ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_147_lprefix_Ocoinduct,axiom,
% 2.70/1.48 ! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
% 2.70/1.48 ( ( X4 @ X @ Xa )
% 2.70/1.48 => ( ! [X3: coinductive_llist @ A,Xa3: coinductive_llist @ A] :
% 2.70/1.48 ( ( X4 @ X3 @ Xa3 )
% 2.70/1.48 => ( ? [Xs5: coinductive_llist @ A] :
% 2.70/1.48 ( ( X3
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 & ( Xa3 = Xs5 ) )
% 2.70/1.48 | ? [Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
% 2.70/1.48 ( ( X3
% 2.70/1.48 = ( coinductive_LCons @ A @ Xb @ Xs5 ) )
% 2.70/1.48 & ( Xa3
% 2.70/1.48 = ( coinductive_LCons @ A @ Xb @ Ys2 ) )
% 2.70/1.48 & ( ( X4 @ Xs5 @ Ys2 )
% 2.70/1.48 | ( coinductive_lprefix @ A @ Xs5 @ Ys2 ) ) ) ) )
% 2.70/1.48 => ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_148_lprefix_Osimps,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coinductive_lprefix @ A )
% 2.70/1.48 = ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
% 2.70/1.48 ( ? [Xs4: coinductive_llist @ A] :
% 2.70/1.48 ( ( A12
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 & ( A23 = Xs4 ) )
% 2.70/1.48 | ? [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A,X2: A] :
% 2.70/1.48 ( ( A12
% 2.70/1.48 = ( coinductive_LCons @ A @ X2 @ Xs4 ) )
% 2.70/1.48 & ( A23
% 2.70/1.48 = ( coinductive_LCons @ A @ X2 @ Ys5 ) )
% 2.70/1.48 & ( coinductive_lprefix @ A @ Xs4 @ Ys5 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_149_lprefix_Ocases,axiom,
% 2.70/1.48 ! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ A1 @ A22 )
% 2.70/1.48 => ( ( ( A1
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 => ! [Xs3: coinductive_llist @ A] : ( A22 != Xs3 ) )
% 2.70/1.48 => ~ ! [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A,X3: A] :
% 2.70/1.48 ( ( A1
% 2.70/1.48 = ( coinductive_LCons @ A @ X3 @ Xs3 ) )
% 2.70/1.48 => ( ( A22
% 2.70/1.48 = ( coinductive_LCons @ A @ X3 @ Ys3 ) )
% 2.70/1.48 => ~ ( coinductive_lprefix @ A @ Xs3 @ Ys3 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_150_lprefix__code_I2_J,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_151_lmirror__aux_Octr_I1_J,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lnull @ A @ Acc )
% 2.70/1.48 => ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_152_llast__LCons,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 2.70/1.48 ( ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = X ) )
% 2.70/1.48 & ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_llast @ A @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_153_LCons__LCons,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 => ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
% 2.70/1.48 => ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_154_Coinductive__List_Olprefix__nitpick__simps,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coinductive_lprefix @ A )
% 2.70/1.48 = ( ^ [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lfinite @ A @ Xs4 )
% 2.70/1.48 => ( coindu328551480prefix @ A @ Xs4 @ Ys5 ) )
% 2.70/1.48 & ( ~ ( coinductive_lfinite @ A @ Xs4 )
% 2.70/1.48 => ( Xs4 = Ys5 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_155_insertCI,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,B4: set @ A,B2: A] :
% 2.70/1.48 ( ( ~ ( member @ A @ A4 @ B4 )
% 2.70/1.48 => ( A4 = B2 ) )
% 2.70/1.48 => ( member @ A @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_156_insert__iff,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,B2: A,A2: set @ A] :
% 2.70/1.48 ( ( member @ A @ A4 @ ( insert @ A @ B2 @ A2 ) )
% 2.70/1.48 = ( ( A4 = B2 )
% 2.70/1.48 | ( member @ A @ A4 @ A2 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_157_insert__subset,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A2 ) @ B4 )
% 2.70/1.48 = ( ( member @ A @ X @ B4 )
% 2.70/1.48 & ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_158_insert__absorb2,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A2: set @ A] :
% 2.70/1.48 ( ( insert @ A @ X @ ( insert @ A @ X @ A2 ) )
% 2.70/1.48 = ( insert @ A @ X @ A2 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_159_order__refl,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( preorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_160_llast__lappend,axiom,
% 2.70/1.48 ! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lnull @ A @ Ys )
% 2.70/1.48 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( coinductive_llast @ A @ Xs ) ) )
% 2.70/1.48 & ( ~ ( coinductive_lnull @ A @ Ys )
% 2.70/1.48 => ( ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( coinductive_llast @ A @ Ys ) ) )
% 2.70/1.48 & ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( undefined @ A ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_161_lfilter__eq__LConsD,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lfilter @ A @ P @ Ys )
% 2.70/1.48 = ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 2.70/1.48 ( ( Ys
% 2.70/1.48 = ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
% 2.70/1.48 & ( coinductive_lfinite @ A @ Us )
% 2.70/1.48 & ! [X7: A] :
% 2.70/1.48 ( ( member @ A @ X7 @ ( coinductive_lset @ A @ Us ) )
% 2.70/1.48 => ~ ( P @ X7 ) )
% 2.70/1.48 & ( P @ X )
% 2.70/1.48 & ( Xs
% 2.70/1.48 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_162_subsetI,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ! [X3: A] :
% 2.70/1.48 ( ( member @ A @ X3 @ A2 )
% 2.70/1.48 => ( member @ A @ X3 @ B4 ) )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_163_subset__antisym,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
% 2.70/1.48 => ( A2 = B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_164_lfilter__idem,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 2.70/1.48 = ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_165_lfilter__LCons,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( P @ X )
% 2.70/1.48 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
% 2.70/1.48 & ( ~ ( P @ X )
% 2.70/1.48 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_166_lfilter__LNil,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o] :
% 2.70/1.48 ( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_167_lnull__lfilter,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
% 2.70/1.48 = ( ! [X2: A] :
% 2.70/1.48 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ~ ( P @ X2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_168_diverge__lfilter__LNil,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.70/1.48 ( ! [X3: A] :
% 2.70/1.48 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ~ ( P @ X3 ) )
% 2.70/1.48 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.70/1.48 = ( coinductive_LNil @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_169_lfilter__lappend__lfinite,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_170_set__mp,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ( member @ A @ X @ A2 )
% 2.70/1.48 => ( member @ A @ X @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_171_in__mono,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ( member @ A @ X @ A2 )
% 2.70/1.48 => ( member @ A @ X @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_172_subsetD,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ( member @ A @ C2 @ A2 )
% 2.70/1.48 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_173_subsetCE,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ( member @ A @ C2 @ A2 )
% 2.70/1.48 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_174_equalityE,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( A2 = B4 )
% 2.70/1.48 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_175_subset__eq,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) )
% 2.70/1.48 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.70/1.48 ! [X2: A] :
% 2.70/1.48 ( ( member @ A @ X2 @ A6 )
% 2.70/1.48 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_176_equalityD1,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( A2 = B4 )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_177_equalityD2,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( A2 = B4 )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_178_set__rev__mp,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( member @ A @ X @ A2 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( member @ A @ X @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_179_subset__iff,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) )
% 2.70/1.48 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.70/1.48 ! [T: A] :
% 2.70/1.48 ( ( member @ A @ T @ A6 )
% 2.70/1.48 => ( member @ A @ T @ B5 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_180_rev__subsetD,axiom,
% 2.70/1.48 ! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( member @ A @ C2 @ A2 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( member @ A @ C2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_181_subset__refl,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_182_Collect__mono,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.48 ( ! [X3: A] :
% 2.70/1.48 ( ( P @ X3 )
% 2.70/1.48 => ( Q @ X3 ) )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_183_subset__trans,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_184_set__eq__subset,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ^ [Y5: set @ A,Z3: set @ A] : ( Y5 = Z3 ) )
% 2.70/1.48 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 2.70/1.48 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_185_contra__subsetD,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ~ ( member @ A @ C2 @ B4 )
% 2.70/1.48 => ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_186_Collect__mono__iff,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.70/1.48 = ( ! [X2: A] :
% 2.70/1.48 ( ( P @ X2 )
% 2.70/1.48 => ( Q @ X2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_187_lprefix__lfilterI,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o] :
% 2.70/1.48 ( ( coinductive_lprefix @ A @ Xs @ Ys )
% 2.70/1.48 => ( coinductive_lprefix @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_188_lfilter__LCons__seek,axiom,
% 2.70/1.48 ! [A: $tType,P3: A > $o,X: A,L: coinductive_llist @ A] :
% 2.70/1.48 ( ~ ( P3 @ X )
% 2.70/1.48 => ( ( coinductive_lfilter @ A @ P3 @ ( coinductive_LCons @ A @ X @ L ) )
% 2.70/1.48 = ( coinductive_lfilter @ A @ P3 @ L ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_189_lfilter__LCons__found,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( P @ X )
% 2.70/1.48 => ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_190_lfilter__cong,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
% 2.70/1.48 ( ( Xs = Ys )
% 2.70/1.48 => ( ! [X3: A] :
% 2.70/1.48 ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) )
% 2.70/1.48 => ( ( P @ X3 )
% 2.70/1.48 = ( Q @ X3 ) ) )
% 2.70/1.48 => ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.70/1.48 = ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_191_lfilter__id__conv,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.70/1.48 = Xs )
% 2.70/1.48 = ( ! [X2: A] :
% 2.70/1.48 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ( P @ X2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_192_lfinite__lfilterI,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_193_lsorted__lfilterI,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( preorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [Xs: coinductive_llist @ A,P: A > $o] :
% 2.70/1.48 ( ( coindu63249387sorted @ A @ Xs )
% 2.70/1.48 => ( coindu63249387sorted @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_194_lfilter__empty__conv,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.70/1.48 = ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( ! [X2: A] :
% 2.70/1.48 ( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
% 2.70/1.48 => ~ ( P @ X2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_195_lfilter__eq__lappend__lfiniteD,axiom,
% 2.70/1.48 ! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lfilter @ A @ P @ Xs )
% 2.70/1.48 = ( coinductive_lappend @ A @ Ys @ Zs ) )
% 2.70/1.48 => ( ( coinductive_lfinite @ A @ Ys )
% 2.70/1.48 => ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
% 2.70/1.48 ( ( Xs
% 2.70/1.48 = ( coinductive_lappend @ A @ Us @ Vs ) )
% 2.70/1.48 & ( coinductive_lfinite @ A @ Us )
% 2.70/1.48 & ( Ys
% 2.70/1.48 = ( coinductive_lfilter @ A @ P @ Us ) )
% 2.70/1.48 & ( Zs
% 2.70/1.48 = ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_196_llast__LNil,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
% 2.70/1.48 = ( undefined @ A ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_197_llast__linfinite,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ~ ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_llast @ A @ Xs )
% 2.70/1.48 = ( undefined @ A ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_198_dual__order_Oantisym,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [B2: A,A4: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ B2 @ A4 )
% 2.70/1.48 => ( ( ord_less_eq @ A @ A4 @ B2 )
% 2.70/1.48 => ( A4 = B2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_199_dual__order_Otrans,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [B2: A,A4: A,C2: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ B2 @ A4 )
% 2.70/1.48 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 2.70/1.48 => ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_200_linorder__wlog,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( linorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [P: A > A > $o,A4: A,B2: A] :
% 2.70/1.48 ( ! [A5: A,B3: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A5 @ B3 )
% 2.70/1.48 => ( P @ A5 @ B3 ) )
% 2.70/1.48 => ( ! [A5: A,B3: A] :
% 2.70/1.48 ( ( P @ B3 @ A5 )
% 2.70/1.48 => ( P @ A5 @ B3 ) )
% 2.70/1.48 => ( P @ A4 @ B2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_201_dual__order_Orefl,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_202_order__trans,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( preorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A,Z: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.70/1.48 => ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_203_order__class_Oorder_Oantisym,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [A4: A,B2: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A4 @ B2 )
% 2.70/1.48 => ( ( ord_less_eq @ A @ B2 @ A4 )
% 2.70/1.48 => ( A4 = B2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_204_ord__le__eq__trans,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [A4: A,B2: A,C2: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A4 @ B2 )
% 2.70/1.48 => ( ( B2 = C2 )
% 2.70/1.48 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_205_ord__eq__le__trans,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [A4: A,B2: A,C2: A] :
% 2.70/1.48 ( ( A4 = B2 )
% 2.70/1.48 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.70/1.48 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_206_antisym__conv,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [Y: A,X: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ Y @ X )
% 2.70/1.48 => ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 = ( X = Y ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_207_le__cases3,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( linorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A,Z: A] :
% 2.70/1.48 ( ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.70/1.48 => ( ( ( ord_less_eq @ A @ Y @ X )
% 2.70/1.48 => ~ ( ord_less_eq @ A @ X @ Z ) )
% 2.70/1.48 => ( ( ( ord_less_eq @ A @ X @ Z )
% 2.70/1.48 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.70/1.48 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.70/1.48 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 2.70/1.48 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.70/1.48 => ~ ( ord_less_eq @ A @ Z @ X ) )
% 2.70/1.48 => ~ ( ( ord_less_eq @ A @ Z @ X )
% 2.70/1.48 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_208_order_Otrans,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [A4: A,B2: A,C2: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A4 @ B2 )
% 2.70/1.48 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 2.70/1.48 => ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_209_le__cases,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( linorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A] :
% 2.70/1.48 ( ~ ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_210_eq__refl,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( preorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A] :
% 2.70/1.48 ( ( X = Y )
% 2.70/1.48 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_211_linear,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( linorder @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_212_antisym,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Y: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ X @ Y )
% 2.70/1.48 => ( ( ord_less_eq @ A @ Y @ X )
% 2.70/1.48 => ( X = Y ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_213_eq__iff,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( order @ A @ ( type2 @ A ) )
% 2.70/1.48 => ( ( ^ [Y5: A,Z3: A] : ( Y5 = Z3 ) )
% 2.70/1.48 = ( ^ [X2: A,Y3: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ X2 @ Y3 )
% 2.70/1.48 & ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_214_ord__le__eq__subst,axiom,
% 2.70/1.48 ! [A: $tType,B: $tType] :
% 2.70/1.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.70/1.48 & ( ord @ A @ ( type2 @ A ) ) )
% 2.70/1.48 => ! [A4: A,B2: A,F: A > B,C2: B] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A4 @ B2 )
% 2.70/1.48 => ( ( ( F @ B2 )
% 2.70/1.48 = C2 )
% 2.70/1.48 => ( ! [X3: A,Y2: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.70/1.48 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.70/1.48 => ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_215_ord__eq__le__subst,axiom,
% 2.70/1.48 ! [A: $tType,B: $tType] :
% 2.70/1.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.70/1.48 & ( ord @ A @ ( type2 @ A ) ) )
% 2.70/1.48 => ! [A4: A,F: B > A,B2: B,C2: B] :
% 2.70/1.48 ( ( A4
% 2.70/1.48 = ( F @ B2 ) )
% 2.70/1.48 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.70/1.48 => ( ! [X3: B,Y2: B] :
% 2.70/1.48 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 2.70/1.48 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.70/1.48 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_216_order__subst2,axiom,
% 2.70/1.48 ! [A: $tType,C: $tType] :
% 2.70/1.48 ( ( ( order @ C @ ( type2 @ C ) )
% 2.70/1.48 & ( order @ A @ ( type2 @ A ) ) )
% 2.70/1.48 => ! [A4: A,B2: A,F: A > C,C2: C] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A4 @ B2 )
% 2.70/1.48 => ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
% 2.70/1.48 => ( ! [X3: A,Y2: A] :
% 2.70/1.48 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.70/1.48 => ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.70/1.48 => ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_217_order__subst1,axiom,
% 2.70/1.48 ! [A: $tType,B: $tType] :
% 2.70/1.48 ( ( ( order @ B @ ( type2 @ B ) )
% 2.70/1.48 & ( order @ A @ ( type2 @ A ) ) )
% 2.70/1.48 => ! [A4: A,F: B > A,B2: B,C2: B] :
% 2.70/1.48 ( ( ord_less_eq @ A @ A4 @ ( F @ B2 ) )
% 2.70/1.48 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 2.70/1.48 => ( ! [X3: B,Y2: B] :
% 2.70/1.48 ( ( ord_less_eq @ B @ X3 @ Y2 )
% 2.70/1.48 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 2.70/1.48 => ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_218_le__fun__def,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType] :
% 2.70/1.48 ( ( ord @ B @ ( type2 @ B ) )
% 2.70/1.48 => ( ( ord_less_eq @ ( A > B ) )
% 2.70/1.48 = ( ^ [F2: A > B,G2: A > B] :
% 2.70/1.48 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_219_le__funI,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType] :
% 2.70/1.48 ( ( ord @ B @ ( type2 @ B ) )
% 2.70/1.48 => ! [F: A > B,G: A > B] :
% 2.70/1.48 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.70/1.48 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_220_le__funE,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType] :
% 2.70/1.48 ( ( ord @ B @ ( type2 @ B ) )
% 2.70/1.48 => ! [F: A > B,G: A > B,X: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.70/1.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_221_le__funD,axiom,
% 2.70/1.48 ! [B: $tType,A: $tType] :
% 2.70/1.48 ( ( ord @ B @ ( type2 @ B ) )
% 2.70/1.48 => ! [F: A > B,G: A > B,X: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.70/1.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_222_mk__disjoint__insert,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,A2: set @ A] :
% 2.70/1.48 ( ( member @ A @ A4 @ A2 )
% 2.70/1.48 => ? [B6: set @ A] :
% 2.70/1.48 ( ( A2
% 2.70/1.48 = ( insert @ A @ A4 @ B6 ) )
% 2.70/1.48 & ~ ( member @ A @ A4 @ B6 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_223_subset__insertI2,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,B2: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_224_subset__insertI,axiom,
% 2.70/1.48 ! [A: $tType,B4: set @ A,A4: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A4 @ B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_225_insert__commute,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Y: A,A2: set @ A] :
% 2.70/1.48 ( ( insert @ A @ X @ ( insert @ A @ Y @ A2 ) )
% 2.70/1.48 = ( insert @ A @ Y @ ( insert @ A @ X @ A2 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_226_subset__insert,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ~ ( member @ A @ X @ A2 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( insert @ A @ X @ B4 ) )
% 2.70/1.48 = ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_227_insert__eq__iff,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,A2: set @ A,B2: A,B4: set @ A] :
% 2.70/1.48 ( ~ ( member @ A @ A4 @ A2 )
% 2.70/1.48 => ( ~ ( member @ A @ B2 @ B4 )
% 2.70/1.48 => ( ( ( insert @ A @ A4 @ A2 )
% 2.70/1.48 = ( insert @ A @ B2 @ B4 ) )
% 2.70/1.48 = ( ( ( A4 = B2 )
% 2.70/1.48 => ( A2 = B4 ) )
% 2.70/1.48 & ( ( A4 != B2 )
% 2.70/1.48 => ? [C4: set @ A] :
% 2.70/1.48 ( ( A2
% 2.70/1.48 = ( insert @ A @ B2 @ C4 ) )
% 2.70/1.48 & ~ ( member @ A @ B2 @ C4 )
% 2.70/1.48 & ( B4
% 2.70/1.48 = ( insert @ A @ A4 @ C4 ) )
% 2.70/1.48 & ~ ( member @ A @ A4 @ C4 ) ) ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_228_insert__absorb,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,A2: set @ A] :
% 2.70/1.48 ( ( member @ A @ A4 @ A2 )
% 2.70/1.48 => ( ( insert @ A @ A4 @ A2 )
% 2.70/1.48 = A2 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_229_insert__ident,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ~ ( member @ A @ X @ A2 )
% 2.70/1.48 => ( ~ ( member @ A @ X @ B4 )
% 2.70/1.48 => ( ( ( insert @ A @ X @ A2 )
% 2.70/1.48 = ( insert @ A @ X @ B4 ) )
% 2.70/1.48 = ( A2 = B4 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_230_Set_Oinsert__mono,axiom,
% 2.70/1.48 ! [A: $tType,C3: set @ A,D: set @ A,A4: A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A4 @ C3 ) @ ( insert @ A @ A4 @ D ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_231_Set_Oset__insert,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A2: set @ A] :
% 2.70/1.48 ( ( member @ A @ X @ A2 )
% 2.70/1.48 => ~ ! [B6: set @ A] :
% 2.70/1.48 ( ( A2
% 2.70/1.48 = ( insert @ A @ X @ B6 ) )
% 2.70/1.48 => ( member @ A @ X @ B6 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_232_insertI2,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,B4: set @ A,B2: A] :
% 2.70/1.48 ( ( member @ A @ A4 @ B4 )
% 2.70/1.48 => ( member @ A @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_233_insertI1,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,B4: set @ A] : ( member @ A @ A4 @ ( insert @ A @ A4 @ B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_234_insertE,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,B2: A,A2: set @ A] :
% 2.70/1.48 ( ( member @ A @ A4 @ ( insert @ A @ B2 @ A2 ) )
% 2.70/1.48 => ( ( A4 != B2 )
% 2.70/1.48 => ( member @ A @ A4 @ A2 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_235_insert__subsetI,axiom,
% 2.70/1.48 ! [A: $tType,X: A,A2: set @ A,X4: set @ A] :
% 2.70/1.48 ( ( member @ A @ X @ A2 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ X4 @ A2 )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X4 ) @ A2 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_236_lset__lappend__lfinite,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( coinductive_lfinite @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_237_lsorted__LCons_H,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord @ A @ ( type2 @ A ) )
% 2.70/1.48 => ! [X: A,Xs: coinductive_llist @ A] :
% 2.70/1.48 ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
% 2.70/1.48 = ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( ord_less_eq @ A @ X @ ( coinductive_lhd @ A @ Xs ) )
% 2.70/1.48 & ( coindu63249387sorted @ A @ Xs ) ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_238_Un__iff,axiom,
% 2.70/1.48 ! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) )
% 2.70/1.48 = ( ( member @ A @ C2 @ A2 )
% 2.70/1.48 | ( member @ A @ C2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_239_UnCI,axiom,
% 2.70/1.48 ! [A: $tType,C2: A,B4: set @ A,A2: set @ A] :
% 2.70/1.48 ( ( ~ ( member @ A @ C2 @ B4 )
% 2.70/1.48 => ( member @ A @ C2 @ A2 ) )
% 2.70/1.48 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_240_Un__subset__iff,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) @ C3 )
% 2.70/1.48 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.70/1.48 & ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_241_Un__insert__left,axiom,
% 2.70/1.48 ! [A: $tType,A4: A,B4: set @ A,C3: set @ A] :
% 2.70/1.48 ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A4 @ B4 ) @ C3 )
% 2.70/1.48 = ( insert @ A @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_242_Un__insert__right,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,A4: A,B4: set @ A] :
% 2.70/1.48 ( ( sup_sup @ ( set @ A ) @ A2 @ ( insert @ A @ A4 @ B4 ) )
% 2.70/1.48 = ( insert @ A @ A4 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_243_lhd__lappend,axiom,
% 2.70/1.48 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.70/1.48 ( ( ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( coinductive_lhd @ A @ Ys ) ) )
% 2.70/1.48 & ( ~ ( coinductive_lnull @ A @ Xs )
% 2.70/1.48 => ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
% 2.70/1.48 = ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_244_Un__mono,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,C3: set @ A,B4: set @ A,D: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ D )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_245_Un__least,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,C3: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.70/1.48 => ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
% 2.70/1.48 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) @ C3 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_246_Un__upper1,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_247_Un__upper2,axiom,
% 2.70/1.48 ! [A: $tType,B4: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_248_Un__absorb1,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 => ( ( sup_sup @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 = B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_249_Un__absorb2,axiom,
% 2.70/1.48 ! [A: $tType,B4: set @ A,A2: set @ A] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
% 2.70/1.48 => ( ( sup_sup @ ( set @ A ) @ A2 @ B4 )
% 2.70/1.48 = A2 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_250_subset__Un__eq,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( ord_less_eq @ ( set @ A ) )
% 2.70/1.48 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.70/1.48 ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
% 2.70/1.48 = B5 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_251_Un__left__commute,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
% 2.70/1.48 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
% 2.70/1.48 = ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_252_Un__left__absorb,axiom,
% 2.70/1.48 ! [A: $tType,A2: set @ A,B4: set @ A] :
% 2.70/1.48 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B4 ) )
% 2.70/1.48 = ( sup_sup @ ( set @ A ) @ A2 @ B4 ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(fact_253_Un__commute,axiom,
% 2.70/1.48 ! [A: $tType] :
% 2.70/1.48 ( ( sup_sup @ ( set @ A ) )
% 2.70/1.48 = ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.70/1.48 ! [A7: $tType,A8: $tType] :
% 2.70/1.48 ( ( preorder @ A8 @ ( type2 @ A8 ) )
% 2.70/1.48 => ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.70/1.48 ! [A7: $tType,A8: $tType] :
% 2.70/1.48 ( ( order @ A8 @ ( type2 @ A8 ) )
% 2.70/1.48 => ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_fun___Orderings_Oord,axiom,
% 2.70/1.48 ! [A7: $tType,A8: $tType] :
% 2.70/1.48 ( ( ord @ A8 @ ( type2 @ A8 ) )
% 2.70/1.48 => ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
% 2.70/1.48 ! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
% 2.70/1.48 ! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
% 2.70/1.48 ! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
% 2.70/1.48 preorder @ $o @ ( type2 @ $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.70/1.48 linorder @ $o @ ( type2 @ $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 2.70/1.48 order @ $o @ ( type2 @ $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
% 2.70/1.48 ord @ $o @ ( type2 @ $o ) ).
% 2.70/1.48
% 2.70/1.48 thf(help_If_3_1_T,axiom,
% 2.70/1.48 ! [P: $o] :
% 2.70/1.48 ( ( P = $true )
% 2.70/1.48 | ( P = $false ) ) ).
% 2.70/1.48
% 2.70/1.48 thf(help_If_2_1_T,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Y: A] :
% 2.70/1.48 ( ( if @ A @ $false @ X @ Y )
% 2.70/1.48 = Y ) ).
% 2.70/1.48
% 2.70/1.48 thf(help_If_1_1_T,axiom,
% 2.70/1.48 ! [A: $tType,X: A,Y: A] :
% 2.70/1.48 ( ( if @ A @ $true @ X @ Y )
% 2.70/1.48 = X ) ).
% 2.70/1.48
% 2.70/1.48 thf(conj_0,conjecture,
% 2.70/1.48 ( ( lMirro427583474mirror @ a @ ( coinductive_LCons @ a @ x @ xs ) )
% 2.70/1.48 = ( coinductive_LCons @ a @ x @ ( coinductive_lappend @ a @ ( lMirro427583474mirror @ a @ xs ) @ ( coinductive_LCons @ a @ x @ ( coinductive_LNil @ a ) ) ) ) ) ).
% 2.70/1.48 ------------------------
% 2.84/1.50 % (7370)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7148 for (2999ds/2Mi)
% 2.84/1.50 % (7370)Instruction limit reached!
% 2.84/1.50 % (7370)------------------------------
% 2.84/1.50 % (7370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.50 % (7370)Termination reason: Unknown
% 2.84/1.50 % (7370)Termination phase: shuffling
% 2.84/1.50
% 2.84/1.50 % (7370)Memory used [KB]: 1279
% 2.84/1.50 % (7370)Time elapsed: 0.002 s
% 2.84/1.50 % (7370)Instructions burned: 2 (million)
% 2.84/1.50 % (7370)------------------------------
% 2.84/1.50 % (7370)------------------------------
% 2.84/1.50 % (7373)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7148 for (2999ds/18Mi)
% 2.84/1.51 % (7372)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_7148 for (2999ds/275Mi)
% 2.84/1.51 % (7373)Instruction limit reached!
% 2.84/1.51 % (7373)------------------------------
% 2.84/1.51 % (7373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.51 % (7373)Termination reason: Unknown
% 2.84/1.51 % (7373)Termination phase: shuffling
% 2.84/1.51
% 2.84/1.51 % (7373)Memory used [KB]: 1663
% 2.84/1.51 % (7373)Time elapsed: 0.008 s
% 2.84/1.51 % (7373)Instructions burned: 19 (million)
% 2.84/1.51 % (7373)------------------------------
% 2.84/1.51 % (7373)------------------------------
% 2.84/1.51 % (7367)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7148 for (2999ds/183Mi)
% 2.84/1.51 % (7368)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_7148 for (2999ds/4Mi)
% 2.84/1.51 % (7371)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_7148 for (2999ds/2Mi)
% 2.84/1.51 % (7369)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_7148 for (2999ds/27Mi)
% 2.84/1.51 % (7371)Instruction limit reached!
% 2.84/1.51 % (7371)------------------------------
% 2.84/1.51 % (7371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.51 % (7371)Termination reason: Unknown
% 2.84/1.51 % (7371)Termination phase: shuffling
% 2.84/1.51
% 2.84/1.51 % (7371)Memory used [KB]: 1279
% 2.84/1.51 % (7371)Time elapsed: 0.004 s
% 2.84/1.51 % (7371)Instructions burned: 3 (million)
% 2.84/1.51 % (7371)------------------------------
% 2.84/1.51 % (7371)------------------------------
% 2.84/1.51 % (7374)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7148 for (2999ds/3Mi)
% 2.84/1.51 % (7368)Instruction limit reached!
% 2.84/1.51 % (7368)------------------------------
% 2.84/1.51 % (7368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.51 % (7368)Termination reason: Unknown
% 2.84/1.51 % (7368)Termination phase: shuffling
% 2.84/1.51
% 2.84/1.51 % (7368)Memory used [KB]: 1407
% 2.84/1.51 % (7368)Time elapsed: 0.004 s
% 2.84/1.51 % (7368)Instructions burned: 5 (million)
% 2.84/1.51 % (7368)------------------------------
% 2.84/1.51 % (7368)------------------------------
% 2.84/1.51 % (7374)Instruction limit reached!
% 2.84/1.51 % (7374)------------------------------
% 2.84/1.51 % (7374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.51 % (7374)Termination reason: Unknown
% 2.84/1.51 % (7374)Termination phase: shuffling
% 2.84/1.51
% 2.84/1.51 % (7374)Memory used [KB]: 1279
% 2.84/1.51 % (7374)Time elapsed: 0.002 s
% 2.84/1.51 % (7374)Instructions burned: 3 (million)
% 2.84/1.51 % (7374)------------------------------
% 2.84/1.51 % (7374)------------------------------
% 2.84/1.52 % (7375)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_7148 for (2999ds/37Mi)
% 2.84/1.52 % (7378)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_7148 for (2999ds/1041Mi)
% 2.84/1.52 % (7369)Instruction limit reached!
% 2.84/1.52 % (7369)------------------------------
% 2.84/1.52 % (7369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.52 % (7369)Termination reason: Unknown
% 2.84/1.52 % (7369)Termination phase: Property scanning
% 2.84/1.52
% 2.84/1.52 % (7369)Memory used [KB]: 1663
% 2.84/1.52 % (7369)Time elapsed: 0.018 s
% 2.84/1.52 % (7369)Instructions burned: 27 (million)
% 2.84/1.52 % (7369)------------------------------
% 2.84/1.52 % (7369)------------------------------
% 2.84/1.53 % (7376)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_7148 for (2999ds/15Mi)
% 2.84/1.53 % (7377)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7148 for (2999ds/3Mi)
% 2.84/1.53 % (7375)Instruction limit reached!
% 2.84/1.53 % (7375)------------------------------
% 2.84/1.53 % (7375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.53 % (7375)Termination reason: Unknown
% 2.84/1.53 % (7375)Termination phase: shuffling
% 2.84/1.53
% 2.84/1.53 % (7375)Memory used [KB]: 2046
% 2.84/1.53 % (7375)Time elapsed: 0.012 s
% 2.84/1.53 % (7375)Instructions burned: 37 (million)
% 2.84/1.53 % (7375)------------------------------
% 2.84/1.53 % (7375)------------------------------
% 2.84/1.53 % (7377)Instruction limit reached!
% 2.84/1.53 % (7377)------------------------------
% 2.84/1.53 % (7377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.53 % (7377)Termination reason: Unknown
% 2.84/1.53 % (7377)Termination phase: shuffling
% 2.84/1.53
% 2.84/1.53 % (7377)Memory used [KB]: 1407
% 2.84/1.53 % (7377)Time elapsed: 0.004 s
% 2.84/1.53 % (7377)Instructions burned: 3 (million)
% 2.84/1.53 % (7377)------------------------------
% 2.84/1.53 % (7377)------------------------------
% 2.84/1.53 % (7372)First to succeed.
% 2.84/1.54 % (7372)Refutation found. Thanks to Tanya!
% 2.84/1.54 % SZS status Theorem for DTF2THF_7148
% 2.84/1.54 % SZS output start Proof for DTF2THF_7148
% 2.84/1.54 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.84/1.54 thf(type_def_7, type, coinductive_llist: ($tType) > $tType).
% 2.84/1.54 thf(type_def_8, type, set: ($tType) > $tType).
% 2.84/1.54 thf(type_def_9, type, a: $tType).
% 2.84/1.54 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 2.84/1.54 thf(func_def_1, type, set: ($tType) > $tType).
% 2.84/1.54 thf(func_def_2, type, itself: ($tType) > $tType).
% 2.84/1.54 thf(func_def_3, type, a: $tType).
% 2.84/1.54 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.84/1.54 thf(func_def_5, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.84/1.54 thf(func_def_6, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.84/1.54 thf(func_def_7, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.84/1.54 thf(func_def_8, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.84/1.54 thf(func_def_9, type, coindu328551480prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.84/1.54 thf(func_def_10, type, coinductive_gen_lset: !>[X0: $tType]:(set @ X0 > coinductive_llist @ X0 > set @ X0)).
% 2.84/1.54 thf(func_def_11, type, coinductive_lappend: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_12, type, coinductive_lfilter: !>[X0: $tType]:((X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_13, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.84/1.54 thf(func_def_14, type, coinductive_llast: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.84/1.54 thf(func_def_15, type, coinductive_LCons: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_16, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_17, type, coindu1259883913_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > $o) > (X0 > coinductive_llist @ X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 2.84/1.54 thf(func_def_18, type, coinductive_lhd: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.84/1.54 thf(func_def_19, type, coinductive_lnull: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.84/1.54 thf(func_def_20, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 2.84/1.54 thf(func_def_21, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 2.84/1.54 thf(func_def_22, type, coinductive_lprefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.84/1.54 thf(func_def_23, type, coindu1478340336prefix: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > $o)).
% 2.84/1.54 thf(func_def_24, type, coindu63249387sorted: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.84/1.54 thf(func_def_25, type, coindu1441602521_llist: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > X1) > (X0 > X0) > X0 > coinductive_llist @ X1)).
% 2.84/1.54 thf(func_def_26, type, undefined: !>[X0: $tType]:(X0)).
% 2.84/1.54 thf(func_def_27, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.84/1.54 thf(func_def_28, type, lMirro427583474mirror: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_29, type, lMirro999291890or_aux: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_30, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.84/1.54 thf(func_def_31, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.84/1.54 thf(func_def_32, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.84/1.54 thf(func_def_33, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.84/1.54 thf(func_def_34, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.84/1.54 thf(func_def_35, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.84/1.54 thf(func_def_36, type, x: a).
% 2.84/1.54 thf(func_def_37, type, xs: coinductive_llist @ a).
% 2.84/1.54 thf(func_def_55, type, sK0: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.84/1.54 thf(func_def_56, type, sK1: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_57, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.84/1.54 thf(func_def_58, type, sK3: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.84/1.54 thf(func_def_59, type, sK4: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_60, type, sK5: !>[X0: $tType]:(coinductive_llist @ X0 > X0)).
% 2.84/1.54 thf(func_def_61, type, sK6: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.84/1.54 thf(func_def_63, type, ph8: !>[X0: $tType]:(X0)).
% 2.84/1.54 thf(f886,plain,(
% 2.84/1.54 $false),
% 2.84/1.54 inference(subsumption_resolution,[status(thm)],[f874,f859])).
% 2.84/1.54 thf(f859,plain,(
% 2.84/1.54 ( ! [X0 : $tType,X2 : X0,X3 : coinductive_llist @ X0,X1 : coinductive_llist @ X0] : (((coinductive_LCons @ X0 @ X2 @ (coinductive_lappend @ X0 @ (lMirro999291890or_aux @ X0 @ coinductive_LNil @ X0 @ X3) @ (coinductive_LCons @ X0 @ X2 @ X1))) = (lMirro999291890or_aux @ X0 @ X1 @ (coinductive_LCons @ X0 @ X2 @ X3)))) )),
% 2.84/1.54 inference(cnf_transformation,[status(thm)],[f749])).
% 2.84/1.54 thf(f749,plain,(
% 2.84/1.54 ! [X0 : $tType,X1 : coinductive_llist @ X0,X2 : X0,X3 : coinductive_llist @ X0] : ((coinductive_LCons @ X0 @ X2 @ (coinductive_lappend @ X0 @ (lMirro999291890or_aux @ X0 @ coinductive_LNil @ X0 @ X3) @ (coinductive_LCons @ X0 @ X2 @ X1))) = (lMirro999291890or_aux @ X0 @ X1 @ (coinductive_LCons @ X0 @ X2 @ X3)))),
% 2.84/1.54 inference(rectify,[status(thm)],[f13])).
% 2.84/1.54 thf(f13,axiom,(
% 2.84/1.54 ! [X0 : $tType,X16 : coinductive_llist @ X0,X6 : X0,X4 : coinductive_llist @ X0] : ((lMirro999291890or_aux @ X0 @ X16 @ (coinductive_LCons @ X0 @ X6 @ X4)) = (coinductive_LCons @ X0 @ X6 @ (coinductive_lappend @ X0 @ (lMirro999291890or_aux @ X0 @ coinductive_LNil @ X0 @ X4) @ (coinductive_LCons @ X0 @ X6 @ X16))))),
% 2.84/1.54 file('/export/starexec/sandbox2/tmp/tmp.Mre2OmccfH/DTF2THF_7148.p',fact_12_lmirror__aux__LCons)).
% 2.84/1.54 thf(f874,plain,(
% 2.84/1.54 ((coinductive_LCons @ a @ x @ (coinductive_lappend @ a @ (lMirro999291890or_aux @ a @ coinductive_LNil @ a @ xs) @ (coinductive_LCons @ a @ x @ coinductive_LNil @ a))) != (lMirro999291890or_aux @ a @ coinductive_LNil @ a @ (coinductive_LCons @ a @ x @ xs)))),
% 2.84/1.54 inference(definition_unfolding,[status(thm)],[f862,f837,f837])).
% 2.84/1.54 thf(f837,plain,(
% 2.84/1.54 ( ! [X0 : $tType] : ((lMirro427583474mirror @ X0 = (lMirro999291890or_aux @ X0 @ coinductive_LNil @ X0))) )),
% 2.84/1.54 inference(cnf_transformation,[status(thm)],[f18])).
% 2.84/1.54 thf(f18,axiom,(
% 2.84/1.54 ! [X0 : $tType] : (lMirro427583474mirror @ X0 = (lMirro999291890or_aux @ X0 @ coinductive_LNil @ X0))),
% 2.84/1.54 file('/export/starexec/sandbox2/tmp/tmp.Mre2OmccfH/DTF2THF_7148.p',fact_17_lmirror__def)).
% 2.84/1.54 thf(f862,plain,(
% 2.84/1.54 ((lMirro427583474mirror @ a @ (coinductive_LCons @ a @ x @ xs)) != (coinductive_LCons @ a @ x @ (coinductive_lappend @ a @ (lMirro427583474mirror @ a @ xs) @ (coinductive_LCons @ a @ x @ coinductive_LNil @ a))))),
% 2.84/1.54 inference(cnf_transformation,[status(thm)],[f725])).
% 2.84/1.54 thf(f725,plain,(
% 2.84/1.54 ((lMirro427583474mirror @ a @ (coinductive_LCons @ a @ x @ xs)) != (coinductive_LCons @ a @ x @ (coinductive_lappend @ a @ (lMirro427583474mirror @ a @ xs) @ (coinductive_LCons @ a @ x @ coinductive_LNil @ a))))),
% 2.84/1.54 inference(flattening,[status(thm)],[f269])).
% 2.84/1.54 thf(f269,negated_conjecture,(
% 2.84/1.54 ~((lMirro427583474mirror @ a @ (coinductive_LCons @ a @ x @ xs)) = (coinductive_LCons @ a @ x @ (coinductive_lappend @ a @ (lMirro427583474mirror @ a @ xs) @ (coinductive_LCons @ a @ x @ coinductive_LNil @ a))))),
% 2.84/1.54 inference(negated_conjecture,[status(cth)],[f268])).
% 2.84/1.54 thf(f268,conjecture,(
% 2.84/1.54 ((lMirro427583474mirror @ a @ (coinductive_LCons @ a @ x @ xs)) = (coinductive_LCons @ a @ x @ (coinductive_lappend @ a @ (lMirro427583474mirror @ a @ xs) @ (coinductive_LCons @ a @ x @ coinductive_LNil @ a))))),
% 2.84/1.54 file('/export/starexec/sandbox2/tmp/tmp.Mre2OmccfH/DTF2THF_7148.p',conj_0)).
% 2.84/1.54 % SZS output end Proof for DTF2THF_7148
% 2.84/1.54 % (7372)------------------------------
% 2.84/1.54 % (7372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.54 % (7372)Termination reason: Refutation
% 2.84/1.54
% 2.84/1.54 % (7372)Memory used [KB]: 6396
% 2.84/1.54 % (7372)Time elapsed: 0.029 s
% 2.84/1.54 % (7372)Instructions burned: 45 (million)
% 2.84/1.54 % (7372)------------------------------
% 2.84/1.54 % (7372)------------------------------
% 2.84/1.54 % (7366)Success in time 0.05 s
%------------------------------------------------------------------------------